Ye Zhou Lawrence Livermore National Laboratory, Livermore, California

This work was performed under the auspices of the U.S. Department of Energy by the University of CaliforniaLawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551-0808

Cambridge Workshop. 1

A perspective on turbulent flows: Cascade dynamics in rotating flows

Ye Zhou

Lawrence Livermore National Laboratory, Livermore, California

UCRL-PRES-205961

Cambridge workshop

2010. 2

Turbulent flows subject to strong rotation occur in many important applications

• Rotating turbomachinery; geophysical problems involving the rotation of the Earth

• Example: rotation suppresses the growth of turbulent mixing layers induced by Rayleigh-Taylor instability

Carnevale, Orlandi, Zhou, and Kloosterziel, JFM

RT mixing layer without rotation

RT mixing layer with strong rotation

Cambridge workshop

2010. 3

Rotation has subtle, yet profound effects on the fundamental properties of the energy cascade

• But rotation does not even enter the kinetic energy equation

• The physics of rotating turbulent flows must be better understood:

– energy transfer process

– dissipation

– anisotropy

• This improved understanding should be incorporated into models:

– subgrid models for large-eddy simulations (LES)

– Reynolds averaged Navier-Stokes (RANS) models

Cambridge workshop

2010. 4

The first step in our approach is to consider how strong rotation modifies the energy spectrum

• A strong similarity between magnetohydrodynamic turbulence and isotropic turbulence subject to solid body rotation has been noted (Zhou, Phys. Fluids)

– For MHD turbulence, Kraichnan (Phys. Fluids) argued that the propagation of Alfvénic fluctuations disrupts phase relations, thereby decreasing energy transfer

– Uniform rotation causes plane waves to propagate with phase speed 2Ωkz/k

• This argument is also analogous to that given by Herring (Meteorol. Atmos. Phys., vol. 38, 106, 1988) for stratified flows

Cambridge workshop

2010. 5

A rotation modified energy spectrum E(k) can be obtained from a phenomenological argument

• Kolmogorov’s theory cannot be directly applied to complex problems with imposed time scales

• The time scale for decay of triple correlations, 3, is crucial to turbulent spectral transfer

• We assume that 3 results from local interactions determined by the rotation time scale, Ω=1/Ω, not by the nonlinear time scale nl

• Because energy is conserved by nonlinear interactions and a local cascade has been assumed, dimensional analysis leads to

where k is the wavenumber and ε is the energy dissipation rate

)()( 24 kEkk3

Cambridge workshop

2010. 6

The rotation modified energy spectrum is supported by both experiments and simulations

22/1)()( kCkE

A direct substitution of 3 =1/Ω results in the energy spectrum for turbulence subject to strong rotation (Zhou, Phys. Fluids) :

Yeung and Zhou, Phys. Fluids Baroud et al., Phys. Rev. Lett., 88, 114501 (2002)

Experiment with Ro=0.06 and Reλ=360

DNS with forcing at

small k and strong

rotation

Cambridge workshop

2010. 7

The phenomenology can be made more precise by appealing to DIA and other closure theories

• For a constant energy flux steady state, the DIA inertial range energy balance (Kraichnan, 1971) is:

where the integration operators are defined by

and = t-s denotes time difference. The time integrals will take the form

With the ansatz (Rubinstein and Zhou)

The leading order solution of the DIA energy balance equation reduces to

),(),(),()(),(),(),()(2)(

00

qQpQkGkPkQqQpGpPdkPII mrnsijjrsirnsmrsimn

0 0,,kk kqp

dddkIqpk

qp

0 0,,kk kqp

dddkIqpk

qp

)///()( 1 kΩkqΩqpΩpqp,k,

22/1)()( kCkE

)/,()( 2 kkΩkfkkQ

Cambridge workshop

2010. 8

A generalized time scale leads to a spectrum intermediate between the “-5/3” and “-2” spectra

3/53/23 )( :)( kCkEk Knl

1

)(

11

3 knl

22/13 )()( :)(

kCkEk

Schematic of generalized spectral law

(Zhou, Phys. Fluids

DNS with forcing at large k

(Smith and Waleffe, Phys. Fluids

No rotation

rotation

Cambridge workshop

2010. 9

Scaling analysis shows that a -2 spectrum leads to an eddy viscosity with 1/Ω dependence

2/1

ijijWW

(Zhou;

Mahalov and Zhou)

The eddy viscosity ~ 1/Ω • Speziale pointed out in turbulent flows with a mean velocity gradient in a rotating frame, the effect of rotation appears in conjunction with the mean vorticity (Gatski and Speziale, 1993)

•The rotation rate, Ω, is replaced by

where

kkijijijW

Cambridge workshop 2010. 10

Generalized eddy viscosity for both strong and weak rotation can be constructed in Padé form

Closure provides perturbation expansions of both the time scale and energy spectrum:

1. Strong rotation limit:

2. Weak rotation limit (Shimomura and Yoshizawa):

3. Generalized eddy viscosity form

which reduces to the correct limits for both strong and weak rotation

Note: K – kinetic energy (Rubinstein and Zhou)

