Equations for averaged variables - Turbulence in the Atmosphere - page 72

(b) Write the term involving the viscous stress as the sum of a divergence
and a dissipative term.
(c) Write the remaining terms as the sum of a divergence and a remainder,
and integrate the equation over a volume. Interpret the result.
(d) Interpret the volume-integrated equation for steady flow in a pipe. What
can you conclude about the role of the turbulent flux term?
(e) Repeat (a)-(d) for a conserved scalar.
3.2 Derive the equation for the evolution of mean vorticity and interpret the
terms.
3.3 Derive the equation for the evolution of the mean gradient of a conserved
scalar and interpret the terms.
3.4 Interpret the kinematic Reynolds stress uw in channel flow (Figure 3.1)
formally in terms of an average over realiz atio ns. Indicate the arguments of
u and w . Then explain how we determine uw from time series measured at
a point, again indicating arguments. Explain physically when and why this
is justifiable.
3.5 Write the Poisson equation for mean pressure by taking the divergence of
the mean momentum equation (3.6) .
3.6 Write the vertical mean momentum equation for steady flow in a channel
(Subsection 3.2.1) .
3.7 Sketch instantaneous and ensemble-averaged effluent plumes from a stack.
What are their important differences?
3.8 Explain why we say that “turbulent diffusion” is partly real, partly virtual.
3.9 Evaluate the lateral equation of mean motion in turbulent channel flow
(Figure 3.1) .
3.10 Prove the averaging rules in Eqs. (3.28) .
3.11 Show with an example that the local average, Eq. (3.29) , does not satisfy
averaging rule (2.6) .
3.12 Show that the local average, Eq. (3.29) , commutes with both derivatives.
3.13 Explain why the space-averaging operation defined in Eq. (3.30) removes
eddies that are much smaller than the cube side h and minimally affects those
that are much larger than h .
3.14 Do the expansion in Eq. (3.41) and show that it produces the rhs of the
equation.
References
Corrsin, S., 1961: Turbulent flow. Am. Sci ., 49 , 300-325.
Deardorff, J. W., 1970: A numerical study of three-dimensional turbulent channel flow at
large Reynolds numbers. J. Fluid Mech. , 41 , 453-480.
Kundu, Pijush K., 1990: Fluid Mechanics . San Diego: Academic Press.