Geoscience Reference
In-Depth Information
1.9.1 The conceptual model
Kolmogorov's seminal work enables us to view turbulence as a nonlinear system
of interacting eddies that dynamically determines some of its properties and takes
others from its fluid-mechanical environment.
The velocity scale
u
and the spatial scale
of the energy-containing turbulent
eddies are of the order of, but typically smaller than, the velocity and length scales
of the mean flow. The turbulence Reynolds number
R
t
=
u/ν
being large, these
energy-containing eddies are essentially inviscid. They determine the viscous dissi-
pation rate, the rate of conversion of turbulence kinetic energy into internal energy;
it is independent of the value of the viscosity (and, as
Holmes
et al
.
(
1996
) suggest,
even independent of the mechanism of the dissipation).
Since the rates of production and dissipation of turbulence kinetic energy per
unit mass (TKE) depend only on
u
and
, on dimensional grounds they are of order
u
3
/
. But the dissipation process itself is a viscous one, so the velocity and length
scales of the eddies in which it occurs do depend on
ν
. By Kolmogorov's 1941
hypotheses the velocity scale
υ
and the length scale
η
of the dissipative motion are
η
(ν)
1
/
4
.
Turbulence obtains its kinetic energy by direct transfer from the mean flow;
in equilibrium it loses kinetic energy at that rate through the viscous dissipa-
tion into internal energy occurring in its smallest eddies. The turbulence field
adjusts the size and intensity of those dissipative eddies in order to achieve the
required energy dissipation rate. This viscous dissipation rate is proportional to the
third power of flow speed, and the fluid heating it causes can be important in hur-
ricanes (
Bister and Emanuel
,
1998
) and more generally in storms (
Businger and
Businger
,
2001
).
(ν
3
/)
1
/
4
,υ
=
η(, ν)
∼
=
υ(,ν)
∼
Questions on key concepts
1.1 Explain the physical mechanism by which turbulence increases surface
fluxes. Can you give some insight into the environmental implications of
this?
1.2 Write out the components of the Navier-Stokes equation in turbulent flow.
1.3 What do we mean by a
conserved
scalar? Write out its equation in turbulent
flow. Can a conserved scalar mix? Explain.
1.4 What are vortex stretching and vortex tilting? Why are they so important in
turbulence?
1.5 What do we mean by the “velocity and length scales”
u
and
of a turbulent
flow? To which set of eddies do they refer? Why do the dissipative eddies
have their own scales? How are the two sets of scales related?
