A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations,International Journal for Numerical Methods in Fluids

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A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2024-03-12 , DOI: 10.1002/fld.5276
A. Brunel ₁ , R. Herbin ₂ , J.‐C. Latché ₁

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We propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

中文翻译：

低阶非一致面心离散化动量对流算子的中心有限体积近似

我们在本文中提出了用于四边形或广义六面体网格上的流体流动模拟的动量对流算子的离散化。空间离散化由低阶非相容 Rannacher-Turek 有限元执行：标量未知数与网格单元相关，而速度未知数与边或面相关。动量对流算子是有限体积类型的，其表达式是通过两步技术导出的，如 MUSCL 方案中所示：计算暂定通量，此处使用速度的中心近似值，并使用该通量的限制单调性论证。限制过程是代数类型的，因为它不调用任何斜率重建，并且独立于单元的几何形状。导出的离散对流算子适用于恒定或可变密度流，因此可以在不可压缩或可压缩流的方案中实现。为了实现这一目标，我们推导了计算的离散模拟（使用速度、其组成部分之一、密度，并假设质量平衡成立）并讨论了该结果的两个应用：首先，我们获得了不可压缩流和正压可压缩流的半隐式时间方案；其次，我们建立了一个一致的、半隐式的时间方案，该方案基于内部能量平衡而不是总能量的离散化。通过对不可压缩纳维-斯托克斯方程、正压和完全可压缩纳维-斯托克斯方程以及可压缩欧拉方程的数值测试来评估所提出的离散对流算子的性能。

更新日期：2024-03-12

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