Improved Numerical Modeling for Perturbations in Homogeneous and Stratified Flows - Advances in Control and Automation of Water Systems - page 144

8.1 introduction
If on the surface of the deep pool filled with water to create a disturbance, then the
surface of the water will begin to propagate. Their origin is explained by the fact that
the fluid particles are located near the cavity. They create disturbances which will seek
to fill the cavity under the influence of gravity. The development of this phenomenon
is led to the spread of waves on the water. This chapter formulated the perturbations in
homogeneous and stratified flows by improved numerical modeling.
One of the problems in the study of fluid flow in plumbing systems is the behav-
ior of stratified fluid in the channels. Mostly steady flows initially are ideal, then the
viscous and turbulent fluid in the pipes. The origin of propagation due to disturbance
in deep pool which filled with water is explained by the fact that the fluid particles
are located near the cavity. They create disturbances which will seek to fill the cav-
ity under the influence of gravity. The development of this phenomenon is led to the
spread of waves on the water. The fluid particles in such a wave do not move up and
down around in circles. The waves of water are neither longitudinal nor transverse.
They seem to be a mixture of both. The radius of the circles varies with depth of mov-
ing fluid particles. They reduce to as long as they do not become equal to zero. If we
analyze the propagation velocity of waves on water, it will be reveal that the velocity
of waves depends on length of waves.

8.2 mAteriAls And metHods
The speed of long waves is proportional to the square root of the acceleration of grav-
ity multiplied by the wavelength v

F =l . The cause of these waves is the force of
gravity. For short waves the restoring force due to surface tension force, and therefore
the speed of these waves is proportional to the square root of the private. The numera-
tor of which is the surface tension, a nd in t he denominator--the product of the wave-
length to the density of water

g

v F = s lr .
Suppose there is a channel with a constant slope bottom, extending to infinity
along the axis Ox . And let the feed in a field of gravity flows, incompressible fluid. It
is assumed that the fluid is devoid of internal friction. Friction neglects on the sides
and bottom of the channel. The liquid level is above the bottom of the channel h . A
small quantity compared with the characteristic dimensions of the flow, the size of the
bottom roughness, and so on. Let

/

=x+ , where h --ordinate denotes the free
surface of the liquid (Figure 8.1). Free liquid surface h (Figure 8.1), which is in equi-
librium in the gravity field is flat. As a result of any external influence, liquid surface
in a location removed from its equilibrium position. There is a movement spreading
across the entire surface of the liquid in the form of waves, called gravity. They are
caused by the action of gravity field. This type of waves occurs mainly on the liquid
surface. They capture the inner layers, the deeper for the smaller liquid surface [1].
We assume that the fluid flow is characterized by a spatial variable x and time
dependent t . Thus, it is believed that the fluid velocity u has a nonzero component
u which will be denoted by u (other components can be neglected) in addition, the
level of h depends only on x and t .

h h

0