turn without destroying the flow equilibrium. Curved vanes break up the flow into a series of
small channels and since the superelevation is directly proportional to the channel width, each
small channel has a smaller superelevation. If the bend is not properly shaped or designed a
hydraulic jump may occur or the cross-waves can be amplified. There are design methods in the
literature (Rouse 1950, Ippen 1950, Chow 1959). In most cases a physical model study is
recommended.
2.8 GRADUALLY VARIED FLOW
2.8.1 Introduction
Thus far, two types of steady flow have been considered. They are uniform flow and rapidly
varied nonuniform flow. In uniform flow, acceleration forces are zero and energy is converted to
heat as a result of viscous forces within the flow. There are no changes in cross-section or flow
direction, and the depth (called normal depth) is constant. In rapidly varied flow, changes in
cross-section, direction, or depth take place in relatively short distances; acceleration forces are
not zero; and viscous forces can be neglected (at least as a first approximation).
Different conditions prevail for each of these two types of steady flow. In steady uniform flow, the
slope of the bed, the slope of the water surface and the slope of the energy gradeline are all
parallel and are equal to the head loss divided by the length of the channel in which the loss
occurred. In rapidly varied flow through short streamlined transitions, resistance is neglected and
changes in depth due to acceleration are dominant. In this section, a third type of steady flow is
considered. In this type of flow, changes in depth and velocity take place slowly over large
distances, resistance to flow dominates and acceleration forces are neglected. This type of flow
is called gradually varied flow.
In gradually varied flow, the actual flow depth y is either larger or smaller than the normal depth yo
and either larger or smaller than the critical depth yc. The water surface profiles, which are often
called backwater curves, depend on the magnitude of the actual depth of flow y in relation to the
normal depth yo and the critical depth yc. Normal depth yo is the depth of flow that would exist for
steady-uniform flow as determined using the Manning or Chezy velocity equations, and the
critical depth is the depth of flow when the Froude number equals 1.0. Reasons for the depth
being different than the normal depth are changes in slope of the bed, changes in cross-section,
obstruction to flow and imbalances between gravitational forces accelerating the flow and shear
forces retarding the flow.
In working with gradually varied flow, the first step is to determine of the general characteristics of
the water surface and what type of backwater curve would exist. The second step is to perform
the numerical computations to determine the elevation of the water surface or depth of flow.
2.8.2 Classification of Flow Profiles
The classification of flow profiles is obtained by analyzing the change of the various terms in the
total head equation in the x-direction. The total head is:
2.50
Previous Page
