Geoscience Reference
In-Depth Information
(Fig. 4.21; see also Section 6.2). In 1904, Prandtl pro-
posed to completely neglect the influence of viscosity away
from the boundary, with the momentum transport being
entirely achieved by eddies. He later made the simplest
possible assumption about the decay rate, that it decreases
as the inverse of distance from the bed, that is, d
u
/d
z
4.5.3 Summary of eddy motions and turbulent
boundary layer structure
We may summarize the above discussion by dividing the
turbulent boundary layer into two rather distinct zones:
1
An inner zone close to the bed with its upper boundary
between the top of the viscous sublayer and logarithmic
region of the turbulent boundary layer. The zone is distin-
guished by (a) being the site of most turbulence production,
(b) containing low- and high-speed fluid streaks that alternate
across the flow, and (c) the lift-up of low-speed streaks in areas
of high local shear near the upper boundary.
2
An outer zone extending up to the flow free surface.
The outer zone and (a) provides the source of the high-
speed fluid of the sweep phase near its lower boundary
(b) contains large vortices near the area of burst break-up
that are disseminated through the outer zone and may
reach the surfaces as “boils” of turbulence.
1/
z
. This assumption leads to (Cookie 11) a logarithmic
relationship for the turbulent velocity as a function of dis-
tance from the boundary that is fully supported by experi-
mental evidence (Figs 4.21 and 4.22) (Cookie 12). The
complete form of this relationship was proposed in 1925
and is commonly called the von Karman-Prandtl equation
or the “
law of the wall
.” The lower part of the
logarithmic
layer
merges into the viscous sublayer via a
buffer zone
where the majority of turbulent stresses are both gener-
ated and dissipated in turbulent flows. The high rates of
both generation and destruction of turbulent kinetic
energy in this area of turbulent flow leads it to be termed
as the
equilibrium layer
.
4.5.4 The simple physics of turbulence - origin of
Reynolds' stresses
hv
Reynolds' approach to the statistical study of turbulence
(Section 3.11; Cookie 8) defined time-mean turbulent
accelerations. Modern high-speed supercomputers track-
ing particle movement in turbulent flows reveal astonish-
ing instantaneous accelerations, up to 1,500
g
, due to
eddy motions. The fact that time-mean turbulent fluctua-
tions exist means that turbulence cannot be random, or
else the positive and negative combinations would cancel
each other out. Further, mean turbulent acceleration
requires a net force to produce it; the only candidate from
the equations of motion is the pressure force. The results
of turbulent flow measurements suggest high contribu-
tions to local
Reynolds' stresses
from burst and sweep
hv
hv
Low-speed
streaks
sense of
vortex stretching
Fig. 4.19
Sketch of major hairpin vortices (hv) that dominate near-
bed turbulent flows.
Water surface
5
5
5
4
4
4
Rough
(9 mm pebbles)
3
Intermediate
(0.2 mm sand)
3
3
2
Smooth
2
2
1
1
1
0
0
0
0
5
10
15
0
5
10
15
0
5
10
15
Mean streamwise velocity, cm s
-1
Fig. 4.20
Velocity boundary layer profiles for turbulent channel flows of water over smooth, intermediate, and rough boundaries. Reynolds
number constant.
Search WWH ::
Custom Search