International audienceComparison exercises have been carried out by different research teams to study the sensitivity of the natural convection occurring in a vertical asymmetrically heated channel to four sets of open boundary conditions. The dimensionless parameters have been chosen so that a return flow exists at the outlet. On the whole, results provided by the partners are in good agreement; benchmark solutions are then defined for each of the boundary conditions. Whilst the local and average Nusselt numbers based on the entrance temperature do not depend much on conditions applied in the aperture sections, the net fluid flow rates crossing the channel and the characteristics of the recirculation cells are highly influenced. But we proved that these modifications of flow patterns do not alter significantly the fluid flow rates leaving the channel through the exit section

A new interpolation technique to deal with fluid-porous media interfaces for topology optimization of heat transfer. Computers and Fluids, Abstract This paper proposes a new interpolation technique based on density approach to solve topology optimization problems for heat transfer. Problems are modeled under the assumptions of steady-state laminar flow using the incompressible Navier-Stokes equations coupled to the convection-diffusion equation through the Boussinesq approximation. The governing equations are discretized using finite volume elements and topology optimization is performed using adjoint sensitivity analysis. Material distribution and effective conductivity are interpolated by two sigmoid functions respectively h τ (α) and k τ (α) in order to provide a continuous transition between the solid and the fluid domains. Comparison with standard interpolation function of the literature (RAMP function) shows a smaller transition zone between the fluid and the solid thereby, avoiding some regularization techniques. In order to validate the new method, numerical applications are investigated on some cases from the literature, namely the single pipe and the bend pipe. Lastly, as two new parameters are introduced thanks to the interpolation functions, we study their impact on results of the optimization problem. The study shows that the proposed technique is a viable approach for designing geometries and fluid-porous media interfaces are well-defined. 19 method, known as the Brinkman penalization, leads to a problem where flow 20 and (almost) non-flow regions are developed by allowing interpolation be-21 tween the lower and upper value of permeability. Generally, authors used 22 the density interpolation function proposed by Borrvall and Petersson [4] 23 or a reformulated version of their convex and q-parametrized interpolation 24function. The parameter q > 0 is a penalty parameter that is used to con-25 trol the level of 'gray' in the optimal design. However, authors had also 26 experienced problems with locally optimal solutions. Therefore, they con-27 sidered a two-steps solution procedure where the problem was first solved 28 with a small penalty value of q = 0.01 for example and then the result is 29 used as initial case for the problem with a penalty value of q = 0.1 [4, 8] 30 or q = 1 [15]. The mathematical foundation of the interpolation of α was 31 further investigated by Evgrafov [14] where the limiting cases of pure fluid 32 and solid were included. Brinkman approach has since been used for several 33 problems as transport problem [28], reactive [32] and transient flows [33, 3], 34 fluid-structure interaction [36] and also flows driven by body forces [37]. 35 A variation of the approach is presented by Guest and Prevost [2]. They 36 proposed to regularize the solid-fluid structure by treating the material phase 37 as a porous medium where fluid flow is governed by Darcy's law. In their 38 approach, flows through voids are governed by Stokes flow and, when the 39 solid phase is impermeable, discrete n...

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Topology optimization for fluid flow aims at finding the location of a porous medium minimizing a cost functional under constraints given by the Navier–Stokes equations. The location of the porous media is usually taken into account by adding a penalization term [Formula: see text], where [Formula: see text] is a kinematic viscosity divided by a permeability and [Formula: see text] is the velocity of the fluid. The fluid part is obtained when [Formula: see text] while the porous (solid) part is defined for large enough [Formula: see text] since this formally yields [Formula: see text]. The main drawback of this method is that only solid that does not let the fluid to enter, that is perfect solid, can be considered. In this paper, we propose to use the porosity of the media as optimization parameter hence to minimize some cost function by finding the location of a porous media. The latter is taken into account through a singular perturbation of the Navier–Stokes equations for which we prove that its weak-limit corresponds to an interface fluid-porous medium problem modeled by the Navier–Stokes–Darcy equations. This model is then used as constraint for a topology optimization problem. We give necessary condition for such problem to have at least an optimal solution and derive first order necessary optimality condition. This paper ends with some numerical simulations, for Stokes flow, to show the interest of this approach.

This paper deals with the finite element approximation of the Darcy-Brinkman-Forchheimer equation, involving a porous media with spatially-varying porosity, with mixed boundary condition such as inhomogeneous Dirichlet and traction boundary conditions. We first prove that the considered problem has a unique solution if the source terms are small enough. The convergence of a Taylor-Hood finite element approximation using a finite element interpolation of the porosity is then proved under similar smallness assumptions. Some optimal error estimates are next obtained when assuming the solution to the Darcy-Brinkman-Forchheimer model are smooth enough. We end this paper by providing a fixedpoint method to solve iteratively the discrete non-linear problems and with some numerical experiments to make more precise the smallness assumptions on the source terms and to illustrate the theoretical convergence results.