Plane Turbulent Wall Jets on Rough Boundaries with Limited Tailwater

Combining the results of a laboratory study of plane turbulent wall jets on rough boundaries with shallow tailwater, with the results of an earlier work of Rajaratnam on wall jets on rough boundaries with deep tailwater, this paper attempts to describe the effects of boundary roughness and tailwater depth on the characteristics of plane turbulent wall jets on rough beds, which are important in the field of hydraulic engineering. The time-averaged axial velocity profiles at different sections in the wall jet were found to be similar, with some difference from the profile of the classical plane wall jet. The normalized boundary layer thickness $\delta /b$, where $b$ is the length scale of the velocity profile, was equal to 0.35 for wall jets on rough boundaries compared to 0.16 for the classic wall jet. Two stages were seen to exist in the decay of the maximum velocity $u_m$ as well as in the growth of the length scale, with the first stage corresponding to that of deep tailwater and the second stage to shallow tailwater. In the first stage, the decay of the maximum velocity $u_m$ at any section in terms of the velocity $u_0$ at the slot, with the longitudinal distance $x$ in terms of $L$ which is the distance where $u_m=0.5U_0$, was described by one general function, for smooth as well as rough boundaries. The length scale $L$ in terms of slot width decreases as the relative roughness of the boundary increases. The onset of the second stage was not affected significantly by the bed roughness. The growth rate of the length scale $b$ of the wall jet increased from 0.076 for a smooth boundary to about 0.125 for a relative roughness $k_s/b_0$ in the range of 0.25 to 0.50, where $k_s$ is the equivalent sand roughness and $b_0$ is the thickness of the jet at the slot.