Geoscience Reference
In-Depth Information
cascade of kinetic energy from large scales to small (
Chapters 6
,
7
). This cascade
dynamically “couples” all the eddies in a turbulent flow.
Equation (1.35)
shows
that the ratio of energy-containing and dissipative eddy sizes increases as the 3/4
power of the large-eddy Reynolds number
R
t
, so a turbulent flow of large
R
t
has
a huge number of interacting eddies. This has thwarted all attempts to solve the
turbulence equations analytically. Even at relatively small
R
t
the “bookkeeping”
for these interactions overwhelms even our largest computers.
1.7 Numerical modeling of turbulent flows
Today's numerical calculations of large-
R
t
turbulent flows (our daily weather fore-
casts, for example) do not use the basic fluid equations. Instead they use approximate
forms of the averaged equations first derived byOsborneReynolds (
1895
). Reynolds
averaged over a region of space surrounding a point; the average over an ensemble
of realizations of the flow was introduced later. For now we shall not be specific
about the type of averaging; it can be time, space, or ensemble averaging. If the
average commutes with differentiation (as we shall see, most averages do) we can
write the averaged form of the Navier-Stokes
equation (1.26)
as
∂u
i
∂t
+
∂u
i
u
j
∂x
j
1
ρ
∂p
∂x
i
,
=−
(1.38)
with the overbar denoting the average. We have assumed the averaged viscous term
is negligible
(Problem 1.8)
. In writing
(1.38)
we have used the zero-diverg
ence
property of
u
j
to bring it into the derivative. B
ut the
averaging has created a
u
i
u
j
term in
(1.38)
; in turbule
nt flo
w it differs from
u
i
u
j
(Problem 1.17)
and, hence, is
an unknown. If we write
u
i
u
j
as
u
i
u
j
+
u
i
u
j
−
u
i
u
j
=
τ
ij
ρ
u
i
u
j
=
u
i
u
j
−
,
(1.39)
the averaged
equation (1.38)
becomes
∂u
i
∂t
+
∂u
i
u
j
∂x
j
1
ρ
∂p
∂x
i
+
1
ρ
∂τ
ij
∂x
j
.
=−
(1.40)
Averaging the Navier-Stokes equation has produced
Eq. (1.40)
for the average
velocity field, but the equation contains a new term involving a turbulent stress
τ
ij
.
We'll see that this is a subtly different quantity for ensemble and space averages,
but in each case it is typically called the
Reynolds stress
. Experience has shown that
in turbulent flow it is always important.
