Computation of Steady Incompressible Flows in Unbounded Domains

TL;DR

This paper introduces a spectrally-accurate computational method for steady incompressible flows in unbounded 2D domains, effectively handling the slow decay at infinity without domain truncation, validated across various Reynolds numbers.

Contribution

The paper develops a novel spectral approach that accurately computes steady Navier-Stokes flows in unbounded domains, overcoming challenges posed by slow decay and discontinuities.

Findings

Method accurately captures flow behavior at infinity.

Validated for Reynolds numbers up to two orders of magnitude.

Results agree with theoretical predictions and literature data.

Abstract

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background "skeleton" field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favourably against known theoretical predictions and the data available in the literature.