We derive a homogenized macroscopic model for fluid flows over ordered homogeneous porous surfaces. The unconfined free flow is described by the Navier–Stokes equation, and the Darcy equation governs the seepage flow within the porous domain. Boundary conditions that accurately capture mass and momentum transport at the contact surface between these two domains are derived using the multiscale homogenization technique. In addition to obtaining the generalized version of the widely used Beavers–Joseph slip condition for tangential velocities, the present work provides an accurate formulation for the transpiration velocity and pressure jump at fluid–porous interfaces; these two conditions are essential for handling two- and three-dimensional flows over porous media. All the constitutive parameters appearing in the interface conditions are computed by solving a set of Stokes problems on a much smaller computational domain, making the formulations free of empirical parameters. The tensorial form of the proposed interface conditions makes it possible to handle flows over isotropic, orthotropic, and anisotropic media. A subset of interface conditions, derived for porous media, can be used to model flows over rough walls. The accuracy of the proposed macroscopic model is numerically quantified for flows over porous and rough walls by comparing the results from our homogenized model with those obtained from geometry-resolved microscopic simulations.

中文翻译：

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粗糙和多孔表面上流动的高阶均质化边界条件

我们推导出了流体在有序均匀多孔表面上流动的均匀宏观模型。无侧限自由流动由纳维-斯托克斯方程描述，达西方程控制多孔域内的渗流。使用多尺度均质化技术推导出在这两个域之间的接触面上准确捕获质量和动量传输的边界条件。除了获得广泛使用的用于切向速度的 Beavers-Joseph 滑移条件的广义版本之外，本工作还为流体 - 多孔界面处的蒸腾速度和压力跳跃提供了准确的公式；这两个条件对于处理多孔介质上的二维和三维流动至关重要。出现在界面条件中的所有本构参数都是通过在更小的计算域上求解一组斯托克斯问题来计算的，从而使公式没有经验参数。所提出的界面条件的张量形式可以处理各向同性、正交各向异性和各向异性介质上的流动。为多孔介质导出的界面条件子集可用于模拟粗糙壁上的流动。通过将我们的均质模型的结果与几何解析微观模拟的结果进行比较，对多孔和粗糙壁上的流动，所提出的宏观模型的准确性进行了数值量化。所提出的界面条件的张量形式可以处理各向同性、正交各向异性和各向异性介质上的流动。为多孔介质导出的界面条件子集可用于模拟粗糙壁上的流动。通过将我们的均质模型的结果与几何解析微观模拟的结果进行比较，对多孔和粗糙壁上的流动，所提出的宏观模型的准确性进行了数值量化。所提出的界面条件的张量形式可以处理各向同性、正交各向异性和各向异性介质上的流动。为多孔介质导出的界面条件子集可用于模拟粗糙壁上的流动。通过将我们的均质模型的结果与几何解析微观模拟的结果进行比较，对多孔和粗糙壁上的流动，所提出的宏观模型的准确性进行了数值量化。