英文摘要 | This paper studies the traveling wave solutions of nonlinear water waves via asymptotic analysis and numerical calculation, under two kinds of physical effects of linear shear current and horizontal electric field. The focus of the research is on the impact of these physical effects on the water wave systems, especially on the dispersion relation and linear stability, as well as the resulting various new types of nonlinear wave
phenomenons. For the hydroelastic waves under a linear shear current, classical Stokes expansion is applied to calculate the third-order approximation of periodic travelling waves. On this basis, we can obtain the approximate particle trajectories by solving the equations of particle motion. For the internal solitary waves under a horizontal electric field, we develop the operator expansion method of nonlinear water wave theory,
and use the asymptotic expansion method of Dirichlet-Neumann (DN) operator under long wave approximation to obtain weak nonlinear models and strong nonlinear models. Theoretical analysis and numerical calculation of these models can help us to obtain qualitative or quantitative understanding of the solitary waves. In the aspect of numerical calculation, we use conformal map and boundary integral method to transform the original free boundary problem into the nonlinear integral equations on a fixed boundary, and apply the fast Fourier transform (FFT) and finite difference scheme to obtain the travelling wave solutions respectively by Newton iterative method. The numerical calculation shows a variety of colorful results. We find that the linear shear current can apparently modify the dispersion relation, and gives rise to a new kind of hydraulic jump with strong dispersion, and makes the hydroelastic wave develop into an overhanging profile. In addition, the linear shear current also leads to ”cat’s eyes” structures, whose envelopes represent a special kind of trajectories, on which the particles have a net displacement in the vertical direction. On the other hand, the horizontal electric field shows a suppression of Rayleigh-Taylor (RT) instability. In some special cases, we can obtain a new kind of solitary waves, which is strongly affected by the modification of dispersion relation due to the electric field. In addition, we also calculate a hydraulic jump similar to those appearing in hydroelastic waves. |
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