The Study On Sloshing Characteristics Of Continuous Stratified Liquid Based On The WKB Method

In this paper,we choose to establish the(x,y)cartesian coordinate system is established to deal with the two-dimensional sloshing problem of the stratified liquid in the rectangular storage tank,and the(r,θ,z)cylindrical coordinate system to deal with the three-dimensional sloshing problem of the stratified liquid in the cylindrical tank.The main problem we solved is:when the Brunt-Vaisala frequency N is considered as a slowly varying function,we use the WKB method to solve the governing equations describing the sloshing characteristics of the stratified liquid.The system of equations finally becomes a second-order non-homogeneous ODE bound-ary value problem with variable coefficients through linearization,Laplace transform,series expansion and other means.We use nonlinear transformation to transform the second-order vari-able coefficient differential equation into a form that does not contain the first-order derivative term,and then apply the WKB method to the corresponding homogeneous equation.We use ex-ponential approximation expansion to give the basic solutions,use variation of parameters and the Green function method to find the special solutions of the non-homogeneous equations.Then we get the WKB solutions of the equations,and verify the rationality of the WKB method.Fi-nally,we analyze the velocity field、the curl field and the free surface of the stratified liquid sloshing in the rectangular tank.In the first chapter,we mainly introduce the concept of liquid sloshing,the significance of studying the characteristics of liquid sloshing,and the current research status and latest progress of liquid sloshing at home and abroad.In the second chapter,we study the sloshing phenomenon of stratified liquid in a square tank.Starting from the sloshing problem of the stratified fluid in a square storage tank,we start to establish a mathematical model.We first establish a rectangular coordinate system,and then use the orthogonality of the sine function to gradually sort out the governin control equations and boundary conditions.we solve the boundary value problem of differential equations.First,we use nonlinear transformation f=u exp((?)dy)and power series expansion,truncating the result to give the first-order approximate WKB solution of the homogeneous equation.Then use the constant variation method and the Green function method to give the special solutions of the corresponding non-homogeneous equations,and obtain the general solution of the governing equation describing the sloshing characteristics of the layered liquid in the three-dimensional cylindrical tank,and then use the inverse transformationu(y)=f(y)exp(?).to get the approximate analytical solution of f.Finally,we verify the effectiveness of the WKB method.Furthermore,the velocity field,curl field and free surface of sloshing problem are given.In the third chapter,we start from the specific physical model of cylindrical storage tank,use conservation equations to gradually linearize,and establish the governing equations and boundary conditions of the sloshing problem.Then the Laplace transform is used to simplify the model,and the boundary value problem of the WKB analytical solution that needs to be obtained is given.In the fourth chapter,the model of cylinder storage tank is expanded by Bessel series,then we use the method in Chapter second to solve the boundary value problem of the second-order nonhomogeneous differential equation with variable coefficients corresponding to the mathe-matical model.Because the process corresponds to different variable coefficients,we need to perform a new validation of the WKB method.Finally,the solution curves are given for different slow variable functions.In the fifth chapter,we summarize the content of the full text and look forward to the results obtained.