Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T11:26:28.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Lyapunov dimension of elastic turbulence

Published online by Cambridge University Press:  06 June 2017

Emmanuel Lance Christopher VI M. Plan Open the ORCID record for Emmanuel Lance Christopher VI M. Plan [Opens in a new window] ,
Anupam Gupta ,
Dario Vincenzi  and
John D. Gibbon
Emmanuel Lance Christopher VI M. Plan*
Affiliation:
Université Côte d’Azur, CNRS, LJAD, 06108 Nice, France
Anupam Gupta
Affiliation:
FERMaT, Université de Toulouse, CNRS, INPT, INSA, UPS, 31062 Toulouse, France
Dario Vincenzi
Affiliation:
Université Côte d’Azur, CNRS, LJAD, 06108 Nice, France
John D. Gibbon
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Non-Newtonian Flows: Viscoelasticity
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , R4
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amarouchene, Y. & Kellay, H. 2002 Polymers in 2D turbulence: suppression of large scale fluctuations. Phys. Rev. Lett. 89, 104502.Google Scholar
Barrett, J. W. & Süli, E. 2011 Existence and equilibration of global weak solutions to kinetic models for dilute polymers I. Math. Models Meth. Appl. Sci. 21, 1211–1289.Google Scholar
Berti, S., Bistagnino, A., Boffetta, G., Celani, A. & Musacchio, S. 2008 Two-dimensional elastic turbulence. Phys. Rev. E 77, 055306(R).Google Scholar
Berti, S. & Boffetta, G. 2010 Elastic waves and transition to elastic turbulence in a two-dimensional viscoelastic Kolmogorov flow. Phys. Rev. E 82, 036314.Google Scholar
Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2005 The viscoelastic Kolmogorov flow: eddy-viscosity and linear stability. J. Fluid Mech. 523, 161–170.Google Scholar
Boffetta, G., Celani, A. & Musacchio, S. 2003 Two-dimensional turbulence of dilute polymer solutions. Phys. Rev. Lett. 91, 034501.Google Scholar
Burghelea, T., Segre, E. & Steinberg, V. 2004 Statistics of particle pair separations in the elastic turbulent flow of a dilute polymer solution. Europhys. Lett. 68, 529–535.Google Scholar
Burghelea, T., Segre, E. & Steinberg, V. 2007 Elastic turbulence in von Karman swirling flow between two disks. Phys. Fluids 19, 053104.CrossRefGoogle Scholar
Constantin, P. 1987 Collective L ∞ -estimates for families of functions with orthonormal derivatives. Indiana Univ. Math. J. 36, 603–616.Google Scholar
Constantin, P. & Foias, C. 1985 Global Lyapunov exponents, Kaplan–Yorke formulas and the dimension of the attractors for 2D Navier–Stokes equations. Commun. Pure Appl. Maths 38, 1–27.Google Scholar
Constantin, P., Foias, C. & Temam, R. 1988 On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284–296.CrossRefGoogle Scholar
Constantin, P. & Kliegl, M. 2012 Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress. Arch. Rat. Mech. Anal. 206, 725–740.Google Scholar
Doering, C. R. & Gibbon, J. D. 1991 Note on the Constantin–Foias–Temam attractor dimension estimate for two-dimensional turbulence. Physica D 48, 471–480.CrossRefGoogle Scholar
Doering, C. R. & Gibbon, J. D. 1995 Applied Analysis of the Navier–Stokes Equations. Cambridge University Press.Google Scholar
Elgindi, T. M. & Rousset, F. 2015 Global regularity for some Oldroyd-B type models. Commun. Pure Appl. Math. 68, 2005–2021.Google Scholar
El-Kareh, A. W. & Leal, L. G. 1989 Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion. J. Non-Newtonian Fluid Mech. 33, 257–287.CrossRefGoogle Scholar
Fouxon, A. & Lebedev, V. 2003 Spectra of turbulence in dilute polymer solutions. Phys. Fluids 15, 2060–2072.Google Scholar
Gibbon, J. D. & Titi, E. 1997 Attractor dimension and small length scale estimates for the three-dimensional Navier–Stokes equations. Nonlinearity 10, 109–119.Google Scholar
Grilli, M., Vázquez-Quesada, A. & Ellero, M. 2013 Transition to turbulence and mixing in a viscoelastic fluid flowing inside a channel with a periodic array of cylindrical obstacles. Phys. Rev. Lett. 110, 174501.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.Google Scholar
Groisman, A. & Steinberg, V. 2001 Efficient mixing at low Reynolds numbers using polymer additives. Nature 410, 905–908.Google Scholar
Gupta, A. & Pandit, R. 2017 Melting of a non-equilibrium vortex crystal in a fluid film with polymers: elastic versus fluid turbulence. Phys. Rev. E 95, 033119.Google Scholar
Gupta, A., Perlekar, P. & Pandit, R. 2015 Two-dimensional homogeneous isotropic fluid turbulence with polymer additives. Phys. Rev. E 91, 033013.Google Scholar
Hu, X. & Lin, F. 2016 Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data. Commun. Pure Appl. Maths 69, 372–404.Google Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282.CrossRefGoogle Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.Google Scholar
Lei, Z., Masmoudi, N. & Zhou, Y. 2010 Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248, 328–341.Google Scholar
Mitchell, J., Lyons, K., Howe, A. M. & Clarke, A. 2016 Viscoelastic polymer flows and elastic turbulence in three-dimensional porous structures. Soft Matt. 12, 460–468.Google Scholar
Oldroyd, J. G. 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
Renardy, M. 2006 A comment on smoothness of viscoelastic stresses. J. Non-Newtonian Fluid Mech. 138, 204–205.Google Scholar
Robinson, J. C. 2001 Infinite-dimensional Dynamical Systems. Cambridge University Press.Google Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.Google Scholar
Sureshkumar, R. & Beris, A. N. 1995 Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newtonian Fluid Mech. 60, 53–80.Google Scholar
Thomases, B. 2011 An analysis of the effect of stress diffusion on the dynamics of creeping viscoelastic flow. J. Non-Newtonian Fluid Mech. 166, 1221–1228.Google Scholar
Thomases, B. & Shelley, M. 2007 Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids 19, 103103.Google Scholar
Thomases, B. & Shelley, M. 2009 Transition to mixing and oscillations in a Stokesian viscoelastic flow. Phys. Rev. Lett. 103, 094501.CrossRefGoogle Scholar
Thomases, B., Shelley, M. & Thiffeault, J.-L. 2011 A Stokesian viscoelastic flow: transition to oscillations and mixing. Physica D 240, 1602–1614.Google Scholar
Vaithianathan, T., Robert, A., Brasseur, J. G. & Collins, L. R. 2006 An improved algorithm for simulating three-dimensional, viscoelastic turbulence. J. Non-Newtonian Fluid Mech. 140, 3–22.Google Scholar