基于σ坐标变换下的Navier-Stokes方程数值模拟

Numerical Simulation of Incompressible Navier-Stokes Equations with σ Coordinate Transformations

研究了在二维水槽带非线性自由面边界条件的Navier-Stokes方程的数值解.通过σ坐标变换对不规则的水槽液体区域变换为一个规则的正方形区域,运用交错网格技术,采用较为稳定的Crank-Nicolson格式,建立流场变量的耦合迭代的算法,求解了粘性不可压缩的Navier-Stokes方程的数值解.数值模拟了水平激励和垂直激励下的自由面波高的值.数值结果表明,与无粘的Euler方程的数值解比较,粘性效果造成的自由面波的衰减非常明显.

A numerical solution algorithm of Navier-Stokes equation with nonlinear free Surface boundary condition is developed. A finite difference method is developed and is applied to solve Navier-Stokes equation in a two dimensional tank. A staggered mesh system is employed and a a coordinate transformation is applied. A Crank-Nicolson scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition. The Crank-Nicolson method is an unconditionally stable and implicit numerical scheme with second-order accuracy in both time and space. In the final, the numerical solutions are presented and agree well with the analytic solution and previously published results.