A Second Order Penalized Direct Forcing for Hybrid Cartesian/Immersed Boundary Flow Simulations
Résumé
Flows around complex stationary/moving solids take an important place in life-science context or in many engineering applications. Usually, these problems are solved by body-fitted approaches on unstructured meshes with boundary conditions directly imposed on the domain boundary. Another way is using immersed boundary (IB) techniques: the physical domain is immersed in a fixed fictitious one of simpler geometry on Cartesian grids. It allows to use efficient, fast and accurate numerical methods avoiding the tedious task of re-meshing in case of time varying geometry. In contrast, one needs specific methods to take into account the IB conditions (IBC). Here, we propose a second order penalized direct forcing method for unsteady incompressible flows with Dirichlet's IBC. It consists in adding a penalized forcing term to the initial problem, applied only on Cartesian nodes near the IB, in order to bring back the variable to the imposed one. Regarding Navier-Stokes solvers using a projection scheme, the forcing term is distributed both in the velocity prediction and in the correction equations. It leads to a natural way to prescribe the pressure boundary conditions around obstacles. Numerical experiments, performed for laminar flows around static/moving solids, assess the validity and illustrate the ability of our method, showing in particular a quadratic convergence rate.
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