Viscous Flow in Domains with Corners: Numerical Artifacts, their Origin and Removal

TL;DR

This paper investigates numerical artifacts in viscous flow simulations near corners, identifies their origin related to eigensolutions, and proposes a method to eliminate these issues for more accurate results.

Contribution

It introduces a novel method utilizing eigensolution knowledge to remove multivaluedness and singularities in numerical viscous flow computations near corners.

Findings

Numerical artifacts are caused by eigensolutions near corners.

The proposed method effectively removes pressure singularities.

Validated on Stokes and Navier-Stokes flows with free surfaces.

Abstract

We show that an attempt to compute numerically a viscous flow in a domain with a piece-wise smooth boundary by straightforwardly applying well-tested numerical algorithms (and numerical codes based on their use, such as COMSOL Multiphysics) can lead to spurious multivaluedness and nonintegrable singularities in the distribution of the fluid's pressure. The origin of this difficulty is that, near a corner formed by smooth parts of the piece-wise smooth boundary, in addition to the solution of the inhomogeneous problem, there is also an eigensolution. For obtuse corner angles this eigensolution (a) becomes dominant and (b) has a singular radial derivative of velocity at the corner. A method is developed that uses the knowledge about the eigensolution to remove multivaluedness and nonintegrability of the pressure. The method is first explained in the simple case of a Stokes flow in a corner region and then generalised for the full-scale unsteady Navier-Stokes flow in a domain with a free surface.