Aerodynamic Drag Prediction Using Unstructured Mesh Solvers

1. VKI Lecture Series, February 3-7, 2003 Aerodynamic Drag Prediction Using Unstructured Mesh Solvers Dimitri J. Mavriplis National Institute of Aerospace Hampton, Virginia, USA

2. VKI Lecture Series, February 3-7, 2003 Overview Introduction Physical model fidelity Grid resolution and discretization issues Designing an efficient unstructured mesh solver for computational aerodynamics Drag prediction using unstructured mesh solvers Conclusions and future work

3. VKI Lecture Series, February 3-7, 2003 Overview Introduction Importance of Drag Prediction Suitability of Unstructured Mesh Approach Physical model fidelity Inviscid Flow Analysis Coupled Inviscid-Viscous Methods Large-Eddy Simulations (LES and DES)

4. VKI Lecture Series, February 3-7, 2003 Overview Grid resolution and discretization issues Choice of discretization and effect of dissipation Cell centered vs. vertex based Effect of discretization variations on drag prediction Grid resolution requirements Choice of element type Grid resolution issues Grid convergence

5. VKI Lecture Series, February 3-7, 2003 Overview Designing an efficient unstructured mesh solver for computational aerodynamics Discretization Solution Methodologies Efficient Hardware Usage

6. VKI Lecture Series, February 3-7, 2003 Overview Drag prediction using unstructured mesh solvers Wing-body cruise drag Incremental effects: engine installation drag High-lift flows Conclusions and Future Work

7. VKI Lecture Series, February 3-7, 2003 Introduction Importance of Drag Prediction Cruise: fuel burn, range, etc… High-lift: Mechanical simplicity, noise High accuracy requirements Absolute or incremental: 1 drag count Specialized computational methods Wide range of scales Thin boundary layers Transition

8. VKI Lecture Series, February 3-7, 2003 Introduction Issues centric to unstructured mesh approach Advantages and drawbacks over other approaches Accuracy, efficiency State-of-the art in aerodynamic predictions De-emphasize non-method specific issues Validation/ verification Drag integration

9. CFD Perspective on Meshing Technology Sophisticated Multiblock Structured Grid Techniques for Complex Geometries

10. CFD Perspective on Meshing Technology Sophisticated Overlapping Structured Grid Techniques for Complex Geometries

11. VKI Lecture Series, February 3-7, 2003 Unstructured Grid Alternative Connectivity stored explicitly Single Homogeneous Data Structure

12. VKI Lecture Series, February 3-7, 2003 Characteristics of Both Approaches Structured Grids Logically rectangular Support dimensional splitting algorithms Banded matrices Blocked or overlapped for complex geometries Unstructured grids Lists of cell connectivity, graphs (edge,vertices) Alternate discretizations/solution strategies Sparse Matrices Complex Geometries, Adaptive Meshing More Efficient Parallelization

13. VKI Lecture Series, February 3-7, 2003 Unstructured Meshes for Aerodynamics Computational aerodynamics rooted in structured methods High accuracy and efficiency requirements Unstructured mesh methods 2 to 4 times more costly Mitigated by extra structured grid overhead Block structured Overset mesh Parallelization Accuracy considerations Validation studies, experience Unstructured mesh solvers potentially more efficient than structured mesh alternatives with equivalent accuracy

14. VKI Lecture Series, February 3-7, 2003 Physical Model Fidelity State-of-the-art in drag prediction: RANS Entire suite of tools available to designer Useful to examine capabilities of other tools Lower fidelity – lower costs Numerous rapid tradeoff studies Higher fidelity – higher costs Fewer detailed analyses Situate RANS tools within this suite

15. VKI Lecture Series, February 3-7, 2003 Physical Model Requirements(Unstructured Mesh Methods)

16. VKI Lecture Series, February 3-7, 2003 Unstructured Mesh Euler Solvers Inviscid flow unstructured mesh solvers well established – robust No viscous effects No turbulence/transition modeling Isotropic meshes Good commercial isoptropic mesh generators Good convergence properties

