Computing hydrodynamic eigenmodes of channel flow with slip -- A highly accurate algorithm

TL;DR

This paper introduces a highly accurate, open-source algorithm for computing hydrodynamic eigenmodes in channel flow with slip, enabling precise CFD validation and analysis of flow dynamics beyond standard precision.

Contribution

The authors present a fast, automated, and rigorously verified open-source method for calculating hydrodynamic eigenmodes with high accuracy, including special case root-finding and asymptotic analysis.

Findings

Confirmed asymptotic power laws for eigenmodes

Validated root-finding in all special cases

Provided a publicly available code with test cases

Abstract

The transient start-up flow solution with slip is a useful tool to verify Computational Fluid Dynamics (CFD) simulations. However, a highly accurate, open-source black box solution does not seem to be available. Our method provides a fast, automated, and rigorously verified open-source implementation that can compute the hydrodynamic eigenmodes of a two-dimensional channel flow beyond the standard floating-point precision. This allows for a very accurate computation of the corresponding Fourier series solution. We prove that all roots are found in all special cases for the general flow problem with different slip lengths on the channel walls. The numerical results confirm analytically derived asymptotic power laws for the leading hydrodynamic eigenmode and the characteristic timescale in the limiting cases of small and large slip. The code repository including test cases is publicly available here https://git.rwth-aachen.de/fricke/start-up-flow