An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

• We develop an octree based finite difference method for incompressible viscous flows.
• The method is shown to be stable, second order accurate and low-dissipative.
• We assess the performance of the solver given several 3D benchmark problems.
• Previously unknown stability issues for staggered grid octree methods are addressed.

The paper introduces a finite difference solver for the unsteady incompressible Navier–Stokes equations based on adaptive cartesian octree grids. The method extends a stable staggered grid finite difference scheme to the graded octree meshes. It is found that a straightforward extension is prone to produce spurious oscillatory velocity modes on the fine-to-coarse grids interfaces. A local linear low-pass filter is shown to reduce much of the bad influence of the interface modes on the accuracy of numerical solution. We introduce an implicit upwind finite difference approximation of advective terms as a low dissipative and stable alternative to semi-Lagrangian methods to treat the transport part of the equations. The performance of method is verified for a set of benchmark tests: a Beltrami type flow, the 3D lid-driven cavity and channel flows over a 3D square cylinder.