A Multi-layer Boussinesq-type Wave-current Equations

There are waves and various currents in offshore areas,such as tidal currents,coastal currents,and rips.The currents not only have spatial distribution differences in horizontal direction but also have vertical distribution differences along water depth,in other words,the currents are not uniform currents in vertical direction.The vertical distribution of currents causes the influences of wave-current interaction varying in different water depths,which means the wave energy will re-distribute along water depth because of vertically non-uniform currents.Therefore,when designing coastal building or researching offshore topography evolution,the effect of currents,vertical distribution on wave propagation and deformation need to be considered.Boussinesq equations have good non-linear performance,and these type of equations is simple enough for practical application,So this research chooses Boussinesq equations to consider wave and non-uniform currents interaction problems.On the basis of Nwogu's[5]Boussinesq equation that uses the velocity u? at a specific water depth Z? as the velocity variable.this paper continues deriving Nwogu's equation to?4,and extend this 1-layer equation to N-layer equations.Then by adding a non-uniform current that has an arbitrary vertical distribution uc(z)into the equation,we get a set of multi-layer high-order Boussinesq equations,which has O(?4)dispersion,and can consider the problems of wave and non-uniform currents interaction.Base on the multi-layer equations,the wave-current coupling and decoupling equations are given respectively,and we use finite difference method(FDM)to build wave-current interaction simulator.In terms of dispersion,The equations,dispersion is accurate to O(?4)(? is relative wavenumber)when arbitrary multi-layer water depths are adopted and approximately accurate to O(?6)when a set of optional depth layers are chosen.In terms of nonlinearity,the equation is completely nonlinear when the dispersion is accurate to O(?2),and when the dispersion is accurate to O(?4),the nonlinearity is accurate to O(?).To verifing the multi-layer Boussinesq equation model,we use the model to calculate following pure wave motion and wave with uniform current motion.For pure wave motion,we simulate wave propagation under a constant water depth flume,compare the calculated wave velocity amplitude with the Stokes theoretical solution to verify the model's velocity profile calculation accuracy.Then we simulate wave propagation under Zhang' s[66]submerged breakwater and two kinds of horizontal two-dimensional complex terrain[69][70],and compare the calculated result with experimental values to verify the dispersion and nonlinearity of the equation.For wave with uniform current motion,we firstly calculate the coexistence of waves and uniform currents under a constant water flume to verify the numerical stability of the model Then we simulate wave and weak uniform current under Luth's[71]submerged breakwater and wave-blocking under Chen's[72]submerged breakwater.In order to apply this multi-layer Boussinesq model to calculate the interaction between waves and non-uniform currents,some interaction conditions between waves and currents with given vertical distribution are simulated.We calculate wave propergation in linear shear currents,power shear currents and Swan's[74]experimental currents,and the influence of the above currents on the wavelength and wave velocity amplitude is analyzed.