17. VKI Lecture Series, February 3-7, 2003 Example: Euler Solution of DLR-F4 Wing-body Configuration 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS

18. VKI Lecture Series, February 3-7, 2003 Example: Euler Solution of DLR-F4 Wing-body Configuration 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS

19. VKI Lecture Series, February 3-7, 2003 Example: Euler Solution of DLR-F4 Wing-body Configuration 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS 1.65 million vertices

20. VKI Lecture Series, February 3-7, 2003 Euler vs. RANS Solution 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS

21. VKI Lecture Series, February 3-7, 2003 Euler vs. RANS Solution Exclusion of viscous effects Boundary layer displacement Incorrect shock location Incorrect shock strength Supercritical wing sensitive to viscous effects Euler solution not useful for transonic cruise drag prediction

22. VKI Lecture Series, February 3-7, 2003 Coupled Euler-Boundary Layer Approach Incorporate viscous effects to first order Boundary layer displacement thickness More accurate shock strength/location Retain efficiency of Euler solution approach Isotropic tetrahedral meshes Fast, robust convergence

23. VKI Lecture Series, February 3-7, 2003 Coupled Euler-Boundary Layer Approach Stripwise 2-dimensional boundary layer 18 stations on wing alone Interpolate from unstructured surface mesh Transpiration condition for simulated BL displacement thickness

24. VKI Lecture Series, February 3-7, 2003 Euler vs. RANS Solution 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS

25. VKI Lecture Series, February 3-7, 2003 Euler-IBL vs. RANS Solution 235,000 vertex mesh (ICEMCFD tetra) Fully tetrahedral mesh Convergence in 50 cycles (multigrid) 3 minutes on 8 Pentiums 50 times faster than RANS

26. VKI Lecture Series, February 3-7, 2003 Coupled Euler-Boundary Layer Approach

27. VKI Lecture Series, February 3-7, 2003 Coupled Euler-Boundary-Layer Approach Vastly improved over Euler alone Correct shock strength, location Accurate lift Reasonable drag More sophisticated coupling possible 25 times faster than RANS Neglibible IBL compute time Convergence dominated by coupling Parameter studies Design optimization

28. VKI Lecture Series, February 3-7, 2003 LES and DES Methods RANS failures for separated flows Good cruise design involves minimal separation Off design, high-lift LES or DES as alternative to turbulence modeling inadequacies LES: compute all scales down to inertial range Based on universality of inertial range DES: hybrid LES/RANS (near wall) Reduced cost

29. VKI Lecture Series, February 3-7, 2003 LES and DES: Notable Successes European LESFOIL program Marie and Sagaut: LES about airfoil near stall DES for massively separated aerodynamic flows Strelets 2001, Forsythe 2000, 2001, 2003 Two to ? Orders of magnitude more expensive than RANS Predictive ability for accurate drag not established RANS methods state-of-art for foreseeable future

30. VKI Lecture Series, February 3-7, 2003 Grid Resolution and Discretization Issues Choice of discretization and effect of dissipation (intricately linked) Cells versus points Discretization formulations Grid resolution requirements Choice of element type Grid resolution issues Grid convergence

31. VKI Lecture Series, February 3-7, 2003 Cell Centered vs Vertex-Based Tetrahedral Mesh contains 5 to 6 times more cells than vertices Hexahedral meshes contain same number of cells and vertices (excluding boundary effects) Prismatic meshes: cells = 2X vertices Tetrahedral cells : 4 neighbors Vertices: 20 to 30 neighbors on average

32. VKI Lecture Series, February 3-7, 2003 Cell Centered vs Vertex-Based On given mesh: Cell centered discretization: Higher accuracy Vertex discretization: Lower cost Equivalent Accuracy-Cost Comparisons Difficult Often based on equivalent numbers of surface unknowns (2:1 for tet meshes) Levy (1999) Yields advantage for vertex-discretization

