带耗散自由面势流方程的Crank-Nicolson有限差分方法数值模拟

Numerical simulation of Crank-Nicolson finite difference method for potential flow equations with a dissipative free surface

该文建立了带耗散自由面势流方程组Crank-Nicolson有限差分方法。通过坐标变换将不规则的水槽区域变换为一个规则的正方形区域,在计算区域构造交错网格,对非线性势流方程组建立Crank-Nicolson隐格式的有限差分方法,设计了流场变量的耦合迭代的算法。数值求解了在无耗散情况下的自由面数值解与解析解的误差,数值解吻合Frandsen的解析解。数值模拟了在不同耗散系数下的自由运动,水平激励和垂直激励下的自由面波高和速度场。数值实验表明自由振动和垂直激励振动随着耗散系数的增加波高和流场速度都减小。在水平激励下的自由面波高和速度场随耗散系数的增加而减小,当波高和速度衰减到一定程度后,自由面保持稳定的波高和速度场,自由面波高出现了耗散现象。

A implicit scheme finite difference method of the Crank-Nicolson type for the 2D nonlinear potential flow equations is proposed with a dissipative free surface. The irregular tank is mapped onto a fixed square domain divided by rectangular cells through a proper mapping function. A staggered mesh system is adopted and a coupling iterative algorithm is presented in a 2D tank. The results show that the numerical solutions agree with the Frandsen＇s analytical solutions of the potential flow equations without dissipative free surface. The paper presents numerical results of the wave elevations and velocity field with free motion, horizontal excited motion and vertical excited motion. Experimental results show that the wave elevation and velocity field will be decreased as the dissipative parameters increase for the free oscillation and vertical excited oscillation. The wave elevation and the velocity field for the horizontal excited oscillation will be reduced as the dissipative parameters increase. The wave elevation and velocity field will be kept stability when the excited force equals the dissipative force, and the wave elevation of the free surface behaves dissipative phenomenon.