Geoscience Reference
In-Depth Information
3
Equations for averaged variables
3.1 Introduction
We saw in
Chapter 1
that turbulent flows dissipate their kinetic energy into internal
energy at an average rate (per unit mass)
u
3
/
,where
u
and
are the velocity
and length scales of the energy-containing eddies. Since the turbulent kinetic energy
per unit mass is of order
u
2
, this implies that if its production mechanism were shut
off turbulence would decay in a time of order
u
2
/
∼
∼
/u
, about one large-eddy
turnover time. That is surprisingly fast. You wouldn't easily ride a bicycle or drive
a car having that much friction.
The decay of the energy-containing eddies in this scenario is caused not by their
viscous friction (their large Reynolds number
R
t
u/ν
makes that negligible),
but by the
energy cascade
. As we'll discuss in
Chapter 6
, this involves the full
range of eddies, energy-containing to dissipative. The cascade begins in the large
eddies, which drain kinetic energy from the mean flow and transfer it to smaller
ones through eddy-eddy interactions; it terminates in the smallest eddies, which
convert kinetic energy into internal energy through viscous friction.
Any direct numerical solution of the turbulent fluid equations must therefore
resolve the entire eddy scale range. Because the length scales at the extremes of
this range are in the ratio
/η
=
R
3
/
4
R
9
/
4
, this requires of order
(/η)
3
∼
∼
t
computational grid points. Since
R
t
values are on the order of 10
8
in the atmospheric
boundary layer and perhaps 10
10
in supercell thunderstorms, they would require
on the order of 10
18
and 10
22
grid points, respectively. Today's computers allow
only 10
10
-10
11
grid points. The
R
t
values in large-vehicle aerodynamics, while
smaller than these geophysical values, are still too large to allow direct numerical
simulation of the drag-producing turbulent flow around them.
Ensemble or space averaging can drastically reduce these computational require-
ments. With some exceptions, such as very near surfaces, the ensemble-mean field
is resolvable on a relatively coarse numerical grid. We'll see that space averaging
t
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