A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries

An immersed boundary method for the incompressible Navier–Stokes equations in irregular domains is developed using a local ghost cell approach. This method extends the solution smoothly across the boundary in the same direction as the discretization it will be used for. The ghost cell value is determined locally for each irregular grid cell, making it possible to treat both sharp corners and thin plates accurately. The time stepping is done explicitly using a second order Runge–Kutta method. The spatial derivatives are approximated by finite difference methods on a staggered, Cartesian grid with local grid refinements near the immersed boundary. The WENO scheme is used to treat the convective terms, while all other terms are discretized with central schemes. It is demonstrated that the spatial accuracy of the present numerical method is second order. Further, the method is tested and validated for a number of problems including uniform flow past a circular cylinder, impulsively started flow past a circular cylinder and a flat plate, and planar oscillatory flow past a circular cylinder and objects with sharp corners, such as a facing square and a chamfered plate.