46 HYDRAULIC JUMP IN NON-RECTANGULAR CHANNELS 



few pipe diameters downstream from the jump. If enough air is not 

 supplied above the jump, the pipe may fill with water and the jump 

 cease to exist. The high velocity filaments will then continue down- 

 stream for a considerable distance. For an extremely high velocity, 

 water vapor could presumably take the place of the air, with the jump 

 forming as though air were being admitted. Limitations of the appa- 

 ratus prevented testing this hypothesis. The condition would be unde- 

 sirable in practice, because of the vibration and cavitation that would 

 probably develop. 



PROBLEMS 



403. A trapezoidal channel with bottom width of 10 feet and side slopes of 4 

 vertical to 3 horizontal carries 500 c.f.s. flowing at a depth of 2 feet. If a jump 

 forms in the channel, what will be the depth after the jump? 



404. In experiments on the jump in a circular conduit 0.492 foot in diameter, 

 the depth before the jump was 0.200 foot and the discharge was 0.667 c.f.s. The 

 observed pressure head after the jump was 0.619 foot, measured above the center 

 line of the pipe. Are these data in agreement with the momentum theory? 



Effect of non-uniform velocity distribution. In the development of 

 the relation between the alternate depths of flow at constant total 

 head, and of the momentum theory giving the depths before and after 

 a hydraulic jump, the velocity distribution has been assumed to be 

 uniform throughout each section. This assumption gives results that 

 are substantially in accordance with experimental results, if the dis- 

 tribution is like that found in uniform or gradually varied flow. If 

 the distribution is markedly non-uniform, or if back currents or large 

 eddies exist, the assumption may lead to conclusions considerably at 

 variance with the facts. Such conditions are frequently found at sudden 

 changes in cross section, particularly at enlargements. 



Figure 404 shows the velocity through 



a differential element dA of a cross sec- 

 tion perpendicular to the axis of a flow- 



[^j;]; ing stream. In general, the velocity 



P .Q. through dA would not be parallel to 



the axis of the stream. Let v represent 

 the resultant velocity at dA and Vn = v cos 6 its component normal to 

 the cross section, 6 being measured from the normal pointing down- 

 stream. The average velocity across the entire section is represented 

 by V, as before. The downstream flow through dA is equal to the 

 component Vn times the area of the infinitesimal element, or 



dQ = VndA 



