## Classificatin of Open Channel Flow

Consider the steady, uniform flow of an inviscid, incompressible liquid in an open channel with a rectangular cross section of constant width. If the width is and the depth is the volume flow rate where is the speed of the liquid. is constant from the continuity equation, since the liquid does not accumulate in any part of the channel. If is the volume flow rate per unit width then We apply Bernoulli's equation along a streamline in the free surface of the liquid in a flat bottomed channel, giving where is the atmospheric pressure, assumed constant. Since the density is also constant we can subtract from each side, giving Dividing by gives where is called the specific energy.

If is constant then so or The last equation is a cubic in h, so may have 3 distinct real roots or one. If there are three roots for this particular equation, one of the roots is negative, so physically meaningless, so for given and there are either one or two possible values of There are three possible cases.

1. If two flows are possible at different depths and corresponding to speeds and are possible. and since The first is described as shallow and fast or supercritical, the second as deep and slow or subcritical.

2. If the flow is unique. The liquid flows with depth and speed and are the critical speed and critical depth respectively.

3. If then no flow is possible.

All the cases are illustrated below. Since the critical depth occurs at a minimum of the specific energy function, we can differentiate and equate to zero to find  The minimum value of the specific energy is then 