Environmental Engineering Reference
In-Depth Information
will show a very unsteady flow behaviour. If the storage capacity in a canal
reach is small, the adaptation of the canal to the new boundary conditions
may be fast and the flow may pass through a series of almost steady state.
This adaptation also depends on the capability of the flow to accelerate
or decelerate. If the adaptation of the velocity of the flow particles occurs
quickly, then the network will pass through a series of nearly steady states;
otherwise, the flow in the network will clearly show an unsteady behaviour.
An example is the flow through a structure in an irrigation canal.
Usually this flow is assumed to be steady. The discharge through the
structure at any moment depends on the water level. Although the water
level may vary rapidly, the discharge generated will respond more or
less instantaneously. The immediate response of the flow to the changing
boundary conditions is due to lack of storage between the upstream and
downstream section and the relatively small water mass to be accelerated
or decelerated. Generally a hydraulic structure is a steady flow element
in a network and how the network as a whole will behave depends on the
flow characteristics of the other elements in the network.
2.9 BASIC DIFFERENTIAL EQUATIONS FOR GRADUALLY
VARIED UNSTEADY FLOW
Unsteady flow equations will describe the dependent variables, namely
the velocity
v
and water depth
y
, as function of the independent variables
x
and
t
. The definition of the two dependent variables requires the
formulation of two equations to solve them. In open channel flow, they
are usually based upon the mass (volume) and momentum conservation.
The conservation of mass assumes that the fluid is incompressible and
that the density is constant. The conservation of mass leads to the
continuity equation.
Gradually varied unsteady flow refers to an unsteady flow in which
the curvature of the wave profile is mild; the change of depth with time
is gradual; the vertical acceleration of the water particles is negligible
in comparison with the total acceleration and the effect of the boundary
friction can not be neglected. Figure 2.14 gives a schematized wave in the
x
,
t
and
y
direction as an example of unsteady flow.
Only a few gradually varied, unsteady flow problems can be solved
analytically; most problems require a numerical solution. The gradually
varied unsteady flow can be described by the de St. Venant equations,
which consist of the continuity and the dynamic equation. Unsteady flow
in open channels is assumed to be a one-dimensional flow with straight and
parallel flow lines. The dynamic equation includes the change of velocity
v
with time and consequently the acceleration, which produces the forces
and causes the energy losses in the flow.
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