aiaa 2002-5531 observations on cfd simulation uncertainities
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
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
• 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
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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
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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
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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”
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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
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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
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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
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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
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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
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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
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Discretization Error
)( 1 ppexactk hOhff (Richardson’s Extrapolation)
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Discretization Error
The approximations to the exact value of “nozzle efficiency” and “p” depend on the grid levels used in the estimations.
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Discretization Error
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.
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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
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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.)
Discretization Error
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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.
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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.
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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
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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
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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.
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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
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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
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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
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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%
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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
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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)