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1.14 Two geometrically identical turbulent flows, one using water and the other
air, have the same
u
and
. By what factor do their total rates of dissipation
of kinetic energy differ?
1.15 If
S
ij
and
A
ij
are symmetric and antisymmetric tensors, respectively, show
that their contraction
S
ij
A
ij
vanishes.
1.16 Resolve the paradox that
is independent of
ν
but the expression for it
(Problem 1.4)
involves
ν
.
1.17 Explain why, in connection with
Eq. (1.40)
,
u
i
u
j
is different from
u
i
u
j
in a
turbulent flow.
1.18 Derive the vorticity
equation (1.28)
from the Navier-Stokes
equation (1.26)
.
Hint: use the vector identities
· ∇
− ∇
· ∇
ω
×
u
=
(
u
)
u
(
u
·
u)/
2and
(
u
)
u
=
+ ∇
u)/
2.
1.19 Derive
Eq. (1.43)
for the ratio of the Taylor and Kolmogorov microscales.
1.20 Do
Problem 1.4
for steady flow in a pipe, integrating between two cross
sections. Interpret your result.
1.21 Show that the total time derivative need not commute with other derivatives.
ω
×
u
(
u
·
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