1. 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
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
complex fluid dynamics problem
affordable for making multiple runs
To compare the magnitude and importance of each source of uncertainty
We show that uncertainties from different sources interact
To demonstrate how much uncertainty an informed CFD user may have from similar flow simulations

4. Uncertainty Sources
Physical Modeling Uncertainty
PDEs describing the flow (Euler, Thin-Layer N-S, Full N-S, etc.)
boundary conditions and initial conditions
auxiliary physical models (turbulence models, thermodynamic models, etc.)
Discretization Error
Originates from the numerical replacement of PDEs and continuum boundary conditions with algebraic equations
Consistency and Stability
Spatial (grid) resolution and temporal resolution
Iterative Convergence Error
Programming Errors

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. 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 (mostly used options in CFD applications)
Upwind-biased 3rd order accurate Roe-Flux scheme
Flux-limiters: Min-Mod and Van Albada
Turbulence models (typical models used in turbulent flows) :
Spalart-Allmaras (Sp-Al)
k- (Wilcox, 1998 version) with Sarkar’s
compressibility correction

7. 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 2 is the typical grid level used in CFD applications:
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

8. 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
Throat height

9. Uncertainty on Nozzle Efficiency

10. Uncertainty on Nozzle Efficiency
Neglect grid 1 results, use Sp-Al, grid 4 results as the comparator to obtain the percentage values
Strong Shock
Weak Shock
Maximum variation in nozzle efficiency
10% 4%
Maximum magnitude of the discretization error at grid level 2 6%
3.5%
Maximum value of relative uncertainty due to the selection of turbulence model
9% (grid 4)
2% (grid 2)

11. Approximation of 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

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 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.

13. 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.

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

15. The flow structure has significant effect on the grid convergence
Conclusions
For attached flows without shocks (or with weak shocks), informed CFD users can obtain reasonably accurate results
They are more likely to 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 (more significant for the separated flow).
The flow structure has significant effect on the grid convergence
It is difficult to isolate physical modeling uncertainties from numerical errors and uncertainties from different sources interact especially in the simulation of flows with separation

16. Conclusions
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%)
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)