Dynamics of a slowly-varying sand bed in a circular pipe
Résumé
The long wave-length dynamics and stability of a bed of sand occupying the lower segment of a circular pipe are studied analytically up to first-order in the small parameter characterizing the slope of the bed. The bed is assumed to be at rest, with at most a thin sand layer (the bedload) moving at the sheared interface. When the sand bed is plane, with depth independent of position z along the axis of the pipe, the velocity of the liquid is known from previous studies of stratified laminar flow of two Newtonian liquids (the lower one with infinite viscosity representing the sand bed). When the depth of the sand bed varies with z, secondary flows develop in the cross-sectional (x, y) plane, and these are computed numerically, assuming that the sand bed remains a straight horizontal line in the cross-sectional plane. The mean shear stress acting on the perturbed sand bed is then determined both from the computed secondary flows and by means of the averaged equations of Luchini and Charru. The latter approach requires knowledge only of the flow over the unperturbed, flat sand bed, combined with an accurate approximation of the distribution of the perturbed stresses between the pipe wall and the sand bed. The perturbed stresses determined by the two methods agree well with each other. Using these stresses, it is then possible to apply standard theories of bed stability to determine the balance between the destabilizing effect of inertial (out-of-phase) stresses and the stabilizing effects of gravity and relaxation of the particle flux, and various examples are considered.
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