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It is the inclusion of the channel slope and friction that allow us to decide which of the two depths is correct.

**Gradually varied flow **

In the previous section of rapidly varied flow
little mention was made of losses due to friction or the influence of the bed
slope. It was assumed that frictional losses were insignificant - this is
reasonable because rapidly varied flow occurs over a very short distance.
However when it comes to long distances they become very important, and as
gradually varied flow occurs over long distances we will consider friction
losses here.

In the section on specific energy it was noted
that there are two depth possible in steady flow for a given discharge at any
point in the channel. (One is super-critical the other depth sub-critical.) The
solution of the Manning equation results in only one depth - the
normal depth.

It is the inclusion of the channel slope and friction
that allow us to decide which of the two depths is correct. i.e. the channel
slope and friction determine whether the uniform flow in the channel is sub or
super-critical.

The
procedure is

i. Calculate
the normal depth from Manning's equation

ii. Calculate
the critical depth from equation

The
normal depth may be greater, less than or equal to the critical depth.

For a given channel and roughness
there is only one slope that will give the normal depth equal to the critical
depth. This slope is known as the critical slope ( Sc ).

If the slope is less than Sc the
normal depth will be greater than critical depth and the flow will be
sub-critical flow. The slope is termed mild .

If the slope is greater than Sc
the normal depth will be less than critical depth and the flow will be
super-critical flow. The slope is termed steep .

**Problem of critical slope calculation**

We have
Equation that gives normal depth and equation that given critical depth
Rearranging these in terms of Q and equating gives

Tags : civil - Applied Hydraulic Engineering: Gradually Varied Flow

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