Turbulent boundary-layer effects on transient wave propagation in shallow water

The depth-integrated continuity and momentum equations developed by Liu & Orfila are extended to include the effects of turbulent bottom boundary layer. The eddy viscosity model is employed in the boundary layer, in which the eddy viscosity is assumed to be a power function of the vertical elevation from the bottom. The leading-order effects of the turbulent boundary layer appear as a convolution integral in the depth-integrated continuity equation because of the boundary-layer displacement. The bottom stress is also expressed as a convolution integral of the depth-averaged horizontal velocity. For simple harmonic progressive waves, the analytical expression for the phase shift between the bottom stress and the depth-averaged velocity is obtained. The analytical solutions for the solitary wave damping rate due to a turbulent boundary layer are also derived. Prandtl's one-seventh power law is described in detail as an example.