A mechanism for streamwise localisation of nonlinear waves in shear flows

Mellibovsky, Fernando; Meseguer, Alvaro

doi:10.1017/jfm.2015.440

Cited by 29 publications

(39 citation statements)

References 52 publications

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“…A pseudo-arclength Newton-Krylov-Poincaré continuation method for computing periodic solutions has been implemented (Kelley, 2003;Mellibovsky & Meseguer, 2015). Stable and unstable periodic orbits are obtained using this method.…”

Section: (A) Numerical Methodsmentioning

confidence: 99%

Extensional channel flow revisited: a dynamical systems perspective

et al. 2017

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Extensional self-similar flows in a channel are explored numerically for arbitrary stretching–shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching–shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier–Stokes solutions.

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Section: (A) Numerical Methodsmentioning

confidence: 99%

Extensional channel flow revisited: a dynamical systems perspective

et al. 2017

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“…They have a turning point significantly lower than the spatially extended traveling wave T W T S [24]. In even longer domains, they turn into fully localized states that are relative periodic orbits, generated out of subharmonic instabilities of the spatially extended TS waves [25,26]. Figure 2a) shows a bifurcation diagram for the modulated TS-waves and the spatially extended traveling wave from which they bifurcate.…”

Section: Spatially Extended and Localized Tollmien-schlichting Wavesmentioning

confidence: 99%

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

Zammert

Eckhardt

2016

Journal of Turbulence

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The transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. In contrast to other shear flows, PPF has a number of different coherent states that are relevant for the transition. We here discuss the different states, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux, the ones for constant pressure drop are always higher. The Tollmien-Schlichting waves bifurcate subcritically from the laminar profile at Re = 5772 and reach down to Re = 2609 (at a different optimal wave length). Their localized counter part bifurcates at the even lower value Re = 2334. Three dimensional exact solutions appear at much lower Reynolds numbers. We describe one exact solutions that is spanwise localized and has a critical Reynolds number of 316. Comparison to plane Couette flow suggests that this is likely the lowest Reynolds number for exact coherent structures in PPF. Streamwise localized versions of this state require higher Reynolds numbers, with the lowest bifurcation occurring near Re = 1018.

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“…The strong peak in the EIT spectrum seen in Figure 1 corresponds to a wavelength of 5, so here we report computations of nonlinear TS wave in a 2D domain with this length. The upper branch of this solution family is linearly stable in 2D at Re = 3000 [21][22][23] and easily captured with DNS using the linear TS mode as the initial condition. In Newtonian flow, the solution family exists at this wavelength down to Re ≈ 2800.…”

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Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence

et al. 2019

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Simulations of elastoinertial turbulence (EIT) of a polymer solution at low Reynolds number are shown to display localized polymer stretch fluctuations. These are very similar to structures arising from linear stability (Tollmien-Schlichting (TS) modes) and resolvent analyses: i.e., critical-layer structures localized where the mean fluid velocity equals the wavespeed. Computation of selfsustained nonlinear TS waves reveals that the critical layer exhibits stagnation points that generate sheets of large polymer stretch. These kinematics may be the genesis of similar structures in EIT.Turbulent drag reduction is an important and puzzling phenomenon in the non-Newtonian flow of complex fluids. Addition of polymers or micelle-forming surfactants to a liquid can lead to dramatic reductions in energy dissipation during turbulent flow while having a negligible effect on laminar flow[1].In Newtonian channel or pipe flow, transition to turbulence occurs by a so-called subcritical or "bypass" transition mechanism as flow rate, measured nondimensionally by Reynolds number, Re, increases: turbulence is initiated by finite-amplitude perturbations to the laminar flow profile, while the laminar flow remains linearly stable. While channel flow exhibits a two-dimensional linear instability leading to so-called Tollmien-Schlichting (TS) waves, the critical Reynolds number Re = 5772 is much higher than that observed for transition, so these are not traditionally viewed as playing an important role in Newtonian transition.For flowing polymer solutions under some conditions (low concentration, short polymer relaxation times), transition to turbulence occurs via the usual bypass transition. With further increase in Re, drag reduction sets in, and the flow eventually approaches the so-called maximum drag reduction (MDR) asymptote, an upper bound on the degree of drag reduction that is insensitive to the details of the fluid.Under other conditions, flow transitions directly from laminar flow into the MDR regime, and can do so at a Reynolds number where the flow would remain laminar if Newtonian [2][3][4][5]. Recent experiments and simulations [6][7][8] suggest that turbulence in this regime has structure very different from Newtonian, denoting it as "elastoinertial turbulence" (EIT). Choueiri et al.[4] experimentally observed that at transitional Reynolds numbers and increasing polymer concentration, turbulence is first suppressed, leading to relaminarization, and then reinitiated with an EIT structure and a level of drag corresponding to MDR. Therefore, there are actually two distinct types of turbulence in polymer solutions, one that is suppressed by viscoelasticity, and one that is promoted.The present work reports computations and analysis that elucidate the mechanisms underlying EIT. We show that EIT at low Re has highly localized polymer stress fluctuations. Surprisingly, these strongly resemble linear Tollmien-Schlichting modes as well as the most strongly amplified fluctuations from the laminar state. Furthermore, the kinematic...

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A mechanism for streamwise localisation of nonlinear waves in shear flows

Cited by 29 publications

References 52 publications

Extensional channel flow revisited: a dynamical systems perspective

Extensional channel flow revisited: a dynamical systems perspective

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence

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