A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

We present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing Immersed Boundary Methods; i.e. the proposed algorithms calculate and apply the force-density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use MAC (Marker And Cell) algorithm for solving the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of force-density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of force-density. We validate the proposed algorithms via the classical benchmark problem of flow past cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement. keywords: viscous fluid, moving rigid structures, Cartesian grid, direct-forcing, Immersed Boundary Method, MAC scheme, incompressible Navier-Stokes equations