Manufactured Solutions for (U)RANS solvers L. Eça M. Hoekstrapowers/vv.presentations/eca.pdf · 2011. 10. 20. · • Build Manufactured Solutions (MS) for Code Verification of (U)RANS

Manufactured Solutions

for

(U)RANS solvers

L. Eça

M. Hoekstra

Contents

1. Motivation

2. Manufactured solutions properties

3. Examples of manufactured solutions

4. Check of the manufactured solutions

5. Final Remarks

• 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

• 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

• 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=

• 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


• 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:


Φand,,~ ων k

• 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)


Lkk −


• 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

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



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



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



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



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


• “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



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



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



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


• 2-D periodic flow

71 10==ν

LURe


• 3-D flow

71 10==ν

LURe


• 3-D periodic flow

71 10==ν

LURe

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


Equally spaced gridsStretched grids


Stretched grids Observed order of accuracy

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”

Documents

Manufactured Solutions for (U)RANS solvers L. Eça M. Hoekstrapowers/vv.presentations/eca.pdf · 2011. 10. 20. · • Build Manufactured Solutions (MS) for Code Verification of (U)RANS