# 2. Reference Information

## 2.1. Publications

This section contains FUN3D-related publications and presentations. The first section presents the publications in bibliography style. The titles are links to entries in a subsequent section that contains the publication title, a link to an online copy, and the publication’s abstract.

### Publication Citations

Toward a Comprehensive Model of Jet Noise Using an Acoustic Analogy, Steven A. Miller, AIAA Journal, Vol. 52, No. 10, pp. 2143-2164, October 2014.

FUN3D Manual: 12.5, Robert T. Biedron, Joseph M. Derlaga, Peter A. Gnoffo, Dana P. Hammond, William T. Jones, Bil Kleb, Elizabeth M. Lee-Rausch, Eric J. Nielsen, Michael A. Park, Christopher L. Rumsey, James L. Thomas, and William A. Wood, NASA-TM-2014-218520, 2014.

Evaluation of Linear, Inviscid, Viscous, and Reduced-Order Modeling Aeroelastic Solutions of the AGARD 445.6 Wing Using Root Locus Analysis, Walter A. Silva, Pawel Chwalowski, and Boyd Perry III, International Journal of Computational Fluid Dynamics, 2014.

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces, Mark H. Carpenter, Travis C. Fisher, Eric J. Nielsen, and Steven H. Frankel, Accepted for publication in SIAM Journal on Scientific Computing.

Application of the FUN3D Solver to the 4th AIAA Drag Prediction Workshop, E.M. Lee-Rausch, D.P. Hammond, E.J. Nielsen, S.Z. Pirzadeh, and C.L. Rumsey, AIAA Journal of Aircraft, Vol. 51, No. 4, pp. 1149-1160, July 2014.

Feature-Based Grid Adaption for the Study of Dynamic Stall, Kyle Hord and Yongsheng Lian, AIAA-2014-2997, June 2014.

Towards Full Aircraft Airframe Noise Prediction: Detached Eddy Simulations, Mehdi R. Khorrami and Raymond E. Mineck, AIAA-2014-2480, June 2014.

Computed and Experimental Flutter/LCO Onset for the Boeing Truss-Braced Wing Wind-Tunnel Model, Robert E. Bartels, Robert C. Scott, Christie J. Funk, Timothy J. Allen, and Bradley W. Sexton, AIAA-2014-2446, June 2014.

Grid-Adapted FUN3D Computations for the Second High Lift Prediction Workshop, E.M. Lee-Rausch, C.L. Rumsey, and M.A. Park, AIAA-2014-2395, June 2014.

Simulation of a Variety of Wings Using a Reynolds Stress Model, K.B. Thompson and H.A. Hassan, AIAA-2014-2192, June 2014.

Uncertainty Quantification and Certification Prediction of Low-Boom Supersonic Aircraft Configurations, Thomas K. West IV, Bryan W. Reuter, Eric L. Walker, Bil Kleb, and Michael A. Park, AIAA-2014-2139, June 2014.

Summary and Statistical Analysis of the First AIAA Sonic Boom Prediction Workshop, Michael A. Park and John M. Morgenstern, AIAA-2014-2006, June 2014.

Overset Adaptive Strategies for Complex Rotating Systems, Rajiv Shenoy, PhD Thesis, Georgia Institute of Technology, May 2014.

FUN3D Manual: 12.4, Robert T. Biedron, Joseph M. Derlaga, Peter A. Gnoffo, Dana P. Hammond, William T. Jones, Bil Kleb, Elizabeth M. Lee-Rausch, Eric J. Nielsen, Michael A. Park, Christopher L. Rumsey, James L. Thomas, and William A. Wood, NASA-TM-2014-218179, 2014.

Noise Generated by an Airfoil Located in the Wake of a Circular Cylinder, Man Zhang and Abdelkader Frendi, Presented at the 5th Symposium on Hybrid RANS-LES Methods, Texas A&M University, 19-21 March, 2014.

Sonic Boom Mitigation Through Aircraft Design and Adjoint Methodology, Sriram K. Rallabhandi, Eric J. Nielsen, and Boris Diskin, AIAA Journal of Aircraft, Vol. 51, No. 2, pp. 502-510, March/April 2014.

Unstructured Overset Mesh Adaptation with Turbulence Modeling for Unsteady Aerodynamic Interactions, Rajiv Shenoy, Marilyn J. Smith, and Michael A. Park, AIAA Journal of Aircraft, vol. 51 no. 1, January/February 2014.

Analytical Correlation of a Flexible Empennage Wind Tunnel Flutter Test at High Transonic Mach Number, F. von Knoblauch, R. Moreno, P.F. Taylor, and J. Newsom, AIAA-2014-0676, January 2014.

The NASA High Speed ASE Project: Computational Analyses of a Low-Boom Supersonic Configuration, Walter A. Silva, Antonio De La Garza, Scott Zink, Elias G. Bounajem, J. Christopher Johnson, Michael Buonanno, Mark D. Sanetrik, Seung Y. Yoo, George Kopasakis, David M. Christhilf, Pawel Chwalowski, AIAA-2014-0675, January 2014.

Evaluation of Linear, Inviscid, Viscous, and Reduced-Order Modeling Aeroelastic Solutions of the AGARD 445.6 Wing Using Root Locus Analysis, Walter A. Silva, Boyd Perry III, Pawel Chwalowski, AIAA-2014-0496, January 2014.

Multi-point Adjoint-Based Design of Tilt-Rotors in a Noninertial Reference Frame, William T. Jones, Eric J. Nielsen, Elizabeth M. Lee-Rausch, Cecil W. Acree, Jr., AIAA-2014-0290, January 2014.

Unsteady Aerodynamic Validation Experiences from the Aeroelastic Prediction Workshop, Jennifer Heeg, Pawel Chwalowski, AIAA-2014-0203, January 2014.

Specialized CFD Grid Generation Methods for Near-Field Sonic Boom Prediction, Michael A. Park, Richard L. Campbell, Alaa Elmiligui, Susan E. Cliff, Sudheer N. Nayani, AIAA-2014-0115, January 2014.

Evaluation of Multigrid Solutions for Turbulent Flows, Boris Diskin, Hiroaki Nishikawa, AIAA-2014-0082, January 2014.

An Efficient Actuating Blade Model for Unsteady Rotating System Wake Simulations,, C.E. Lynch, D.T. Prosser, and M.J. Smith, Computers and Fluids, Vol. 92, pp. 138-150, March 2014.

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces, Mark H. Carpenter, Travis C. Fisher, Eric J. Nielsen, and Steven H. Frankel, NASA-TM-2013-218039, 2013.

CFD Analysis and Design Optimization of Flapping Wing Flows, Martin Jones, PhD Thesis, North Carolina A&T State University, Summer 2013.

Coupled CFD/CSD Analysis of Rotor Blade Structural Loads with Experimental Validation, Steven J. Massey, Andrew R. Kreshock, Martin K. Sekula, AIAA-2013-3158, June 2013.

Developing an Accurate CFD Based Gust Model for the Truss Braced Wing Aircraft, Robert E. Bartels, AIAA-2013-3044, June 2013.

Application of Adjoint Methodology to Supersonic Aircraft Design Using Reversed Equivalent Areas, Sriram K. Rallabhandi, AIAA-2013-2663, June 2013.

Functional Equivalence Acceptance Testing of FUN3D for Entry, Descent, and Landing Applications, Peter A. Gnoffo, William A. Wood, Bil Kleb, Stephen J. Alter, Chris Glass, Jose Padilla, Dana Hammond, and Jeffery A. White, AIAA-2013-2558, June 2013.

Numerical Simulation of the Aircraft Wake Vortex Flowfield, Nash’at N. Ahmad, Fred H. Proctor, R. Brad Perry, AIAA-2013-2552, June 2013.

Adjoint-Based Shape and Kinematics Optimization of Flapping Wing Propulsive Efficiency, Martin Jones, Nail K. Yamaleev, AIAA-2013-2472, June 2013.

Discrete Adjoint-Based Design for Unsteady Turbulent Flows on Dynamic Overset Unstructured Grids,, Eric J. Nielsen and Boris Diskin, AIAA Journal, Vol. 51, No. 6, pp. 1355-1373, June 2013.

Towards a Comprehensive Model of Jet Noise using an Acoustic Analogy and Steady RANS Solutions, Steven A. E. Miller, AIAA-2013-2278, May 2013.

Aeroacoustic Simulation of Nose Landing Gear on Adaptive Unstructured Grids with FUN3D, Veer N. Vatsa, Mehdi R. Khorrami, Michael A. Park, David P. Lockard, AIAA-2013-2071, May 2013.

The Effects of Surfaces on the Aerodynamics and Acoustics of Jet Flows, Matthew J. Smith and Steven A. E. Miller, AIAA-2013-2041, May 2013.

FUN3D Airload Predictions for the Full-Scale UH-60A Airloads Rotor in a Wind Tunnel, Elizabeth M. Lee-Rausch and Robert T. Biedron, Presented at the AHS 69th Annual Forum, Phoenix, AZ, May 2013.

Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018 to 2020 Period, John Morgenstern, Nicole Norstrud, Jack Sokhey, Steve Martens, and Juan J. Alonso, NASA/CR-2013-217820, February 2013.

FUN3D Analyses in Support of the First Aeroelastic Prediction Workshop, Pawel Chwalowski, Jennifer Heeg, Carol D. Wieseman, Jennifer P. Florance, AIAA-2013-0785, January 2013.

Summary of the 2008 NASA Fundamental Aeronautics Program Sonic Boom Prediction Workshop, Michael A. Park, Michael J. Aftosmis, Richard L. Campbell, Melissa B. Carter, Susan E. Cliff, Linda S. Bangert, AIAA-2013-649, January 2013.

Full Configuration Low Boom Model and Grids for 2014 Sonic Boom Prediction Workshop, John M. Morgenstern, Michael Buonanno, Frank Marconi, AIAA-2013-647, January 2013.

CFL3D, FUN3D, and NSU3D Contributions to the Fifth Drag Prediction Workshop, Michael A. Park, Kelly R. Laflin, Mark S. Chaffin, Nicholas Powell, David W. Levy, AIAA-2013-50, January 2013.

Directivity Effects of Shaped Plumes from Plug Nozzles, James D. Chase, G. Andres Garzon, D. Papamoschou AIAA-2013-0008, January 2013.

Recent Advances in Agglomerated Multigrid, Hiroaki Nishikawa, Boris Diskin, James L. Thomas, and Dana P. Hammond, AIAA-2013-863, January 2013.

Development, Verification and Use of Gust Modeling in the NASA Computational Fluid Dynamics Code FUN3D, Robert E. Bartels, NASA/TM-2012-217771, October 2012.

Production Level CFD Code Acceleration for Hybrid Many-Core Architectures, Austen C. Duffy, Dana P. Hammond, and Eric J. Nielsen, NASA/TM-2012-217770, October 2012.

Computational Aeroelastic Analysis of the Ares I Crew Launch Vehicle During Ascent, Robert E. Bartels, Pawel Chwalowski, Steven J. Massey, Jennifer Heeg, and Raymond E. Mineck, Journal of Spacecraft and Rockets, Vol. 49, No. 4, pp. 651-658, July/August 2012.

Flexible Launch Vehicle Stability Analysis Using Steady and Unsteady Computational Fluid Dynamics, Robert E. Bartels, Journal of Spacecraft and Rockets, Vol. 49, No. 4, pp. 644-650, July/August 2012.

Adjoint-Based Optimization of the Flapping Wing Performance, Martin Jones and Nail Yamaleev, ICCFD7-2403, Presented at the Seventh International Conference on Computational Fluid Dynamics, Big Island, Hawaii, July 9-13, 2012.

Sonic Boom Mitigation Through Aircraft Design and Adjoint Methodology, Sriram K. Rallabhandi, Eric J. Nielsen, and Boris Diskin, AIAA-2012-3220, June 2012.

N+2 Low Boom Wind Tunnel Model Design and Validation, John M. Morgenstern, Michael Buonanno, and Nicole Nordstrud, AIAA-2012-3217, June 2012.

Aerodynamic Impacts of Helicopter Blade Erosion Coatings, Mark E. Calvert and Tin-Chee Wong, AIAA-2012-2914, June 2012.

NASA Trapezoidal Wing Computations Including Transition and Advanced Turbulence Modeling, C.L. Rumsey and E.M. Lee-Rausch, AIAA-2012-2843, June 2012.

Radiation Coupling with the FUN3D Unstructured-Grid CFD Code, William A. Wood, AIAA-2012-2741, June 2012.

Boundary Layer Stability Analysis of the Mean Flows Obtained Using Unstructured Grids, Wei Liao, Mujeeb R. Malik, Elizabeth M. Lee-Rausch, Fei Li, Eric J. Nielsen, Pieter G. Buning, Chau-Lyan Chang, and Meelan Choudhari, AIAA-2012-2690, June 2012.

Aeroacoustic Simulation of a Nose Landing Gear in an Open Jet Facility using FUN3D, Veer N. Vatsa, David P. Lockard, Mehdi R. Khorrami, and Jan-Renee Carlson, AIAA-2012-2280, June 2012.

Unsteady Reynolds-Averaged Navier-Stokes-Based Hybrid Methodologies for Rotor-Fuselage Interaction,, Eliot W. Quon, Marilyn J. Smith, Glen R. Whitehouse, and Dan Wachspress, AIAA Journal of Aircraft, Vol. 49, No. 3, pp. 961-965, May/June 2012.

An Assessment of CFD/CSD Prediction State-of-the-Art Using the HART II International Workshop Data, Marilyn Smith, Joon Lim, Berend van der Wall, James Baeder, Robert Biedron, D. Douglas Boyd Jr., Buvana Jayaraman, Sung Jung and Byung-Young Min, Presented at the AHS 68th Annual Forum, Fort Worth, TX, May 2012.

An Examination of Unsteady Airloads on a UH-60A Rotor: Computation versus Measurement, Robert T. Biedron and Elizabeth M. Lee-Rausch, Presented at the AHS 68th Annual Forum, Fort Worth, TX, May 2012.

Rotating Hub Drag Prediction Methodology, Matthew J. Hill and Matthew E. Louis, Presented at the AHS Future Vertical Lift Aircraft Design Conference, San Francisco, CA, January 18-20, 2012.

The Effect of a Gust on the Flapping Wing Performance, Martin Jones and Nail K. Yamaleev, AIAA-2012-1080, January 2012.

Exploration of the Physics of Hub Drag, Vrishank Raghav, Rajiv Shenoy, Felipe T. Ortega, Narayanan Komerath, and Marilyn Smith, AIAA-2012-1070, January 2012.

Continuing Validation of Computational Fluid Dynamics For Supersonic Retropropulsion, Daniel G. Schauerhamer, Kerry A. Trumble, Bil Kleb, Jan-Renee Carlson, and Karl T. Edquist, AIAA-2012-0864, January 2012.

Effects of Mesh Regularity on Accuracy of Finite-Volume Schemes, Boris Diskin and James L. Thomas, AIAA-2012-0609, January 2012.

Discrete Adjoint-Based Design for Unsteady Turbulent Flows on Dynamic Overset Unstructured Grids, Eric Nielsen and Boris Diskin, AIAA-2012-0554, January 2012.

Inflow/Outflow Boundary Conditions with Application to FUN3D, Jan-Renee Carlson, NASA/TM-2011-217181, October 2011.

Computational Investigation of Hub Drag Deconstruction from Model to Full Scale, Rajiv Shenoy, Marlin Holmes, Marilyn Smith, and Narayanan Komerath, ERF2011_165, Presented at the 37th European Rotorcraft Forum, Milan, Italy, September 13-15, 2011.

Extension and Exploration of a Hybrid Turbulence Model on Unstructured Grids, C. Eric Lynch and Marilyn J. Smith, AIAA Journal, Vol. 49, No. 11, pp. 2585-2590, November 2011, doi: 10.2514/1.J051177.

Analysis of Effectiveness of Phoenix Entry Reaction Control System, Artem A. Dyakonov, Christopher E. Glass, Prasun N. Desai, and John W. Van Norman, Journal of Spacecraft and Rockets, Vol. 48, No. 5, pp. 746-755.

Computational Analysis of the G-III Laminar Flow Glove, Mujeeb Malik, Wei Liao, Elizabeth Lee-Rausch, Fei Li, Meelan Choudhari, and Chau-Lyan Chang, AIAA-2011-3525, June 2011.

Toward Supersonic Retropropulsion CFD Validation, Bil Kleb, D. Guy Schauerhamer, Kerry Trumble, Emre Sozer, Michael Barnhardt, Jan-Renee Carlson, and Karl Edquist, AIAA-2011-3490, June 2011.

Sonic Boom Adjoint Methodology and its Applications, Sriram K. Rallabhandi, AIAA-2011-3497, June 2011.

Low Boom Configuration Analysis with FUN3D Adjoint Simulation Framework, Michael A. Park, AIAA-2011-3337, June 2011.

Development and Application of Parallel Agglomerated Multigrid Methods for Complex Geometries, Hiroaki Nishikawa and Boris Diskin, AIAA-2011-3232, June 2011.

Code-to_Code Comparison of CFD/CSD Simulations for a Helicopter Rotor in Forward Flight, Jasim Ahmad and Robert T. Biedron. AIAA-2011-3819, June 2011.

Preliminary Computational Analysis of the HIRENASD Configuration in Preparation for the Aeroelastic Prediction Workshop, Pawel Chwalowski, Jennifer P. Florance, Jennifer Heeg, Carol D. Wieseman, and Boyd Perry III, IFASD-2011-108, Presented at the 2011 International Forum on Aeroelasticity and Structural Dynamics, June 2011.

Application of FUN3D Solver for Aeroacoustics Simulation of a Nose Landing Gear Configuration, Veer N. Vatsa, David P. Lockard, and Mehdi R. Khorrami, AIAA-2011-2820, June 2011.

Integrated Design of an Active Flow Control System Using a Time-Dependent Adjoint Method, Eric J. Nielsen and W.T. Jones, Special thematic issue of Mathematical Modeling of Natural Phenomena devoted to modern trends in computational aerodynamics, Vol. 6, No. 3, 2011, pp. 141-165.

Massively Parallel Algorithms for CFD Simulation and Optimization on Heterogeneous Many-Core Architectures, Austen C. Duffy, PhD Thesis, Florida State University, Spring 2011.

Computation of UH-60A Airloads Using CFD/CSD Coupling On Unstructured Meshes, Robert T. Biedron and Elizabeth M. Lee-Rausch, Presented at the AHS 67th Annual Forum, Virginia Beach, VA, May 2011.

Hierarchical Variable Fidelity Methods for Rotorcraft Aerodynamic Design and Analysis, Eliot W. Quon, Marilyn J. Smith, Glen R. Whitehouse, and Daniel A. Wachspress, Presented at the AHS 67th Annual Forum, Virginia Beach, VA, May 2011.

Unstructured Overset Grid Adaptation for Rotorcraft Aerodynamic Interactions, Rajiv Shenoy and Marilyn J. Smith, Presented at the AHS 67th Annual Forum, Virginia Beach, VA, May 2011.

Comparison of Node-Centered and Cell-Centered Unstructured
Finite-Volume Discretizations: Inviscid Fluxes,
Boris Diskin and James L. Thomas, AIAA Journal, Vol. 49,
No. 4, pp. 836-854, April 2011, doi: 10.2514/1.53963; see also AIAA-2010-1079.
**Note that published journal version has many incorrect equations. The preprint linked here on the FUN3D site is correct.**

Supersonic Retro-Propulsion Experimental Design for Computational Fluid Dynamics Model Validation, Scott A. Berry, Christopher T. Laws, William L. Kleb, Matthew N. Rhode, Courtney Spells, Andrew C. McCrea, Kerry A. Trumble, D. Guy Schauerhamer, and William L. Oberkampf, IEEE AC-1499, March 2011.

Analysis of Navier-Stokes Codes Applied to Supersonic Retro-Propulsion Wind Tunnel Test, Kerry A. Trumble, Daniel G. Schauerhamer, William L. Kleb, Jan-Renee Carlson, and Karl T. Edquist, IEEE AC-1471, March 2011.

CFD Verification of Supersonic Retropropulsion for a Central and Peripheral Configuration, Christopher E. Cordell, Jr., Ian G. Clark, and Robert D. Braun, IEEE AC-1190, March 2011.

Advanced CFD Methods for Wind Turbine Analysis, C. Eric Lynch, PhD Thesis, Georgia Tech, January 2011.

A Quasi-steady Flexible Launch Vehicle Stability Analysis Using Steady CFD with Unsteady Aerodynamic Enhancement, Robert E. Bartels, AIAA-2011-1114, January 2011.

Numerical Study Comparing RANS and LES Approaches on a Circulation Control Airfoil, Christopher L. Rumsey and Takafumi Nishino, AIAA-2011-1179, January 2011.

FUN3D and CFL3D Computations for the First High Lift Prediction Workshop, Michael A. Park, Elizabeth M. Lee-Rausch, and Christopher L. Rumsey, AIAA-2011-936, January 2011.

Optimization of a 2-D Flap Geometry Using Matlab and FUN3D, Gregory D. Howe, AIAA-2011-823, January 2011.

Application of CFD in the Design of Flow Control Concepts for a Ducted-Fan Configuration, W. Kelly Londenberg and O. John Ohanian III, Presented at the AHS International Powered Lift Conference, October 5-7, 2010, Philadelphia, PA.

Feature-Based and Output-Based Grid Adaptation Study for Hypersonic Propulsive Deceleration Jet Flows, Hicham Alkandry, Michael A. Park, William L. Kleb, and Iain D. Boyd, Presented at the 19th International Meshing Roundtable, Chattanooga, Tennessee, October 4-6, 2010.

A Critical Study of Agglomerated Multigrid Methods for Diffusion on Highly-Stretched Grids, James L. Thomas, Boris Diskin, and Hiroaki Nishikawa, Computers and Fluids, vol. 41, September 2010.

Validation of an Output-Adaptive, Tetrahedral Cut-Cell Method for Sonic Boom Prediction, Michael A. Park and David L. Darmofal, AIAA Journal, Vol. 48, No. 9, 2010, pp. 1928-1945.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Viscous Fluxes, Boris Diskin, James L. Thomas, Eric J. Nielsen, Hiroaki Nishikawa, and Jeffrey A. White, AIAA Journal, Vol. 48, No. 7, July 2010.

Comparison of Inviscid and Viscous Aerodynamic Predictions of Supersonic Retropropulsion Flowfields, Ashley M. Korzun, Christopher E. Cordell, Jr., and Robert D. Braun, AIAA-2010-5048, June 2010.

An Initial Assessment of Navier-Stokes Codes Applied to Supersonic Retro-Propulsion, Kerry A. Trumble, D. Guy Schauerhamer, William Kleb, Jan-Renee Carlson, Pieter G. Buning, and Karl Edquist, AIAA 2010-5047, June 2010.

Application of the FUN3D Unstructured-Grid Navier-Stokes Solver to the 4th AIAA Drag Prediction Workshop Cases, Elizabeth M. Lee-Rausch, Dana P. Hammond, Eric J. Nielsen, Shahyar Z. Pirzadeh, and Christopher L. Rumsey, AIAA-2010-4551, June 2010.

Development and Application of Agglomerated Multigrid Methods for Complex Geometries, Hiroaki Nishikawa, Boris Diskin, and James L. Thomas, AIAA-2010-4731, June 2010.

Description of a Website Resource for Turbulence Modeling Verification and Validation, Christopher L. Rumsey, Brian R. Smith, and George P. Huang, AIAA-2010-4742, June 2010.

Reduced-Order Models for the Aeroelastic Analysis of Ares Launch Vehicles, Walter A. Silva, Veer N. Vatsa, and Robert T. Biedron, AIAA-2010-4375, June 2010.

Computational Aeroelastic Analysis of the Ares Launch Vehicle During Ascent, Robert E. Bartels, Pawel Chwalowski, Steven J. Massey, Jennifer Heeg, Carol D. Wieseman, and Raymond E. Mineck, AIAA-2010-4374, June 2010.

Computational Aeroelastic Analysis of Ares Crew Launch Vehicle Bi-Modal Loading, Steven J. Massey and Pawel Chwalowski, AIAA-2010-4373, June 2010.

FUN3D Grid Refinement and Adaptation Studies for the Ares Launch Vehicle, Robert E. Bartels, Veer Vatsa, Jan-Renee Carlson, Mike Park, and Raymond E. Mineck, AIAA-2010-4372, June 2010.

Assessment of Hybrid RANS/LES Turbulence Models for Aeroacoustics Applications, Veer N. Vatsa and David P. Lockard, AIAA-2010-4001, June 2010.

Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids, Eric J. Nielsen, Boris Diskin, and Nail K. Yamaleev, AIAA Journal, Vol. 48, No. 6, 2010, pp. 1195-1206.

Investigation of Mixed Element Hybrid Grid-Based CFD Methods for Rotorcraft Flow Analysis, Glen R. Whitehouse, Alexander H. Boschitsch, Marilyn J. Smith, C. Eric Lynch, and Richard E. Brown, Presented at the AHS 66th Annual Forum, Phoenix, AZ, May 2010.

Analysis of CFD Modeling Techniques over the MV-22 Tiltrotor, Jennifer Abras and Robert Narducci, Presented at the AHS 66th Annual Forum, Phoenix, AZ, May 2010.

Local-in-Time Adjoint-Based Method for Design Optimization of Unsteady Flows, Nail K. Yamaleev, Boris Diskin, and Eric Nielsen, Journal of Computational Physics, Vol. 229, 2010, pp. 5394-5407.

Notes on Accuracy of Finite-Volume Discretization Schemes on Irregular Grids, Boris Diskin and James L. Thomas, Applied Numerical Mathematics, vol. 60, 2010, pp. 224-226.

Critical Study of Agglomerated Multigrid Methods for Diffusion, Hiroaki Nishikawa, Boris Diskin, and James L. Thomas, AIAA Journal, vol. 48 no. 4, April 2010.

Adjoint-Based Design of Rotors in a Noninertial Reference Frame, Eric J. Nielsen, Elizabeth M. Lee-Rausch, and William T. Jones, AIAA Journal of Aircraft, vol. 47 no. 2, March/April 2010.

Mitigation of Dynamic Stall Using Small Controllable Devices, Tin-Chee Wong, Presented at the AHS Aeromechanics Specialists Conference, San Francisco, CA, January 2010.

Updates to Multi-Dimensional Flux Reconstruction for Hypersonic Simulations on Tetrahedral Grids, Peter A. Gnoffo, AIAA-2010-1271, January 2010.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes, Boris Diskin and James L. Thomas, AIAA-2010-1079, January 2010.

CFD Assessment of Aerodynamic Degradation of a Subsonic Transport Due to Airframe Damage, Neal T. Frink, Shahyar Z. Pirzadeh, Harold L. Atkins, Sally A. Viken, and Joseph H. Morrison, AIAA-2010-500, January 2010.

Re-evaluation of an Optimized Second Order Backward Difference (BDF2OPT) Scheme for Unsteady Flow Applications, Veer N. Vatsa, Mark H. Carpenter, and David P. Lockard, AIAA-2010-0122, January 2010.

Turbulent Output-Based Anisotropic Adaptation, Michael A. Park and Jan-Renee Carlson, AIAA-2010-0168, January 2010.

Output Based Grid Adaptation for Viscous Flow, Julie C. Andren and Michael A. Park, Presented at the 18th International Meshing Roundtable, Salt Lake City, Utah, October 25-28, 2009.

A Critical Study of Agglomerated Multigrid Methods for Diffusion, Hiroaki Nishikawa, Boris Diskin, and James L. Thomas, AIAA 2009-4138, 19th AIAA Computational Fluid Dynamics 22-25 June 2009, San Antonio, Texas

Development of Unsteady Aerodynamic and Aeroelastic Reduced-Order Models Using the FUN3D Code, Walter A. Silva, Veer N. Vatsa, and Robert T. Biedron, IFASD-2009-030, Presented at the 2009 International Forum on Aeroelasticity and Structural Dynamics, June 2009.

Consistency, Verification, and Validation of Turbulence Models for Reynolds-Averaged Navier-Stokes Applications, Chris L. Rumsey, EUCASS2009-7, Presented at the 3rd European Conference for Aerospace Sciences, 2009.

Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids, Eric J. Nielsen, Boris Diskin, and Nail K. Yamaleev, AIAA-2009-3802, June 2009. See also AIAA Journal, Vol. 48, No. 6, June 2010.

Aerodynamic Interference Due to MSL Reaction Control System, Artem A. Dyakonov, Mark Schoenenberger, William I. Scallion, John W. Van Norman, Luke A. Novak, and Chun Y. Tang, AIAA-2009-3915, June 2009.

Aerodynamic Challenges for the Mars Science Laboratory Entry, Descent and Landing, Mark Schoenenberger, Artem Dyakonov, Pieter Buning, William I. Scallion, and John W. Van Norman, AIAA-2009-3914, June 2009.

Ducted-Fan Force and Moment Control via Steady and Synthetic Jets, Osgar John Ohanian III, Etan D. Karni, W. Kelly Londenberg, Paul A. Gelhausen, and Daniel J. Inman, AIAA-2009-3622, June 2009.

Computational Fluid Dynamics Validation of a Single Central Nozzle Supersonic Retropropulsion Configuration, Christopher E. Cordell, Jr. and Robert D. Braun, Georgia Institute of Technology AE8900 Report, May 2009.

Enhancement of Aeroelastic Rotor Airload Prediction Methods, Jennifer Abras, PhD Thesis, Georgia Tech, May 2009.

Computational Aeroelasticity of Rotating Wings with Deformable Airfoils, Smith Thepvongs, James R. Cook, Carlos E.S. Cesnik, and Marilyn J. Smith, 65th Annual AHS Forum, May 2009.

Simulation of an Isolated Tiltrotor in Hover with an Unstructured Overset-Grid RANS Solver, Elizabeth Lee-Rausch and Robert T. Biedron, 65th Annual AHS Forum, May 2009.

Adjoint-Based Design of Rotors Using the Navier-Stokes Equations in a Noninertial Reference Frame, Eric J. Nielsen, Elizabeth Lee-Rausch, and William T. Jones, 65th Annual AHS Forum, May 2009. See also AIAA Journal of Aircraft, Vol. 47, No. 2, March/April 2010, pp.638-646.

Calibration of a Unified Flux Limiter for Ares-Class Launch Vehicles from Subsonic to Supersonic Speeds, Veer N. Vatsa and Jeff A. White. Paper presented at 56th JANNAF Propulsion Meeting, Las Vegas, Nevada, April 14-17, 2009. (Note: This publication has security restrictions which preclude the manuscript from being included in full here.)

Recent Enhancements To The FUN3D Flow Solver For Moving-Mesh Applications, Robert T. Biedron and James L. Thomas, AIAA-2009-1360, January 2009.

Local-in-time Adjoint-based Method for Design Optimization of Unsteady Compressible Flows, N.K. Yamaleev, B. Diskin, and E.J. Nielsen, AIAA-2009-1169, January 2009.

Multi-Dimensional, Inviscid Flux Reconstruction for Simulation of Hypersonic Heating on Tetrahedral Grids, Peter A. Gnoffo, AIAA-2009-599, January 2009.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations. Part I: Viscous Fluxes, Boris Diskin, James L. Thomas, Eric J. Nielsen, Hiroaki Nishikawa, and Jeffrey A. White, AIAA-2009-0597, January 2009.

A Computational Study of the Aerodynamics and Aeroacoustics of a Flatback Airfoil Using Hybrid RANS-LES, Christopher Stone, Matthew Barone, C. Eric Lynch, and Marilyn J. Smith, AIAA-2009-0273, January 2009.

Analysis of Effectiveness of Phoenix Entry Reaction Control System, Artem A. Dyakonov, Christopher E. Glass, Prasun N. Desai, and John W. Van Norman, AIAA-2008-7220, August 2008.

Hybrid RANS-LES Turbulence Models on Unstructured Grids, C. Eric Lynch and Marilyn J. Smith, AIAA-2008-3854, June 2008.

Application of FUN3D and CFL3D to the Third Workshop on CFD Uncertainty Analysis, Chris L. Rumsey and James L. Thomas, NASA TM-2008-215537, November 2008.

Uncertainty Analysis of Computational Fluid Dynamics Via Polynomial Chaos, Rafael A. Perez, PhD Thesis, Virginia Tech, September 2008.

Anisotropic Output-Based Adaptation with Tetrahedral Cut Cells for Compressible Flows, Michael A. Park, PhD Thesis, Massachusetts Institute of Technology, September 2008.

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction, Michael A. Park and David L. Darmofal, AIAA-2008-6594, August 2008.

The Impact of Advanced Airfoils on Rotor Hover Performance, Tin-Chee Wong, AIAA-2008-7342, August 2008.

Rotor Airloads Prediction Using Unstructured Meshes and Loose CFD/CSD Coupling, Robert T. Biedron, Elizabeth Lee-Rausch, AIAA-2008-7341, August 2008.

Application of the FUN3D CFD Code to ARES I, ADAC2 Configurations, Veer N. Vatsa, Raymond E. Mineck, and Robert T. Biedron. Paper presented at 55th JANNAF Propulsion Meeting, Newton, Massachusetts, May 12-16, 2008. (Note: This publication has security restrictions which preclude the manuscript from being included in full here.)

Prediction of Launch Vehicle Aerodynamics Using a Node Based Unstructured Grid Solver, Veer N. Vatsa, Robert T. Biedron, and Raymond E. Mineck. Paper presented at 55th JANNAF Propulsion Meeting, Newton, Massachusetts, May 12-16, 2008. (Note: This publication has security restrictions which preclude the manuscript from being included in full here.)

Development of Advanced Computational Aeroelasticity Tools at NASA Langley Research Center, R.E. Bartels, NATO RTO Specialists Meeting on Advanced Aeroelasticity AVT-154, Paper 003, May 3-6, 2008.

An Examination of Engine Effects on Helicopter Aeromechanics, David M. O’Brien, Jr., Mark E. Calvert, Steven L. Butler, Presented at AHS Specialists’ Conference on Aeromechanics, January 2008.

Parallel Anisotropic Tetrahedral Adaptation, Michael A. Park, David L. Darmofal, AIAA-2008-917, January 2008.

Towards Verification of Unstructured-Grid Solvers, James L. Thomas, Boris Diskin, Christopher L. Rumsey, AIAA Journal, Vol. 46, No. 12, pp. 3070-3079, 2008. Also see AIAA-2008-666, January 2008.

Simulation of Stagnation Region Heating in Hypersonic Flow on Tetrahedral Grids, Peter A. Gnoffo, AIAA-2007-3960, June 2007.

Advances in Rotorcraft Simulations with Unstructured CFD, Jennifer N. Abras, C. Eric Lynch, and Marilyn J. Smith, Presented at the AHS 63rd Annual Forum, Virginia Beach, VA, May 2007.

Accuracy Analysis for Mixed-Element Finite-Volume Discretization Schemes, Boris Diskin, James Thomas, NIA Report No. 2007-08.

Breakthrough Advantage in Computational Fluid Dynamics with the IBM System Blue Gene Solution, IBM white paper by Srini Chari, September 2006.

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations, Peter A. Gnoffo, AIAA-2006-1090, January 2006.

Aerothermodynamic Analyses of Towed Ballutes, Peter A. Gnoffo, Greg Buck, James N. Moss, Rena Rudavsky, Eric Nielsen, Karen Berger, William T. Jones, AIAA-2006-3771, June 2006.

Parallel, Gradient-Based Anisotropic Mesh Adaptation for Re-entry Vehicle Configurations, Karen L. Bibb, Peter A. Gnoffo, Michael A. Park, William T. Jones, AIAA-2006-3579, June 2006.

Blade Contour Deformation and Helicopter Performance, Mark E. Calvert, Tin-Chee Wong, James A. O’ Malley III, AIAA-2006-3167, June 2006.

Computational Analysis of Dual Radius Circulation Control Airfoils, E.M. Lee-Rausch, V.N. Vatsa, C.L. Rumsey, AIAA-2006-3012, June 2006.

Investigation of Effect of Dynamic Stall and Its Alleviation on Helicopter Performance and Loads, T.-C. Wong, J.A. O’Malley III, D.M. O’Brien Jr., Presented at 62nd Annual AHS Forum, Phoenix, AZ, May 2006.

Analysis of Computational Modeling Techniques for Complete Rotorcraft Configurations, Dave O’Brien, PhD Thesis, Georgia Tech, May 2006.

Using An Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design, Eric J. Nielsen and Michael A. Park, AIAA Journal, Vol.44, No. 5, pp. 948-953, May 2006.

Validation of 3D Adjoint Based Error Estimation and Mesh Adaptation for Sonic Boom Prediction, William T. Jones, Eric J. Nielsen, Michael A. Park, AIAA-2006-1150, Jan 2006.

Sonic Boom Computations for Double-Cone Configuration Using CFL3D, FUN3D and Full-Potential Codes, Osama Kandil and Isik A. Ozcer, AIAA-2006-0414, Jan 2006.

Adjoint-Based Algorithms for Adaptation and Design Optimization on Unstructured Grids, Eric J. Nielsen, Invited Lecture at 3rd East-West High-Speed Flowfield Conference, Beijing, China, October 2005.

Computational Methods for Stability and Control (COMSAC): The Time Has Come, Robert M. Hall, Robert T. Biedron, Douglas N. Ball, David R. Bogue, James Chung, Bradford E. Green, Matthew J. Grismer, Gregory P. Brooks, and Joseph R. Chambers, AIAA-2005-6121, August 2005.

Simulation of Unsteady Flows Using an Unstructured Navier-Stokes Solver on Moving and Stationary Grids, Robert T. Biedron, Veer N. Vatsa, and Harold L. Atkins, AIAA-2005-5093, June 2005.

Computational Simulations and the Scientific Method, Bil Kleb and Bill Wood, AIAA-2005-4873, June 2005.

Application of Parallel Adjoint-Based Error Estimation and Anisotropic Grid Adaptation for Three-Dimensional Aerospace Configurations, Elizabeth M. Lee-Rausch, Michael A. Park, William T. Jones, Dana P. Hammond, Eric J. Nielsen, AIAA-2005-4842, June 2005.

Parallel Adaptive Solvers in Compressible PETSc-FUN3D Simulations, Sanjukta Bhowmick, Dinesh K. Kaushik, Lois C. McInnes, Boyana Norris, and Padma Raghavan, Proceedings of the Parallel CFD 2005 Conference, Baltimore, May 2005.

Analysis of Rotor-Fuselage Interactions Using Various Rotor Models, David M. O’Brien, Jr. and Marilyn J. Smith, AIAA-2005-0468, January 2005.

Using An Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design, Eric J. Nielsen and Michael A. Park, AIAA-2005-0491, January 2005.

Efficient Construction of Discrete Adjoint Operators on Unstructured Grids by Using Complex Variables, Eric J. Nielsen and William L. Kleb, AIAA Journal, Vol.44, No. 4, pp. 827-836. See also AIAA-2005-0324, January 2005.

Navier-Stokes Computations of Longitudinal Forces and Moments for a Blended Wing Body, S. Paul Pao, Robert T. Biedron, Michael A. Park, C. Michael Fremaux, and Dan D. Vicroy, AIAA-2005-0045, January 2005.

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation, Michael A. Park, AIAA Journal, Vol. 42, No. 9, 2004, pp. 1854-1862.

Aerodynamic Shape Optimization Based on Free-Form Deformation, Jamshid A. Samareh, AIAA 2004-4630, 2004.

Evaluation of Isolated Fuselage and Rotor-Fuselage Interaction Using CFD, Thomas Renaud, David O’Brien, Marilyn Smith, and Mark Potsdam, American Helicopter Society 60th Annual Forum, Baltimore, MD, June 7-10, 2004.

