Equations of Motion II

1. 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

2. So now we have …… X-momentum Equation: What are we missing????

3. 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

4. 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

5. 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”

6. 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:

7. 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.

8. Reynolds Averaging Time u=<u>+u’ v=<v>+v’ w=<w>+w’ P=<P>+P’ By definition: <u’> = 0 But <u’w’> ≠ 0

9. 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:

10. X-Momentum Equation: Acceleration terms becomes: Pressure terms becomes: Coriolis term becomes: Friction terms becomes:

11. X-Momentum Equation: What about advective terms? For example: This one does not! These terms go away!

12. 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!!

13. 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

14. 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)

15. x-momentum advective terms Coriolis Friction acceleration Pressure gradient y-momentum z-momentum