多孔底面轴对称二相重力流的数值模拟

对于具有多孔介质底面的轴对称二相重力流，引进基于浅水近似的控制方程和相应的边界条件，采用贴体坐标变换使运动边界问题化为固定边界问题，提出了基于特征插值并结合使用梯形积分公式和Newton-Raphson迭代法在时间和空间都具有二阶精度的数值边界条件．为检验格式的性能和避免编写程序时可能出现的错误，对类似的方程构造了一类精确解．在空间上采用了二步Lax格式、二阶TVD格式、三阶ENO格式及五阶WENO格式，在时间上采用了二阶及三阶的TVD-Runge-Kutta方法对该问题进行数值模拟．数值结果表明，在解的光滑区域，这几种格式的精度都很高，但是在大梯度区，二步Lax格式将会产生强烈的数值振荡，且振荡不会随网格宽度的减小而减小，而其他3种格式将不会或仅会产生幅度要小得多的数值振荡，且振荡会随网格宽度的减小而趋向于零．对实际应用目的来说，结合使用二阶TVD-Runge-Kutta方法的二阶TVD格式是一个经济而又适当的选择．

Numerical Simulation of Axisymmetrical Two-phase Gravity Currents over a Porous Substrate

Based on shallow-water approximations the governing equations and corresponding boundary conditions for axisymmetrical two-phase gravity currents over a porous substrate are introduced. By use of a body fitted coordinate transformation, the moving boundary problem is reduced to a fixed one. Based on characteristic interpolations combined with trapezoidal quadrate and Newton-Raphson iteration method, some numerical boundary conditions are proposed, which have second order accuracy both in space and time. In order to check the abilities of the numerical schemes and to avoid possible mistakes in coding, a series of exact solutions for similar equations are constructed. The numerical simulations of the problem are conducted by use of two-step Lax scheme, second order TVD scheme, third order ENO scheme and fifth order WENO scheme in space discretization and by use of second and third order TVD-Runge-Kutta method in time discretization. Numerical results show that in smooth regions of the solution, all of the schemes have high accuracy, but in lager gradient regions of the solution the two-step Lax scheme will produce violent numerical oscillations which will not reduce when the grid spacing decrease, whereas the other three schemes will not or will only produce much smaller ones which will approach to zero when the grid spacing reduces to zero. For practical application purpose the second order TVD scheme combined with the second order TVD-Runge-Kutta method is an economical and suitable choice.