Modal stability analysis of viscoelastic channel and pipe flows using a well-conditioned spectral method

TL;DR

This study uses a well-conditioned spectral method to analyze the modal stability of viscoelastic pipe and channel flows, challenging previous claims of linear instability and highlighting the importance of numerical accuracy.

Contribution

It demonstrates the linear stability of viscoelastic pipe and channel flows across various parameters using a stable spectral method, correcting prior instability claims.

Findings

Viscoelastic pipe flow is linearly stable across a broad parameter range.

Plane Poiseuille flow remains stable where previous studies suggested instability.

Identification of spurious modes that can be mistaken for true eigenmodes.

Abstract

Modal stability analysis provides information about the long-time growth or decay of small-amplitude perturbations around a steady-state solution of a dynamical system. In fluid flows, exponentially growing perturbations can initiate departure from laminar flow and trigger transition to turbulence. Although flow of a Newtonian fluid through a pipe is linearly stable for very large values of the Reynolds number ($Re \sim 10^7$), a transition to turbulence often occurs for $Re$ as low as $1500$. When a dilute polymer solution is used in the place of a Newtonian fluid, the transitional value of the Reynolds number decreases even further. Using the spectral collocation method and Oldroyd-B constitutive equation, Garg et al. (Phys. Rev. Lett. 121:024502, 2018) claimed that such a transition in viscoelastic fluids is related to linear instability. Since differential matrices in the collocation method become ill-conditioned when a large number of basis functions is used, we revisit this problem using the well-conditioned spectral integration method. We show modal stability of viscoelastic pipe flow for a broad range of fluid elasticities and polymer concentrations, including cases considered by Garg et al. Similarly, we find that plane Poiseuille flow is linearly stable for cases where Garg et al. report instability. In both channel and pipe flows, we establish the existence of spurious modes that diverge slowly with finer discretization and demonstrate that these can be mistaken for grid-independent modes if the discretization is not fine enough.