Introduction - Turbulence in the Atmosphere - page 24

(a) By differentiating this temperature equation with respect to x show that
∂T/∂x does not depend on r . Then derive the solution (1.11) that we
wrote for this equation.
(b) Show that the wall heat flux H satisfies

∂T
∂x .
1.3 Define a time average for use in Eq. (1.9) . Show that it commutes with
differentiation.
1.4 Dot the Navier-Stokes equation (1.26) with velocity to form a kinetic
energy equation. Where possible write terms as divergences. Integrate the
equation over the entire volume of a turbulent flow, assuming the velocity
vanishes on the bounding surface, and thereby eliminate them. Show that the
volume-integrated kinetic energy must decay with time. Identify the decay
mechanism.
1.5 Suppose in Problem 1.4 we applied a body-force field to the fluid. When can
the volume-integrated balance of kinetic energy now be steady in time? Inter-
pret the steady energy balance in terms of the first law of thermodynamics.
What can you conclude about the role of viscous dissipation in turbulence?
1.6 It has been found that the viscous dissipation rate of kinetic energy per unit
mass is of order u 3 / . Using the expression for viscous dissipation found
in Problem 1.4 , show that the velocity and length scales of the dissipative
eddies cannot be u and .
1.7 Write the rate of viscous dissipation per unit mass as the scalar product of
the viscous stress tensor and the strain-rate tensor.
1.8 Explain why the viscous term in the averaged Navier-Stokes equation (1.38)
is negligible. What restrictions must you place on the averaging scale for this
to be true in the case of spatial averaging?
1.9 How does the number of grid points needed to calculate a turbulent flow
directly from the governing equations depend on R t ?
1.10 If a cumulus cloud is turbulent, why does it appear “frozen”?
1.11 Derive the equation for the evolution of the gradient of a conserved scalar.
Can its production term operate in two-dimensional turbulence?
1.12 Show that

Du ave ρc p
4

=−

H

Dab

Dt = a Db Dt + b Da Dt .

Use this property to show that the dot product of the gradient of a conserved
scalar and vorticity is a conserved scalar.
1.13 Show that if c 1 ( x ,t) and c 2 ( x ,t) are solutions of Eq. (1.31) ,then c 1 +

c 2 is
also a solution. How is this used in determining the dispersion of effluents in
the lower atmosphere? Why is it not valid for the Navier-Stokes equation?