Flow over Shallow Bump

Figure 4.5: Flow over a shallow bump.

Consider a shallow stream of depth $H$ , uniform width, and uniform flow velocity $V$ . Suppose that there is a very shallow bump of height $d\ll H$ on the (horizontal) bed of the stream, as shown in Figure 4.5. Suppose, further, that, at the point where the stream passes over the top of the bump, its velocity is $v$ , and its surface rises a height $h\ll H$ above the unperturbed surface.

Fluid continuity yields

$$H\,V = (H+h-d)\,v.$$ (4.29)

Furthermore, application of Bernoulli's equation to a streamline lying on the surface of the water (where the pressure is atmospheric) gives

$$g\,H + \frac{1}{2}\,V^{\,2} = g\,(H+h)+\frac{1}{2}\,v^{\,2}.$$ (4.30)

The previous equation reduces to

$$v^{\,2}= V^{\,2}-2\,g\,h.$$ (4.31)

Eliminating $v$ between Equations (4.29) and (4.31), we obtain

$$H^{\,2}\,V^{\,2} = (H+h-d)^{\,2}\,(V^{\,2}-2\,g\,h),$$ (4.32)

which can be rearranged to give

$$V^{\,2}\left[(H+h-d)^2-H^{\,2}\right]=2\,g\,h\,(H+h-d)^2,$$ (4.33)

or

$${\rm Fr}^{\,2}\left[\left(1+\frac{h-d}{H}\right)^2-1\right] = 2\,\frac{h}{H}\left(1+\frac{h-d}{H}\right)^{\,2}.$$ (4.34)

Here,

$${\rm Fr} = \frac{V}{\sqrt{g\,H}}$$ (4.35)

is the Froude number of the unperturbed flow. Finally, given that $h/H\ll 1$ and $d/H\ll 1$ , Equation (4.34) reduces to

$$h\simeq \frac{d}{1-1/{\rm Fr}^{\,2}}.$$ (4.36)

It follows, from the previous expression, that if the flow is super-critical, so that ${\rm Fr}>1$ , then $h$ is positive. On the other hand, if the flow is sub-critical, so that ${\rm Fr}<1$ , then $h$ is negative. Thus, if a super-critical shallow stream passes over a very shallow bump on its bed then the surface of the stream becomes slightly elevated. On the other hand, if a sub-critical stream passes over such a bump then the surface of the stream becomes slightly depressed. A similar effect occurs when there is a narrowing of the channel in the horizontal direction. A more sophisticated version of the previous calculation, which does not necessarily assume that the stream is shallow, can be found in Section 11.10.