Computing Viscous Flow Along a 3D Open Tube Using the Immerse Interface Method

TL;DR

This paper extends a numerical immersed interface method to simulate 3D viscous flow in open tubular structures, accurately capturing sharp solution features with second-order convergence, thus enabling efficient and precise modeling of fluid-structure interactions in complex geometries.

Contribution

The study introduces a 3D immersed interface method for open tubular flow in axisymmetric coordinates, improving computational efficiency and accuracy over previous 2D approaches.

Findings

Method preserves sharp solution jumps and derivatives.

Achieves second-order convergence in space and time.

Successfully models elastic wall deformation effects.

Abstract

In a companion study \cite{patterson2020computing2D}, we present a numerical method for simulating 2D viscous flow through an open compliant closed channel, drive by pressure gradient. We consider the highly viscous regime, where fluid dynamics is described by the Stokes equations, and the less viscous regime described by the Navier-Stokes equations. In this study, we extend the method to 3D tubular flow. The problem is formulated in axisymmetric cylindrical coordinates, an approach that is natural for tubular flow simulations and that substantially reduces computational cost. When the elastic tubular walls are stretched or compressed, they exert forces on the fluid. These singular forces introduce unsmoothness into the fluid solution. As in the companion 2D study \cite{patterson2020computing2D}, we extend the immersed interface method to an open tube, and we compute solution to the model equations using the resulting method. Numerical results indicate that this new method preserves sharp jumps in the solution and its derivatives, and converges with second-order accuracy in both space and time.