Unified formulation of the momentum-weighted interpolation for collocated variable arrangements
Unified formulation of the momentum-weighted interpolation for collocated variable arrangements
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并置变量排列动量加权插值的统一表述
DOI：
10.1016/j.jcp.2018.08.030
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发表时间：
2018
影响因子：
4.1
通讯作者：
Bartholomew P
中科院分区：
文献类型：
--
作者：
Bartholomew P
动量加权插值 (MWI) 是一种广泛使用的离散方法，用于在具有并置变量排列的网格上模拟不可压缩和低马赫数流动时防止压力-速度解耦。尽管 MWI 很受欢迎，但目前还没有统一、一致的表述。在这项工作中，根据物理一致的论点对 MWI 的各个项进行了深入分析，设计了一个离散化程序，在此基础上提出了结构化和非结构化网格上流动的 MWI 的统一公式，包括动量方程中不连续源项的扩展以及密度的不连续变化。如所提供的分析和数值结果所示，MWI 对压力场实施低通滤波器，从而抑制振荡解。此外，如果 MWI 是从动量方程一致导出的，那么 MWI 引入的动能数值耗散会在空间中收敛于三阶，并且与时间步无关。在存在源项的情况下，可以通过仔细选择插值系数来塑造压力场上的低通滤波器，以确保滤波器仅作用于与流体运动相关的驱动压力梯度，这对于数值解的准确性至关重要。为此，提出了源项的力平衡离散化，其精确匹配压力梯度的离散化并保留施加到流动的力。使用不可压缩和低马赫数流动的代表性测试案例，包括具有不连续源项的流动和具有大密度比的两相流，新提出的MWI公式与现有公式相比具有优势，并且被证明可以显着减少甚至消除解算误差，同时提高具有大密度比的流动的稳定性。
Momentum-weighted interpolation (MWI) is a widely used discretisation method to prevent pressure–velocity decoupling in simulations of incompressible and low Mach number flows on meshes with a collocated variable arrangement. Despite its popularity, a unified and consistent formulation of the MWI is not available at present. In this work, a discretisation procedure is devised following an in-depth analysis of the individual terms of the MWI, derived from physically consistent arguments, based on which a unified formulation of the MWI for flows on structured and unstructured meshes is proposed, including extensions for discontinuous source terms in the momentum equations as well as discontinuous changes of density. As shown by the presented analysis and numerical results, the MWI enforces a low-pass filter on the pressure field, which suppresses oscillatory solutions. Furthermore, the numerical dissipation of kinetic energy introduced by the MWI is shown to converge with third order in space and is independent of the time-step, if the MWI is derived consistently from the momentum equations. In the presence of source terms, the low-pass filter on the pressure field can be shaped by a careful choice of the interpolation coefficients to ensure the filter only acts on the driving pressure gradient that is associated with the fluid motion, which is shown to be vitally important for the accuracy of the numerical solution. To this end, a force-balanced discretisation of the source terms is proposed, that precisely matches the discretisation of the pressure gradients and preserves the force applied to the flow. Using representative test cases of incompressible and low Mach number flows, including flows with discontinuous source terms and two-phase flows with large density ratios, the newly proposed formulation of the MWI is favourably compared against existing formulations and is shown to significantly reduce, or even eliminate, solution errors, with an increased stability for flows with large density ratios.
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