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

The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier-Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to some other similar methods, there is not 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 and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The classical explicit fourth-order Runge-Kutta method is used in time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increase the time accuracy. The method is employed in simulation 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 show accuracy and efficiency of the method.