Sediment Movement in Alluvial Rivers - River Dynamics and Integrated River Management - page 276

eddies is dissipated locally into thermal energy due to the viscosity of water. These processes describe
completely the way flow energy is transformed.
The energy taken in unit time from an element of water at y from the bed is given as follows
b wJusu

J J (5.20)

From the balance of the forces acting on the element abcd in Fig. 5.17, one can write
d

§ W ·

W dd yx W d x J dd x s 0

¨ ¸

d y

© ¹

In reduced form

d W J

s 0

d y

Substituting this relation into Eq. (5.20), one gets

d
d

W

b wu y

(5.21)

A part of the energy taken from the water body is lost in overcoming the resistance at a given location,
as shown in Fig. 5.17(b). The water element deforms due to the forces acting on it; after a time step d t ,
the water body abcd has deformed to ab'c'd . The work done during the deformation is equal to the
product of the shear stress

d u

dW and displacement . Hence in a unit time step, the energy loss

dd ut dd yt

d y

for a unit water body at the point y above the bed in overcoming local resistance is
d
d

u

w W (5.22)

s

y

if the high-order
terms are neglected. If the terms in the equation are divided by the volume of the water body d x d y , the
energy transmitted from a unit water body at the point y above the river bed to the bed in a unit time step
is obtained:

The energy transmitted to the stream bed in a unit time is d( W xu WW d)d( x u d) u

d ()
d

w

y W

u

(5.23)

In the main flow region, part of the energy taken from each layer is lost in overcoming the local resistance,
and the surplus energy is transmitted to the boundary through the gradient of W u . However, the local energy
near the boundary is not sufficient to overcome the local resistance, and must be supplemented from the
main flow region. From the view point of the energy balance, it can be expressed as

www

(5.24)

b s t

or

d W d u d

(5.25)

u W () W u

d y d y d y

Equation (5.25) is the total differential of W , and the above derivation shows that all terms in the
equation have a definite physical significance.
The vertical distributions of w b , w t , and w s are shown in Fig. 5.18. The maximum of the mechanical
energies provided in all flow layers for overcoming the resistance occurs at the water surface, and the
minimum value of zero occurs at the stream bottom. In contrast, the energy loss due to overcoming the
local resistance is zero at the water surface and has its maximum value at the stream bottom. Thus, the
energy of the flow is mostly in the main flow region, but the loss is concentrated near the boundary. The