Coralic Vedran, Colonius Tim
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA.
J Comput Phys. 2014 Oct 1;274:95-121. doi: 10.1016/j.jcp.2014.06.003.
We develop a shock- and interface-capturing numerical method that is suitable for the simulation of multicomponent flows governed by the compressible Navier-Stokes equations. The numerical method is high-order accurate in smooth regions of the flow, discretely conserves the mass of each component, as well as the total momentum and energy, and is oscillation-free, it does not introduce spurious oscillations at the locations of shockwaves and/or material interfaces. The method is of Godunov-type and utilizes a fifth-order, finite-volume, weighted essentially non-oscillatory (WENO) scheme for the spatial reconstruction and a Harten-Lax-van Leer contact (HLLC) approximate Riemann solver to upwind the fluxes. A third-order total variation diminishing (TVD) Runge-Kutta (RK) algorithm is employed to march the solution in time. The derivation is generalized to three dimensions and nonuniform Cartesian grids. A two-point, fourth-order, Gaussian quadrature rule is utilized to build the spatial averages of the reconstructed variables inside the cells, as well as at cell boundaries. The algorithm is therefore fourth-order accurate in space and third-order accurate in time in smooth regions of the flow. We corroborate the properties of our numerical method by considering several challenging one-, two- and three-dimensional test cases, the most complex of which is the asymmetric collapse of an air bubble submerged in a cylindrical water cavity that is embedded in 10% gelatin.
我们开发了一种激波捕捉和界面捕捉数值方法，适用于模拟由可压缩纳维-斯托克斯方程控制的多组分流动。该数值方法在流动的光滑区域具有高阶精度，离散地守恒各组分的质量以及总动量和能量，并且无振荡，即在激波和/或物质界面处不会引入虚假振荡。该方法属于戈东诺夫型，利用五阶有限体积加权基本无振荡（WENO）格式进行空间重构，并使用哈滕-拉克斯-范里尔接触（HLLC）近似黎曼求解器来迎风计算通量。采用三阶总变差递减（TVD）龙格-库塔（RK）算法推进时间上的解。推导推广到三维和非均匀笛卡尔网格。利用两点四阶高斯求积法则来构建单元内部以及单元边界处重构变量的空间平均值。因此，该算法在流动的光滑区域在空间上具有四阶精度，在时间上具有三阶精度。我们通过考虑几个具有挑战性的一维、二维和三维测试案例来证实我们数值方法的特性，其中最复杂的是浸没在嵌入10%明胶的圆柱形水腔中的气泡的不对称坍塌。
