The equilibrium of grains on the bed of a stream

Abstract

A fluid flowing over a loose flat granular bed tends to move the grains forward, and the conditions under which movement begins depend, not only upon the speed, but also upon the nature of the motion in the stream itself: viscous steady motion, steady inviscid motion, and turbulent motion act somewhat differently, and the speed necessary to dislodge grains differs appreciably. Like any solid boundary, the loose granular bed exerts a drag upon the fluid, and the accompanying shear stress is transmitted from bed to fluid almost wholly by the more prominent grains in the uppermost layer. Each such grain transmits a small force, and the manner in which it does so, again depends upon the type of motion, though it is now the local motion which is concerned, rather than that of the main stream. Two extreme cases may be distinguished. At slow speeds and with small grains the pressure at the front of the grain does not appreciably exceed that at its rear, and the force applied is the resultant of viscous stresses acting tangentially as in figure 2. At high speeds and with large grains such tangential drag becomes relatively unimportant compared with the drag due to pressure differences: the pressure distribution is not symmetrical but is less over the downstream half of the grain, so that when integrated it gives rise to a resultant force or form drag whose component resists the motion. Both at low speeds and at high speeds, if the main stream be turbulent, the force applied to the grain fluctuates irregularly. In the low-speed case, if the main stream be steady, the force on the grain is also steady. In the highspeed case, whatever the state of the main stream, eddies are shed from the back of the grains, and grains farther downstream are at least subjected to the cumulative effect of the pulsations from upstream.