In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase (uniform density and viscosity) and multiphase (variable density and viscosity) flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness/instability. It is made possible by including the penalty term in the pressure Poisson equation (PPE), which was not included in previous works that solved VP incompressible Navier-Stokes equations using the projection method. We show how and where the Brinkman penalty term enters the PPE by re-deriving the projection algorithm for the VP method. Solver scalability under grid refinement is demonstrated, i.e., convergence is achieved with the same number of iterations regardless of the problem size. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Various values of body's permeability, denoted $\kappa$, are considered. Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small values of $\kappa$. Error saturation occurs when $\kappa$ is extremely small, but the convergence rate of the solver does not degrade. The solver converges faster as $\kappa$ decreases, contrary to prior experience. A multiphase fluid-structure interaction (FSI) case is also simulated to evaluate the solver's performance (in terms of its number of iterations). The convergence rates remain robust in the multiphase case as well.

翻译：本文提出了一种基于投影方法的预条件策略，用于求解体积惩罚（VP）不可压与低马赫数纳维-斯托克斯方程。该投影预条件器能够实现单相（均匀密度与黏度）及多相（变密度与变黏度）流动场景中速度-压力耦合方程组的整体求解。在该方法中，惩罚力被隐式处理，即使取任意大数值也不会影响求解器的收敛速率或引发数值刚性/不稳定性。这得益于将惩罚项纳入压力泊松方程（PPE），而此前采用投影法求解VP不可压纳维-斯托克斯方程的研究均未包含此项。通过重新推导VP方法的投影算法，我们展示了布林克曼惩罚项如何及在何处进入PPE。证明了求解器在网格细化下的可扩展性，即无论问题规模大小，均能以相同迭代次数实现收敛。采用单相场景中的解析解确定惩罚解的空间精度，并考虑了不同渗透率值（记作κ）。对于足够小的κ值，速度与压力解均达到二阶逐点精度。当κ极小时会出现误差饱和，但求解器收敛速率不会退化。与既往经验相反，本求解器随κ减小而收敛更快。同时模拟了多相流固耦合（FSI）案例以评估求解器性能（以迭代次数衡量）。在多相情况下，收敛速率同样保持稳健。