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Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence

Published online by Cambridge University Press:  10 May 2019

Anupam Gupta  and
Dario Vincenzi Open the ORCID record for Dario Vincenzi [Opens in a new window]
Anupam Gupta
Affiliation:
Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Dario Vincenzi*
Affiliation:
Université Côte d’Azur, CNRS, LJAD, 06100 Nice, France
*
†Email address for correspondence: [email protected]
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Abstract

Elastic turbulence is a chaotic regime that emerges in polymer solutions at low Reynolds numbers. A common way to ensure stability in numerical simulations of polymer solutions is to add artificially large polymer-stress diffusion. In order to assess the accuracy of this approach in the elastic turbulence regime, we compare numerical simulations of the two-dimensional Oldroyd-B and FENE-P models sustained by a cellular force with and without artificial diffusion. We find that artificial diffusion can have a dramatic effect even on the large-scale properties of the flow and we show some of the spurious phenomena that may arise when artificial diffusion is used.

JFM classification

Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 405 - 418
Copyright
© 2019 Cambridge University Press 

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References

Abed, W. M., Whalley, R. D., Dennis, D. J. C. & Poole, R. J. 2016 Experimental investigation of the impact of elastic turbulence on heat transfer in a serpentine channel. J. Non-Newtonian Fluid Mech. 231, 68–78.Google Scholar
Balci, N., Thomases, B., Renardy, M. & Doering, C. R. 2011 Symmetric factorization of the conformation tensor in viscoelastic fluid models. J. Non-Newtonian Fluid Mech. 166, 546–553.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
Bird, R. B., Hassager, O., Armstrong, R. C. & Curtiss, C. F. 1987 Dynamics of Polymeric Liquids, vol. II. Wiley.Google Scholar
van Buel, R., Schaaf, C. & Stark, H. 2018 Elastic turbulence in two-dimensional Taylor–Couette flows. Europhys. Lett. 124, 14001.Google Scholar
Cardoso, O., Marteau, D. & Tabeling, P. 1994 Quantitative experimental study of the free decay of quasi-two-dimensional turbulence. Phys. Rev. E 49, 454–461.Google Scholar
Dallas, V., Vassilicos, J. C. & Hewitt, G. F. 2010 Strong polymer-turbulence interactions in viscoelastic turbulent channel flow. Phys. Rev. E 82, 066303.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.Google Scholar
Fattal, R. & Kupferman, R. 2004 Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123, 281–285.Google Scholar
Fouxon, A. & Lebedev, V. 2003 Spectra of turbulence in dilute polymer solutions. Phys. Fluids 15, 2060–2072.Google Scholar
Garg, H., Calzavarini, E., Mompean, G. & Berti, S. 2018 Particle-laden two-dimensional elastic turbulence. Eur. Phys. J. E 41, 115.Google Scholar
Gotoh, K. & Yamada, M. 1984 Instability of a cellular flow. J. Phys. Soc. Japan 53, 3395–3398.Google Scholar
Graham, M. D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26, 101301.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
Groisman, A. & Steinberg, V. 2004 Elastic turbulence in curvilinear flows of polymer solutions. New J. Phys. 6, 29.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
Gupta, A. & Pandit, R. 2017 Melting of a nonequilibrium 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
Hameduddin, I., Meneveau, C., Zaki, T. A. & Gayme, D. F. 2018 Geometric decomposition of the conformation tensor in viscoelastic turbulence. J. Fluid Mech. 842, 395–427.Google Scholar
Jin, S. & Collins, L. R. 2007 Dynamics of dissolved polymer chains in isotropic turbulence. New J. Phys. 9, 360.Google Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.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.Google Scholar
Liu, N. & Khomami, B. 2013 Elastically induced turbulence in Taylor–Couette flow: direct numerical simulation and mechanistic insight. J. Fluid Mech. 737, R4.Google Scholar
Liu, Y. & Steinberg, V. 2014 Single polymer dynamics in a random flow. Macromol. Symp. 337, 34–43.Google Scholar
Min, T., Yoo, J. Y. & Choi, H. 2001 Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows. J. Non-Newtonian Fluid Mech. 100, 27–47.Google Scholar
Oldroyd, J. G. 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
Perlekar, P., Mitra, D. & Pandit, R. 2006 Manifestations of drag reduction by polymer additives in decaying, homogeneous, isotropic turbulence. Phys. Rev. Lett. 97, 264501.Google Scholar
Perlekar, P., Mitra, D. & Pandit, R. 2010 Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence with polymer additives. Phys. Rev. E 82, 066313.Google Scholar
Plan, E. L. C. VI M., Gupta, A., Vincenzi, D. & Gibbon, J. D. 2017 Lyapunov dimension of elastic turbulence. J. Fluid Mech. 822, R4.Google Scholar
Poole, R. J., Budhiraja, B., Cain, A. R. & Scott, P. A. 2012 Emulsification using elastic turbulence. J. Non-Newtonian Fluid Mech. 177–178, 15–18.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T & Flannery, B. P. 2007 Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press.Google Scholar
Ray, S. S. & Vincenzi, D. 2016 Elastic turbulence in a shell model of polymer solution. Europhys. Lett. 114, 44001.Google Scholar
Robert, A., Vaithianathan, T., Collins, L. R. & Brasseur, J. G. 2010 Polymer-laden homogeneous shear-driven turbulent flow: a model for polymer drag reduction. J. Fluid Mech. 657, 189–226.Google Scholar
Rothstein, D., Henry, E. & Gollub, J. P. 1999 Persistent patterns in transient chaotic fluid mixing. Nature 401, 770–772.Google Scholar
Sid, S., Terrapon, V. E. & Dubief, Y. 2018 Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids 3, 011301(R).Google Scholar
Shekar, A., McMullen, R. M., Wang, S.-N., McKeon, B. J. & Graham, M. D. 2019 Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett. 122, 124503.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. 2009 Transition to mixing and oscillations in a Stokesian viscoelastic flow. Phys. Rev. Lett. 103, 094501.Google 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
Traore, B., Castelain, C. & Burghelea, T. 2015 Efficient heat transfer in a regime of elastic turbulence. J. Non-Newtonian Fluid Mech. 223, 62–76.Google Scholar
Vaithianathan, T. & Collins, L. R. 2003 Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. J. Comput. Phys. 187, 1–21.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
Vaithianathan, T., Robert, A., Brasseur, J. G. & Collins, L. R. 2007 Polymer mixing in shear-driven turbulence. J. Fluid Mech. 585, 487–497.Google Scholar
Valente, P. C., da Silva, C. B. & Pinho, F. T. 2014 The effect of viscoelasticity on the turbulent kinetic energy cascade. J. Fluid Mech. 760, 39–62.Google Scholar
Watanabe, T. & Gotoh, T. 2013 Kinetic energy spectrum of low-Reynolds-number turbulence with polymer additives. J. Phys. Conf. Ser. 454, 012007.Google Scholar
Watanabe, T. & Gotoh, T. 2014 Power-law spectra formed by stretching polymers in decaying isotropic turbulence. Phys. Fluids 26, 035110.Google Scholar
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