Homogenization-based design of microstructured membranes: wake flows past permeable shells

Abstract

A formal framework to characterize and control/optimize the flow past permeable membranes by means of a homogenization approach is proposed and applied to the wake flow past a permeable cylindrical shell. From a macroscopic viewpoint, a Navier-like effective stress jump condition is employed to model the presence of the membrane, in which the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip numbers multiplied by the stresses. Regarding the particular geometry considered here, a characterization of the steady flow for several combinations of constant filtrability and slip numbers shows that the flow morphology is dominantly influenced by the filtrability and exhibits a recirculation region that moves downstream of the body and eventually disappears as this number increases. A linear stability analysis further shows the suppression of vortex shedding as long as large values of the filtrability number are employed. In the control/optimization phase, specific objectives for the macroscopic flow are formulated by adjoint methods. A homogenization-based inverse procedure is proposed to obtain the optimal constrained microscopic geometry from macroscopic objectives, which accounts for fast variations of the filtrability and slip profiles along the membrane. As a test case for the proposed design methodology, a cylindrical membrane is designed to maximize the resulting drag coefficient.

Permeability sets the linear path instability of buoyancy-driven disks

“…For and , the average velocity across the membrane is and thus the Reynolds number referred to this velocity and to the rectangle size is . Hence, the assumption of negligible inertia is reasonable for the considered cases of and Reynolds number (Nield & Bejan 2013), already shown in Ledda et al (2021) for the two-dimensional case and confirmed by the reasonable agreement of this section. Thanks to the generality of the homogenized model (2.3), the same value of permeability can be obtained by tuning the value of with a different geometry, e.g. by reducing the height of the rectangle one can obtain larger permeabilities and thus reduce the value of , improving the accuracy of the method at large permeabilities.…”

“…When and , the fluid can flow through the membrane but cannot slip tangentially. However, such a situation is unlikely to happen since actual permeable microstructures are always characterized by non-zero values of permeability and slip (Ledda et al 2021).…”

“…Following Ledda et al (2021) an almost-solid case is given by . Figure 4 shows the curves which represent the iso-level of zero growth rate for different modes, i.e.…”

“…Homogenization provides insights into the modelling of flow through porous structures, rigorously establishing a link between the microstructure and its macroscopic effect on the large-scale flow (Hornung 1997; Zampogna & Bottaro 2016; Lācis & Bagheri 2017). Homogenized models recover the macroscopic effects of the permeability and provide a formal framework to obtain these permeability properties from a microscopic structure, and vice versa (Ciuti et al 2021; Ledda et al 2021). Therefore, the homogenized model does not suffer from the limitations stemming from the choice of a particular microscopic geometry.…”

“…Thin permeable objects can be modelled as microstructured membranes, enabling the use of a homogenized model, which states that the velocity at the membrane is proportional to the difference of stresses across the membrane itself (Zampogna & Gallaire 2020). The same model was validated in Ledda et al (2021) for .…”