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motions; more than 70 percent of the Reynolds stresses in
turbulent flows are due to these events, the majority
produced close to the wall in the equilibrium layer. Use of a
simple quadrant diagram (Fig. 4.23) brings out the essential
contrasts between burst and sweep turbulent interactions
and enables us to get the signs of the stresses. Turbulent
bursts are quadrant 2 events because they involve injections
of slower-than-mean horizontal velocity fluid upward (i.e.
instantaneous
u
values are negative and
w
velocities are posi-
tive). The instantaneous average product,
u
turbulent stresses dominate (Fig. 4.24). Just how much
may be appreciated from Bradshaw's argument. For
boundary layer flow, the rate of ch
an
ge of velocity with
height
h
through
t
he flow is of order
u
h
, so that the mean
viscous stress is
u
h
. If the flow has a realistic mea
n
veloc-
ity fluctu
a
tion,
u
, of say 10 percent of the mean, , then
and th
e
Reynolds stress per unit volume of fluid
is of order . The ratio of the
m
ag
ni
tude of turbu-
lent to viscous stress, that is, is therefore
0.01
Re
. For a geophysical flow,
Re
may easily be greater
than 10
5
(Section 4.2) and so turbulent stresses com-
pletely dominate the boundary layer.
u
u
0.1
u
0.01
u
2
0.01
u
2
u
h
, is thus nega-
tive and this determines that the Reynolds' stress (
w
) is
overall positive. It is the same for sweep motions, but here
the motions are quadrant 4 events, with faster-than-average
downward motion giving the negative product.
u
w
4.5.6
Turbulence over roughened beds
4.5.5
To illustrate the importance of turbulent stresses
Experimental results of great practical interest have been
conducted into turbulence over roughened boundaries.
Increasing boundary roughness causes increasing boundary
layer velocities and therefore mean boundary shear stress, as
Notwithstanding the viscous stress contribution in the
viscous sublayer, through most of any turbulent flow field,
1000
10
Rough
Smooth
Smooth
boundary
log layer
100
1.0
top vsl
y
+
=
u
+
10
0.1
viscous
sublayer
0
5
10
15
Note: extreme
thinness of vsl
0
5
10
15
20
25
Mean streamwise velocity, cm s
-1
Mean dimensionless velocity,
u
+
Fig. 4.21
A plot of log height versus mean velocity reveals zones
where: (1) velocity varies as a function of log height, where
Prandtl's “law of the wall” operates and (2) velocity increases
linearly with height, defining the viscous sublayer (vsl).
Fig. 4.22
The viscous sublayer is destroyed by the rough boundary.
Here the height is nondimensionalized in
z
units. Mean velocity is
also dimensionless, expressed in
u
units.
(a)
(b)
w
Point of
measurement
Bursts
w
w
2
1
u
-
u
-
u
u
3
4
Sweeps
-
u
u
-
w
-
w
-
w
Fig. 4.23
(a) The four possible combinations of velocity fluctuations for 2D flows. (b) A quadrant of possible fluctuation possibilities. Random
turbulence would generate equal likelihood of events 1-4 and no net Reynolds' acceleration. In fact, quadrants 2 and 4 dominate, giving
deceleration and implying a “structure” to turbulence.
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