Modal stability analysis of viscoelastic channel and pipe flows using a wellconditioned spectral method
Abstract
Modal stability analysis provides information about the longtime growth or decay of smallamplitude perturbations around a steadystate 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 OldroydB 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 illconditioned when a large number of basis functions is used, we revisit this problem using the wellconditioned 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 gridindependent modes if the discretization is not fine enough.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.02980
 Bibcode:
 2019arXiv191102980H
 Keywords:

 Physics  Fluid Dynamics;
 Mathematics  Analysis of PDEs;
 Mathematics  Dynamical Systems
 EPrint:
 We are withdrawing the manuscript because the boundary conditions on stress fluctuations used in the pipeflow calculations may be too restrictive, causing unstable modes to be missed