Implementation of an Immersed Boundary Method for a fourth-order Finite Volume Scheme

Based on the work of Arpiruk Hokpunna, 2010, Markus Uhlmann, 2005

A second-order and fourth-order discretization schemes for the incompressible Navier- Stokes equations on staggered grids are implemented. Two spatial dimensions are considered and the discretizations are carried out for Cartesian uniform and non-uniform grids. Implementations are then verified and the order of accuracy of the adopted spatial and temporal discretization schemes is validated. Next, an Immersed Boundary method (IBM) with a direct forcing strategy is integrated with both flow solvers to simulate fluid-solid interaction problems. The continuous IB method employs a smoothed approximation of the Dirac delta function with a specific order of accuracy to smear (regularize) the immersed boundary forces over the adjacent fluid cells. The accuracy of the solver then becomes dependent on the order at which the coupling quantities are interpolated. In the literature, there exist different second-order regularized delta-function variants which are well-suited to use with the second-order finite volume solver. However, an appropriate fourth-order regularized delta function is not available and its detailed derivation is presented. At last, several numerical tests with smooth and non-smooth velocity fields are conducted to observe the rate of convergence achieved of both IBM solvers.