TABLE OF CONTENTS A flow control is any feature that imposes a relationship between the flow depth and discharge in a channel. A critical flow section, for instance, is a flow control, since at this section F r = 1.0. Likewise, various hydraulic structures such as weirs and gates will control the flow. Normal flow may be viewed as a flow control also because a normal flow equation like Equation 3.26 describes a depth-discharge relationship. In the absence of other flow controls, the flow in an open channel tends to become normal. However, where present, the other controls will pull the flow away from the normal flow conditions. The flow depth varies between two flow controls. Such a non-uniform flow is called gradually-varied flow if the changes in the flow depth are gradual. This chapter is devoted to steady, gradually-varied flow.
To obtain an expression for gradually-varied flow, let us recall Equation 2.9, defining the total energy head, H, as
where z b = elevation of the channel bottom, y = flow depth, V = average cross-sectional velocity, and g = gravitational acceleration. Now, recalling the definition of specific energy given in Equation 2.11 as
Equation 4.1 can be expressed as
Let us differentiate both sides of Equation 4.3 with respect to x to obtain
where x is the displacement in the flow direction. By definition, S f = ? dH/dx, and S 0 = ? dz b /dx.
