不连续Galerkin无发散两相流解算器中速度场的斜率限制
Slope limiting the velocity field in a discontinuous Galerkin divergence free two-phase flow solver
摘要：
求解纳维-斯托克斯方程时，密度场包含一个大的尖锐的不连续性-如水/空气自由表面-是数值上的挑战。对流不稳定性引起吉布斯振荡，迅速破坏解决方案。我们调查使用的速度场的斜率限制器，以克服这个问题的方式，不妥协的质量守恒性质。方程离散使用内部罚间断伽辽金有限元方法，是发散自由的机器精度。一个斜率限制器，专门用于准确发散自由（螺线管）领域，并用于说明在获得对流稳定的领域，也正是螺线管的困难。从这方面吸取的经验教训，适用于构建一个更简单的方法的基础上使用现有的标量斜率限制器应用于每个速度分量。我们通过数值例子表明，这两个斜率限制方法是如何大大优于天真的非限制方法。这些方法可以解决具有高密度比和高雷诺数的困难两相问题-典型的海洋和近海水/空气模拟-以一种保持质量并阻止吉布斯现象引起的无限能量增长的方式。
英文摘要：
Solving the Navier-Stokes equations when the density field contains a large sharp discontinuity---such as a water/air free surface---is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass conservation properties. The equations are discretised using the interior penalty discontinuous Galerkin finite element method that is divergence free to machine precision. A slope limiter made specifically for exactly divergence free (solenoidal) fields is presented and used to illustrated the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component. We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density ratios and high Reynolds numbers---typical for marine and offshore water/air simulations---in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.
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