Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation. II. Fourth, sixth and eighth orders
In the numerical simulation of medium-length surface waves in the framework of nonlinear dispersive (NLD) models, an increased accuracy of reproducing the characteristics of the simulated processes is required. A number of works (Kirby (2016), e.g.) describe approaches to improve the known NLD-models. In particular, NLD-models of the fourth order of the long-wave approximation have been proposed and, based on a comparison of numerical results with experimental data, their high accuracy has been demonstrated (Ataie-Ashtiani and Najafi-Jilani (2007); Zhou and Teng (2010)). In these new models, the horizontal component of the velocity vector of the threedimensional (FNPF-) model of potential flows at a certain surface located between the bottom and the free boundary was chosen as the velocity vector. The result was a very cumbersome form of equations. In addition, the laws of conservation of mass and momentum do not hold for these models. The main result of this work is the derivation of a two-parameter fully nonlinear weakly dispersive (mSGN4) model of the fourth order of the long-wave approximation, which is a generalization of the well-known Serre-Green-Naghdi (SGN) second order model. In the derivation, the velocity averaged over the thickness of the liquid layer was used. The assumption about the potentiality of the three-dimensional flow was used only at the stage of closing the model. The movement of the bottom is taken into account. For the derived model, the law of conservation of mass is satisfied, and the law of conservation of total momentum is satisfied in the case of a horizontal stationary bottom. The equations of the mSGN4-model are invariant under the Galilean transformation and are presented in a compact form similar to the equations of gas dynamics. The dispersion relation of the mSGN4-model has the fourth order of accuracy in the long wave region and satisfactorily approximates the dispersion relation of the FNPF-model in the short wave region. Moreover, with a special choice of the values of the model parameters, an increased accuracy of approximating the dispersion relation of the FNPF-model at long waves (sixth or eighth order) is achieved. Analysis of the deviations of the values of the phase velocity of the mSGN4 model from the values of the “reference” speed of the FNPF model in the entire wavelength range showed that the most preferable is the mSGN4 model with the parameter values corresponding to the Pad’e approximant (2,4).