1. Grid Quality and Resolution Issues from the Drag Prediction Workshop Series The DPW Committee Dimitri Mavriplis : University of Wyoming USA J. Vassberg, E. Tinoco, M. Mani : The Boeing Company USA O. Brodersen, B. Eisfeld: DLR Braunschweig, GERMANY R. Wahls, J. Morrison: NASA Langley Research Center, USA T. Zickhur, D. Levy: Cessna Aircraft Co. USA M. Murayama: Japan Aerospace Exploration Agency, JAPAN

2. Motivation DPW Series –Assess State-of-art for Transonic Cruise Drag Prediction using RANS methods DPW I: Anaheim CA, June 2001 DPW II: Orlando FL, June 2003 DPW III: San Francisco CA, June 2006 DPW IV: June 2009 –Considerable scatter in results particularly for cases with flow separation (off-design) –Emerging Consensus Discretization errors are (a) dominant source of error

3. Motivation DPW focused increasingly on assessing discretization/grid induced errors –DPW I: Single grid study –DPW II: Grid convergence study (3 grids) –DPW III: All results examined in context of grid convergence study (3 or 4 grids) Implications –Dominant discretization errors preclude accurate assessment of other errors Turbulence/transition modeling

4. Motivation DPW demonstrated grid convergence for some codes mostly for attached flow cases Separated flow cases much more difficult to obtain grid independent results Scatter often does not decrease with increasing grid resolution Contradictory grid convergence results –Different grid families converge to different results

5. Overview Overview of DPW test cases DPW Gridding Guidelines Discussion of gridding issues –Grid Resolution –Grid Convergence –Grid Quality Possible improvements Conclusions

6. DLRF4-F6 Test Cases (DPW I,II,III) Wing-Body Configuration Transonic Flow Mach=0.75, Incidence = 0 degrees, Reynolds number=3,000,000

7. DPW III Series Cases Designed fairing to suppress flow separation (Vassberg et al. AIAA 2005-4730)

8. DPW III Series Cases 2 closely related simple wing geometries –Well behaved flow –Enhanced grid refinement study (4 grids)

9. General Gridding Guidelines Grid Resolution Guidelines –BL Region Y+ < 1.0, 2/3, 4/9, 8/27 (Coarse,Med,Fine,VeryFine) 2 cell layers constant spacing at wall Growth rates < 1.25 –Far Field: 100 chords –Local Spacings (Medium grid) Chordwise: 0.1% chord at LE/TE Spanwise spacing: 0.1% semispan at root/tip Cell size on Fuselage nose, tail: 2.0% chord –Trailing edge base: 8,12,16,24 cells across TE Base (Coarse,Med,Fine,Veryfine) Grid Convergence Sequences –X3 increase in resolution per refinement –Maintain same family of grids in sequence

10. Overset Meshes (DPW III)

12. Structured Multi-Block Wing-Body Grids Constructed with Boeing Zeus/Advancing Front Method

13. Typical Wing Grid H-H Topology Embedded Blunt Trailing Edge Grid Block

14. VGRID : Wing Body (~40M pts)

15. VGRID : Wing Alone (~30M pts)

16. DPW Submitted Grids Wide variety of grid types and constructions Grid topology and type affects local resolution Compliance with guidelines not evaluated precisely Large data-base of high-quality aero grids made available

17. DPW I RESULTS (circa 2001) Drag polar for single grid resolution

18. DPW II RESULTS (circa 2003) Drag vs number of grid points (Wing-body alone)

19. DPW III RESULTS (2006) Idealized drag vs grid index factor (N -2/3 ) –Wing-body and Wing-body+fairing

20. Grid Related Experiences from DPW Grid Resolution Grid Convergence Grid Quality

21. Grid Resolution Always need more –DPW I: ~ 3M pts –DPW III: ~ 40M pts –Interim/Follow-on studies/DPW4: > 100M pts –Grid convergence studies point to need for > 10 9 pts Wide range of scales present in aerodynamics –Highly variable: Far field ~100 MAC Trailing edge ~.01 MAC –Anisotropic: –Boundary Layer Y + =1: ~ 10 -6 MAC

22. Grid Resolution Wide range of scales requires: –Intuition or rule-based grid generation –Anisotropic in Boundary Layer (and spanwise) –Codified in DPW guidelines Effect of Grid Resolution is Complex –Direct effect on surface profiles is small –Indirect effect can be large Location of separation Integration of small differences  Lift, Drag, Moment

23. W1 Grid Convergence Study CP at station 5:

24. W1 Grid Convergence Study CP at station 5:

25. W1 Grid Convergence Study CP at station 5:

26. W1 Grid Convergence Study CP at station 5:

27. Effect of Normal Spacing in BL Inadequate resolution under-predicts skin friction –Direct influence on drag prediction –Indirect influence: Wrong separation prediction

28. 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

29. Grid Resolution Separated flow cases more demanding and often contradictory experiences

30. Grid Resolution Side-of-Body Separation increases with grid resolution –Boeing: Overset –Boeing: Unstructured –DLR: Unstructured Side-of-Body Separation constant with grid resolution –Boeing: Block Structured –JAXA: Block Structured, Unstructured Trailing edge separation grows with grid res: –UW : Unstructured (NSU3D) Trailing edge separation constant with grid res: –JAXA: Structured, Unstructured –Boeing: Overset Experimentation with much finer grids required to understand behavior…