33. VKI Lecture Series, February 3-7, 2003 Cell Centered vs Vertex-Based Both approaches have advantages/drawbacks Methods require substantially different grid resolutions for similar accuracy Factor 2 to 4 possible in grid requirements Important for CFD practitioner to understand these implications

34. VKI Lecture Series, February 3-7, 2003 Example: DLR-F4 Wing-body (AIAA Drag Prediction Workshop)

35. VKI Lecture Series, February 3-7, 2003 Illustrative Example: DLR-F4 NSU3D: vertex-based discretization Grid : 48K boundary pts, 1.65M pts (9.6M cells) USM3D: cell-centered discretization Grid : 50K boundary cells, 2.4M cells (414K pts) Uses wall functions NSU3D: on cell centered type grid Grid: 46K boundary cells, 2.7M cells (470K pts)

36. VKI Lecture Series, February 3-7, 2003 Cell versus Vertex Discretizations Similar Lift for both codes on cell-centered grid Baseline NSU3D (finer vertex grid) has lower lift

37. VKI Lecture Series, February 3-7, 2003 Cell versus Vertex Discretizations Pressure drag Wall treatment discrepancies NSU3D : cell centered grid High drag, (10 to 20 counts) Grid too coarse for NSU3D Inexpensive computation USM3D on cell-centered grid closer to NSU3D on vertex grid

38. VKI Lecture Series, February 3-7, 2003 Grid Resolution and Discretization Issues Choice of discretization and effect of dissipation (intricately linked) Cells versus points Discretization formulations Grid resolution requirements Choice of element type Grid resolution issues Grid convergence

39. VKI Lecture Series, February 3-7, 2003 Discretization Governing Equations: Reynolds Averaged Navier-Stokes Equations Conservation of Mass, Momentum and Energy Single Equation turbulence model (Spalart-Allmaras) Convection-Diffusion – Production Vertex-Based Discretization 2nd order upwind finite-volume scheme 6 variables per grid point Flow equations fully coupled (5x5) Turbulence equation uncoupled

40. VKI Lecture Series, February 3-7, 2003 Spatial Discretization Mixed Element Meshes Tetrahedra, Prisms, Pyramids, Hexahedra Control Volume Based on Median Duals Fluxes based on edges Single edge-based data-structure represents all element types

41. Upwind Discretization

42. VKI Lecture Series, February 3-7, 2003 Matrix Artificial Dissipation

43. VKI Lecture Series, February 3-7, 2003 Entropy Fix L matrix: diagonal with eigenvalues: u, u, u, u+c, u-c Robustness issues related to vanishing eigenvalues Limit smallest eigenvalues as fraction of largest eigenvalue: |u| + c u = sign(u) * max(|u|, d(|u|+c)) u+c = sign(u+c) * max(|u+c|, d(|u|+c)) u – c = sign(u -c) * max(|u-c|, d(|u|+c))

44. VKI Lecture Series, February 3-7, 2003 Entropy Fix u = sign(u) * max(|u|, d(|u|+c)) u+c = sign(u+c) * max(|u+c|, d(|u|+c)) u – c = sign(u -c) * max(|u-c|, d(|u|+c)) d = 0.1 : typical value for enhanced robustness d = 1.0 : Scalar dissipation - L becomes scaled identity matrix T |L| T-1 becomes scalar quantity Simplified (lower cost) dissipation operator Applicable to upwind and art. dissipation schemes

45. VKI Lecture Series, February 3-7, 2003 Discretization Formulations Examine effect of discretization type and parameter variations on drag prediction Effect on drag polars for DLR-F4: Matrix artificial dissipation Dissipation levels Entropy fix Low order blending Upwind schemes Gradient reconstruction Entropy fix Limiters

46. Effect of Artificial Dissipation Level Increased accuracy through lower dissipation coef. Potential loss of robustness