CFD: A Castle in the Sand?, Bil Kleb and Bill Wood, AIAA-2004-2627, June 2004.

Computational Aerothermodynamic Simulation Issues on Unstructured Grids, Peter A. Gnoffo and Jeffery A. White, AIAA-2004-2371, June 2004.

An Implicit, Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids, Eric J. Nielsen, James Lu, Michael A. Park, and David L. Darmofal, Computers and Fluids, Vol. 33, No. 9, pp. 1131-1155.

Transonic Drag Prediction on a DLR-F6 Transport Configuration Using Unstructured Grid Solvers, Elizabeth M. Lee-Rausch, Neal T. Frink, Dimitri J. Mavriplis, Russ D. Rausch and William E. Milholen, AIAA-2004-0554, January 2004.

Team Software Development for Aerothermodynamic and Aerodynamic Analysis and Design, N.M. Alexandrov, H.L. Atkins, K.L. Bibb, R.T. Biedron, M.H. Carpenter, P.A. Gnoffo, D.P. Hammond, W.T. Jones, W.L. Kleb, E.M. Lee-Rausch, E.J. Nielsen, M.A. Park, V.V. Raman, T.W. Roberts, J.L. Thomas, V.N. Vatsa, S.A. Viken, J.A. White, W.A. Wood, NASA TM-2003-212421, November 2003.

Computational Fluid Dynamics Technology for Hypersonic Applications, Peter A. Gnoffo, AIAA/ICAS International Air & Space Symposium and Exposition, Dayton, Ohio, AIAA 2003-3259, July 14-17, 2003.

Anisotropic Grid Adaptation for Functional Outputs: Application to Two-Dimensional Viscous Flows, David Venditti and David Darmofal, Journal of Computational Physics, Vol. 187, p. 22-46, 2003. (preprint form)

Collaborative Software Development in Support of Fast Adaptive AeroSpace Tools, William L. Kleb, Eric J. Nielsen, Peter A. Gnoffo, Michael A. Park, William A. Wood, AIAA-2003-3978, June 2003.

Three-Dimensional Turbulent RANS Adjoint-Based Error Correction, Michael A. Park, AIAA-2003-3849, June 2003.

CFD Sensitivity Analysis of a Drag Prediction Workshop Wing/Body Transport Configuration, E.M. Lee-Rausch, P. G. Buning, J. H. Morrison, M. A. Park, S. M. Rivers, C. L. Rumsey, AIAA-2003-3400, June 2003.

Exploring XP for Scientific Research, Willam A. Wood and William L. Kleb, IEEE Software, Vol. 20, No. 3, May/June 2003.

An Implicit, Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids, Eric Nielsen, James Lu, Mike Park, and Dave Darmofal, Computers and Fluids, Vol. 33, No. 9, pp. 1131-1155, November 2004. See also AIAA-03-0272.

The Efficiency of High Order Temporal Schemes, Mark Carpenter, Sally Viken, and Eric Nielsen, AIAA-03-0086, January 2003.

Flow Control Analysis on the Hump Model with RANS Tools, Sally Viken, Veer Vatsa, Chris Rumsey, and Mark Carpenter, AIAA-03-0218, January 2003.

Grid Adaptation for Functional Outputs of Compressible Flow Simulations, David A. Venditti, PhD. Dissertation, Massachusetts Institute of Technology, June, 2002.

Opportunities for Breakthroughs in Large-Scale Computational Simulation and Design, Langley FAAST Team, NASA-TM-211747, June 2002.

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation, Michael A. Park, AIAA-2002-3286, June 2002.

Three-Dimensional Effects on Multi-Element High Lift Computations, Christopher L. Rumsey, Elizabeth M. Lee-Rausch, Ralph D. Watson, AIAA-02-0845, January 2002.

Isolating Curvature Effects in Computing Wall-Bounded Turbulent Flows, Christopher L. Rumsey, Thomas B. Gatski, W. Kyle Anderson, and Eric J. Nielsen, Int. J. Heat and Fluid Flow, Vol 22, 2001, pp.573-582.

Factorizable Upwind Schemes: The Triangular Unstructured Grid Formulation, David Sidilkover and Eric J. Nielsen, AIAA-01-2575, June 2001.

Latency, Bandwidth, and Concurrent Issue Limitations in High-Performance CFD, William D. Gropp, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith. Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 2001.

A Scientific Data Management System for Irregular Applications, Jaechun No, Rajeev Thakur, Dinesh Kaushik, Lori Freitag, and Alok Choudhary, Proceedings of the Eighth International Workshop on Solving Irregular Problems in Parallel (Irregular 2001), April 2001.

High-Performance Parallel Implicit CFD, William D. Gropp, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith. Parallel Computing, Vol. 27, No. 4, March 2001.

Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes, Eric J. Nielsen and W. Kyle Anderson, AIAA Journal, Vol.40, No. 6, pp. 1155-1163. See also AIAA-01-0596, January 2001.

Understanding the Parallel Scalability of An Implicit Unstructured Mesh CFD Code, William D. Gropp, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith, Proceedings of the 7th International Conference on High Performance Computing (HiPC ‘2000), Bangalore, India, December, 2000, pp. 395-404.

Performance Modeling and Tuning of an Unstructured Mesh CFD Application, William D. Gropp, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith, Proceedings of SC 2000, IEEE Computer Society, 2000.

First-Order Model Management with Variable-Fidelity Physics Applied to Multi-Element Airfoil Optimization, N. M. Alexandrov, E. J. Nielsen, R. M. Lewis, and W. K. Anderson, AIAA-00-4886, September 2000.

Application of Adjoint Optimization Method to Multi-Element Rotorcraft Airfoils, Mark S. Chaffin, Presented at the American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco, CA, 2000.

Numerical Prediction of the Interference Drag of a Streamlined Strut Intersecting a Surface in Transonic Flow, P. A. Tetrault, PhD. Dissertation, Virginia Polytechnic Institute and State University, January, 2000.

Efficient Parallelization of an Unstructured Grid Solver: A Memory-Centric Approach, Dinesh K. Kaushik and David E. Keyes, Proceedings of the International Conference on Parallel CFD, Istanbul, Turkey, June 1999.

Implementation of a Parallel Framework for Aerodynamic Design Optimization on Unstructured Meshes, E.J. Nielsen, W.K. Anderson, and D.K. Kaushik, Presented at the 11th International Parallel CFD Conference, Williamsburg, Virginia, May 1999.

Sensitivity Analysis for the Navier-Stokes Equations on Unstructured Meshes Using Complex Variables, W. Kyle Anderson, James C. Newman, David L. Whitfield, and Eric J. Nielsen, AIAA-99-3294, June, 1999. (See also AIAA J. Vol. 39, No. 1, 2001, pp. 56-63).

Towards Realistic Performance Bounds for Implicit CFD Codes, William D. Gropp, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith, Proceedings of the Parallel CFD ‘99 Conference, Williamsburg, May 1999, pp. 241-248.

Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation, W. Kyle Anderson and V. Venkatakrishnan, Computers and Fluids, Vol. 28, Nos. 4-5, pp. 443-480.

Multiblock Approach for Calculating Incompressible Fluid Flows on Unstructured Grids, C. Sheng, D.L. Whitfield, and W. K. Anderson, AIAA Journal, Vol. 37, No. 2, pp. 169-176, 1999.

An O(Nm2) Plane Solver for the Compressible Navier-Stokes Equations, J. L. Thomas, D. L. Bonhaus, W. K. Anderson, C. L. Rumsey, and R. T. Biedron, AIAA 99-0785, January, 1999.

Three-Dimensional Incompressible Navier-Stokes Flow Computations About Complete Configurations Using a Multiblock Unstructured Grid Approach, C. Sheng, D. Hyams, K. Sreenivas, A. Gaither, D. Marcum, D. Whitfield, and W.K. Anderson, AIAA 99-0778, January, 1999.

Achieving High Sustained Performance in an Unstructured Mesh CFD Application, W.K. Anderson, W.D. Gropp, D.K. Kaushik, D.E. Keyes, and B.F. Smith, Bell Prize Award Paper, Special Category, in Proceedings of SC’99 1999.

Prospects for CFD on Petaflops Systems, David E. Keyes, Dinesh K. Kaushik, and Barry F. Smith, in CFD Review 1998, M. Hafez, et al eds, World Scientific, Singapore, pp. 1079-1096. Also published as ICASE Report Number 97-93, December, 1997 and in the IMA Lecture Series on ‘Parallel Solution of Partial Differential Equations’, edited by Peter Bjorstad and Mitchell Luskin, Vol. 120, Springer, 2000, pages 247-277.

A Higher Order Accurate Finite Element Method for Viscous Compressible Flows, D. L. Bonhaus, PhD. Dissertation, Virginia Polytechnic Institute and State University, December, 1998.

Aerodynamic Design Sensitivities on an Unstructured Mesh Using the Navier-Stokes Equations and a Discrete Adjoint Formulation, E. J. Nielsen, PhD. Dissertation, Virginia Polytechnic Institute and State University, December, 1998.

Multidisciplinary Sensitivity Derivatives Using Complex Variables, J. C. Newman, W. K. Anderson, and D. L. Whitfield, MSSU-COE-ERC-98-08 (Mississippi State University), July 1998.

Newton-Krylov-Schwarz Methods for Aerodynamics Problems: Compressible and Incompressible Flows on Unstructured Grids, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith. Submitted to Proceedings of the Eleventh International Conference on Domain Decomposition Methods, Greenwich, July 1998.

Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations, Eric J. Nielsen and W. Kyle Anderson. AIAA 98-4809, September, 1998. (See also AIAA Journal Vol. 37, No. 11, 1999, pp. 1411-1419.)

The Use of Heavy Gas for Increased Reynolds Numbers in Transonic Wind Tunnels, J. B. Anders, W. K. Anderson, and A. V. Murthy. AIAA 98-2882, June, 1998.

Porting FUN3D to Distributed Memory Parallelism, David E. Keyes, Dinesh K. Kaushik, Barry F. Smith, and W. Kyle Anderson. Parallel Computing Research, Vol. 5, No. 4, CRPC, Fall 1997.

On the Interaction of Architecture and Algorithm in the Domain-Based Parallelization of an Unstructured Grid Incompressible Flow Code, Dinesh K. Kaushik, David E. Keyes, and Barry F. Smith. Proceedings of the Tenth International Conference on Domain Decomposition Methods, Boulder, CO, August 1997, pages 311-319.

Airfoil Design on Unstructured Grids for Turbulent Flows, W. Kyle Anderson, Daryl L. Bonhaus, Submitted for Publication June, 1997.

Aerodynamic Design on Unstructured Grids for Turbulent Flows, W. Kyle Anderson, Daryl L. Bonhaus, NASA Technical Memorandum 112867, June, 1997.

A Multiblock Approach for Calculating Incompressible Fluid Flows on Unstructured Grids, Chunhua Sheng, David L. Whitfield, and W. Kyle. Anderson, AIAA 97-1866, June, 1997.

Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation, W. Kyle. Anderson, and V. Venkatakrishnan, AIAA 97-0643, January, 1997.

Navier-Stokes Computations and Experimental Comparisons for Multielement Airfoil Configurations, W. K. Anderson, and Daryl L. Bonhaus, J. Aircraft, Vol. 32, No. 6, pp. 1246-1253, Nov. 1995. (See also AIAA 93-0645.)

Application of Newton-Krylov Methodology to A Three Dimensional Unstructured Euler Code, E. Nielsen, W. K. Anderson, R. Walters, and D. Keyes, AIAA 95-1733-CP, June, 1995.

Parallel Algorithms of Newton-Krylov-Schwarz Type, David E. Keyes, ICASE Research Quarterly, Vol. 4, No. 1, March 1995.

An Upwind Multigrid Method for Solving Viscous Flows on Unstructured Triangular Meshes, Daryl L. Bonhaus, M.S. Thesis, George Washington University, Aug. 1993.

Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids, W. K. Anderson, Russ D. Rausch, and Daryl L. Bonhaus, AIAA 95-1740 (J. Comp. Phys. Vol. 128, 1996, pp. 391-408).

An Implicit Upwind Algorithm for Computing Turbulent Flows on Unstructured Grids, W. K. Anderson, and Daryl L. Bonhaus, Computers and Fluids, Vol. 23, No. 1. pp. 1-21, 1994.

Grid Generation and Flow Solution Method for Euler Equations on Unstructured Grids, W. K. Anderson, NASA TM 4295, April 1992.

### Publication Abstracts and Documents

**Toward a Comprehensive Model of Jet Noise Using an
Acoustic Analogy**
(8.2 MB PDF)

An acoustic analogy is developed to predict the noise from jet flows. It contains two source models that independently predict the noise from turbulence and shock wave shear layer interactions. The acoustic analogy is based on the Euler equations and separates the sources from propagation. Propagation effects are taken into account by approximating the vector Greenâ€™s function of the linearized Euler equations with the use of a locally parallel mean flow assumption. A statistical model of the two-point cross correlation of the velocity fluctuations is used to describe the turbulence. The acoustic analogy attempts to take into account the correct scaling of the sources for a wide range of nozzle pressures and temperature ratios. It does not make assumptions regarding fine- or large-scale turbulent noise sources, self- or shear noise, or convective amplification. The acoustic analogy is partially informed by three- dimensional steady Reynolds-averaged Navierâ€“Stokes solutions that include the nozzle geometry. The predictions are compared with experiments of jets operating subsonically through supersonically and at unheated and heated temperatures. Predictions generally capture the scaling of both mixing noise and broadband shock-associated noise for the conditions examined, but some discrepancies remain, which are due to the accuracy of the steady Reynolds- averaged Navierâ€“Stokes turbulence model closure, the equivalent sources, and the use of a simplified vector Greenâ€™s function solver of the linearized Euler equations using a locally parallel mean flow.

**FUN3D Manual: 12.5**
(1.6 MB PDF)

This manual describes the installation and execution of FUN3D version 12.5, including optional dependent packages. FUN3D is a suite of computational fluid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables efficient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status.

**Evaluation of Linear, Inviscid, Viscous, and Reduced-Order
Modeling Aeroelastic Solutions of the AGARD 445.6 Wing Using Root Locus Analysis**
(3.4 MB PDF)

Reduced-order modelling (ROM) methods are applied to the Computational Fluid Dynamics (CFD)-based aeroelastic analysis of the AGARD 445.6 wing in order to gain insight regarding well-known discrepancies between the aeroelastic analyses and the experimental results. The results presented include aeroelastic solutions using the inviscid Computational Aeroelasticity Programmeâ€“Transonic Small Disturbance (CAP-TSD) code and the FUN3D code (Euler and Navierâ€“Stokes). Full CFD aeroelastic solutions and ROM aeroelastic solutions, computed at several Mach numbers, are presented in the form of root locus plots in order to better reveal the aeroelastic root migrations with increasing dynamic pressure. Important conclusions are drawn from these results including the ability of the linear CAP-TSD code to accurately predict the entire experimental flutter boundary (repeat of analyses performed in the 1980s), that the Euler solutions at supersonic conditions indicate that the third mode is always unstable, and that the FUN3D Navierâ€“Stokes solutions stabilize the unstable third mode seen in the Euler solutions.

**Entropy Stable Spectral Collocation Schemes for the
Navier-Stokes Equations: Discontinuous Interfaces**
(0.6 MB PDF)

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial- based spectral collocation element methods of arbitrary order, for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements, but conserve entropy for the Euler equations and are entropy stable for the Navier-Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy.

**Feature-Based Grid Adaption for the Study of Dynamic Stall**
(3.1 MB PDF)

In this paper we study dynamic stall phenomenon of a pitching NACA0015 airfoil using feature based grid adaption technique. The fluid solver is based on FUN3D which solves the three-dimensional, compressible, unsteady Reynolds-Averaged Naiver-Stokes equations. The one equation Spalart-Allmaras is used a the turbulence closure. The governing equations are discretized spatially using second-order finite volume methods and temporally using an optimized second order backward difference scheme. The grid adaption is based the anisotropic tetrahedral adaptation approach in which grid is adapted to match a desired quality via an anisotropic metric calculated throughout the simulation. Particularly we track vorticity throughout the pitching cycle and adapt the grid in areas where vorticity is damped. Comparisons are also made with results using uniformly refined grids. Results suggest feature-based adaptation has potential in refining the mesh in the wake of the airfoil, allowing vorticity to be carried out several chords behind the airfoil without excessive dissipation. Our study also shows that great care must be placed in allowing the grid to be adapted in the vicinity of the airfoil as grid resolution can be lost.

**Towards Full Aircraft Airframe Noise Prediction: Detached Eddy
Simulations**
(5.0 MB PDF)

Results from a computational study on the aeroacoustic characteristics of an 18%-scale, semi-span Gulfstream aircraft model are presented in this paper. NASAâ€™s FUN3D unstructured compressible Navier-Stokes solver was used to perform steady and unsteady simulations of the flow field associated with this high-fidelity aircraft model. Solutions were obtained for free-air at a Mach number of 0.2 with the flap deflected at 39o, with the main gear off and on (the two baseline configurations). Initially, the study focused on accurately predicting the prominent noise sources at both flap tips for the baseline configuration with deployed flap only. Building upon the experience gained from this initial effort, subsequent work involved the full landing configuration with both flap and main landing gear deployed. For the unsteady computations, we capitalized on the Detached Eddy Simulation capability of FUN3D to capture the complex time-dependent flow features associated with the flap and main gear. To resolve the noise sources over a broad frequency range, the tailored grid was very dense near the flap inboard and outboard tips and the region surrounding the gear. Extensive comparison of the computed steady and unsteady surface pressures with wind tunnel measurements showed good agreement for the global aerodynamic characteristics and the local flow field at the flap inboard tip. However, the computed pressure coefficients indicated that a zone of separated flow that forms in the vicinity of the outboard tip is larger in extent along the flap span and chord than measurements suggest. Computed farfield acoustic characteristics from a FW-H integral approach that used the simulated pressures on the model solid surface were in excellent agreement with corresponding measurements.

**Computed and Experimental Flutter/LCO Onset for the Boeing
Truss-Braced Wing Wind-Tunnel Model**
(8.1 MB PDF)

This paper presents high fidelity Navier-Stokes simulations of the Boeing Subsonic Ultra Green Aircraft Research truss-braced wing wind-tunnel model and compares the results to linear MSC.Nastran flutter analysis and preliminary data from a recent wind-tunnel test of that model at the NASA Langley Research Center Transonic Dynamics Tunnel. The simulated conditions under consideration are zero angle of attack, so that structural nonlinearity can be neglected. It is found that, for Mach numbers greater than 0.78, the linear flutter analysis predicts flutter onset dynamic pressure below that of the Navier-Stokes analysis and significantly below that found in the wind tunnel test. Furthermore, the wind-tunnel test revealed that the majority of the high structural dynamics cases were wing limit cycle oscillation (LCO) rather than flutter. Most Navier-Stokes simulated cases were also LCO rather than hard flutter. There is dip in the wind-tunnel test flutter/LCO onset in the Mach 0.76 âˆ’ 0.80 range. Conditions tested above that Mach number exhibited no aeroelastic instability at the dynamic pressures reached in the tunnel. The linear flutter analyses do not show a flutter/LCO dip. The Navier-Stokes simulations reveal a slight dip in onset at Mach 0.82, then a significantly higher onset dynamic pressure at Mach 0.90. The Navier-Stokes simulations indicate a mild LCO onset at Mach 0.82, then a more rapidly growing instability at Mach 0.86 and 0.90. Finally, the modeling issues and their solution related to the use of a beam and pod finite element model to generate the Navier-Stokes structure mode shapes are discussed.

**Grid-Adapted FUN3D Computations for the Second High Lift
Prediction Workshop**
(3.2 MB PDF)

Contributions of the unstructured Reynolds-averaged Navier-Stokes code FUN3D to the 2nd AIAA CFD High Lift Prediction Workshop are described, and detailed comparisons are made with experimental data. Using workshop-supplied grids, results for the clean wing configuration are compared with results from the structured code CFL3D Using the same turbulence model, both codes compare reasonably well in terms of total forces and moments, and the maximum lift is similarly over-predicted for both codes compared to experiment. By including more representative geometry features such as slat and flap brackets and slat pressure tube bundles, FUN3D captures the general effects of the Reynolds number variation, but under-predicts maximum lift on workshop-supplied grids in comparison with the experimental data, due to excessive separation. However, when output-based, off-body grid adaptation in FUN3D is employed, results improve considerably. In particular, when the geometry includes both brackets and the pressure tube bundles, grid adaptation results in a more accurate prediction of lift near stall in comparison with the wind-tunnel data. Furthermore, a rotation-corrected turbulence model shows improved pressure predictions on the outboard span when using adapted grids.

**Simulation of a Variety of Wings Using a Reynolds Stress
Model**
(5.1 MB PDF)

The Wilcox 2006 stress-Ï‰ model, a Reynolds stress model (RSM), implemented in both the NASA Langley codes FUN3D and CFL3D have been used to study a number of 2-D and 3-D cases. This study continues the assessments of the stress-Ï‰ model by simulating the flow over two wings: the DPW-W1 and the DLR-F11 wings. Using FUN3D, which uses unstructured grids, and CFL3D, which uses structured grid, the results were compared to solvers employing one- and two-equation turbulence models and experimental data. In general, in situations where experimental data is available, the stress- Ï‰ model performs as well or better than one- and two-equation models.

**Uncertainty Quantification and Certification Prediction of
Low-Boom Supersonic Aircraft Configurations**
(7.0 MB PDF)

The primary objective of this work was to develop and demonstrate a process for accurate and efficient uncertainty quantification and certification prediction of low-boom, supersonic, transport aircraft. High-fidelity computational fluid dynamics models of multiple low-boom configurations were investigated including the Lockheed Martin SEEB-ALR body of revolution, the NASA 69â—¦ Delta Wing, and the Lockheed Martin 1021-01 configuration. A nonintrusive polynomial chaos surrogate modeling approach was used for reduced computational cost of propagating mixed, inherent (aleatory) and model-form (epistemic) uncertainty from both the computation fluid dynamics model and the near-field to ground level propagation model. A methodology has also been introduced to quantify the plausibility of a design to pass a certification under uncertainty. Results of this study include the analysis of each of the three configurations of interest under inviscid and fully turbulent flow assumptions. A comparison of the uncertainty outputs and sensitivity analyses between the configurations is also given. The results of this study illustrate the flexibility and robustness of the developed framework as a tool for uncertainty quantification and certification prediction of low-boom, supersonic aircraft.

**Summary and Statistical Analysis of the First AIAA Sonic Boom
Prediction Workshop**
(3.9 MB PDF)

A summary is provided for the First AIAA Sonic Boom Workshop held 11 January 2014 in conjunction with AIAA SciTech 2014. Near-field pressure signatures extracted from computational fluid dynamics solutions are gathered from nineteen participants representing three countries for the two required cases, an axisymmetric body and simple delta wing body. Structured multiblock, unstructured mixed-element, unstructured tetrahedral, overset, and Cartesian cut-cell methods are used by the participants. Participants provided signatures computed on participant generated and solution adapted grids. Signatures are also provided for a series of uniformly refined workshop provided grids. These submissions are propagated to the ground and noise measures are computed. This allows the grid convergence of a noise measure and a validation metric (difference norm between computed and wind tunnel measured near-field signatures) to be studied for the first time. A statistical analysis is also presented for these measures. An optional configuration includes fuselage, wing, tail, flow-through nacelles, and blade sting. This full configuration exhibits more variation in eleven submissions than the sixty submissions provided for each required case. Recommendations are provided for potential improvements to the analysis methods and a possible subsequent workshop.

**FUN3D Manual: 12.4**
(1.4 MB PDF)

This manual describes the installation and execution of FUN3D version 12.4, including optional dependent packages. FUN3D is a suite of computational fluid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables efficient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status.

**Noise Generated by an Airfoil Located in the Wake of a Circular
Cylinder**
(0.5 MB PDF)

In this paper, a problem involving noise radiation from a bluff body is solved numerically using a hybrid RANS-LES method. In particular the problem of noise radiated by an airfoil leading edge located in the wake of a circular cylinder is addressed. Our results compare well to experimental measurements and other CFD computations. It is found that the hybrid RANS-LES method is able to resolve enough turbulent scales to compute the nearfield noise spectra and the directivity pattern. Our CFD results indicate that the coherent structures are responsible for the peak Strouhal number in the spectra.

**Sonic Boom Mitigation Through Aircraft Design and
Adjoint Methodology**
(8.1 MB PDF)

This paper presents a novel approach to design of the supersonic aircraft outer mold line by optimizing a Aweighted loudness-based objective of the sonic-boom signature predicted on the ground. The optimization process uses the sensitivity information obtained by coupling the discrete adjoint formulations for the augmented Burgers equation and computational-fluid-dynamics equations. This coupled formulation links the loudness of the ground boom signature to the aircraft geometry, thus allowing efficient shape optimization for the purpose of minimizing the loudness. The accuracy of the adjoint-based sensitivities is verified against sensitivities obtained using an independent complex-variable approach. The adjoint-based optimization methodology is applied to a configuration previously optimized using alternative state-of-the-art optimization methods and produces additional loudness reduction. The results of the optimizations are reported and discussed.

**Application of the FUN3D Solver to the 4th AIAA Drag Prediction
Workshop**
(7.6 MB PDF)

FUN3D Navierâ€“Stokes solutions were computed for the 4th AIAA Drag Prediction Workshop grid-convergence study, downwash study, and Reynolds-number study on a set of node-based mixed-element grids. All of the baseline tetrahedral grids were generated with the VGRID (developmental) advancing-layer and advancing-front grid-generation software package following the gridding guidelines developed for the workshop. With maximum grid sizes exceeding 100 million nodes, the grid-convergence study was particularly challenging for the node-based unstructured grid generators and flow solvers. At the time of the workshop, the super-fine grid with 105 million nodes and 600 million tetrahedral elements was the largest grid known to have been generated using VGRID. FUN3D Version 11.0 has a completely new pre- and postprocessing paradigm that has been incorporated directly into the solver and functions entirely in a parallel, distributed-memory environment. This feature allowed for practical preprocessing and solution times on the largest unstructured-grid size requested for the workshop. For the constant-lift grid-convergence case, the convergence of total drag is approximately second-order on the finest three grids. The variation in total drag between the finest two grids is only two counts. At the finest grid levels, only small variations in wing and tail pressure distributions are seen with grid refinement. Similarly, a small wing side-of-body separation also shows little variation at the finest grid levels. Overall, the FUN3D results compare well with the structured-grid code CFL3D. For the grid-convergence case, the FUN3D total and component forces/moments are within one standard deviation of the workshop core solution medians and are very close to the median values especially at the finest grid levels. The FUN3D downwash study and Reynolds-number study results also compare well with the range of results shown in the workshop presentations.

**Unstructured Overset Mesh Adaptation with Turbulence
Modeling for Unsteady Aerodynamic Interactions**
(18.7 MB PDF)

Schemes for anisotropic grid adaptation for dynamic overset simulations are presented. These approaches permit adaptation over a periodic time window in a dynamic flowfield so that an accurate evolution of the unsteady wake may be obtained, as demonstrated on an unstructured flow solver. Unlike prior adaptive schemes, this approach permits grid adaptation to occur seamlessly across any number of grids that are overset, excluding only the boundary layer to avoid surface manipulations. A demonstration on a rotor/fuselage-interaction configuration includes correlations with time-averaged and instantaneous fuselage pressures, and wake trajectories. Additionally, the effects of modeling the flow as inviscid and turbulent are reported. The ability of the methodology to improve these predictions is confirmed, including a vortex/fuselage-impingement phenomenon that has before now not been captured by computational simulations. The adapted solutions exhibit dependency based on the choice of the feature to form the adaptation indicator, indicating that there is no single best practice for feature-based adaptation across the spectrum of rotorcraft applications.

**Overset Adaptive Strategies for Complex Rotating Systems**
(22.3 MB PDF)

The resolution of the complex physics of rotating configurations is critical for any engineering analysis that requires multiple frames of reference. Two well-known applications are in the rotorcraft and wind energy industries. Rotor wake impingement from rotor-fuselage and wind turbine-tower interactions impact structural and acoustic characteristics. Additionally, parasite drag resulting from rotorcraft hubs may result in severe limitations on forward flight vehicle performance. Complex turbulent wakes from rotors and hubs impinging on downstream empennage can create adverse aeroelastic behavior and can affect handling qualities. Numerical simulations of these flows require state-of-the-art Navier Stokes methods using dynamic overset grids. However, many current methods typically used in industry result in wakes that dissipate essential features. In order to address these concerns, two advancements are introduced in this thesis.

Feature-based grid adaptation on dynamic overset grids has been developed and demonstrated with an unstructured Navier Stokes solver. The unique feature of the adaptation technique is that it is applied globally on the overset grid system except within the boundary layer. In concert with grid adaptation, an efficient parallelized search algorithm for solution interpolation over massively distributed systems has been created. This results in cost-effective interpolation that retains the numerical order of accuracy and has been verified in both space and time. The improvements have been demonstrated for rotor-fuselage interaction and a generic rotating hub. Detailed analysis of convergence of the methodology and sensitivity of the results to relevant parameters have also been included.

**Analytical Correlation of a Flexible Empennage Wind
Tunnel Flutter Test at High Transonic Mach Number**
(10.2 MB PDF)

The current paper presents the correlation of a high fidelity computational aeroelastic analysis of a flexible empennage model at high transonic Mach number to wind tunnel flutter test data acquired from NASA Langley Research Center’s Transonic Dynamics Tunnel (TDT).

**The NASA High Speed ASE Project: Computational
Analyses of a Low-Boom Supersonic Configuration**
(8.1 MB PDF)

A summary of NASA’s High Speed Aeroservoelasticity (ASE) project is provided with a focus on a low-boom supersonic configuration developed by Lockheed-Martin and referred to as the N+2 configuration. The summary includes details of the computational models developed to date including a linear finite element model (FEM), linear unsteady aerodynamic models, structured and unstructured CFD grids, and discussion of the FEM development including sizing and structural constraints applied to the N+2 configuration. Linear results obtained to date include linear mode shapes and linear utter boundaries. In addition to the tasks associated with the N+2 configuration, a summary of the work involving the development of AeroPropulsoServoElasticity (APSE) models is also discussed.

**Evaluation of Linear, Inviscid, Viscous, and
Reduced-Order Modeling Aeroelastic Solutions of the
AGARD 445.6 Wing Using Root Locus Analysis**
(3.0 MB PDF)

Reduced-order modeling (ROM) methods are applied to the CFD-based aeroelastic anal- ysis of the AGARD 445.6 wing in order to gain insight regarding well-known discrepancies between the aeroelastic analyses and the experimental results. The results presented include aeroelastic solutions using the inviscid CAP-TSD code and the FUN3D code (Euler and Navier-Stokes). Full CFD aeroelastic solutions and ROM aeroelastic solutions, com- puted at several Mach numbers, are presented in the form of root locus plots in order to better reveal the aeroelastic root migrations with increasing dynamic pressure. Important conclusions are drawn from these results including the ability of the linear CAP-TSD code to accurately predict the entire experimental flutter boundary (repeat of analyses performed in the 1980’s), that the Euler solutions at supersonic conditions indicate that the third mode is always unstable, and that the FUN3D Navier-Stokes solutions stabilize the unstable third mode seen in the Euler solutions.

**Multi-point Adjoint-Based Design of Tilt-Rotors in a
Noninertial Reference Frame**
(0.6 MB PDF)

Optimization of tilt-rotor systems requires the consideration of performance at multiple design points. In the current study, an adjoint-based optimization of a tilt-rotor blade is considered. The optimization seeks to simultaneously maximize the rotorcraft figure of merit in hover and the propulsive efficiency in airplane-mode for a tilt-rotor system. The design is subject to minimum thrust constraints imposed at each design point. The rotor flowfields at each design point are cast as steady-state problems in a noninertial reference frame. Geometric design variables used in the study to control blade shape include: thickness, camber, twist, and taper represented by as many as 123 separate design variables. Performance weighting of each operational mode is considered in the formulation of the composite objective function, and a build up of increasing geometric degrees of freedom is used to isolate the impact of selected design variables. In all cases considered, the resulting designs successfully increase both the hover figure of merit and the airplane-mode propulsive efficiency for a rotor designed with classical techniques.

**Unsteady Aerodynamic Validation Experiences from the
Aeroelastic Prediction Workshop**
(7.9 MB PDF)

The AIAA Aeroelastic Prediction Workshop (AePW) was held in April 2012, bringing together communities of aeroelasticians, computational fluid dynamicists and experimentalists. The extended objective was to assess the state of the art in computational aeroelastic methods as practical tools for the prediction of static and dynamic aeroelastic phenomena. As a step in this process, workshop participants analyzed unsteady aerodynamic and weakly-coupled aeroelastic cases. Forced oscillation and unforced system experiments and computations have been compared for three configurations. This paper emphasizes interpretation of the experimental data, computational results and their comparisons from the perspective of validation of unsteady system predictions. The issues examined in detail are variability introduced by input choices for the computations, post-processing, and static aeroelastic modeling. The final issue addressed is interpreting unsteady information that is present in experimental data that is assumed to be steady, and the resulting consequences on the comparison data sets.

**Specialized CFD Grid Generation Methods for
Near-Field Sonic Boom Prediction**
(4.4 MB PDF)

Ongoing interest in analysis and design of low sonic boom supersonic transports requires accurate and efficient Computational Fluid Dynamics (CFD) tools. Specialized grid generation techniques are employed to predict near-field acoustic signatures of these configurations. A fundamental examination of grid properties is performed including grid alignment with flow characteristics and element type. The issues affecting the robustness of cylindrical surface extrusion are illustrated. This study will compare three methods in the extrusion family of grid generation methods that produce grids aligned with the freestream Mach angle. These methods are applied to configurations from the First AIAA Sonic Boom Prediction Workshop.

**Evaluation of Multigrid Solutions for Turbulent Flows**
(21.4 MB PDF)

A multigrid methodology has been recently developed in NASA solver, FUN3D, and successfully applied for a wide range of turbulent flows, from simple two-dimensional geometries to realistic three-dimensional configurations. The methodology is applicable to structured- and unstructured-grid solutions and includes both regular and agglomerated coarser meshes. Significant speed-ups over single-grid computations have been demonstrated. In the current work, we report on further enhancements of the relaxation scheme, and a detailed evaluation of the solver performance in computing benchmark turbulent flows. For those benchmark computations, multigrid solutions are compared with the corresponding single-grid solutions in terms of time-to-solution characteristics measured in the same computing environment. Implementation strategies on grids of various types are also discussed.

**An Efficient Actuating Blade Model for Unsteady Rotating
System Wake Simulations**
(3.2 MB PDF)

This paper describes an innovative, efficient actuating blade model to capture the unsteady motion of a rotating system within Computational Fluid Dynamics (CFD) methods, with application to wind turbine blades. Each blade planform is modeled via a cloud of sources that move independently during the simulation to provide rotation of the blade as well as optional motion such as blade flexibility (aeroelasticity) and active controls (flaps, morphing, adaptive shapes). The model can be implemented into structured or unstructured methods that span the gamut from full potential to large eddy simulations (LES), and it does not require the use of overset grids. A key feature of this model is the development of a highly efficient parallelized kd-tree algorithm to determine the interactions between actuator sources and grid nodes. Computational evaluation of the method successfully demonstrates its capability to predict root and tip vortex location and strength compared to an overset Navier-Stokes methodology on an identical background grid, and further improvements in the solution are shown by the use of grid adaptation.

**Entropy Stable Spectral Collocation Schemes for the Navier-Stokes
Equations: Discontinuous Interfaces**
(0.6 MB PDF)

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.

**CFD Analysis and Design Optimization of Flapping Wing Flows**
(4.1 MB PDF)

The main objectives of this research work are to perform the CFD analysis of the 3-D flow around a flapping wing in a gusty environment and to optimize its kinematics and shape to maximize the performance. The effects of frontal, side, and downward wind gusts on the aerodynamic characteristics of a rigid wing undergoing insect-based flapping motion are analyzed numerically. The turbulent, low-Reynolds-number flow near a flapping wing is governed by the 3-D unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the Spalart-Allmaras turbulence model. The governing equations are solved using a second-order node-centered finite volume method on a hexahedral mesh that rigidly moves along with the wing. Our numerical results show that a centimeter-scale wing considered is susceptible to strong downward wind gusts. In the case of frontal and side gusts, the flapping wing can alleviate the gust effect if the gust velocity is less than or comparable to the wing tip velocity. The second objective is to optimize the wing kinematics and shape to improve its aerodynamic characteristics. To our knowledge, this is the first attempt to perform high-fidelity combined optimization of flapping wing kinematics and shape in 3-D unsteady turbulent flows. For our optimization studies, an adjoint-based gradient method using the method of Lagrange multipliers is employed to minimize an objective functional with the 3D URANS and grid equations as constraints. It has been shown that some unsteady phenomena such as the clap and fling mechanism found in use by flying insects (e.g., a wasp Encarsaria formosa, or greenhouse white-fly Trialeurodes vaporariorium), maximize the wing propulsive efficiency. These results indicate that the time-dependent adjoint-based optimization method is an efficient tool for design of a new generation of micro air vehicles.

**Coupled CFD/CSD Analysis of Rotor Blade Structural Loads with
Experimental Validation**
(11.8 MB PDF)

An unsteady Reynolds averaged Navier-Stokes analysis loosely coupled with a comprehensive rotorcraft code for blade trim and aeroelastic effects is presented for a second-generation Active Twist Rotor. High fidelity Navier-Stokes results are compared to lifting-line theory based comprehensive rotorcraft code calculations and wind tunnel data. Results indicate that the CFD/CSD solutions are mesh converged and in very good agreement with flapwise bending moments for both the low and high advance ratio cases presented. The accuracy of the predicted rotor torque is also very good across the full sweep of advance ratio cases available for comparison with data.

**Developing an Accurate CFD Based Gust Model for the Truss
Braced Wing Aircraft**
(3.8 MB PDF)

The increased flexibility of long endurance aircraft having high aspect ratio wings necessitates attention to gust response and perhaps the incorporation of gust load alleviation. The design of civil transport aircraft with a strut or truss-braced high aspect ratio wing furthermore requires gust response analysis in the transonic cruise range. This requirement motivates the use of high fidelity nonlinear computational fluid dynamics (CFD) for gust response analysis. This paper presents the development of a CFD based gust model for the truss braced wing aircraft. A sharp-edged gust provides the gust system identification. The result of the system identification is several thousand time steps of instantaneous pressure coefficients over the entire vehicle. This data is filtered and downsampled to provide the snapshot data set from which a reduced order model is developed. A stochastic singular value decomposition algorithm is used to obtain a proper orthogonal decomposition (POD). The POD model is combined with a convolution integral to predict the time varying pressure coefficient distribution due to a novel gust profile. Finally the unsteady surface pressure response of the truss braced wing vehicle to a one-minus-cosine gust, simulated using the reduced order model, is compared with the full CFD.

**Application of Adjoint Methodology to Supersonic Aircraft
Design Using Reversed Equivalent Areas**
(1.2 MB PDF)

This paper presents an approach to shape an aircraft to equivalent area based objectives using the discrete adjoint approach. Equivalent areas can be obtained either using reversed augmented Burgers equation or direct conversion of off-body pressures into equivalent area. Formal coupling with CFD allows computation of sensitivities of equivalent area objectives with respect to aircraft shape parameters. The exactness of the adjoint sensitivities is veried against derivatives obtained using the complex step approach. This methodology has the benet of using designer-friendly equivalent areas in the shape design of low-boom aircraft. Shape optimization results with equivalent area cost functionals are discussed and further rened using ground loudness based objectives.

