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

Bibliography

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

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.

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

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.

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.)

Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation, W. Kyle. Anderson, and V. Venkatakrishnan, AIAA 97-0643, January, 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 on Unstructured Grids for Turbulent Flows, W. Kyle Anderson, Daryl L. Bonhaus, NASA Technical Memorandum 112867, June, 1997.

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

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.

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.)

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.

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.

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.

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.

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

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.

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).

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.

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.

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.

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.

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

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.

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

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

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

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

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

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

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.

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

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.

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)

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.

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.

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.

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.

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

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

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.

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.

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. See also AIAA-2005-0491, January 2005.

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

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.

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

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.

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

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

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

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.

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

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

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

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.

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

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.)

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.)

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

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

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

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

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

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.

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

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

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.

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.

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

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.

A Critical Study of Agglomerated Multigrid Methods for Diffusion, Hiroaki Nishikawa, Boris Diskin, and James L. Thomas, AIAA-2009-4138, 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.

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.

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.

Abstracts and Documents

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.

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.

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 Upwind Multigrid Method for Solving Viscous Flows on Unstructured Triangular Meshes (0.9 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.

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.

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.

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 rela tionship 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 sen sitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

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 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.

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.

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.

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.

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.

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.

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.

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.

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 discretiza-tions. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Aerodynamic Design Optimization Using the Navier-Stokes Equations (10.9 MB PDF)

Much effort has recently focused on developing a gradient-based design optimization capability based on the Navier-Stokes equations. The presentation will introduce the fundamental components necessary to achieve such a tool and the inherent difficulties that arise with each. The unstructured-grid analysis code that forms the foundation of the current work is described, followed by a discussion on a discrete adjoint-based approach to sensitivity analysis and grid adaptation. Several mesh movement schemes will also be presented.

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.

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 increase in 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.

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.

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.

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.

Ongoing Research Into Numerical Simulation of Fluid Flows Utilizing Software Development Practices (1.7 MB PDF Presentation, 78 KB PDF Handout)

How is software complexity managed when the required infrastructure is increasing? CFD packages are continuing to evolve into more and more complex systems to handle more classes of problems. They require larger teams to assemble and maintain. One way to address this complexity is with modern programming practices.

How is discretization error impacting the solution? Local error estimates for the discretization error have been used to describe where increased grid resolution is required to improve a solution. These methods have missed the connection between the impact of local errors on global output quantities and how these local errors are transported. The adjoint solution provides the critical connection between local errors and global outputs as well as how errors are transported.

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.

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.

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.

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.

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.

Adjoint-Based Algorithms for Adaptation and Design Optimization on Unstructured Grids (3.2 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 nonexistent in today’s large-scale CFD simulations. An overview of adjoint-based algorithm work at NASA Langley will be presented, with examples demonstrating its potential impact on complex computational problems related to design optimization as well as mesh adaptation.

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.

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.

Investigation of Effect of Dynamic Stall and Its Alleviation on Helicopter Performance and Loads

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.

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.

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.

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.

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 accomodate 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.

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 numb er 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.

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.

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 imp ortance 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.

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.

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 1000­1 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

A Critical Study of Agglomerated Multigrid Methods for Diffusion (1.7 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 highly 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. Actual cycle results are verified using quantitative analysis methods in which parts of the cycle are replaced by their idealized counterparts.

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.

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.

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.

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-Renee Carlson

Computational Aerosciences Branch, NASA Langley

• Implementation and solution verification of jet flows.
• Arbitrary geometry flowfield data extraction and probing.
• Theory manual production.

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.

Mike Park

Computational Aerosciences Branch, NASA Langley

• Implemented MPI communication infrastructure
• 3D adjoint-based error estimation and distributed grid adaptation connected directly to CAD
• Currently pursuing MIT Ph.D. on viscous cut-cell methods

Chris Rumsey

Computational Aerosciences Branch, NASA Langley

• Transonic turbulence modeling.
• CFL3D developer/maintainer.
• Resident CGNS expert.

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

Jennifer Abras

Jennifer is working towards her PhD at Georgia Tech (2007) 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

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

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.

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

Hicham Alkandry

Hicham is a 2009 LARSS intern from the University of Michigan who is applying FUN3D to Orion aftbody hypersonic flows with active RCS jets.

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.

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

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.

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 24222 2007-04-10 15:01:41Z kleb $'

  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 24222 2007-04-10 15:01:41Z kleb $'

  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 24222 2007-04-10 15:01:41Z kleb $'

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

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.

LAURA User’s Manual

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/m3^
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

• as a function of iteration count
• as a function of CPU time

Contour Plots

• Temperature
• Pressure

Stagnation Streamline Plots

• Temperature
• Pressure
• Density

Boundary Layer Profiles at Stagnation Point

• Temperature
• Density

Boundary Layer Velocity Profiles

• 15 Deg station
• 15 Deg station
• 45 Deg station
• 45 Deg station

Surface Plots

• Pressure coefficient
• Heating rate
• Skin friction

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

• as a function of iteration count
• as a function of CPU time

Contour Plots

• Temperature
• Pressure

Stagnation Streamline Plots

• Temperature
• Pressure
• Density
• Effective gamma
• Mass fractions
• Mass fractions

Boundary Layer Profiles at Stagnation Point

• Temperature
• Density
• Pressure
• Mach Number
• Effective Gamma

Surface Plots

• Pressure
• Heating rate
• Skin friction

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Contour Plot:

• View temperature contours

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:

• View temperature profile
• View density profile

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Contour Plot:

• View temperature contours

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:

• View temperature profile
• View density profile

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Contour Plot:

• View temperature contours

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:

• View temperature profile
• View density profile

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Contour Plot:

• View temperature contours Tannehill model

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Contour Plot:

• View Translational-rotational temperature contours
• View Vibrational-electronic temperature contours

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

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:

• View convergence history versus iteration count
• View convergence history versus CPU time

Surface Distributions:

• View surface heating distribution

Profiles Across Stagnation Streamline:

• Mass fraction profiles
• Mass fraction profiles