Sediment Movement in Alluvial Rivers - River Dynamics and Integrated River Management - page 265

5.1.3 Velocity Profile
Velocity profile— The foregoing reasoning helps us understand the mechanism of momentum exchange
between adjacent layers in a moving fluid. In Fig. 5.5(c), the adjacent surface area is A 0 , the relative
velocity between adjacent layers is u' and the fluctuating cross-current velocity is X'. If this cross-current
velocity were uniformly distributed over the surface A 0 , then the mass transported from the lower layer B
to the upper layer A per unit of time would be UX 'A 0. If the particles thus transported into layer A become
thoroughly mixed with particles already in layer A and are brought to move axially with the velocity of
layer A , the momentum exchange exerts a shear force F on the surface of layer A in the direction of the
motion:

F U Au cc (5.7)

0

The shear stress per unit area would then be

U cc (5.8)

In reality u c and X c vary continuously at any point with in th e flow, and the shear stress W is proportional
to the average value of the product of u ' and X',

W F / Au

0

. In a river, the velocity generally increases
with distance from the bed. If the direction of u' is taken as positive when it coincides with the flow
direction, and the direction of X c is positive when it is directed upward from the bed, then the sign of u c
is always opposite to that of

W Ucc u

X c For such a system, the preceding equation should be rewritten as:
u (5.9)

This is the conventional form for turbulent shear stress. Since the nineteenth century, much of the study
concerning turbulence has been an attempt to transform the pattern of velocity fluctuation into a function
of the time-averaged velocity and the consequent establishment of a relation between the distribution of
velocity and stress.
In 1925 Prandtl first simulated the momentum exchange on a macro-scale in an effort to explain the
mixing phenomenon induced by turbulence in water flow; and, he, thus, established the mixing-length
theory for turbulent flow. He assumed

W Ucc

d u

X c

c

~ ul (5.10)

d y

where l is called the Prandtl mixing length. Then Eq. (5.9) can be written as

2 dd
dd

uu

WU l (5.11)

yy

In analogy to the equation for laminar flow, the equation is rewritten as
d
d

u
y

WK (5.12)

t

in which

2 d
d

u

KU l

t

y

is called the eddy viscosity. If the Reynolds number of the flow is not much larger than its critical value,
so that turbulent and viscous effects are both important, the shear between fluid layers is the sum of the
shears due to turbulence and viscosity, i.e.

d u

y

WPK ( ) d (5.13)

t