The method of fundamental solutions for Brinkman flows. Part I. Exterior domains

Abstract

The method of fundamental solutions (MFS) is developed for solving numerically the Brinkman flow in the porous medium outside obstacles of known or unknown shapes. The MFS uses the fundamental solution of the Brinkman equation as in the boundary element method (BEM), but the single-layer representation is desingularized by moving the boundary sources to fictitious points outside the solution domain. In the case of unbounded flow past obstacles, these source points are placed in the domain inside the obstacle on a contracted fictitious pseudo-boundary. When the obstacle is known, then the fluid flow in porous media problem is direct, linear and well-posed. In the case of Brinkman flow in the porous medium outside an infinitely long circular cylinder, the MFS numerical solution is found to be in very good agreement with the available analytical solution. However, when the obstacle is unknown and has to be determined from fluid velocity measurements at some points inside the fluid, the resulting problem becomes inverse, nonlinear and ill-posed. The $\hbox {MATLAB}^{\copyright }$ optimization toolbox routine lsqnonlin is employed for minimizing the least-squares gap between the computed and measured fluid velocity which is further penalized with extra smoothness regularization terms in order to overcome the instability of the solution. For proper choices of the regularization parameters involved, accurate and stable numerical reconstructions are achieved for various star-shaped obstacles.