**Functional Equivalence Acceptance Testing of FUN3D for Entry, Descent, and Landing Applications**
(3.0 MB PDF)

The functional equivalence of the unstructured grid code FUN3D to the the structured grid code LAURA (Langley Aerothermodynamic Upwind Relaxation Algorithm) is documented for applications of interest to the Entry, Descent, and Landing (EDL) community. Examples from an existing suite of regression tests are used to demonstrate the functional equivalence, encompassing various thermochemical models and vehicle configurations. Algorithm modifications required for the node-based unstructured grid code (FUN3D) to reproduce functionality of the cell-centered structured code (LAURA) are also documented. Challenges associated with computation on tetrahedral grids versus computation on structured-grid derived hexahedral systems are discussed.

**Numerical Simulation of the Aircraft Wake Vortex Flowfield**
(0.8 MB PDF)

The near-wake vortex flowfield from a NACA0012 half-wing was simulated using a fully unstructured Navier-Stokes flow solver in three dimensions at a chord Reynolds number of 4.6 million and a Mach number of approximately 0.15. Several simulations were performed to examine the effects of boundary conditions, mesh resolution, and turbulence scheme on the formation of wingtip vortex and its downstream propagation. The standard Spalart-Allmaras turbulence model was compared with the Dacles-Mariani and Spalart-Shur corrections for rotation and curvature effects. The simulation results were evaluated using the data from an experiment performed at NASA Ames’ 32in x 48in low speed wind tunnel.

**Adjoint-Based Shape and Kinematics Optimization of
Flapping Wing Propulsive Efficiency**
(0.9 MB PDF)

Optimization of the 3-D unsteady viscous flow near a flapping wing is performed using a time-dependent adjoint-based methodology developed in [AIAA 2008-5857 and AIAA J. Vol.48, No.6, pp.1195-1206, 2010]. Sensitivities of the thrust and propulsive efficiency to wing shape and kinematic parameters are computed using the time-dependent discrete adjoint formulation. The unsteady discrete adjoint equations required for calculation of the sensitivity derivatives are integrated backward in time over the entire interval of interest. The gradient of the objective functional obtained using the adjoint formulation is then used to update the values of shape and kinematic design variables. The efficiency of this adjoint-based methodology is demonstrated by optimizing shape and kinematics of a wing undergoing insect-based flapping motion. Our numerical results show that the highest improvement in the thrust and propulsive efficiency is obtained by using the combined optimization of wing shape and kinematics.

**Discrete Adjoint-Based Design for Unsteady Turbulent
Flows on Dynamic Overset Unstructured Grids**
(8.5 MB PDF)

A discrete adjoint-based design methodology for unsteady turbulent flows on three-dimensional dynamic overset unstructured grids is formulated, implemented, and verified. The methodology supports both compressible and incompressible flows and is amenable to massively parallel computing environments. The approach provides a general framework for performing highly efficient and discretely consistent sensitivity analysis for problems involving arbitrary combinations of overset unstructured grids that may be static, undergoing rigid or deforming motions, or any combination thereof. General parent-child motions are also accommodated, and the accuracy of the implementation is established using an independent verification based on a complex-variable approach. The methodology is used to demonstrate aerodynamic optimizations of a wind-turbine geometry, a biologically inspired flapping wing, and a complex helicopter configuration subject to trimming constraints. The objective function for each problem is successfully reduced, and all specified constraints are satisfied.

**Towards a Comprehensive Model of Jet Noise using an
Acoustic Analogy and Steady RANS Solutions**
(9.4 MB PDF)

An acoustic analogy is developed to predict the noise from jet flows. It contains two source models that independently predict the noise from turbulence and shock wave shear layer interactions. The acoustic analogy is based on the Euler equations and separates the sources from propagation. Propagation effects are taken into account by calculating the vector Greenâ€™s function of the linearized Euler equations. The sources are modeled following the work of Tam and Auriault, Morris and Boluriaan, and Morris and Miller. A statistical model of the two-point cross-correlation of the velocity fluctuations is used to describe the turbulence. The acoustic analogy attempts to take into account the correct scaling of the sources for a wide range of nozzle pressure and temperature ratios. It does not make assumptions regarding fine- or large-scale turbulent noise sources, self- or shear-noise, or convective amplification. The acoustic analogy is partially informed by three-dimensional steady Reynolds-Averaged Navier-Stokes solutions that include the nozzle geometry. The predictions are compared with experiments of jets operating subsonically through supersonically and at unheated and heated temperatures. Predictions generally capture the scaling of both mixing noise and BBSAN for the conditions examined, but some discrepancies remain that are due to the accuracy of the steady RANS turbulence model closure, the equivalent sources, and the use of a simplified vector Greenâ€™s function solver of the linearized Euler equations.

**Aeroacoustic Simulation of Nose Landing Gear on Adaptive
Unstructured Grids with FUN3D**
(7.6 MB PDF)

Numerical simulations have been performed for a partially-dressed, cavity-closed nose landing gear configuration that was tested in NASA Langley’s closed-wall Basic Aerodynamic Research Tunnel (BART) and in the University of Florida’s open-jet acoustic facility known as the UFAFF. The unstructured-grid flow solver FUN3D, developed at NASA Langley Research center, is used to compute the unsteady flow field for this configuration. Starting with a coarse grid, a series of successively finer grids were generated using the adaptive gridding methodology available in the FUN3D code. A hybrid Reynolds-averaged Navier-Stokes/large eddy simulation (RANS/LES) turbulence model is used for these computations. Time-averaged and instantaneous solutions obtained on these grids are compared with the measured data. In general, the correlation with the experimental data improves with grid refinement. A similar trend is observed for sound pressure levels obtained by using these CFD solutions as input to a FfowcsWilliams-Hawkings noise propagation code to compute the farfield noise levels. In general, the numerical solutions obtained on adapted grids compare well with the hand-tuned enriched fine grid solutions and experimental data. In addition, the grid adaption strategy discussed here simplifies the grid generation process, and results in improved computational efficiency of CFD simulations.

**The Effects of Surfaces on the Aerodynamics and
Acoustics of Jet Flows**
(5.4 MB PDF)

Aircraft noise mitigation is an ongoing challenge for the aeronautics research community. In response to this challenge, low-noise aircraft concepts have been developed that exhibit situations where the jet exhaust interacts with an airframe surface. Jet flows interacting with nearby surfaces manifest a complex behavior in which acoustic and aerodynamic char- acteristics are altered. In this paper, the variation of the aerodynamics, acoustic source, and far-field acoustic intensity are examined as a large flat plate is positioned relative to the nozzle exit. Steady Reynolds-Averaged Navier-Stokes solutions are examined to study the aerodynamic changes in the field-variables and turbulence statistics. The mixing noise model of Tam and Auriault is used to predict the noise produced by the jet. To validate both the aerodynamic and the noise prediction models, results are compared with Particle Image Velocimetry (PIV) and free-field acoustic data respectively. The variation of the aerodynamic quantities and noise source are examined by comparing predictions from various jet and flat plate configurations with an isolated jet. To quantify the propulsion airframe aeroacoustic installation effects on the aerodynamic noise source, a non-dimensional number is formed that contains the flow-conditions and airframe installation parameters.

**FUN3D Airload Predictions for the Full-Scale UH-60A Airloads
Rotor in a Wind Tunnel**
(2.4 MB PDF)

An unsteady Reynolds-Averaged Navier-Stokes solver for unstructured grids, FUN3D, is used to compute the rotor performance and airloads of the UH-60A Airloads Rotor in the National Full-Scale Aerodynamic Complex (NFAC) 40- by 80-foot Wind Tunnel. The flow solver is loosely coupled to a rotorcraft comprehensive code, CAMRAD-II, to account for trim and aeroelastic deflections. Computations are made for the 1-g level flight speed-sweep test conditions with the airloads rotor installed on the NFAC Large Rotor Test Apparatus (LRTA) and in the 40- by 80-ft wind tunnel to determine the influence of the test stand and wind-tunnel walls on the rotor performance and airloads. Detailed comparisons are made between the results of the CFD/CSD simulations and the wind tunnel measurements. The computed trends in solidity-weighted propulsive force and power coefficient match the experimental trends over the range of advance ratios and are comparable to previously published results. Rotor performance and sectional airloads show little sensitivity to the modeling of the wind-tunnel walls, which indicates that the rotor shaft-angle correction adequately compensates for the wall influence up to an advance ratio of 0.37. Sensitivity of the rotor performance and sectional airloads to the modeling of the rotor with the LRTA body/hub increases with advance ratio. The inclusion of the LRTA in the simulation slightly improves the comparison of rotor propulsive force between the computation and wind tunnel data but does not resolve the difference in the rotor power predictions at mu = 0.37. Despite a more precise knowledge of the rotor trim loads and flight condition, the level of comparison between the computed and measured sectional airloads/pressures at an advance ratio of 0.37 is comparable to the results previously published for the high-speed flight test condition.

**Advanced Concept Studies for Supersonic Commercial
Transports Entering Service in the 2018 to 2020 Period**
(24.9 MB PDF)

Lockheed Martin Aeronautics Company (LM), working in conjunction with General Electric Global Research (GE GR), Rolls-Royce Liberty Works (RRLW), and Stanford University, herein presents results from the “N+2 Supersonic Validations” contract’s initial 22 month phase, addressing the NASA solicitation “Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018 to 2020 Period.” This report version adds documentation of an additional three month low boom test task. The key technical objective of this effort was to validate integrated airframe and propulsion technologies and design methodologies. These capabilities aspired to produce a viable supersonic vehicle design with environmental and performance characteristics. Supersonic testing of both airframe and propulsion technologies (including LM3: 97-023 low boom testing and April-June nozzle acoustic testing) verified LM’s supersonic low-boom design methodologies and both GE and RRLW’s nozzle technologies for future implementation. The N+2 program is aligned with NASA’s Supersonic Project and is focused on providing system-level solutions capable of overcoming the environmental and performance/efficiency barriers to practical supersonic flight. NASA proposed “Initial Environmental Targets and Performance Goals for Future Supersonic Civil Aircraft”. The LM N+2 studies are built upon LM’s prior N+3 100 passenger design studies. The LM N+2 program addresses low boom design and methodology validations with wind tunnel testing, performance and efficiency goals with system level analysis, and low noise validations with two nozzle (GE and RRLW) acoustic tests.

**FUN3D Analyses in Support of the First Aeroelastic Prediction
Workshop**
(8.1 MB PDF)

This paper presents the computational aeroelastic results generated in support of the first Aeroelastic PredictionWorkshop for the Benchmark Supercritical Wing (BSCW) and the HIgh REynolds Number AeroStructural Dynamics (HIRENASD) configurations and compares them to the experimental data. The computational results are obtained using FUN3D, an unstructured grid Reynolds-averaged Navier-Stokes solver developed at NASA Langley Research Center. The analysis results for both configurations include aerodynamic coefficients and surface pressures obtained for steady-state or static aeroelastic equilibrium (BSCW and HIRENASD, respectively) and for unsteady flow due to a pitching wing (BSCW) or a modally-excited wing (HIRENASD). Frequency response functions of the pressure coefficients with respect to displacement are computed and compared with the experimental data. For the BSCW, the shock location is computed aft of the experimentallylocated shock position. The pressure distribution upstream of this shock is in excellent agreement with the experimental data, but the pressure downstream of the shock in the separated flow region does not match as well. For HIRENASD, very good agreement between the numerical results and the experimental data is observed at the mid-span wing locations.

**Summary of the 2008 NASA Fundamental Aeronautics
Program Sonic Boom Prediction Workshop**
(6.1 MB PDF)

The Supersonics Project of the NASA Fundamental Aeronautics Program organized an internal sonic boom workshop to evaluate near-field sonic boom prediction capability at the Fundamental Aeronautics Annual Meeting in Atlanta, Georgia on October 8, 2008. Workshop participants computed sonic boom signatures for three non-lifting bodies and two lifting configurations. A cone-cylinder, parabolic, and quartic bodies of revolution comprised the non-lifting cases. The lifting configurations were a simple 69-degree delta wing body and a complete low-boom transport configuration designed during the High Speed Research Project in the 1990s with wing, body, tail, nacelle, and boundary layer diverter components. The AIRPLANE, Cart3D, FUN3D, and USM3D flow solvers were employed with the ANET signature propagation tool, output-based adaptation, and a priori adaptation based on freestream Mach number and angle of attack. Results were presented orally at the workshop. This article documents the workshop, results, and provides context on previously available and recently developed methods.

**Full Configuration Low Boom Model and Grids for 2014 Sonic
Boom Prediction Workshop**
(1.9 MB PDF)

A conceptual supersonic transport design, identified as 1021-01, was developed for the NASA N+2 Supersonic Validations program. It was designed to produce very low sonic boom. A wind tunnel model was fabricated and tested to validate the predicted low sonic boom. An efficient “spatial averaging” measurement technique was used to handle distortions endemic to low sonic boom wind tunnel measurement, resulting in measurements precise enough to match predicted ground loudness within 1 PLdB. It was decided this model and data would make a good case for the 2014 Sonic Boom Prediction Workshop. Model development details and flow prediction challenges encountered during development are illustrated. Test oil flow visualization is shown to guide analyses, especially with regard to viscous boundary layer modeling. Viscosity was not important at full-scale but was found to be important at wind tunnel model scale (1/125). Geometry and grid files are expected to be available on the workshop website by 31 January 2013.

**CFL3D, FUN3D, and NSU3D Contributions to the
Fifth Drag Prediction Workshop**
(4.9 MB PDF)

Results presented at the Fifth Drag Prediction Workshop using CFL3D, FUN3D, and NSU3D are described. These are calculations on the workshop provided grids and drag adapted grids. The NSU3D results have been updated to reflect an improvement to skin friction calculation on skewed grids. FUN3D results generated after the workshop are included for custom participant generated grids and a grid from a previous workshop. Uniform grid refinement at the design condition shows a tight grouping in calculated drag, where the variation in the pressure component of drag is larger than the skin friction component. At this design condition, A fine-grid drag value was predicted with a smaller drag adjoint adapted grid via tetrahedral adaption to a metric and mixed-element subdivision. The buffet study produced larger variation than the design case, which is attributed to large differences in the predicted side-of-body separation extent. Various modeling and discretization approaches had a strong impact on predicted side-of-body separation. This large wing root separation bubble was not observed in wind tunnel tests indicating that more work is necessary in modeling wing root juncture flows to predict experiments.

**Directivity Effects of Shaped Plumes from Plug Nozzles**
(2.5 MB PDF)

A set of various shaped nozzles designed by Aerion Corporation were investigated on a reduced scale facility at University of California Irvine (UCI). Initial results on some of the configurations show great potential in directing the noise away from the relevant azimuth positions and producing significant reduction in noise levels. Flow surveys and numerical analysis have been performed to better understand the mechanisms which produce such improvement.

**Recent Advances in Agglomerated Multigrid**
(5.7 MB PDF)

We report recent advancements of the agglomerated multigrid methodology for complex flow simulations on fully unstructured grids. An agglomerated multigrid solver is applied to a wide range of test problems from simple two-dimensional geometries to realistic three-dimensional configurations. The solver is evaluated against a single-grid solver and, in some cases, against a structured-grid multigrid solver. Grid and solver issues are identified and overcome, leading to significant improvements over single-grid solvers.

**Development, Verification and Use of Gust Modeling in
the NASA Computational Fluid Dynamics Code FUN3D**
(1.6 MB PDF)

The increased flexibility of long endurance aircraft having high aspect ratio wings necessitates attention to gust response and perhaps the incorporation of gust load alleviation. The design of civil transport aircraft with a high aspect ratio strut or truss braced wing furthermore requires gust response analysis in the transonic cruise range. This requirement motivates the use of high fidelity nonlinear computational fluid dynamics (CFD) for gust response analysis. This paper presents the implementation of gust modeling capability in the CFD code FUN3D. The gust capability is verified by computing the response of an airfoil to a sharp edged gust. This result is compared with the theoretical result. The present simulations will be compared with other CFD gust simulations. This paper also serves as a users manual for FUN3D gust analyses using a variety of gust profiles. Finally, the development of an auto-regressive moving-average (ARMA) reduced order gust model using a gust with a Gaussian profile in the FUN3D code is presented. ARMA simulated results of a sequence of one-minus-cosine gusts is shown to compare well with the same gust profile computed with FUN3D. Proper orthogonal decomposition (POD) is combined with the ARMA modeling technique to predict the time varying pressure coefficient increment distribution due to a novel gust profile. The aeroelastic response of a pitch/plunge airfoil to a gust environment is computed with a reduced order model, and compared with a direct simulation of the system in the FUN3D code. The two results are found to agree very well.

**Production Level CFD Code Acceleration for
Hybrid Many-Core Architectures**
(0.6 MB PDF)

In this work, a novel graphics processing unit (GPU) distributed sharing model for hybrid many-core architectures is introduced and employed in the acceleration of a production-level computational fluid dynamics (CFD) code. The latest generation graphics hardware allows multiple processor cores to simultaneously share a single GPU through concurrent kernel execution. This feature has allowed the NASA FUN3D code to be accelerated in parallel with up to four processor cores sharing a single GPU. For codes to scale and fully use resources on these and the next generation machines, codes will need to employ some type of GPU sharing models presented in this work. Findings include the effects of GPU sharing on overall performance. A discussion of the inherent challenges that parallel unstructured CFD codes face in accelerator-based computing environments is included, with considerations for future generation architectures. This work was completed by the author in August 2010, and reflects the analysis and results of the time.

**Computational Aeroelastic Analysis of the Ares I
Crew Launch Vehicle During Ascent**
(1.5 MB PDF)

Static and dynamic aeroelastic analyses have been performed for the Ares I crew launch vehicle during atmospheric ascent. It is shown that, through the transonic speed range, there is a rapid change in the static aeroelastic center-of-pressure increment with increasing Mach number. The greatest sensitivity to grid resolution is observed through the transonic range. Dynamic aeroelastic analyses are also performed to assess the aeroelastic stability of the launch vehicle. Flexible dynamic linearized quasi-steady analyses using steady rigid line loads are compared with fully coupled aeroelastic time-marching computational fluid dynamic analyses. There are significant differences between the methods through the transonic Mach number range. The largest difference is at Mach 1. At that Mach number, the linearized quasi-steady method produces strong damping in modes 1 and 2. The unsteady computational aeroelastic method indicates that the first mode is significantly undamped, while mode 2 is strongly damped. The cause of the disparity in damping between modes 1 and 2 is also investigated. A vehicle with no protuberances other than rings produced damping values in modes 1 and 2 that were nearly identical. It is shown that the disparity in damping of modes one and two is due to asymmetric placement of protuberances around the vehicle circumference.

**Flexible Launch Vehicle Stability Analysis Using
Steady and Unsteady Computational Fluid Dynamics**
(1.4 MB PDF)

Launch vehicles frequently experience a reduced stability margin through the transonic Mach number range. This reduced stability margin can be caused by the aerodynamic undamping one of the lower-frequency flexible or rigid-body modes. Analysis of the behavior of a flexible vehicle is routinely performed with quasi-steady aerodynamic line loads derived from steady rigid aerodynamics. However, a quasi-steady aeroelastic stability analysis can be unconservative at the critical Mach numbers, where experiment or unsteady computational aeroelastic analysis show a reduced or even negative aerodynamic damping. A method of enhancing the quasi-steady aeroelastic stability analysis of a launch vehicle with unsteady aerodynamics is developed that uses unsteady computational fluid dynamics to compute the response of selected lower-frequency modes. The response is contained in a time history of the vehicle line loads. A proper orthogonal decomposition of the unsteady aerodynamic line-load response is used to reduce the scale of data volume and system identification is used to derive the aerodynamic stiffness, damping, and mass matrices. The results are compared with the damping and frequency computed from unsteady computational aeroelasticity and from a quasi-steady analysis. The results show that incorporating unsteady aerodynamics in this way brings the enhanced quasi-steady aeroelastic stability analysis into close agreement with the unsteady computational aeroelastic results.

**Adjoint-based Optimization of the Flapping Wing
Performance**
(800 KB PDF)

A time-dependent adjoint-based methodology developed in [AIAA 2008-5857 and AIAA J. Vol.48, No.6, pp.1195-1206, 2010] is used for optimization of the 3-D unsteady turbulent flow near a flapping wing. The sensitivities of the thrust coefficient to wing kinematic parameters are computed using the time-dependent discrete adjoint formulation. The unsteady discrete adjoint equations required for calculation of the sensitivity derivatives are integrated backward in time. The gradient of the objective functional computed using the adjoint formulation is then used to update the values of the kinematic design variables. The efficiency of this time-dependent optimization methodology is demonstrated by maximizing the performance of a wing undergoing insect-based flapping motion. Our numerical results show that the wing thrust coefficient and propulsive efficiency have been significantly increased after the optimization.

**Sonic Boom Mitigation Through Aircraft Design and Adjoint
Methodology**
(1.3 MB PDF)

This paper presents a novel approach to design of the supersonic aircraft outer mold line (OML) by optimizing the A-weighted loudness of sonic boom signature predicted on the ground. The optimization process uses the sensitivity information obtained by coupling the discrete adjoint formulations for the augmented Burgers Equation and Computational Fluid Dynamics (CFD) equations. This coupled formulation links the loudness of the ground boom signature to the aircraft geometry thus allowing efficient shape optimization for the purpose of minimizing the impact of loudness. The accuracy of the adjoint-based sensitivities is verified against sensitivities obtained using an independent complex-variable approach. The adjoint based optimization methodology is applied to a configuration previously optimized using alternative state of the art optimization methods and produces additional loudness reduction. The results of the optimizations are reported and discussed.

**N+2 Low Boom Wind Tunnel Model Design and Validation**
(6.8 MB PDF)

A conceptual supersonic transport design was developed for the NASA N+2 Supersonic Validations program and was designed to produce very low sonic boom. A wind tunnel model was fabricated and tested to validate the predicted low sonic boom. Measurements matched the predictions with extremely good precision.

**Aerodynamic Impacts of Helicopter Blade Erosion Coatings**
(2.4 MB PDF)

The United States Army helicopter fleet experiences deformation of rotor blade contours from sand erosion and the implementation of technologies to protect against it. An investigation was performed to determine the effect of a typical erosion protection coating on the main rotor performance of a UH-60A Blackhawk utility helicopter. Computational fluid dynamics was used to calculate aerodynamic coefficients for representative coated airfoil sections. Hover analyses were performed to evaluate the impact of the coated airfoils on main rotor performance. The results show that airfoil erosion protection coatings can cause a decrease in the rotor’s aerodynamic efficiency.

**NASA Trapezoidal Wing Computations Including Transition and
Advanced Turbulence Modeling**
(5.1 MB PDF)

Flow about the NASA Trapezoidal Wing is computed with several turbulence models by using grids from the first High Lift Prediction Workshop in an effort to advance understanding of computational fluid dynamics modeling for this type of flowfield. Transition is accounted for in many of the computations. In particular, a recently-developed 4-equation transition model is utilized and works well overall. Accounting for transition tends to increase lift and decrease moment, which improves the agreement with experiment. Upper surface flap separation is reduced, and agreement with experimental surface pressures and velocity profiles is improved. The predicted shape of wakes from upstream elements is strongly influenced by grid resolution in regions above the main and flap elements. Turbulence model enhancements to account for rotation and curvature have the general effect of increasing lift and improving the resolution of the wing tip vortex as it convects downstream. However, none of the models improve the prediction of flap surface pressures near the wing tip.

**Radiation Coupling with the FUN3D Unstructured-Grid CFD Code**
(0.4 MB PDF)

The HARA radiation code is fully-coupled to the FUN3D unstructured-grid CFD code for the purpose of simulating high-energy hypersonic flows. The radiation energy source terms and surface heat transfer, under the tangent slab approximation, are included within the fluid dynamic flow solver. The Fire II flight test, at the Mach-31 1643-second trajectory point, is used as a demonstration case. Comparisons are made with an existing structured-grid capability, the LAURA/HARA coupling. The radiative surface heat transfer rates from the present approach match the benchmark values within 6%. Although radiation coupling is the focus of the present work, convective surface heat transfer rates are also reported, and are seen to vary depending upon the choice of mesh connectivity and FUN3D flux reconstruction algorithm. On a tetrahedral-element mesh the convective heating matches the benchmark at the stagnation point, but under-predicts by 15% on the Fire II shoulder. Conversely, on a mixed-element mesh the convective heating over-predicts at the stagnation point by 20%, but matches the benchmark away from the stagnation region.

**Boundary Layer Stability Analysis of the Mean Flows Obtained
Using Unstructured Grids**
(3.6 MB PDF)

Boundary-layer stability analyses of mean flows extracted from unstructured-grid Navier-Stokes solutions have been performed. A procedure has been developed to extract mean flow profiles from the FUN3D unstructured-grid solutions. Extensive code-to-code validations have been performed by comparing the extracted mean flows as well as the corresponding stability characteristics to the predictions based on structured-grid solutions. Comparisons are made on a range of problems from a simple flat plate to a full aircraft configuration – a modified Gulfstream-III with a natural laminar flow glove. The future aim of the project is to extend the adjoint-based design capability in FUN3D to include natural laminar flow and laminar flow control by integrating it with boundary-layer stability analysis codes, such as LASTRAC.

**Aeroacoustic Simulation of a Nose Landing Gear in an Open Jet Facility using FUN3D**
(1.7 MB PDF)

Numerical simulations have been performed for a partially-dressed, cavity-closed nose landing gear configuration that was tested in NASA Langley’s closed-wall Basic Aerodynamic Research Tunnel (BART) and in the University of Florida’s open-jet acoustic facility known as UFAFF. The unstructured-grid flow solver, FUN3D, developed at NASA Langley Research center is used to compute the unsteady flow field for this configuration. A hybrid Reynolds-averaged Navier-Stokes/large eddy simulation (RANS/LES) turbulence model is used for these computations. Time-averaged and instantaneous solutions compare favorably with the measured data. Unsteady flowfield data obtained from the FUN3D code are used as input to a Ffowcs Williams-Hawking noise propagation code to compute the sound pressure levels at microphones placed in the farfield. Significant improvement in predicted noise levels is obtained when the flowfield data from the open jet UFAFF simulations is used as compared to the case using flowfield data from the closed-wall BART configuration.

**Unsteady Reynolds-Averaged Navier-Stokes-Based Hybrid
Methodologies for Rotor-Fuselage Interaction**
(3.3 MB PDF)

**An Assessment of CFD/CSD Prediction State-of-the-Art
Using the HART II International Workshop Data**
(5.6 MB PDF)

Over the past decade, there have been significant advancements in the accuracy of rotor aeroelastic simulations with the application of computational fluid dynamics methods coupled with computational structural dynamics codes (CFD/CSD). The HART II International Workshop database, which includes descent operating conditions with strong blade-vortex interactions (BVI), provides a unique opportunity to assess the ability of CFD/CSD to capture these physics. In addition to a baseline case with BVI, two additional cases with 3/rev higher harmonic blade root pitch control (HHC) are available for comparison. The collaboration during the workshop permits assessment of structured, unstructured, and hybrid overset CFD/CSD methods from across the globe on the dynamics, aerodynamics, and wake structure. Evaluation of the plethora of CFD/CSD methods indicate that the most important numerical variables associated with most accurately capturing BVI are a two-equation or detached eddy simulation (DES)-based turbulence model and a sufficiently small time step. An appropriate trade-off between grid fidelity and spatial accuracy schemes also appears to be important for capturing BVI on the advancing rotor disk. Overall, the CFD/CSD methods generally fall within the same accuracy; cost-effective hybrid Navier-Stokes/Lagrangian wake methods tend to correlate less accurately with experiment and have larger data scatter than the full CFD/CSD methods for most parameters evaluated. The importance of modeling the fuselage is observed, and other requirements are discussed.

**An Examination of Unsteady Airloads on a UH-60A Rotor: Computation versus Measurement**
(0.6 MB PDF)

An unsteady Reynolds-averaged Navier-Stokes solver for unstructured grids is used to simulate the flow over a UH-60A rotor. Traditionally, the computed pressure and shear stresses are integrated on the computational mesh at selected radial stations and compared to measured airloads. However, the corresponding integration of experimental data uses only the pressure contribution, and the set of integration points (pressure taps) is modest compared to the computational mesh resolution. This paper examines the difference between the traditional integration of computed airloads and an integration consistent with that used for the experimental data. In addition, a comparison of chordwise pressure distributions between computation and measurement is made. Examination of this unsteady pressure data provides new opportunities to understand differences between computation and flight measurement.

**Rotating Hub Drag Prediction Methodology**
(0.7 MB PDF)

Hub drag reduction has long been a sought after technology goal. This is because modern rotary-wing hubs can account for up to 30% of the drag of an aircraft. However, achieving realizable benefits on production aircraft requires a significant demonstrable improvement in present design and analysis tools. Bell Helicopter is currently engaged in a four year NRTC/CRI project with one of the principal criteria for success being the identification of a physics-based analysis methodology capable of accurately predicting drag on realistic hub geometries. The results from the fourth and final year of the project are described herein and concern the applications to rotating hubs. Details offered include a description of a recent Bell aircraft wind tunnel test used for correlation, the grid generation paradigm used, and the correlation between measured and computed data for both pitch and yaw orientations, and future work planned.

**The Effect of a Gust on the Flapping Wing Performance**
(0.5 MB PDF)

The effect of a wind gust on the aerodynamic characteristics of a rigid wing undergoing insect-based flapping motion is studied numerically. The turbulent flow near the flapping wing is described by the 3-D unsteady compressible Reynolds-Averaged Navier-Stokes equations with the Spalart-Allmaras turbulence model. The governing equations are solved using a second-order node-centered finite volume scheme on a hexahedral body-fitted grid that rigidly moves along with the wing. A low-Mach-number preconditioner is used to accelerate the convergence at each time step. The effects of wind gust direction with respect to a wing orientation are investigated. Our numerical results show that the centimeter-scale wing considered in the present study is susceptible to strong downward wind gusts. In the case of frontal and side gusts, the flapping wing can alleviate the gust effect if the gust velocity is less or comparable with the wing tip velocity. For all cases considered, the thrust coefficient returns to its original baseline profile within one full stroke after the gust is removed, thus indicating that the flapping wing can effectively recover from wind gust fluctuations.

**Exploration of the Physics of Hub Drag**
(3.4 MB PDF)

Main rotor hubs contribute significantly to the helicopter parasite drag. Due to the complex geometry of typical hub configurations, large uncertainties in loads and wake flowfield phenomena inhibit the development of improved concepts at the design stage. Experimental and computational evaluations of the near wake of a generic, subscale complex rotor hub are explored. Prior efforts have shown that state of the art computational fluid dynamics (CFD) simulations are accurate enough to predict drag characteristics of the complex hub. Velocity profiles and turbulence spectra measurements in the hub near wake (0.5 – 1.0 hub diameters) correlate well with CFD when proper fidelity of grid, time step, and turbulence model are chosen. The wake spectral content is characterized by broadband turbulence in addition to large scale spectral content energizing the flowfield. Current results substantiate earlier findings of the role played by Magnus effect due to bodies of rotation in forward flight.

**Continuing Validation of Computational Fluid Dynamics For
Supersonic Retropropulsion**
(31.5 MB PDF)

A solid step in the validation of Computational Fluid Dynamics (CFD) for Supersonic Retropropulsion is shown through the comparison of three Navier-Stokes solvers (DPLR, FUN3D, and OVERFLOW) and wind tunnel test results. The test was designed specifically for CFD validation and was conducted in the NASA Langley Research Center supersonic 4’x4’ Unitary Plan Wind Tunnel. The test includes variations in the number of nozzles, Mach and Reynolds numbers, thrust coefficient, and angles of orientation. Code-to-code and code-to-test comparisons are encouraging and possible error sources are discussed.

**Effects of Mesh Regularity on Accuracy of Finite-Volume
Schemes**
(0.4 MB PDF)

The effects of mesh regularity on the accuracy of unstructured node-centered finite-volume discretizations are considered. The focus of this paper is on an edge-based approach that uses unweighted least-squares gradient reconstruction with a quadratic fit. Gradient errors and discretization errors for inviscid and viscous fluxes are separately studied according to a previously introduced methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing-layer viscous grids. The meshes within these classes range from regular to extremely irregular including meshes with random perturbation of nodes. The inviscid scheme is nominally third-order accurate on general triangular meshes. The viscous scheme is a nominally second-order accurate discretization that uses an average-least-squares method. The results have been contrasted with previously studied schemes involving other gradient reconstruction methods such as the Green-Gauss method and the unweighted least-squares method with a linear fit. Recommendations are made concerning the inviscid and viscous discretization schemes that are expected to be least sensitive to mesh regularity in applications to turbulent flows for complex geometries.

**Discrete Adjoint-Based Design for Unsteady Turbulent Flows on
Dynamic Overset Unstructured Grids**
(5.9 MB PDF)

A discrete adjoint-based design methodology for unsteady turbulent flows on three-dimensional dynamic overset unstructured grids is formulated, implemented, and verified. The methodology supports both compressible and incompressible flows and is amenable to massively parallel computing environments. The approach provides a general framework for performing highly efficient and discretely consistent sensitivity analysis for problems involving arbitrary combinations of overset unstructured grids which may be static, undergoing rigid or deforming motions, or any combination thereof. General parent-child motions are also accommodated, and the accuracy of the implementation is established using an independent verification based on a complex-variable approach. The methodology is used to demonstrate aerodynamic optimizations of a wind turbine geometry, a biologically-inspired flapping wing, and a complex helicopter configuration subject to trimming constraints. The objective function for each problem is successfully reduced and all specified constraints are satisfied.

**Inflow/Outflow Boundary Conditions with Application to FUN3D**
(2.1 MB PDF)

Several boundary conditions that allow subsonic and supersonic flow into and out of the computational domain are discussed. These boundary conditions are demonstrated in the FUN3D computational fluid dynamics (CFD) code which solves the three-dimensional Navier-Stokes equations on unstructured computational meshes. The boundary conditions are enforced through determination of the flux contribution at the boundary to the solution residual. The boundary conditions are implemented in an implicit form where the Jacobian contribution of the boundary condition is included and is exact. All of the flows are governed by the calorically perfect gas thermodynamic equations. Three problems are used to assess these boundary conditions. Solution residual convergence to machine zero precision occurred for all cases. The converged solution boundary state is compared with the requested boundary state for several levels of mesh densities. The boundary values converged to the requested boundary condition with approximately second-order accuracy for all of the cases.

**Computational Investigation of Hub Drag Deconstruction from
Model to Full Scale**
(8.1 MB PDF)

Parasite drag on rotorcraft can become a crucial factor in forward flight, especially for high speed flight. Prior evaluations of the ability of computational methods to predict hub drag have focused on the ability of these solvers to match model scale experimental data, but the codes have not been examined for full scale conditions. Using an unstructured computational method, the sources of hub drag on a moderately complex model are deconstructed and examined for model and full scale configurations and flight conditions. Correlations with a model-scale wind tunnel test and theoretical data are provided to confirm the appropriateness of the initial grid. Unlike prior efforts, grid adaptation across the overset meshes permits grid refinement where needed and minimizes the grid cost. It has been observed that for the moderately complex hub evaluated that grids developed for the model scale analysis can not be applied directly to a full-scale analysis. Deconstruction of the drag illustrates that evaluation of the Reynolds number for each component to evaluate its impact on drag, as well as consideration of changes in the interference effects, are required when scaling results from model to full scale, even for static (nonrotating) configurations. Velocity scaling for rotating hubs must also be considered; scaling to similar advance ratio rather than rotor angular velocity appears to be more appropriate. Estimation of the interference drag for rotating hubs must consider the Magnus effect, which appear to directly influence the nonlinearities observed in scaling the drag.

**Extension and Exploration of a Hybrid Turbulence
Model on Unstructured Grids**
(8.9 MB PDF)

**Analysis of Effectiveness of Phoenix Entry Reaction Control
System**
(8.0 MB PDF)

Interaction between the external flowfield and the reaction control system thruster plumes of the Phoenix capsule during entry has been investigated. The analysis covered rarefied, transitional, hypersonic and supersonic flight regimes. Performance of pitch, yaw and roll control authority channels was evaluated, with specific emphasis on the yaw channel due to its low nominal yaw control authority. Because Phoenix had already been constructed and its reaction control system could not be modified before flight, an assessment of reaction control system efficacy along the trajectory was needed to determine possible issues and to make necessary software changes. Effectiveness of the system at various regimes was evaluated using a hybrid direct simulation Monte-Carlo computational fluid dynamics technique, based on direct simulation Monte-Carlo analysis code and general aerodynamic simulation program, the Langley aerothermal upwind relaxation algorithm code, and the fully unstructured 3-D code. Results of the analysis at hypersonic and supersonic conditions suggest a significant aeroreaction control system interference which reduced the efficacy of the thrusters and could likely produce control reversal. Very little aeroreaction control system interference was predicted in rarefied and transitional regimes. A recommendation was made to the project to widen controller system deadbands to minimize (if not eliminate) the use of reaction control system thrusters through hypersonic and supersonic flight regimes, where their performance would be uncertain.

**Computational Analysis of the G-III Laminar Flow Glove**
(1.4 MB PDF)

Under NASA’s Environmentally Responsible Aviation Project, flight experiments are planned with the primary objective of demonstrating the Discrete Roughness Elements (DRE) technology for passive laminar flow control at chord Reynolds numbers relevant to transport aircraft. In this paper, we present a preliminary computational assessment of the Gulfstream-III (G-III) aircraft wing-glove designed to attain natural laminar flow for the leading-edge sweep angle of 34.6 deg. Analysis for a flight Mach number of 0.75 shows that it should be possible to achieve natural laminar flow for twice the transition Reynolds number ever achieved at this sweep angle. However, the wing-glove needs to be redesigned to effectively demonstrate passive laminar flow control using DREs. As a by-product of the computational assessment, effect of surface curvature on stationary crossflow disturbances is found to be strongly stabilizing for the current design, and it is suggested that convex surface curvature could be used as a control parameter for natural laminar flow design, provided transition occurs via stationary crossflow disturbances.

**Toward Supersonic Retropropulsion CFD Validation**
(35.0 MB PDF)

This paper begins the process of verifying and validating computational fluid dynamics (CFD) codes for supersonic retropropulsive flows. Four CFD codes (DPLR, FUN3D, OVERFLOW, and US3D) are used to perform various numerical and physical modeling studies toward the goal of comparing predictions with a wind tunnel experiment specifically designed to support CFD validation. Numerical studies run the gamut in rigor from code-to-code comparisons to observed order-of-accuracy tests. Results indicate that this complex flowfield, involving time-dependent shocks and vortex shedding, design order of accuracy is not clearly evident. Also explored is the extent of physical modeling necessary to predict the salient flowfield features found in high-speed Schlieren images and surface pressure measurements taken during the validation experiment. Physical modeling studies include geometric items such as wind tunnel wall and sting mount interference, as well as turbulence modeling that ranges from a RANS (Reynolds-Averaged Navier-Stokes) 2-equation model to DES (Detached Eddy Simulation) models. These studies indicate that tunnel wall interference is minimal for the cases investigated; model mounting hardware effects are confined to the aft end of the model; and sparse grid resolution and turbulence modeling can damp or entirely dissipate the unsteadiness of this self-excited flow.

