CHAPTER VII. RELATION OF CAPACITY TO VELOCITY. 



PRELIMINARY CONSIDERATIONS. 



The work of stream traction is accomplished 

 by the movement of water along the bed of the 

 channel. For that reason the system of water 

 movements and water velocities near the bed 

 is intimately related to the load or capacity for 

 load. In certain parts of this paper and in 

 the writings of some other investigators use is 

 made of the term "bed velocity," or its equiva- 

 lent, but the term has no satisfactory defini- 

 tion. The difficulties which are encountered 

 in this connection have to do also with the 

 vertical velocity curve. 



In all the streams with which we are here 

 concerned the flow is eddying or turbulent. 

 At any point the direction of motion and the 



FIGITRE 52. Vertical velocity curve, drawn to illustrate its theoretic 

 character near the stream's bed. OD is the origin of velocities. 



velocity are constantly changing. If a mean 

 be taken of the instantaneous forward com- 

 ponents of velocity the components parallel 

 to the axis of the stream it gives for the 

 point a mean velocity coordinate with the 

 mean velocity for the cross section obtained 

 by dividing the discharge by the sectional area. 

 It will bo observed that the mean at a point is 

 a mean with respect to time, while the sec- 

 tional mean is primarily a mean with respect to 

 space. The mean at a point, as thus defined, 

 being called V p , the vertical velocity curve 

 may be defined as the curve obtained by plot- 

 ting the values of V p for any vertical of the 

 current in their relation to depth. As commonly 

 drawn by hydraulic engineers, it terminates 

 downward at some distance from the origin of 

 velocities, OD say at B in figure 52 connoting 

 a finite velocity for the water in actual con- 



tact with the bed. This implication contra- 

 venes a theorem of hydrodynamics that the 

 velocity at contact with the wall of a con- 

 duit is either zero or indefinitely small. The 

 theorem is believed to have been established 

 experimentally by the work of J. L. M. 

 Poiseuille * and is generally accepted. In the 

 direct study of the velocities of streams instru- 

 mental observation is not carried from surface 

 to bed, but ceases at some point, C, and the 

 drawing of the curve below that point is a 

 matter of inference. The inference accordant 

 with the hydrodynamic principle is that the 

 curve changes its course below C and reaches 

 the origin at or near O. 1 This inference accords 

 also with our observations in connection with 

 the study of saltation (see p. 29); and those 

 observations suggest likewise that the curve is 

 materially modified by the resistances to the 

 current involved in the work of saltation. 



It thus appears that in the region with which 

 traction studies are specially concerned the 

 range of V p is great. The work of traction 

 depends on a system of velocities and nob on a 

 single one, and there is no individual value of 

 V p with special claim to the title "bed veloc- 

 ity." It would be possible to define bed 

 velocity as the value of V p at some particular 

 distance from the bed or at a distance consti- 

 tuting some particular fraction of the depth of 

 current; but such a definition would be hard to 

 apply. 



However smooth a stream bed of debris may 

 be in its general aspect, it is never smooth as 

 regards details. Figure 53 gives an ideal pro- 



FIGURE 53. Ideal profile of a stream bed composed of debris grains. 



tile, the intersection of a bed by a vertical plane. 

 Not only are there salients and reentrants, but 

 some of the reentrants communicate with the 

 voids within the mass of dfibris. In many of 



> See Lamb's Hydrodynamics, 3d ed., p. 544, 1906. 



' See Cunningham, -Allan, Hydraulic experiments at Roorkee, p. 46, 

 1 S75, and Inst. Civil Eng. Proc., vol. 71, p. 23, 1882, where he discusses 

 t he horizontal velocity curve; and Von Wagner, idem, p. 90. 



155 



