Geoscience Reference
In-Depth Information
2.9 Use the Kolmogorov scaling of
Subsection 2.5.4
to show that viscous stresses
are largest in the dissipative eddies.
2.10 A thin, sinuous plume with instantaneous concentration
c
occasionally “hits”
a concentration sensor. If the average duration of the plume in the sensor is
d
and the average duration between hits is
D
, estimate the mean concentration
and the concentration variance. What are the implications for measuring
mean concentration at the edge of a plume?
2.11 Show that
Eq. (2.66)
holds even for
r
=
and for
r
=
η
.
2.12 Using the inequalities
u
σ
u
−
2
u
σ
u
+
2
v
σ
v
v
σ
v
≥
0
,
≥
0
,
prove that the correlation coefficient is bounded by 1.0 in magnitude.
2.13 Discuss the relative rates of convergence of line, area, and volume averages
to the ensemble average under homogeneous conditions.
2.14 A direct numerical simulation of a turbulent boundary layer is done at a
Reynolds number
R
h
=
10
4
. It is claimed to be representative of
the atmospheric boundary layer. Calculate
R
h
for an atmospheric boundary
layer with
h
=
U
∞
h/ν
10 m s
−
1
. Given the difference in
R
h
values, how could the claim be justifiable? What is the kinematic viscosity
needed if the flow had the atmospheric
h
and
U
∞
values but
R
h
10
3
mand
U
∞
=
=
10
4
?
To what fluid would that roughly correspond? Does it seem plausible that
a boundary layer in that fluid could be a good model of the atmospheric
boundary layer?
2.15 A layer of fluid of depth
h
in the vertical is subject to the sudden applica-
tion of a horizontal velocity
U
at its lower surface. Write the equation of
motion for the fluid, assuming the problem has no horizontal variations.
Estimate the time required for the vertical profile of fluid velocity to reach
a constant value. Under what conditions would the fluid become turbulent?
Estimate the time in that case.
2.16 Before Kolmogorov published his 1941 hypotheses about the dissipative
scales in turbulence,
Taylor
(
1935
) had shown that
=
νu
2
/λ
2
,
with
λ
a
length scale defined through the behavior of the autocorrelation function at
the origin
(Part III)
. (In his honor
λ
is now called the Taylor microscale.)
He interpreted
λ
as the size of the dissipative eddies. Using
∼
νu
2
/λ
2
,
determine the Reynolds number
u(λ)λ/ν
of
λ
-sized eddies,
u(λ)
being the
velocity characteristic of
λ
-sized eddies. Would you say they are directly
influenced by viscosity? Develop an expression for
λ/η
. Was Taylor correct
in interpreting
λ
as the size of the dissipative eddies?
∼
