The dynamics of vortical perturbations in viscoelastic shear flows

Abstract

Polymer additives have a striking impact on vortical motions in transitional and fully turbulent flows. Coherent structures which are familiar from Newtonian dynamics are altered in both strength and size while new, elasto-inertial phenomenology also emerges. In this work three canonical problems are used to explain the formation and evolution of particular flow structures in terms of the coupling between vorticity and polymer torque. Viscoelastic streak amplification is investigated first by forcing laminar Couette flow with a row of streamwise rolls. The streak response is shown to be determined by the ratio of the polymer relaxation time to the disturbance diffusion timescale in the solvent. When the timescales are disparate, quasi-Newtonian and elastic dynamics can be distinguished. When the timescales are commensurate, the fluid supports the propagation of vorticity waves whose reflection and superposition result in spanwise-travelling streaks which re- energise cyclically. In the second problem the receptivity of viscoelastic Couette flow to surface roughness on the lower wall is examined. The vorticity response is classified using a phase diagram, which is parameterised by the ratios of the channel depth and of the vorticity waves’ critical layer depth to the wavelength of the surface waviness. In shallow flows the bulk streamline distortion matches the topography of the lower wall, while large vorticity fluctuations are contained in an upper-wall solvent boundary layer. In deep channels, vorticity fluctuations are generated by a kinematic amplification mechanism at the critical layer. Finally, the evolution of a weak Gaussian vortex in a viscoelastic shear flow is computed to examine the dynamics of spanwise vorticity. Vorticity wave propagation along the tensioned mean streamlines causes the vortex to split into two as it is reoriented by the background flow. In addition, the analysis identifies the existence of a ‘reverse-Orr’ mechanism, whereby the vorticity amplifies as structures are tipped forward with the shear. The vorticity-polymer torque framework provides a unifying framework for studying these coherent motions. It also provides a grounding for understanding more complex, multiscale flows, such as elastic and inertial turbulence at low- and high-Reynolds numbers.