1. 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
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
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. Drag polar results from 1st AIAA Drag Prediction Workshop (Hemsch, 2001)

4. 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. Transonic Diffuser Problem
Weak shock case (Pe/P0i=0.82)
experiment
Pe/P0i
CFD
Strong shock case (Pe/P0i=0.72)
Pe/P0i
streamlines
Separation bubble
Contour variable: velocity magnitude

6. 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. 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 3rd 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. Grids Used in the Computations
y/ht
A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles)
Grid level
Mesh Size (number of cells) 1
40 x 25 2
80 x 50 3
160 x 100 4
320 x 200 5
640 x 400
Grid 2 is the typical grid level used in CFD applications

9. Nozzle efficiency
Nozzle efficiency (neff ), a global indicator of CFD results:
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

10. Uncertainty in Nozzle Efficiency

11. Uncertainty in Nozzle Efficiency
Strong Shock
Weak Shock
Maximum value of
the total variation in nozzle efficiency
10% 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)

12. Discretization Error by Richardson’s Extrapolation
error coefficient
order of the method
a measure of grid spacing
grid level
Turbulence model
Pe/P0i
estimate of p (observed order of accuracy)
estimate of (neff)exact
Grid level
Discretization error (%)
Sp-Al
0.72
(strong shock)
1.322 1
14.298 2
6.790 3
2.716 4
1.086
0.82
(weak shock)
1.578
8.005
3.539
1.185
0.397
k- 
1.656
4.432
1.452
0.461
0.146

13. 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. Error in Geometry Representation

15. 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. Error in Geometry Representation
In nozzle efficiency, the uncertainty due to the error in geometry representation is not as large as the uncertainties originating from the other sources
negligible for the strong shock case
can be up to 1.4% for the weak shock case

17. Downstream Boundary Condition

18. Downstream Boundary Condition
Extending the geometry or changing the exit pressure ratio affect:
location and strength of the shock
size of the separation bubble

19. 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

20. 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)