aiaa 2002-5531 9 th aiaa/issmo symposium on mao, 09/05/2002, atlanta, ga 0 aiaa 2002-5531...

AIAA 2002-5531

9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 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

AIAA 2002-5531


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

AIAA 2002-5531


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

AIAA 2002-5531


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

AIAA 2002-5531


x/ht

y/h

t

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.00.0

1.0

2.0

3.0

4.0grid 2ext (90x50 cells)

x/ht

y/h

t

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.00.0

1.0

2.0

3.0

4.0grid 2 (80x50 cells)

Transonic Diffuser Problem (primary case)

“strong shock”

“weak shock”

AIAA 2002-5531


Transonic Airfoil Problem

x/c

y/c

0.0 0.5 1.0

-0.5

0.0

0.5

368 x64 cells (grid 3)

• 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

AIAA 2002-5531


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

AIAA 2002-5531


Grids Used in the Computations

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

Transonic diffuser (original geometry)

Grid level Mesh Size (number of cells)

1 92 x 16

2 184 x 32

3 368 x 64

4 736 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

AIAA 2002-5531


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

esi

eieff HH

HHn

0

0

ydyhyuyH oi

y

i

i

)()()(0

0

ydyhyuyH e

y

e

e

)()()(0

ydyhyuyH es

y

es

e

)()()(0

1

0

)()(

i

eoipes P

yPTcyh

th

yy

Throat height

AIAA 2002-5531


Orthogonal Distance Error, En

A measure of error in wall pressures between the experiment and the curve representing the CFD results

Output Variables (2)

exp

1

exp

N

dE

N

ii

n

2exp

2 ))()(()(min icixxxi xPxPxxdexitinlet

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

AIAA 2002-5531


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

AIAA 2002-5531


Discretization Error

)( 1 ppexactk hOhff (Richardson’s Extrapolation)

AIAA 2002-5531



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

AIAA 2002-5531



Pe/P0i

ne

ff

0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.840.700

0.725

0.750

0.775

0.800

0.825

0.850

0.875

0.900

grid 3, Sp-Al, Min-Mod

grid 1, Sp-Al, Min-Mod

grid 2, k-,Min-Mod

grid 1, k-, Min-Mod

grid 2, Sp-Al,Min-Mod

grid 3, k-, Min-Mod

Noise error small compared to the systematic discretization error between each grid level. However, this can be important in a gradient-based optimization.

AIAA 2002-5531


h

CL

1 2 3 4 5 6 7 8 9 100.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

RAE 2822, Sp-Al, Min-Mod

=3.19o, Mach=0.75, Rec=6.2 x 106

=0.0o, Mach=0.30, Rec=6.2 x 106

Case Grid level

CL CD (drag counts)

Mach =0.3,

=0.0 deg,

Re=6.2x106

1 0.15940 191

2 0.19694 104

3 0.20546 85

4 0.20550 83

Mach =0.75,

=3.19 deg,

Re=6.2x106

1 0.68992 353

2 0.75094 298

3 0.77889 295

4 0.79341 302

Discretization ErrorComplexity 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

AIAA 2002-5531


Cycle

CL

0 5000 10000 150000.00

0.05

0.10

0.15

0.20

0.25

0.30

grid 1, with Min-Mod limiter

grid 2, no limiter

=0.0o

Mach=0.30

Inviscid

RAE 2822, Roe, 3rd order Upwind

grid 2, with Min-Mod limiter

grid 1, no limiter

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.)


AIAA 2002-5531



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.

AIAA 2002-5531


Error in Geometry Representation

x/ht

y/h

t

-1.0 -0.5 0.0 0.5 1.0 1.5 2.00.98

0.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

modified experimentaldata points

upper wall contour obtainedwith the analytical equation

upper wall contour of the modified-wallgeometry (cubic-spline fit to the modified data points)

upper wall contour of the modified-wallgeometry (cubic-spline fit to the data points)

x/ht

P/P

0i

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Pe/P0i=0.72Top Wall

experiment

Sp-Al, Min-Mod, grid 2mw , wall contourfrom modified experimental data

Sp-Al, Min-Mod, grid 2,wall contour from equattion

Sp-Al, Min-Mod, grid 2mw , wall contourfrom experimental data

• 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.

AIAA 2002-5531


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

AIAA 2002-5531


Turbulence Models

Turbulence model Grid level

nozzle efficiency

k- w/ Sarkar’s Comp. Correct.

1 0.8113

2 0.79362

3 0.78543

k- w/o Sarkar’s Comp. Correct

1 0.78117

2 0.75434

3 0.74271

Sp-Al

1 0.81827

2 0.76452

3 0.73535

Effect of the Sarkar’s compressibility correction on the nozzle efficiency

Turbulence model Grid level

nozzle efficiency

k- w/ Sarkar’s Comp. Correct.

1 0.86563

2 0.84093

3 0.83271

k- w/o Sarkar’s Comp. Correct

1 0.86494

2 0.83561

3 0.82465

Sp-Al

1 0.87577

2 0.83956

3 0.82048

Strong shock Weak Shock

AIAA 2002-5531


Turbulence Models

Strong shock Weak Shock

Effect of the Sarkar’s compressibility correction on the wall pressure

x/ht

P/P

0i

-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.000.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00Pe/P0i=0.72Top WallMin-Mod, grid 2

k-, w/o Sarkar Comp. Cor.

experiment

Sp-Al

k-, w/ Sarkar Comp. Cor.

x/ht

P/P

0i

-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.000.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00Pe/P0i=0.82Top WallMin-Mod, grid 2

k-, w/o Sarkar Comp. Cor.

experiment

Sp-Al

k-, w/ Sarkar Comp. Cor.

AIAA 2002-5531


Downstream Boundary Condition

x/ht

y/h

t

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.7468

x/ht

y/h

t

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.72

x/ht

y/h

t

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

Extended geomtry, Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.72

Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72

Extended geomtry, Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.7468

x/ht

P/P

0i

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Top Wall

experiment

Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.72

Sp-Al, Van Albada, grid 3

Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.7468

Extending the geometry or changing the exit pressure ratio affect:

• location and strength of the shock

• size of the separation bubble

AIAA 2002-5531


Uncertainty on Nozzle Efficiency

ne

ff

0.700

0.720

0.740

0.760

0.780

0.800

0.820

0.840

0.860

0.880

0.900

original geometry 0.72 0.82modified-wall geometry 0.72 0.82extended geometry 0.7468 0.8368extended geometry 0.72 0.82

1 2 1Van Alb.

k-Min-Mod

k-

3 4 2 3 14 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4Min-Mod

Sp-AlVan Alb.Sp-Al

Min-Modk-

Van Alb.k-

Min-ModSp-Al

Van Alb.Sp-Al

B

A

A

BPe/P0i

grid level:

• Nozzle efficiency as a global indicator of CFD results

• Cloud of the results that a reasonably informed user may obtain from CFD calculations

AIAA 2002-5531



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

AIAA 2002-5531



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%

AIAA 2002-5531


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

AIAA 2002-5531


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)

aiaa 2002-5531 9 th aiaa/issmo symposium on mao, 09/05/2002, atlanta, ga 0 aiaa 2002-5531...

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