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The present paper is devoted to implementation of the immersed boundarytechnique into the Fourier pseudo-spectral solution of the vorticity-velocityformulation of the two-dimensional incompressible Navier--Stokes equations. Theimmersed boundary conditions are implemented via direct modification of theconvection and diffusion terms, and therefore, in contrast to many othersimilar methods, there is not an explicit external forcing function in thepresent formulation. The desired immersed boundary conditions are approximatedon some regular grid points, using different orders (up to second-order)polynomial extrapolations. At the beginning of each timestep, the solenoidalvelocities (also satisfying the desired immersed boundary conditions), areobtained and fed into a conventional pseudo-spectral solver, together with amodified vorticity. The zero-mean pseudo-spectral solution is employed, andtherefore, the method is applicable to the confined flows with zero meanvelocity and vorticity, and without mean vorticity dynamics. In comparison tothe classical Fourier pseudo-spectral solution, the method needs ${\calO}(4(1+\log N)N)$ 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 timeintegration, and the boundary conditions are set at the beginning of eachsub-step, in order to increasing the time accuracy. The method is applied tosome fixed and moving boundary problems, with different orders of boundaryconditions; and in this way, the accuracy and performance of the method areinvestigated and compared with the classical Fourier pseudo-spectral solutions.

An Immersed Boundary Fourier Pseudo-spectral Method for Simulation of Confined Two-dimensional Incompressible Flows