Kinetic Flux Vector Splitting Method For Simulating Shallow Water Wave

This thesis covers two main parts: the theoretical research of KFVS method, and application of KFVS method in engineering, the simulation of shallow water wave and dam-breaking problems.1.1 Boltzmann EquationFor study the imbalance state of molecule system, Boltzmann proposed the famous control-equations of molecule movement in 1872, which was called Boltzmann Equation. This equation describes the distribution of molecule velocity with time and space change. Bhatnagar-Gross-Krood proposed In equation (1), F is the sum of outside force, m is molecule mass, Q ( f )is collided function. BGK model assume that the average influence of collision affects distributed function f . So BGK model can be written as:collide-model method in 1954, which was later called BGK Model. It is simple and can reflect the influence of molecule moving to velocity distribution function well. It is an applied model. If f stands for the molecule distribution function, then f is defined by velocityc , location x and timet . The Boltzmann Equation is:In equation (2), q is equilibrium velocity distributed function,τis collided time. So BGK Boltzmann Equation can be written as:1.2 KFVS methodThinking about two-dimension flows, the zero-order and first-order are:the integral of the fourth and fifth terms in equations (4),(5),(6) are zero. In equations (5),(6):Here, g is gravity acceleration,h is water height. S fand Sσare coefficients of resistances derived from friction and slope. So equations (4),(5),(6) can be rewritten as:The following will deduce the KFVS equation from gas molecule movement.The conserve form of incompressible and inviscid gas equation is: for space dimension, P stands for pressure,ρstands for density,ρEstands for total energy. Split the vector into two parts: positive and negative, and sum them respectively:In equation (9), vector , integral equation (9). If set: and according to gas state equation add the volume force to the mass conserve equation right side, get rid of the energy equation in gas kinetic equations, then the Kinetic Flux VectorSplitting equations that can be applied to liquid flows is gained. It is same with equation (7).2. ExampleAt the first part, one-dimension flat-bottomed and uneven shallow water flows are computed. In flat-bottomed flows, the depths of water at x = 0.5m and x = 0.8m are presented and are compared with exact value. The errors are between 1.4% and 1.5%. When it comes to uneven flows, mixture flows with shock wave and without shock wave are computed. The results are similar with ones of other methods.At the second part, two-dimension dam-breaking problems are concerned. There is a barrier at the center of riverway. The folws near the barrier are complexed, but the KFVS method also gains the characters that other numerical methods have gained.The simulations of one-dimension and two-dimension flows prove that the KFVS method is a reliable and high precision numerical method.