Transfer of mass and momentum at rough and porous surfaces

Abstract

The surface texture of materials plays a critical role in wettability, turbulence and transport phenomena. In order to design surfaces for these applications, it is desirable to characterise non-smooth and porous materials by their ability to exchange mass and momentum with flowing fluids. While the underlying physics of the tangential (slip) velocity at a fluid-solid interface is well understood, the importance and treatment of normal (transpiration) velocity and normal stress is unclear. We show that, when slip velocity varies at an interface above the texture, a non-zero transpiration velocity arises from mass conservation. The ability of a given surface texture to accommodate for a normal velocity of this kind is quantified by a transpiration length. We further demonstrate that normal momentum transfer gives rise to a pressure jump. For a porous material, the pressure jump can be characterised by so called resistance coefficients. By solving five Stokes problems, the introduced measures of slip, transpiration and resistance can be determined for any anisotropic non-smooth surface consisting of regularly repeating geometric patterns. The proposed conditions are a subset of effective boundary conditions derived from formal multi-scale expansion. We validate and demonstrate the physical significance of the effective conditions on two canonical problems -a lid-driven cavity and a turbulent channel flow, both with non-smooth bottom surfaces. † Email address for correspondence: ugis@mech.kth.se arXiv:1812.09401v2 [physics.flu-dyn]

Effective boundary conditions at a rough wall: a high-order homogenization approach

“…Two different approaches have been presented to derive effective boundary conditions which go beyond the usual Navier-slip paradigm, to model regularly microstructured walls without the need to numerically resolve fine-scale near-wall details. The techniques employed, inspired by previous studies [7,22], differ in the details of the small-scale formulation but produce the same results. They are based on matching the outer flow solution, which only depends on macroscopic spatial variables and time, to the inner flow state, which is assumed to depend on both smalland large-scale variables.…”

“…It is important to stress, again, that the choice y ¼ 0 is just one among infinitely many other possibilities. A different choice of the fictitious wall is always possible and, for example, Lācis et al [22] typically set y slightly above the upper rim of the roughness elements.…”

“…The terms in (58-59) with underbraces correspond to those used previously [3,22] to assess the effect of wall transpiration for the case of a turbulent flow in a channel with regularly corrugated walls. However, for things to be formally correct, all of the terms Oð 2 Þ in Eq.…”

“…Also in this case, the Navier slip condition was recovered to first order. More recent analyses based on multiscale homogenization were conducted by Jiménez-Bolaños and Vernescu [20], Zampogna et al [21] and Lācis et al [22]. The latter study was the only one to push the development to second order, albeit only for V , deriving a transpiration condition at a fictitious wall.…”

“…In the next section the problem is formulated mathematically, following the approach initiated by Lācis et al [22]. A related, but different, strategy believed to lead faster to accurate results is described in Sect.…”