Sub-Critical and Super-Critical Flow

Consider a shallow stream of depth $h$ , uniform width, and uniform flow velocity $v$ , that is fed from a deep reservoir whose surface lies a height $H$ above the (horizontal) bed of the stream. Here, $H$ is usually referred to as the head height. Assuming that the water in the reservoir is effectively stationary, application of Bernoulli's equation to a streamline lying on the surface of the water (where the pressure is atmospheric) yields

$$H = h + \frac{v^{\,2}}{2\,g}.$$ (4.18)

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

$$Q = h\,v.$$ (4.19)

The previous two equations can be combined to give

$$H =F(v),$$ (4.20)

where

$$F(v) = \frac{Q}{v} + \frac{v^{\,2}}{2\,g}.$$ (4.21)

It is easily demonstrated that the function $F(v)$ attains its minimum value,

$$F_{\rm c} = \frac{3}{2}\left(\frac{Q^{\,2}}{g}\right)^{1/3},$$ (4.22)

when $v=v_c$ , where

$$v_c = (Q\,g)^{1/3}.$$ (4.23)

We conclude that, as long as $H>(3/2)\,(Q^{\,2}/g)^{1/3}$ , 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, $v_-$ , which is such that $v_-<v_c$ . In the other, the stream flows at a relatively fast velocity, $v_+$ , which is such that $v_+>v_c$ . The corresponding depths are $h_-= Q/v_-$ and $h_+= Q/v_+$ , respectively.

It is helpful to introduce the dimensionless Froude number,

$${\rm Fr} = \frac{v}{\sqrt{g\,h}}.$$ (4.24)

(See Section 1.15.) Note that $\sqrt{g\,h}$ is the characteristic propagation velocity of a gravity wave in shallow water of depth $h$ . (See Section 11.4.) Hence, if ${\rm Fr}<1$ then the stream's flow velocity falls below the wave speed--such flow is termed sub-critical. On the other hand, if ${\rm Fr}>1$ then the flow velocity exceeds the wave speed--such flow is termed super-critical.

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

$$H = G({\rm Fr}),$$ (4.25)

where

$$G({\rm Fr}) = \left(\frac{Q^{\,2}}{g}\right)^{1/2}\left(\frac{1}{{\rm Fr}^{\,2/3}}+\frac{{\rm Fr}^{\,4/3}}{2}\right).$$ (4.26)

It is easily demonstrated that $G({\rm Fr})$ attains its minimum value

$$G_{\rm c} = \frac{3}{2}\left(\frac{Q^{\,2}}{g}\right)^{1/3},$$ (4.27)

when ${\rm Fr}={\rm Fr}_{\rm c}$ , where

$${\rm Fr}_{\rm c} = 1.$$ (4.28)

Hence, we again conclude that, as long as $H>(3/2)\,(Q^{\,2}/g)^{1/3}$ , 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., ${\rm Fr}<1$ ), whereas the larger velocity is super-critical (i.e., ${\rm Fr}>1$ ).