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# On Three-dimensional Rotating Turbulence - PowerPoint PPT Presentation

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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Presentation Transcript

Turbulence

Shiyi Chen

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

z

y

x

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

Non-dimensionalized N-S equation:

Rossby #

Ekman #

Engineering Applications

Turbomachinery (i.e. centrifugal

Pumps)

Geophysical Fluid Dynamics

Solid boundary is present

Dimensionless form:

If and

(geostrophic flow)

3D rotating flow is

two-dimensionalized.

Taking a curl , we have

Dynamic Taylor-Proudman Theorem

• For turbulence,

• Coriolis force and nonlinear term?

In k-space:

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)

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)

Solution:

Resonant condition:

“averaged equation”:

Conservation of energy and helicity in each triad gives:

(general)

From resonant condition and triad condition :

Combined with

How energy is transferred among different modes?

Note:

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

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,

w

v

q

p

k

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

k

p

e

* Energy exchange only happens between two 3D modes!

p

q

Since

Using “averaged equation”

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 .

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

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.

DNS with hyperviscousity

Forcing:

Energy injected at

• 3D rotating flow

• 2. 2D turbulence

• 3. 2D passive scalar

Energy injection scales

t

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

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

t

2D

* Inverse cascade for the vertically averaged horizontal velocity

2D

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

• The difference decreases

• as Rossby number decreases.

* 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.