**Sonic Boom Adjoint Methodology and Its Applications**
(6.0 MB PDF)

This paper presents an approach to predict the sensitivity of the sonic boom ground signatures by numerically solving the augmented Burgers’ equation along with its discrete adjoint. The discrete adjoint equations are derived and solved. The exactness of the adjoint sensitivities is verified against derivatives obtained using the complex variable approach. Under- and off-track ground signature sensitivities to different design variables may be obtained efficiently. The formulation of the coupling between boom adjoint and CFD adjoint is derived and discussed. This formulation represents the first time in literature that boom propagation and CFD are formally coupled for the purpose of obtaining gradients of a ground based objective with respect to the aircraft shape design variables. The coupled formulation is effective in calculating discretely accurate sensitivities, and should be an extremely useful tool in the design of supersonic cruise low-boom aircraft.

**Low Boom Configuration Analysis with FUN3D Adjoint Simulation Framework**
(2.6 MB PDF)

Off-body pressure, forces, and moments for the Gulfstream Low Boom Model are computed with a Reynolds Averaged Navier Stokes solver coupled with the Spalart-Allmaras (SA) turbulence model. This is the first application of viscous output-based adaptation to reduce estimated discretization errors in off-body pressure for a wing body configuration. The output adaptation approach is compared to an a priori grid adaptation technique designed to resolve the signature on the centerline by stretching and aligning the grid to the freestream Mach angle. The output-based approach produced good predictions of centerline and off-centerline measurements. Eddy viscosity predicted by the SA turbulence model increased significantly with grid adaptation. Computed lift as a function of drag compares well with wind tunnel measurements for positive lift, but predicted lift, drag, and pitching moment as a function of angle of attack has significant differences from the measured data. The sensitivity of longitudinal forces and moment to grid refinement is much smaller than the differences between the computed and measured data.

**Development and Application of Parallel Agglomerated
Multigrid Methods for Complex Geometries**
(4.3 MB PDF)

We report further progress in the development of agglomerated multigrid techniques for fully unstructured grids in three dimensions. Following the previous studies that identified key elements to grid-independent multigrid convergence for a model equation, and that demonstrated impressive speed-up in single-processor computations for a model diffusion equation, inviscid flows, and Reynolds-averaged Navier-Stokes (RANS) simulations for realistic geometries, we now present a parallelized agglomerated multigrid technique for 3D complex geometries. We demonstrate a robust parallel fully-coarsened agglomerated multigrid technique for the Euler, the Navier-Stokes, and the RANS equations for 3D complex geometries, incorporating the following key developments: consistent and stable coarse-grid discretizations, a hierarchical agglomeration scheme, and line-agglomeration/relaxation using prismatic-cell discretizations in the highly-stretched grid regions. A significant speed-up in computer time over state-of-art large-scale computations is demonstrated for RANS simulations over 3D realistic geometries.

**Code-to-Code Comparison of CFD/CSD Simulation for a Helicopter Rotor in Forward Flight**(1.9 MB PDF)

Two unsteady Reynolds-averaged Navier-Stokes solvers are used to compute the rotor airloads on the UH-60A rotorcraft at several flight conditions across the flight envelope. One code, OVERFLOW, solves the flow equations using either structured grids, or a combination of structured and Cartesian grids. The other solver, FUN3D, uses unstructured grids. Both flow solvers are coupled to the same rotorcraft comprehensive code (CAMRAD II) in order to account for trim and aeroelastic deflections, and both utilize the same loose coupling scheme to transfer data between the flow solver and the comprehensive code. In the process of performing the code-to-code comparison, several small but important details are examined that may sometimes be overlooked when comparing results from rotorcraft simulations using different codes. Computed normal force, pitching moment, and chord force are compared between codes, and also with flight data.

**Preliminary Computational Analysis of the HIRENASD Configuration
in Preparation for the Aeroelastic Prediction Workshop**
(9.1 MB PDF)

This paper presents preliminary computational aeroelastic analysis results generated in preparation for the first Aeroelastic Prediction Workshop (AePW). These results were produced using FUN3D software developed at NASA Langley and are compared against the experimental data generated during the HIgh REynolds Number AeroStructural Dynamics (HIRENASD) Project. The HIRENASD wind-tunnel model was tested in the European Transonic Windtunnel in 2006 by Aachen University’s Department of Mechanics with funding from the German Research Foundation. The computational effort discussed here was performed (1) to obtain a preliminary assessment of the ability of the FUN3D code to accurately compute physical quantities experimentally measured on the HIRENASD model and (2) to translate the lessons learned from the FUN3D analysis of HIRENASD into a set of initial guidelines for the first AePW, which includes test cases for the HIRENASD model. This paper compares the computational and experimental results obtained at Mach 0.8 for a Reynolds number of 7 million based on chord, corresponding to the HIRENASD test conditions No. 132 and No. 159. Aerodynamic loads and static aeroelastic displacements are compared at two levels of the grid resolution. Harmonic perturbation numerical results are compared with the experimental data using the magnitude and phase relationship between pressure coefficients and displacement. A dynamic aeroelastic numerical calculation is presented at one wind-tunnel condition in the form of the time history of the generalized displacements. Additional FUN3D validation results are also presented for the AGARD 445.6 wing data set. This wing was tested in the Transonic Dynamics Tunnel and is commonly used in the preliminary benchmarking of computational aeroelastic software.

**Application of FUN3D Solver for Aeroacoustics Simulation of a
Nose Landing Gear Configuration**
(2.4 MB PDF)

Numerical simulations have been performed for a nose landing gear configuration corresponding to the experimental tests conducted in the Basic Aerodynamic Research Tunnel at NASA Langley Research Center. A widely used unstructured grid code, FUN3D, is examined for solving the unsteady flow field associated with this configuration. A series of successively finer unstructured grids has been generated to assess the effect of grid refinement. Solutions have been obtained on purely tetrahedral grids as well as mixed element grids using hybrid RANS/LES turbulence models. The agreement of FUN3D solutions with experimental data on the same size mesh is better on mixed element grids compared to pure tetrahedral grids, and in general improves with grid refinement.

**Integrated Design of an Active Flow Control System Using a
Time-Dependent Adjoint Method**
(2.6 MB PDF)

An exploratory study is performed to investigate the use of a time-dependent discrete adjoint methodology for design optimization of a high-lift wing configuration augmented with an active flow control system. The location and blowing parameters associated with a series of jet actuation orifices are used as design variables. In addition, a geometric parameterization scheme is developed to provide a compact set of design variables describing the wing shape. The scaling of the implementation is studied using several thousand processors and it is found that asynchronous file operations can greatly improve the overall performance of the approach in such massively parallel environments. Three design examples are presented which seek to maximize the mean value of the lift coefficient for the coupled system, and results demonstrate improvements as high as 27% relative to the lift obtained with non-optimized actuation. This lift gain is more than three times the incremental lift provided by the non-optimized actuation.

**Massively Parallel Algorithms for CFD Simulation and
Optimization on Heterogeneous Many-Core Architectures**
(4.9 MB PDF)

In this dissertation we provide new numerical algorithms for use in conjunction with simulation based design codes. These algorithms are designed and best suited to run on emerging heterogeneous computing architectures which contain a combination of traditional multicore processors and new programmable many-core graphics processing units (GPUs). We have developed the following numerical algorithms (i) a new Multidirectional Search (MDS) method for PDE constrained optimization that utilizes a Multigrid (MG) strategy to accelerate convergence, this algorithm is well suited for use on GPU clusters due to its parallel nature and is more scalable than adjoint methods (ii) a new GPU accelerated point implicit solver for the NASA FUN3D code (unstructured Navier-Stokes) that is written in the Compute Unified Device Architecture (CUDA) language, and which employs a novel GPU sharing model, (iii) novel GPU accelerated smoothers (developed using PGI Fortran with accelerator compiler directives) used to accelerate the multigrid preconditioned conjugate gradient method (MGPCG) on a single rectangular grid, and (iv) an improved pressure projection solver for adaptive meshes that is based on the MGPCG method which requires fewer grid point calculations and has potential for better scalability on heterogeneous clusters. It is shown that a multigrid - multidirectional search (MGMDS) method can run up to 5.5X faster than the MDS method when used on a one dimensional data assimilation problem. It is also shown that the new GPU accelerated point implicit solver of FUN3D is up to 5.5X times faster than the CPU version and that the solver can perform up to 40% faster on a single GPU being shared by four CPU cores. It is found that GPU accelerated smoothers for the MGPCG method on uniform grids can run over 2X faster than the non-accelerated versions for 2D problems, and that the new MGPCG pressure projection solver for adaptive grids is up to 4X faster than the previous MG algorithm.

**Computation of UH-60A Airloads Using CFD/CSD Coupling On
Unstructured Meshes**
(2.3 MB PDF)

An unsteady Reynolds-averaged Navier-Stokes solver for unstructured grids is used to compute the rotor airloads on the UH-60A helicopter at high-speed and high thrust conditions. The flow solver is coupled to a rotorcraft comprehensive code in order to account for trim and aeroelastic deflections. Simulations are performed both with and without the fuselage, and the effects of grid resolution, temporal resolution and turbulence model are examined. Computed airloads are compared to flight data.

**Hierarchical Variable Fidelity Methods for Rotorcraft Aerodynamic
Design and Analysis**
(6.2 MB PDF)

A coupling framework has been developed for unstructured computational fluid dynamics (CFD) solvers to allow for simple activation of a variety of wake solvers. In addition, the interface has been parallelized and extended to support dynamic, overset meshes in rotating frames. Wake capture and performance calculations demonstrate the validity of the framework. Demonstration cases include an oscillating wing, the hovering Caradonna-Tung rotor and ROBIN rotor-fuselage interaction for multiple coupling variations between CHARM, VorTran-M, and FUN3D. These results illustrate that the hybrid methods can provide more accurate results with reduced grid sizes for various applications. The modification of only one solver at each incremental level of analysis permits the user to identify the source of changes in solution results.

**Unstructured Overset Grid Adaptation for Rotorcraft Aerodynamic
Interactions**
(3.8 MB PDF)

A new adaptation strategy is presented that permits time-dependent anisotropic adaptation for dynamic overset simulations. The current development permits adaptation to be executed over a periodic time window in a dynamic flow field so that an accurate evolution of the unsteady wake may be obtained within a single unstructured methodology. Unlike prior adaptive schemes, this approach permits grid adaptation to occur seamlessly across any number of grids that are overset, excluding only the boundary layer to avoid surface manipulations. Demonstrations on rotor-fuselage interactions, including flow field physics, time-averaged and instantaneous fuselage pressures, and wake trajectories are included. The ability of the methodology to improve these predictions without user intervention is confirmed, including simulations that confirm physics that have before now, not been captured by computational simulations. The adapted solutions exhibit dependency based on the choice of the flow field feature-based metric and the number of adaptation cycles, indicating that there is no single best practice for feature-based adaptation across the spectrum of rotorcraft applications.

**Comparison of Node-Centered and Cell-Centered Unstructured
Finite-Volume Discretizations: Inviscid Fluxes**
(700 KB PDF)

Nominally second-order cell-centered and node-centered approaches are compared for unstructured finite volume discretization of inviscid fluxes in two dimensions. Three classes of grids are considered: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing-layer grids. The classes contain regular and irregular grids, including mixed-element grids and grids with random perturbations of nodes. Complexity, accuracy, and convergence of defect-correction iterations are studied. Deficiencies of specific schemes, such as instability, accuracy degradation, and/or poor convergence of defect-correction iterations, have been observed in computations and confirmed in analysis. All schemes may produce large relative gradient-reconstruction errors on grids with perturbed nodes. On advancing-layer grids, a local approximate-mapping technique based on the distance function restores gradient-reconstruction accuracy and fast convergence of defect-correction iterations. Among the considered scheme, the best cell-centered and node-centered schemes, which are low-complexity, stable, robust, and uniformly second-order-accurate, are recommended.

**Supersonic Retro-Propulsion Experimental Design for Computational Fluid Dynamics Model Validation**
(7.7 MB PDF)

The development of supersonic retro-propulsion, an enabling technology for heavy payload exploration missions to Mars, is the primary focus for the present paper. A new experimental model, intended to provide computational fluid dynamics model validation data, was recently designed for the Langley Research Center Unitary Plan Wind Tunnel Test Section 2. Pre-test computations were instrumental for sizing and refining the model, over the Mach number range of 2.4 to 4.6, such that tunnel blockage and internal flow separation issues would be minimized. A 5-in diameter 70-deg sphere-cone forebody, which accommodates up to four 4:1 area ratio nozzles, followed by a 10-in long cylindrical aftbody was developed for this study based on the computational results. The model was designed to allow for a large number of surface pressure measurements on the forebody and aftbody. Supplemental data included high-speed Schlieren video and internal pressures and temperatures. The run matrix was developed to allow for the quantification of various sources of experimental uncertainty, such as random errors due to run-to-run variations and bias errors due to flow field or model misalignments. Some preliminary results and observations from the test are presented, although detailed analyses of the data and uncertainties are still on going.

**Analysis of Navier-Stokes Codes Applied to Supersonic Retro-Propulsion Wind Tunnel Test**
(6.5 MB PDF)

This paper describes the pre-test analysis of three Navier-Stokes codes applied to a Supersonic Retro-Propulsion (SRP) wind tunnel test. Advancement of SRP as a technology hinges partially on the ability of computational methods to accurately predict vehicle aerodynamics during the SRP phase of atmospheric descent. A wind tunnel test at the Langley Unitary Plan Wind Tunnel was specifically designed to validate Navier-Stokes codes for SRP applications. The test consisted of a 5-inch diameter, 70- degree sphere-cone forebody with cylindrical afterbody, with four configurations spanning 0 to 4 jets. Test data include surface pressure (including high-frequency response), flowfield imagery, and internal pressure and temperature measurements. Three computational fluid dynamics (CFD) codes (DPLR, FUN3D, and OVERFLOW) are exercised for both single and multiple-nozzle configurations for a range of Mach (M) numbers and thrust coefficients. Comparisons to test data will be used to evaluate accuracy, identify modeling shortcomings, and gain insight into the computational requirements necessary for computing these complex flows.

**CFD Verification of Supersonic Retropropulsion for a Central and Peripheral Configuration**
(1.8 MB PDF)

Supersonic retropropulsion (SRP) is a potential enabling technology for deceleration of high mass vehicles at Mars. Previous sub-scale testing, performed during the 1960s and 1970s to explore and characterize various decelerator technologies, focused on SRP configurations with a single central nozzle located along the axis of a vehicle; however, some multiple nozzle configurations were examined. Only one of these tests showed a peripheral configuration with nozzles outboard of the vehicle centerline. Recent computational efforts have been initiated to examine the capability of computational fluid dynamics (CFD) to capture the complex SRP flow fields. This study assesses the accuracy of a CFD tool over a range of thrust conditions for both a central and peripheral configuration. Included is a discussion of the agreement between the CFD simulations and available wind tunnel data as well as a discussion of computational impacts on SRP simulation.

**Advanced CFD Methods for Wind Turbine Analysis**
(13 MB PDF)

Horizontal-axis wind turbines operate in a complex, inherently unsteady aerodynamic environment. The flow over the blades is dominated by 3-D effects, particularly during stall, which is accompanied by massive flow separation and vortex shedding. There is always bluff-body shedding from the turbine nacelle and support structure which interacts with the rotor wake. In addition, the high aspect ratios of wind turbine blades make them very flexible, leading to substantial aeroelastic deformation of the blades, altering the aerodynamics. Finally, when situated in a wind farm, turbines must operate in the unsteady wake of upstream neighbors. Though computational fluid dynamics (CFD) has made significant inroads as a research tool, simple, inexpensive methods, such as blade element momentum theory, are still the workhorses in wind turbine design and aeroelasticity applications. These methods are unable to accurately predict rotor loads near the edges of the operating envelope.

In this work, a range of unstructured grid CFD techniques for predicting wind turbine loads and aeroelasticity has been developed and applied to the NREL Unsteady Aerodynamics Experiment Phase VI rotor. First, a kd-tree based nearest neighbor search algorithm was used to improve the computational efficiency of an approximate unsteady actuator blade method. This method was then shown to predict root and tip vortex locations and strengths similar to an overset method, but without the computational expense of modeling the blade surfaces. A hybrid Reynolds-averaged Navier-Stokes / Large Eddy Simulation (HRLES) turbulence model was extended to an unstructured grid framework and demonstrated to improve predictions of unsteady loading and shedding frequency in massively separated cases. For aeroelastic predictions, a methodology for tight coupling between an unstructured CFD solver and a computational structural dynamics tool was developed. Finally, time-accurate overset rotor simulations of a complete turbine—-blades, nacelle, and tower—-were conducted using both RANS and HRLES turbulence models. The HRLES model was able to accurately predict rotor loads when stalled. In yawed flow, excellent correlations of mean blade loads with experimental data were obtained across the span, and wake asymmetry and unsteadiness were also well-predicted.

**A Quasi-steady Flexible Launch Vehicle Stability Analysis
Using Steady CFD with Unsteady Aerodynamic Enhancement**
(0.5 MB PDF)

Launch vehicles frequently experience a reduced stability margin through the transonic Mach number range. This reduced stability margin is caused by an undamping of the aerodynamics in one of the lower frequency flexible or rigid body modes. Analysis of the behavior of a flexible vehicle is routinely performed with quasi-steady aerodynamic line loads derived from steady rigid computational fluid dynamics (CFD). However, a quasi-steady aeroelastic stability analysis can be unconservative at the critical Mach numbers where experiment or unsteady computational aeroelastic (CAE) analysis show a reduced or even negative aerodynamic damping. This paper will present a method of enhancing the quasi-steady aeroelastic stability analysis of a launch vehicle with unsteady aerodynamics. The enhanced formulation uses unsteady CFD to compute the response of selected lower frequency modes. The response is contained in a time history of the vehicle line loads. A proper orthogonal decomposition of the unsteady aerodynamic line load response is used to reduce the scale of data volume and system identification is used to derive the aerodynamic stiffness, damping and mass matrices. The results of the enhanced quasi-static aeroelastic stability analysis are compared with the damping and frequency computed from unsteady CAE analysis and from a quasi-steady analysis. The results show that incorporating unsteady aerodynamics in this way brings the enhanced quasi-steady aeroelastic stability analysis into close agreement with the unsteady CAE analysis.

**Numerical Study Comparing RANS and LES Approaches on a
Circulation Control Airfoil**
(6.1 MB PDF)

A numerical study over a nominally two-dimensional circulation control airfoil is performed using a large eddy simulation code and two Reynolds-averaged Navier-Stokes codes. Different Coanda jet blowing conditions are investigated. In addition to investigating the influence of grid density, a comparison is made between incompressible and compressible flow solvers. The incompressible equations are found to yield negligible differences from the compressible equations up to at least a jet exit Mach number of 0.64. The effects of different turbulence models are also studied. Models that do not account for streamline curvature effects tend to predict jet separation from the Coanda surface too late, and can produce non-physical solutions at high blowing rates. Three different turbulence models that account for streamline curvature are compared with each other and with large eddy simulation solutions. All three models are found to predict the Coanda jet separation location reasonably well, but one of the models predicts specific flow field details near the Coanda surface prior to separation much better than the other two. All Reynolds-averaged Navier-Stokes computations produce higher circulation than large eddy simulation computations, with different stagnation point location and greater flow acceleration around the nose onto the upper surface. The precise reasons for the higher circulation are not clear, although it is not solely a function of predicting the jet separation location correctly.

**FUN3D and CFL3D Computations for the First High Lift Prediction
Workshop**
(1.4 MB PDF)

Two Reynolds-averaged Navier-Stokes codes were used to compute flow over the NASA Trapezoidal Wing at high lift conditions for the 1st AIAA CFD High Lift Prediction Workshop, held in Chicago in June 2010. The unstructured-grid code FUN3D and the structured-grid code CFL3D were applied to several different grid systems. The effects of code, grid system, turbulence model, viscous term treatment, and brackets were studied. The SST model on this configuration predicted lower lift than the Spalart-Allmaras model at high angles of attack; the Spalart-Allmaras model agreed better with experiment. Neglecting viscous cross-derivative terms caused poorer prediction in the wing tip vortex region. Output-based grid adaptation was applied to the unstructured-grid solutions. The adapted grids better resolved wake structures and reduced flap flow separation, which was also observed in uniform grid refinement studies. Limitations of the adaptation method as well as areas for future improvement were identified.

**Optimization of 2-D Flap Geometry Using Matlab and Fun3D**
(900 KB PDF)

This paper describes work done in the process of creating a workable system for the optimization of two-element high-lift airfoil design based on a fixed “cruise configuration” baseline. Methods were developed to define airfoil flap geometry, automatically create and run unstructured computational meshes based on this geometry, and to iteratively optimize this geometry. Validation cases are presented based on different optimization algorithms and parameters. Significant work is also presented on the attempt to characterize the design space of this problem in order to better understand the performance of different optimization routines.

**Application of CFD in the Design of Flow Control Concepts for a
Ducted-Fan Configuration**
(5.4 MB PDF)

Leading and trailing edge flow control concepts were investigated for control of small ducted-fan unmanned aerospace vehicles. These concepts have the potential to augment vehicle controllability while decreasing dependence on conventional control surfaces. Steady state and unsteady CFD analyses were utilized in the design and analysis of Synthetic Jet Actuators, narrowing the concepts that were built and wind tunnel tested. Steady state analyses of leading-edge blowing concepts oriented 45 degrees against the local flow, 5 degrees inside the duct lip created a 60% change in pitching moment. Unsteady analyses of trailing edge blowing concepts revealed that blowing can significantly affect the flow over the Coanda trailing-edge surface. When the vehicle is at an angle of attack, partial circumferential blowing resulted in nearly the same effects on pitching moment as full circumferential blowing, thereby reducing experimental fabrication costs. Both of these predicted effects were later verified in wind tunnel testing.

**Feature-Based and Output-Based Grid Adaptation Study for
Hypersonic Propulsive Deceleration Jet Flows**
(2.5 MB PDF)

The size requirements for conventional aerodynamic decelerators (parachutes) used to slow Mars entry vehicles during atmospheric descent are becoming infeasible due to the increasing mass and landing site altitude of future missions. One alternative is propulsive decelerator (PD) jets. The use of PD jets, however, involves complex flow interactions that are still not well understood. Computational fluid dynamics (CFD) is currently being investigated as a tool for predicting these flow interactions. However, manually generating appropriate grids for these flows is difficult and time consuming because of the complexity of the flowfield. Therefore, automatic grid adaptation techniques present an attractive option to accurately capture the flow features and interactions. This study compares the grids and solutions for hypersonic PD jet flows using feature-based and output-based grid adaptation techniques.

**A Critical Study of Agglomerated Multigrid Methods for
Diffusion on Highly-Stretched Grids**
(1.2 MB PDF)

Agglomerated multigrid methods for unstructured grids are studied critically for solving a model diffusion equation on highly-stretched grids typical of practical viscous simulations, following a previous work focused on isotropic grids. Different primal elements, including prismatic and tetrahedral elements in three dimensions, are considered. The components of an efficient node-centered full-coarsening multigrid scheme are identified and assessed using quantitative analysis methods. Fast grid-independent convergence is demonstrated for mixed-element grids composed of tetrahedral elements in the isotropic regions and prismatic elements in the highly-stretched regions. Implicit lines natural to advancing-layer/advancing-front grid generation techniques are essential elements of both relaxation and agglomeration. On agglomerated grids, consistent average-least-square discretizations augmented with edge-directional gradients to increase h-ellipticity of the operator are used. Simpler (edge-terms-only) coarse-grid discretizations are also studied and shown to produce grid-dependent convergence

~~-~~only effective on grids with minimal skewing.

**Validation of an Output-Adaptive, Tetrahedral Cut-Cell
Method for Sonic Boom Prediction**
(16 MB PDF)

A cut-cell approach to computational fluid dynamics that uses the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated for an adaptive method to control error in a specified output. The adaptive method is applied to sonic boom prediction by specifying an integral of offbody pressure signature as the output. These predicted signatures are compared to wind-tunnel measurements to validate the method for sonic boom prediction. Accurate midfield sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly refined, shock-aligned anisotropic grids are produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provides stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut cells with an output-based adaptive scheme automates the volume grid generation task after a triangular mesh is generated for the cut surface.

**Comparison of Inviscid and Viscous Aerodynamic Predictions of Supersonic Retropropulsion Flowfields**
(19 MB PDF)

Supersonic retropropulsion, or the initiation of a retropropulsion phase at supersonic freestream conditions, is an enabling decelerator technology for high-mass planetary entries at Mars. The current knowledge on supersonic retropropulsion is largely derived from exploratory development efforts prior to the Viking missions in the 1960s and early 1970s, predominantly sub-scale wind tunnel testing. Little literature exists on analytical and computational modeling approaches for supersonic aerodynamic-propulsive interactions at moderate thrust levels and flight-relevant conditions. This investigation presents a discussion of the relevant flow physics to provide insight into the effectiveness of inviscid and viscous computational analysis approaches in consistently and accurately capturing the relevant flow physics. Preliminary computational results for a blunt body with two retropropulsion configurations are compared with experimental data for the location of prominent flow features and surface pressure distributions. This work is intended to provide an initial discussion of the challenges facing the computational simulation of supersonic retropropulsion flowfields.

**An Initial Assessment of Navier-Stokes Codes Applied to Supersonic Retro-Propulsion**
(12 MB PDF)

This paper describes an initial evaluation of Navier-Stokes computational fluid dynamics codes applied to the problem of Supersonic Retro-Propulsion flowfield prediction. A few cases with existing wind tunnel data were selected to evaluate Navier-Stokes codes and build experience running flowfield simulations. Three codes (DPLR, FUN3D, and OVERFLOW) have been exercised for both single- and multi-nozzle configurations for a range of Mach numbers and thrust coefficients, all at zero degrees angle-of-attack. Comparisons of surface pressure and flow structure have been used to evaluate the codes and identify modeling strengths and weaknesses. In addition, lessons learned about grid generation, grid adaptation, and solution advancement are reported for each code.

**Application of the FUN3D Unstructured-Grid Navier-Stokes Solver
to the 4th AIAA Drag PredictionWorkshop Cases**
(1.8 MB PDF)

FUN3D Navier-Stokes solutions were computed for the 4th AIAA Drag Prediction Workshop grid convergence study, downwash study, and Reynolds number study on a set of node-based mixed-element grids. All of the baseline tetrahedral grids were generated with the VGRID (developmental) advancing-layer and advancing-front grid generation software package following the gridding guidelines developed for the workshop. With maximum grid sizes exceeding 100 million nodes, the grid convergence study was particularly challenging for the node-based unstructured grid generators and flow solvers. At the time of the workshop, the super-fine grid with 105 million nodes and 600 million elements was the largest grid known to have been generated using VGRID. FUN3D Version 11.0 has a completely new pre- and post-processing paradigm that has been incorporated directly into the solver and functions entirely in a parallel, distributed memory environment. This feature allowed for practical pre-processing and solution times on the largest unstructured-grid size requested for the workshop. For the constant-lift grid convergence case, the convergence of total drag is approximately second-order on the finest three grids. The variation in total drag between the finest two grids is only 2 counts. At the finest grid levels, only small variations in wing and tail pressure distributions are seen with grid refinement. Similarly, a small wing side-of-body separation also shows little variation at the finest grid levels. Overall, the FUN3D results compare well with the structured-grid code CFL3D. The FUN3D downwash study and Reynolds number study results compare well with the range of results shown in the workshop presentations.

**Development and Application of Agglomerated Multigrid Methods for
Complex Geometries**
(9.1 MB PDF)

We report progress in the development of agglomerated multigrid techniques for fully unstructured grids in three dimensions, building upon two previous studies focused on efficiently solving a model diffusion equation. We demonstrate a robust fully-coarsened agglomerated multigrid technique for 3D complex geometries, incorporating the following key developments: consistent and stable coarse-grid discretizations, a hierarchical agglomeration scheme, and line-agglomeration/relaxation using prismatic-cell discretizations in the highly-stretched grid regions. A significant speed-up in computer time over state-of-art single-grid computations is demonstrated for a model diffusion problem, the Euler equations, and the Reynolds-averaged Navier-Stokes equations for 3D realistic complex geometries.

**Description of a Website Resource for Turbulence Modeling
Verification and Validation**
(2.4 MB PDF)

The activities of the Turbulence Model Benchmarking Working Group – which is a subcommittee of the American Institute of Aeronautics and Astronautics (AIAA) Fluid Dynamics Technical Committee – are described. The group’s main purpose is to establish a web-based repository for Reynolds-averaged Navier-Stokes turbulence model documentation, including verification and validation cases. This turbulence modeling resource has been established based on feedback from a survey on what is needed to achieve consistency and repeatability in turbulence model implementation and usage, and to document and disseminate information on new turbulence models or improvements to existing models. The various components of the website are described in detail: description of turbulence models, turbulence model readiness rating system, verification cases, validation cases, validation databases, and turbulence manufactured solutions. An outline of future plans of the working group is also provided.

**Reduced-Order Models for the Aeroelastic Analysis of Ares Launch
Vehicles**
(3.8 MB PDF)

This document presents the development and application of unsteady aerodynamic, structural dynamic, and aeroelastic reduced-order models (ROMs) for the ascent aeroelastic analysis of the Ares I-X flight test and Ares I crew launch vehicles using the unstructured-grid, aeroelastic FUN3D computational fluid dynamics (CFD) code. The purpose of this work is to perform computationally-efficient aeroelastic response calculations that would be prohibitively expensive via computation of multiple full-order aeroelastic FUN3D solutions. These efficient aeroelastic ROM solutions provide valuable insight regarding the aeroelastic sensitivity of the vehicles to various parameters over a range of dynamic pressures.

**Computational Aeroelastic Analysis of the Ares Launch
Vehicle During Ascent**
(1.9 MB PDF)

This paper presents the static and dynamic computational aeroelastic (CAE) analyses of the Ares crew launch vehicle (CLV) during atmospheric ascent. The influence of launch vehicle flexibility on the static aerodynamic loading and integrated aerodynamic force and moment coefficients is discussed. The ultimate purpose of this analysis is to assess the aeroelastic stability of the launch vehicle along the ascent trajectory. A comparison of analysis results for several versions of the Ares CLV will be made. Flexible static and dynamic analyses based on rigid computational fluid dynamic (CFD) data are compared with a fully coupled aeroelastic time marching CFD analysis of the launch vehicle.

**Computational Aeroelastic Analysis of Ares Crew Launch Vehicle
Bi-Modal Loading**
(19.0 MB PDF)

A Reynolds averaged Navier-Stokes analysis, with and without dynamic aeroelastic effects, is presented for the Ares I-X launch vehicle at transonic Mach numbers and flight Reynolds numbers for two grid resolutions and two angles of attack. The purpose of the study is to quantify the force and moment increment imparted by the sudden transition from fully separated flow around the crew module – service module junction to that of the bi-modal flow state in which only part of the flow reattaches. The bi-modal flow phenomenon is of interest to the guidance, navigation and control community because it causes a discontinuous jump in forces and moments. Computations with a rigid structure at zero angle of attack indicate significant increases in normal force and pitching moment. Dynamic aeroelastic computations indicate the bi-modal flow state is insensitive to vehicle flexibility due to the resulting deflections imparting only very small changes in local angle of attack. At an angle of attack of 2.56, the magnitude of the pitching moment increment resulting from the bi-modal state nearly triples, while occurring at a slightly lower Mach number. Significant grid induced variations between the solutions indicate that further grid refinement is warranted.

**FUN3D Grid Refinement and Adaptation Studies for the Ares
Launch Vehicle**
(14.4 MB PDF)

This paper presents grid refinement and adaptation studies performed in conjunction with computational aeroelastic analyses of the Ares crew launch vehicle (CLV). The unstructured grids used in this analysis were created with GridTool and VGRID while the adaptation was performed using the Computational Fluid Dynamic (CFD) code FUN3D with a feature based adaptation software tool. GridTool was developed by ViGYAN, Inc. while the last three software suites were developed by NASA Langley Research Center. The feature based adaptation software used here operates by aligning control volumes with shock and Mach line structures and by refining/de-refining where necessary. It does not redistribute node points on the surface. This paper assesses the sensitivity of the complex flow field about a launch vehicle to grid refinement. It also assesses the potential of feature based grid adaptation to improve the accuracy of CFD analysis for a complex launch vehicle configuration. The feature based adaptation shows the potential to improve the resolution of shocks and shear layers. Further development of the capability to adapt the boundary layer and surface grids of a tetrahedral grid is required for significant improvements in modeling the flow field.

**Comparison of Node-Centered and Cell-Centered Unstructured
Finite-Volume Discretizations: Viscous Fluxes**
(740 KB PDF)

Finite-volume discretization schemes for viscous fluxes on general grids are compared using node-centered and cell-centered approaches. The grids range from regular grids to highly irregular grids, including random perturbations of the grid nodes. Accuracy and complexity are studied for four nominally second-order accurate schemes: a node-centered scheme and three cell-centered schemes (a node-averaging scheme and two schemes using least-squares face-gradient reconstruction). The two least-squares schemes use either a nearest-neighbor or an adaptive-compact stencil at a face. The node-centered and least-squares schemes have similarly low levels of complexity. The node-averaging scheme has the highest complexity and can fail to converge to the exact solution when clipping of the node-averaged values is used. On highly anisotropic grids, typical of those encountered in grid adaptation, the least-squares schemes, the node-averaging scheme without clipping, and the node-centered scheme demonstrate similar second-order accuracies per degree of freedom. On anisotropic grids over a curved body, typical of turbulent flow simulations, the node-centered scheme is second-order accurate. The node-averaging scheme may degenerate on mixed-element grids. The least-squares schemes have to be amended to maintain second-order accuracy by either introducing a local approximate mapping or modifying the stencil to reflect the direction of strong coupling. Overall, the accuracies of the node-centered and the best cell-centered schemes are comparable at an equivalent number of degrees of freedom on isotropic and curved anisotropic grids. On stretched, randomly perturbed grids in a rectangular geometry, both gradient and discretization errors for all schemes are orders of magnitude higher than corresponding errors on regular grids.

**Assessment of Hybrid RANS/LES Turbulence Models for
Aeroacoustics Applications**
(2.0 MB PDF)

Predicting the noise from aircraft with exposed landing gear remains a challenging problem for the aeroacoustics community. Although computational fluid dynamics (CFD) has shown promise as a technique that could produce high-fidelity flow solutions, generating grids that can resolve the pertinent physics around complex configurations can be very challenging. Structured grids are often impractical for such configurations. Unstructured grids offer a path forward for simulating complex configurations. However, few unstructured grid codes have been thoroughly tested for unsteady flow problems in the manner needed for aeroacoustic prediction. A widely used unstructured grid code, FUN3D, is examined for resolving the near field in unsteady flow problems. Although the ultimate goal is to compute the flow around complex geometries such as the landing gear, simpler problems that include some of the relevant physics, and are easily amenable to the structured grid approaches are used for testing the unstructured grid approach. The test cases chosen for this study correspond to the experimental work on single and tandem cylinders conducted in the Basic Aerodynamic Research Tunnel (BART) and the Quiet Flow Facility (QFF) at NASA Langley Research Center. These configurations offer an excellent opportunity to assess the performance of hybrid RANS/LES turbulence models that transition from RANS in unresolved regions near solid bodies to LES in the outer flow field. Several of these models have been implemented and tested in both structured and unstructured grid codes to evaluate their dependence on the solver and mesh type. Comparison of FUN3D solutions with experimental data and numerical solutions from a structured grid flow solver are found to be encouraging.

**Discrete Adjoint-Based Design Optimization of Unsteady
Turbulent Flows on Dynamic Unstructured Grids**
(3.5 MB PDF)

An adjoint-based methodology for design optimization of unsteady turbulent flows on dynamic unstructured grids is described. The implementation relies on an existing unsteady three-dimensional unstructured grid solver capable of dynamic mesh simulations and discrete adjoint capabilities previously developed for steady flows. The discrete equations for the primal and adjoint systems are presented for the backward-difference family of time-integration schemes on both static and dynamic grids. The consistency of sensitivity derivatives is established via comparisons with complex-variable computations. The current work is believed to be the first verified implementation of an adjoint-based optimization methodology for the true time-dependent formulation of the Navier-Stokes equations in a practical computational code. Large-scale shape optimizations are demonstrated for turbulent flows over a tilt-rotor geometry and a simulated aeroelastic motion of a fighter jet.

**Investigation of Mixed Element Hybrid Grid-Based CFD
Methods for Rotorcraft Flow Analysis**
(900 KB PDF)

Accurate first-principles flow prediction is essential to the design and development of rotorcraft, and while current numerical analysis tools can, in theory, model the complete flow field, in practice the accuracy of these tools is limited by various inherent numerical deficiencies. An approach that combines the first-principles physical modeling capability of CFD schemes with the vortex preservation capabilities of Lagrangian vortex methods has been developed recently that controls the numerical diffusion of the rotor wake in a grid-based solver by employing a vorticity-velocity, rather than primitive variable, formulation. Coupling strategies, including variable exchange protocols are evaluated using several unstructured, structured, and Cartesian-grid Reynolds Averaged Navier-Stokes (RANS)/Euler CFD solvers. Results obtained with the hybrid grid-based solvers illustrate the capability of this hybrid method to resolve vortex-dominated flow fields with lower cell counts than pure RANS/Euler methods.

**Analysis of CFD Modeling Techniques over the MV-22 Tiltrotor**
(1.4 MB PDF)

The V-22 Osprey is a tiltrotor aircraft designed to operate under a wide range of flight conditions. Its outer mold line geometry is aerodynamically complex in part because aerodynamic considerations were not primary influential factors for the major features of the aircraft. As mission requirements change and additional devices are added to the aircraft, questions regarding the aerodynamic impact must be answered. While many of these questions can be adequately answered using lower-fidelity methods, some situations require the use of higher-fidelity analysis. Computational fluid dynamics (CFD) is a tool that has been used frequently to answer aerodynamic questions associated with the V-22. However, the complexity of the aircraft makes this analysis challenging. Using unstructured grids is one way of reducing the lead time required to setup the simulation as unstructured grids lend themselves to modeling complex geometries. This paper compares independent OVERFLOW and FUN3D CFD analyses of the MV-22 tiltrotor in airplane mode over a range of angles of attack, and compares these results to data from a recent high-angle-of-attack wind tunnel test run in the Boeing V/STOL Wind Tunnel. The results lend insight into the choice of grid structure, near-body vortex generation, and numerical methodology, and reveal that care must be taken when setting up the CFD model as well as identifying any numerical phenomena that could be considered code specific.

**Local-in-Time Adjoint-Based Method for Design Optimization
of Unsteady Flows**
(200 KB PDF)

We present a new local-in-time discrete adjoint-based methodology for solving design optimization problems arising in unsteady aerodynamic applications. The new methodology circumvents storage requirements associated with the straightforward implementation of a global adjoint-based optimization method that stores the entire flow solution history for all time levels. This storage cost may quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and to approximate the solution of the unsteady adjoint equations and the sensitivity derivative as a combination of the corresponding local quantities computed on each time subinterval. Since each subinterval contains relatively few time levels, the storage cost of the local-in-time method is much lower than that of the global methods, thus making the time-dependent adjoint optimization feasible for practical applications. Another attractive feature of the new technique is that the converged solution obtained with the local-in-time method is a local extremum of the original optimization problem. The new method carries no computational overhead as compared with the global implementation of adjoint-based methods. The paper presents a detailed comparison of the global- and local-in-time adjoint-based methods for design optimization problems governed by the unsteady compressible 2-D Euler equations.

