A Model for Flow Over Two-Dimensional Bed Forms

When non-cohesive sediment is set in motion by a unidirectional flow, waves of sand often result; moreover, these waves typically are asymmetrical with steep lee faces that produce flow separation. Behind each wave, a wake region forms which grows in height and decays in strength with distance downstream, producing a near-bottom acceleration. The no-slip condition at the bed, however, requires an internal boundary layer to form beneath the wake region which retards the flow there. The interaction of these flow processes produces a local maximum in the boundary shear stress downstream of a bump, even over an otherwise flat bed. Because the erosion rate is proportional to the stress divergence for bed load and weak suspended load transport, erosion will occur upstream of this point and desposition will occur downstream of it, thus influencing the bottom shape. Herein, the equations of motion are solved for the accelerating internal boundary layer beneath a wake. The resulting velocity and boundary shear stress fields are shown to agree well with laboratory data from flow over a negative step as well as with field data on flow over a large riverine sand wave.