Geoscience Reference
In-Depth Information
Problems
5.1
Deriv
e a
nd interpret the flow-integrated budget of full scalar variance
C
2
+
c
2
.
5.2
Derive, scale, and interpret the conservation equation for the covariance of
two conserved scalars. Is there a complication if their molecular diffusivities
differ? (Hint: use the form in
Eq. (5.64)
.)
5.3
Explain how you would derive a conservation equation for
.
5.4
One cannot mount
w
and
c
sensors at
the
same point. This raises the ques-
tion: when measuring the scalar flux
wc
in the surface layer, is it better to
mount the
c
sensor just above the
w
sensor, or vice-versa?
†
To answer the
questi
on, derive a ra
te
equation for
the difference of the displaced covari-
ances,
w(z)c(z
d)c(z),
where
d
is the sensor separ
atio
n in the
vertical. Assume that you are in the “constant flux” layer where
wc
is inde-
pendent of
z
and a
ssu
me that each of these displaced covariances is smaller
in magnitude than
wc
. Use the WET model.
5.5 Derive the fluctuating vorticity equation.
5.6 Use the result of
Problem 5.5
to derive the conservation equation for the
variance of fluctuating vorticity. Scale it; what are the two leading terms?
5.7 Derive the conservation equation for mean helicity, the mean of the dot
product of fluctuating vorticity and fluctuating velocity.
5.8 Write the TKE balance for
grid turbulence
- the flow downstream of a
turbulence-producing grid in a wind tunnel. Assume homogeneity in the
normal plane.
5.9 Write the conservation equation for horizontal scalar flux in a horizontally
homogeneous boundary layer. It has been found that its turbulent trans-
port and molecular terms are negligible. What then are its production and
destruction terms? Can you explain why this budget provides a good test of
a fluctuating pressure sensor (
Wilczak and Bedard
,
2004
)?
5.10 Explain why the second term in the first group on the right side of
(5.51)
was
interpreted as “the turbulent flux of joint kinetic energy.”
5.11 Interpret the pressure transport term in
Eq. (5.42)
physically.
5.12 Show that the pressure covariance in the scalar flux budget
(5.53)
can be
scaled as the product of a kinematic pressure gradient of order
u
2
/
and a
scalar fluctuation of order
s
.
5.13 Work out
Eq. (5.46)
, showing that the second-derivative term can be
neglected as claimed.
5.14 Derive and scale the conservation equation for scalar-gradient variance.What
are its two leading terms?
−
d)
−
w(z
−
†
This problem is based on the paper by
Kristensen
et al.
(
1997
).
