Simultaneous solution of incompressible Navier-Stokes flows on multiple surfaces

TL;DR

This paper introduces a hybrid finite element method for solving incompressible Navier-Stokes flows simultaneously on multiple curved surfaces within a bulk domain, using level set representations and stabilization techniques.

Contribution

It presents a novel hybrid finite element approach that handles multiple surfaces implicitly defined by level sets within a bulk domain, enabling accurate and higher-order convergence.

Findings

Good agreement with independent solutions on individual surfaces

Higher-order convergence rates for smooth solutions

Effective stabilization techniques ensure inf-sup condition compliance

Abstract

A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier-Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary. The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor-Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.