AIAA 2002-5531

9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 1

AIAA 2002-5531OBSERVATIONS 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. HaftkaUniversity of Florida

Gainesville, FL

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization

4-6 September 2002Atlanta, GA

AIAA 2002-5531

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

• CFD results have an associated uncertainty, originating from different sources

• Sources and magnitudes of the uncertainty important to assess the accuracy of the results

AIAA 2002-5531

9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 3

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

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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

• known as “Sajben Transonic Diffuser” in CFD validation studies

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9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 5

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 8.00.0

0.5

1.0

1.5

2.0

2.5

y/h

t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Transonic Diffuser Problem

y/h

t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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 8.00.0

0.5

1.0

1.5

2.0

2.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P/P0i

P/P0i

Strong shock case (Pe/P0i=0.72)

Weak shock case (Pe/P0i=0.82)

Separation bubble

experiment CFD

Contour variable: velocity magnitude

streamlines

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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

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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 (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

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9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 8

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

A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles)

y/ht

Grid 2

Grid 2 is the typical grid level used in CFD applications

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

esi

eieff HH

HHn

0

0

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Uncertainty in Nozzle Efficiency

+ ++ + + +

+ + ++ + + + +

+ + + + +

+

+

x x x x x x x x x x x xx

xx

xx

xx

x

x

++ + + +

+ + + ++ + + + +

+ + +

+ + ++

++

x x x x x x x x x x x x x x x xx

xx

xx

x

x

o

o

o

o

+

xo

+

xo +

+

x

x

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 1grid 2

grid 4grid 3

grid 5

Sp-Al k-

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Discretization Error by Richardson’s Extrapolation

)( 1 ppexactk hOhff

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 0.71950

1 14.298

2 6.790

3 2.716

4 1.086

Sp-Al

0.82

(weak shock)

1.578 0.81086

1 8.005

2 3.539

3 1.185

4 0.397

k- 0.82

(weak shock)

1.656 0.82889

1 4.432

2 1.452

3 0.461

4 0.146

order of the method

a measure of grid spacing grid level

error coefficient

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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

y/h

t

-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

1.00

1.10

1.20

1.30

1.40

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

upper wall contour obtainedwith the analytical equation

Error in Geometry Representation

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Error in Geometry Representation

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.

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Downstream Boundary Condition

x/hty/

ht

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

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

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

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

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

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Downstream Boundary Condition

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

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the total variation in nozzle efficiency 10% 4%

the difference between grid level 2 and grid level 4

6%

(Sp-Al)

3.5%

(Sp-Al)

the relative uncertainty due to the selection of turbulence model

9%

(grid 4)

2%

(grid 2)

the uncertainty due to the error in geometry representation

0.5%

(grid 3, k-)

1.4%

(grid 3, k-)

the uncertainty due to the change in exit boundary location

0.8%

(grid 3, Sp-Al)

1.1%

(grid 2, Sp-Al)

Uncertainty Comparison in Nozzle EfficiencyStrong Shock

Weak ShockMaximum value of

AIAA 2002-5531

9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 17

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

• Difficult to isolate physical modeling uncertainties from numerical errors

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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

• Discretization error and turbulence models are dominant sources of uncertainty in nozzle efficiency and they are larger for the strong shock case

• We should asses the contribution of CFD uncertainties to the MDO problems that include the simulation of complex flows