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Equations of Motion II

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Equations of Motion II

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

So now we have ……

X-momentum Equation:

What are we missing????

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:

Momentum Equations are now:

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.

Reynolds Averaging

Time

u=<u>+u’

v=<v>+v’

w=<w>+w’

P=<P>+P’

By definition:

<u’> = 0

But

<u’w’> ≠ 0

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:

X-Momentum Equation:

Acceleration terms becomes:

Pressure terms becomes:

Coriolis term becomes:

Friction terms becomes:

X-Momentum Equation:

What about advective terms?

For example:

This one does not!

These terms go away!

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)

x-momentum

advective terms

Coriolis

Friction

acceleration

Pressure gradient

y-momentum

z-momentum