Geoscience Reference
In-Depth Information
numerically. Interpret the cases
g
and
g
. How would you
choose
g
?
6.6 George Crunch found a three-dimensional, time-dependent numerical model
on the Web but doesn't understand it fully. George can see that the model
uses an eddy diffusivity to close the equations, but there is no indication
whether the model deals with an ensemble mean or a spatial average. How
should he be able to tell by examining the eddy diffusivity?
6.7 Explain the ideas underlying LES.
6.8 Assume that the subgrid-scale stress tensor in the resolvable-scale velocity
equation (6.60) is modeled like the Newtonian viscous stress tensor but with
constant “eddy” viscosity. Show that this model does give a qualitatively
correct rate of interscale energy transfer. How can it be made quantitatively
correct?
6.9 Carry out the integral in
Eq. (6.12)
.
6.10 Derive the conservation equation for the evolution of the mean-squared gra-
dient of a conserved scalar and identify the two leading terms. Which is the
principal source? Principal sink?
6.11 Show that ensemble averaging commutes with spatial filtering.
6.12 Show that
G
and
T
for a running-mean filter,
Eqs. (6.9)
and
(6.5)
,area
Fourier transform pair.
6.13 Prove that the scalar field in
Subsection 6.3.5
can be homogeneous and
steady, as claimed.
6.14 Explain where the scale of the filter in LES would ideally be placed
relative to
.
6.15 Show that we can neglect the molecular diffusion term in
Eq. (6.33)
.
6.16 There is no viscous dissipation in LES. Why? Use the momentum equation
in
Eq. (6.64)
and a decomposition of its variables into ensemble-mean and
fluctuating parts to derive a TKE equation for LES. What takes the place of
the viscous dissipation term? Explain.
6.17 Show that
Eq. (6.57)
results from adding
Eqs. (6.54)
and
(6.56)
.
6.18 Osborne Reynolds used spatial averages. Do the “Reynolds averaging
rules” apply to them? Do they apply in any limit? Discuss.
6.19 Explain why high effective Reynolds numbers can easily be attained in LES
of the atmospheric boundary layer but can be difficult to attain for severe
storms.
6.20 If you wanted ensemble-mean statistics from LES of homogeneous
boundary-layer flows, how would you produce them? Explain.
6.21 How would you achieve randomness in LES? (Hint: revisit our definition of
randomness,
Chapter 1
.)
6.22 Show that reaching the
Re
-independent range with LES requires
.
