A Fast Immersed Boundary Fourier Pseudo-spectral Method for Simulation of theIncompressible Flows

The present paper is devoted to implementation of the immersed boundary technique into the Fourierpseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressibleNavier-Stokes equations. The immersed boundary conditions are implemented via direct modificationof the convection and diffusion terms, and therefore, in contrast to some other similar methods, there isnot an explicit external forcing function in the present formulation. At the beginning of each time step,the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained andfed into a conventional pseudo-spectral solver, together with a modified vorticity. The classical explicitfourth-order Runge-Kutta method is used in time integration, and the boundary conditions are set at thebeginning of each sub-step, in order to increase the time accuracy. The method is employed insimulation of some different test cases, including the flow behind impulsively started circular cylinder,oscillating circular cylinder in fluid at rest and insect-like flapping wing motion. The results showaccuracy and efficiency of the method.