Open Channel Flow   Gradually Varying Flow  (continued)

 

Recall that the Froude number, Fr, is an important parameter governing the flow in open channels.

Fr = V/Cwave = V/√gy  where V is the flow velocity, y is the depth, and g is the acceleration of gravity.

 

For Fr > 1  the flow is supercritical (and unstable).  For  Fr < 1 the flow is subcritical.

The figure below depicts a channel with the length along the channel taken as horizontal and

the depth, y, along the channel taken as vertical (since slopes are typically very small).  Let  So

denote the slope of the channel.  Then  So  =  (z1 – z2 )/ L  and   z1 – z2  =  So L.

 

                                 

 

 

For conservation of energy:

 

                   P1/γ  +  V12/2g  +  z1  =   P2/γ  +  V22/2g  +  z2  +  hL

 

where  Pi  is the hydrostatic pressure at station  i  .  So  P1  =  γy1  and  P2  =  γy2 

 

   So                  y1 +  V12/2g  +  z1  =   y2  +  V22/2g  +  z2  +  hL

 

Define  Sf, the ”friction slope” such that   hL = Sf L.  Then the energy equation becomes:

 

                           y1 +  V12/2g   =   y2  +  V22/2g  +  ( Sf  – So ) L

 

In equilibrium (no loss of energy)   Sf  = So .    Gravity adds energy while friction

removes it.

                    Specific Energy  =  y  +  V2 / 2g    (definition)  i.e.  E1  =  E2 + ( Sf  – So ) L

 

 

Click here to continue discussion.

 

 

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