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On Three-dimensional Rotating Turbulence. Shiyi Chen Collaborator: Q. Chen, G. Eyink, D. Holm. z. y. x. For solid-body rotating flow*,. Governing equations. In general, N-S :. Coriolis force. Centrifugal force.

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On Three-dimensional Rotating

Turbulence

Shiyi Chen

Collaborator: Q. Chen, G. Eyink, D. Holm

z

y

x


For solid-body rotating flow*,

Governing equations

In general, N-S :

Coriolis

force

Centrifugal

force

* All quantities here are relative to the rotating frame.

* Centrifugal force

* Coriolis force

(angular momentum)

* Nonlinear term


Governing equations

Non-dimensionalized N-S equation:

Rossby #

Ekman #


Climate prediction

Engineering Applications

Turbomachinery (i.e. centrifugal

Pumps)

Geophysical Fluid Dynamics

Solid boundary is present


Classic Taylor-Proudman theorem

Dimensionless form:

If and

(geostrophic flow)

3D rotating flow is

two-dimensionalized.

Taking a curl , we have


Dynamic taylor proudman theorem

Which one dominates?

Dynamic Taylor-Proudman Theorem

  • For turbulence,

  • Coriolis force and nonlinear term?



Helical representation of n s eq

q

k

p

  • Nonrotating:

  • Rotating:

Helical Representation of N-S Eq.

3D mode/fast mode:

2D mode/slow mode:


Helical waves and Inertial waves

Or

In k-space,

Eigenmodes

(Inertial waves)

Greenspan (1969)


O(1)

0

3D nonrotating

turbulence

Our interest:

1)How does 3D flow become two-dimensional ?

2) Resonant triadic interactions’ role in the two-dimensionalization when .

Research Scope:

3D homogeneous turbulence

( no boundary effects, small Rossby number)



3D turbulence under rapid rotation

Resonant condition:

“averaged equation”:


Helical representation of n s eq1
Helical Representation of N-S Eq.

Conservation of energy and helicity in each triad gives:

And triadic energy transfer function:

(general)


3d turbulence under rapid rotation1
3D turbulence under rapid rotation

From resonant condition and triad condition :

Combined with

How energy is transferred among different modes?

Note:


Three resonant triadic interactions

k

q

p

k

q

p

k

p

1. “fast-fast-fast” interactions

2. “fast-slow-fast” interactions 3. “slow-slow-slow” interactions

q

4. Slow-Fast-Fast


“Fast-fast-fast” resonant triadic interactions

Instability assumption (Waleffe92): energy transfer is

from the mode whose coefficient is opposite to the other two.

One transfer function is negative and the other two are

positive.

If we normalize three wave numbers by the middle one,


1

w

v

“Fast-fast-fast” resonant triadic interactions


k

q

p

k

“Fast-fast-fast” resonant triadic interactions

“Fast-fast-fast” resonant triadic interactions tends to drive flow quasi-2D.


q

k

p

“Fast-slow-fast” resonant triadic interactions

e

* Energy exchange only happens between two 3D modes!


k

p

“Slow-slow-slow” resonant triadic interactions

q

Since

Using “averaged equation”


Dynamic Taylor-Proudman Theorem(2D-3C)

Let

“slow-slow-slow” resonant interactions split into two parts:

1.

(2D N-S)

2.

(2D passive scalar)

Note: Emid & Majda(1996); Mahalov & Zhou(1996)

  • “slow-slow-slow” resonant triadic interactions can split into

  • 2D turbulence and 2D passive scalar as .


Passive scalar T(x,y)

Dynamic Taylor-Proudman Theorem(2D-3C)


Numerical plans

Whether resonant interaction is responsible for flow two

dimensionalization as ?

1) “slow-slow-slow” interactions.

3D simulation under rotation

2D nonrotating turbulence

(2D-3C)

Passive scalar T(x,y)

2) “slow-fast-fast” non-resonant interactions disappear?

3) “fast-fast-fast” resonant interactions.


Numerical schemes

DNS with hyperviscousity

Forcing:

Energy injected at

  • 3D rotating flow

  • 2. 2D turbulence

  • 3. 2D passive scalar



Inverse energy cascade

Energy injection scales

t






“Fast-fast-fast” triadic interactions

* Fast-mode energy from “fast-fast-fast” triadic interactions tends to accumulates at small


Passive scalar T(x,y)

3D averaged field compared with the solution of 2D-3C equations


Dynamics of

t

2D

* Inverse cascade for the vertically averaged horizontal velocity


Dynamics of :

2D

  • behaves like 2D passive scalar when Rossby number decreases.



3D averaged field compared with the solution of 2D-3C equations

  • The difference decreases

  • as Rossby number decreases.


“Slow-fast-fast” triadic interactions equations

* Energy from non-resonant triads into small wavenumbers decreases with Rossby number.


Conclusions equations

  • The rate of two-dimensionalization of 3D rotating flow decreases when Rossby number decreases.

  • Slow-mode energy spectrum approaches

    and its energy flux is closer to a constant.

  • The vertically averaged velocity and the solution of 2D-3C equations converge as

  • The energy flux from non-resonant triads into small in the 2D plane decreases as

  • The fast-mode energy is transferred toward the 2D plane, consistent with the consequence of “fast-fast-fast” resonant triadic interactions.


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