Parallel computing of incompressible flow around multiple moving objects on Cartesian grids using tree-topological local mesh refinement

Bio-inspired flow around moving objects, such as a fish during undulatory swimming, generates fine vortex structures and can be computationally costly. To address the time cost associated with such computations, we propose the use of a tree-topological local mesh refinements (TLMR) method. The TLMR subdivides structured Cartesian grids inside target cuboid-shaped regions, to resolve finer flow features. The method utilizes a fractional-step method to solve incompressible Navier-Stokes equations discretized using a finite-difference formulation. An immersed boundary method was used to resolve complex internal boundaries. For improved convergence speed and accuracy, the solver simultaneous computes the momentum equation on all TLMR blocks, while computing the Poisson equation recursively from the coarsest block to the finer ones. When the blocks of the same TLMR level were connected, a parallel Schwarz method was used to iteratively solve both the momentum and Poisson equations. We use canonical flow problems featuring intricate geometry and kinematic patterns to demonstrate that the TLMR algorithm can save computational time by over 80%, and convergence studies found the algorithm to be second-order accurate. The algorithm also shows potential for the incorporation of high-performance computing (HPC) and parallel adaptive mesh refinement (AMR) so that the target region of grid refinement could be determined automatically without a priori human input.