Geoscience Reference
In-Depth Information
14.16 Turbulent
free convection
can be established between parallel horizontal
plates, the bottom one heated and the top one cooled. The mean velocity is
zero. Why is this flow inhomogeneous in the vertical? When can it approach
homogeneity in the horizontal? In the latter case could it also have statistical
symmetry about a vertical axis? In this case show that statistics in a horizontal
plane follow isotropy but with the separation vector restricted to lie in the
plane.
14.17 Use
Eqs. (14.24
)and
(14.25)
to express the two leading terms in the vorticity
variance equation
(Problem 5.6)
in terms of streamwise derivatives of
u
1
under the assumption of local isotropy.
14.18 Write the isotropic forms of the terms representing the rate of molecular
destruction of stress and scalar flux.
14.19 Show that under the hypothesis of local isotropy the rates of molecular
destruction of the TKE components are equal.
14.20 Explain why the rate of buoyant production of T
KE vani
shes under isotropy.
14.21 Under the local isotropy hypothesis the tensor
u
i,j
θ
,k
,asacovarianceof
derivatives, might be expected to have the isotropic form
αδ
ij
δ
k
+
βδ
ik
δ
j
+
γδ
i
δ
jk
. Find two constraints on this tensor and use them to solve for two of
the coefficients in terms of the third.
14.22 Consider steady isotropic turbulence with zero mean velocity and mean
vorticity.
(a) Write the equation for fluctuating vorticity
ω
i
. Use it to derive the
vorticity variance budget.
(b) Identify the turbulent transport term in that budget and explain why it
is zero.
(c) Interpret the two remaining terms physically. Write them as velocity
derivative moments, and use local isotropy and
Eqs. (14.24
)and
(14.25)
to express them as moments of streamwise derivatives of streamwise
velocity. Use the physics underlying this budget to explain why
∂u/∂x
has nonzero skewness.
14.23 Derive the conservation equation for
(θ
,
1
)
3
in the ABL,
Eq. (14.35)
.
14.24 Derive the conservation equation for
(θ
,
3
)
3
in the ABL,
Eq. (14.36)
.
14.25 Carry out the scaling of
Eq. (14.32)
and show that the leading-order form is
Eq. (14.34)
.
References
Antonia, R. A., F. Anselmet, and A. J. Chambers, 1986: Local-isotropy assessment in a
turbulent plane jet.
J. Fluid Mech.
,
163
, 365-391.
Antonia, R. A., J. Kim, and L. W. B. Browne, 1991: Some characteristics of small-scale
turbulence in a turbulent duct flow.
J. Fluid Mech.
,
233
, 369-388.
