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????.

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Equations of Motion II

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Equations of motion ii

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


Equations of motion ii

So now we have ……

X-momentum Equation:

What are we missing????


Equations of motion ii

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


Equations of motion ii

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


Equations of motion ii

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”


Equations of motion ii

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:


Equations of motion ii

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.


Equations of motion ii

Reynolds Averaging

Time

u=<u>+u’

v=<v>+v’

w=<w>+w’

P=<P>+P’

By definition:

<u’> = 0

But

<u’w’> ≠ 0


Equations of motion ii

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:


Equations of motion ii

X-Momentum Equation:

Acceleration terms becomes:

Pressure terms becomes:

Coriolis term becomes:

Friction terms becomes:


Equations of motion ii

X-Momentum Equation:

What about advective terms?

For example:

This one does not!

These terms go away!


Equations of motion ii

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!!


Equations of motion ii

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


Equations of motion ii

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)


Equations of motion ii

x-momentum

advective terms

Coriolis

Friction

acceleration

Pressure gradient

y-momentum

z-momentum


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