Sub-Critical and Super-Critical Flow

Let be the flow rate per unit width of the stream, which is assumed to be fixed. It follows that

The previous two equations can be combined to give

where

It is easily demonstrated that the function attains its minimum value,

when , where

We conclude that, as long as , Equation (4.20) possesses two possible solutions that are consistent with a given head height and flow rate. In one solution, the stream flows at a relatively slow velocity, , which is such that . In the other, the stream flows at a relatively fast velocity, , which is such that . The corresponding depths are and , respectively.

It is helpful to introduce the dimensionless *Froude number*,

(See Section 1.15.) Note that is the characteristic propagation velocity of a gravity wave in shallow water of depth . (See Section 11.4.) Hence, if then the stream's flow velocity falls below the wave speed--such flow is termed

We can combine Equations (4.18), (4.19), and (4.24) to give

(4.25) |

where

(4.26) |

It is easily demonstrated that attains its minimum value

(4.27) |

when , where

(4.28) |

Hence, we again conclude that, as long as , there are two possible flow velocities of the stream (parameterized by two different Froude numbers) that are consistent with a given head height and flow rate. However, it is now clear that the smaller velocity is sub-critical (i.e., ), whereas the larger velocity is super-critical (i.e., ).