高密度比多相流不可压Navier-Stokes方程的鲁棒求解
A robust incompressible Navier-Stokes solver for high density ratio multiphase flows
摘要：
本文提出了一个强大的，自适应的数值模拟方案，高密度比和高剪切多相流的局部细化笛卡尔网格，适应不断变化的接口和高涡度的轨道区域。该算法结合了接口捕获水平集方法与变系数不可压缩Navier-Stokes求解器，证明稳定解决高达六个数量级的材料对比度。离散化方法确保了速度和压力的二阶逐点精度，并采用了几种物理边界处理方法，包括速度和牵引力边界条件。本文包括几个测试案例，证明了准确性和算法的可扩展性的流量求解器的顺序。为了确保在高密度和粘度比的存在下的数值方案的稳定性，我们采用了一致的处理质量和动量输运的保守形式的离散方程。这种一致性是通过求解一个额外的质量平衡方程，我们近似通过一个强大的稳定性保持龙加-库塔时间积分器，并采用相同的质量通量（从质量方程获得）的离散动量方程。该格式采用高阶总变差递减（TVD）和满足限制器的对流有界性准则（CBC）来避免输运密度场的数值波动。高阶有界对流输运是在逐维的基础上进行的，这使得该方案易于实现。我们还表明，通过几个测试情况下，缺乏一致的质量和动量输运在非保守的配方，这是在实践中常用的，或使用非CBC满足限制器可以产生非常大的数值误差和非常差的精度对流占主导地位的高密度比流。
英文摘要：
This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows.
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