Tollmien-Schlichting route to elastoinertial turbulence in channel flow

Abstract

Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J. Fluid Mech. 893, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number Re = 3000, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, Wi and increasing Reynolds number from 3000-10000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. We show that this transition is directly tied to the strength of the TS critical layer fluctuations and can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS wave exceeds 1 2 . At Re = 10000, unlike 3000, the Newtonian TS attractor evolves continuously into the EIT state as Wi is increased from zero to about 13. We describe how the structure of the flow and stress fields changes, highlighting in particular a "sheet-shedding" process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.

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“…However, the fact that there is no analog (linear or nonlinear) of the TSmode in the Newtonian pipe-flow spectrum, and that the centermode remains the least stable one even in the weakly elastic regime [78], suggests that the TS-analog-based subcritical mechanism may not be obviously applicable to pipe Poiseuille flow; more work is clearly required in this regard. The recent subcritical continuation of the unstable center mode, in viscoelastic channel flow, to a nonlinear EIT structure [56] implies that subcritical mechanisms based on the centermode might also be operative in certain regions of Re-W i-β space, and thus the relevance of the centermode might extend outside of the linearly unstable regions indicated; 4 Given that recent experimental evidence points to EIT and MDR states being one and the same, for low to moderate Re values, it is worth mentioning here that the 2D TS-wave-analogs recently proposed to underlie EIT [57,58,91] stand in sharp contrast to an earlier interpretation that regarded the MDR regime as corresponding to a hibernating state of turbulence [86,99] comprising 3D so-called edge-state solutions (lying on the basin boundary between the laminar fixed point and the turbulent attractor in an appropriate phase space). Such states already exist in Newtonian turbulence, and their frequency of occurrence is thought to be progressively enhanced with increasing polymer concentration (although, the hibernating periods have been found to be strongly box-size dependent [29]).…”

“…Figures 10 and 11 illustrate the various possible transition scenarios in the W i-Re plane, for different fixed β, for pipe and plane Poiseuille flows. In these schematic illustrations, we bring together ideas based on the section above, on the centermode instability, and other hypotheses based on earlier nonmodal and nonlinear analyses (see sections 6 and 7 respectively for a detailed discussion); we also comment briefly on a recent independent line of work by Graham and coworkers that proposes a new subcritical route to EIT based on elastoinertial TS-wave analogs [57,58,91]. In the aforementioned figures, the linearly unstable regions in the interior of the W i-Re plane correspond to the domain of the elastoinertial centermode instability, and are depicted using colored lines for different β.…”

“…In regions of the Re-W i-β space where the centermode is linearly stable, and the originally Newtonian ECS are stabilized by elasticity, novel subcritical mechanisms are expected to dominate the transition process. For plane Poiseuille flow, at moderate Re, recent work [57,58,91] has identified a nonlinear mechanism based on elastoinertial wall modes closely related to the stable Newtonian TS mode (although still disconnected from it in phase space until a Re of 10 4 ). Such a pathway could be especially relevant in a direct transition between the Newtonian and elastoinertial turbulent states (as in Fig 11b), with the nearwall coherent structures in the former state acting as possible seeds for the aforementioned TS-wave analogs 4 .…”

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“…Finally, through the evolution of the flow structures and stress structures, they described how the individual sheets associated with critical-layer dynamics break up to form the layered multisheet-structure characteristics of EIT (see Shekar et al. 2021).…”

“…Regarding the origin of EIT, recent observations of linear instability indicate that EIT is possibly related with the 'centre-mode' instability (Garg et al 2018;Page, Dubief & Kerswell 2020;Chaudhary et al 2021;Khalid et al 2021) or the 'wall mode (Tollmien-Schlichting (TS) mode)' instability (Chaudhary et al 2019;Shekar et al 2019Shekar et al , 2020Shekar et al , 2021 in a different parameter space. Garg et al (2018) first found that viscoelastic pipe flow is linearly unstable at Re (e.g.…”