12/3 )(11

~

O

2/1322 )/(1 ~ kOkE

)(1 ~ 3/23/1 Ok

)/(1 ~ 3/23/13/53/2 kOkE

2/1

2

2

/1

1

KC

KC


Strong rotation leads to reduced energy transfer with external energy input to large-scales

0

42/3

3 )(15

35

4dkkEkS

• The skewness factor (Batchelor)

Yeung and Zhou, Phys. Fluids

dissipation

Kinetic energy

Ro=0.0039

Ro=0.0195

DNS with forcing at small k and strong rotation


The response of isotropic turbulence under strong rotation is revealed through spectral dynamics

)()()()(),(

kkkkk

ijijijijij FDT

t

tE

The evolution equation for the energy spectrum tensor is


Ro=0.0195Ro=∞


Anisotropy in the energy transfer is clearly seen


Inverse cascade

Transfer function component parallel to Ω

Transfer function component perpendicular to Ω


RANS models must satisfy two major consistency conditions in rotating flows

In rapidly rotating flows, based on the previous discussion:

• Cascade is disturbed by suppressing phase coherence

• Two-dimensionalization and its often associated laminarization

Therefore, two conditions must be met at Ω→

1. The dissipation rate ε→0

1. The eddy viscosity K2/ε →0

(Speziale, Younis, Rubinstein, and Zhou, Phys. Fluids)

∞


Analytical theory of the destruction term in the dissipation rate equations can be derived

Many RANS models do not satisfy these two conditions. For example,

Launder, Reece and Rodi model:

ε is not affected by rotation, so that the first condition is not met

Launder, Priddin, and Sharma model:

the eddy viscosity is decreased by having ε increased

• Once the inertial range theory is given, rotation effects arise naturally in the dissipation rate equation

– The -2 energy spectrum– The energy flux function as given by closure


The procedure of Schiestel offers an alternative dissipation rate equation for rotating flows

– The large-scale energy and the inertial range energy are linked through a self-similarity assumption

• Energy becomes trapped in the largest scales of motion, where it undergoes purely viscous decay

• The large-scale and inertial range energies evolve independently

• Thus, the kinetic energy and transfer in the inertial range vanish, but the kinetic energy and the integral scale both approach constants in the absence of viscosity

Rubinstein and Zhou, Comput. Math. Appl., 46, 633, 2003


In the long time limit, K is a constant and ε =0, indicating that the energy transfer to small scales has stopped

• The solution for decaying turbulence is

tCtCCK

K

20

2

2

0

0

exp ,1exp1

• In the long time limit, K is a constant and ε =0, indicating that the energy transfer to small scales has stopped

• This rotation dependence is in agreement with the Bardina model (JFM, 1985)


Two distinctive dissipation range spectra may exist for rotating flow

2/32/32/1

2/3

2/34/3

4/3

,

cosRe

cos~

O

dd

R

k

kO

• Matching the appropriate Kolmogorov and wave frequency

• The ordering parameter is

• This parameter is related to Ro (micro)

2/133 cos

k

• Normal ordering, If Od,Ω >> 1: Kolmogorov scaling is recovered at small k

• Inverted ordering, If Od,Ω << 1:

Rotation effects are important in the dissipation range


DIA provides a starting point for studying the dissipation range energy spectrum

)( 2

3

2

qO

kqkq zz

qk

)()()(),,(),,(4

1)(2 22 pQkQqQqpkbkqpkkQk dpdq

q

q

p

p

k

kqpk zzz

2

1),,(

Rubinstein and Zhou, 1999

The DIA energy balance equation in the dissipation range is

• Under the inverted ordering

• The dissipation range dynamics is dominated by interactions q<< p ≈k (Kraichnan, JFM, 1959)

• The resonance condition gives • For q near zero

)(),(1 2

2

22

2

222 kQk

k

Qq

k

QqkqqqQdqq

z

zz

Note that is << 2

2

zk

Q

2

2

k

Q


Energy moves toward the horizontal plane inwavenumber space, where dissipation is possible

• The -2 spectrum is established at moderate Ro or large Ro Re1/2

when Kolmogorov scaling is recovered at small inertial range, and an isotropic dissipation range exists

• At asymptotically low Ro or Ro Re1/2 small, the argument leading to a -2 scaling breaks down. These conditions can exist in decaying turbulence

• The dissipation range dynamics depends on Ω; it isnot exactly two-dimensional.


Summary and conclusion

• We have investigated the effects of strong rotation on cascade dynamics energy transfer process

• Rotation-modified energy transfer provides the starting point for our model development

• An eddy viscosity that depends on rotation rate is derived

• Two constraints on RANS models under strong rotation are obtained:

– Dissipation rate must vanish

– Eddy viscosity must vanish

• Applications to engineering flows, based on the energy spectrum have shown promise (Thangam, Wang, and Zhou, Phys. Fluids)

• We are interested in the stratified turbulence and stratified rotating turbulent flows

Ye Zhou Lawrence Livermore National Laboratory, Livermore, California

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