Methods For Analysing Nonlinear Waves In Shallow Water Under Hamiltonian System

The researches on water waves are essential to hydrodynamics and the engineering projects about rivers, lakes and seas etc. The shallow water wave, which is characterized by much more larger scale along the horizontal coordinate than the vertical one, is an important one sort in problems of water waves. The governing equations for shallow water waves have typical hyperbolic form. Therefore, the basic methods, both analytical and numerical, and the charactors of solutions to the governing equations, are of great value in both scientific researches and engineering applications. However, due to the complexity of wave problems in shallow water and the difficulties of solving equations, too many assumptions have to be made while studying the waves. In the assumptions, the solution of the equations may not ensure that the results agree with the physical phenomenon. Whereas, the Hamiltonian system in mechanics can provide a set of methods and ideas. In the system, the defects can be avoided. The governing equations of waves in shallow water are obtained through modern mathematical method, which maintains the conservation properties.In this paper, the shallow water waves in single- and two-layered fluids with free surface are discussed. The Hamiltonian function is constructed by the energy in shallow water and further Hamiltonian canonical equations are derived. The solutions of the equations satisfy the symplectic relation through variational principle. Analytical solutions and traveling wave solutions are obtained according to linear equations and nonlinear equations respectively. Finite element method is introduced to numerically solve the higher-order Hamiltonian canonical equations. At the same time, Characteristic Line-Euler scheme is proposed to solve the dam break problem which is based on the hyperbolic character of the equations. The basic theory is used to simulate the hydrodynamics process of ship tank in the ship lifting system of the Three Gorges.The main researching contents are given as following.(1) Method of Hamiltonian System for Shallow Water Waves of single-layered fluid.Two small parameters,αandβ, are introduced for expanding the velocity potential in one-dimension, axisymmetric and two-dimensional space respectively. Thus, Hamiltonian canonical equations of every orders are obtained through the variation of the Hamiltonian function which is constructed through the expansion. The zeroth order equations for linear problems and nonlinear higher order equations for nonlinear problems are shown respectively. For linear equations of zeroth-order approximation, The symplectic space are eshtablished and eigensolutions satisfy the adjoint relationship of symplectic orthogonality in the space. In addition, a kind of symplectic scheme is shown and the influence of derivation for boundary conditions is studied. The traveling wave solution, which is expressed by elliptic function, is obtained by solving the equations of the first-order approximation. The boundary condition of wave source for nonlinear equations is proposed according to the character of the traveling waves. A numerical method is established which combines finite element method, reverse time-domain method, nonlinear equations and their corresponding boundary conditions to study the propagation, reflection and scattering of water waves. The procedure of forming tsunami is numerically simulated to reveal the rule.(2) Construction and Analytical Method of Hamiltonian System for Shallow Water Waves of two-layered fluidsThe Hamiltonian function is constructed by basic variables and dual variables through the expansion of the velocity potential on free surface and interface aid the character of two-layered fluid. The Hamiltonian canonical equations of shallow two-layered fluid for varies orders approximation are derived. The symplectic space is eshtablished for linear problems. In the space, the basic solutions and solutions of traveling waves are obtained and the expression of two wave speeds at the same time. The wavelength is decided by not only the incident frequency but also the depth of fluids, the proportion of the thickness of each fluids and the ratio of density. It is revealed through further investigation that the beat phenomenon and special harmonic waves can appear under certain circumstance. The traveling wave solution of the nonlinear equations, in the first-order Hamiltonian approximation, can be expressed by elliptic functions. The wave amplitude, wavelength and wave pattern are decided by the boundary conditions and the parameters of liquids. Numerical analysis shows that the propagation of waves can be caused by dynamic boundary conditions. The generation and pattern of waves are employed to controlling frequencies and amplitudes of driving plate on the boundary. The methods and ideas in this investigation are also useful in multiple-layered fluids.(3) Characteristic Line-Euler scheme for dam break problemThe dam break problem is a typical Riemann problem. Characteristic Line-Euler scheme that based on Riemann invariants and characteristic line is proposed for the problem with a discontinuous interface in this paper. Using the hyperbolicity of dual equations, the high order difference scheme can be introduced to catch the Riemann invariants along the characteristic line. Then interpolate under Euler meshes. The interpolation method is upwind scheme of the high order and the direction of the interpolation can be confirmed by the sign of eigenvalues. The numerical results show that the scheme is stable and the numerical dispersion does not appear. Besides, the result is obviously more accurate by the method than by characteristic line method.(4) Hydrodynamic simulation of the Three Gorges Ship LiftBased on the previous investigations of generalized Hamitonian principle under non-conservative system, generalized forces are introduced to construct the generalized Hamiltonian equations. With finite element method, a new numerical method of studying the ship lifting system of the Three Gorges is proposed. The coupling activity of ship and water waves is investigated and discussed. A method of moving grids and a method of discontinuous interface treatment are applied also. In the method, the moving grid is simplified by combining establishing moving coordinate on ship and moving and abandoning meshes along the boundary. The discontinuous interface along the ship's side is handled by restoring two velocities in each node along the side of the ship. The numerical results prove the effectiveness of the method. Through numerical simulation, the rules and conclusion are given for the design of ship lifting system.