1. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 0 AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINITIES 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 Increasingly being used in multidisciplinary design and optimization (MDO) problems Different levels of fidelity (from linear potential solvers to RANS codes) CFD results have a certain level of 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 Objective of the Paper To illustrate different sources of uncertainty in CFD simulations, by a careful study of 2-D, turbulent, transonic flow In a converging-diverging channel (primary case) Around a transonic airfoil To compare the magnitude and importance of each source of uncertainty Use different turbulence models, grid densities and flux-limiters Use modified geometries and boundary conditions

4. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 3 Uncertainty Sources Physical Modeling Uncertainty –PDEs describing the flow (Euler, Thin-Layer N-S, Full N-S, etc.) –Boundary and initial conditions (B.C and I.C) –Auxiliary physical models (turbulence models, thermodynamic models, etc.) Discretization Error –Originates from the Numerical replacement of PDEs and continuum B.C with algebraic equations Consistency and Stability Spatial (grid) and temporal resolution Iterative Convergence Error Programming Errors

5. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 4 Transonic Diffuser Problem (primary case) “strong shock” “weak shock”

6. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 5 Transonic Airfoil Problem RAE 2822 Airfoil Test case: Re c =6.2 x 10 6, Mach=0.75,  =3.19  (AGARD case 10) Test case: Re c =6.2 x 10 6, Mach=0.30,  =0.0  Test case: Inviscid, Mach=0.30,  =0.0 

7. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 6 Computational Modeling General Aerodynamic Simulation Program (GASP) –Reynolds-averaged, 3-D, finite volume Navier-Stokes (N-S) code Inviscid fluxes calculated by upwind-biased 3 rd (nominal) order spatially accurate Roe-flux scheme –Flux-limiters: Min-Mod and Van Albada In viscous runs, full N-S equations are solved –Turbulence models: Spalart-Allmaras (Sp-Al) k-  (Wilcox, 1998 version) with Sarkar’s compressibility correction Implicit time integration to reach steady-state solution with Gauss- Seidel algorithm

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 Transonic diffuser (original geometry) Grid levelMesh Size (number of cells) 192 x 16 2184 x 32 3368 x 64 4736 x 128 RAE 2822 Airfoil A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles) Typical grid levels used in CFD applications: For transonic diffuser case : Grid level 2 For RAE 2822 case: Grid level 3

9. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 8 Output Variables (1) Nozzle efficiency, n eff 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 Throat height

10. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 9 Orthogonal Distance Error, E n A measure of error in wall pressures between the experiment and the curve representing the CFD results Output Variables (2) P c : Wall pressure obtained from CFD calculations P exp : Experimental Wall Pressure Value N exp : Number of experimental data points d i : Orthogonal distance from the i th experimental data point to P c (x) curve

11. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 10 Uncertainty Sources Studied In transonic diffuser case, uncertainty in CFD simulations has been studied in terms of five contributions: 1.Iterative convergence error 2.Discretization error 3.Error in geometry representation 4.Turbulence model 5.Changing the downstream boundary condition Numerical uncertainty Physical modeling uncertainty

12. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 11 Discretization Error (Richardson’s Extrapolation)

13. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 12 Discretization Error The approximations to the exact value of “nozzle efficiency” and “p” depend on the grid levels used in the estimations.

14. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 13 Discretization Error Noise error small compared to the systematic discretization error between each grid level. However, this can be important in a gradient-based optimization.

15. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 14 CaseGrid level CLCD (drag counts) Mach =0.3,  =0.0 deg, Re=6.2x10 6 10.15940191 20.19694104 30.2054685 40.2055083 Mach =0.75,  =3.19 deg, Re=6.2x10 6 10.68992353 20.75094298 30.77889295 40.79341302 Discretization Error Complexity level of the flow structure affects the grid convergence RAE case, Mach =0.3,  =0.0 deg, Re=6.2x10 6 : Attached flow RAE case, Mach =0.75,  =3.19 deg, Re=6.2x10 6 : Shock-induced separation region

16. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 15 3.8 % difference in CL between the cases with and without the limiter at grid level 2 (RAE 2822, inviscid, Mach=0.3, and  =0.0 deg.) Discretization Error

17. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 16 Discretization Error Major observations on the discretization errors: For transonic diffuser cases and the RAE 2822 case with flow separation, grid convergence is not achieved with grid levels that have moderate mesh sizes. Shock-induced flow separation has significant effect on the grid convergence Discretization error magnitudes are different for the cases with different turbulence models. The magnitude of numerical errors are affected by the physical models used.

18. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 17 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.

19. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 18 Turbulence Models Compare orthogonal distance error calculated downstream of the shock at grid level 4 for each case Difficult to isolate the numerical errors from the physical uncertainties For each flow condition, highest accuracy obtained with a different turbulence model In some cases, physical modeling uncertainties may cancel each other, and the closest result to the experiment can be obtained at intermediate grid levels

20. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 19 Turbulence Models Turbulence modelGrid level nozzle efficiency k-  w/ Sarkar’s Comp. Correct. 10.8113 20.79362 30.78543 k-  w/o Sarkar’s Comp. Correct 10.78117 20.75434 30.74271 Sp-Al 10.81827 20.76452 30.73535 Effect of the Sarkar’s compressibility correction on the nozzle efficiency Turbulence modelGrid level nozzle efficiency k-  w/ Sarkar’s Comp. Correct. 10.86563 20.84093 30.83271 k-  w/o Sarkar’s Comp. Correct 10.86494 20.83561 30.82465 Sp-Al 10.87577 20.83956 30.82048 Strong shockWeak Shock

21. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 20 Turbulence Models Strong shockWeak Shock Effect of the Sarkar’s compressibility correction on the wall pressure

22. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 21 Downstream Boundary Condition Extending the geometry or changing the exit pressure ratio affect: location and strength of the shock size of the separation bubble

23. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 22 Uncertainty on Nozzle Efficiency Nozzle efficiency as a global indicator of CFD results Cloud of the results that a reasonably informed user may obtain from CFD calculations

24. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 23 Uncertainty on Nozzle Efficiency Major observations on the uncertainty in nozzle efficiency for the strong shock case The maximum variation is about 10% (original geometry) Magnitude of the discretization error is larger than that of the weak shock case. This error can be up to 6% at grid level 2. Depending on the grid level used, relative uncertainty due to the selection of turbulence model can be larger than the discretization error (can be as large as 9% at grid level 4) Contribution of the error in geometry representation to the overall uncertainty negligible compared to the other sources of uncertainty

25. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 24 Uncertainty on Nozzle Efficiency Major observations on the uncertainty in nozzle efficiency for the weak shock case The maximum variation is about 4% (original geometry) The maximum value of the discretization error is 3.5% The maximum value of the relative uncertainty due to the selection of turbulence model is 2% Nozzle efficiency values more sensitive to the exit boundary conditions. The difference between the results of the original geometry and the extended geometry can be as large as 7% depending on the exit pressure ratio used. Contribution of the error in geometry representation to the overall uncertainty can be up to 1.5%

26. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 25 Conclusions For attached flows without shocks (or with weak shocks), informed CFD users can obtain reasonably accurate results More likely to get large errors for the cases with strong shocks and substantial separation For transonic diffuser cases and the RAE 2822 case with flow separation, grid convergence is not achieved with grid levels that have moderate mesh sizes. The shock induced flow separation has significant effect on the grid convergence The magnitudes of numerical errors are influenced by the physical models (turbulence models) used. Difficult to isolate physical modeling uncertainties from numerical errors

27. AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 26 Conclusions Depending on the flow structure, highest accuracy is obtained with a different turbulence model In some cases, physical modeling uncertainties may cancel each other, and the closest result to the experiment can be obtained at intermediate grid levels 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%) discretization error can be up to 6% at grid level 2 (strong shock) relative uncertainty due to the selection of the turbulence model can be as large as 9% (strong shock) changing the boundary condition can give 7% difference (weak shock)