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首页> 外文期刊>Journal of Computational Physics >A collocated method for the incompressible Navier-Stokes equations inspired by the Box scheme
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A collocated method for the incompressible Navier-Stokes equations inspired by the Box scheme

机译：Box方案启发的不可压缩Navier-Stokes方程的并置方法

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摘要

We present a new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity-pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier-Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.
机译：我们提出了一种新的有限差分数值方法，使用在逻辑笛卡尔网格上的空间并置离散化来解决不可压缩的Navier-Stokes方程。该方法与Box方案有一些共同的方面，并且受到Box方案的启发。它对与半隐式时间积分结合的空间导数使用居中的二阶精度有限差分逼近。通过离散化方程的原始速度-压力形式来构造所提出的方法，以确保离散的质量和动量守恒。连续性方程在并置的位置精确地执行（以提高机器的精度），而动量方程则以交错的方式求值。该公式可避免在与压力相关的嵌入式椭圆问题中出现杂散压力模式。该方法在速度和压力方面在空间和时间上均显示出统一的精度顺序。另外，Navier-Stokes方程的非线性对流项的斜对称形式将动能的离散守恒性提高到了无粘性极限，达到了时间积分器的截断误差量级。制定了该方法以适应不同类型的边界条件。全周期，周期性通道，流入流出和盖驱动腔;始终确保全球的大规模保护。这种有限差分公式的一个新颖方面是，如有限元方法那样，使用等式的弱形式来推导边界附近的离散化。所制造解决方案的方法用于执行求解器的精度分析和验证。为了评估本文提出的新方法的适用性，模拟了四个现实的流动问题，并将结果与​​文献中的结果进行了比较。这些情况包括盖子驱动的空腔，朝后的台阶，Kovasznay流动和充分发展的湍流通道。

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