Applying the immersed boundaries to the incompressible fluid flows using inverse source problem

A new method is suggested for imposing the non-regular no-slip/no-penetration conditions, on the two-dimensional incompressible viscous fluid flows. The method employs a new immersed boundary technique in the vorticity–stream function formulation of the Navier–Stokes equations. By semi-implicit time discretization, the vorticity–stream function equations are decomposed to the Helmholtz and Poisson equations at each time step. Then, these direct problems are converted to inverse ones and the solution domain is changed from a multiply connected domain to a simply connected one, where the inverse problems are solved. The method is flexible as it can be applied easily to the complex and moving geometries. Moreover, since the problem is solved on Cartesian grids, the fast Helmholtz and Poisson solvers are employed. In order to demonstrate the capability of the method, it is employed for analyses of a couple of stationary and moving boundary problems. First, the computational performance and the convergence rate of the solver are assessed using the method of manufactured solution. The results showed a first-order convergence rate with the computational efforts that scales by $$\left( {N\,\log \,N} \right)$$ . Then, as a classical non-regular fixed boundary problem, the flow around a stationary circular cylinder is studied. For the moving boundary problems, the method is applied to the problems of the oscillating cylinder in a quiescent fluid and two rotating cylinders placed in a channel. The results show good agreements with the available data in the literature.