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A new method is proposed for implementing the no-slip/no-penetration conditions on the irregular immersed boundaries in the vorticity-streamfunction formulation of the incompressible viscous fluid flow. Time integration is performed using a semi-implicit method such that in each time step the vorticity-streamfunction equations are changed to a Helmholtz and a Poisson’s equation. Some singular source terms are added to the right hand sides of these equations, in the solid region, such that the desired boundary conditions can be satisfied. The singular source terms are found, using the inverse problems method, such that the desired boundary conditions of the vorticity-streamfunction equations be satisfied. Since the fast Poisson’s solvers are used, the method is high performance, with the computational effort of O(NlogN); and it is also flexible because it can be applied easily to the complex geometries. The method is applied in simulation of the fluid flow around a square solid obstacle, placed in a channel, and the agreement of the results with the other benchmark results are shown.

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In the present paper, a new penalization method is proposed for implementation of the rigid surfaces on the Navier-Stokes equations in the vorticity-stream function formulation. In this method, a rigid body is considered as a region in the fluid flow, where the time is stopped. Therefore, by stopping the fluid particles, this region plays the role of a rigid body. In this regard, a new transformation is introduced and applied to the governing equations and a set of modified equations are obtained. Then, in the modified equations, the time dilation of the solid region is approached to infinity, while the time dilation of the fluid region remains In the article, the physical and mathematical properties of modified equations are investigated and satisfaction of the no-slip and no-penetration conditions are justified. Then, a suitable numerical algorithm is presented for solving the modified equations. In the proposed algorithm, the modified equation is time integrated via the Crank–Nicolson method, and the spatial discretization with the second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, the sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The results show that the no-slip and no-penetration conditions are satisfied accurately, while the flow fields are also high level of accuracy.