**Notes on Accuracy of Finite-Volume Discretization Schemes on
Irregular
Grids**
(0.3 MB PDF)

These notes rebut some overreaching conclusions of Svard et al., (2008) concerning relations between truncation errors and discretization errors on irregular grids. Convergence of truncation errors severely degrades on general irregular grids. Such degradation does not necessarily imply a less than design-order convergence of discretization errors.

**Critical Study of Agglomerated Multigrid Methods for
Diffusion**
(5.6 MB PDF)

Agglomerated multigrid techniques used in unstructured-grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit. The studied target-grid discretizations and discretizations used on agglomerated grids are typical of current node-centered formulations. Agglomerated multigrid convergence rates are presented using a range of two- and three-dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids. Two agglomeration techniques are used within an overall topology-preserving agglomeration framework. The results show that a multigrid with an inconsistent coarse-grid scheme using only the edge derivatives (also referred to in the literature as a thin-layer formulation) provides considerable speedup over single-grid methods, but its convergence can deteriorate on highly skewed grids. A multigrid with a Galerkin coarse-grid discretization using piecewise-constant prolongation and a heuristic correction factor is slower and also can be grid dependent. In contrast, nearly grid-independent convergence rates are demonstrated for a multigrid with consistent coarse-grid discretizations. Convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two-grid cycle are replaced by their idealized counterparts.

**Adjoint-Based Design of Rotors in a Noninertial
Reference Frame**
(2.8 MB PDF)

Optimization of rotorcraft flowfields using an adjoint method generally requires a time-dependent implementation of the equations. The current study examines an intermediate approach in which a subset of rotor flowfields are cast as steady problems in a noninertial reference frame. This technique permits the use of an existing steady-state adjoint formulation with minor modifications to perform sensitivity analyses. The formulation is valid for isolated rigid rotors in hover or where the freestream velocity is aligned with the axis of rotation. Discrete consistency of the implementation is demonstrated by using comparisons with a complex-variable technique, and a number of single and multipoint optimizations for the rotorcraft figure of merit function are shown for varying blade collective angles. Design trends are shown to remain consistent as the grid is refined.

**Mitigation of Dynamic Stall Using Small Controllable Devices**
(2.5 MB PDF)

The unsteady, compressible Reynolds-averaged Navier-Stokes equations, based on an unstructured-grid approach with one-equation Spalart-Allmaras, and two-equation Menter shear-stress-transport turbulence models, have been used to investigate flow over oscillating airfoils. The dynamic stall characteristics of the Boeing VR-7 airfoil without controllable devices were computed and compared with experimental data. Two actively controllable devices in the form of trailing-edge flap and leading-edge slat were analyzed for the same airfoil to mitigate dynamic stall effects. The addition of a trailing-edge flap on a VR-7 airfoil with sinusoidal motion about flap hinge opposite to the main oscillating airfoil can delay stall and reduce negative peak pitching moment. The addition of a stationary or moving leading-edge slat on a VR-7 airfoil completely eliminates the development of a dynamic vortex and enhances lift.

**Updates to Multi-Dimensional Flux Reconstruction for
Hypersonic Simulations on Tetrahedral Grids**
(1.9 MB PDF)

The quality of simulated hypersonic stagnation region heating with tetrahedral meshes is investigated by using an updated three-dimensional, upwind reconstruction algorithm for the inviscid flux vector. An earlier implementation of this algorithm provided improved symmetry characteristics on tetrahedral grids compared to conventional reconstruction methods. The original formulation however displayed quantitative differences in heating and shear that were as large as 25% compared to a benchmark, structured-grid solution. The primary cause of this discrepancy is found to be an inherent inconsistency in the formulation of the flux limiter. The inconsistency is removed by employing a Green- Gauss formulation of primitive gradients at nodes to replace the previous Gram-Schmidt algorithm. Current results are now in good agreement with benchmark solutions for two challenge problems: (1) hypersonic flow over a three-dimensional cylindrical section with special attention to the uniformity of the solution in the spanwise direction and (2) hypersonic flow over a three-dimensional sphere. The tetrahedral cells used in the simulation are derived from a structured grid where cell faces are bisected across the diagonal resulting in a consistent pattern of diagonals running in a biased direction across the otherwise symmetric domain. This grid is known to accentuate problems in both shock capturing and stagnation region heating encountered with conventional, quasi-one-dimensional inviscid flux reconstruction algorithms. Therefore the test problems provide a sensitive indicator for algorithmic effects on heating. Additional simulations on a sharp, double cone and the shuttle orbiter are then presented to demonstrate the capabilities of the new algorithm on more geometrically complex flows with tetrahedral grids. These results provide the first indication that pure tetrahedral elements utilizing the updated, three-dimensional, upwind reconstruction algorithm may be used for the simulation of heating and shear in hypersonic flows in upwind, finite volume formulations.

**Comparison of Node-Centered and Cell-Centered Unstructured
Finite-Volume Discretizations: Inviscid Fluxes**
(0.5 MB PDF)

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. Regular and irregular grids are considered, including mixed-element grids and grids with random perturbations of nodes. Complexity, accuracy, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes, NC and NC-WLSQ, with respective unweighted and weighted least-square methods for gradient reconstruction and six cell-centered schemes – two node-averaging schemes with and without clipping and four schemes that employ different stencils for least-square gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the least-square stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The node-centered solutions converge with second order on regular and/or triangular grids and with first order on perturbed quadrilateral and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and small second-order discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors are comparable for all schemes except the CC-NA scheme; the latter may produce large discretization errors explained by node-averaging degeneration. Accurate gradients can be reconstructed by least-square methods with a local approximate mapping; without mapping, only the NC-WLSQ scheme provides accurate gradients. Defect-correction iterations may diverge for the CC-NA and NC-WLSQ schemes; the iterations converge fast for the CC-SA and CC-FA schemes on all grids, and converge slower for other schemes on at least some irregular grids.

**CFD Assessment of Aerodynamic Degradation of a Subsonic
Transport Due to Airframe Damage**
(5.9 MB PDF)

A computational study is presented to assess the utility of two NASA unstructured Navier-Stokes flow solvers for capturing the degradation in static stability and aerodynamic performance of a NASA General Transport Model (GTM) due to airframe damage. The approach is to correlate computational results with a substantial subset of experimental data for the GTM undergoing progressive losses to the wing, vertical tail, and horizontal tail components. The ultimate goal is to advance the probability of inserting computational data into the creation of advanced flight simulation models of damaged subsonic aircraft in order to improve pilot training. Results presented in this paper demonstrate good correlations with slope-derived quantities, such as pitch static margin and static directional stability, and incremental rolling moment due to wing damage. This study further demonstrates that high- fidelity Navier-Stokes flow solvers could augment flight simulation models with additional aerodynamic data for various airframe damage scenarios.

**Re-evaluation of an Optimized Second Order Backward
Difference (BDF2OPT) Scheme for Unsteady Flow Applications**
(3.3 MB PDF)

Recent experience in the application of an optimized, second-order, backward-difference (BDF2OPT) temporal scheme is reported. The primary focus of the work is on obtaining accurate solutions of the unsteady Reynolds-averaged Navier-Stokes equations over long periods of time for aerodynamic problems of interest. The baseline flow solver under consideration uses a particular BDF2OPT temporal scheme with a dual-timestepping algorithm for advancing the flow solutions in time. Numerical difficulties are encountered with this scheme when the flow code is run for a large number of time steps, a behavior not seen with the standard second-order, backward-difference, temporal scheme. Based on a stability analysis, slight modifications to the BDF2OPT scheme are suggested. The performance and accuracy of this modified scheme is assessed by comparing the computational results with other numerical schemes and experimental data.

**Turbulent Output-Based Anisotropic Adaptation**
(4.4 MB PDF)

Controlling discretization error is a remaining challenge for computational fluid dynamics simulation. Grid adaptation is applied to reduce estimated discretization error in drag or pressure integral output functions. To enable application to high O(10

^{7}) Reynolds number turbulent flows, a hybrid approach is utilized that freezes the near-wall boundary layer grids and adapts the grid away from the no slip boundaries. The hybrid approach is not applicable to problems with under resolved initial boundary layer grids, but is a powerful technique for problems with important off-body anisotropic features. Supersonic nozzle plume, turbulent flat plate, and shock-boundary layer interaction examples are presented with comparisons to experimental measurements of pressure and velocity. Adapted grids are produced that resolve off-body features in locations that are not known a priori.

**Output Based Grid Adaptation for Viscous Flow**
(900 KB PDF)

Output (adjoint) based adaptation is a method that has been used to automate the unstructured grid generation task of inviscid and two-dimensional (2D) turbulent flow simulation. This project challenges existing three dimensional (3D) techniques to produce strongly anisotropic grids for modeling the boundary layer on a flat plate. Elements with large face angles can cause difficulties for the diffusion operator employed in this study, so a range of Reynolds numbers are explored to quantify the accuracy of the solutions on the output adapted grids. Laminar and turbulent flows are simulated on an extruded flat plate to exercise the 3D algorithm on a case with a know solution. Regular, fully adapted, and hybrid unstructured tetrahedral grids are used to examine the ability of different grid constructions to reproduce Blasius or empirically derived velocity profiles. The hybrid approach retained the original regular tetrahedral grid near the flat plate and adapted the grid outside of this region. This approach produced similar velocity profiles to the regular grids and better results than the fully adaptive cases for the highest Reynolds number cases.

**A Critical Study of Agglomerated Multigrid Methods for
Diffusion**
(1.6 MB PDF)

Agglomerated multigrid techniques used in unstructured-grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit. The studied target-grid discretizations and discretizations used on agglomerated grids are typical of current node-centered formulations. Agglomerated multigrid convergence rates are presented using a range of two- and three-dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids. Two agglomeration techniques are used within an overall topology-preserving agglomeration framework. The results show that multigrid with an inconsistent coarse-grid scheme using only the edge terms (also referred to in the literature as a thin-layer formulation) provides considerable speedup over single-grid methods but its convergence deteriorates on finer grids. Multigrid with a Galerkin coarse-grid discretization using piecewise-constant prolongation and a heuristic correction factor is slower and also grid-dependent. In contrast, grid-independent convergence rates are demonstrated for multigrid with consistent coarse-grid discretizations. Convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two-grid cycle are replaced by their idealized counterparts.

**Development of Unsteady Aerodynamic and Aeroelastic
Reduced-Order Models
Using the FUN3D Code**
(2.0 MB PDF)

Recent significant improvements to the development of CFD-based unsteady aerodynamic reduced-order models (ROMs) are implemented into the FUN3D unstructured flow solver. These improvements include the simultaneous excitation of the structural modes of the CFD-based unsteady aerodynamic system via a single CFD solution, minimization of the error between the full CFD and the ROM unsteady aerodynamic solution, and computation of a root locus plot of the aeroelastic ROM. Results are presented for a viscous version of the two-dimensional Benchmark Active Controls Technology (BACT) model and an inviscid version of the AGARD 445.6 aeroelastic wing using the FUN3D code.

**Computational Fluid Dynamics Validation of a Single
Central Nozzle Supersonic Retropropulsion Configuration**
(1.6 MB PDF)

Supersonic retropropulsion provides an option that can potentially enhance drag characteristics of high mass entry, descent, and landing systems. Preliminary flow field and vehicle aerodynamic characteristics have been found in wind tunnel experiments; however, these only cover specific vehicle configurations and freestream conditions. In order to generate useful aerodynamic data that can be used in a trajectory simulation, a quicker method of determining vehicle aerodynamics is required to model supersonic retropropulsion effects. Using computational fluid dynamics, flow solutions can be determined which yield the desired aerodynamic information. The flow field generated in a supersonic retropropulsion scenario is complex, which increases the difficulty of generating an accurate computational solution. By validating the computational solutions against available wind tunnel data, the confidence in accurately capturing the flow field is increased, and methods to reduce the time required to generate a solution can be determined. Fun3D, a computational fluid dynamics code developed at NASA Langley Research Center, is capable of modeling the flow field structure and vehicle aerodynamics seen in previous wind tunnel experiments. Axial locations of the jet terminal shock, stagnation point, and bow shock show the same trends which were found in the wind tunnel, and the surface pressure distribution and drag coefficient are also consistent with available data. The flow solution is dependent on the computational grid used, where a grid which is too coarse does not resolve all of the flow features correctly. Refining the grid will increase the fidelity of the solution; however, the calculations will take longer if there are more cells in the computational grid.

**Consistency, Verification, and Validation of Turbulence
Models for
Reynolds-Averaged Navier-Stokes Applications**
(3.5 MB PDF)

In current practice, it is often difficult to draw firm conclusions about turbulence model accuracy when performing multi-code CFD studies ostensibly using the same model because of inconsistencies in model formulation or implementation in different codes. This paper describes an effort to improve the consistency, verification, and validation of turbulence models within the aerospace community through a website database of verification and validation cases. Some of the variants of two widely-used turbulence models are described, and two independent computer codes (one structured and one unstructured) are used in conjunction with two specific versions of these models to demonstrate consistency with grid refinement for several representative problems. Naming conventions, implementation consistency, and thorough grid resolution studies are key factors necessary for success.

**Discrete Adjoint-Based Design Optimization of Unsteady
Turbulent Flows on Dynamic Unstructured Grids**
(2.0 MB PDF)

An adjoint-based methodology for design optimization of unsteady turbulent flows on dynamic unstructured grids is described. The implementation relies on an existing unsteady three-dimensional unstructured grid solver capable of dynamic mesh simulations and discrete adjoint capabilities previously developed for steady flows. The discrete equations for the primal and adjoint systems are presented for the backward-difference family of time-integration schemes on both static and dynamic grids. The consistency of sensitivity derivatives is established via comparisons with complex-variable computations. The current work is believed to be the first verified implementation of an adjoint-based optimization methodology for the true time-dependent formulation of the Navier-Stokes equations in a practical computational code. Large-scale shape optimizations are demonstrated for turbulent flows over a tiltrotor geometry and a simulated aeroelastic motion of a fighter jet.

**Aerodynamic Interference Due to MSL Reaction Control
System**
(3.2 MB PDF)

An investigation of effectiveness of the reaction control system (RCS) of Mars Science Laboratory (MSL) entry capsule during atmospheric flight has been conducted. MSL is designed to fly a lifting actively guided entry with hypersonic bank maneuvers therefore, an understanding of RCS effectiveness is required. In the course of the study several jet configurations were evaluated using Langley Aerothermal Upwind Relaxation Algorithm (LAURA) code, Data Parallel Line Relaxation (DPLR) code, Fully Unstructured 3D (FUN3D) code and an Overset Grid Flow solver (OVERFLOW) code. Computations indicated that some of the proposed configurations might induce aero-RCS interactions, sufficient to impede and even overwhelm the intended control torques. It was found that the maximum potential for aero-RCS interference exists around peak dynamic pressure along the trajectory. Present analysis largely relies on computational methods. Ground testing, flight data and computational analyses are required to fully understand the problem. At the time of this writing some experimental work spanning range of Mach number 2.5 through 4.5 has been completed and used to establish preliminary levels of confidence for computations. As a result of the present work a final RCS configuration has been designed such as to minimize aero-interference effects and it is a design baseline for MSL entry capsule.

**Aerodynamic Challenges for the Mars Science Laboratory
Entry, Descent and Landing**
(6.7 MB PDF)

An overview of several important aerodynamics challenges new to the Mars Science Laboratory (MSL) entry vehicle are presented. The MSL entry capsule is a 70-deg sphere-cone based on the original Mars Viking entry capsule. Due to payload and landing accuracy requirements, MSL will be flying at the highest lift-to-drag ratio of any capsule sent to Mars (L/D=0.24). The capsule will also be flying a guided entry, performing bank maneuvers, a first for Mars entry. The system’s mechanical design and increased performance requirements require an expansion of the MSL flight envelope beyond those of historical missions. In certain areas, the experience gained by Viking and other recent Mars missions can no longer be claimed as heritage information. New analysis and testing is required to ensure the safe flight of the MSL entry vehicle. The challenge topics include: hypersonic gas chemistry and laminar-versus turbulent flow effects on trim angle, a general risk assessment of flying at greater angles-of-attack than Viking, quantifying the aerodynamic interactions induced by a new reaction control system and a risk assessment of recontact of a series of masses jettisoned prior to parachute deploy. An overview of the analysis and tests being conducted to understand and reduce risk in each of these areas is presented. The need for proper modeling and implementation of uncertainties for use in trajectory simulation has resulted in a revision of prior models and additional analysis for the MSL entry vehicle. The six degree-of-freedom uncertainty model and new analysis to quantify roll torque dispersions are presented.

**Ducted-Fan Force and Moment Control via Steady and Synthetic
Jets**
(12.0 MB PDF)

The authors have explored novel applications of synthetic jet actuators for leading and trailing edge flow control on ducted fan vehicles. The synthetic jets on the duct are actuated asymmetrically around the circumference to produce control forces and moments. These forces and moments could be utilized as flight control effectors for combating wind gusts or reducing control surface allocation required for trimmed flight. Synthetic jet component design, vehicle integration, CFD modeling, and wind tunnel experimental results are presented with a comparison to steady blowing. The flow control concepts demonstrated production of aerodynamic forces and moments on a ducted fan, although some cases required high flow rate steady blowing to create significant responses. Attaining high blowing momentum coefficients from synthetic jets is challenging since the time-averaged velocity is only a function of the outstroke: from bench test experiments it was seen that the time-averaged velocity was roughly one fourth of the peak velocity observed during the outstroke. The synthetic jets operated at lower blowing momentum coefficients than the steady jets tested, and in general the ducted fan application required more flow control authority than the synthetic jets could impart. However, synthetic jets were able to produce leading edge separation comparable to that obtained from steady jets with much higher blowing coefficients.

**Enhancement of Aeroelastic Rotor Airload Prediction Methods**
(41.0 MB PDF)

The accurate prediction of rotor airloads is a current topic of interest in the rotorcraft community. The complex nature of this loading makes this problem especially difficult. Some of the issues that must be considered include transonic effects on the advancing blade, dynamic stall effects on the retreating blade, and wake vortex interactions with the blades, fuselage, and other components. There are numerous codes to perform these predictions, both aerodynamic and structural, but until recently each code has refined either the structural or aerodynamic aspect of the analysis without serious consideration to the other, using only simplified modules to represent the physics. More recent work has concentrated on coupling CFD and CSD computations to be able to use the most accurate codes available to combine the best of the structural and the aerodynamic codes. However, CFD codes are the most computationally expensive codes available, and although combined CFD and CSD methods are shown to give the most accurate predictions available today, the additional accuracy must be deemed worth the time required to perform the computations.

The objective of the research is to both evaluate and extend a range of prediction methods comparing accuracy and computational expense. This range covers many methods where the highest accuracy method shown is a delta loads coupling between an unstructured CFD code and a comprehensive code, and the lowest accuracy is found through a free wake and comprehensive code coupling using simplified 2D aerodynamics. From here, methods to improve the efficiency and accuracy of the CFD code are considered through implementation of grid adaptation and low Mach number preconditioning methods. Applying grid adaptation allow coarser grids to be used where high gradients in the physics are not present, reserving the denser areas for more interesting regions. For steady-state problems, clustering of the grid provides better wake resolution behind the actuator disk. This method is proven to work for the steady-state equations, but its application to rotor flows using the time-accurate equations still needs to be tested. Low Mach number preconditioning is also an efficiency and an accuracy improvement which allows the CFD code to work for a wider range of Mach numbers within a single simulation. There are many cases, especially for rotor flows, where the range of Mach numbers contained in the flow field cover both the incompressible and compressible regimes. Thus, applying the compressible equations to the entire flow field results in governing equations with high stiffness matrices. The preconditioning reduces the numerical stiffness and thus improves the quality of the results. This improved quality is demonstrated through low speed rotor-fuselage simulations.

Further efficiency improvements are obtained by modifying the codes used in the analysis to include more simplified methods. On the aerodynamic side, a coupling between a CFD code and a prescribed rigid motion module has been completed, and on the structural side a coupling between a CSD code and a combination of a 2D airfoil theory and a free wake code is shown. It is found that the rigid motion method is more appropriately applied where blade elasticity is not significant, and the CSD method is far more efficient than CFD methods, but with a penalty in accuracy. The exact formulation of the 2D aerodynamic model used in the CSD code is discussed, as are efficiency improvements to improve the speed of the free wake code. The advantages of the computationally expensive free wake code are tested against a faster dynamic inflow model, and show that there are improvements when using the more accurate wake formulation. A comparison of these methods evaluates the advantages and consequences of each combination, including the types of physics that each method is able to, or not able to, capture through examination of how closely each method matches flight test data.

**Computational Aeroelasticity of Rotating Wings with
Deformable
Airfoils**
(935 KB PDF)

This paper presents a simulation for high-fidelity aeroelastic analysis of rotating wings with camber-wise structural flexibility and embedded actuators. An unstructured Reynolds-Averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solver is coupled with a non-linear structural dynamics analysis. The CFD solution uses overset grids to combine the stationary and moving frames of reference. The structural formulation expands the conventional one-dimensional beam representation with additional degrees-of-freedom to capture plate-like cross-sectional deformations while allowing an arbitrary distribution of active and passive materials in the cross section. Motion and forces on the non-coincident fluid and structural grids are transferred using a finite-element-based interpolation, along with a least-squares fit for extrapolations. Trim and convergence to periodic response are assisted by a low-order analysis that is also discussed. Finally, as an initial verification of the implementations, results from the low-order and CFD-based solutions are compared for a rigid-airfoil rotor in forward flight.

**Simulation of an Isolated Tiltrotor in Hover with an
Unstructured Overset-Grid RANS Solver**
(2.1 MB PDF)

An unstructured overset-grid Reynolds Averaged Navier-Stokes (RANS) solver, FUN3D, is used to simulate an isolated tiltrotor in hover. An overview of the computational method is presented as well as the details of the overset-grid systems. Steady-state computations within a noninertial reference frame define the performance trends of the rotor across a range of the experimental collective settings. Results are presented to show the effects of off-body grid refinement and blade grid refinement. The computed performance and blade loading trends show good agreement with experimental results and previously published structured overset-grid computations. Off-body flow features indicate a significant improvement in the resolution of the first perpendicular blade vortex interaction with background grid refinement across the collective range. Considering experimental data uncertainty and effects of transition, the prediction of figure of merit on the baseline and refined grid is reasonable at the higher collective range- within 3 percent of the measured values. At the lower collective settings, the computed figure of merit is approximately 6 percent lower than the experimental data. A comparison of steady and unsteady results show that with temporal refinement, the dynamic results closely match the steady-state noninertial results which gives confidence in the accuracy of the dynamic overset-grid approach.

**Adjoint-Based Design of Rotors Using the Navier-Stokes
Equations
in a Noninertial Reference Frame**
(3.0 MB PDF)

Optimization of rotorcraft flowfields using an adjoint method generally requires a time-dependent implementation of the equations. The current study examines an intermediate approach in which a subset of rotor flowfields are cast as steady problems in a noninertial reference frame. This technique permits the use of an existing steady-state adjoint formulation with minor modifications to perform sensitivity analyses. The formulation is valid for isolated rigid rotors in hover or where the freestream velocity is aligned with the axis of rotation. Discrete consistency of the implementation is demonstrated using comparisons with a complex-variable technique, and a number of single- and multi-point optimizations for the rotorcraft figure of merit function are shown for varying blade collective angles. Design trends are shown to remain consistent as the grid is refined.

**Recent Enhancements To The FUN3D Flow Solver
For Moving-Mesh Applications**
(10.3 MB PDF)

An unsteady Reynolds-averaged Navier-Stokes solver for unstructured grids has been extended to handle general mesh movement involving rigid, deforming, and overset meshes. Mesh deformation is achieved through analogy to elastic media by solving the linear elasticity equations. A general method for specifying the motion of moving bodies within the mesh has been implemented that allows for inherited motion through parent-child relationships, enabling simulations involving multiple moving bodies. Several example calculations are shown to illustrate the range of potential applications. For problems in which an isolated body is rotating with a fixed rate, a noninertial reference-frame formulation is available. An example calculation for a tilt-wing rotor is used to demonstrate that the time-dependent moving grid and noninertial formulations produce the same results in the limit of zero time-step size.

**Local-in-time Adjoint-based Method for Design Optimization of
Unsteady
Compressible Flows**
(0.2 MB PDF)

We develop a new local-in-time adjoint-based method for minimization of flow matching functionals subject to the 2-D unsteady compressible Euler equations. The new methodology is aimed at circumventing the memory requirements associated with conventional time-dependent adjoint-based optimization methods that require the flow solutions to be available at all time levels. These storage requirements quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and approximate the solution of the global-in-time adjoint equations and the global sensitivity derivative as the combination of the corresponding local quantities computed on each subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for design optimization of unsteady subsonic and supersonic flows around a bump. Our numerical results show that the local-in-time method converges to the optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

**Multi-Dimensional, Inviscid Flux Reconstruction for
Simulation of Hypersonic
Heating on Tetrahedral Grids**
(10.2 MB PDF)

The quality of simulated hypersonic stagnation region heating on tetrahedral meshes is investigated by using a three-dimensional, upwind reconstruction algorithm for the inviscid flux vector. Two test problems are investigated: hypersonic flow over a three-dimensional cylinder with special attention to the uniformity of the solution in the spanwise direction and hypersonic flow over a three-dimensional sphere. The tetrahedral cells used in the simulation are derived from a structured grid where cell faces are bisected across the diagonal resulting in a consistent pattern of diagonals running in a biased direction across the otherwise symmetric domain. This grid is known to accentuate problems in both shock capturing and stagnation region heating encountered with conventional, quasi-one-dimensional inviscid flux reconstruction algorithms. Therefore the test problem provides a sensitive test for algorithmic effects on heating. This investigation is believed to be unique in its focus on three-dimensional, rotated upwind schemes for the simulation of hypersonic heating on tetrahedral grids. This study attempts to fill the void left by the inability of conventional (quasi-one-dimensional) approaches to accurately simulate heating in a tetrahedral grid system. Results show significant improvement in spanwise uniformity of heating with some penalty of ringing at the captured shock. Issues with accuracy near the peak shear location are identified and require further study.

**Comparison of node-centered and cell-centered unstructured
finite-volume discretizations. Part I: viscous fluxes**
(2.0 MB PDF)

Discretization of the viscous terms in current finite-volume unstructured-grid schemes are compared using node-centered and cell-centered approaches in two dimensions. Accuracy and efficiency are studied for six nominally second-order accurate schemes: a node-centered scheme, cell-centered node-averaging schemes with and without clipping, and cell-centered schemes with unweighted, weighted, and approximately mapped least-square face gradient reconstruction. The grids considered range from structured (regular) grids to irregular grids composed of arbitrary mixtures of triangles and quadrilaterals, including random perturbations of the grid points to bring out the worst possible behavior of the solution. Two classes of tests are considered. The first class of tests involves smooth manufactured solutions on both isotropic and highly anisotropic grids with discontinuous metrics, typical of those encountered in grid adaptation. The second class concerns solutions and grids varying strongly anisotropically over a curved body, typical of those encountered in high-Reynolds number turbulent flow simulations. Results from the first class indicate the face least-square methods, the node-averaging method without clipping, and the node-centered method demonstrate second-order convergence of discretization errors with very similar accuracies per degree of freedom. The second class of tests are more discriminating. The node-centered scheme is always second order with an accuracy and complexity in linearization comparable to the best of the cell-centered schemes. In comparison, the cell-centered node-averaging schemes are less accurate, have a higher complexity in linearization, and can fail to converge to the exact solution when clipping of the node-averaged values is used. The cell-centered schemes using least-square face gradient reconstruction have more compact stencils with a complexity similar to that of the node-centered scheme. For simulations on highly anisotropic curved grids, the least-square methods have to be amended either by introducing a local mapping of the surface based on a distance function commonly available in practical schemes or modifying the scheme stencil to reflect the direction of strong coupling. The major conclusion is that accuracies of the node centered and the best cell-centered schemes are comparable at equivalent number of degrees of freedom.

**A Computational Study of the Aerodynamics and Aeroacoustics of
a Flatback Airfoil Using Hybrid RANS-LES**
(1.6 MB PDF)

This work compares the aerodynamic and aeroacoustic predictions for flatback airfoil geometries obtained by applying advanced turbulence modeling simulation techniques within Computational Fluid Dynamics (CFD) methods that resolve the Reynolds-Averaged Navier-Stokes (RANS) equations of motion. These flatback airfoil geometries are designed for wind turbine applications. Results from different CFD codes using hybrid RANS-LES and RANS turbulence simulations are correlated and include analysis with experimental data. These data comparisons include aerodynamic and a limited amount of aeroacoustic results. While the mean lift prediction remains relatively insensitive across many simulation techniques and parameters, the mean drag prediction is dependent on both the grid and turbulence simulation method. Aeroacoustic predictions obtained from post-processing of the airfoil surface pressure agree reasonably well with experimental data when consistent boundary layer tripping is used for both the simulation and experimental configuration.

**Analysis of Effectiveness of Phoenix Entry Reaction Control
System**
(2.8 MB PDF)

Interaction between the external flowfield and the reaction control system (RCS) thruster plumes of the Phoenix capsule during entry has been investigated. The analysis covered rarefied, transitional, hypersonic and supersonic flight regimes. Performance of pitch, yaw and roll control authority channels was evaluated, with specific emphasis on the yaw channel due to its low nominal yaw control authority. Because Phoenix had already been constructed and its RCS could not be modified before flight, an assessment of RCS efficacy along the trajectory was needed to determine possible issues and to make necessary software changes. Effectiveness of the system at various regimes was evaluated using a hybrid DSMC-CFD technique, based on DSMC Analysis Code (DAC) code and General Aerodynamic Simulation Program (GASP), the LAURA (Langley Aerothermal Upwind Relaxation Algorithm) code, and the FUN3D (Fully Unstructured 3D) code. Results of the analysis at hypersonic and supersonic conditions suggest a significant aero-RCS interference which reduced the efficacy of the thrusters and could likely produce control reversal. Very little aero-RCS interference was predicted in rarefied and transitional regimes. A recommendation was made to the project to widen controller system deadbands to minimize (if not eliminate) the use of RCS thrusters through hypersonic and supersonic flight regimes, where their performance would be uncertain.

**Hybrid RANS-LES Turbulence Models on Unstructured Grids**
(1.7 MB PDF)

This work evaluates the ability of a hybrid Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) turbulence method to accurately predict the physics of an unsteady separated flow field in an unstructured legacy RANS computational fluid dynamics code. The hybrid method consists of a blending of the k-w SST RANS model with a one-equation LES model for the subgrid-scale turbulent kinetic energy (ksgs). Unstructured grids provide better resolution of complex geometries which is the motivation for extending this method. Correlations include theoretical data, experimental data and computational results with RANS turbulence models.

**Application of FUN3D and CFL3D to the Third Workshop on
CFD Uncertainty Analysis**
(1.7 MB PDF)

Two Reynolds-averaged Navier-Stokes computer codes—one unstructured and one structured—are applied to two workshop cases (for the 3rd Workshop on CFD Uncertainty Analysis, held at Instituto Superior Tecnico, Lisbon, in October 2008) for the purpose of uncertainty analysis. The Spalart-Allmaras turbulence model is employed. The first case uses the method of manufactured solution and is intended as a verification case. In other words, the CFD solution is expected to approach the exact solution as the grid is refined. The second case is a validation case (comparison against experiment), for which modeling errors inherent in the turbulence model and errors/uncertainty in the experiment may prevent close agreement. The results from the two computer codes are also compared. This exercise verifies that the codes are consistent both with the exact manufactured solution and with each other. In terms of order property, both codes behave as expected for the manufactured solution. For the backward facing step, CFD uncertainty on the finest grid is computed and is generally very low for both codes (whose results are nearly identical). Agreement with experiment is good at some locations for particular variables, but there are also many areas where the CFD and experimental uncertainties do not overlap.

**Uncertainty Analysis of Computational Fluid Dynamics Via
Polynomial Chaos**
(9.9 MB PDF)

The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs.

**Anisotropic Output-Based Adaptation with Tetrahedral Cut Cells for
Compressible Flows**
(12.1 MB PDF)

Anisotropic, adaptive meshing for flows around complex, three-dimensional bodies remains a barrier to increased automation in computational fluid dynamics. Two specific advances are introduced in this thesis. First, a finite-volume discretization for tetrahedral cut-cells is developed that makes possible robust, anisotropic adaptation on complex bodies. Through grid refinement studies on inviscid flows, this cut-cell discretization is shown to produce similar accuracy as boundary-conforming meshes with a small increase in the degrees of freedom. The cut-cell discretization is then combined with output-based error estimation and anisotropic adaptation such that the mesh size and shape are controlled by the output error estimate and the Hessian (i.e. second derivatives) of the Mach number, respectively. Using a parallel implementation, this output-based adaptive method is applied to a series of sonic boom test cases and the automated ability to correctly estimate pressure signatures at several body lengths is demonstrated starting with initial meshes of a few thousand control volumes. Second, a new framework for adaptation is introduced in which error estimates are directly controlled by removing the common intermediate step of specifying a desired mesh size and shape. As a result, output error control can be achieved without the ad-hoc selection of a specific field (such as Mach number) to control anisotropy, rather anisotropy in the mesh naturally results from both the primal and dual solutions. Furthermore, the direct error control extends naturally to higher-order discretizations for which the use of a Hessian is no longer appropriate to determine mesh shape. The direct error control adaptive method is demonstrated on a series of simple test cases to control interpolation error and discontinuous Galerkin finite element output error. This new direct method produces grids with less elements but the same accuracy as existing metric-based approaches.

**Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom
Prediction**
(3.7 MB PDF)

A cut-cell approach to Computational Fluid Dynamics (CFD) that utilizes the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated, which permits adaptation based on improving the calculation of a specified output (off-body pressure signature) in supersonic inviscid flow. These predicted signatures are compared to wind tunnel measurements on and off the configuration centerline 10 body lengths below the model to validate the method for sonic boom prediction. Accurate mid-field sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly-refined, shock-aligned anisotropic grids were produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provided stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut-cells with an output-based adaptive scheme completely automated this accurate prediction capability after a triangular mesh is generated for the cut surface. This automation drastically reduces the manual intervention required by existing methods.

**The Impact of Advanced Airfoils on Rotor Hover Performance**
(1.7 MB PDF)

Unsteady, compressible Reynolds-averaged Navier-Stokes equation based on an unstructured-grid approach with the Spalart-Allmaras one-equation turbulence model has been used to investigate flow over stationary and oscillating airfoils. The dynamic stall characteristics of Boeing VR-7 airfoils with and without slats were computed and compared with experimental data. Tunnel walls were included in the simulation to investigate the blockage effect on the aerodynamic characteristics. The computed dynamic stall characteristics correlate well with experimental data. In general, the results of airfoil with slat show improvement in the lift characteristics and suppress the negative pitching moment in all dynamic cases. Furthermore, a helicopter hover performance code is used to quantify performance gains using the computed static characteristics of the advanced airfoils. The addition of VR-7 with half size slat airfoil to the baseline rotor blade shows a hover performance gain.

**Rotor Airloads Prediction Using Unstructured Meshes and Loose
CFD/CSD Coupling**
(6 MB PDF)

The FUN3D unsteady Reynolds-averaged Navier-Stokes solver for unstructured grids has been modified to allow prediction of trimmed rotorcraft airloads. The trim of the rotorcraft and the aeroelastic deformation of the rotor blades are accounted for via loose coupling with the CAMRAD II rotorcraft computational structural dynamics code. The set of codes is used to analyze the HART-II Baseline, Minimum Noise and Minimum Vibration test conditions. The loose coupling approach is found to be stable and convergent for the cases considered. Comparison of the resulting airloads and structural deformations with experimentally measured data is presented. The effect of grid resolution and temporal accuracy is examined.

**Development of Advanced Computational Aeroelasticity Tools at
NASA Langley Research Center**
(900 KB PDF)

NASA Langley Research Center has continued to develop its long standing computational tools to address new challenges in aircraft and launch vehicle design. This paper discusses the application and development of those computational aeroelastic tools. Four topic areas will be discussed: 1) Modeling structural and flow field nonlinearities; 2) Integrated and modular approaches to nonlinear multidisciplinary analysis; 3) Simulating flight dynamics of flexible vehicles; and 4) Applications that support both aeronautics and space exploration.

**An Examination of Engine Effects on Helicopter
Aeromechanics**
(800 KB PDF)

An engine modeling capability has been implemented into a Reynolds Averaged, Navier-Stokes based computational fluid dynamics code to assist in examining engine effects on helicopter aeromechanics. The procedure involves coupling a one-dimensional engine program to the flow solver through inlet and exhaust boundary conditions. Rotor influence is approximated with a time-averaged actuator disk model, which has a trim procedure capable of including fuselage loads. This simulation capability is found to provide useful insight for investigating aeromechanics problems that have been observed due to engine induced effects. In particular, this paper shows that this capability enables the visualization of the engine exhaust plume, provides estimates of the engine impact on helicopter trim, and assists in understanding the impact of various exhaust concepts.

**Parallel Anisotropic Tetrahedral Adaptation**
(3.7 MB PDF)

An adaptive method that robustly produces high aspect ratio tetrahedra to a general 3D metric specification without introducing hybrid semi-structured regions is presented. The grid operators and higher-level logic is described with their respective domain-decomposed parallelizations. An tetrahedral adaptation scheme is demonstrated for 1000—1 anisotropy in a simple cube geometry. This form of adaptation is applicable to more complex domain boundaries via a cut-cell approach as demonstrated by a parallel 3D supersonic simulation of a complex fighter aircraft. To avoid the assumptions and approximations required to form a metric to specify adaptation, an approach is introduced that directly evaluates interpolation error. The grid is adapted to reduce and equidistribute this interpolation error calculation without the use of an intervening anisotropic metric. Direct interpolation error adaptation is illustrated for fifth-order elements in 1D and linear and quadratic tetrahedra in 3D.

**Towards Verification of Unstructured-Grid Solvers**
(700 KB PDF)

New methodology for verification of finite-volume computational methods using unstructured grids is presented. The discretization order properties are studied in computational windows, easily constructed within a collection of grids or a single grid. Tests are performed within each window and address a combination of problem-, solution-, and discretization/grid-related features affecting discretization error convergence. The windows can be adjusted to isolate particular elements of the computational scheme, such as the interior discretization, the boundary discretization, or singularities. Studies can use traditional grid-refinement computations within a fixed window or downscaling, a recently-introduced technique in which computations are made within windows contracting toward a focal point of interest. Grids within the windows are constrained to be consistently refined, allowing a meaningful assessment of asymptotic error convergence on unstructured grids. Demonstrations of the method are shown, including a comparative accuracy assessment of commonly-used schemes on general mixed grids and the identification of local accuracy deterioration at boundary intersections. Recommendations to enable attainment of design-order discretization errors for large-scale computational simulations are given.

**Simulation of Stagnation Region Heating in Hypersonic Flow
on Tetrahedral
Grids**
(6.9 MB PDF)

Hypersonic flow simulations using the node based, unstructured grid code FUN3D are presented. Applications include simple (cylinder) and complex (towed ballute) configurations. Emphasis throughout is on computation of stagnation region heating in hypersonic flow on tetrahedral grids. Hypersonic flow over a cylinder provides a simple test problem for exposing any flaws in a simulation algorithm with regard to its ability to compute accurate heating on such grids. Such flaws predominantly derive from the quality of the captured shock. The importance of pure tetrahedral formulations are discussed. Algorithm adjustments for the baseline Roe / Symmetric, Total-Variation-Diminishing (STVD) formulation to deal with simulation accuracy are presented. Formulations of surface normal gradients to compute heating and diffusion to the surface as needed for a radiative equilibrium wall boundary condition and finite catalytic wall boundary in the node-based unstructured environment are developed. A satisfactory resolution of the heating problem on tetrahedral grids is not realized here; however, a definition of a test problem, and discussion of observed algorithm behaviors to date are presented in order to promote further research on this important problem.

