Geoscience Reference
In-Depth Information
6.6 Explain physically why molecular diffusion terms are absent in the filtered
Navier-Stokes and scalar conservation equations. What takes over their
important role in the variance budgets?
6.7 Explain why ensemble-averaged equations have no terms involving deriva-
tives in a homogeneous direction, but the space-averaged equations do. Why
is this important?
6.8 In what limit should the conservation equations for subfilter-scale fluxes,
Eqs. (6.71)
and
(6.73)
, become their ensemble-averaged counterparts from
Chapter 5
?
6.9 Outline and explain the concepts underlying LES. Contrast it to second-
moment modeling.
6.10 Explain why LES is considered to be an inherently more reliable form of
turbulence modeling than that based on the second-moment equations of
Chapter 5
.
6.11 Compare the computational requirements for LES and second-moment mod-
eling in the same problem - a horizontally homogeneous boundary layer, for
example. What can you conclude from this?
6.12 In view of your result in
Question 6.11
, discuss how LES might be used to
improve second-moment modeling. In what applications might this approach
be useful?
6.13 Explain the advantages of placing the LES cutoff scale in the inertial
subrange.
Problems
6.1
Prove that the running-mean filter in one dimension has the filter function
given in
Eq. (6.9)
.
6.2
Show that for the filter
(6.1)
∂u
∂x
r
∂u
r
∂x
;
=
i.e., that it commutes with spatial differentiation.
6.3
Prove that the wave-cutoff filter in one dimension satisfies the constraint
(6.21)
.
6.4
Derive and interpret the equations for the variance of the resolvable and
subfilter-scale parts of a conserved scalar. Discuss the variance budget in the
inertial subrange of spatial scales.
6.5
Discuss the difference between
, the scale of the spatial filter, and
g
,
the mesh size of the spatial grid on which the filtered equations are solved
