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## PowerPoint Slideshow about ' Equations of Motion II' - toviel

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Lecture 09

OEAS-604

October 24, 2011

- Outline:
- Friction and molecular flux of momentum
- Navier Stokes Equation
- Reynolds Averaging
- Reynolds Stress
- Reynolds-average Equations of Motion

Remember from the conservation of salt/heat that we can represent the flux due to molecular diffusion.

Where κc is the diffusion coefficient for the substance C

z

Just like there are molecular fluxes of salt and heat, there are molecular fluxes of momentum

Friction is caused by the momentum flux of molecular motion.

μ -- Dynamic viscosity 1×10-3 kgm-1s-1

Kinematic viscosity 1×10-6 m2s-1

This is usually called a “stress”

z

z

Just like the conservation of salt, we are not interested in the flux of salt as much as we are interested in the divergence in salt flux.

Remember if flux ~ dS/dz, then the flux divergence ~ d2S/dz2

So, for the momentum equation, we are interested in the “Stress-divergence”

Stress is second-order tensor, so there are nine components:

Friction is caused by the momentum flux of molecular motion. In x-direction there are 3 stress:

Dynamic viscosity 1×10-3 kgm-1s-1

Kinematic viscosity 1×10-6 m2s-1

Stress-divergence in x-direction then can be represented as:

Plus continuity

- There are now four equations:
- x-momentum
- y-momentum
- z-momentum
- continuity

- … and four unknowns:
- u
- v
- w
- Pressure

Can be solved numerically, but only for very small Reynolds numbers.

We Can Apply this “Reynolds Averaging” to the Equation of Motion.

For example, the first term becomes

Where the angled brackets indicate averaging in time

By definition:

Acceleration terms becomes:

Pressure terms becomes:

Coriolis term becomes:

Friction terms becomes:

In x-direction, Reynolds averaging of advective terms gives:

(1)

Reynolds averaging of continuity gives:

So, this is also true:

(2)

Add (2) to (1):

This is what’s left over.

Don’t Worry About the Details of all this Math!!

Reynolds-averaged X-Momentum Equation :

Scaling:

How big is u’ v’ and w’?

Over what distance does the flow change in the x-direction?

The y-direction?

The z-direction?

~ 0.10 m/s

~ 10s of km

~ 10s of km

~ 10s of m

(0.10)2

(0.10)2

=10-6 s-1

=10-3 s-1

10000

10

This term is called the “Reynolds Stress”, where angle bracket indicate averaging over some time when the mean flow is not changing

Reynolds stress is a turbulent flux of momentum:

Where Az is an eddy viscosity (m2/s).

So, there is an acceleration when there is a divergence in flux (or stress)

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