31. Grid Convergence Increased focus of DPW Series For second-order accurate method, error should decrease as O(h 2 ) –Define average cell size h as: N -1/3 N=number of grid pts –Drag vs N -2/3 should plot as straight line –Project to y-axis to get continuum value

32. Importance of Grid Convergence Agreement on initial grid (DPW I) gets worse (Lee-Rausch et al. AIAA-2003-3400)

33. Grid Convergence Grids must come from same “family” –Self-similar topologically –Same relative variations of resolution Achieved through IJK factors for structured grids Requires global grid spacing factor for unst. grids Boundary layer growth must be taken into account Not clear how well all grids meet these requirements –Most likely represents state-of-art Perform grid convergence at fixed Lift or fixed incidence conditions ?

34. Grid convergence for attached flow cases Inconsistent behavior for separated flow case –Separation bubble grows with grid resolution Grid Convergence (Overflow)

35. More consistent grid convergence at fixed C L Grid Convergence (Wing Alone)

36. W1-W2 Grid Convergence Study (NSU3D Unstructured) Apparently uniform grid convergence

37. W1-W2 Results Discrepancy between results on 2 different families of grids (both generated with VGRID)

38. W1-W2 Results Removing effect of lift-induced drag : Results on both grid families converge consistently –Consistent grid convergence at fixed CL instead of alpha

39. Grid Quality Distinguish grid quality from grid resolution –Relative distribution of resolution –Topology –Element type/shape –Aspect ratio –Orthogonality (BL, hybrid) Grid quality is (should be) constant for self-similar family of grids used for grid convergence study

40. Two Unstructured Grid Topologies 65 million pt grid 72 million pt grid High Resolution grids for DLR-F6 (DPW II) using NSU3D solver

41. Grid Convergence on Topology #1 Drag is grid converging Sensitivity to dissipation decreases as expected

42. 65M pt mesh Results 10% drop in C L at AoA=0 o : closer to experiment Drop in C D : further from experiment Same trends at Mach=0.3 Little sensitivity to dissipation

43. Grid Convergence Grid convergence apparent using self-similar family of grids Large discrepancies possible across grid families –Sensitive areas Separation, Trailing edge Pathological cases ? Would grid families converge to same result limit of infinite resolution ? –i.e. Do we have consistency ? –Due to element types ?, Aspect ratio ? Possible ways forward: –Higher order discretizations –Adjoint-based error estimation

44. hp-adaptive DG Li Wang and Dimitri Mavriplis Adjoint solution, Λ (2) Mach number contours Adjoint-Based Spatial Error Estimation + AMR  Adjoint Solution : Green’s Function for Objective (Lift) Change in Lift for Point sources of Mass/Momentum Error in objective ~ Adjoint. Residual (approx. solution)  Predicts objective value for new solution (on finer mesh)  Cell-wise indicator of error in objective (only)

45. hp-adaptive DG Li Wang and Dimitri Mavriplis h-refinement for target functional of lift Fixed discretization order of p = 1 Final h-adapted mesh (8387 elements)Close-up view of the final h-adapted mesh

46. hp-adaptive DG Li Wang and Dimitri Mavriplis Comparison between h-refinement and uniform mesh refinement Error convergence history vs. degrees of freedom Functional Values and Corrected Values h-refinement for target functional of lift

47. Complex Geometry: Vehicle Stage Separation(CART3D/inviscid) Top View Side View Initial mesh contains only 13k cells Final meshes contain between 8M to 20M cells Initial Mesh

48. Pressure Contours M ∞ =4.5, α=0°

49. Minimal refinement of inter-stage region Gap is highly refined Overall, excellent convergence of functional and error estimate Cutaway view of inter-stage

50. Unsteady Problems Total error in solution Temporal error (discretization/resolution) Spatial error (discretization/resolution) Flow Algebraic error MeshOtherFlowMeshOtherFlowMeshOther Solution of time-dependent adjoint: backwards integration in time Disciplinary adjoint inner product with disciplinary residual

51. Interaction of isentropic vortex with slowly pitching NACA0012 Mach number = 0.4225 Reduced frequency = 0.001 Center of pitch is quarter chord Functional is Time-integrated functional 8,600 elements

52. Unsteady Adjoint Error Estimation Density contours of initial condition

53. Comparison of adapted temporal domain Temporal Error Adaptation

54. Algebraic Error Adaptation Adapted Flow/Mesh convergence tolerances:

55. Adjoint-Based Refinement Results Error in Lift versus CPU Time  Uniform cost is only finest solution cost  Adaptive cost is all solutions (+ adjoint cost)  Corrected value provides further improvement

56. Conclusions Grid related issues are dominant error source in drag prediction Grid and solver interaction is complex Drive toward much higher resolution Grid quality difficult to assess Inconsistent grid convergence results point to possible inconsistent errors : O(1) All these issues much more prevalent for separated flows

57. Conclusions A posteriori error estimation –Criteria for adaptive meshing –Gradient based –Adjoint based for specific outputs Looks promising –Extends to temporal, algebraic error from different disciplines –Combine spatial, temporal, algebraic… –Still a linearization about current solution –Slow to production –Inconsistent errors not resolvable with AMR A priori error estimation –Grid quality metrics –Approximation error of test functions