Numerical Simulation Of Nonlinear Water Wave

This dissertation is devoted to simulate the numerical solutions of the nonlinearpartial differential equations in a two dimensional tank. In this dissertation, the numer-ical solutions of the wave elevation on free surface are investigated in a inviscid andviscous fluid tank. In 1847, Stokes investigated the oscillatory for the nonlinear poten-tial flow equations and firstly presented the second order approximation formulae of thevelocity potential and wave elevation on free surface with Fourier series. The numeri-cal solutions of nonlinear equations have gained more and more attention in the fieldsof science computations and engineering applications[1-100]. The present thesis dealswith the nonlinear partial differential equations relating to the following three nonlin-ear viscous and inviscid fluid flow models, inviscid potential flow equations, inviscidincompressible Euler equations and viscous incompressible Navier-Stokes equations.Our research works mainly focus the numerical solutions of the three nonlinear partialdifferential equations in a two dimensional tank. A finite difference method based onflfl coordinate transformations is used to solve the wave elevations of the potential flowequations. For the Euler equations, the finite difference method developed by BangfuhChen is used to simulate the wave elevations of the standing wave and large amplitudewave. Finally, the finite difference method developed by Bangfuh Chen based on flfl co-ordinate transformations is used to simulate the numerical solutions for the simplifieddimensionless Navier-Stokes equations.In Chapter 1, we introduce the backgrounds and the main results for three kindsnonlinear equations. In Chapter 2, we investigate new numerical solutions for non-linear potential flow equations with a finite difference method. Applying this methodto calculate the wave elevations in a two dimensional tank, we analyze the wave el-evation error values on free surface between the numerical solutions and the analyticsolutions with two oscillation motions, free oscillation and horizontal excited motion.In Chapter 3, a finite difference method based on flfl coordinate transformations is usedto calculate the Euler equations and to calculate the wave elevation on free surface with the free oscillation motion, the horizontal oscillation motion and the vertical oscilla-tion motion,the wave elevation error values are given. In Chapter 4, we simulate thelarge amplitude wave on free surface with the free oscillation, the horizontal excitedmotion and the vertical excited motion, and we compare the numerical solutions withthe analytic solutions and previous results with different excited frequencies and tankparameters. In Chapter 5, based on flfl coordinate transformations, a finite differencemethod is used to solve the wave elevation for simplified dimensionless Navier-Stokesequations in a two dimensional viscous fluid tank,we give the horizontal and verti-cal excited motions and compare the numerical solutions with the inviscid numericalsolutions of Euler equations.