166 HYDRAULICS AND ITS APPLICATIONS 



clings to the crest over its whole width. In some cases a mass of 

 turbulent water separates the nappe from the weir crest. Under such 

 circumstances Bazin found that the discharge is the same as from a thin 

 weir with drowned nappe, and is given by 



Q = K' b H*, where K' = K j '878 + '128 ~ 



P being the depth of the approach channel below the weir crest, and K 

 being the coefficient for a thin-crested weir freely discharging under the 

 same head. 



ART. 57. RATIONAL FORMULA FOR FLOW OVER BROAD-CRESTED WEIRS. 



As was first pointed out by Dr. W. C. Vnwin, a rational formula for flow 

 over a weir of this type may be deduced if the crest be assumed to be so 

 wide that the filaments form a parallel stream of thickness t (Fig. 88^ 

 before leaving the crest, and that in this stream the pressure at any 

 point is that statically corresponding to its depth. We then have the 

 velocity at the surface and at every point in this stream = ^/ 2 g (H t), 



while Q = b t J 2 g (H t). 



The value of t will adjust itself to give a maximum discharge, and this 



theoretical value for maximum flow may be determined by equating ^- ^ 



cl t 



2 



to zero. This gives t = -^ H , and substituting this value, we get Q = 



385 b s/ 2 g H 2 as the maximum possible discharge. Writing this in 

 the usual form Q = ~C b *J~^Tg H*, we get C = '573, and K = 3'087. 



o 



This method of treatment becomes more rational if account be taken of 

 the fact that in a parallel stream flowing in an open channel, the distri- 

 bution of velocity over any vertical is not uniform, being a maximum at 

 or near the surface and a minimum at the bottom. 



Experiments show that the ratio of the mean velocity over the section 

 of such a stream to the maximum surface velocity, while varying with 

 the depth, width, and roughness of the bottom of the channel, lies 

 between the limits *82 and "87 for such surfaces and depths as are 

 common on the crest of such weirs, this ratio increasing with the depth 

 of water. 



Assuming, as is practically the case, that the maximum surface velocity 

 in the case of the weir is equal to V 2 g (H t), the mean velocity will 

 equal k V 2 g (H t), and the discharge will be given by 