47. Effect of Entropy Fix for Artificial Dissipation Scheme Insensitive to small values of d=0.1, 0.2 High drag values for large d and scalar scheme

48. VKI Lecture Series, February 3-7, 2003 Effect of Artificial Dissipation

49. Effect of Low-Order Dissipation Blending for Shock Capturing Lift and drag relatively insensitive Generally not recommended for transonics

50. Comparison of Discretization Formulation (Art. Dissip vs. Grad. Rec.) Least squares approach slightly more diffusive Extremely sensitive to entropy fix value

51. Effect of Limiters on Upwind Discretization Limiters reduces accuracy, increase robustness Less sensitive to non-monotone limiters

52. VKI Lecture Series, February 3-7, 2003 Effect of Discretization Type

53. Effect of Element Type Right angle tetrahedra produced in boundary layer regions Highly stretched elements for efficiency Non obtuse angle requirement for accuracy Semi-structured tetrahedra combinable into prisms Prism elements of lower complexity (fewer edges) No significant accuracy benefit (Aftosmis et. Al. 1994 in 2D)

54. Effect of Element Type in BL Region Little overall effect on accuracy Potential differences between two codes

55. VKI Lecture Series, February 3-7, 2003 Grid Resolution Issues Possibly greatest impediment to reliable RANS drag prediction Promise of adaptive meshing held back by development of adequate error estimators Unstructured mesh requirement similar to structured mesh requirements 200 to 500 vertices chordwise (cruise) Lower optimal spanwise resolution

56. VKI Lecture Series, February 3-7, 2003 Illustration of Spanwise Stretching (VGRIDns, c/o S. Pirzadeh, NASA Langley) Factor of 3 savings in grid size

57. Effect of Normal Spacing in BL Inadequate resolution under-predicts skin friction Direct influence on drag prediction

58. Effect of Normal Resolution for High-Lift (c/o Anderson et. AIAA J. Aircraft, 1995) Indirect influence on drag prediction Easily mistaken for poor flow physics modeling

59. Grid Convergence (2D Euler) Lift converges as h2 Drag vanishes in continuous limit

60. VKI Lecture Series, February 3-7, 2003 Grid Convergence Seldom achieved for 3D RANS Wide range of scales: 109 in AIAA DPW grid High stretching near wall/wake regions Good initial mesh required (even if adaptive) Prohibitive Cost in 3D Each refinement: 8:1 cost 4:1 accuracy improvement (2nd order scheme) Emphasis: User expertise, experience Verification, validation, error estimation

61. VKI Lecture Series, February 3-7, 2003 Designing an Efficient Unstructured Mesh Solver for Aerodynamics Discretization Efficient solution techniques Multigrid Efficient hardware utilization Vector Cache efficiency Parallelization

62. VKI Lecture Series, February 3-7, 2003 Discretization Mostly covered previously Vertex-based discretization Matrix-based artificial dissipation k2=1.0, d=0.1 No low order blending of dissipation (k1 = 0.0) Hybrid Elements Prismatic elements in boundary layer Single edge based data-structure

63. VKI Lecture Series, February 3-7, 2003 Discretization Edge-based data structure Building block for all element types Reduces memory requirements Minimizes indirect addressing / gather-scatter Graph of grid = Discretization stencil Implications for solvers, Partitioners

64. VKI Lecture Series, February 3-7, 2003 Spatially Discretized Equations Integrate to Steady-state Explicit: Simple, Slow: Local procedure Implicit Large Memory Requirements Matrix Free Implicit: Most effective with matrix preconditioner Multigrid Methods

65. VKI Lecture Series, February 3-7, 2003 Multigrid Methods High-frequency (local) error rapidly reduced by explicit methods Low-frequency (global) error converges slowly On coarser grid: Low-frequency viewed as high frequency