**Advances in Rotorcraft Simulations with Unstructured CFD**
(2.4 MB PDF)

Advances have been made in the development of unstructured solver methods suitable for fixed wing analyses, as well as rotary-wing applications, including interaction aerodynamics. This paper demonstrates the ability of an unstructured overset solver, FUN3D, to resolve the viscous compressible equations of motion for rotor-fuselage interactions and rotor-alone configurations. The solver is capable of modeling fully-articulated rotors (prior work was limited to rotation and flapping) and CFD-CSD loosely-coupled solutions including trim, using overset approaches. Initial results with the UH60A rotor indicate that the unstructured solver provides similar loading to its structured solver counterparts. Additional development of efficient domain connectivity interface routines are warranted to provide the ability to perform tightly-coupled rotor simulations.

**Accuracy Analysis for Mixed-Element Finite-Volume Discretization
Schemes**
(900 KB PDF)

A new computational analysis tool, downscaling (DS) test, has been introduced and applied for studying the convergence rates of truncation and discretization errors of finite-volume discretization (FVD) schemes on general unstructured grids. The study corrects a misconception that the discretization accuracy of FVD schemes on irregular grids is directly linked to convergence of truncation errors. The DS test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the DS test analysis. It was shown that the discretization accuracy of a common node-centered FVD scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for certain mixed-element grids. Alternative node-centered schemes have been presented and demonstrated to provide second and third order accuracies on general mixed-element grids. The local accuracy deterioration at intersections of tangency and inflow/outflow boundaries has been demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions has been demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.

**Breakthrough Advantage in Computational Fluid Dynamics with the
IBM System Blue Gene Solution**
(900 KB PDF)

Over the last 40 years, the use of Computational Fluid Dynamics (CFD) has increased by several orders of magnitude in industry and research laboratories, largely due to the impressive advances in computing architectures as well as in the algorithmic techniques created to exploit these architectures. Each major innovation in the computing industry has directly enabled CFD practitioners to solve more realistic and complex engineering design and simulation problems, resulting in better products faster. The investment by the community in adapting CFD applications to take advantage of newer computing architectures has paid off handsomely.

The IBM Blue Gene is the first system in a generation of innovative, ultrascalable architectures that will enable CFD engineers to make significant improvements in the understanding and the solution of some of the most complex problems in engineering design. For the first time, it will be possible to solve complex problems in turbulence that are interdisciplinary, multi-scale, and with multi-body interactions on very complex geometries. These problems require very high-resolution models to gain profound engineering insights that were previously not possible. However, this entails a continuing investment by the CFD developers, mathematicians, and computer scientists to develop new algorithms and applications for ultrascalable parallel computing environments. This investment will be protected as newer Blue Gene systems become available. As before, the payoff will far outweigh this investment.

IBM has expanded its five year Blue Gene collaboration with the Lawrence Livermore Laboratory to the CFD community. This collaborative investment has produced impressive early results for some of the most challenging CFD problems in industry and research institutions. The scalability and performance obtained from these simulations on the IBM Blue Gene are unsurpassed yet affordable, and easily accessible. More importantly, the solution of previously intractable CFD problems has resulted in breakthrough engineering insights. The direct numerical simulation of turbulence for realistic engineering configurations is now – for the first time - plausible.

**Semi-Analytic Reconstruction of Flux in Finite Volume
Formulations**
(850 KB PDF)

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger’s Equation as well as a conventional, upwind discretization using Roe’s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe’s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe’s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

**Aerothermodynamic Analyses of Towed Ballutes**
(12.6 MB PDF)

A ballute (balloon-parachute) is an inflatable, aerodynamic drag device for application to planetary entry vehicles. Two challenging aspects of aerothermal simulation of towed ballutes are considered. The first challenge, simulation of a complete system including inflatable tethers and a trailing toroidal ballute, is addressed using the unstructured-grid, Navier-Stokes solver FUN3D. Auxiliary simulations of a semi-infinite cylinder using the rarefied flow, Direct Simulation Monte Carlo solver, DSV2, provide additional insight into limiting behavior of the aerothermal environment around tethers directly exposed to the free stream. Simulations reveal pressures higher than stagnation and corresponding large heating rates on the tether as it emerges from the spacecraft base flow and passes through the spacecraft bow shock. The footprint of the tether shock on the toroidal ballute is also subject to heating amplification. Design options to accommodate or reduce these environments are discussed. The second challenge addresses time-accurate simulation to detect the onset of unsteady flow interactions as a function of geometry and Reynolds number. Video of unsteady interactions measured in the Langley Aerothermodynamic Laboratory 20-Inch Mach 6 Air Tunnel and CFD simulations using the structured grid, Navier-Stokes solver LAURA are compared for flow over a rigid spacecraft-sting-toroid system. The experimental data provides qualitative information on the amplitude and onset of unsteady motion which is captured in the numerical simulations. The presence of severe unsteady fluid-structure interactions is undesirable and numerical simulation must be able to predict the onset of such motion.

**Parallel, Gradient-Based Anisotropic Mesh Adaptation
for Re-entry Vehicle Configurations**
(6.0 MB PDF)

Two gradient-based adaptation methodologies have been implemented into the FUN3D-refine-GridEx infrastructure. A spring-analogy adaptation which provides for nodal movement to cluster mesh nodes in the vicinity of strong shocks has been extended for general use within FUN3D, and is demonstrated for a 70-degree sphere cone at Mach 2. A more general feature-based adaptation metric has been developed for use with the adaptation mechanics available in FUN3D, and is applicable to any unstructured tetrahedral flow solver. The basic functionality of general adaptation is explored through a case of flow over the forebody of a 70-degree sphere cone at Mach 6. A practical application for Mach 10.0 flow over an Apollo capsule computed with the FELISA flow solver is given to compare the adaptive mesh refinement with uniform mesh refinement. The examples of the paper demonstrate that the gradient-based adaptation capability as implemented can give an improvement in solution quality.

**Blade Contour Deformation and Helicopter Performance**
(1.6 MB PDF)

The United States Army helicopter fleet is experiencing deformation of rotor blade contours from sand erosion and the implementation of technologies to reduce it. An investigation was performed to determine the effect of these types of degradations on the tail rotor performance of Apache attack helicopters. Computational fluid dynamics was used to calculate aerodynamic coefficients for representative deformed airfoil sections. A hover analysis code was used to evaluate the impact of the damaged airfoils on the tail rotor performance. The results show that airfoil erosion can lead to a significant reduction in the maximum thrust available from worn tail rotors.

**Computational Analysis of Dual Radius Circulation Control
Airfoils**
(6.2 MB PDF)

The goal of the work is to use multiple codes and multiple configurations to provide an assessment of the capability of RANS solvers to predict circulation control dual radius airfoil performance and also to identify key issues associated with the computational predictions of these configurations that can result in discrepancies in the predicted solutions. Solutions were obtained for the Georgia Tech Research Institute (GTRI) dual radius circulation control airfoil and the General Aviation Circulation Control (GACC) dual radius airfoil. For the GTRI-DR airfoil, two-dimensional structured and unstructured grid computations predicted the experimental trend in sectional lift variation with blowing coefficient very well. Good code-to-code comparisons between the chordwise surface pressure coefficients and the solution streamtraces also indicated that the detailed flow characteristics were matched between the computations. For the GACC-DR airfoil, two-dimensional structured and unstructured grid computations predicted the sectional lift and chordwise pressure distributions accurately at the no blowing condition. However at a moderate blowing coefficient, although the code-to-code variation was small, the differences between the computations and experiment were significant. Computations were made to investigate the sensitivity of the sectional lift and pressure distributions to some of the experimental and computational parameters, but none of these could entirely account for the differences in the experimental and computational results. Thus, CFD may indeed be adequate as a prediction tool for dual radius CC flows, but limited and difficult to obtain two-dimensional experimental data prevents a confident assessment at this time.

**Investigation of Effect of Dynamic Stall and Its Alleviation
on Helicopter
Performance and Loads**
(1.6 MB PDF)

The static and dynamic stall characteristics of VR-7 baseline and two modified airfoils were computed and compared with available experimental data. The unsteady, compressible Reynolds-averaged Navier-Stokes equations based on an unstructured-grid approach with the one-equation Spalart-Allmaras turbulence model has been used to investigate flow over these airfoils in stationary and oscillating conditions. The baseline VR-7 results correlate well with static test data; the computed dynamic results of the VR-7 show a large negative pitching moment and drag observed in the hysteresis curves and agree fairly well with dynamic test data. An optimization technique was used to modify the upper surface of VR-7 airfoil with the cost function of minimized drag while maintaining a specified lift. The computed static and dynamic characteristics of the modified airfoils at low Mach numbers show improvement in the static characteristics and a large reduction of the negative pitching moment in the dynamic case. The effect on helicopter performance and loads are analyzed using a comprehensive analysis code with the computed static and dynamic characteristics of VR-7 and modified VR-7 airfoils for aerodynamics.

**Analysis of Computational Modeling Techniques for Complete
Rotorcraft
Configurations**
(22.5 MB
PDF)

Helicopters and tilt-rotor aircraft exhibit complex aerodynamic phenomena resulting from an unsteady, vortical wake generated by the rotating blades. The complex nature of the rotor wake makes it difficult to obtain accurate predictions of the flow with the traditional forms of analysis. Since the vehicle aerodynamics have a direct influence on performance, handling, and the loads on the structure, the inability to obtain accurate airloads can potentially affect the design of the entire system. This ultimately leads to increased life-cycle costs to own and operate the vehicle.

Computational fluid dynamics (CFD) provides the helicopter designer with a powerful tool for identifying problematic aerodynamics. Through the use of CFD, design concepts can be analyzed in a virtual wind tunnel long before a physical model is ever created. Traditional CFD analysis tends to be a time consuming process, where much of the effort is spent generating a high quality computational grid. Recent increases in computing power and memory have created renewed interest in alternative grid schemes such as unstructured grids, which facilitate rapid grid generation by relaxing restrictions on grid structure.

Three rotor models have been incorporated into a popular fixed-wing unstructured CFD solver to increase its capability and facilitate availability to the rotorcraft community. The benefit of unstructured grid methods is demonstrated through rapid generation of high fidelity configuration models. The simplest rotor model is the steady state actuator disk approximation. By transforming the unsteady rotor problem into a steady state one, the actuator disk can provide rapid predictions of performance parameters such as lift and drag.

The actuator blade and overset blade models provide a depiction of the unsteady rotor wake, but incur a larger computational cost than the actuator disk. The actuator blade model is convenient when the unsteady aerodynamic behavior needs to be investigated, but the computational cost of the overset approach is too large. The overset or chimera method allows the blades loads to be computed from first principles and therefore provides the most accurate prediction of the rotor wake for the models investigated. The physics of the flow fields generated by these models for rotor / fuselage interactions are explored, along with efficiencies and limitations of each methodology.

**Using An Adjoint Approach to Eliminate Mesh
Sensitivities in Computational Design**
(0.5 MB PDF)

An adjoint algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced. The method eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables, an expense that has hindered large-scale design optimization for practical applications. The effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation, followed by an explicit matrix-vector product whose cost scales with the number of design variables and the resolution of the parameterized surface grid. The methodology augments the current practice of using adjoints solely for the flowfield and leads to a dramatic computational savings. The accuracy of the implementation is established, and several sample design optimizations are shown.

**Validation of 3D Adjoint Based Error
Estimation and Mesh Adaptation for Sonic Boom Prediction**
(15.8 MB PDF)

A procedure used to validate a 3-D mesh adaptation scheme based on adjoint-based error estimation with application to sonic boom propagation is described. The method is based on a cost function formulation that integrates the near-field pressure differential over a prescribed surface. The uncertainty in the computation of this cost function is used to drive automatic h-r mesh adaptation such that errors in the functional are reduced without human intervention. The primary configurations used to validate the technique are a family of simple cone-cylinder geometries for which experimental data is available. Computed results for inviscid flow at Mach numbers of 1.26 and 1.41 are presented at various distances in the near-field up to 20 body lengths. These results are compared against the available test data and show good agreement.

**Sonic Boom Computations for Double-Cone Configuration
Using CFL3D, FUN3D and Full-Potential Codes**
(1.9 MB PDF)

Development of highly accurate computational codes for both near-field and far-field sonic boom problem is the focus of this paper. The structured grid CFL3D code is modified using a new, highly accurate grid-adaptation and shock-fitting scheme for supersonic near-field domain prediction. The modified CFL3D code is applied to a double-cone configuration at Mach numbers of 1.26 and 1.41. Because of its sophisticated grid adaptation methodology, the unstructured-grid FUN3D is also used for the near-field computations at Mach number of 1.26. The computed near-field results are compared with the available experimental data. The FUN3D code results and the CFL3D results at an interface located at h/L = 2 (altitude height/body length) are used to generate input data for the highly efficient, far-field, structured-grid full-potential (FP) code. The relative errors for the velocity components, at the interface h/L = 2, between the results of the unstructured-grid FUN3D code and the results of the structured-grid FP far-field code are computed and presented. Next, the FP Far-field code is used to advance the solution from h/L = 2 to h/L = 6, 10 and 18 and the results are compared with those obtained from matching FP with FUN3D, matching FP with the modified CFL3D, and the experimental data. The interface results have also been advanced to a farfield location at h/L = 40. The conclusion of this study is that the FUN3D code is highly accurate for near-field and far-field computations. The grid adaptation and shock fitting scheme has to be used in the FP code and CFL3D code for obtaining highly accurate results.

**Adjoint-Based Algorithms for Adaptation and Design
Optimization on Unstructured Grids**
(2.8 MB PDF)

Schemes based on discrete adjoint algorithms present several exciting opportunities for significantly advancing the current state of the art in computational fluid dynamics. Such methods provide an extremely efficient means for obtaining discretely consistent sensitivity information for hundreds of design variables, opening the door to rigorous, automated design optimization of complex aerospace configurations using the Navier-Stokes equations. Moreover, the discrete adjoint formulation provides a mathematically rigorous foundation for mesh adaptation and systematic reduction of spatial discretization error. Error estimates are also an inherent by-product of an adjoint-based approach, valuable information that is virtually non-existent in today’s large-scale CFD simulations.

An overview of adjoint-based algorithm work at NASA Langley Research Center is presented, with examples demonstrating the potential impact on complex computational problems related to design optimization as well as mesh adaptation.

**Computational Methods for Stability and Control (COMSAC): The
Time Has Come**
(18.7 MB PDF)

Powerful computational fluid dynamics (CFD) tools have emerged that appear to offer significant benefits as an adjunct to the experimental methods used by the stability and control community to predict aerodynamic parameters. The decreasing costs for and increasing availability of computing hours are making these applications increasingly viable as time goes on and the cost of computing continues to drop. This paper summarizes the efforts of four organizations to utilize high-end computational fluid dynamics (CFD) tools to address the challenges of the stability and control arena. General motivation and the backdrop for these efforts will be summarized as well as examples of current applications.

**Simulation of Unsteady Flows Using an
Unstructured Navier-Stokes Solver on Moving and Stationary Grids**
(2.1 MB PDF)

We apply an unsteady Reynolds-averaged Navier-Stokes (URANS) solver for unstructured grids to time-dependent problems on both moving and stationary grids. Example problems considered are relevant to active flow control and stability and control. Computational results are presented using the Spalart-Allmaras turbulence model and are compared to experimental data. The effect of grid and time-step refinement are examined.

**Computational Simulations and the Scientific Method**
(1.5 MB PDF)

As scientific simulation software becomes more complicated, the scientific-software implementor’s need for component tests from new model developers becomes more crucial. The community’s ability to follow the basic premise of the Scientific Method requires independently repeatable experiments, and model innovators are in the best position to create these test fixtures. Scientific software developers also need to quickly judge the value of the new model relative to other models, i.e., the new model’s cost-to-benefit ratio in terms of gains provided by the new model and risks such as implementation time and software quality.

This paper asks two questions. The first is whether other scientific software developers would find published component tests useful, and the second is whether model innovators think publishing test fixtures is a feasible approach.

**Application of Parallel Adjoint-Based Error
Estimation and Anisotropic Grid Adaptation for
Three-Dimensional Aerospace Configurations**
(18.1 MB PDF)

This paper demonstrates the extension of error estimation and adaptation methods to parallel computations enabling larger, more realistic aerospace applications and the quantification of discretization errors for complex 3-D solutions. Results were shown for an inviscid sonic-boom prediction about a double-cone configuration and a wing/body segmented leading edge (SLE) configuration where the output function of the adjoint was pressure integrated over a part of the cylinder in the near field. After multiple cycles of error estimation and surface/field adaptation, a significant improvement in the inviscid solution for the sonic boom signature of the double cone was observed. Although the double-cone adaptation was initiated from a very coarse mesh, the near-field pressure signature from the final-adapted mesh compared very well with the wind-tunnel data which illustrates that the adjoint-based error estimation and adaptation process requires no a priori refinement of the mesh. Similarly, the near-field pressure signature for the SLE wing/body sonic boom configuration showed a significant improvement from the initial coarse mesh to the final adapted mesh in comparison with the wind tunnel results. Error estimation and field adaptation results were also presented for the viscous transonic drag prediction of the DLR-F6 wing/body configuration, and results were compared to a series of globally refined meshes. Two of these globally refined meshes were used as a starting point for the error estimation and field-adaptation process where the output function for the adjoint was the total drag. The field-adapted results showed an improvement in the prediction of the drag in comparison with the finest globally refined mesh and a reduction in the estimate of the remaining drag error. The adjoint-based adaptation parameter showed a need for increased resolution in the surface of the wing/body as well as a need for wake resolution downstream of the fuselage and wing trailing edge in order to achieve the requested drag tolerance. Although further adaptation was required to meet the requested tolerance, no further cycles were computed in order to avoid large discrepancies between the surface mesh spacing and the refined field spacing.

**Parallel Adaptive Solvers in Compressible PETSc-FUN3D
Simulations**
(0.3 MB PDF)

We consider parallel, three-dimensional transonic Euler flow using the PETSc-FUN3D application, which employs pseudo-transient Newton-Krylov methods. Solving a large, sparse linear system at each nonlinear iteration dominates the overall simulation time for this fully implicit strategy. This paper presents a polyalgorithmic technique for adaptively selecting the linear solver method to match the numeric properties of the linear systems as they evolve during the course of the nonlinear iterations. Our approach combines more robust, but more costly, methods when needed in particularly challenging phases of solution, with cheaper, though less powerful, methods in other phases. We demonstrate that this adaptive, polyalgorithmic approach leads to improvements in overall simulation time, is easily parallelized, and is scalable in the context of this large-scale comptuational fluid dynamics application.

**Analysis of Rotor-Fuselage Interactions Using Various Rotor
Models**
(4.0 MB PDF)

Accurate prediction of the rotor and fuselage interaction is essential for the design and analysis of modern rotorcraft. A variety of Navier-Stokes based methodologies have been employed in the past to simulate these effects. The purpose of this study is to examine the merits of some of the simplified techniques of modeling the rotor and their influence on the physics of the overall rotor/fuselage interaction problem. Specifically, a constant actuator disk, varying actuator disk, and blade element actuator disk are considered. The computational results are compared with wind tunnel data obtained on various rotorcraft models. The constant actuator disk is found to be inadequate for most applications, but can be easily improved upon by allowing for pressure variations about the blade radius and azimuth.

**Using An Adjoint Approach to Eliminate Mesh Sensitivities
in Computational Design**
(4.0 MB PDF)

An algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced. The method is based on an adjoint approach and eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables, an expense that has hindered practical large-scale design optimization using discrete adjoint methods. The effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation, followed by an explicit matrix-vector product scaling with the number of design variables and the resolution of the parameterized surface grid. The accuracy of the implementation is established and dramatic computational savings obtained using the new approach are demonstrated using several test cases. Sample design optimizations are also shown.

**Efficient Construction of Discrete Adjoint Operators
on Unstructured Grids by Using Complex Variables**
(4.7 MB PDF AIAA Paper, 1.5 MB PDF Readable
Version)

A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.

**Navier-Stokes Computations of Longitudinal Forces and Moments
for a Blended Wing Body**
(6.5 MB PDF)

The object of this paper is to investigate the feasibility of applying CFD methods to aerodynamic analyses for aircraft stability and control. The integrated aerodynamic parameters used in stability and control, however, are not necessarily those extensively validated in the state of the art CFD technology. Hence, an exploratory study of such applications and the comparison of the solutions to available experimental data will help to assess the validity of the current computation methods. In addition, this study will also examine issues related to wind tunnel measurements such as measurement uncertainty and support interference effects. Several sets of experimental data from the NASA Langley 14×22-Foot Subsonic Tunnel and the National Transonic Facility are presented. Two Navier-Stokes flow solvers, one using structured meshes and the other unstructured meshes, were used to compute longitudinal static stability derivatives for an advanced Blended Wing Body configuration over a wide range of angles of attack. The computations were performed for two different Reynolds numbers and the resulting forces and moments are compared with the above mentioned wind tunnel data.

**Adjoint-Based, Three-Dimensional Error Prediction and Grid
Adaptation**
(1.2 MB PDF)

Engineering computational fluid dynamics analysis and design applications often focus on output functions, such as lift or drag. Errors in these output functions are generally unknown, and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solution of the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user-specified error tolerance. This procedure for estimation and adaptation to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a CAD surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with many more grid points.

**Aerodynamic Shape Optimization Based on Free-Form
Deformation**
(0.3 MB PDF)

This paper presents a free-form deformation technique suitable for aerodynamic shape optimization. Because the proposed technique is independent of grid topology, we can treat structured and unstructured computational fluid dynamics grids in the same manner. The proposed technique is an alternative shape parameterization technique to trivariate volume technique. It retains the flexibility and freedom of trivariate volumes for CFD shape optimization, but it uses a bivariate surface representation. This reduces the number of design variables by an order of magnitude, and it provides a much better control for surface shape changes. The proposed technique is simple, compact, and efficient. The analytical sensitivity derivatives are independent of the design variables and are easily computed for use in a gradient-based optimization. The paper includes the complete formulation and aerodynamics shape optimization results.

**Evaluation of Isolated Fuselage and Rotor-Fuselage Interaction
Using CFD**
(3.1 MB PDF)

The US Army Aeroflightdynamics Directorate (AFDD), the French Office National d’Etudes et de Recherches Aerospatiales (ONERA) and the Georgia Institute of Technology (GIT) are working under the United States/France Memorandum of Agreement on Helicopter Aeromechanics to study rotorcraft aeromechanics issues of interest to both nations. As a task under this agreement, a comparative study of the Dauphin 365N helicopter has been undertaken to analyze the capabilities and weaknesses of state-of-the-art computational fluid dynamics (CFD) codes, with the aim of fuselage performance prediction and investigation of rotor-fuselage interaction. Three CFD flow solvers applied on three meshes provide similar results in terms of pressure coefficient. Force predictions vary somewhat. This paper presents details on the grid sensitivity and the low Mach number preconditioning influence. The importance of taking into account the wind tunnel strut and the rotor hub is shown. The pressure coefficients along top and bottom centerlines of the fuselage are in good agreement with the experiment except in the area aft of the hub. There remains a discrepancy between the computed forces and the experimental data due in part to modeling inaccuracies. Rotor-fuselage interactions are performed using uniform and non-uniform actuator disk models in order to simulate the rotor downwash.

**CFD: A Castle in the Sand?**
(700 KB PDF)

The computational simulation community is not routinely publishing independently verifiable tests to accompany new models or algorithms. A survey reveals that only 22% of new models published are accompanied by tests suitable for independently verifying the new model. As the community develops larger codes with increased functionality, and hence increased complexity in terms of the number of building block components and their interactions, it becomes prohibitively expensive for each development group to derive the appropriate tests for each component. Therefore, the computational simulation community is building its collective castle on a very shaky foundation of components with unpublished and unrepeatable verification tests. The computational simulation community needs to begin publishing component-level verification tests before the tide of complexity undermines its foundation.

**Computational Aerothermodynamic Simulation Issues on
Unstructured Grids**
(2.2 MB PDF)

The synthesis of physical models for gas chemistry and turbulence from the structured grid codes LAURA and VULCAN into the unstructured grid code FUN3D is described. A directionally Symmetric, Total Variation Diminishing (STVD) algorithm and an entropy fix (eigenvalue limiter) keyed to local cell Reynolds number are introduced to improve solution quality for hypersonic aeroheating applications. A simple grid-adaptation procedure is incorporated within the flow solver. Simulations of flow over an ellipsoid (perfect gas, inviscid), Shuttle Orbiter (viscous, chemical nonequilibrium) and comparisons to the structured grid solvers LAURA (cylinder, Shuttle Orbiter) and VULCAN (flat plate) are presented to show current capabilities. The quality of heating in 3D stagnation regions is very sensitive to algorithm options—in general, high aspect ratio tetrahedral elements complicate the simulation of high Reynolds number, viscous flow as compared to locally structured meshes aligned with the flow.

**An Implicit, Exact Dual Adjoint Solution Method for
Turbulent Flows on Unstructured Grids**
(1.3 MB PDF)

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.

**Transonic Drag Prediction on a DLR-F6 Transport Configuration
Using
Unstructured Grid Solvers**
(48 MB PDF)

A second international AIAA Drag Prediction Workshop (DPW-II) was organized and held in Orlando Florida on June 21-22, 2003. The primary purpose was to investigate the code-to-code uncertainty, address the sensitivity of the drag prediction to grid size and quantify the uncertainty in predicting nacelle/pylon drag increments at a transonic cruise condition. This paper presents an in-depth analysis of the DPW-II computational results from three state-of-the-art unstructured grid Navier-Stokes flow solvers exercised on similar families of tetrahedral grids. The flow solvers are USM3D—a tetrahedral cell-centered upwind solver, FUN3D—a tetrahedral node-centered upwind solver, and NSU3D—a general element node-centered central-differenced solver.

For the wing/body, the total drag predicted for a constant-lift transonic cruise condition showed a decrease in code-to-code variation with grid refinement as expected. For the same flight condition, the wing/body/nacelle/pylon total drag and the nacelle/pylon drag increment predicted showed an

increasein code-to-code variation with grid refinement. Although the range in total drag for the wing/body fine grids was only 5 counts, a code-to-code comparison of surface pressures and surface restricted streamlines indicated that the three solvers were not all converging to the same flow solutions—different shock locations and separation patterns were evident. Similarly, the wing/body/nacelle/pylon solutions did not appear to be converging to the same flow solutions.

Overall, grid refinement did not consistently improve the correlation with experimental data for either the wing/body or the wing/body/nacelle pylon configuration. Although the absolute values of total drag predicted by two of the solvers for the medium and fine grids did not compare well with the experiment, the incremental drag predictions were within 3 counts of the experimental data. The correlation with experimental incremental drag was not significantly changed by specifying transition. Although the sources of code-to-code variation in force and moment predictions for the three unstructured grid codes have not yet been identified, the current study reinforces the necessity of applying multiple codes to the same application to assess uncertainty.

**Team Software Development for Aerothermodynamic and
Aerodynamic Analysis and
Design**
(0.5 MB PDF)

A collaborative approach to software development is described. The approach employs the agile development techniques: project retrospectives, Scrum status meetings, and elements of Extreme Programming to efficiently develop a cohesive and extensible software suite. The software product under development is a fluid dynamics simulator for performing aerodynamic and aerothermodynamic analysis and design. The functionality of the software product is achieved both through the merging, with substantial rewrite, of separate legacy codes and the authorship of new routines. Examples of rapid implementation of new functionality demonstrate the benefits obtained with this agile software development process. The appendix contains a discussion of coding issues encountered while porting legacy FORTRAN 77 code to FORTRAN 95, software design principles, and a FORTRAN 95 coding standard.

**Computational Fluid Dynamics Technology for Hypersonic
Applications**
(0.3 MB PDF)

Several current challenges in computational fluid dynamics and aerothermodynamics for hypersonic vehicle applications are discussed. Example simulations are presented from code validation and code benchmarking efforts to illustrate capabilities and limitations. Opportunities to advance the state-of-art in algorithms, grid generation and adaptation, and code validation are identified. Highlights of diverse efforts to address these challenges are then discussed. One such effort to re-engineer and synthesize the existing analysis capability in LAURA, VULCAN, and FUN3D will provide context for these discussions. The critical (and evolving) role of agile software engineering practice in the capability enhancement process is also noted.

**Anisotropic Grid Adaptation for Functional Outputs:
Application to
Two-Dimensional Viscous Flows**
(5.8 MB PDF)

An anisotropic, unstructured grid adaptive method is presented for improving the accuracy of functional outputs of viscous, compressible flow simulations for general discretizations. The procedure merges output error control with Hessian-based anisotropic grid adaptation. An adjoint formulation is used to relate the estimated functional error to the local residual errors of both the primal and adjoint solutions. This relationship allows local error contributions to be used as indicators in a grid adaptive method designed to produce specially tuned grids for accurately estimating the chosen functional. Element stretching and orientation information is obtained from interpolation error estimates for linear triangular finite elements. The proposed adaptive method is implemented using a standard second-order upwind finite volume discretization, although the procedure is applicable to other types of discretizations such as the finite element method. A series of airfoil test cases, including separated, high-lift flows, are presented to demonstrate the approach; the functionals considered are the lift and drag coefficients. The proposed adaptive method is shown to be superior in terms of reliability and output accuracy relative to pure Hessian-based adaptation.

**Collaborative Software Development in Support of Fast
Adaptive
AeroSpace Tools (FAAST)**
(0.4 MB PDF)

A collaborative software development approach is described. The software product is an adaptation of proven computational capabilities combined with new capabilities to form the Agency’s next generation aerothermodynamic and aerodynamic analysis and design tools. To efficiently produce a cohesive, robust, and extensible software suite, the approach uses agile software development techniques; specifically, project retrospectives, the Scrum status meeting format, and a subset of Extreme Programming’s coding practices are employed. Examples are provided which demonstrate the substantial benefits derived from employing these practices. Also included is a discussion of issues encountered when porting legacy FORTRAN 77 code to FORTRAN 95 and a FORTRAN 95 coding standard.

**Three-Dimensional Turbulent RANS Adjoint-Based Error Correction**
(9.8 MB PDF)

Engineering problems commonly require functional outputs of computational fluid dynamics (CFD) simulations with specified accuracy. These simulations are performed with limited computational resources. Computable error estimates offer the possibility of quantifying accuracy on a given mesh and predicting a fine grid functional on a coarser mesh. Such an estimate can be computed by solving the flow equations and the associated adjoint problem for the functional of interest. An adjoint-based error correction procedure is demonstrated for transonic inviscid and subsonic laminar and turbulent flow. A mesh adaptation procedure is formulated to target uncertainty in the corrected functional and terminate when error remaining in the calculation is less than a user-specified error tolerance. This adaptation scheme is shown to yield anisotropic meshes with corrected functionals that are more accurate for a given number of grid points then isotropic adapted and uniformly refined grids.

**CFD Sensitivity Analysis of a Drag Prediction Workshop Wing/Body
Transport Configuration**
(5.4 MB PDF)

The current work revisits calculations for the First AIAA Drag Prediction Workshop (DPW-1) configuration and uses a grid convergence study to evaluate the quantitative effects of discretization error on the code-to-code variation of forces and moments. Four CFD codes commonly used at NASA Langley Research Center are used in the study: CFL3D and OVERFLOW are structured grid codes, and NSU3D and FUN3D are unstructured grid codes. Although the drag variation reported in the summary of DPW-1 results was for the constant-lift cruise condition, the focus of the current grid convergence study is a constant angle-of-attack condition (Alpha=0 deg) near the same cruise lift in order to maintain identical boundary conditions for all of the CFD codes. Forces and moments were computed on the standard DPW-1 structured overset and node-based unstructured grids and the results were compared for the required transonic drag polar case. The range in total drag predicted using the workshop standard grids at Alpha=0 deg was 14 counts. The variation of drag in terms of standard deviation was 6 counts. Additional calculations at Alpha=0 deg were performed on the two families of structured and unstructured grids to evaluate the variation in forces and moments with grid refinement. The structured grid refinement study was inconclusive because of difficulties computing on the fine grid. The grid refinement study for the unstructured grid codes showed an increase in variation of forces and moments with grid refinement. However, all of the unstructured grid results were not definitively in the range of asymptotic grid convergence. The study indicated that certain numerical schemes (center vs. upwind, thin-layer vs. full viscous) or other code-to-code differences may have a larger effect than previously thought on grid sizes considered to be “medium” or “fine” by current standards.

**Exploring XP for Scientific Research**
(900 KB PDF)

Extreme Programming, as an agile programming methodology, focuses on delivering business value. In the realm of exploratory, long-term, small-scale research projects, prioritizing near-term tasks relative to their business or scientific value can be difficult. Assigning even a qualitative monetary value can be particularly challenging for government research in enabling fields for which business markets have not yet developed. The conflict between near-term value and long-term research objectives leads to a culture clash when applying basic XP practices. We decided to explore this culture clash when the Langley Creativity and Innovation Office solicited bids for exploring non-traditional methodologies for aerospace engineering research. C&I was looking for a way to produce extraordinary gains in productivity or enable entirely new applications. We submitted a bid and received one-year funding to perform a short prototyping assessment of XP at the NASA Langley Research Center. We conducted the project using a GNU/Linux operating system, the Emacs integrated development environment, and the Ruby programming language. We had prior experience programming related algorithms for the advection-diffusion equation using Fortran but no experience in team software development, object-oriented design, unit testing, or programming with Ruby.

**An Implicit, Exact Dual Adjoint Solution Method for
Turbulent Flows on Unstructured Grids**
(1.3 MB PDF)

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.

**The Efficiency of High Order Temporal Schemes**
(1.4 MB PDF)

A comparison of four temporal integration techniques is presented in the context of a general purpose aerodynamics solver. The study focuses on the temporal efficiency of high-order schemes, relative to the Backward Differentiation Formulae (BDF2) scheme. The high order algorithms used include the third-order BDF3 scheme, the fourth-order Modified Extended BDF (MEBDF4) scheme, and the fourth-order Explicit, Singly Diagonally Implicit Runge-Kutta (ESDIRK4) scheme.

Design order convergence is observed for all schemes. Specifically, second-, third-, and fourth-order accuracy for the BDF2, BDF3, and MEBDF4 schemes, while the ESDIRK4 scheme converges initially at a fourth-order rate but the order reduces down to third-order at high precisions. Very little advantage is observed with high-order schemes over the popular BDF2 scheme at accuracy tolerances of 10-3 or less. The MEBDF4 scheme is a possible practical alternative to BDF2 in aerodynamic applications at high precision levels.

**Flow Control Analysis on the Hump Model with RANS Tools**
(4.7 MB PDF)

A concerted effort is underway at NASA Langley Research Center to create a benchmark for Computational Fluid Dynamic (CFD) codes, both unstructured and structured, against a data set for the hump model with actuation. The hump model was tested in the NASA Langley 0.3-m Transonic Cryogenic Tunnel. The CFD codes used for the analyses are the FUN2D (Full Unstructured Navier-Stokes 2-Dimensional) code, the structured TLNS3D (Thin-Layer Navier-Stokes 3-Dimensional) code, and the structured CFL3D code, all developed at NASA Langley. The current investigation uses the time-accurate Reynolds-Averaged Navier-Stokes (RANS) approach to predict aerodynamic performance of the active flow control experimental database for the hump model. Two-dimensional computational results verified that steady blowing and suction and oscillatory suction/blowing can be used to significantly reduce the separated flow region on the model. Discrepancies do exist between the CFD results and experimental data in the region downstream of the slot with the largest differences in the oscillatory cases. Overall, the structured CFD codes exhibited similar behavior with each other for a wide range of control conditions, with the unstructured FUN2D code showing moderately different results in the separated flow region for the suction and oscillatory cases.

**Grid Adaptation for Functional Outputs of Compressible Flow
Simulations**
(25.1 MB PDF)

An error correction and grid adaptive method is presented for improving the accuracy of functional outputs of compressible flow simulations. The procedure is based on an adjoint formulation in which the estimated error in the functional can be directly related to the local residual errors of both the primal and adjoint solutions. This relationship allows local error contributions to be used as indicators in a grid adaptive method designed to produce specially tuned grids for accurately estimating the chosen functional. The method is applied to two-dimensional inviscid and viscous (laminar) flows using standard finite volume discretizations, and to scalar convection-diffusion using a Galerkin finite element discretization.

Isotropic h-refinement is used to iteratively improve the grids in a series of subsonic, transonic, and supersonic inviscid test cases. A commonly-used adaptive method that employs a curvature sensor based on measures of the local interpolation error in the solution is implemented to comparatively assess the performance of the proposed output-based procedure. In many cases, the curvature-based method fails to terminate or produces erroneous values for the functional at termination. In all test cases, the proposed output-based method succeeds in terminating once the prescribed accuracy level has been achieved for the chosen functional.

Output-based adaptive criteria are incorporated into an anisotropic grid-adaptive procedure for laminar Navier-Stokes simulations. The proposed method can be viewed as a merging of Hessian-based adaptation with output error control. A series of airfoil test cases are presented for Reynolds numbers ranging from 5,000 to 100,000. The proposed adaptive method is shown to compare very favorably in terms of output accuracy and computational efficiency relative to pure Hessian-based adaptation.

**Opportunities for Breakthroughs in Large-Scale Computational
Simulation
and Design**
(0.2 MB PDF)

Opportunities for breakthroughs in the large-scale computational simulation and design of aerospace vehicles are presented. Computational fluid dynamics tools to be used within multidisciplinary analysis and design methods are emphasized. The opportunities stem from speedups and robustness improvements in the underlying unit operations associated with simulation (geometry modeling, grid generation, physical modeling, analysis, etc.). Further, an improved programming environment can synergistically integrate these unit operations to leverage the gains. The speedups result from reducing the problem setup time through geometry modeling and grid generation operations, and reducing the solution time through the operation counts associated with solving the discretized equations to a sufficient accuracy. The opportunities are addressed only at a general level here, but an extensive list of references containing further details is included. The opportunities discussed are being addressed through the Fast Adaptive Aerospace Tools (FAAST) element of the Advanced Systems Concept to Test (ASCoT) and the 3rd Generation Reusable Launch Vehicles (RLV) projects at NASA Langley Research Center. The overall goal is to enable greater inroads into the design process with large-scale simulations.

**Adjoint-Based, Three-Dimensional Error Prediction and Grid
Adaptation**
(4 MB PDF)

Engineering computational fluid dynamics (CFD) analysis and design applications focus on output functions (e.g., lift, drag). Errors in these output functions are generally unknown and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solving the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user specified error tolerance. This procedure for estimating and adapting to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a Computer Aided Design (CAD) surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with ten times as many grid points.

**Three-Dimensional Effects on Multi-Element High Lift
Computations**
(5.4 MB PDF)

In an effort to discover the causes for disagreement between previous 2-D computations and nominally 2-D experiment for flow over the 3-element McDonnell Douglas 30P-30N airfoil configuration at high lift, a combined experimental/CFD investigation is described. The experiment explores several different side-wall boundary layer control venting patterns, documents venting mass flow rates, and looks at corner surface flow patterns. The experimental angle of attack at maximum lift is found to be sensitive to the side wall venting pattern: a particular pattern increases the angle of attack at maximum lift by at least 2 degrees. A significant amount of spanwise pressure variation is present at angles of attack near maximum lift. A CFD study using 3-D structured-grid computations, which includes the modeling of side-wall venting, is employed to investigate 3-D effects on the flow. Side-wall suction strength is found to affect the angle at which maximum lift is predicted. Maximum lift in the CFD is shown to be limited by the growth of an off-body corner flow vortex and consequent increase in spanwise pressure variation and decrease in circulation. The 3-D computations with and without wall venting predict similar trends to experiment at low angles of attack, but either stall too early or else overpredict lift levels near maximum lift by as much as 5%. Unstructured-grid computations demonstrate that mounting brackets lower the lift levels near maximum lift conditions.

