Rapidly Varying Flow - Hydraulic Jumps

 

Example:   The figure below shows the flow under a sluice gate where  y1 = 2 inches and

V1 = 17ft/sec.    Determine the depth of flow downstream of the jump and the head loss

across the jump.

                           

                                       

Strategy:  Calculate the Froude Number at the upstream station.  If  Fr1 > 1, then the flow is

unstable and results in energy dissipation producing a hydraulic jump.  Then apply the following

relations to determine the downstream depth and energy loss across the jump:

 

                                               ( y2 /y1 )   =   [ -1  +  √( 1 + 8 Fr12)] / 2  

and

                                                    hL / y1   =   Fr12/2  [ 1 – (y1/y2)2] + [1– (y2/ y1)]    

 

    Fr1  =  17/√ 32.2)(2/12)  =  7.3    (therefore supercritical flow upstream)

 

        ( y2 /y1 )   =   [ -1  +  √( 1 + 8 (7.3)2)] / 2   =  9.88       y2  =  19.8 inches   (result)

 

 Next calculate the head loss.           y2 /y1   =  9.88   ,       y1/y2  =  0.1011

 

So               hL / y1   =   7.32/2  [ 1 – (0.1011)2] + [1– 9.4)]   =  17.76

 

                           hL  =   35.5  in  or  2.96 ft               (result)

 

Initial Specific energy  =  y1 + V12/2g  =  2/12 + (17)2/64.4  =  4.65 ft

 

       hL / E1  =  2.96/4.65  =  0.64    Hydraulic jump dissipates 64% of incoming energy.

 

Click here to examine the specific energy diagram for this example.

 


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