Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

A reverse transition route from inertial to elasticity-dominated turbulence in viscoelastic Taylor–Couette flow

A high-order transition route from inertial to elasticity-dominated turbulence (EDT) in Taylor–Couette flows of polymeric solutions has been discovered via direct numerical simulations. This novel two-step transition route is realized by enhancing the extensional viscosity and hoop stresses of the polymeric solution via increasing the maximum chain extension at a fixed polymer concentration. Specifically, in the first step inertial turbulence is stabilized to a laminar flow much like the modulated wavy vortex flow. The second step destabilizes this laminar flow state to EDT, i.e. a spatially smooth and temporally random flow with a $-3.5$ scaling law of the energy spectrum reminiscent of elastic turbulence. The flow states involved are distinctly different to those observed in the reverse transition route from inertial turbulence via a relaminarization of the flow to elasto-inertial turbulence in parallel shear flows, underscoring the importance of polymer-induced hoop stresses in realizing EDT that are absent in parallel shear flows.

Numerical Computation of Instabilities and Internal Waves from In Situ Measurements via the Viscous Taylor–Goldstein Problem

We explore numerical methods for the stability analysis of stratified, parallel shear flows considering the effects of small-scale turbulence represented by eddy viscosity and diffusivity. The result is an extension of the classical Taylor–Goldstein problem applicable to oceanic and atmospheric flows. Solutions with imaginary frequency describe shear and convective instabilities, whereas those with real frequency represent internal gravity waves. Application to large observational datasets can involve considerable computation and therefore requires a compromise between speed and accuracy. We compare several numerical methods to identify optimal approaches to various problems.

A conjecture on the least stable mode for the energy stability of plane parallel flows

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

Extension of classical stability theory to viscous planar wall-bounded shear flows

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternale et al., Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.