A novel algorithm for fluid dynamics simulation with cell-adaptive Cartesian grids and immersed boundary method

Adaptive Cartesian grids, often accompanied by non-uniform hanging grids, result in non-equidistant node distributions, posing numerical challenges for finite difference discretization. This work introduces a novel cell-based adaptive Cartesian grids solver for compressible flows using a finite difference method, combined with an immersed boundary method to effectively simulate various complex flows. The algorithm selects easily locatable and retrievable neighboring cells around the hanging grid as interpolation templates and employs distance-weighted least squares interpolation to achieve high-precision interpolation. Consequently, equidistant finite difference templates are constructed, enabling high-accuracy finite difference discretization in the transition regions between coarse and fine grids. The advantages of this algorithm include ease of integration with commonly used numerical schemes, flexibility, and implementation convenience. Accuracy tests with the Taylor–Green vortex and vortex convection confirm third-order precision at hanging grids. The algorithm's effectiveness is further demonstrated in laminar flat plate flow, Rayleigh–Taylor instability, cylinder flow, and supersonic flow around a high-aspect-ratio aircraft, showcasing its capability to accurately and efficiently simulate complex flow situations.