A mass and momentum-conservative semi-implicit finite volume scheme for complex non-hydrostatic free surface flows,International Journal for Numerical Methods in Fluids

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A mass and momentum-conservative semi-implicit finite volume scheme for complex non-hydrostatic free surface flows
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2021-06-02 , DOI: 10.1002/fld.5017
Davide Ferrari ₁ , Michael Dumbser ₁

Affiliation

In this article, a novel mass and momentum conservative semi-implicit method is presented for the numerical solution of the incompressible free-surface Navier–Stokes equations. This method can be seen as an extension of the semi-implicit mass-conservative scheme presented by Casulli. The domain is covered by the fluid, by potential solid obstacles, and by the surrounding void via a scalar volume fraction function for each phase, according to the so-called diffuse interface approach. The semi-implicit finite volume discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure. The nonlinearity on the diagonal of the system stems from the nonlinear definition of the volume, while the remaining linear part of the pressure system is symmetric and at least positive semi-definite. Hence, the pressure can be efficiently obtained with the family of nested Newton-type techniques recently introduced and analyzed by Brugnano and Casulli. The time step size is only limited by the flow speed and eventually by the velocity of moving rigid obstacles contained in the computational domain, and not by the gravity wave speed. Therefore, the method is efficient also for low Froude number flows. Moreover the scheme is formulated to be locally and globally conservative: for this reason it fits well in the presence of shock waves, too. In the special case of only one grid cell in vertical direction, the proposed scheme automatically reduces to a mass and momentum conservative discretization of the shallow water equations. The proposed method is first validated against the exact solution of a set of one-dimensional Riemann problems for inviscid flows. Then, some computational results are shown for non-hydrostatic flow problems and for a simple fluid-structure interaction problem.

中文翻译：

复杂非静水自由表面流动的质量和动量守恒半隐式有限体积方案

在本文中，提出了一种新的质量和动量保守半隐式方法用于不可压缩自由表面 Navier-Stokes 方程的数值解。这种方法可以看作是 Casulli 提出的半隐式质量守恒方案的扩展。根据所谓的扩散界面方法，通过每个相的标量体积分数函数，该域被流体、潜在的固体障碍物和周围的空隙覆盖。质量和动量方程的半隐式有限体积离散化导致压力的温和非线性系统。系统对角线上的非线性源于体积的非线性定义，而压力系统的其余线性部分是对称的，至少是半正定的。因此，使用 Brugnano 和 Casulli 最近引入和分析的一系列嵌套牛顿型技术可以有效地获得压力。时间步长仅受流速限制，最终受计算域中包含的移动刚性障碍物的速度限制，不受重力波速度的限制。因此，该方法对于低弗劳德数流也是有效的。此外，该方案被制定为局部和全局保守：因此，它也适用于存在冲击波的情况。在垂直方向只有一个网格单元的特殊情况下，所提出的方案自动简化为浅水方程的质量和动量保守离散化。所提出的方法首先针对无粘性流的一组一维黎曼问题的精确解进行了验证。然后，显示了非静水流问题和简单流固耦合问题的一些计算结果。

更新日期：2021-08-09

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