Weakening the effect of boundaries: `diffusion-free' boundary conditions as a `do least harm' alternative to Neumann

TL;DR

This paper explores 'diffusion-free' boundary conditions as a less harmful alternative to Neumann conditions, demonstrating their effectiveness in reducing boundary layer effects and enabling more efficient numerical simulations in fluid dynamics.

Contribution

The paper introduces and illustrates the use of diffusion-free boundary conditions as a novel alternative to traditional boundary conditions in fluid flow problems.

Findings

Diffusion-free boundary conditions weaken boundary layers more than stress-free conditions.

They are effective in inertial wave eigenvalue problems in rotating fluids.

These conditions facilitate more efficient numerical simulations by reducing boundary layer effects.

Abstract

In this note, we discuss a poorly known alternative boundary condition to the usual Neumann or `stress-free' boundary condition typically used to weaken boundary layers when diffusion is present but very small. These `diffusion-free' boundary conditions were first developed (as far as the authors know) in 1995 (Sureshkumar & Beris, J. Non-Newtonian Fluid Mech., vol 60, 53-80, 1995) in viscoelastic flow modelling but are worthy of general consideration in other research areas. To illustrate their use, we solve two simple ODE problems and then treat a PDE problem - the inertial wave eigenvalue problem in a rotating cylinder, sphere and spherical shell for small but non-zero Ekman number $E$. Where inviscid inertial waves exist (cylinder and sphere), the viscous flows in the Ekman boundary layer are $O(E^{1/2})$ weaker than for the corresponding stress-free layer and fully $O(E)$ weaker than in a non-slip layer. These diffusion-free boundary conditions can also be used with hyperdiffusion and provide a systematic way to generate as many further boundary conditions as required. The weakening effect of this boundary condition could allow precious numerical resources to focus on other areas of the flow and thereby make smaller, more realistic values of diffusion accessible to simulations.