The flow of cohesionless grains in fluids

Abstract Part I. The results of previous experiments (Bagnold 1954) on the stresses set up in a uniform gravity-free dispersion of solid grains when uniformly sheared in a fluid are applied to the nonuniform case of grain flow over a gravity bed, assuming the results are quantitatively applicable to any sufficiently thin shear layer. It is found that if the bed is composed entirely of potentially mobile grains a stress-equilibrium relation at the bed surface can be defined whereby the magnitude of a certain ‘bed load’ of grains in transit over unit bed area is given in terms of the applied tangential stress. The bed load is independent both of the existence of any additional suspended load and of the degree of dispersion of the grains. The state of internal fluid motion enters as a single experimental constant. From a consideration of the stability of this equilibrium relation it is possible to predict the conditions under which an initially plane bed surface should become rippled; and general quantitative agreement is found with experimental data both for wind-blown and water-driven grains. Primary and secondary bed rippling are distinguished. The magnitude of the ‘form-drag’ due to primary bed ripples can be calculated. That due to secondary ripples is definable as an experimental constant. The gravity-free experiments disclosed that the shear resistance of a grain dispersion may vary as the square or the first power of the rate of shear, analogously to that of a true fluid, according to the value of a number G analogous to a Reynolds number. The square law is followed when the effects of grain inertia dominate over those of fluid viscosity. Assuming that the phenomenon of ‘saltation’ as observed over a gravity bed is an inertia effect, the conditions for saltation are predictable. The results again agree quantitatively with observation. Part II. The resistance offered by the grains to their displacement along the flow is shown to be proportional to their normal immersed weight component. And their measurable mass transport rate is hereby proportional to the rate of useful work done in transporting them. On this basis separate expressions are found for the transport rates of the bed load and suspended load, in terms of the applied tangential stress and of a tangential and a normal relative velocity respectively. When conditions are restricted to those of the ‘stream case’ these velocities become constant for any given system, being functions of an appropriate constant mean drag coefficient. The bed-load transport-rate expression found gives magnitudes, and variations of magnitude with grain size, in agreement with the experimental data for wind-blown sand. Agreement is also found for water-driven grains in open channels from the threshold of movement up to a certain value of the applied stress. The experimental rates are then found to increase suddenly. This increase is attributed to the development of an additional suspended load. The abrupt development of a suspended load may be explained as due to a change in the nature of the fluid turbulence when the stationary boundary becomes occluded from the fluid flow by a concentrated layer of moving bed-load grains. The assumption that under these new moving-boundary conditions the available fluid energy derived from shearing over the bed is equally apportioned between bed-load and suspended-load transport work leads to values for the suspended-load transport rate which agree closely with the experimental data. A critical relation emerges between the gravity slope of the bed, the fall velocity and the mean transport velocity of the suspended grains at which their transport may become very large. Conditions are examined under which the steady transport may be possible of grains of heterogeneous size or density. Part III. When the fluid flow is non-inertial (laminar) and the grain flow is also non-inertial the semi-empirical relations found previously for the internal stresses are such that both viscosity and shear rate can be eliminated, and a differential equation obtained whose solution gives the grain concentration in terms only of distance from the bed and of the applied tangential stress. It appears that with constant applied stress (unlimited flow depth) the degree of grain dispersion greatly exceeds that to be expected in turbulent fluid flow. But when the applied stress diminishes linearly with distance from the bed boundary a possible solution gives constant grain concentration throughout the flow. This appears to explain certain experimental results, including the behaviour of ‘slurries’. The effect is examined of a fixed or partially fixed bed on the grain flow in a turbulent fluid. The effect may be pronounced in the case of suspended grains. Under certain clearly definable conditions a loose grain bed must cease to remain stationary. And if the fluid flow above is turbulent the whole grain bed should flow at constant maximum concentration, underneath the flow proper and separated from it by a moving-bed surface interface at which the concentration is discontinuous. This explains phenomena sometimes found under river torrents. The factors giving rise to and limiting the development of bed features (dunes) on a bigger scale than ripples are examined. Dune formation appears as an inherent tendency of the grain flow alone, which may or may not be inhibited by the conditions of the fluid flow.