66. VKI Lecture Series, February 3-7, 2003 Multigrid Correction Scheme(Linear Problems)

67. Multigrid for Unstructured Meshes Generate fine and coarse meshes Interpolate between un-nested meshes Finest grid: 804,000 points, 4.5M tetrahedra Four level Multigrid sequence

68. VKI Lecture Series, February 3-7, 2003 Geometric Multigrid Order of magnitude increase in convergence Convergence rate equivalent to structured grid schemes Independent of grid size: O(N)

69. VKI Lecture Series, February 3-7, 2003 Agglomeration vs. Geometric Multigrid Multigrid methods: Time step on coarse grids to accelerate solution on fine grid Geometric multigrid Coarse grid levels constructed manually Cumbersome in production environment Agglomeration Multigrid Automate coarse level construction Algebraic nature: summing fine grid equations Graph based algorithm

70. VKI Lecture Series, February 3-7, 2003 Agglomeration Multigrid Agglomeration Multigrid solvers for unstructured meshes Coarse level meshes constructed by agglomerating fine grid cells/equations

71. Agglomeration Multigrid

72. VKI Lecture Series, February 3-7, 2003 Agglomeration MG for Euler Equations Convergence rate similar to geometric MG Completely automatic

73. VKI Lecture Series, February 3-7, 2003 Anisotropy Induced Stiffness Convergence rates for RANS (viscous) problems much slower than inviscid flows Mainly due to grid stretching Thin boundary and wake regions Mixed element (prism-tet) grids Use directional solver to relieve stiffness Line solver in anisotropic regions

74. VKI Lecture Series, February 3-7, 2003 Directional Solver for Navier-Stokes Problems Line Solvers for Anisotropic Problems Lines Constructed in Mesh using weighted graph algorithm Strong Connections Assigned Large Graph Weight (Block) Tridiagonal Line Solver similar to structured grids

75. VKI Lecture Series, February 3-7, 2003 Multigrid Line Solver Convergence DLR-F4 wing-body, Mach=0.75, 1o, Re=3M Baseline Mesh: 1.65M pts

76. Multigrid Insensitivity to Mesh Size High-Lift Case: Mach=0.2, 10o, Re=1.6M

77. Implementation on Parallel Computers Intersected edges resolved by ghost vertices Generates communication between original and ghost vertex Handled using MPI and/or OpenMP Portable, Distributed and Shared Memory Architectures Local reordering within partition for cache-locality

78. VKI Lecture Series, February 3-7, 2003 Partitioning Graph partitioning must minimize number of cut edges to minimize communication Standard graph based partitioners: Metis, Chaco, Jostle Require only weighted graph description of grid Edges, vertices and weights taken as unity Ideal for edge data-structure Line solver inherently sequential Partition around line using weighted graphs

79. VKI Lecture Series, February 3-7, 2003 Partitioning Contract graph along implicit lines Weight edges and vertices Partition contracted graph Decontract graph Guaranteed lines never broken Possible small increase in imbalance/cut edges

80. VKI Lecture Series, February 3-7, 2003 Partitioning Example 32-way partition of 30,562 point 2D grid Unweighted partition: 2.6% edges cut, 2.7% lines cut Weighted partition: 3.2% edges cut, 0% lines cut

81. Parallel Scalability (MPI) 24.7M pts, Cray T3E

82. Parallel Scalability 3M pts, Origin 2000

83. VKI Lecture Series, February 3-7, 2003 Drag Prediction Using Unstructured Mesh Solvers Absolute drag for transonic wing-body AIAA drag prediction workshop (June 2001) Incremental effects DLR engine installation drag study High lift flows Large scale 3D simulation (NSU3D) Experience base in 2D

84. AIAA Drag Prediction Workshop (2001) Transonic wing-body configuration Typical cases required for design study Matrix of mach and CL values Grid resolution study Follow on with engine effects (2003)

