Sediment Movement in Alluvial Rivers - River Dynamics and Integrated River Management - page 263

Inertial force acting on a unit water volume is proportional to U U 2 /L , where U is the average velocity
over the cross section and L is a representative dimension that usually taken to be the diameter of pipe (in
pipe flow). The viscous force, behaving as an internal force binding the water molecules, can reduce their
fluidity and attenuate a disturbance. The viscous force acting on a unit water volume is proportional to
P U/L 2 . The turbulence phenomenon essentially depends on the balance between these two forces. The
ratio of the inertial force to the viscous force acting on a unit volume of water constitutes a dimensionless
number called the Reynolds number ( Re ):

(5.2)
in which d is the diameter of a pipe. In the experiments conducted by Reynolds, the flow was laminar if
the Reynolds number was less than about 2,000, and it always became turbulent if the Reynolds number
exceeded a value in the range 10,000-12,000, and the flow is in a transitional range if the Reynolds
number is between the two values. Later researchers found that laminar flow could be maintained for
Reynolds numbers as high as 50,000 for very smooth pipes and very quiescent initial conditions.
Define the hydraulic radius, R , as the ratio of the cross sectional flow area, A, to the wetted perimeter P ,
which is given for a pipe flow by:

Re Ud / v

1

2

S d

A 1

4

R d (5.3)

Pd S 4

According to the definition the hydraulic radius for an open channel flow is, assuming a simple river
channel with a rectangular cross section, given by:

A h

R h

PB h

| (5.4)

2

in which B is the width of the channel, h is the depth of the flow, and B >> h . For alluvial rivers, the width
is usually a hundred times larger than the depth, and, therefore, the hydraulic radius is approximately equal
to the average water depth. Applying Eqs. (5.2)-(5.4) the Reynolds number for an open channel flow is
given by:

Ud 4 Uh

Re (5.5)

Q Q

With this definition the critical values for laminar and turbulent flows are the same for an open channel
flow as for a pipe flow.
For water flows in alluvial rivers, the Reynolds number exceeds 12,000 and is always turbulent. Only
for hyperconcentrated flows, which the kinematical viscosity may be 100-1,000 times larger than that of
clear water, the flow may be laminar.
Turbulent flow consists of numerous eddies of various sizes. In order to explain how local disturbances
can induce eddies, one can envision a surface of separation in a non-viscous fluid with different velocities
on the two sides of the surface, as shown in Fig. 5.5(a). If the streamlines on the separation surface are
deformed or bent for some reason, as shown in Fig. 5.5(b), the velocity is higher, and the pressure lower,
at places where the streamlines are more concentrated. The situation is just the opposite where
streamlines are more widely dispersed. As a consequence, the bending of streamlines is intensified, as
shown in Fig. 5.5(c). Ultimately eddies are produced, as shown in Fig. 5.5(d). In alluvial rivers, not only
does the flow separate from dune crests, but it also produces local small-scale separations in the flow
around individual protruding sediment particles, and eddies can form at any of these separation surfaces.
Furthermore, the velocity gradient in open channel flow is generally large near the boundary. Hence, the
entire perimeter of a river bed is a source of turbulence.