Numerical Simulation Of Dam-break Flood Wave And Focusing Shock Wave In ESWL By Using Space-time Conservation Element And Solution Element Method

In this Dissertation, dam-break flood wave and the focusing of underwater shock wave in ESWL (Extracorporeal Shock Wave Lithotripsy) where the underwater shock wave is the most successful medical application are studied by using a novel numerical scheme (space-time Conservation Element and Solution Element method). They belong to two different waves respectively: shallow water wave and underwater shock wave. Their physical forms are different, but all of them belong to the hyperbolic wave. Therefore, they can be mathematically described by uniform formal equations and are numerical simulated.Dams in the rivers are benefited to human, such as sluice, prevent and control flood, irrigation, generate electricity and so on. Dikes in the rivers control river flowing and prevent human from flooding. Flooding due to the failure of a dam or dike with man-made or natural causes has potentially disastrous consequences to human life and property. It can reasonably determine the standards of preventing or controlling flood and measures of avoiding danger in the dam or dyke. It is important to the forecast and determination of the dam-break flood flux, water level, wave height, velocity, the front of dam-break and the arrived time at different locations along the channel. This problem has occupied the attention of researchers as well as practicing engineers for several decades.The observational data of natural dam-break flows are difficult to obtain because of its sudden, violent and disastrous event. Mathematically, the dam-break problem is commonly described by the shallow water equations. One feature of these equations is the formation of surge bore which is the rapidly varying discontinuous flow whether the initial values are continuous. In the dam-break flood wave problem, the initial values are discontinuous, which constitutes the Riemann problem of shallow water equations. It is an important basis for validating the numerical method which can capture the dam-break bore waves accurately without nonphysical distortion and numerical oscillation, specifically, in the supercritical-flow problem. Because the movement of free surface flows is similar the propagation of compressible fluids flows, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of hydrodynamics with dam-break bore waves.Extracorporeal shock wave lithotripsy (ESWL) is the most common treatment of kidney stone disease because of the treatment is non-invasive and the stone can be effectively broken down into small fragments. Hundreds of underwater shock waves are generated outside the patient's body and focused on the kidney stone. The stone are broken down into small fragments because of the high energy of the focusing shock waves and the cavitations. Generally, the water generates a cloud of cavitating bubbles because of the negative pressures component of an ESWL shock wave. Although effective in breaking kidney stones, ESWL can also cause significant short- and long-term damage to the kidneys. Damage has been observed on both cellular and systemic level. A common side effect of a lithotripsy treatment is the presence of blood in the urine (hematuria), chronic hemorrhage, hematoma in kidney parenchyma, kidney edema and so on. It is important that various solutions designed to maximize stone comminution and minimize tissue damage in extracorporeal shock wave lithotripsy. Both the determination of the dynamical focus and the generation and the evolvement of the negative pressures play very important roles in design and clinical ESWL.Shock wave is a nonlinear wave, which is not valid to the linear wave law of Snell refection. In numerical simulating the evolution of underwater shock wave, water is compressible, inviscid and non-heat exchange. The characteristic of water can be represented by Tait's state equation. Mathematically, the focusing shock wave problem in ESWL can be described by the Euler equations with Tait's state equation, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of underwater focusing shock wave.Shallow water equations and Euler equations can be mathematically described by a uniform formal time-dependent set of nonlinear partial differential conservation equations of hyperbolic type. Dam-break flood wave and underwater shock wave are the physical solution of shallow water equations and Euler equations respectively. The development of the numerical scheme to solve the nonlinear partial differential hyperbolic equations is a hot topic in studying the nonlinear problems.The space-time conservation element and solution element (CE/SE) method, originally proposed by Chang in 1995, is a novel numerical framework for solving the problem of hyperbolic conservation laws. Now this new method is already become to one of primary methods in the second CFD scheme in NASA.The CE/SE method is not an incremental improvement of a previously existing CFD method, and it is different in both concept and methodology from well-established traditional numerical methods (such as finite difference method, finite element method, finite volume method etc.). The CE/SE method has many nontraditional features. Firstly, space and time are unified and treated on the same footing, and by the introduction of conservation element and solution element, both local and global flux conservations in space and time in stead of in space only are enforced. Secondly, all flow variables and their spatial derivatives are considered as individual unknowns to be solved for simultaneously at each grid point, its accuracy is higher than well-established traditional numerical methods in the same grids. Thirdly, it is conceptually simple and robust, neither Riemann solver nor technique based on characteristics are involved. Last, this new scheme is easily extended to solve multidimensional problem without flux-splitting. The CE/SE method can be used to simulate the general problem, it also solve the discontinuous problem with shock wave or dam-break and so on. Until now, this new scheme doesn't use to numerical study the problem of dam-break flood wave and the problem of the focusing of underwater shock wave.Firstly, the shallow water equations are discretized by the Zhang's modified CE/SE method. One-dimensional numerical model and two-dimensional numerical model on structured meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied in the ideal conditions and the 2-D simple computational regions. Secondly, the CE/SE method is modified and a new constructing method on the 2-D unstructured mixed meshes is developed by author. The new modified CE/SE method is used to solve the hyperbolic conservation equations. The reconstructing method not only maintains all the features, but also is easy to study the problems on the complicated computational regions. Thirdly, the shallow water equations are discretized by the new modified CE/SE method. Two-dimensional numerical model on the 2-D unstructured mixed meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied on the complicated computational regions. Fourthly, the 2-D axisymmetrical Euler equations are discretized by the new modified CE/SE method and numerical model on the 2-D unstructured mixed meshes are constructed to study the generation and the evolvement of the negative pressures in spherical piezoelectric emitters ESWL. Last, using Chisnell-Chester-Whitham (CCW) geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically. All codes of the numerical models are written by author with FORTRAN90 to implement function. It is successful that the problem of dam-break flood wave and the problem of the focusing of underwater shock wave in ESWL are simulated by using the CE/SE method.The computational results of dam-break flood wave's examples show that many advantages of the CE/SE method such as simple, robust, high efficiency, high accuracy and high resolution of dam-break. It shows that the proposed model can satisfactorily describe the whole process and flow characteristics of the dam-break wave and possesses high computation accuracy. It is proved that the CE/SE method is a new and high accuracy numerical method for studying dam-break flood wave.In study of the ESWL, a robust model for the solution of focusing of underwater shock wave in ESWL has been described. Using the modified space-time Conservation Element and Solution Element method on unstructured meshes, the two-dimensional compressible inviscid Euler equations with Tait's equation of state for water are solved. The pressure field of underwater shock wave in spherical piezoelectric emitters ESWL is simulated successfully in the first. The computational results show that the negative pressure is generated because of the diffraction in the edge of ESWL. The pressure near of the focus of ESWL is the negative pressure after the positive pressure, which shows that the cavitations are inevitable for the focusing of underwater shock wave in spherical piezoelectric emitters ESWL. The cavitations must be considered in clinical and designing ESWL. Secondly, another model for the track of the front of the underwater shock wave has been described, using CCW geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically. The computational results show that the dynamic focus in spherical piezoelectric emitters ESWL is different from geometric focus in general. The correct location of the factual focus plays very important roles in clinical ESWL.