Geoscience Reference
In-Depth Information
14.2 On the basis of local isotropy, what can you say about the individual com-
ponents of the molecular destruction rates of scalar-scalar covariance and
scalar variance?
14.3 For a coordinate transformation caused by rotation of the
x
1
and
x
2
axes by
an angle
α
about the
x
3
axis, write the transformation equations in terms of
α
. Then rewrite them in terms of the direction cosines.
14.4 Show that
δ
ij
is unchanged under coordinate rotation and reflection and
hence is an isotropic tensor.
14.5 Show that
ij k
jik
. Why does this imply that
ij k
is unchanged under
coordinate rotation but not under reflection?
14.6 Show that a diagonal second-order tensor does not change under a coordinate
rotation if and only if its diagonal ele
ments
are equal.
14.7 Using the definition of vorticity, relate
ω
i
ω
j
to
M
ij km
. Use the isotropic form
of
M
ij km
to d
evelo
p an expression for
ω
i
ω
j
.
14.8 Define
a
ik
=
=−
δ
ik
q
2
/
3. Show that the pressure covariance is its only
global sink. What is its source term?
14.9 What is the form of the mean velocity gradient
∂U
i
/∂x
j
under isotropy? (Use
continuity.)
14.10 The
structure function C
(
r
) for a conserved scalar
c
in homogeneous
turbulence is defined as
u
i
u
k
−
r
))
2
.
C(
r
)
=
(c(
x
)
−
c(
x
+
What is the dependence of
C
on separation
r
in an isotropic field? How
would you expect it to behave in a real turbulence field at separation distances
(a)oforder
and (b) much less than
?
14.11 Show that the squared and averaged continuity equation is consistent with
local isotropy.
14.12 Write the isotropic forms of
∂c
∂x
i
∂c
∂x
j
∂c
∂x
k
,
∂c
∂x
i
∂c
∂x
j
∂c
∂x
k
∂c
∂x
l
.
14.13 Use local isotropy to determine the forms of the two leading terms in the
equation for conservation of scalar-gradient variance
(Problem 5.14)
.
14.14 What constraints must an added body-force term in the Navier-Stokes
equation satisfy in order that numerically calculated turbulence be isotropic?
14.15 Rewrite the molecular terms in the equation for the evolution of mean helic-
ity
(Problem 5.7)
. Under the local-isotropy assumption is mean helicity
destroyed molecularly?
