UCRL-PRES-205961
A perspective on turbulent flows:
Cascade dynamics in rotating flows
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 California
Lawrence 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
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
RT mixing
layer without
rotation
RT mixing
layer with
strong rotation
Carnevale, Orlandi, Zhou, and Kloosterziel, JFM
Cambridge
workshop
2010. 2
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. 3
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. 4
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
2

3(k)k4E
(k
)
where k is the wavenumber and ε is the energy dissipation rate
Cambridge
workshop
2010. 5
The rotation modified energy spectrum is
supported by both experiments and simulations
A direct substitution of 3 =1/Ω results in the energy spectrum for
turbulence subject to strong rotation (Zhou, Phys. Fluids) :

2
E
(
k
)
C
(

)1/2k

DNS with
forcing at
small k
and strong
rotation
Yeung and Zhou, Phys. Fluids
Experiment
with Ro=0.06
and Reλ=360
Baroud et al., Phys. Rev. Lett., 88,
114501 (2002)
Cambridge
workshop
2010. 6
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:
 

 




I

I
P
(
k
)
d
2
P
(
p
)
G
(
p
,
)
Q
(
q
,
)
Q
(
k
,
)

P
(
k
)
G
(
k
,
)
Q
(
p
,
)
Q
(
q
,
)






where the integration operators are defined by




imn
rs
m
ns
ir jrs
ij
ns
m
0
0

I

 dk

k

k
0
d
p
d
q
k

p

q
,p
,
q

k
0

I

 dk

k

k
0
d
p
d
q
k

p

q
,p
,
q

k
0
and  = t-s denotes time difference. The time integrals will take the form


1

(
k,
p,
q
)


(

p

Ω
/
p


q

Ω
/
q


k

Ω
/
k

)
With the ansatz (Rubinstein and Zhou)



2
Q
(
k
)

k
f
(
k
,
Ω

k
/
k

)
The leading order solution of the DIA energy balance equation reduces to
E
(
k
)
C
(

) k

1
/2 
2
Cambridge
workshop
2010. 7
A generalized time scale leads to a spectrum
intermediate between the “-5/3” and “-2” spectra
1
1
1


3 nl(k) 
Schematic of
generalized
spectral law

(
k
)

:E
(
k
)

C
k

(
k
)

:E
(
k
)

C
(


)k
3
nl
2
/
3

5
/
3
K
3


No
rotation
(Zhou, Phys.
Fluids
1
/
2
2
rotation
DNS with
forcing at
large k
(Smith and
Waleffe, Phys.
Fluids
Cambridge
workshop
2010. 8
Scaling analysis shows that a -2 spectrum leads
to an eddy viscosity with 1/Ω dependence
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
W W 
ij
1/ 2
ij
where
(Zhou;
Mahalov and Zhou)
W




ij
ij
kij
k
Cambridge
workshop
2010. 9
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:
1
1

3
/2 


~
1

O
(

)




2
2
3
1
/
2


E
~
k
1

O
(
k
/

)
2. Weak rotation limit (Shimomura and
Yoshizawa):

1
/3
2
/3



~

k
1

O
(

)


2
/
3

5
/
3
/
3
2
/
3


E
~
k
1

O
(

/1
k
)
3. Generalized eddy viscosity form
which reduces to the correct limits for
both strong and weak rotation
Note: K – kinetic energy
Zhou)
(Rubinstein and
1
/
2
 1 
K


C

 
2



1

C

K
/


2
Cambridge
workshop
2010. 10
Strong rotation leads to reduced energy transfer
with external energy input to large-scales
DNS with forcing at small k
and strong rotation
Kinetic energy
•
The skewness factor (Batchelor)


3
/
2 

4
15

 
4

S
  
k
E
(
k
)
dk
3



35
  
0

Ro=0.0195
dissipation
Yeung and Zhou, Phys. Fluids
Ro=0.0039
Cambridge
workshop
2010. 11
The response of isotropic turbulence under strong
rotation is revealed through spectral dynamics
The evolution equation for the energy spectrum tensor is

E
(
k
,
t
)
ij

T
(
k
)


(
k
)

D
(
k
)

F
(
k
)
ij
ij
ij
ij

t
Ro=∞
Yeung and Zhou, Phys. Fluids
Ro=0.0195
Cambridge
workshop
2010. 12
Anisotropy in the energy transfer is clearly seen
Inverse
cascade
Transfer function component
parallel to Ω
Transfer function component
perpendicular to Ω
Yeung and Zhou, Phys. Fluids
Cambridge
workshop
2010. 13
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
1.
The eddy viscosity
ε→0
 K2/ε →0
(Speziale, Younis, Rubinstein, and Zhou, Phys. Fluids)
Cambridge
workshop
2010. 14
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
Cambridge
workshop
2010. 15
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
Cambridge
workshop
2010. 16
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



 


K


0

1

exp

C

t

1
,

exp

C

t
2

2
 
K
0 C
0

2
• 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)
Cambridge
workshop
2010. 17
Two distinctive dissipation range spectra may
exist for rotating flow
• Matching the appropriate Kolmogorov and wave frequency
• The ordering parameter is
• This parameter is related to Ro (micro)
1
/2
3
3


cos



k

  


3
/4


k


3
/2
d




O

~
cos

d
,
3
/4 3
/2

k



 
1
/2
R
 cos


Re

O
3
/2

3
/2
• 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
Cambridge
workshop
2010. 18
DIA provides a starting point for studying the
dissipation range energy spectrum

The DIA energy balance equation in the dissipation range is
1
2
2


2
k
Q
(
k
)

dpdq

(
k
,
p
,
q
)
k
b
(
k
,
p
,
q
)
Q
(
q
)
Q
(
k
)

Q
(
p
)

4
• Under the inverted ordering
k

q
2
2


q


k
q

O
(
q
)
z
z

3
k

• The dissipation range dynamics is dominated by interactions q<< p
≈k (Kraichnan, JFM, 1959)

1k
z p
z q
z

(
k
,
p
,
q
)






kpq

• The resonance condition gives
2



• For q near zero


2
2

2
1

Q

Q
2
22
q
dq
Q
(
q
,
q
)
q
k
q

q

k
Q
(
k
)





z

2z 2



k
k
 
z


Rubinstein and Zhou, 1999
 2Q
 2Q
2
Note that k 2 is <<

k
z

Cambridge
workshop
2010. 19
Energy moves toward the horizontal plane in
wavenumber 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 is
not exactly two-dimensional.
Cambridge
workshop
2010. 20
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
Cambridge
workshop
2010. 21