Flow over natural or engineered surfaces: an adjoint homogenization perspective

Abstract

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need for ad hoc parameters. In particular, notable results reported herein include:(i)an original formulation to describe filtration in porous media in the presence of inertial effects;(ii)the microscopic and macroscopic equations needed to characterize flows through poroelastic media;(iii)an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;(iv)the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and(v)the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

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

“…An even more difficult test case for the condition on the vertical velocity at Y ¼ 0 would be represented by a three-dimensional turbulent wall flow, because of the presence of violent near-wall events, such as ejections and sweeps. As shown by Bottaro [3], the transpiration condition seems adequate in a turbulent channel flow when ¼ 0:2 and the friction Reynolds number is 180; however, a better In Fig. 10 a close-up view of the U velocity component is shown near the surface where the effective conditions are enforced, highlighting the fact that the approximations made do a good job at representing the physics near the roughness.…”

“…Challenging test cases for the conditions given above are envisaged in future work, including in particular turbulent wall flows. It is indeed a nice and unexpected surprise to observe [3,22] that a model of the rough surface based on only the leading order, nontrivial terms for the velocity components (i.e. those including only the factors K, m 1 and m 2 in Eqs.…”

“…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.…”

“…From that moment on, the no-slip condition gained (almost) universal acceptance. Configurations for which a slip condition remained in use included the triple-line flow (for example, to describe the leading edge motion of a liquid drop sliding down an incline), the flow of rarified gases, for example in micro-fluidic devices (in this case Navier's condition is often associated to Maxwell's name), or the flow over micro-corrugated surfaces, eventually impregnated with a lubricant fluid (for a recent review the reader is referred to [3]). All of these exceptional cases share the peculiarity that the continuum description of the flow either breaks down or becomes too difficult/expensive to be resolved, e.g.…”

A nonlinear asymptotic model for the inertial flow at a fluid-porous interface

“…The momentum transport at the interface between a fluid and a permeable region is present in a large variety of industrial applications (dendritic solidification of multicomponent mixtures, fuel cell, oil recovery, separation processes, etc), but also in environmental situations or for water resources (surface or subsurface hydrology, interaction atmosphere/canopy, benthic boundary layers, etc) or biological systems (bone growth, biofilms, cell proliferation, etc). We refer to the recent paper [1] and the references therein for many examples of application. Due to the different characteristic length scales involved in these heterogeneous configurations, the momentum transport analysis is often performed at the macroscopic scale where the concept of interface is actually related to the nature of the average representation.…”