There are many situations in computational fluid dynamics which require the definition of source terms in the Navier–Stokes equations. These source terms not only allow to model the physics of interest but also have a strong impact on the reliability, stability, and convergence of the numerics involved. Therefore, sophisticated numerical approaches exist for the description of such source terms. In this paper, we focus on the source terms present in the Navier–Stokes or Euler equations due to porous media—in particular the Darcy–Forchheimer equation. We introduce a method for the numerical treatment of the source term which is independent of the spatial discretization and based on linearization. In this description, the source term is treated in a fully implicit way whereas the other flow variables can be computed in an implicit or explicit manner. This leads to a more robust description in comparison with a fully explicit approach. The method is well suited to be combined with coarse-grid-CFD on Cartesian grids, which makes it especially favorable for accelerated solution of coupled 1D–3D problems. To demonstrate the applicability and robustness of the proposed method, a proof-of-concept example in 1D, as well as more complex examples in 2D and 3D, is presented.

中文翻译：

---

稳态 CFD 中多孔介质的半隐式处理

在计算流体动力学中有许多情况需要在 Navier-Stokes 方程中定义源项。这些源项不仅允许对感兴趣的物理进行建模，而且对所涉及数值的可靠性、稳定性和收敛性有很大影响。因此，存在用于描述此类源术语的复杂数值方法。在本文中，我们关注由于多孔介质而出现在 Navier-Stokes 或 Euler 方程中的源项，尤其是 Darcy-Forchheimer 方程。我们介绍了一种独立于空间离散化并基于线性化的源项数值处理方法。在此描述中，源项以完全隐式方式处理，而其他流变量可以以隐式或显式方式计算。与完全显式的方法相比，这会导致更健壮的描述。该方法非常适合与笛卡尔网格上的粗网格 CFD 结合使用，这使得它特别有利于加速解决耦合的 1D-3D 问题。为了证明所提出方法的适用性和鲁棒性，提供了一个 1D 中的概念验证示例，以及更复杂的 2D 和 3D 示例。