**Isolating Curvature Effects in Computing Wall-Bounded Turbulent
Flows**
(0.4 MB PDF)

An adjoint optimization method is utilized to design an inviscid outer wall shape required for a turbulent flow field solution of the So-Mellor convex curved wall experiment using the Navier-Stokes equations. The associated cost function is the desired pressure distribution on the inner wall. Using this optimized wall shape with a Navier-Stokes method, the abilities of various turbulence models to simulate the effects of curvature without the complicating factor of streamwise pressure gradient are evaluated. The one-equation Spalart-Allmaras (SA) turbulence model overpredicts eddy viscosity, and its boundary layer profiles are too full. A curvature-corrected version of this model improves results, which are sensitive to the choice of a particular constant. An explicit algebraic stress model does a reasonable job predicting this flow field. However, results can be slightly improved by modifying the assumption on anisotropy equilibrium in the model’s derivation. The resulting curvature-corrected explicit algebraic stress model (EASM) possesses no heuristic functions or additional constants. It slightly lowers the computed skin friction coefficient and the turbulent stress levels for this case, in better agreement with experiment. The effect on computed velocity profiles is minimal.

**Factorizable Upwind Schemes: The Triangular Unstructured Grid
Formulation**
(0.8 MB PDF)

The upwind factorizable schemes for the equations of fluid was introduced recently. They facilitate achieving the Textbook Multigrid Efficiency (TME) and are expected also to result in the solvers of unparalleled robustness. The approach itself is very general. Therefore, it may well become a general framework for the large-scale Computational Fluid Dynamics. In this paper we outline the triangular grid formulation of the factorizable scheme. The derivation is based on the fact that the factorizable schemes can be expressed entirely using vector notation, without explicitly mentioning a particular coordinate frame. We describe the resulting discrete scheme in detail and present some computational results verifying the basic properties of the scheme/solver.

**Latency, Bandwidth, and Concurrent Issue Limitations in
High-Performance CFD**
(0.1 MB PDF)

To achieve high performance, a parallel algorithm needs to effectively utilize the memory subsystem and minimize the communication volume and the number of network transactions. These issues gain further importance on modern architectures, where the peak CPU performance is increasing much more rapidly than the memory or network performance. In this paper, we present some performance enhancing techniques that were employed on an unstructured mesh implicit solver. Our experimental results show that this solver adapts reasonably well to the high memory and network latencies.

**A Scientific Data Management System for Irregular Applications**
(0.1 MB PDF)

Many scientific applications are I/O intensive and generate large data sets, spanning hundreds or thousands of “files.” Management, storage, efficient access, and analysis of this data present an extremely challenging task. We have developed a software system, called Scientific Data Manager (SDM), that uses a combination of parallel file I/O and database support for high-performance scientific data management. SDM provides a high-level API to the user and, internally, uses a parallel file system to store real data and a database to store application-related metadata. In this paper, we describe how we designed and implemented SDM to support irregular applications. SDM can efficiently handle the reading and writing of data in an irregular mesh, as well as the distribution of index values. We describe the SDM user interface and how we have implemented it to achieve high performance. SDM makes extensive use of MPI-IO’s noncontiguous collective I/O functions. SDM also uses the concept of a history file to optimize the cost of the index distribution using the metadata stored in database. We present performance results with two irregular applications, a CFD code called FUN3D and a Rayleigh-Taylor instability code, on the SGI Origin2000 at Argonne National Laboratory.

**High-Performance Parallel Implicit CFD**
(0.2 MB PDF)

Fluid dynamical simulations based on finite discretizations on (quasi-) static grids scale well in parallel, but execute at a disappointing percentage of per-processor peak floating point operation rates without special attention to layout and access ordering of data. We document both claims from our experience with an unstructured grid CFD code that is typical of the state of the practice at NASA. These basic performance characteristics of PDE-based codes can be understood with surprisingly simple models, for which we quote earlier work, presenting primarily experimental results. The performance models and experimental results motivate algorithmic and software practices that lead to improvements in both parallel scalability and per node performance. This snapshot of ongoing work updates our 1999 Bell Prize-winning simulation on ASCI computers.

**Recent Improvements in Aerodynamic Design Optimization On
Unstructured Meshes**
(0.9 MB PDF)

Recent improvements in an unstructured-grid method for large-scale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.

**Understanding the Parallel Scalability of An
Implicit Unstructured Mesh CFD Code**
(0.1 MB PDF)

In this paper, we identify the scalability bottlenecks of an unstructured grid CFD code (PETSc-FUN3D) by studying the impact of several algorithmic and architectural parameters and by examining different programming models. We discuss the basic performance characteristics of this PDE code with the help of simple performance models developed in our earlier work, presenting primarily experimental results. In addition to achieving good per-processor performance (which has been addressed in our cited work and without which scalability claims are suspect) we strive to improve the implementation and convergence scalability of PETSc-FUN3D on thousands of processors.

**Performance Modeling and Tuning of an Unstructured Mesh CFD
Application**
(0.1 MB PDF)

This paper describes performance tuning experiences with a three-dimensional unstructured grid Euler flow code from NASA, which we have reimplemented in the PETSc framework and ported to several large-scale machines, including the ASCI Red and Blue Pacific machines, the SGI Origin, the Cray T3E, and Beowulf clusters. The code achieves a respectable level of performance for sparse problems, typical of scientific and engineering codes based on partial differential equations, and scales well up to thousands of processors. Since the gap between CPU speed and memory access rate is widening, the code is analyzed from a memory-centric perspective (in contrast to traditional flop-orientation) to understand its sequential and parallel performance. Performance tuning is approached on three fronts: data layouts to enhance locality of reference, algorithmic parameters, and parallel programming model. This effort was guided partly by some simple performance models developed for the sparse matrix-vector product operation.

**First-Order Model Management with Variable-Fidelity Physics Applied
to Multi-Element Airfoil Optimization**
(0.8 MB PDF)

First-order approximation and model management is a methodology for a systematic use of variable-fidelity models or approximations in optimization. The intent of model management is to attain convergence to high-fidelity solutions with minimal expense in high-fidelity computations. The savings in terms of computationally intensive evaluations depends on the ability of the available lower-fidelity model or a suite of models to predict the improvement trends for the high-fidelity problem. Variable-fidelity models can be represented by data-fitting approximations, variable-resolution models, variable-convergence models, or variable physical fidelity models. The present work considers the use of variable-fidelity physics models. We demonstrate the performance of model management on an aerodynamic optimization of a multi-element airfoil designed to operate in the transonic regime. Reynolds-averaged Navier-Stokes equations represent the high-fidelity model, while the Euler equations represent the low-fidelity model. An unstructured mesh-based analysis code FUN2D evaluates functions and sensitivity derivatives for both models. Model management for the present demonstration problem yields fivefold savings in terms of high-fidelity evaluations compared to optimization done with high-fidelity computations alone.

**Application of Adjoint Optimization Method to Multi-Element
Rotorcraft
Airfoils**
(2.3 MB PDF)

An adjoint optimization method coupled with an unstructured Navier-Stokes code was applied to the case of a multi-element rotorcraft airfoil. The combined optimization tool was used to reduce the drag of the airfoil at high Mach numbers and low angles of attack without significantly reducing the maximum lift at low Mach numbers and high angles of attack.

**Implementation of a Parallel Framework for Aerodynamic Design
Optimization on Unstructured Meshes**
(1.8 MB PDF)

A parallel framework for performing aerodynamic design optimizations on unstructured meshes is described. The approach utilizes a discrete adjoint formulation which has previously been implemented in a sequential environment and is based on the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. Here, only the inviscid terms are treated in order to develop a basic foundation for a multiprocessor design methodology. A parallel version of the adjoint solver is developed using a library of MPI-based linear and nonlinear solvers known as PETSc, while a shared-memory approach is taken for the mesh movement and gradient evaluation codes. Parallel efficiencies are demonstrated and the linearization of the residual is shown to remain valid.

**Sensitivity Analysis for the Navier-Stokes Equations on
Unstructured
Meshes Using Complex Variables**
(0.3 MB PDF)

The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. Although a step size parameter is required, the numerical derivatives are not subject to subtractive cancellation errors and therefore exhibit true second-order accuracy as the step size is reduced. As a result, this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude. This behavior is in contrast to the use of finite differences, which suffer from inaccuracies due to subtractive cancellation errors. In addition, the complex-variable procedure is easily implemented into existing codes.

**Towards Realistic Performance Bounds for Implicit CFD Codes**
(0.1 MB PDF)

The performance of scientific computing applications often achieves a small fraction of peak performance. In this paper, we discuss two causes of performance problems – insufficient memory bandwidth and a suboptimal instruction mix – in the context of a complete, parallel, unstructured mesh implicit CFD code. These results show that the performance of our code and of similar implicit codes is limited by the memory bandwidth of RISC-based processor nodes to as little as 10% of peak performance for some critical computational kernels. Limits on the number of basic operations that can be performed in a single clock cycle also limit the performance of “cache-friendly” parts of the code.

**Numerical Prediction of the Interference Drag of a Streamlined
Strut Intersecting a Surface in Transonic Flow**
(23.5 MB PDF)

In transonic flow, the aerodynamic interference that occurs on a strut-braced wing airplane, pylons, and other applications is significant. The purpose of this work is to provide relationships to estimate the interference drag of wing-strut, wing-pylon, and wing-body arrangements. Those equations are obtained by fitting a curve to the results obtained from numerous Computational Fluid Dynamics (CFD) calculations using state-of-the-art codes that employ the Spalart-Allmaras turbulence model.

In order to estimate the effect of the strut thickness, the Reynolds number of the flow, and the angle made by the strut with an adjacent surface, inviscid and viscous calculations are performed on a symmetrical strut at an angle between parallel walls. The computations are conducted at a Mach number of 0.85 and Reynolds numbers of 5.3 and 10.6 million based on the strut chord. The interference drag is calculated as the drag increment of the arrangement compared to an equivalent two-dimensional strut of the same cross-section. The results show a rapid increase of the interference drag as the angle of the strut deviates from a position perpendicular to the wall. Separation regions appear for low intersection angles, but the viscosity generally provides a positive effect in alleviating the strength of the shock near the junction and thus the drag penalty. When the thickness-to-chord ratio of the strut is reduced, the flowfield is disturbed only locally at the intersection of the strut with the wall. This study provides an equation to estimate the interference drag of simple intersections in transonic flow.

In the course of performing the calculations associated with this work, an unstructured flow solver was utilized. Accurate drag prediction requires a very fine grid and this leads to problems associated with the grid generator. Several challenges facing the unstructured grid methodology are discussed: slivers, grid refinement near the leading edge and at the trailing edge, grid convergence studies, volume grid generation, and other practical matters concerning such calculations.

**Efficient Parallelization of an Unstructured Grid Solver: A
Memory-Centric Approach**
(0.1 MB PDF)

For an unstructured grid computational fluid dynamics computation typical of many large-scale partial dirential equations requiring implicit treatment, we describe coding practices that lead to high implementation efficiency for standard computational and communication kernels, in both uniprocessor and parallel senses. Moreover, a family of Newton-like preconditioned Krylov algorithms whose convergence rate degrades only slightly with increasing parallel granularity, relying primarily on sparse Jacobian-vector multiplications, can be expressed in terms of these kernels. A combination of the three (uniprocessor performance, parallel scalability, and algorithmic scalability) is required for overall high performance on the largest scale problems that a given generation of parallel platforms supports.

**Aerodynamic Design Optimization on Unstructured Grids with a
Continuous Adjoint Formulation**
(1.1 MB PDF)

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

**Multiblock Approach for Calculating Incompressible Fluid
Flows on Unstructured Grids**
(700 KB PDF)

A multiblock approach is presented for solving two-dimensional incompressible turbulent flows on unstructured grids. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme, which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. A multiblock strategy to the solution procedure is introduced, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements while not increasing the CPU time. Results presented show that the current multiblock algorithm requires 73% less memory than the single-block algorithm.

**A Higher Order Accurate Finite Element Method for Viscous
Compressible
Flows**
(0.9 MB PDF)

The Streamline Upwind/Petrov-Galerkin (SU/PG) method is applied to higher-order finite-element discretizations of the Euler equations in one dimension and the Navier-Stokes equations in two dimensions. The unknown flow quantities are discretized on meshes of triangular elements using triangular Bezier patches. The nonlinear residual equations are solved using an approximate Newton method with a pseudotime term. The resulting linear system is solved using the Generalized Minimum Residual algorithm with block diagonal preconditioning.

The exact solutions of Ringleb flow and Couette flow are used to quantitatively establish the spatial convergence rate of each discretization. Examples of inviscid flows including subsonic flow past a parabolic bump on a wall and subsonic and transonic flows past a NACA 0012 airfoil and laminar flows including flow past a a flat plate and flow past a NACA 0012 airfoil are included to qualitatively evaluate the accuracy of the discretizations. The scheme achieves higher order accuracy without modification. Based on the test cases presented, significant improvement of the solution can be expected using the higher-order schemes with little or no increase in computational requirements. The nonlinear system also converges at a higher rate as the order of accuracy is increased for the same number of degrees of freedom; however, the linear system becomes more difficult to solve. Several avenues of future research based on the results of the study are identified, including improvement of the SU/PG formulation, development of more general grid generation strategies for higher order elements, the addition of a turbulence model to extend the method to high Reynolds number flows, and extension of the method to three-dimensional flows. An appendix is included in which the method is applied to inviscid flows in three dimensions. The three-dimensional results are preliminary but consistent with the findings based on the two-dimensional scheme.

**Aerodynamic Design Sensitivities on an Unstructured Mesh Using
the
Navier-Stokes Equations and a Discrete Adjoint Formulation**
(7.2 MB PDF)

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated, and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences and a complex-variable approach. Several simplifying approximations to the complete linearization of the residual are also presented. A first-order approximation to the dependent variables is implemented in the adjoint and design equations, and the effect of a “frozen” eddy viscosity and neglecting mesh sensitivity terms is also examined. The resulting derivatives from these approximations are all shown to be inaccurate and often of incorrect sign. However, a partially-converged adjoint solution is shown to be sufficient for computing accurate sensitivity derivatives, yielding a potentially large cost savings in the design process. The convergence rate of the adjoint solver is compared to that of the flow solver. For inviscid adjoint solutions, the cost is roughly one to four times that of a flow solution, whereas for turbulent computations, this ratio can reach as high as ten. Sample optimizations are performed for inviscid and turbulent transonic flows over an ONERA M6 wing, and drag reductions are demonstrated.

**An O(Nm ^{2}) Plane Solver for the Compressible
Navier-Stokes Equations**
(0.2 MB PDF)

A hierarchical multigrid algorithm for efficient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in two ways: a Full Approximation Scheme (FAS) for a nonlinear residual equation and a Correction Scheme (CS) for a linearized defect correction implicit equation. Multigrid analyses which include the effect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-line-implicit algorithms are compared in terms of efficiency. The analyses indicate that full multigrid efficiency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous flow simulations on highly stretched meshes.

**Three-Dimensional Incompressible Navier-Stokes Flow Computations
About Complete Configurations Using a Multiblock Unstructured Grid**
(1.0 MB PDF)

A multiblock unstructured grid approach is presented for solving three-dimensional incompressible inviscid and viscous turbulent flows about complete configurations. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work employs a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements by a factor of 5 over the single grid algorithm while maintaining a similar convergence behavior. The numerical accuracy of solutions is assessed by comparing with the experimental data for a submarine with stem appendages and a high-lift configuration.

**Achieving High Sustained Performance in an Unstructured Mesh CFD
Application**
(0.1 MB PDF)

This paper highlights a three-year project by an interdisciplinary team on a legacy F77 computational fluid dynamics code, with the aim of demonstrating that implicit unstructured grid simulations can execute at rates not far from those of explicit structured grid codes, provided attention is paid to data motion complexity and the reuse of data positioned at the levels of the memory hierarchy closest to the processor, in addition to traditional operation count complexity. The demonstration code is from NASA and the enabling parallel hardware and (freely available) software toolkit are from DOE, but the resulting methodology should be broadly applicable, and the hardware limitations exposed should allow programmers and vendors of parallel platforms to focus with greater encouragement on sparse codes with indirect addressing. This snapshot of ongoing work shows a performance of 15 microseconds per degree of freedom to steady-state convergence of Euler flow on a mesh with 2.8 million vertices using 3072 dual-processor nodes of Sandia’s “ASCI Red” Intel machine, corresponding to a sustained floating-point rate of 0.227 Tflop/s.

**Prospects for CFD on Petaflops Systems**
(0.4 MB PDF)

With teraflops-scale computational modeling expected to be routine by 2003-04, under the terms of the Accelerated Strategic Computing Initiative (ASCI) of the U.S. Department of Energy, and with teraflops-capable platforms already available to a small group of users, attention naturally focuses on the next symbolically important milestone, computing at rates of 1015 floating point operations per second, or \petaflop/s”. For architectural designs that are in any sense extrapolations of today’s, petaflops-scale computing will require approximately one-million-fold instruction-level concurrency. Given that cost-effective one-thousand-fold concurrency is challenging in practical computational fluid dynamics simulations today, algorithms are among the many possible bottlenecks to CFD on petaflops systems. After a general outline of the problems and prospects of petaflops computing, we examine the issue of algorithms for PDE computations in particular. A back-of-the-envelope parallel complexity analysis focuses on the latency of global synchronization steps in the implicit algorithm. We argue that the latency of synchronization steps is a fundamental, but addressable, challenge for PDE computations with static data structures, which are primarily determined by grids. We provide recent results with encouraging scalability for parallel implicit Euler simulations using the Newton-Krylov-Schwarz solver in the PETSc software library. The prospects for PDE simulations with dynamically evolving data structures are far less clear.

**Multidisciplinary Sensitivity Derivatives Using Complex Variables**
(0.1 MB PDF)

A new method for computing single and multidisciplinary sensitivity derivatives using complex variables has been developed. Extremely accurate derivatives are computed from high fidelity aerodynamic, structural, and aero-structural analysis. This report briefly reviews the various techniques to obtain single discipline sensitivity derivatives and how they may be used to evaluate multidisciplinary derivatives. The advantages and disadvantages of the complex variable approximation are compared with these existing techniques. It is shown that this new method has all the advantages of existing direct-discrete approaches and the finite-difference approximation, while avoiding some of their shortcomings. In addition, existing software can be easily modified to incorporate this technique, which makes it a valuable tool for multidisciplinary applications. To demonstrate the accuracy of the complex variable approximation, a low aspect ratio ONERA M6 wing, that has been used in previous optimization studies, is examined. Aerodynamic, structural, and aero-structural sensitivity derivatives have been computed for a variety of design variables. Design variables appropriate for both aerodynamic and structural optimization have been selected.

**Newton-Krylov-Schwarz Methods for Aerodynamics Problems:
Compressible and Incompressible Flows on Unstructured Grids**
(0.2 MB PDF)

We review and extend to the compressible regime an earlier parallelization of an implicit incompressible unstructured Euler code, and solve for flow over an M6 wing in subsonic, transonic, and supersonic regimes. While the parallelization philosophy of the compressible case is identical to the incompressible, we focus here on the nonlinear and linear convergence rates, which vary in different physical regimes, and on comparing the performance of currently important computational platforms.

**Aerodynamic Design Optimization on Unstructured Meshes Using the
Navier-Stokes Equations**
(5.1 MB PDF)

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.

**The Use of Heavy Gas for Increased Reynolds Numbers in Transonic
Wind
Tunnels**
(0.3 MB PDF)

The use of a high molecular weight test gas to increase the Reynolds number range of transonic wind tunnels is explored. Modifications to a small transonic wind tunnel are described and the real gas properties of the example heavy gas (sulfur hexafluoride) are discussed. Sulfur hexafluoride is shown to increase the test Reynolds number by a factor of more than 2 over air at the same Mach number. Experimental and computational pressure distributions on an advanced supercritical airfoil configuration at Mach 0.7 in both sulfur hexafluoride and nitrogen are presented. Transonic similarity theory is shown to be partially successful in transforming the heavy gas results to equivalent nitrogen (air) results, provided the correct definition of gamma is used.

**Porting FUN3D to Distributed Memory Parallelism**
(0.1 MB PDF)

While much research in parallel computer science is oriented toward language, software, and architectural environments to support future code, porting valuable legacy codes to contemporary parallel environments remains an important objective. This is particularly true since parallelism is the only means by which massive amounts of memory can be cost-effectively brought to bear on legacy applications that need to outgrow the vector Crays for which they were created. NASA’s mission to support computational design and optimization requires that computational fluid dynamics (CFD) and other types of analyses be routinely extended to higher fidelity models on finer grids. Moreover, since optimization depends on derivatives of the solutions to the analysis problem, it is advantageous to employ analysis techniques that work directly with the Jacobian. This focuses considerable interest on parallel implicit algorithms for elliptic PDE-based simulations of all kinds.

**On the Interaction of Architecture and Algorithm in the
Domain-Based Parallelization of an Unstructured Grid Incompressible Flow Code**
(0.2 MB PDF)

The convergence rates and, therefore, the overall parallel efficiencies of additive Schwarz methods are often notoriously dependent on subdomain granularity. Except when effective coarse-grid operators and intergrid transfer operators are known, so that optimal multilevel preconditioners can be constructed, the number of iterations to convergence and the communication overhead per iteration tend to increase with granularity for elliptically-controlled problems, for either fixed or memory-scaled problem sizes.

In practical large-scale applications, however, the convergence rate degradation of fine-grained single-level additive Schwarz is sometimes not as serious as the scalar, linear elliptic theory would suggest. Its effects are mitigated by several factors, including pseudo-transient nonlinear continuation and dominant intercomponent coupling that can be captured exactly in a point-block ILU preconditioner. We illustrate these claims with encouraging scalabilities for a legacy unstructured-grid Euler flow application code, parallelized with the pseudo-transient Newton-Krylov-Schwarz algorithm using the PETSc library. We note some impacts on performance of the horizontal (distributed) and vertical (hierarchical) aspects of the memory system and consider architecturally motivated algorithmic variations for their amelioration.

**Airfoil Design on Unstructured Grids for Turbulent Flows**
(1.2 MB PDF)

This paper is similar to the one above but it is the one I submitted for publication. This paper does not have the description of the user interface but it does include some mesh sensitivity information that I left out of the previous paper. The test cases are also different and there are some derivatives relevant to multielement airfoils. It probably has some typos fixed as well.

**Aerodynamic Design on Unstructured Grids for Turbulent Flows**
(0.6 MB PDF)

An aerodynamic design algorithm for turbulent flows using unstructured grids is described. The current approach uses adjoint (costate) variables to obtain derivatives of the cost function. The solution of the adjoint equations is obtained by using an implicit formulation in which the turbulence model is fully coupled with the flow equations when solving for the costate variables. The accuracy of the derivatives is demonstrated by comparison with finite-difference gradients and a few sample computations are shown. In addition, a user interface is described that significantly reduces the time required to set up the design problems. Recommendations on directions of further research into the Navier-Stokes design process are made.

**A Multiblock Approach for Calculating Incompressible Fluid Flows
on
Unstructured Grids**
(0.3 MB PDF)

A multiblock approach is presented for solving two-dimensional incompressible turbulent flows on unstructured grids. The artificial compressibility form of the governing equations is solved by a vertex-centered, finite-volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work introduces a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements while not increasing the CPU time. Results presented in this work shows that a current multiblock algorithm requires 70% less memory than the single block algorithm.

**Aerodynamic Design Optimization on Unstructured Grids with a
Continuous
Adjoint Formulation**
(1.7 MB PDF)

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

**Navier-Stokes Computations and Experimental Comparisons for
Multielement Airfoil Configurations**
(0.7 MB PDF)

A two-dimensional unstructured Navier-Stokes code is utilized for computing the flow around multi-element airfoil configurations. Comparisons are shown for a landing configuration with an advanced-technology flap. Grid convergence studies are conducted to assess inaccuracies caused by inadequate grid resolution. Although adequate resolution is obtained for determining the pressure distributions, further refinement is needed to sufficiently resolve the velocity profiles at high angles of attack.

For the advanced flap configuration, comparisons of pressure distributions and lift are made with experimental data. Here, two flap riggings and two Reynolds numbers are considered. In general, the trends caused by variations in these quantities are well predicted by the computations, although the angle of attack for maximum lift is overpredicted.

**Application of Newton-Krylov Methodology to A Three Dimensional
Unstructured Euler Code**
(0.6 MB PDF)

A Newton-Krylov scheme is applied to an unstructured Euler code in both two and three dimensions. A simple and computationally efficient means of differencing residual of perturbed solutions is presented that allows consistent levels of convergence to be obtained, independent of the mesh size. Results are shown for subsonic and transonic flow over an airfoil that indicate the Newton-Krylov method can be effective in accelerating convergence over a baseline scheme provided the initial conditions are sufficiently close to the root to allow the fast convergence associated with Newton’s method. Two methodologies are presented to accomplish this requirement. Comparisons are made between two methods for forming the matrix-vector product used in the GMRES algorithm. These include a matrix-free finite-difference approach as well as a formulation that allows exact calculation of the matrix-vector product. The finite-difference formulation requires slightly more computer time than the exact method, but has less stringent memory requirements. Lastly, three-dimensional results are shown for an isolated wing as well as for a complex-geometry helicopter configuration.

**Parallel Algorithms of Newton-Krylov-Schwarz Type**
(0.1 MB PDF)

Parallel implicit solution methods are increasingly important in aerodynamics and other fields leading to large nonlinear systems with sparse Jacobians. Several trends contribute to their importance. Multidisciplinary analysis and optimization require rapidly achievable low residual solutions, since individual component codes are often iterated and their results differenced for sensitivities. Problems possessing multiple space or time scales motivate implicit algorithms, and arise frequently in locally adaptive contexts and in dynamical contexts such as aeroelasticity. Meanwhile, the demand for resolution and prompt turnaround forces consideration of parallelism, and, for cost effectiveness, the high-latency, low-bandwidth parallelism available from workstation clusters. An ICASE program in Newton-Krylov-Schwarz (NKS) solvers responds to this need, in collaboration with academia, national laboratories (NASA and DOE), and industry (Boeing and UTRC).

**An Upwind Multigrid Method for Solving Viscous Flows on
Unstructured Triangular Meshes**
(1.7 MB PDF)

A multigrid algorithm is combined with an upwind scheme for solving the two-dimensional Reynolds-averaged Navier-Stokes equations on triangular meshes resulting in an efficient, accurate code for solving complex flows around multiple bodies. The relaxation scheme uses a backward-Euler time difference and relaxes the resulting linear system using a red-black procedure. Roe’s flux-splitting scheme is used to discretize convective and pressure terms, while a central difference is used for the diffusive terms. The multigrid scheme is demonstrated for several flows around single and multi-element airfoils, including inviscid, laminar and turbulent flows. The results show an appreciable speedup of the scheme for inviscid and laminar flows, and dramatic increases in efficiency for turbulent cases, especially those on increasingly refined grids.

**Implicit/Multigrid Algorithms for Incompressible
Turbulent Flows on Unstructured Grids**
(2.1 MB PDF)

An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned Generalized Minimal Residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that uses non-nested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

**An Implicit Upwind Algorithm for Computing Turbulent Flows on
Unstructured Grids**
(1.1 MB PDF)

An implicit, Navier-Stokes solution algorithm is presented for the computation of turbulent flow on unstructured grids. The inviscid fluxes are computed using an upwind algorithm and the solution is advanced in time using a backward-Euler time-stepping scheme. At each time step, the linear system of equations is approximately solved with a point-implicit relaxation scheme. This methodology provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.

Results are shown for subsonic flow over a NACA 0012 airfoil and for transonic flow over a RAE 2822 airfoil exhibiting a strong upper-surface shock. In addition, results are shown for 3-element and 4-element airfoil configurations. For the calculations, two one-equation turbulence models are utilized. For the NACA 0012 airfoil, a pressure distribution and force data are compared with other computational results as well as with experiment. Comparisons of computed pressure distributions and velocity profiles with experimental data are shown for the RAE airfoil and for the 3-element configuration. For the 4-element case, comparisons of surface pressure distributions with experiment are made. In general, the agreement between the computations and the experiment is good.

**Grid Generation and Flow Solution Method for Euler Equations on
Unstructured Grids**
(845 KB PDF)

A grid generation and flow solution algorithm for the Euler equations on unstructured grids is presented. The grid generation scheme, which utilizes Delaunay triangulation, generates the field points for the mesh based on cell aspect ratio and allows clustering of grid points near solid surfaces. The flow solution method is an implicit algorithm in which the linear set of equations arising at each time step is solved using a Gauss-Seidel procedure that is completely vectorizable. In addition, a study is conducted to examine the number of sub-iterations required for good convergence of the overall algorithm. Grid generation results are shown in two dimensions for an NACA 0012 airfoil as well as a two-element configuration. Flow solution results are shown for a two-dimensional flow over the NACA 0012 airfoil and for a two-element configuration in which the solution has been obtained through an adaptation procedure and compared with an exact solution. Preliminary three-dimensional results are also shown in which the subsonic flow over a business jet is computed.

## 2.2. Presentations and Other Materials

Seeking Reality in the Future of Aeronautical Simulation, article in ScienceDaily, June 2014. ( Link 3.0 MB PDF )

Bearing Heavy Loads, Engineering Notebook article in Aerospace America, June 2014. ( 0.3 MB PDF )

NASA X television program focusing on Environmentally Responsible Aviation project. Featured CFD simulations were performed using FUN3D. February 2014. (ERA Project )

Efficient Physics-Based Analysis and Design for Complex Aerospace Configurations, Eric Nielsen and Dana Hammond, Demo for NASA booth at Supercomputing 2013. (Demo Overview Flash Demo 0.6 MB PDF Poster )

In-Situ Exploration of Large Scale CFD with FUN3D and VisIt, Bill Jones and Eric Nielsen, Invited Presentation at 50th HPC User Forum, Boston, September 2013. (Flash)

Adjoint-Based Optimization of Flapping Wing Flows, Martin Jones and Nail Yamaleev, NIA CFD Seminar, June 2013. ( Flash )

CFL3D and FUN3D Analysis of HiLiftPW-2 Workshop Cases, Elizabeth M. Lee-Rausch and Christopher L. Rumsey, June 2013. ( 3.3 MB PDF Talk )

Recent Algorithm Development and Application Efforts Using FUN3D, Eric Nielsen, Seminar at University of Louisville, November 30, 2012. (Flash)

Recent Advances in Agglomerated MultiGrid. Hiroaki Nishikawa, Boris Diskin, James L. Thomas, and Dana P. Hammmond. AIAA Aerospace Sciences Meeting, January 9, 2013, Grapevine, TX. ( 7.6 MB PDF Talk )

Adjoint-Based Design for Complex Aerospace Configurations, Eric Nielsen and Boris Diskin, Demo for NASA booth at Supercomputing 2012. (Media Sheet Demo Overview Flash Demo 12.6 MB PDF Poster )

From a Roar to a Whisper: Making Modern Aircraft Quieter, Mehdi Khorrami, Demo for NASA booth at Supercomputing 2012. (Demo Overview 12.7 MB PDF Poster )

Supersonic Retropropulsion for Mars Entry, Michael Wright and Guy Schauerhamer, Demo for NASA booth at Supercomputing 2012. (Demo Overview 12.3 MB PDF Poster )

Discrete Adjoint-Based Design for Unsteady Turbulent Flows On Dynamic Overset Mixed-Element Grids, Eric J. Nielsen and Boris Diskin, 11th Symposium on Overset Composite Grids and Solution Technology, Dayton, OH, October 2012. (Flash)

Sensitivity Analysis for Chaotic Turbulent Flows, Patrick Blonigan, LARSS Final Presentation, August 2012. ( 5.6 MB PDF Talk )

Recent Algorithm Development and Application Efforts Using FUN3D, Eric Nielsen, Seminar at North Carolina A&T State University, July 19, 2012. (Flash)

Adjoint-Based Optimization of the Flapping Wing Performance, Martin Jones and Nail Yamaleev, 7th International Conference on Computational Fluid Dynamics (ICCFD7), Hawaii, 9-13 July 2012. (Flash)

Drag Prediction Workshop-V, Nick Powell, 5th AIAA CFD Drag Prediction Workshop, New Orleans, LA, 23-24 June 2012. ( 4.2 MB PDF Talk )

FUN3D and CFL3D Calculations for the Fifth AIAA Drag Prediction Workshop, Mike Park, 5th AIAA CFD Drag Prediction Workshop, New Orleans, LA, 23-24 June 2012. ( 5.5 MB PDF Talk )

Application of Adaptive Gridding in FUN3D for Simulation of Flow over a Nose Landing Gear. Veer N. Vatsa, Mehdi R. Khorrami, and David P. Lockard. BANC II Workshop 7-8 June 2012, Colorado Springs, CO. ( 5.3 MB PDF Talk )

Multidisciplinary Design Optimization: What Remains to Be Done, Natalia Alexandrov, Keynote presentation at 8th AIAA Multidisciplinary Design Optimization Specialist Conference, Honolulu, April 2012. (Flash)

Recent Algorithm Development and Application Efforts Using FUN3D, Eric Nielsen, Presentation for Air Force Research Laboratory, Dayton, March 2012. (Flash)

Overview of Recent FUN3D Code Development and Applications for the Subsonic Rotary Wing Project, Eric Nielsen, Bob Biedron, Dana Hammond, Bill Jones, Beth Lee-Rausch, Mike Park, and Jim Thomas, Presentation for Subsonic Rotary Wing Project at 2012 Annual Fundamental Aeronautics Meeting, Cleveland, March 2012. (Flash)

Hypersonic Flow Simulations on Tetrahedral Grids Using FUN3D, Peter Gnoffo and William Wood, Presentation for Hypersonics Project at 2012 Annual Fundamental Aeronautics Meeting, Cleveland, March 2012. (Flash)

Adjoint-Based Algorithms for Complex Aerodynamic Flows in Large-Scale Computational Environments, Eric Nielsen, Michael Park, and Dana Hammond, presented in minisymposium “Challenges in Massively Parallel Simulations Using Unstructured Meshes” at 14th SIAM Conference on Parallel Processing for Scientific Computing, Savannah, February 2012. (Flash)

East Bound and Down, Loaded Up and Truckin’ , Jon Riley, Article in Digital Manufacturing Report, February 6, 2012.

The Effect of a Gust on the Flapping Wing Performance, Martin Jones and Nail Yamaleev, AIAA-2012-1080, Nashville, January 2012. (Flash)

Exploration of the Physics of Hub Drag, Vrishank Raghav, Rajiv Shenoy, Felipe T. Ortega, Narayanan Komerath, and Marilyn Smith, AIAA-2012-1070, Nashville, January 2012. (Flash)

Continuing Validation of Computational Fluid Dynamics For Supersonic Retropropulsion, Daniel G. Schauerhamer, Kerry A. Trumble, Bil Kleb, Jan-Renee Carlson, and Karl T. Edquist, AIAA-2012-0864, Nashville, January 2012. (Flash)

Discrete Adjoint-Based Design for Unsteady Turbulent Flows on Dynamic Overset Unstructured Grids, Eric Nielsen, AIAA-2012-0554, Nashville, January 2012. (Flash)

Simulation of Jet Plumes for Orion Launch Abort, Jan-Renee Carlson, FUN3D demo for NASA booth at Supercomputing 2011. ( Demo Overview 700 KB PDF Poster )

UH-60A Blackhawk Helicopter Aerodynamics, Beth Lee-Rausch and Bob Biedron, FUN3D demo for NASA booth at Supercomputing 2011. ( Demo Overview 600 KB PDF Poster )

Supersonic Retropropulsion for Mars Entry, Kerry Trumble, Demo for NASA booth at Supercomputing 2011. ( Demo Overview )

Aerodynamic Design Optimization for Unsteady Flows and Dynamic Geometries Using Discrete Adjoint Methods, Eric Nielsen, ACDL Seminar Presentation at MIT, Cambridge, September 2011. (Flash)

Adjoint-Based Design Optimization Using FUN3D and Sculptor, Joint webinar with Optimal Solutions, August 10, 2011. (Archived WebEx Recording) (6.9 MB PDF)

Libmo – Software Library for Motion Simulation, Poster presented at 2011 DoD High Performance Computing Modernization Program Users Group Conference by Nathan Prewitt, June 2011. (1 MB jpg)

High-Fidelity Simulation of Landing Gear Noise , HPCSource Newsletter, April 2011.

BMI Uses Jaguar to Overhaul Long-Haul Trucks , Oak Ridge Leadership Computing Facility, March 2, 2011.

Supercomputer Helps Make Energy Efficient Trucks , WBIR-TV 10 local news report, NBC affiliate in Knoxville, TN, March 2011.