85. VKI Lecture Series, February 3-7, 2003 Cases Run Baseline grid: 1.6 million points Full drag Polars for Mach=0.5,0.6,0.7,0.75,0.76,0.77,0.78,0.8 Total = 72 cases Medium grid: 3 million points Full drag polar for each Mach number Total = 48 cases Fine grid: 13 million points Drag polar at mach=0.75 Total = 7 cases

86. VKI Lecture Series, February 3-7, 2003 Sample Solution (1.65M Pts) Mach=0.75, CL=0.6, Re=3M 2.5 hours on 16 Pentium IV 1.7GHz

87. VKI Lecture Series, February 3-7, 2003 Observed Flow Flow Details Mach = 0.75, CL=0.6 Separation in wing root area Post shock and trailing edge separation

88. VKI Lecture Series, February 3-7, 2003 Typical Simulation Characteristics Y+ < 1 over most of wing surfaces Multigrid convergence < 500 cycles

89. VKI Lecture Series, February 3-7, 2003 Lift vs Incidence at Mach = 0.75 Lift values overpredicted Increased lift with additional grid resolution

90. VKI Lecture Series, February 3-7, 2003 Drag Polar at Mach = 0.75 Grid resolution study Good comparison with experimental data

91. Comparison with Experiment Grid Drag Values Incidence Offset for Same CL

92. Surface Cp at 40.9% Span Aft shock location results in lift overprediction Matching CL condition produces low suction peak Adverse effect on predicted moments

93. Drag Polars at other Mach Numbers Grid resolution study Discrepancies at Higher Mach/CL Conditions

94. Drag Rise Curves Grid resolution study Discrepancies at Higher Mach/CL Conditions

95. Structured vs Unstructured Drag Prediction (AIAA workshop results) Similar predictive ability for both approaches More scatter for structured methods More submissions/variations for structured methods

96. VKI Lecture Series, February 3-7, 2003 Absolute Drag Prediction (AIAA DPW 2001) Unstructured mesh capabilities comparable to other methods Lift overprediction tainted assessment of overall results Absolute drag prediction not within 1 count 10 to 20 counts Poorer agreement at high Mach, CL (separation) Grid convergence not established Better results possible with extensive validation Potentially better success for incremental effects

97. VKI Lecture Series, February 3-7, 2003 Timings on Various Architectures

98. VKI Lecture Series, February 3-7, 2003 Cases Run on ICASE Cluster 120 Cases (excluding finest grid) About 1 week to compute all cases

99. VKI Lecture Series, February 3-7, 2003 Incremental Effects Absolute drag prediction to 1 count not yet feasible in general Incremental effects potentially easier to capture Cancellation of drag bias in non-critical regions Important in design study tradeoffs Pre-requisite for automated design optimization Engine installation drag prediction DLR study (tau unstructured grid code) (Broderson and Sturmer AIAA-2001-2414)

100. VKI Lecture Series, February 3-7, 2003 DLR-F6 Configuration Similar to DLR-F4 Wing aspect ratio: 9.5 Sweep: 27.1 degrees Twin engine (flow through nacelles) Test as wing-body alone Test 3 different nacelle positions Two nacelle types (not included herein) Subject of 2nd AIAA Drag Prediction Workshop (June 2003)

101. DLR-F6 Nacelle Positions

102. VKI Lecture Series, February 3-7, 2003 DLR tau Unstructured Solver Similar to NSU3D solver Vertex discretization Artificial dissipation Scaled scalar dissiption Agglomeration multigrid Spalart Allmaras turbulence model Productionalized adaptive meshing capability

103. VKI Lecture Series, February 3-7, 2003 DLR tau Unstructured Solver Productionalized adaptive meshing capability 3 levels of adaptive meshing employed Refinement based on flow-field gradients Wing-body grids Initial: 2.9 million points Final: 5.5 million points Wing-body nacelle-pylon grids Initial: 4.5 million points Final: 7.5 million points

