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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

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

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

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

The previous two equations can be combined to give

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

where

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

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

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

when $ v=v_c$ , where

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

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

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

where

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

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

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

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

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


next up previous
Next: Flow over Shallow Bump Up: Incompressible Inviscid Flow Previous: Flow Through an Orifice
Richard Fitzpatrick 2016-01-22