A volume of solid implicit forcing immersed boundary method for solving incompressible Navier-Stokes equations in complex domain

Abstract

In this study, a new implicit forcing immersed boundary (IFIB) method is proposed to solve incompressible viscous fluid flow problems involving complex domains. In the conventional immersed boundary (IB) method, a forcing term computed from the volume of solid (VOS) is added to the incompressible Navier-Stokes equations in order to satisfy the velocity condition within the embedded solid body. However, the velocity boundary condition and the divergence-free condition are enforced at different time levels, i.e. intermediate and new time levels. Penetration of streamlines into the stationary solid body is visible as the velocity boundary condition inside the solid body is not strictly enforced at the new time level. In the current work, the proposed IFIB method can ensure velocity field which satisfies both the velocity boundary condition and the divergence-free condition at the same (new) time level. This is accomplished by solving the pressure equation and calculating the forcing term simultaneously (implicitly). A modified pressure Poisson equation (MPPE) is derived in order to couple the pressure and the forcing terms by treating the forcing term as part of the source term of MPPE. Also, a new cell-based method is proposed to compute the VOS for getting a better parallel efficiency. The accuracy of the present IFIB method is then demonstrated by solving several benchmark problems. No penetration of streamlines has been found in the solid body.