Improving convergence of volume penalized fluidsolid interactions
Abstract
Boundary conditions on arbitrary geometries are a common issue in simulating partial differential equations. The conventional approach is to discretize on a grid conforming to the geometry. However grid construction is challenging, and this difficulty is compounded for evolving domains. Several methods instead augment the equations themselves to implicitly enforce the boundary conditions. This paper examines the Volume Penalty Method, which approximates Dirichlet boundary conditions in the Navier Stokes equations with rapid linear damping (nondimensional time scale $\eta$) inside the object. This technique is proven to converge to the true solution, and also leads to simple volumeintegral force and torque calculations. Unfortunately, previous analysis showed convergence of only $\mathcal{O}(\eta^{1/2})$. We analyze the source of this error using matched asymptotic expansions and show that it stems from a displacement length, proportional to a Reynolds number Re dependent boundary layer of size $\mathcal{O}(\eta^{1/2}\text{Re}^{1/2})$. The relative size of the displacement length and damping time scale lead to the emergence of multiple asymptotic regimes. The key finding is that there is a simple correction that can be efficiently calculated to eliminate the displacement length and promote the accuracy to $\mathcal{O}(\eta)$. This improvement also extends to the force and torque calculations. We demonstrate these findings in 1D planar Poiseuille flow, 2D steady flow past a viscous stagnation point, and 2D unsteady flow past a rotating cylinder, and finally show that Richardson extrapolation can be used with our correction to further improve convergence to $\mathcal{O}(\eta^{2})$.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.11914
 Bibcode:
 2019arXiv190311914H
 Keywords:

 Mathematics  Numerical Analysis
 EPrint:
 40 pages, 12 figures, added reference to GitHub repository of minimal working examples