Abstract

A compact and efficient numerical method is described for studying plane flows of an ideal fluid with a smooth free boundary over a curved and nonuniformly moving bottom. Exact equations of motion in terms of the so-called conformal variables are used. In addition to the previously known applications for shear flows with constant (including zero) vorticity, here a generalization is made to the case of potential flows in uniformly rotating coordinate systems, where centrifugal and Coriolis forces are added to the gravity force. A brief review is given of previous results obtained by this method in a number of physically interesting problems such as modeling of tsunami waves caused by the movement of nonuniform bottom, the dynamics of Bragg (gap) solitons over a spatially periodic bottom profile, the Fermi–Pasta–Ulam (FPU) recurrence phenomenon for waves in a finite pool, the formation of anomalous waves in an opposing nonuniform curent, and the propagation of a solitary wave in a shear current and its runup on a depth difference. In addition, a number of new numerical results are presented concerning the nonlinear dynamics of a free boundary in closed rotating containers partially filled with a fluid—centrifuges of complex shape. In this case, the equations of motion differ in some essential details from those of x-periodic systems.

中文翻译：

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波浪在弯曲的底部上方：复合共形映射的方法

摘要

描述了一种紧凑而有效的数值方法，用于研究在弯曲且非均匀移动的底部上具有平滑自由边界的理想流体的平面流动。使用根据所谓的共形变量的精确运动方程。除了先前已知的具有恒定（包括零）涡度的剪切流的应用之外，在此还对均匀旋转坐标系中的势流情况进行了归纳，其中将离心力和科里奥利力添加到重力中。简要回顾了通过此方法获得的先前在许多物理上有趣的问题中得到的结果，例如海底波建模，这些海啸波是由不均匀底部的运动引起的，布拉格（间隙）孤子在空间周期性底部轮廓上的动力学，有限池中波的费米－帕斯塔－乌拉姆（FPU）复发现象，相对的非均匀电流中异常波的形成以及剪切电流中孤波的传播及其在深度差上的上升。此外，还提供了一些新的数值结果，这些数值结果涉及部分装有流体-离心机的密闭旋转容器中自由边界的非线性动力学。在这种情况下，运动方程在某些基本细节上与 提出了许多新的数值结果，这些数值结果涉及部分装有流体的离心机（形状复杂的离心机）中封闭的旋转容器中自由边界的非线性动力学。在这种情况下，运动方程在某些基本细节上与 提出了一些新的数值结果，这些数值涉及部分装有流体的离心机（形状复杂的封闭离心机）中自由边界的非线性动力学。在这种情况下，运动方程在某些基本细节上与x周期系统。