1. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 0 AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINTIES Serhat Hosder, Bernard Grossman, William H. Mason, and Layne T. Watson Virginia Polytechnic Institute and State University Blacksburg, VA Raphael T. Haftka University of Florida Gainesville, FL 9 th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002 Atlanta, GA

2. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 1 Introduction Computational fluid dynamics (CFD) as an aero/hydrodynamic analysis and design tool CFD being used increasingly in multidisciplinary design and optimization (MDO) problems Different levels of fidelity from linear potential solvers to RANS codes CFD results have an associated uncertainty, originating from different sources Sources and magnitudes of the uncertainty important to assess the accuracy of the results

3. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 2 Drag polar results from 1 st AIAA Drag Prediction Workshop (Hemsch, 2001)

4. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 3 Objective of the Paper Finding the magnitude of CFD simulation uncertainties that a well informed user may encounter and analyzing their sources We study 2-D, turbulent, transonic flow in a converging-diverging channel complex fluid dynamics problem affordable for making multiple runs

5. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 4 Transonic Diffuser Problem Pe/P0i Strong shock case (Pe/P0i=0.72) Weak shock case (Pe/P0i=0.82) Separation bubble experiment CFD Contour variable: velocity magnitude streamlines

6. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 5 Uncertainty Sources (following Oberkampf and Blottner) Physical Modeling Uncertainty PDEs describing the flow Euler, Thin-Layer N-S, Full N-S, etc. boundary conditions and initial conditions geometry representation auxiliary physical models turbulence models, thermodynamic models, etc. Discretization Error Iterative Convergence Error Programming Errors We show that uncertainties from different sources interact

7. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 6 Computational Modeling General Aerodynamic Simulation Program (GASP) A commercial, Reynolds-averaged, 3-D, finite volume Navier-Stokes (N-S) code Has different solution and modeling options. An informed CFD user still “uncertain” about which one to choose For inviscid fluxes (most commonly used options in CFD) Upwind-biased 3 rd order accurate Roe-Flux scheme Flux-limiters: Min-Mod and Van Albada Turbulence models (typical for turbulent flows) Spalart-Allmaras (Sp-Al) k-  (Wilcox, 1998 version) with Sarkar’s compressibility correction

8. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 7 Grids Used in the Computations Grid levelMesh Size (number of cells) 140 x 25 280 x 50 3160 x 100 4320 x 200 5640 x 400 A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles) y/h t Grid 2 Grid 2 is the typical grid level used in CFD applications

9. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 8 Nozzle efficiency Nozzle efficiency (n eff ), a global indicator of CFD results: H 0i : Total enthalpy at the inlet H e : Enthalpy at the exit H es : Exit enthalpy at the state that would be reached by isentropic expansion to the actual pressure at the exit

10. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 9 Uncertainty in Nozzle Efficiency

11. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 10 the total variation in nozzle efficiency10%4% the discretization error 6% (Sp-Al) 3.5% (Sp-Al) the relative uncertainty due to the selection of turbulence model 9% (grid 4) 2% (grid 2) Uncertainty in Nozzle Efficiency Strong Shock Weak Shock Maximum value of

12. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 11 Discretization Error by Richardson’s Extrapolation Turbulence model Pe/P0iestimate of p (observed order of accuracy) estimate of (n eff ) exact Grid level Discretization error (%) Sp-Al 0.72 (strong shock) 1.3220.71950 114.298 26.790 32.716 41.086 Sp-Al 0.82 (weak shock) 1.5780.81086 18.005 23.539 31.185 40.397 k-  0.82 (weak shock) 1.6560.82889 14.432 21.452 30.461 40.146 order of the method a measure of grid spacing grid level error coefficient

13. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 12 Major Observations on the Discretization Errors Grid convergence is not achieved with grid levels that have moderate mesh sizes. For the strong shock case, even with the finest mesh level we can afford, asymptotic convergence is not certain As a consequence of above result, it is difficult to separate physical modeling uncertainties from numerical errors Shock-induced flow separation, thus the flow structure, has a significant effect on grid convergence Discretization error magnitudes are different for different turbulence models. The magnitudes of numerical errors are affected by the physical models chosen.

14. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 13 Error in Geometry Representation

15. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 14 Error in Geometry Representation Upstream of the shock, discrepancy between the CFD results of original geometry and the experiment is due to the error in geometry representation. Downstream of the shock, wall pressure distributions are the same regardless of the geometry used.

16. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 15 Downstream Boundary Condition Extending the geometry or changing the exit pressure ratio affect: location and strength of the shock size of the separation bubble

17. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 16 Conclusions Based on the results obtained from this study, For attached flows without shocks (or with weak shocks), informed users may obtain reasonably accurate results They may get large errors for the cases with strong shocks and substantial separation Grid convergence is not achieved with grid levels that have moderate mesh sizes (especially for separated flows) The flow structure has a significant effect on the grid convergence It is difficult to isolate physical modeling uncertainties from numerical errors

18. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 17 Conclusions Uncertainties from different sources interact, especially in the simulation of flows with separation The magnitudes of numerical errors are influenced by the physical models (turbulence models) used In nozzle efficiency results, range of variation for the strong shock is much larger than the one observed in the weak shock case (10% vs. 4%) the error between grid level 2 and grid level 4 can be up to 6% (strong shock) relative uncertainty due to the selection of the turbulence model can be as large as 9% (strong shock)