Question

A rectangular channel 2 m wide has a bottom slope of 0.01 and a roughness
coefficient of 0.017. A sluice gate 2 m wide is located in the channel. The sluice
gate has a contraction coefficient of 0.61 and a discharge coefficient of 0.58. The
depth of water just upstream from the gate is 2 m. The depth of water just
downstream from the gate is 0.5 m. Determine how far upstream from the gate
the depth of water will be within 1% of the normal depth.

Answer 1

In open channel flows one of the important parameter is "specific energy" and is defined as,

(1)

where is the water depth and is the flow velocity. It is also called as energy grade line (EGL). For a given flow rate, there are two possible states for the same specific energy.

In a simpler case, consider two possible states of specific energy in a rectangular channel of width . The discharge per unit width for this channel is given by,

Eq. (1) can now be written as,
      

For a given channel of constant width, the value of remains constant along the channel although the depth may vary. The variation of with is plotted in specific energy diagram (Fig. 2). From this curve, it is clear that specific energy attains to a minimum value at certain depth for a given . This depth is known as critical depth and it can be obtained by setting in Eq. (2).

Minimum specific energy occurs at,
                                              (3)

The velocity of flow at "critical depth" is known as "critical velocity" and the corresponding discharge is .
Referring to Fig. 2, for , no solution exists and thus the flow is unrealistic. For , there are two possible solutions;

• Large depth with Þ sub-critical flow.
• Small depth with Þ s called super-critical flow.