Two-dimensional vortex structures in the bottom boundary layer of progressive and solitary waves

The two-dimensional vortices characterizing the bottom boundary layer of both progressive and solitary waves, recently discovered by experimental flow visualizations and referred to as vortex tubes, are studied by numerical solution of the governing equations. In the case of progressive waves, the Reynolds numbers investigated belong to the subcritical range, according to Floquet linear stability theory. In such a range the periodic generation of strictly two-dimensional vortex structures is not a self-sustaining phenomenon, being the presence of appropriate ambient disturbances necessary to excite certain modes through a receptivity mechanism. In a physical experiment such disturbances may arise from several coexisting sources, among which the most likely is roughness. Therefore, in the present numerical simulations, wall imperfections of small amplitude are introduced as a source of disturbances for both types of wave, but from a macroscopic point of view the wall can be regarded as flat. The simulations show that even wall imperfections of small amplitude may cause flow instability and lead to the appearance of vortex tubes. These vortices, in turn, interact with a vortex layer adjacent to the wall and characterized by vorticity opposite to that of the vortex tubes. In a first stage such interaction gives rise to corrugation of the vortex layer and this affects the spatial distribution of the wall shear stress. In a second stage the vortex layer rolls up and pairs of counter-rotating vortices are generated, which leave the bottom because of the self-induced velocity.