Code verification and mesh uncertainty
• The goal is to verify that a computer code produces the right solution
to the mathematical model underlying it.
• Tests detailed by Oberkampf and Roy (Section 5.1) include
1.
2.
3.
4.
5.
Simple tests (symmetry, conservation, Gallilean invariance)
Code to code comparisons.
Discretization error quantification.
Convergence tests.
Order of accuracy tests.
• We will focus on 3,4 and 5 using a 2002 paper on comparing CFD
simulations to published test results.
Truncation and discretization errors
u ( x)  u ( x0 ) 
du
2
( x0 )( x  x0 )  O  x  x0  


dx
• In Taylor series
the truncation error is
estimated by the last term.
• We have used this in the integration of the paper helicopter fall
velocity to get the distance travelled h(t  t )  h(t )  V (t )t
• For a given time increment, the truncation error is O(t 2 )
• The truncation error decreases as the square of the time interval.
• The discretization error is the error in the final distance h(tf)
• However, it decreases approximately linearly with the time interval.
Why?
Order of accuracy
• Order of accuracy p is the power in the relationship between
discretization error and interval size h
p
p 1
 h  g p h  O(h
)
• For simple problems like integration of the paper helicopter distance
we can estimate the order of accuracy analytically.
• For most complex problems we have to estimate it from different
meshes, say hcoarse and hfine
p
p
h
coarse
 g p hcoarse
h
fine
 g p h fine
• If we know the errors (from an exact solution) we can solve for p.
Richardson extrapolation
• When we do not know what the error is, we can use Richardson
extrapolation to estimate what our solution will be with an infinitely
p
p 1
fine discretization
f h  f 0  g p h  O (h )
• Now we need f values from three values of h to estimate p and f0
• Unfortunately, the result is exact as h goes to zero, but you often
cannot get near enough.
• In addition there is noise when fine meshes change boundaries
compared to coarse mesh. So it is popular to reduce h by a factor of 2.
Top hat question
• For 4 meshes with h=1,0.5,0.25, and 0.125, the results were, 20, 14.5,
12.5, 11.7. Estimate the converged value and the order
•
•
•
•
f0=11, p=1.5
f0=10.5, p=1
f0=11, p=1
f0=11.2, p=1.5
AIAA 2002-5531
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
Hosder, S. Grossman, B., Haftka, R. T., Mason, W. H., and Watson, L. T. (2006), “Quantitative
Relative Comparison of CFD Simulation Uncertainties for a Transonic Diffuser,” Computers and
Fluids, 35 (10), 1444-1458, December.
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
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
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Drag polar results from 1st AIAA Drag Prediction
Workshop (Hemsch, 2001)
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
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
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Transonic Diffuser Problem
0.9
Weak shock case (Pe/P0i=0.82)
0.7
0.6
experiment
2.5
CFD
Pe/P0i
0.5
0.4
2.0
y/ht
y/ht
0.8
0.3
1.5
1.0
0.5
0.0
-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
x/ht
0.9
Strong shock case (Pe/P0i=0.72)
0.7
0.6
2.5
streamlines
y/ht
2.0
y/ht
0.8
Pe/P0i
0.5
Separation bubble
0.4
0.3
1.5
1.0
0.5
0.0
-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
x/ht
Contour variable: velocity magnitude
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
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
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
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
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Grids Used in the Computations
Grid 2
y/ht
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)
Grid 2 is the typical
grid level used in CFD
applications
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Nozzle efficiency
Nozzle efficiency (neff ), a global indicator of CFD results:
neff
H 0i  H e

H 0 i  H es
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
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Uncertainty in Nozzle Efficiency
0.900
0.875
0.850
Sp-Al
grid 1 +
grid 2
grid 3 x
grid 4 o
grid 5
+
+
+
o
+ + +
+
+
+ +
+ + + + + +
+
+ + +
+ + +
x
+
o
+
+ + + + + + +
x x
+
+ + + + + + +
x
o
+ +
x
x
+ + + +
x
+
x
0.825
neff
k-
0.800
x
x
o
x x
x
x x x x x x
x
x
x x x x
0.775
x
x x
x
x
x x
x
x x x
0.750
0.725
0.700
x
x
x
x
x
x
x
x
x
x
x
o
0.70
0.72
0.74
0.76
0.78
0.80
0.82
0.84
P e/P 0i
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Uncertainty in Nozzle Efficiency
Strong
Shock
Weak
Shock
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)
Maximum value of
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
Discretization Error by Richardson’s Extrapolation
error coefficient
order of the method
f k  f exact   h p  O(h p 1 )
a measure of grid spacing
grid level
Turbulence
model
Sp-Al
Sp-Al
k- 
Pe/P0i
0.72
(strong
shock)
0.82
(weak
shock)
0.82
(weak
shock)
estimate of p (observed
order of accuracy)
1.322
1.578
1.656
estimate of
(neff)exact
Grid
level
Discretization
error (%)
1
14.298
2
6.790
3
2.716
4
1.086
1
8.005
2
3.539
3
1.185
4
0.397
1
4.432
2
1.452
3
0.461
4
0.146
0.71950
0.81086
0.82889
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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AIAA 2002-5531
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.
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA
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Discussion
• What are the key ingredients of the Richardson extrapolation?
• Why do we get non-integer orders?
• Why do we reduce mesh sizes by factors of 2?