Virtual Body-Fitted Grid-Based Immersed Boundary Velocity Correction Method for Simulation of Incompressible Flows with Moving Boundaries

In this work, a virtual body-fitted grid is introduced and combined with the implicit velocity correction-based immersed boundary method (IBM) for simulation of incompressible flows with moving boundaries. The original implicit velocity correction-based IBM to this kind of problem requires the matrix inversion operation to be repeated at each time step, resulting in significant computational resources and time when the Gaussian elimination method is used for solution, whose computational complexity is $\mathcal{O}(M^3).$ The present method introduces a virtual body-fitted grid that moves together with the immersed boundary to overcome the above defect. As with the original implicit velocity correction-based IBM, the fractional step technique, which includes the prediction step and the correction step, is applied in the present method. Note that the correction step is implemented on the virtual grid in the present method instead of the Eulerian mesh in the original method. Since the relative positions between virtual grid points and Lagrangian points are changeless, the matrix in the correction step can be pre-calculated and stored, avoiding the need to update it at every time step. Although, within this approach, three additional steps including the marking virtual grid points and covered Eulerian points, and two interpolations between the Eulerian mesh and the virtual grid must be conducted at each time step, the computational effort is still greatly reduced as the computational complexity of these steps is $\mathcal{O}(M).$ A numerical experiment of flow around a transversely oscillating cylinder is first performed, demonstrating the improved efficiency, especially when the number of Lagrangian points is large. As validation, the flow over a flapping elliptical wing and a fluid-structure interaction (FSI) problem of vortex-induced vibrations of a circular cylinder are simulated. The numerical results are found to be in line with reference results, verifying the ability of the proposed method to simulate complex moving boundary problems.