104. VKI Lecture Series, February 3-7, 2003 Computed Absolute Values Overprediction of lift for all cases Under-prediction of drag for all cases

105. Computed Incremental Values Absolute drag underpredicted by 10-20 counts Installation drag accurate to 1 to 4 counts Similar to variations between wind-tunnel campaigns

106. Effect of (Adaptive) Grid Resolution Absolute drag correlation decreases as grid refined Incremental drag correlation improves as grid refined

107. VKI Lecture Series, February 3-7, 2003 Prediction of Installation Drag Accuracy of absolute drag not sufficient Accurate installation drag (incremental) Changes in drag due to nacelle position detectable to within 1 to 2 counts Enables CFD design-based decisions Design optimization Results from careful validation study More complete study at AIAA DPW 2003

108. High-Lift Flows Complicated flow physics High mesh resolution requirements On body, off body Complex geometries Original driver for unstructured meshes in aerodynamics

109. VKI Lecture Series, February 3-7, 2003 High-Lift Flows Prediction of surface pressures Separation possible at design conditions(landing) Lift, drag and moments CLmax, stall Large 3D high-lift case 2D experience base

110. VKI Lecture Series, February 3-7, 2003 NASA Langley Energy Efficient Transport Complex geometry Wing-body, slat, double slotted flaps, cutouts Experimental data from Langley 14x22ft wind tunnel Mach = 0.2, Reynolds=1.6 million Range of incidences: -4 to 24 degrees

111. VGRID Tetrahedral Mesh 3.1 million vertices, 18.2 million tets, 115,489 surface pts Normal spacing: 1.35E-06 chords, growth factor=1.3

112. Computed Pressure Contours on Coarse Grid Mach=0.2, Incidence=10 degrees, Re=1.6M

113. VKI Lecture Series, February 3-7, 2003 Spanwise Stations for Cp Data Experimental data at 10 degrees incidence

114. VKI Lecture Series, February 3-7, 2003 Comparison of Surface Cp at Middle Station

115. Computed Versus Experimental Results Good drag prediction Discrepancies near stall

116. VKI Lecture Series, February 3-7, 2003 Multigrid Convergence History Mesh independent property of Multigrid

117. VKI Lecture Series, February 3-7, 2003 Parallel Scalability Good overall Multigrid scalability Increased communication due to coarse grid levels Single grid solution impractical (>100 times slower) 1 hour solution time on 1450 PEs

118. VKI Lecture Series, February 3-7, 2003 Two-Dimensional High-Lift Large body of experience in 2D High resolution grids possible 50,000 pts required for Cp on 3 elements Up to 250,000 pts required for CLmax Effect of wake resolution Rapid assessment of turbulence/transition models Ability to predict incremental effects Reynolds number effects Small geometry changes (gap/overlap)

119. Typical Agreement for NSU2D Solver Good CP agreement in linear region of CL curve (Lynch, Potter and Spaid, ICAS 1996)

120. Typical Agreement for NSU2D Solver CLmax overpredicted CLmax Incidence overpredicted by 1 degree (Lynch, Potter and Spaid, ICAS 1996)

121. Effect of Grid Resolution and Dissipation Wake capturing requires fine off-body grid Enhanced by low dissipation scheme More difficult further downstream Slat wake deficit consistently overpredicted

122. Prediction of Incremental Effects Adequate Reynolds number effect prediction Provided no substantial transitional effects Transition is important player Transition models

123. Prediction of Gap/Overlap Effects Change due to 0.25% chord increase in flap gap CL increase at low/high incidences captured CL decrease at intermediate incidence missed Flap separation not captured by turb model

124. VKI Lecture Series, February 3-7, 2003 Status of High Lift Simulation Two-dimensional cases Good predictive ability provided flow physics are captured adequately Turbulence, transition Grid resolution Three dimensional simulations coming of age Grid resolution from 2D studies Extensive validation required

