298 HYDRAULICS AND ITS APPLICATIONS 



Experiments by Mr. E. C. Thntpp 1 on the Thames and the Kennet 

 the former having a mean depth of 7'6 feet and the latter of 2'4 feet, 

 showed that velocities of '665 feet per second in the former, and '64 feet 

 per second in the latter case, were below the critical, and that for all 

 smaller values the velocity was practically proportional to the surface 

 slope. The experiments did not, however, indicate the point at which 

 the velocity-slope law changed. 



Although, owing to the difficulty of measuring such small differences of 

 head as are involved in flow at low velocities, accurate determinations are 

 not practicable, yet the values quoted above show that this critical velocity 

 is immensely high compared with that calculated from Pteynold's formula 

 (p. 55) for a cylindrical pipe of the same hydraulic mean depth. The 

 existence of a critical velocity would at once explain the great discrepancy 

 which in some cases exists between the results of experiments on channels 

 having similar physical characteristics, but with very different velocities 

 of flow. 



ART. 87. FORM OF CHANNEL. 



Since for a channel of given sectional area A t the hydraulic mean 

 depth A -r- P varies with the form of its section, while the resistance to 

 flow increases as A -r- P diminishes, it becomes important to determine 

 what form of channel will give the maximum value of A -f- P for a 

 given value of A t since this will be the channel of maximum discharge 

 for a given slope. Further, if this sectional area is a minimum, the 

 cost of excavation is a minimum, and since in general the perimeter 

 increases with the sectional area, the cost of pitching the faces of the 

 channel is also a minimum. Theoretically, the best form of channel is 

 one in which the bed is a circular arc, s"ince this gives a minimum ratio 

 of wetted perimeter' to sectional area. 



An investigation into the properties of different sections will be simpli- 

 fied if the coefficient C in the formula v = C V . i, be assumed 

 constant for a given surface. On this assumption we have: 



V~j8 

 . i cub ft. per sec. 



A (A \ 



For v to be a maximum, =- must be a maximum, so that d ( ^ J = 



/. P d A - A d P = 0. (1) 



1 "Proc. Inst. O.E.," vol. 171, 1907-8, p. 346. 



