Virtual Body-Fitted Grid-Based Immersed Boundary Velocity Correction Method for Simulation of Incompressible Flows with Moving Boundaries (Communications in Computational Physics, 2025)
In this work, a virtual body-fitted grid is introduced and combined withthe implicit velocity correction-based immersed boundary method (IBM) for simulation of incompressible flows with moving boundaries. The original implicit velocitycorrection-based IBM to this kind of problem requires the matrix inversion operationto be repeated at each time step, resulting in significant computational resources andtime 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 gridthat moves together with the immersed boundary to overcome the above defect. Aswith the original implicit velocity correction-based IBM, the fractional step technique,which includes the prediction step and the correction step, is applied in the presentmethod. Note that the correction step is implemented on the virtual grid in the presentmethod instead of the Eulerian mesh in the original method. Since the relative positions between virtual grid points and Lagrangian points are changeless, the matrix inthe correction step can be pre-calculated and stored, avoiding the need to update itat every time step. Although, within this approach, three additional steps includingthe marking virtual grid points and covered Eulerian points, and two interpolationsbetween 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 ofthese steps is $\mathcal{O}(M).$ A numerical experiment of flow around a transversely oscillatingcylinder 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 vibrationsof a circular cylinder are simulated. The numerical results are found to be in line withreference results, verifying the ability of the proposed method to simulate complexmoving boundary problems.
Keywords: Immersed boundary method; velocity correction; virtual body-fitted grid; incompressible flows; moving boundaries.