Simulating incompressible flows over complex geometries using the shifted boundary method with incomplete adaptive octree meshes

TL;DR

Addresses the challenge of simulating incompressible flows around complex geometries without boundary-fitted meshes by introducing Octree-SBM with an optimal surrogate boundary on incomplete octrees. The method combines a variational multiscale formulation of the incompressible Navier–Stokes equations with a Nitsche-type shifted boundary enforcement on the surrogate boundary and a backflow stabilization strategy, aided by a distance-based closest-point map ${M_h}$ and distance vector ${\boldsymbol{d}}$. It is implemented on the Dendro-KT octree framework with PETSc solvers and a KD-tree distance calculator, and validated on 2D and 3D benchmarks including cylinder, airfoil, sphere, cylinder, nozzle, and gyroid geometries, showing good agreement with reference data in ${C_d}$, Strouhal number, and pressure distributions, as well as robust parallel scalability. The work demonstrates that an optimal surrogate boundary (${\lambda=0.5}$) with incomplete octrees enables boundary-accurate, mesh-generation-free simulations of complex flows, with potential impact on large-scale engineering analyses and design optimization.

Abstract

We extend the shifted boundary method (SBM) to the simulation of incompressible fluid flow using immersed octree meshes. Previous work on SBM for fluid flow primarily utilized two- or three-dimensional unstructured tetrahedral grids. Recently, octree grids have become an essential component of immersed CFD solvers, and this work addresses this gap and the associated computational challenges. We leverage an optimal (approximate) surrogate boundary constructed efficiently on incomplete and adaptive octree meshes. The resulting framework enables the simulation of the incompressible Navier-Stokes equations in complex geometries without requiring boundary-fitted grids. Simulations of benchmark tests in two and three dimensions demonstrate that the Octree-SBM framework is a robust, accurate, and efficient approach to simulating fluid dynamics problems with complex geometries.

Figures (34)

• Figure 1: Surrogate domain ($\tilde{\Omega}_h$), surrogate boundary ($\tilde{\Gamma}_h = \partial \tilde{\Omega}_h$), and true boundary ($\Gamma$)
• Figure 2: The surrogate domain, its boundary, and the distance vector $\boldsymbol{d}$.
• Figure 4: The approaches to determining the value at a point of interest ($\blacksquare$) within the SBM framework.
• Figure 5: Tree diagram for the results section of Octree-SBM for Navier-Stokes.
• Figure 6: Illustrations of two test cases in \ref{['subsec:2D_cylinder']}: (a) Strong BC: a no-slip boundary condition is applied strongly at the nodal points ($\blacksquare$) along the pixelated surrogate boundary (---); (b) SBM BC: a shifted boundary condition is applied weakly at the Gauss points ($\blacksquare$) on the pixelated surrogate boundary (---).

Theorems & Definitions (1)

• Remark