Geoscience Reference
In-Depth Information
5.15 Explain the claim that the tilting production term in
Eq. (5.39)
represents
changes in the magnitude and direction of the flux through its interaction
with the mean velocity gradient.
5.16 Using
u
3
/
, write the Kolmogorov microscale as an order-of-
magnitude expression in
ν
,
u
,and
.
5.17 Using the scaling of
∂c
2
/∂x
, find the dependence of the correlation
coefficient of
c
and
∂c/∂x
on
R
t
. Expla
in t
he dependence physically.
5.18 The horizontal turbulent heat flux
ρc
p
uθ
has been found to be typically
as large or larger than the vertical one. Why do we seldom (if ever) hear
about it?
5.19 Derive the conservation equation for the covariance of the fluctuating gra-
dient of a conserved scalar and fluctuating vorticity. Can you identify its
production term(s)? Destruction term(s)?
5.20 Evaluate the TKE equation
(5.42)
on the stagnation streamline just upstream
of a body of revolution. Can you interpret the terms?
5.21 Explain why in the development leadin
g to
Eq. (5.
49)
t
he eddy diffusivity
K
was written for the deviatoric stress
u
i
u
j
−
∼
2
δ
ij
u
k
u
k
/
3
.
(Hint: consider
the
trace
, the form under contraction of the indices.)
5.22 Consider
Eq. (5.51)
for steady flow in a pipe. Integrate the equation over the
region between two cross sections. Show that all but one of the divergence
terms integrates to zero. Interpret your result.
5.23 Explain physically why the rate of molecular destruction of scalar flux
vanishes, as argued in the Appendix.
5.24 The dependent variable
w(x, t)
in a nonlinear system is governed on the
interval 0
≤
x
≤
L
by
γ
∂
2
w
∂w
∂t
+
w
∂w
∂x
=
β(x,t
;
α)
+
∂x
2
,w
0
,t)
=
w(L, t)
=
0
,γ
a constant
.
β
is a zero-mean, random, stochastic, forcing function.
(a) For a given realization sketch an example of
w(x)
for fixed
t
and
w(t)
for fixed
x
.
(b) Ensemble average the equation. Express the second term on the left side
as a gradient. Write the steady equation. Integrate it once, using the
boundary conditions.
(c) Derive the equation for
w
2
, writing its advection term as a gradient and its
molecular term as the sum of destruction and diffusion terms. Interpret its
steady form. Integrate it from
x
=
0to
x
=
L
, using the given boundary
conditions, and interpret that.
(d) Use the equation and your inferences to sketch the profiles of
w
and
w
2
from 0 to
L
.