Overview of Recent FUN3D Code Developments and Applications, Beth Lee-Rausch, Presentation for Subsonic Rotary Wing Project at 2011 Annual Fundamental Aeronautics Meeting, Cleveland, March 2011. (Flash)

FUN3D General Overview: Research Areas and Applications, Eric Nielsen, Presentation for Sandia National Labs, Albuquerque, March 2011. (Flash)

Pointwise, Gridgen Streamline Ducted Fan, Stator Design, Featured in Focal Point, The Newsletter for Pointwise Users, Volume 14, Issue 2, Winter 2010. (830 KB PDF)

High-Fidelity Simulations of Landing Gear Noise, Mehdi Khorrami, Demo for NASA booth at Supercomputing 2010. ( Media Sheet 6 MB PDF Poster 200 KB PDF Handout )

Computational Scaling for an Unstructured-Grid CFD Solver, Eric Nielsen, FUN3D demo for NASA booth at Supercomputing 2010. (Flash 9.2 MB PDF Poster 200 KB PDF Handout )

Designing a Smart Truck with the Power of Jaguar, Brochure and video produced by Oak Ridge National Laboratory for Supercomputing 2010. (1.1 MB PDF)

Not Your Father’s Hybrid Code: Advancements in CFD-Based Hybrid Methods for a New Millennium, Eliot Quon, Poster presented at the 10th Symposium on Overset Composite Grids and Solution Technology, NASA Ames Research Center, September 20-23, 2010. (2.1 MB PDF)

An Adaptive Mesh Refinement (AMR) Strategy for Static and Dynamic Overset Unstructured Meshes, Rajiv Shenoy, Poster presented at the 10th Symposium on Overset Composite Grids and Solution Technology, NASA Ames Research Center, September 20-23, 2010. (1.5 MB PDF)

Applications and Research Efforts Using FUN3D in HPC Environments, Eric Nielsen and Dana Hammond, Invited Presentation at 37th HPC User Forum, Seattle, September 2010. (Flash)

CFD Code Acceleration on Hybrid Many-Core Architectures, Austen C. Duffy. (2.0 MB PDF)

Output-Based Grid Adaptation Applied to the HiLiftPW-1, Mike Park, Presented at the 1st AIAA CFD High Lift Prediction Workshop, Chicago, June 2010. (3.7 MB PDF)

CFL3D and FUN3D Analysis of HiLiftPW-1 Workshop Cases, Elizabeth M. Lee-Rausch and Christopher L. Rumsey, Presented at the 1st AIAA CFD High Lift Prediction Workshop, Chicago, June 2010. (1.2 MB PDF)

FUN3D Solutions for Nose Landing Gear, Veer N. Vatsa, David P. Lockard, and Mehdi R. Khorrami, Presented at the AIAA Workshop on Benchmark Problems for Airframe Noise Computations (BANC), Stockholm, Sweden, June 7-9, 2010. (3.9 MB PDF)

FUN3D Solutions for Tandem Cylinders, Veer N. Vatsa and David P. Lockard, Presented at the AIAA Workshop on Benchmark Problems for Airframe Noise Computations (BANC), Stockholm, Sweden, June 7-9, 2010. (2.3 MB PDF)

Unstructured CFD for Wind Turbine Analysis, C. Eric Lynch and Marilyn Smith, Presented at the US-Egypt Workshop on Wind Energy Development, Cairo, Egypt, March 22-24, 2010. (5.5 MB PDF)

Overview of NSF Energy for Sustainability Program, highlighting wind turbine analysis, 2009. (3.9 MB PDF)

Adjoint-Based Design Optimization Using FUN3D, Eric Nielsen, Presentation for Subsonic Rotary Wing Project at 2009 Annual Fundamental Aeronautics Meeting, Atlanta, September 2009. (Flash)

Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids, Eric Nielsen, Presentation for Supersonics Project at 2009 Annual Fundamental Aeronautics Meeting, Atlanta, September 2009. (Flash)

In the Wind, Barbara Jewett, NCSA Access Magazine, Summer 2009. (0.3 MB PDF)

Ongoing Research Into Numerical Simulation of Fluid Flows Utilizing Software Development Practices, Michael A. Park, Seminar given at the MIT Aerospace Computational Design Lab (ACDL), Cambridge, Massachusetts, September 2004. (1.7 MB PDF Presentation, 78 KB PDF Handout)

Aerodynamic Design Optimization Using the Navier-Stokes Equations, Eric J. Nielsen, Overview talk given at 18th International Symposium on Mathematical Programming, Copenhagen, Denmark, August 2003. (10.9 MB PDF)

Achieving High Sustained Performance in an Unstructured Mesh CFD Application, Kyle Anderson, William Gropp, Dinesh Kaushik, David Keyes, and Barry Smith, Gordon Bell Prize at Supercomputing 1999. ( 1.7 MB PDF Presentation, 37 KB PDF Argonne News Brief, 25 KB PDF ODU News Brief, 11 KB PDF Lawrence Livermore News Brief, 1999 Gordon Bell Honorees at Lawrence Livermore )

## 2.3. Development Team

### NASA Contributors

#### Ponnampalam (Bala) Balakumar

Flow Physics and Control Branch, NASA Langley

- Transonic turbulence models, especially Reynolds-stress modeling.
- Repaired SST model

#### Karen Bibb

Aerothermodynamics Branch, NASA Langley

- CAD-based mesh adaption, both adjoint and feature based, for hypersonic aerodynamics
- Inviscid hypersonic aerodynamics applications, primarily with FELISA code

#### Bob Biedron

Computational Aerosciences Branch, NASA Langley

- Formerly a primary developer of CFL3D
- Implemented mixed-element infrastructure throughout FUN3D framework
- Current research aimed at mixed-element algorithms, computational stability and control, and moving-mesh applications

#### Jan Carlson

Computational Aerosciences Branch, NASA Langley

- User/Developer

#### Mark Carpenter

Computational Aerosciences Branch, NASA Langley

- Developed and verified time-accurate capabilities
- Newton-Krylov relaxation

#### Peter Gnoffo

Aerothermodynamics Branch, NASA Langley

- Developer of LAURA external hypersonics flow solver
- Implemented real-gas physical and turbulence models of LAURA and VULCAN into FUN3D framework
- Current research aimed at accurate hypersonic predictions using pure-tetrahedral unstructured grids

#### Dana Hammond

Advanced Engineering Environments Branch, NASA Langley

- Expert on distributed computing, grid computing, and client-server applications
- Developed distributed version of FUN3D preprocessor
- Resident computer science expert

#### Bill Jones

Advanced Engineering Environments Branch, NASA Langley

- Principle author of GridEx, a CAD-based interactive software system for the generation of unstructured grids
- Resident expert for computational geometry and CAD-based mesh adaption

#### Bil Kleb

Aerothermodynamics Branch, NASA Langley

- Developed automated complex-variable form of source code with Ruby
- Develops various CASE and refactoring tools.
- Usually serves as Scrum master
- Implemented AUFS and HLLC flux functions
- Constantly pushes team to seek more effective software development practices (XP level 0, CMMI level 5 behavior)

#### Beth Lee-Rausch

Computational Aerosciences Branch, NASA Langley

- Applications expert
- Expanding the boundaries of problem size and complexity

#### Eric Nielsen

Computational Aerosciences Branch, NASA Langley

- Performed full linearization of flow solver and built framework for 3D design
- Extended complete 3D design effort to parallel environment
- Current research aimed at advanced solution algorithms and 3D design studies
- All-around code guru.

#### Hiroaki Nishikawa

National Institute of Aerospace, NASA Langley

- Implemented grid agglomeration for multigrid
- Expert on the analysis of discretizations
- Proponent of all things hyperbolic

#### Mike Park

Computational Aerosciences Branch, NASA Langley

- Implemented MPI communication infrastructure
- 3D adjoint-based error estimation and distributed grid adaptation connected directly to CAD
- Cut-cell discretization

#### Chris Rumsey

Computational Aerosciences Branch, NASA Langley

#### Jim Thomas

Computational Aerosciences Branch, NASA Langley

- Original CFL3D developer
- Performed system-level verification via novel application of the Method of Manufactured Solutions
- Current research aimed at multigrid and relaxation strategies as well as mixed-element discretizations.

#### Veer Vatsa

Computational Aerosciences Branch, NASA Langley

- Wall-function turbulence models
- Time-accurate capability
- CLV applications.

#### Jeff White

Computational Aerosciences Branch, NASA Langley

- Expert on internal hypersonic flows, developer of VULCAN solver
- Implemented advanced turbulence models of VULCAN into FUN3D framework
- Working blunt-body hypersonic stagnation region flows and improving modularity of code architecture

### Other Developers

#### Alejandro Campos

Alejandro is a Ph.D. student in the Department of Aeronautics and Astronautics at Stanford University where he implemented the Algebraic Structure-Based Model (ASBM) in FUN3D.

#### Rajiv Shenoy

Rajiv is currently an MS student at Georgia Tech under Prof. Marilyn Smith’s direction.

- Extended grid adaptation to overset grids
- Interested in feature and adjoint-based adaptation methodologies for time-dependent flows
- Applications of interest include high fidelity Rotor-Fuselage and Wind Turbine Interactions

#### Marilyn Smith

School of Aerospace Engineering, Georgia Institute of Technology

Marilyn is an associate professor at Georgia Tech working in the area of unsteady aerodynamics and aeroelasticity, including rotorcraft, propulsion and fixed-wing applications. For more details, please visit her webpage.

### Interns

#### Joe Derlaga

Joe is a NASA Co-op developing output-based grid adaptation and error estimation techniques. He is simultaneously pursuing a doctorate at Virginia Tech.

### Past Contributors

#### Natalia Alexandrov

Multidisciplinary Tools and Methods Branch, NASA Langley

- Optimization methods for simulation-based design
- Design methods for complex adaptive systems

#### W. Kyle Anderson

Kyle left Langley in 2000. He was responsible for laying the foundations of the FUN2D/3D effort.

- Developed initial versions of FUN2D and FUN3D in the late 1980s as testbeds for unstructured mesh research at NASA Langley.
- Formulated and implemented design methodology using both continuous and discrete adjoint formulations in 2D. Discrete formulation later extended to 3D.
- In conjunction with researchers at Mississippi State University, developed complex variable approach for computing sensitivity derivatives for multidisciplinary applications
- Parallelization of 2D and 3D flow solvers
- PhD adviser to Eric Nielsen

#### Harold Atkins

Computational Aerosciences Branch, NASA Langley

- Resident Discontinuous Galerkin expert
- Implemented non-uniform boundary conditions
- Circulation control applications

#### Bill Wood

Aerothermodynamics Branch, NASA Langley

- Developed automated complex-variable form of FUN3D source code
- Pushes team toward more thorough and automated testing practices

#### Austen Duffy

Austin is a NIA visiting researcher from Florida State University investigating code speed up using the Compute Unified Device Architecture (CUDA) parallel programming language and Graphics Processing Units (GPU). He is finishing his Ph.D. in applied and computational mathematics.

#### Clara Helm

Clara is a 2010 LARSS intern who graduated from Clarkson University in 2010 and is heading to Maryland University in the Fall of 2010. She is improving parallel visualization technology in FUN3D.

#### Chris Cordell

Chris is a 2009 LARSS intern from the Georgia Institute of Technology who is using FUN3D to investigate supersonic retropropulsion for atmospheric reentry.

#### Kyle Thompson

Kyle is a 2013 LARSS intern from the NC State who developed the adjoint solver for the generic gas solver.

#### Hicham Alkandry

Hicham is a 2009 and 2010 LARSS intern from the University of Michigan who is applying FUN3D to Orion aftbody hypersonic flows with active RCS jets and supersonic retropropulsion concepts.

#### Julie Andren

Julie is a 2009 LARSS intern from the Massachusetts Institute of Technology who is verifying the accuracy of FUN3D for laminar and turbulent boundary layers with uniformly refined and adapted grids.

#### Shelly Jiang

Shelly is a 2009 LARSS intern from the University of Michigan who is researching active flow control using FUN3D and CFL3D for circulation control of airfoils.

#### Eric Lynch

Eric finished his PhD at Georgia Tech in 2011 under the direction of Prof. Marilyn Smith, funded by the National Science Foundation.

- Implemented the HRLES turbulence model
- Improved performance of the unsteady actuator blade model
- Implemented tight CFD/CSD coupling between FUN3D and Dymore

#### Jennifer Abras

Jennifer completed her PhD at Georgia Tech in 2009 under Prof. Marilyn Smith’s direction, funded by the U.S. Army Vertical Lift Research Center of Excellence.

- Implemented rotor articulation routines
- Added unsteady LMP and artificial compressibility terms for use in unsteady rotorcraft applications
- Teamed with Bob Biedron and Beth Lee-Rausch to add elastic beam coupling for rotorcraft (DYMORE Solver)

#### Nicholas Burgess

Nick is working towards his MSAE at Georgia Tech (2007) under Prof. Marilyn Smith’s direction, funded by the U.S. Army Vertical Lift Research Center of Excellence.

- Implementing advanced turbulence models and transition capabilities

#### Dave O’Brien

Aeromechanics Division, Aviation Engineering Directorate, U.S. Army

Dave finished his PhD dissertation at Georgia Tech in ‘06, under Prof. Marilyn Smith’s direction, funded by the U.S. Army Rotorcraft Center of Excellence.

- Implemented actuator disk for rotorcraft applications
- Hooked FUN3D to DiRTlib and SUGGAR libraries to address rotor-fuselage interaction

#### Tommy Lambert

Tommy is a 2009 LARSS intern from the Carnegie Mellon University who is researching viscous overset meshes for hypersonic flows with FUN3D, SUGGAR, and DiRTLib.

#### Ved Vyas

Ved is a 2009 LARSS intern from the Carnegie Mellon University who is researching automated, grid metric tensor-based grid generation driven by FUN3D.

#### Shatra Reehal

Shatra is a 2007 Undergraduate Student Research Program (USRP) intern from University of Central Florida who worked on hierarchical partitioning schemes for multicore processors.

#### Kan Yang

Kan is a 2007 Undergraduate Student Research Program (USRP) intern from University of Michigan who worked on improving the LDFSS flux linearization for convergence acceleration.

#### Andrew Sweeney

Andrew is a 2007 LARSS intern from George Washington University who worked on a RESTful interface for CFD

#### Brad Neff

Brad is a 2007 LARSS intern from Wofford College who worked on a RESTful interface for CFD

#### Genny Pang

Genny Pang was a fall 2005 USRP intern from UCLA where she was working on her BS in Mechanical Engineering.

- Developed a CAD model for an RCS jet for the MSL aeroshell using Pro/E, generated grids by using GridEx, and tweaked FUN3D so it could run an under-expanded jet in a supersonic crossflow.

#### Gregory Bluvshteyn

Gregory was a 2005 LARSS intern from New York City College of Technology.

- Implemented database portion of continuous build automation system.
- Developed ruby scripts that served as an interface between business layer and presentation layer.

#### Dan Gerstenhaber

Dan was a 2005 LARSS intern from Indiana University.

- Implemented RSS feed portion of continuous build automation system.
- Developed parser for the build logs generated by FUN3D during the past few years.

#### Geoff Parsons

Geoff was a 2005 LARSS intern from Old Dominion University where he was working on his MS in Computer Science.

- Developed an interface to version control systems (both CVS and SVN) for the automated build system.

#### Rena Rudavsky

Rena was a 2005 LARSS intern from Columbia University where she was working on her BS in History and Mechanical Engineering, with an emphasis on fluid mechanics.

- Developed a parametric CAD model for a tethered ballute configuration by using Pro/E, and performed hypersonic computations for Titan-like entry conditions

## 2.4. F95 Coding Standard

### Style

- Free format with no character past column 80
- Indentation: begin in first column and recursively indent all subsequent blocks by two spaces.
- Start all comments within body of code in first column.
- Use all lowercase characters; however, mixed-case should be used in comments and strings.
- Align trailing continuation ampersands within code blocks.
- No tab characters.
- Name
`end`

s.

### Comments

- For cryptic variable names, state description in a comment immediately preceding declaration or on end of the declaration line.
- For subroutines, functions, and modules, insert a contiguous comment block immediately preceding declaration containing a brief overview followed by an optional detailed description.

### Variable Declarations

- Do not use Fortran intrinsic function names.
- Avoid multi-line variable declarations.
- Declare
`intent`

on all dummy arguments. - Declare the kind for all reals, including literal constants, by using a kind definition module.
- Declare
`dimension`

attribute for all non-scalars. - Line up attributes within variable declaration blocks.
- Any scalars used to define extent must be declared prior to use.
- Declare a variable name only once in a scope, including
`use module`

statements.

### Module Headers

- Declare
`implicit none`

. - Include a public character parameter containing the CVS
`\$Id\$`

tag. - Include a
`private`

statement and explicitly declare public attributes.

### Subroutines and Functions

- The first executable line should be
`continue`

. - Use the
`only`

attribute on all`use`

statements. - Keep
`use`

statements local, i.e., not in the module header. - Group all dummy argument declarations first, followed by local variable declarations.
- All subroutines and functions must be contained within a module.
- To avoid null or undefined pointers, pointers passed through an argument list must be allocated.

### Control Constructs

- Name control constructs (e.g.,
`do`

,`if`

,`case`

) which span a significant number of lines or form nested code blocks. - No numbered do-loops.
- Name loops that contain
`cycle`

or`exit`

statements. - Use
`cycle`

or`exit`

rather than`goto`

. - Use case statements with case defaults rather than if-constructs wherever possible.
- Use F90-style relational symbols, e.g.,
`>=`

rather than`.ge.`

.

### Miscellaneous

- In the interest of efficient execution, consider avoiding:
- assumed-shape arrays
- derived types in low-level computationally intensive numerics
`use`

modules for large segments of data

- Remove unused variables.
- Do not use common blocks or includes.
- Name files the same as the module they contain.
- Have only one module per file—-except for
`visit`

, which has a module plus subroutines in the global namespace.

### Illustrative Example

! Define kinds to use for reals in one place module kind_defs implicit none character (len=*), parameter :: kind_defs_cvs_id = & '$Id: coding_standard.txt 62734 2012-08-16 15:19:04Z wawood $' integer, parameter :: sp=selected_real_kind(P=6) ! single precision integer, parameter :: dp=selected_real_kind(P=15) ! double precision end module kind_defs ! A token module for demonstration purposes module some_other_module implicit none character (len=*), parameter :: some_other_module_cvs_id = & '$Id: coding_standard.txt 62734 2012-08-16 15:19:04Z wawood $' integer, parameter :: some_variable = 1 end module some_other_module ! A collection of transformations which includes ! stretches, rotations, and shearing. This comment ! block will be associated with the module declaration ! immediately following. module transformations implicit none character (len=*), parameter :: transformations_module_cvs_id = & '$Id: coding_standard.txt 62734 2012-08-16 15:19:04Z wawood $' contains ! Computes a stretching transformation.\label{comment} ! ! This stretching is accomplished by moving ! things around and going into a lot of other details ! which would be described here and possibly even ! another "paragraph" following this. ! ! This contiguous comment block will be associated with the ! subroutine or function declaration immediately following. ! It is intended to contain an initial section which gives ! a one or two sentence overview followed by one or more ! "paragraphs" which give a more detailed description. subroutine stretch ( points, x, y, z ) use kind_defs use some_other_module, only: a_variable integer, intent(in) :: points ! component to be transformed real(dp), dimension(points), intent(in) :: x, y real(dp), dimension(points), intent(out) :: z ! transformation result external positive integer :: i continue i = 0 if ( x(1) > 0.0_dp ) then call positive ( points, x, y, z ) else do i = 1, points z(i) = x(i)*x(i) + 1.5_dp * ( real(i) + x(i) )**i & + ( y(i) * real(i) ) * ( x(i)**i + 2.0_dp ) & + 2.5_dp * real(i) + 148.2_dp * a_variable enddo endif end subroutine stretch end module transformations

## 2.5. Hypersonic Benchmarks

When presenting results of a computation made with a finite volume, cell-centered algorithm, one must decide whether to present solutions at the dependent variable location (averaged independent variables from cell corners) or at the independent variable location (averaged dependent variables from cell centers). The second approach is used here. All of the LAURA benchmarks use a plotting convention in which the average value of dependent variables at surrounding cell centers are plotted at the independent variable (x,y,z) mesh point location. Boundaries shared by two blocks are also averaged using the same algorithm as interior points. Dependent variables at other boundary points away from corners are injected with the averaged values of the nearest cell centers above the respective mesh point. Dependent variables at corners not shared by multiple blocks are injected from the nearest cell center.

This convention enables smooth contours across block boundaries and exact preservation of grid files. However, this convention also makes shocks appear to be smeared over more mesh points then actually occurs in the solution and may distort the profiles appearance at a boundary where the averaging algorithm changes abruptly. These effects are symptoms of the plotting algorithm – not of the actual solution. Furthermore, in the case of surface quantities, the actual face centered value on the surface that was used in the finite volume flux computation is presented.

### LAURA Algorithm

Laura was created by Peter Gnoffo of the Aerothermodynamics Branch at NASA Langley.

NASA’s interest in viscous, hypersonic flow field simulation has grown in recent years in anticipation of the design needs for space transportation and exploration over the next three decades. Proposed aero-assisted space transfer vehicles will use the upper layers of planetary atmospheres in hypersonic aerobraking maneuvers. Supersonic combustion ramjet engines are being designed to propel vehicles at hypersonic speeds through the Earth’s atmosphere to orbit. Various concepts for a SSTO vehicle are now being considered. The external flow field surrounding such vehicles, as well as the internal flow field through the scramjet engine and nozzle, can be significantly influenced by thermochemical non-equilibrium processes in the flow. Accurate simulations of these phenomena would provide designers valuable information concerning the aerodynamic and aerothermodynamic character of these vehicles.

Two major challenges exist to the simulation of flow fields in thermochemical non-equilibrium around vehicles traveling at hypersonic velocities through the atmosphere. First, these simulations require modeling of the non-equilibrium processes in the flow; these processes frequently occur at energies in which the models currently lack sufficient experimental or analytic validation. Second, because of the large number of unknowns associated with chemical species and energy modes and because of disparate time scales within the flow field, these simulations require algorithmic innovations to maintain numerical stability and fully exploit supercomputer resources.

Non-equilibrium processes occur in a flow when the time required for a process to accommodate itself to local conditions within some region is of the same order as the transit time across the region. The equations and the models used in this manual for non-equilibrium flow have been documented, and they were substantially derived from the work of Park and Lee. Calibration and validation of the physical models intrinsic to this code have been documented. Other code development and calibration programs (e.g. GASP, Candler, Candler and MacCormack, Park and Yoon, Netterfield, and Coquel et.~al) are now in progress within the area of viscous, hypersonic, reacting gas flow field simulations.

Numerical stability is maintained through an implicit treatment of the governing equations. A great variety of implicit treatments is possible. For problems in which only the steady-state solution is required, one is free to evaluate any element of the difference stencil at any iteration (pseudo-time) level which facilitates the relaxation process. In the most rigorous implicit treatment, all variables in all cells are simultaneously solved at an advanced iteration level, thus requiring the solution of a linearized equation set involving (n x I x J x K) equations where n is the number of unknowns at a cell and I, J, and K are the number of computational cells in the three respective coordinate directions. The various forms of factored implicit schemes and line relaxation methods sequentially solve equation sets involving (n x I), (n x J), and/or (n x K) variables. The point-implicit schemes, as utilized in the present work, sequentially solve equation sets involving n simultaneous, linearized equations. Further simplification is possible in chemical kinetic problems by linearizing contributions to the residual from only the source terms to alleviate problems of disparate chemical time scales, thus resulting in methods which involve no matrix operations.

The essence of the point-implicit strategy is to treat the variables at the cell center of interest implicitly at the advanced iteration level and to use the latest available data from neighbor cells in defining the “left-hand-side” numerics. The success of this approach is made possible by the robust stability characteristics of the underlying upwind difference scheme. Even simulations of thermochemical non-equilibrium flows in a near-equilibrium state can be handled by this approach. The algorithm requires only a single pseudo-time level of storage and is efficiently implemented on vector or parallel processors. Details of the relaxation algorithm, including effects of a gas in thermal and chemical non-equilibrium, are presented herein.

As noted above, there is no requirement to synchronize the evolution of the solution at neighboring points in the single-level-storage point-implicit relaxation strategy. Consequently, algorithm parallelization can be implemented on a subroutine level across several domains without the need to synchronize tasks or restrict parallel code to a “do loop” level. Scalar code and conditional logic do not inhibit parallel efficiency. Dynamic allocation of resources to domains that are slow to converge is enabled in this environment. These capabilities are exploited on CRAY class computers and are discussed in greater detail within this manual.

The code and the user interface are structured to make liberal use of Fortran “include” statements that tailor the resource requirements for each case to a minimum. System requirements vary from standard workstations for many perfect-gas applications to 128 Mw(megaword) in-core memory, 128 Mw of “fast disk” (SSD) memory, and more than 100 CPU hours to obtain a converged solution on a YMP for thermochemical non-equilibrium flow (seven species) over the Space Shuttle with the thin-layer Navier-Stokes equations using a grid of 150×109 x 60.

### Cylinder

Benchmarks in this section examine the hypersonic, viscous flow over a cylinder at a single freestream condition using two thermochemical models: calorically and thermally perfect air and a five-species air model. For the perfect gas case LAURA results are also compared with FUN2D results. These test cases document run time, memory requirements, convergence characteristics, shock-layer/boundary-layer profiles, and surface distributions. All cases were initialized with uniform flow at freestream conditions. The grids used for the runs are available.

#### Grids

A structured grid was constructed for this case using the self-start capability within LAURA. The align-shock option within LAURA was employed to align the outer (inflow) boundary with the captured bow shock and also optimize the distribution of points in the near-wall (boundary-layer) region. A separate grid was generated for the perfect gas case and the 5-species air case since the shock stand-off distance varies considerably between the two. For each case, the adapted grid was used as the starting point for a flow field initialized to freestream conditions, i.e., the grid-forming runs were discarded.

For the 1-meter radius cylinder, the z-axis originates from the stagnation point and is normal to the body, in a direction opposed to the oncoming flow (w_inf = -1) while the x-axis is perpendicular to the z-axis and the solution is generated in the y=0 plane.

The structured grids have 64 cells (65 points) normal to the body. There are 30 equally spaced cells (31 points) along the semi circle from the stagnation point to 90 deg around the cylinder, yielding 60 equally spaced cells on the complete semi-circle forebody surface.

Note: while this flow could have been computed using only half of this domain, this is the default grid topology generated by LAURA. In addition, by using the full domain, solution contamination associated with an axis-singularity boundary condition are avoided.

An unstructured grid was obtained from the structured grid by
merely bisecting each of the quadrilaterals using the program
`p3d2fun.f`

(available upon request).
No attempt was made to alternate the diagonal directions or
produce a symmetrical grid.

Download `cylinder_pg.g`

: Plot3D grid file
used for LAURA perfect gas solution using unformatted (Fortran),
multi-block, 3d-whole options. [ieee binary]

Download `cylinder_pg.fun`

: FUN2D grid
file used for the FUN2D perfect gas solution. [ASCII text]

Download `cylinder_5s.g`

: Plot3D grid file
used for LAURA 5-species air solution using unformatted (Fortran),
multi-block, 3d-whole options. [ieee binary]

#### Flow Conditions

The freestream conditions are as follows:

V_inf: | 5000 m/s |

rho_inf: | .001 kg/m^{3^ } |

T_inf: | 200 K |

T_wall: | 500 K |

Mach Number: | 17.605 |

Reynolds Number: | 376,930 /m |

#### LAURA Modus Operandi

Both cases were run using both point-implicit and line-implicit (new to LAURA.4.7.1) relaxation strategies. In the point-implicit strategy, the solution is marched in alternating k-directions (normal to body) using latest available data from neighboring k-planes (a la Gauss-Seidel). All equations at a point are implicitly coupled using a Jacobian matrix (e.g., a 5×5 matrix for a 3D perfect gas case), enabling unlimited Courant number. The line-implicit strategy fully couples all cells along k-lines using block tri-diagonal relaxation while marching in alternating i-directions (along the body). Convergence histories versus iteration count and CPU time (SGI INDIGO2 R10000) are available for both cases.

After the initial grid was created as a completely separate step described in the Grid section, the flow was initialized to freestream conditions on the aligned grid.

The line-implicit strategy was not sufficiently robust to work from a “cold” start (uniform flow initial conditions). Consequently, the point-implicit strategy was used for the first 500 relaxation sweeps before continuing with either the point-implicit strategy or the line-implicit strategy.

For the first 500 iterations LAURA was run with default values with the exception of using second-order spatial accuracy (the default is first-order) and freezing the flux Jacobians for 10 iterations at a time (the default is unfrozen). During these first 500 iterations, the grid automatically doubles, and then doubles again to the full 64 cells normal to the body according to the value of the L_2 error norm.

For the next 2500 iterations (for a total of 3000 iterations), the code was run with point-implicit relaxation and with line-implicit relaxation to compare convergence rates of the two strategies.

#### FUN2D Modus Operandi

FUN2D was run using the following options/modifications:

- Roe’s Flux Difference Splitting
- Eigenvalue smoothing a la LAURA [added]
- Venkat’s flux limiter (K=0, e.g., minmod)
- Hard-wired extrapolation boundary condition [added]

The code was run with first-order reconstruction for the fluxes for the first 1000 iterations ramping the CFL number from 1 to 10 in the first 100 iteration. Next, second-order fluxes were invoked while the CFL number remained at 10. Due to the minmod flux limiting which is active even in smooth regions of the flow, the L_2 error norm “hangs” after only a few orders of magnitude reduction. Convergence was judged by monitoring skin friction.

#### Perfect Gas Air Numerical Results

`laura`

executable: 3.4 Mb

`fun2d`

executable: 9.5 Mb

##### Convergence Histories

##### Contour Plots

##### Stagnation Streamline Plots

##### Boundary Layer Profiles at Stagnation Point

##### Boundary Layer Velocity Profiles

##### Surface Plots

#### 5-Species Air Numerical Results

`laura`

executable: 6.7 Mb

The bow shock sits closer to the body because heat of formation of non-equilibrium gas constituents takes up a fraction of the freestream kinetic energy, effectively cooling the shock layer and raising its density.

##### Convergence Histories

##### Contour Plots

##### Stagnation Streamline Plots

##### Boundary Layer Profiles at Stagnation Point

##### Surface Plots

### Sphere

Benchmarks in this section examine the hypersonic, viscous flow over a sphere at a single freestream condition using various thermochemical models. These test cases document run time, memory requirements, convergence characteristics, shock-layer and boundary-layer profiles, and surface distributions. A grid file is available for downloading. All cases were initialized with uniform flow at freestream conditions unless otherwise noted.

#### Grid

The grid was constructed using the self-start capability in LAURA. The outer (inflow) boundary was aligned with the captured bow shock and the near wall distribution was adapted using the align-shock option in LAURA. The grid was adapted to the perfect gas case which provides for the largest shock standoff distance and enables the same grid to be used for all test cases. There are 30 equally spaced cells (31 points) along the body from the stagnation point to 90 deg around the sphere. The finest grid has 128 cells (129 points) normal to the body. Axisymmetric and two-dimensional flow in LAURA is accommodated using three-dimensional cells in which the side wall boundary conditions are defined using the appropriate constant or periodic specifications of dependent variables. In the case of two-dimensional flow, side walls are parallel. In the case of axisymmetric flow, side walls form a 5 deg wedge emanating from the axis when viewed from above the stagnation point. The z-axis originates from the stagnation point and is normal to the body, in a direction opposed to the oncoming flow (w_inf = -1). The x-axis is in the radial direction from the axis of symmetry. The solution is generated in the y =0 plane.

Download `sphere3.g`

: Plot3D grid file used for
all LAURA solutions using unformatted, multi-block, 3d-whole options.

Download `readgrid.f`

: Fortran 77 source code
to read `sphere3.g`

and write `sphere2.g`

: a two-dimensional cut through the
y=0 plane.

#### Flow Conditions

V_inf | 5000 m/s |

rho_inf | .001 kg/m3 |

T_inf | 200 K |

T_wall | 500 K |

Mach Number | 17.6 |

Reynolds Number | 376,930 / m |

#### Perfect Gas

These cases were converged using both the point-implicit and line-implicit (New to LAURA.4.7.1) relaxation strategies. In the point-implicit strategy, the solution is marched in alternating k-directions (normal to body) using latest available data (Gauss – Seidel) from neighboring k-planes. All equations at a point are implicitly coupled using a 5×5 Jacobian matrix (for 3D perfect gas), enabling unlimited Courant number. The line-implicit strategy fully couples all cells along k-lines using block tri-diagonal relaxation while marching in alternating i-directions (around body). Convergence histories versus iteration count and CPU time (SGI INDIGO2 R10000) are available for all cases. Details of profiles from the coarse and baseline grids are presented in the fine grid section where results from three grid levels are compared.

##### Coarse Grid – 30×32

Memory required for `laura`

executable: 1.2 Mb

The line-implicit strategy was not sufficiently robust to work from a “cold” start (uniform flow initial conditions). Consequently, the point-implicit strategy was used for the first 500 relaxation sweeps (25 CPU s) before continuing with the line-implicit strategy.

Convergence History:

Contour Plot:

Surface Distributions with Comparison to Other Grid Results:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Shock Layer Profiles Across Stagnation Streamline with Comparison to Other Grid Results:

- View detail temperature profile
- View pressure profile
- View detail pressure profile
- View density profile
- View detail density profile

Boundary Layer Profiles at Stagnation Point with Comparison to Other Grid Results:

##### Baseline Grid – 30×64

Memory required for `laura`

executable: 2.1 Mb

This case was initialized by injecting the 30×32 cell coarse grid solution
into the 30×64 cell file. Convergence history shows point- and line
implicit results but only the line-implicit solutions with error norm less
than 10^{(-9) are presented.}

Convergence History:

Contour Plot:

Surface Distributions with Comparison to Other Grid Results:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Shock Layer Profiles Across Stagnation Streamline with Comparison to Other Grid Results:

- View detail temperature profile
- View pressure profile
- View detail pressure profile
- View density profile
- View detail density profile

Boundary Layer Profiles at Stagnation Point with Comparison to Other Grid Results:

##### Perfect Gas (Fine Grid – 30×128)

Memory required for `laura`

executable: 3.8 Mb

Memory required for `laura`

executable: 2.4 Mb (Point-implicit option
only—Does not include memory for off-diagonal Jacobians)

This case was initialized by injecting the 30×64 baseline grid solution into the 30×128 cell file. Convergence history shows point- and line implicit results but only the line-implicit solutions with error norm less than 10(-9) are presented.

Convergence History:

Contour Plot:

Surface Distributions with Comparison to Other Grid Results:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Shock Layer Profiles Across Stagnation Streamline with Comparison to Other Grid Results:

- View detail temperature profile
- View pressure profile
- View detail pressure profile
- View density profile
- View detail density profile

Boundary Layer Profiles at Stagnation Point with Comparison to Other Grid Results:

Comparison of Point-Implicit Solution after 950 s with Line-Implicit Solution after 650 s (fully converged):

- View surface pressure distribution
- View surface heating distribution
- View skin friction coefficient distribution

##### Equilibrium Air (Baseline Grid – 30×64)

Memory required for `laura`

executable: 2.5 Mb

This case was initialized from freestream conditions. Point-implicit relaxation was used for the first 500 iterations because the line-implicit method is not sufficiently robust to work from cold start. However, line-implicit was used to get faster convergence once the shock layer structure started to take shape. Two sets of curve fits for thermodynamic properties referred to as Tannehill and Vinokur are tested. The Vinokur solution started from a converged Tannehill solution. The bow shock sits closer to the body because heat of formation of equilibrium gas constituents takes up a fraction of the freestream kinetic energy, effectively cooling the shock layer and raising its density. A slight oscillation in pressure at the boundary-layer edge is evident in both solutions. It appears to be related to the behavior of the heat capacity of the gas at conditions encountered at the boundary-layer edge which is also manifested in the plot of effective gamma. Effective gamma is defined here as the equilibrium sound speed squared multiplied by the ratio of local density to pressure.

Convergence History:

Contour Plot:

Surface Distributions:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Shock Layer Profiles Across Stagnation Streamline:

- View detail temperature profile
- View pressure profile
- View detail pressure profile
- View density profile
- View detail density profile
- View Mach number profile
- View effective gamma profile

Boundary Layer Profiles at Stagnation Point:

- View pressure profile
- View temperature profile
- View density profile
- View Mach number profile
- View effective gamma profile

##### 7-Species Non-equilibrium Air (Baseline Grid – 30×64)

Memory required for `laura`

executable: 5.6 Mb

This case was initialized from freestream conditions. Point-implicit relaxation was used for the first 1500 iterations because the line-implicit method is not sufficiently robust to work from cold start. However, line-implicit was used to get faster convergence once the shock layer structure started to take shape. The bow shock sits closer to the body because heat of formation of non-equilibrium gas constituents takes up a fraction of the freestream kinetic energy, effectively cooling the shock layer and raising its density. Still, the shock standoff distance is slightly larger than for the equilibrium air case because full equilibration of all 7 species has not occurred. A slight oscillation in pressure at the boundary-layer edge in the equilibrium case is not evident in the non-equilibrium case. The abrupt oscillation observed in the effective gamma for equilibrium does not manifest itself in the non-equilibrium case because of longer accommodation time. Only a slight change in curvature in the non-equilibrium distribution is observed. Effective gamma is defined here in two ways; (1) as the frozen sound speed squared multiplied by the ratio of local density to pressure, and (2) as the ratio of enthalpy to internal energy. These quantities are compared to the corresponding definition of effective gamma for equilibrium flow, based on equilibrium sound speed. Differences in the sound speed arise from the differences in the way pressure and energy perturbations are accommodated in the gas. The LAURA code uses frozen sound speed as a reference because that is the quantity which is derived from the eigenvalues of the flux Jacobian.

Convergence History:

Contour Plot:

Surface Distributions:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Shock Layer Profiles Across Stagnation Streamline:

- View detail temperature profiles
- View pressure profile
- View detail density profile
- View Mach number profile
- View effective gamma profile
- Mass fraction profiles
- Mass fraction profiles
- Electron number density profile

Boundary Layer Profiles at Stagnation Point:

- View pressure profile
- View temperature profile
- View density profile
- View Mach number profile
- View effective gamma profile

##### 7-Species Non-equilibrium Air—Non-Catalytic Wall (Baseline Grid – 30×64)

This case was initialized from the converged, fully catalytic case described above. Point-implicit relaxation was used for the entire run. The implicit boundary condition for the line-implicit solver is not sufficiently robust. The zero gradient condition for mole fraction converges slowly – even using line relaxation. Note the slow asymptotic convergence of the heating rate still evident when the error norms are of order 10**-7. In the non-catalytic case, the atomic oxygen mass fraction is nearly constant across the boundary layer. The atomic nitrogen still shows significant reduction from boundary-layer edge values. Shuffle reactions in the cooling layer between N and O2 to form NO and O and between N and NO to form O and N2 tend to deplete atomic nitrogen and promote production of atomic oxygen and nitric oxide. Three body collisions required to deplete atomic oxygen are relatively slow. In the fully catalytic case recombination of atomic oxygen at the surface raises the surface heating rate relative to the non-catalytic case. The energy flux is associated with the heat of formation of atomic oxygen which is released back into the system on recombination. The source of atomic oxygen to the surface in the fully catalytic case is driven by diffusion of O from the boundary-layer edge.

Convergence History:

- View convergence history versus iteration count
- View convergence history versus CPU time
- View convergence history of stagnation point heating versus CPU time

Surface Distributions:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Boundary-Layer Profiles Across Stagnation Streamline:

- Mass fraction profiles
- Mass fraction profiles
- Electron number density profile
- View pressure profile
- View temperature profile
- View density profile
- View effective gamma profile

##### 7-Species Non-equilibrium Air—Dunn and Kang Kinetics (Baseline Grid – 30×64)

This case was initialized from the converged, fully catalytic case described previously. Line-implicit relaxation was used for the entire run. This option is engaged by LOCALIZING the file “gas_model_vars.strt” and setting “kmodel = 5”. Differences between the baseline kinetic model of Park (“kmodel = 3”) and this one are very small for these test conditions, with modest changes in temperature and species mass fractions across the inviscid portion of the shock layer.

Convergence History:

Surface Distributions:

- View surface pressure distribution
- View surface heating distribution
- View surface skin friction coefficient distribution

Profiles Across Stagnation Streamline:

- Mass fraction profiles
- Mass fraction profiles
- Electron number density profile
- View pressure profile
- View density profile
- View temperature profile
- Boundary layer density profile
- Boundary layer temperature profile

##### 5-Species Non-equilibrium Air—Park Kinetics (Baseline Grid – 30×64)

Memory required for `laura`

executable: 4.4 Mb

This case was initialized from the converged, fully catalytic case described previously. The subroutine blkout.F was modified to save only the first 5 species of the 7-species solution. The deletion of NO+ and electrons is the only difference with the baseline, non-equilibrium solution. Line-implicit relaxation was used for the entire run. Differences between the 5- and 7-species model can only be seen in trace species concentration; the energy taken up by ionization is small. The 7-species model would only be needed if electron number density were required.

Convergence History:

Surface Distributions:

Profiles Across Stagnation Streamline:

## 2.6. Obsolete Online Manual

During the FUN3D 12.4 release cycle (summer of 2014) the on-line manual was removed now that a PDF version of the manual has been formally released as a NASA TM. While the use of older versions of FUN3D is strongly discouraged, PDF versions of the on-line manual are provided for reference.

( Obsolete Chapter 2: Installation )

( Obsolete Chapter 3: Grid Generation )

( Obsolete Chapter 4: Boundary Conditions )

( Obsolete Chapter 5: Pre- and Post-Processing )

( Obsolete Chapter 6: Analysis )

Today's NASA Official:
Peter Gnoffo, a member of
The FUN3D Development Team

Contact: FUN3D-support@lists.nasa.gov

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