The process of mechanical aeration of water reservoirs is one of the techniques most frequently used to combat eutrophication. It consists in pumping a source of compressed air into the reservoir bottom via injectors in order to create a dynamic, and aerate the water by bringing it into contact with the surface air (see Fig. 1). We focus in this work, on one hand, on the direct problem. This concerns the numerical simulation of the resulting two-phase water–air-bubble flow. Different models can be used to describe this problem [2], [4], [5] and [9]. Using the fact that the water phase is dominant, we used a simplified model in which the water phase is governed by the Navier–Stokes equations and the aeration effects are taken into account through a local boundary condition for the velocity, on the injector holes. Our discretization method is based on a three-dimensional mixed finite element method, View the MathML source [3]. The Uzawa algorithm is used to solve the matrix system obtained.
On the other hand, we also look at the inverse problem: find the optimal injector location generating the best motion in the fluid for aeration purposes. The optimal injector location is characterized as the solution to a topological optimization problem. Topological sensitivity analysis is used to solve this problem [6], [7], [8] and [10]. The main idea is to compute the asymptotic topological expansion with respect to the insertion of an injector.
The paper is organized as follows. The model used, its numerical analysis and a direct numerical simulation are presented in Section 2. Section 3 is devoted to a topological sensitivity analysis for the quasi-Stokes equations. The results obtained are valid for a large class of cost functions. Finally, we illustrate the efficiency of the proposed method by means of a numerical test.