Numerical simulation of surface waves in a basin with moving impermeable boundaries

Представлены результаты, связанные с разработкой вычислительного алгоритма для изучения поверхностных волн в областях с подвижными непроницаемыми границами (подвижные боковые стенки бассейна, подвижные фрагменты дна). Алгоритм основан на конечноразностной аппроксимации на подвижных сетках двумерной модели потенциальных течений несжимаемой жидкости со свободной границей. Предложены новые начальные данные для задачи о движении уединенной волны, неотражающие условия, способ задания нулевого итерационного приближения для расчета потенциала скорости и результаты исследования устойчивости численного метода. Показано, что поверхностные волны, вызванные подводным оползнем, существенно влияют на процесс наката на берег одиночной волны и могут увеличить ее максимальный заплеск. In numerical simulation of fluid flows with a free surface, the most difficult problems are those in which not only the free boundary is mobile, but also some other parts of the boundary of the region occupied by the liquid. For example, these are the problems of surface waves caused by underwater and coastal landslides, the generation of waves by wavemakers in laboratory flumes and tanks, the problem of waves runup on the shore, the interaction of waves with moving wave protection walls and problems addressed moving semisubmerged or fully immersed objects. The purpose of this paper is to analyze the properties and evaluate the capabilities of the computational algorithm based on the finitedifference approximation of the equations of potential fluid flows with a free boundary and designed to study surface waves in a confined basin, part of the impermeable boundary of which can be mobile. The algorithm relies on the use of curvilinear grids that adapt to all parts of the boundary, both moving and stationary. New initial data are proposed for the problem of motion of a solitary or a single wave, consistent with the initial data for the shallow water equations of the second longwave approximation. New nonreflecting conditions have been developed that allow waves to be emitted across the boundary of a flow region with very little reflection. A new initial approximation is proposed for the iterative process of calculating the potential of the velocity vector. By using this approach we significantly reduce the number of iterations at each time step. The original stability condition of the linearized difference scheme is derived. The reasons for the appearance of two peaks in the chronograms of pressure when the long waves of large amplitude roll onto a vertical wall are discussed. The capabilities of the numerical algorithm are demonstrated on the problem of generating waves by a moving wall travelling in the initial part of the flume. The results of the calculations well reproduce the experimental data, in particular, the decrease in the length and increase in the amplitude of the wave when it moves towards the shallow part of the flume, as well as the formation of a dispersion tail as the waves reverse motion after reflection from the vertical wall installed at the end of the flume. The developed algorithm was used to study the process of generation for surface waves by an underwater landslide moving along an uneven bottom, and the interaction of these landslide waves with a single surface wave moving towards the shore. It is shown that the surface waves caused by an underwater landslide significantly affect the process of rolling of a single wave on the shore and it could increase its maximum splash.