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

Abstract

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.

SPGrid

“…Additionally, we have focused on the Linux programming interface for our current implementation; as future work we will produce a port of the SPGrid library that also runs under the Windows operating system. Finally, our smoke simulation scheme was very elementary, and we did not leverage opportunities for more accurate advection schemes, methods to counteract numerical dissipation [Olshanskii et al 2013], and higher order accurate discretizations of the Poisson operator [Losasso et al 2005]. We will address such simulation features, along with an investigation of applying SPGrid to free surface flows in future work.…”

A Unified Approach for Computing Tsunami, Waves, Floods, and Landslides

“…fu.t C t/; p.t C t/g. This is done in the following substeps: (i) The convective terms are approximated using the second order accurate compact upwind method on octree grids from [12] and the momentum equation is solved for an intermediate velocity, while the plasticity terms are treated explicitly [11]; (ii) The intermediate velocity field is projected onto the subspace of discrete divergence free functions and the pressure is updated by the solution of the discrete Poisson equation.…”

“…However, building accurate and stable discretizations on such grids is a challenging task. For the fluid equations we use the second order accurate finite difference/finite volume method with compact nodal stencils developed in [12]. This method is a stable extension of the classical MAC scheme on octree meshes.…”

“…This scheme decouples one time step on convection, diffusion and plasticity, pressure correction, and the level set function advection substeps. Numerical stability and accuracy of the whole method were verified for the case of Newtonian flows in [10,12] by computing analytical solutions and benchmark flows in a cubic cavity and over a 3D cylinder. Computed results for a collapsing water column perfectly match available experimental data.…”

A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

“…When the free surface evolves, the grid is dynamically re ned or coarsened according to the distance to the free boundary. We note that discretizations on octree (quadtree) cartesian grids already enjoyed an employment in free surface ow computations [24,27,28,32,38], yet many aspects of using octree grids for accurate and predictive computations of free surface ows require more thorough study. To discretize divergence, gradient and Laplace operators, we use nite di erence approximations with compact stencils.…”