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Erosion by planar turbulent wall jets

New scaling laws are presented for the spatial variation of the mean velocity and lateral extent of a two-dimensional turbulent wall jet, flowing over a fixed rough boundary. These scalings are analogous to those derived by Wygnanski et al. (1992) for the flow of a wall jet over a smooth boundary. They reveal that the characteristics of the jet depend weakly upon the roughness length associated with the boundary, as confirmed by experimental studies (Rajaratnam 1967). These laws are used in the development of an analytical framework to model the progressive erosion of an initially flat bed of grains by a turbulent jet. The grains are eroded if the shear stress, exerted on the grains at the surface of the bed, exceeds a critical value which is a function of the physical characteristics of the grains. After the wall jet has been flowing for a sufficiently long period, the boundary attains a steady state, in which the mobilizing forces associated with the jet are insufficient to further erode the boundary. The steady-state profile is calculated separately by applying critical conditions along the bed surface for the incipient motion of particles. These conditions invoke a relationship between the mobilizing force exerted by the jet, the weight of the particles and the local gradient of the bed. Use of the new scaling laws for the downstream variation of the boundary shear stress then permits the calculation of the shape of the steady-state scour pit. The predicted profiles are in good agreement with the experimental studies on the erosive action of submerged water and air jets on beds of sand and polystyrene particles (Rajaratnam 1981). The shape of the eroded boundary at intermediate times, before the steady state is attained, is elucidated by the application of a sediment-volume conservation equation. This relationship balances the rate of change of the bed elevation with the divergence of the flux of particles in motion. The flux of particles in motion is given by a semi-empirical function of the amount by which the boundary shear stress exceeds that required for incipient motion. Hence the conservation equation may be integrated to reveal the transient profiles of the eroded bed. There is good agreement between these calculated profiles and experimental observations (Rajaratnam 1981).