Manufactured Solutions for (U)RANS solvers L. Eça M. Hoekstrapowers/vv.presentations/eca.pdf ·...
Transcript of Manufactured Solutions for (U)RANS solvers L. Eça M. Hoekstrapowers/vv.presentations/eca.pdf ·...
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Manufactured Solutions
for
(U)RANS solvers
L. Eça
M. Hoekstra
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Contents
1. Motivation
2. Manufactured solutions properties
3. Examples of manufactured solutions
4. Check of the manufactured solutions
5. Final Remarks
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• Code Verification is the first step of
Verification & Validation
• Code Verification requires error evaluation,
i.e. the knowledge of the exact solution
• Turbulent flows do not have exact solutions
• Method of Manufactured Solutions (MMS)
provides the perfect environment
Motivation
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• Build Manufactured Solutions (MS) for
Code Verification of (U)RANS solvers that
resemble a near-wall incompressible
turbulent flow
• Proposed MS include 2-D and 3-D statistically
steady (RANS) and periodic (URANS) flows
• MS defined in simple domains
Motivation
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• MS based on eddy-viscosity turbulence
models
• The flow field is defined as a function of
the Reynolds number, allowing the choice
of values in the range of 106 to 109,
• Bottom boundary of the domain is a “wall”
• Velocity field is divergence free
Manufactured Solutions Properties
ν
LURe
1=
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• Mean velocity profiles include a “viscous
sub-layer” in the near wall region
• Skin-friction coefficient matches an empirical
correlation for a flat plate boundary-layer
• Flow field tends to a uniform flow with the
increase of the “distance to the wall”
• Pressure field matches typical boundary
conditions of practical applications
Manufactured Solutions Properties
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• Turbulence quantities are defined from
available expressions for “automatic wall
functions” combined with an exponential
decay in the outer region
• Free-stream values are adjustable
• Supported turbulence quantities:
Manufactured Solutions Properties
Φand,,~ ων k
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• Supported eddy-viscosity models:
- One-equation models
Spalart-Allmaras (SPAL), Menter(MNTR,SKL)
- Two-equation models
Wilcox (1998,KWW), TNT, Baseline (BSL)
and SST k-ω
(KSKL)
Manufactured Solutions Properties
Lkk −
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Manufactured Solutions Properties
• Two basic solutions are defined for a
2-D rectangular domain
• Unsteady and 3-D solutions are obtained
from the basic solutions
• For any of the proposed MS, stretched
grids are required to attain the
“asymptotic range” with a reasonable
number of cells
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Examples of Manufactured Solutions
y+
u+
0 2 4 6 8 100
2
4
6
8
10
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
• Mimic of a flat plate turbulent boundary-layer
y+
u+
10-1
100
101
102
103
104
105
1060
5
10
15
20
25
30
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
u+=1/0.41ln(y
+)+5.2
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Examples of Manufactured Solutions
• Mimic of a flat plate turbulent boundary-layer
y+
u+
10-1
100
101
102
103
104
105
1060
5
10
15
20
25
30
35
40
Rex=2.8×10
7
Rex=5.5×10
7
Rex=8.2×10
7
u+=1/0.41ln(y
+)+5.2
y+
u+
0 2 4 6 8 100
2
4
6
8
10
Rex=2.8×10
7
Rex=5.5×10
7
Rex=8.2×10
7
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Examples of Manufactured Solutions
• Mimic of a flat plate turbulent boundary-layer
y+
u+
0 2 4 6 8 100
2
4
6
8
10
Rex=2.8×10
8
Rex=5.5×10
8
Rex=8.2×10
8
y+
u+
10-1
100
101
102
103
104
105
1060
10
20
30
40
50
Rex=2.8×10
8
Rex=5.5×10
8
Rex=8.2×10
8
u+=1/0.41ln(y
+)+5.2
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Examples of Manufactured Solutions
• Mimic of a flat plate turbulent boundary-layer
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6SPAL
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6MNTR
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
SKL
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000SPAL
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000MNTR
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
SKL
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Examples of Manufactured Solutions
• Mimic of a flat plate turbulent boundary-layer
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6KWW, TNTBSL, SKL
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6SST
y+
ν+
0 2 4 6 8 10 12 14 16 18 200
2
4
6
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
KSKL
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000KWW, TNTBSL, SKL
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000SST
y+
ν+
101
102
103
104
105
1060
500
1000
1500
2000
2500
3000
Rex=2.8×10
6
Rex=5.5×10
6
Rex=8.2×10
6
KSKL
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Examples of Manufactured Solutions
• “Separation bubble” added to the flow field
x
y
0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.250 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
Cp
71 10==ν
LURe
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Examples of Manufactured Solutions
• “Separation bubble” added to the flow field
x
y
0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25-0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
Cp
71 10==ν
LURe
-
Examples of Manufactured Solutions
• “Separation bubble” added to the flow field
x
y
0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
Cp
81 10==ν
LURe
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Examples of Manufactured Solutions
• “Separation bubble” added to the flow field
x
y
0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.250 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24
Cp
91 10==ν
LURe
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Examples of Manufactured Solutions
• 2-D periodic flow
71 10==ν
LURe
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Examples of Manufactured Solutions
• 3-D flow
71 10==ν
LURe
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Examples of Manufactured Solutions
• 3-D periodic flow
71 10==ν
LURe
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Check of the Manufactured Solutions
• Finite-differences approximations (2nd order)
of all manufactured quantities (including the
source terms of transport equations) in sets of
21 geometrically similar grids
(801×801 to 51×51)
• Convergence of L∞, L1(mean) and L2(RMS)
norms of the errors checked for four levels
of grid refinement
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Check of the Manufactured Solutions
Equally spaced gridsStretched grids
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Check of the Manufactured Solutions
Stretched grids Observed order of accuracy
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Final Remarks
• Present Manufactured Solutions provide
an excellent framework to perform Code
Verification of (U)RANS solvers based
on eddy-viscosity models
• Solution Verification techniques may be
efficiently tested with current MS due to the
(desired) difficulty to attain the
“asymptotic range”