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AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINITIES

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

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AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINITIES

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  1. AIAA 2002-5531OBSERVATIONS 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 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002 Atlanta, GA

  2. 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. 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. 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. Transonic Diffuser Problem (primary case) “strong shock” “weak shock”

  6. Transonic Airfoil Problem • RAE 2822 Airfoil • Test case:Rec=6.2 x 106, • Mach=0.75, =3.19 • (AGARD case 10) • Test case:Rec=6.2 x 106, • Mach=0.30, =0.0 • Test case: Inviscid, • Mach=0.30, =0.0

  7. Computational Modeling • General Aerodynamic Simulation Program (GASP) • Reynolds-averaged, 3-D, finite volume Navier-Stokes (N-S) code • Inviscid fluxes calculated by upwind-biased 3rd (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. Grids Used in the Computations Transonic diffuser (original geometry) 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. Output Variables (1) Nozzle efficiency, neff H0i: Total enthalpy at the inlet He : Enthalpy at the exit Hes : Exit enthalpy at the state that would be reached by isentropic expansion to the actual pressure at the exit Throat height

  10. Output Variables (2) Orthogonal Distance Error, En A measure of error in wall pressures between the experiment and the curve representing the CFD results Pc : Wall pressure obtained from CFD calculations Pexp:Experimental Wall Pressure Value Nexp: Number of experimental data points di: Orthogonal distance from the ith experimental data point to Pc(x) curve

  11. Uncertainty Sources Studied • In transonic diffuser case, uncertainty in CFD simulations has been studied in terms of five contributions: • Iterative convergence error • Discretization error • Error in geometry representation • Turbulence model • Changing the downstream boundary • condition Numerical uncertainty Physical modeling uncertainty

  12. Discretization Error (Richardson’s Extrapolation)

  13. Discretization Error The approximations to the exact value of “nozzle efficiency” and “p” depend on the grid levels used in the estimations.

  14. 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. Discretization Error • Complexity level of the flow structure affects the grid convergence • RAE case, Mach =0.3,  =0.0 deg, Re=6.2x106 : Attached flow • RAE case, Mach =0.75,  =3.19 deg, Re=6.2x106 : Shock-induced separation region

  16. Discretization Error 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.)

  17. 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. 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. 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. Turbulence Models Effect of the Sarkar’s compressibility correction on the nozzle efficiency Strong shock Weak Shock

  21. Turbulence Models Effect of the Sarkar’s compressibility correction on the wall pressure Strong shock Weak Shock

  22. 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. 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. 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. 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. 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. 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)

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