Internal Wave-Maker for Navier-Stokes Equations Models

The flow motion of incompressible fluid can be described by Navier-Stokes equations with the continuity equation, which requires zero divergence of the velocity vector (i.e., ∂u_i/∂x_i = 0). A new method is developed to generate specific wave trains by using designed mass source functions for the equation of mass conservation, i.e., ∂u_i∂x_i = f(x, t), in the internal flow region. The new method removes the difficulty in specifying incident waves through an inflow boundary with the presence of strong wave reflection. Instead, only the open (radiation) boundary condition is needed in the simulation. By using different source functions, the writers are able to generate various wave trains, including the linear monochromatic wave, irregular wave, Stokes wave, solitary wave, and cnoidal wave. By comparing numerical results with analytical solutions, the writers have shown that the proposed method can accurately generate not only small amplitude waves but also nonlinear waves in both intermediate and shallow water. This method has important applications of simulating wave-current interaction, wave shoaling on a relatively steep slope, and wave-structure interaction where wave reflection is significant.