Brinkmann Model and Double Penalization Method for the Flow Around a Porous Thin Layer

Abstract

in this paper we study a penalization method used to compute the flow of a viscous fluid around a thin layer of porous material. Using a BKW method, we perform an asymptotic expansion of the solution when a little parameter, measuring the thickness of the thin layer and the inverse of the penalization coefficient, tends to zero. We compare then this numerical method with a Brinkman model for the flow around a porous thin layer.

Asymptotic study for Stokes–Brinkman model with jump embedded transmission conditions

“…For instance in [7,5,6] the authors derive WKB expansions with boundary layer terms or thin layer asymptotics to describe penalization methods in the context of viscous incompressible flow.…”

Numerical modelling and passive flow control using porous media

“…For numerical applications we set respectively K = 10 16 and K = 10 À8 in the two regions. When the thickness of the porous layer between the fluid and the solid goes to zero, it is shown in [14] that it is equivalent to solve NavierStokes equations in the fluid with a Robin boundary condition instead of the usual no-slip one. That gives a mathematical relevance of the Beavers and Joseph type boundary conditions seen above.…”

“…m is the Reynolds number based on the mean velocity U and the height of the domain H. These equations can be specified also in the solid region as shown in [1,14]. In the fluid region the permeability coefficient goes to infinity, the penalisation term vanishes and we recover the non-dimensional Navier-Stokes equations.…”

“…This method can be seen as a fictitious domain method which is very easy to implement, robust and efficient. It does need neither to have a mesh fitting the obstacle nor to impose a boundary condition at the boundary of the solid or an interface condition between the porous and the fluid media [1,14,6,35]. Let us note that Brinkman equation is valid only when the porosity of the porous medium is close to one.…”

Coupling active and passive techniques to control the flow past the square back Ahmed body

“…Previous works have shown that a good value for the control is K = 10 À1 [4] related to a high intrinsic permeability medium. The addition of such layers is equivalent to solve the Navier-Stokes equations in the fluid with a Fourier-like boundary condition instead of the no-slip boundary condition [10]. Thus the shear effects in the boundary layer are reduced.…”