In this question, we develop a scaling law for incompressible turbulence.

In the absence of viscosity (or any other form of dissipation) Newton’s law for a fluid takes the form:

a) If the fluid velocities are very small in comparison to the speed of sound (but nevertheless nonzero) what important approximation can we make to the continuity equation? Write down an expression for this simplified form.

b) With the simplification from part a) in hand, show that Newton’s law (the momentum equation shown above) takes the form:

where

c) nviscid (or nearly inviscid) flows are generally turbulent. Imagine that the flow described by the equation in part b) is typified by a velocity scale, U, and then succumbs to instability at its largest length scale, L. To order-of-magnitude approximation, what is the rate,, at which energy is fed into the largest body?

d) The so-called turbulent cascade occurs because turbulent energy cannot remain only at the largest scale L if steady-state is maintained. If energy is deposited into a smaller eddy scale, , what is the typical velocity of an eddy at this scale?

e) In the absence of viscosity, energy would be continually dumped into ever-smaller eddys. Any realistic fluid, however, will have a finite Reynold’s number,

In terms of physical quantities, what is ?

f) If the turbulence is fully subsonic, what can we say about the smallest eddy size, , in comparison to the particle mean free path ?

g) Show, drawing on arguments developed above, that

Solution(incomplete):

a) When the fluid velocities are smaller than the speed of the sound in the medium, you can assume:

b)

c)

d)

so

e) where is the dynamical viscosity.

In terms of microphysical quantities, the kinematic viscosity is the thermal velocity times the mean free path:

f) The eddy size is independent from velocity.

g)