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.

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.

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

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