An Immersed Boundary Fourier Pseudo-spectral Method for Simulation of Confined Two-dimensional 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 many other similar methods, there is not an explicit external forcing function in the present formulation. The desired immersed boundary conditions are approximated on some regular grid points, using different orders (up to second-order) polynomial extrapolations. At the beginning of each timestep, 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 zero-mean pseudo-spectral solution is employed, and therefore, the method is applicable to the confined flows with zero mean velocity and vorticity, and without mean vorticity dynamics. In comparison to the classical Fourier pseudo-spectral solution, the method needs ${\cal O}(4(1+\log N)N)$ more operations for boundary condition settings. Therefore, the computational cost of the method, as a whole, is scaled by ${(N \log N)}$. The classical explicit fourth-order Runge--Kutta method is used for time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increasing the time accuracy. The method is applied to some fixed and moving boundary problems, with different orders of boundary conditions; and in this way, the accuracy and performance of the method are investigated and compared with the classical Fourier pseudo-spectral solutions.