Grid Quality and Resolution Issues from the Drag Prediction Workshop Series

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)

11. 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 > 109 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(h2) • 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 (Overflow) • Grid convergence for attached flow cases • Inconsistent behavior for separated flow case • Separation bubble grows with grid resolution

35. Grid Convergence (Wing Alone) • More consistent grid convergence at fixed CL

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 CL at AoA=0o: closer to experiment • Drop in CD: 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. Adjoint-Based Spatial Error Estimation + AMR Mach number contours Adjoint solution, Λ(2) • 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) Li Wang and Dimitri Mavriplis

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

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

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

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 Algebraic error Spatial error (discretization/resolution) Temporal error (discretization/resolution) Flow Mesh Other Flow Mesh Other Flow Mesh Other • Solution of time-dependent adjoint: backwards integration in time • Disciplinary adjoint inner product with disciplinary residual