The Improvement For The Conventional Immersed Boundary Method And Its Numerical Simulations Of Incompressible Viscous Flows

A new numerical approach is presented in this work to simulate incompressible flows, and it is about the improvement for the conventional immersed boundary method (IBM). It is known that the immersed boundary method uses Cartesian mesh instead of body-fitted girds. Additionally, the effect of rigid body on the surrounding flow is modeled through a forcing term, which is in turn used to correct the surrounding velocity field. It was found that this process is actually an iterative procedure, trying to satisfy the non-slip boundary condition at the solid wall. The major advantage of IBM is that it is easy and efficiency to generate grids. However, IBM suffers two major drawbacks. One is the streamline penetration to the body and the other is the low order of accuracy near the boundary. The cause behind these two drawbacks is actually the fact that the boundary condition in IBM is only approximately satisfied.Inspired from the local domain-free discretization (DFD) method, we combine the ideas of local DFD method and the conventional IBM to present a new approach. In this approach, we first solve governing equations only at regular girds. Then we correct the surrounding velocity field to accurately implement the physical boundary conditions.Finally, we give several numerical examples. In this paper, flows past a fixed circular cylinder and the NACA-0012 airfoil are simulated, also including more complex moving boundary problems, such as unsteady flows around an oscillating circular cylinder under different conditions. Our numerical results agree well with the experimental and numerical results in literature, and this shows that our new approach is very effective.