Partially Resolved Numerical Simulation

CRTI-02-0093RD Project Review MeetingCanadian Meteorological CentreAugust 22-23, 2006

Introduction

Urban Environment in City Scale (~1 km range)• Contains regions of massive separation, recirculation, and

turbulent wakes, etc.

• Conventional RANS approach is unable to correctly simulate above application

• Hybrid RANS/LES approach is needed

Hybrid RANS/LES approaches

Conventional approach• spatially filtered equations are solved in the LES region

• time averaged equations are solved in the RANS region

• Incompatibility of flow properties between two regions occurs

Partially Resolved Numerical Simulation (PRNS)• temporal filtered equations are solved in both the LES and RANS

regions

• A unified simulation approach that spans the spectrum from RANS towards LES and DNS

Partially Resolved Numerical Simulation

Basic Idea• RANS and LES have the same form of filtered transport equation

• A unified model can be developed to determine the degree of modeling required to represent the unresolved turbulent stresses

• This can be achieved in principle through the rescaling of RANS models

12i

i j ij ijj i j

u pu u S

t x x x

PRNS model

Modeling Approach• A PRNS model is obtained by multiplying a dimensionless

resolution control function (FR) to a RANS model:

• Depends on the physical resolution requirement, the model can serve as:

• a RANS model (FR1) where all turbulence scales are modeled, or

• provides no modeling (FR0) where all scales are resolved

• In between these two limits, the model behaves like a LES-type subscale model where only the unresolved scales are modeled

• The fidelity of PRNS depends on the value of FR as well as on the specific formulation of the model

RANSij R ijF 0 1RF where

Formulation of resolution control function (FR)

Speziale (1998):

Batten et al. (2000):

Liu and Shih (2006):

1 expn

R KF l

min ,SGS RANSR

RANS

F

1 43Kl

4 3

R RANSF l 3 2RANSl k

Our Formulation of FR

Hsieh, Lien and Yee (2006):

• Assume energy spectrum E() 2/3-5/3 for i c k

2 32 3 2 3

2 3 2 3 2 3

K

c

K

i

K K i i i

R

K i i i K

E dr r

Fr r

E d

1 1 1, , , ,i c K i c KC l l l

c Kilog

logE()

c i K ir

1 43Kl

3 2il k

2max ,cl u t

0.2C

FR=1 FR=0

Implementation of FR

Method 1:

• FR is only applied in momentum equations to reduce the magnitude of turbulent stresses from RANS-like calculation

• k and are the large scale turbulent energy and dissipation rate

• Adopted by Speziale (1998) and Hsieh, Lien and Yee (2006)

12i

i j ij ijj i j

u pu u S

t x x x

tj k

j j k j

k ku k P

t x x x

1 2t

j kj j j

u C P Ct x x x k

2k t ij ijP S S

2ij R t ijF S

2t C k

Implementation of FR

Method 2:

• FR is applied in momentum and turbulence transport equations to reduce the amplitude of eddy viscosity

• k and are the subscale scale turbulent energy and dissipation rate

• Adopted by Batten et al. (2000) and Liu and Shih (2006)

12i

i j ij ijj i j

u pu u S

t x x x

uj u

j j k j

k ku k P

t x x x

1 2u

j uj j j

u C P Ct x x x k

2u u ij ijP S S

2ij u ijS

2u RF C k

Implementation of FR

Summary

iu

k, t=Ck2/

2ij R t ijF S

Method 1:

Method 2: 2k R t ij ijP F S S

2k t ij ijP S S

Reynolds stresses calculation in PRNS

Reynolds stresses• has contributions from both the resolved and unresolved scales

Reconstruction of modeled Reynolds stresses• To use PRNS calculation on a RANS-type coarse grid, a new

formulation which require the reconstruction of is proposed

• An ad hoc modeled Reynolds stresses is introduced, where the non-linear effect of subscale stresses are absorbed into FR

• The optimal value of the exponent n needs to be determined over a range of flow conditions, and currently n=0.3 is used

r mij ij ijR R R r

ij i j i jR u u u u mij ijR

mijR

m n RANSij R ijR F

Test case

Case 6.2• Fully developed channel flow over a matrix of 250 wall-mounted

cubes (Meinders and Hanjalic, 1999)

• Benchmark problem for the 8th ERCOFTAC workshop (1999)

Ub

y

xz4H 4H

H

x/H=-0.3 0.3 1.3 2.3

x/H=4

1.7

x

y

Ub

y/H=3.4

Results

Numerical results are obtained based on:• 45x45x45 nodes (streamwise by spanwise by vertical)

• Standard k- turbulence model with wall functions are used for URANS and PRNS calculations

Streamwise mean velocity

u/Ub

y/H

-0.5 0 0.5 1 1.50

1

2

3

4

ExptURANSPRNSMILES

x/H=-0.3

u/Ub

y/H

-0.5 0 0.5 1 1.50

1

2

3

4x/H=1.7

u'u'/Ub2

y/H

0 0.025 0.05 0.075 0.1

1

2

3

4

ExptURANSPRNSMILES

x/H=-0.3

u'u'/Ub2

y/H

0 0.025 0.05 0.075 0.1

1

2

3

4x/H=1.7

Reynolds stress (u’u’)

Reynolds stress (w’w’)

w'w'/Ub2

y/H

0 0.025 0.05 0.075 0.1

1

2

3

4

ExptURANSPRNSMILES

x/H=-0.3

w'w'/Ub2

y/H

0 0.025 0.05 0.075 0.1

1

2

3

4x/H=1.7

Resolution control function distribution

FR

y/H

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4x/H=-0.3

FR

y/H

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4x/H=1.7

Resolution control function distribution

x/H

y/H

-2 -1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5RCP

1.00000.92860.85710.78570.71430.64290.57140.50000.42860.35710.28570.21430.14290.07140.0000

Time History

t

u

250 500 750 10001

1.02

1.04

1.06

1.08

1.1

u

PRNS (t=0.05)

t

u

1000 2000 3000 4000 5000

0

0.5

1

1.5 MILES (t=0.01)

at (x,y,z) = (0.5, 1.3, 0)

Energy Spectrum

at (x,y,z) = (0.5, 1.3, 0)

Conclusions

PRNS provides a unified simulation strategy for high Reynolds number complex turbulent flows

Implementation of PRNS to any CFD code that runs URANS simulation is straightforward and requires very minimum modifications

PRNS has been demonstrated in improving the prediction of turbulent mean quantities

References

• Batten, P., Goldberg, U., and Chakravarthy, S. (2000), “Sub-grid turbulence modeling for unsteady flow with acoustic resonance”, AIAA paper 2000-0473.

• Hellsten, A., Rautaheimo, P. (1999). “Workshop on refined turbulence modelling”, Proceedings of the 8th ERCOFTAC/IAHR/COST Workshop, 17–18 June, 1999, Helsinki, Finland. Helsinki University of Technology.

• Meinders E.R. and Hanjalic K. (1999), “Vortex structure and heat transfer in turbulent flow over a wall-mounted matrix of cubes”, Int. J. Heat Fluid Flow, Vol. 20, pp. 255-267.

• Liu N.S. and Shih T.H. (2006), “Turbulence modeling for very large-eddy simulation”, AIAA Journal, Vol. 44, pp. 687-697.

• Speziale, C.G. (1998), “Turbulent modeling for time-dependent RANS and VLES: a review”, AIAA Journal, Vol. 36, pp. 173-184.