A new conservative symmetry-preserving second-order time-accurate PISO-based pressure-velocity coupling for solving the incompressible Navier-Stokes equations on unstructured collocated grids is presented in this paper. This new method for implicit time stepping is an extension of the conservative symmetry-preserving incremental-pressure projection method for explicit time stepping and unstructured collocated meshes of Trias et al. [35]. In order to assess and compare both methods, we have implemented them within one unified solver in the open source code OpenFOAM where we use a Butcher array to prescribe the Runge-Kutta method. Thus, by changing the entries of the Butcher array, explicit and diagonally implicit Runge-Kutta schemes can be combined into one solver. We assess the energy conservation properties of the implemented discretisation methods and the temporal consistency of the selected Runge-Kutta schemes using Taylor-Green vortex and lid-driven cavity flow test cases. Finally, we use a more complex turbulent channel flow test case in order to further assess the performance of the presented new conservative symmetry-preserving incremental-pressure PISO-based method.

Although both implemented methods are based on a symmetry-preserving discretisation, we show they still produce a small amount of numerical dissipation when the total pressure is directly solved from a Poisson equation. When an incremental-pressure approach is used, where a pressure correction is solved from a Poisson equation, both methods are effectively fully-conservative. For high-fidelity simulations of incompressible turbulent flows, it is highly desirable to use fully-conservative methods. For such simulations, the presented numerical methods are therefore expected to have large added value, since they pave the way for the execution of truly energy-conservative high-fidelity simulations in complex geometries. Furthermore, both methods are implemented in OpenFOAM, which is widely used within the CFD community, so that a large part of this community can benefit from the developed and implemented numerical methods.

提出了一种新的保守的，对称的，基于二阶时间精确度的基于PISO的压力-速度耦合方法，用于求解非结构化并置网格上的不可压缩的Navier-Stokes方程。这种用于隐式时间步长的新方法是对Trias等人的用于显式时间步长和非结构化并置网格的保守对称保持增量压力投影方法的扩展。[35]。为了评估和比较这两种方法，我们在开源代码OpenFOAM中的一个统一求解器中实现了它们，在其中我们使用Butcher数组来规定Runge-Kutta方法。因此，通过更改Butcher数组的条目，可以将显式和对角隐式的Runge-Kutta方案组合到一个求解器中。我们使用泰勒-格林涡旋和盖驱动腔流测试案例评估了实施离散化方法的节能特性以及所选龙格-库塔方案的时间一致性。最后，我们使用更复杂的湍流通道测试案例，以进一步评估所提出的基于保守的对称对称增量压力PISO的新方法的性能。

尽管这两种已实现的方法都基于保持对称性的离散化，但是我们证明当直接从泊松方程中求解总压力时，它们仍然会产生少量的数值耗散。当使用增量压力方法时，通过泊松方程求解压力校正，这两种方法实际上都是完全保守的。对于不可压缩湍流的高保真模拟，非常需要使用完全保守的方法。因此，对于这种模拟，期望所提出的数值方法具有较大的附加值，因为它们为在复杂几何形状中执行真正的节能高保真度模拟铺平了道路。此外，这两种方法都是在CFD社区中广泛使用的OpenFOAM中实现的，