125. VKI Lecture Series, February 3-7, 2003 Conclusions and Future Work Cruise drag prediction requires improvement Incremental effects (cruise) to wind tunnel accuracy are feasible High-lift simulations in initial development Higher accuracy, efficiency, reliability Adaptive meshing Error estimation Higher-order methods

126. VKI Lecture Series, February 3-7, 2003 Adaptive Meshing and Error Control Potential for large savings trough optimized mesh resolution Error estimation and control Guarantee or assess level of grid convergence Immense benefit for drag prediction Driver for adaptive process Mechanics of mesh adaptation Refinement criteria and error estimation

127. VKI Lecture Series, February 3-7, 2003 Mechanics of Adaptive Meshing Various well know isotropic mesh methods Mesh movement Spring analogy Linear elasticity Local Remeshing Delaunay point insertion/Retriangulation Edge-face swapping Element subdivision Mixed elements (non-simplicial) Require anisotropic refinement in transition regions

128. VKI Lecture Series, February 3-7, 2003 Subdivision Types for Tetrahedra

129. VKI Lecture Series, February 3-7, 2003 Subdivision Types for Prisms

130. VKI Lecture Series, February 3-7, 2003 Subdivision Types for Pyramids

131. VKI Lecture Series, February 3-7, 2003 Subdivision Types for Hexahedra

132. VKI Lecture Series, February 3-7, 2003 Adaptive Tetrahedral Mesh by Subdivision

133. VKI Lecture Series, February 3-7, 2003 Adaptive Hexahedral Mesh by Subdivision

134. VKI Lecture Series, February 3-7, 2003 Adaptive Hybrid Mesh by Subdivision

135. VKI Lecture Series, February 3-7, 2003 Refinement Criteria Weakest link of adaptive meshing methods Obvious for strong features Difficult for non-local (ie. Convective) features eg. Wakes Analysis assumes in asymptotic error convergence region Gradient based criteria Empirical criteria Effect of variable discretization error in design studies, parameter sweeps

136. VKI Lecture Series, February 3-7, 2003 Adjoint-based Error Prediction Compute sensitivity of global cost function to local spatial grid resolution Key on important output, ignore other features Error in engineering output, not discretization error e.g. Lift, drag, or sonic boom … Captures non-local behavior of error Global effect of local resolution Requires solution of adjoint equations Adjoint techniques used for design optimization

137. VKI Lecture Series, February 3-7, 2003 Adjoint-based Mesh Adaptation Criteria

138. VKI Lecture Series, February 3-7, 2003 Adjoint-based Mesh Adaptation Criteria

139. High-Order Accurate Discretizations Uniform X2 refinement of 3D mesh: Work increase = factor of 8 2nd order accurate method: accuracy increase = 4 4th order accurate method: accuracy increase = 16 For smooth solutions Potential for large efficiency gains Spectral element methods Discontinuous Galerkin (DG) Streamwise Upwind Petrov Galerkin (SUPG)

140. Higher-Order Methods Most effective when high accuracy required Potential role in drag prediction High accuracy requirements Large grid sizes required

141. VKI Lecture Series, February 3-7, 2003 Higher-Order Accurate Discretizations Transfers burden from grid generation to Discretization

142. Spectral Element Solution of Maxwell’s Equations J. Hestahaven and T. Warburton (Brown University)

143. VKI Lecture Series, February 3-7, 2003 Combined H-P Refinement Adaptive meshing (h-ref) yields constant factor improvement After error equidistribution, no further benefit Order refinement (p-ref) yields asymptotic improvement Only for smooth functions Ineffective for inadequate h-resolution of feature Cannot treat shocks H-P refinement optimal (exponential convergence)

144. VKI Lecture Series, February 3-7, 2003 Conclusions Drag prediction is demanding, specialized task Unstructured mesh approach offers comparable accuracy, efficiency with future potential for adaptive meshing advantages Major impediments: Grid convergence Flow physics modeling Continued investment in extensive validation verification required for useful capability