48 HYDRAULIC JUMP IN NON-RECTANGULAR CHANNELS 



values of a is high, indicating a markedly non-uniform distribution, 

 the introduction of this coefficient may cause erroneous results. Con- 

 sider a stream flowing at less than the critical velocity, with partly 

 closed gate obstructing the flow as shown in Fig. 405. At a section 



(^l 



Fig. 405. Water Flowing Under a Gate. 



immediately downstream from the gate, the horizontal component of 

 the velocity is zero over part of the cross section, say half. Assuming 

 that the velocity is uniform over the other half of the section, the 

 values of a and a are 



O.V?-\A , (2Vf-\A 



" = V^A =2 " = V'A = * 



with V representing the average velocity over the entire cross section. 

 It would lead to incorrect results, however, to use this value of a in 

 Bernoulli's theorem at an upstream section, assuming that the velocity 

 head so obtained could be reconverted into pressure or elevation head. 

 The law of conservation of energy still holds, but in this case a large 

 part of the kinetic energy is not recoverable. It will be dissipated by 

 the eddies which are generated at the plane of contact of the high 

 velocity sheet and the stagnant water above, and are carried down- 

 stream with the current. On the other hand, the use of a high value 

 of a for the downstream section in Bernoulli's equation is permissible, 

 if warranted by actual conditions. The coefficient a', however, is 

 chiefly useful in laboratory investigations where the energy loss between 

 the two sections is the unknown to be determined. 



EXERCISE 



Prove that for a non-uniform distribution of velocity, a is greater than unity. 



Channel in which the flow is critical at any stage. For such a channel 

 the velocity head must equal half the average depth at all stages, or 



A. = ^ 



If we assume the total head to remain constant, the level of the water 

 surface may be determined by its distance below the elevation rep- 



