Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fullynonlinearsolution for bi-chromatic progressivewaves inwater of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromaticwave field is assumed to be obtained by thenonlinearinteraction of two monochromaticwave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, andenergy fluxes based on the accurate high-order homotopy seriessolutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, andenergy fluxes before and after theinteraction of the twononlinear monochromaticwave trains are proposed to establish the relationship between the steady-state bi-chromaticwave field and the twononlinear monochromaticwave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromaticwave field, is performed to obtain a sufficiently small standard deviationSd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, andenergy fluxes before and after theinteraction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After theinteraction, the amplitude of the primarywave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of thewa...

Source: Physics of Fluids - December 8, 2016 Category: Physics Authors: Zhen Liu, Zhiliang Lin and Longbin Tao Source Type: research