A Linear Route to Elasto-inertial Turbulence

V. Shankar

Department of Chemical Engineering

Indian Institute of Technology Kanpur, India

http://home.iitk.ac.in/~vshankar

Abstract

Pipe flow of Newtonian fluids is well known to exhibit a transition from the laminar to turbulent regime usually at a Reynolds number ~ 2000, despite being linearly stable at all Reynolds numbers. In stark contrast, we show that pressure-driven pipe flow of a viscoelastic (Oldroyd-B) fluid is linearly unstable to axisymmetric perturbations. The dimensionless groups that govern stability are the Reynolds number Re = ρU_maxR/η, the elasticity number E = λη/(ρR²) and the ratio of solvent to solution viscosity β = η_s/η; here, R is the pipe radius, U_max is the maximum velocity of the base flow, ρ is the fluid density, and λ is the polymer relaxation time. The unstable mode has a phase speed close to U_max over the entire unstable region in the Re-E-β space. In parameter regimes accessible to experiments (e.g. β > 0.6 and E > 0.08), the critical Reynolds number Re_c ~ 400, with the associated eigenfunctions spread out across the pipe cross section. In the asymptotic limit of E(1-β) << 1, but fixed E, the critical Reynolds number for instability and the critical wavenumber diverge as Re_c ~ (E (1-β))^-3/2 and k_c ~ (E(1-β))^-1/2 respectively. The unstable eigenfunction in this limit is localized near the centerline, implying that the unstable mode belongs to a class of viscoelastic `center modes’.

The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the Re-W plane (W = Re E being the Weissenberg number). The prediction of a linear instability is consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of `early turbulence' in the 1970's to the more recent discovery of `elasto-inertial turbulence' (B. Hof and coworkers, Proc. Natl. Acad. Sci., 110, 10557-10562 (2013)), wherein transition is observed at Re much lower than 2000. An analogous center-mode instability is predicted for viscoelastic channel flows over a similar range of parameters. Thus, there is the suggestion of a universal linear mechanism that underlies the onset of turbulence in rectilinear viscoelastic shearing flows for sufficiently elastic dilute polymer solutions,

marking a possible paradigm shift in our understanding of transition in such flows. We will end with a discussion of the possible transition scenarios in the Re-W-β space for viscoelastic shearing flows.

Collaborators:

Prof. Ganesh Subramanian (Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore; theory), Prof. Debopam Das (IIT Kanpur; experiments).

PhD Students from IIT Kanpur: Indresh Chaudhary, Mohammad Khalid, Shailendra Yadav (theory), Bidhan Chandra (experiments).

PhD Student from JNCASR Bangalore: Piyush Garg (theory).

References:

1. P. Garg et al., Phys. Rev. Lett., 121, 024502 (2018).

2. I. Chaudhary et al., arXiv: 2003.09369, under review (2020).

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