Geoscience Reference
In-Depth Information
z
+ dimensionless height units, (
z
+
=
zu
*
/
ν)
20
40
60
80
100
120
140
Total shear stress
1.0
0.8
Reynolds shear stress
0.6
Total shear stress
0.4
Viscous shear stress
0.2
0
0.01
0.02
0.03
0.04
0.05 0.2
0.4
0.6
0.8
1.0
Height,
z
, /flow depth,
h
Fig. 4.24
The total shear stress in a turbulent wall flow is a combination of a viscous stress and a turbulent stress. The former rapidly tends to
zero away from the viscous sublayer, while the latter dominates the flow from the buffer layer outward. The total stress decays to zero in the
outer flow.
we would expect, but the turbulence intensities measured
by and are independent of roughness conditions
beyond a certain height. In the outer layer the intensity
depends solely on boundary distance and shear stress and is
independent of the conditions producing the stress. Closer
to the bed the data separate so that, with increasing bound-
ary roughness, the longitudinal turbulence intensity stays
constant or decreases while the vertical intensity increases.
We can envisage smooth-boundary viscous sublayer fluid
and the fluid trapped between roughness elements as
“passive” reservoirs of low-momentum fluid that are drawn
on during bursting phases. Entrainment of this fluid is
extremely violent in the rough-boundary case, with vertical
upwelling of fluid from between the roughness elements.
Viscous sublayer streaks are poorly developed over rough-
ened boundary flows. Faster deceleration of sweep fluid
causes the decrease of longitudinal turbulent intensity and
the increase of vertical turbulent intensity.
rms
rms
u
w
4.6
Stratified flow
Many geophysical flows occur because of instabilities due
to local density contrasts (Section 4.1). A stratified flow is
one which exhibits some vertical variation in its density
brought about because of heat energy transfer, salinity vari-
ations, or the effects of suspended solids. If the density
increases with height, for example, a desert surface heating
an incoming wind, then the situation is unstable and the
forward transport of fluid is accompanied by turnover
motion that tries to reverse the unstable stratification.
Examples of such
forced convection
are given in Section 4.20.
Here we are interested in the case of a
stable stratification
in which the density decreases vertically, as might be
produced in a wind blowing over a cool surface and being
heated from above (Fig. 4.25). Left to its own devices a
stationary stratified fluid or a purely laminar stratified flow
would simply slowly lose its density contrast by slow
molecular diffusion. Once put into turbulent motion how-
ever, the stratified flow will experience burst-like coherent
motions that try to lift the heavier fluid upward and sweep-
like turbulent motions that will try to carry the lighter
fluid downward. In both cases work must be done by the
flow against resisting buoyancy forces. We might imagine
in the most general way that the turbulent energy may be
either capable or incapable of overturning the buoyancy.
Sea
Fig. 4.25
View of cool sea breeze system (arrow) propagating up-valley. This stratified flow is about 300 m thick advancing right to left at several
meters per second velocity. Note Kelvin-Helmholtz billows at shearing interface indicative of vigorous turbulent mixing (i.e.
Ri
g
0.25).
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