Skip to main content Accessibility help
×
Home

Self-sustained elastoinertial Tollmien–Schlichting waves

Published online by Cambridge University Press:  09 June 2020

Ashwin Shekar
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin–Madison, Madison, WI53706, USA
Ryan M. McMullen
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
Beverley J. McKeon
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
Michael D. Graham
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin–Madison, Madison, WI53706, USA
Corresponding
E-mail address:

Abstract

Direct simulations of two-dimensional plane channel flow of a viscoelastic fluid at Reynolds number $Re=3000$ reveal the existence of a family of attractors whose structure closely resembles the linear Tollmien–Schlichting (TS) mode, and in particular exhibits strongly localized stress fluctuations at the critical layer position of the TS mode. At the parameter values chosen, this solution branch is not connected to the nonlinear TS solution branch found for Newtonian flow, and thus represents a solution family that is nonlinearly self-sustained by viscoelasticity. The ratio between stress and velocity fluctuations is in quantitative agreement for the attractor and the linear TS mode, and increases strongly with Weissenberg number, $\mathit{Wi}$ . For the latter, there is a transition in the scaling of this ratio as $\mathit{Wi}$ increases, and the $\mathit{Wi}$ at which the nonlinear solution family comes into existence is just above this transition. Finally, evidence indicates that this branch is connected through an unstable solution branch to two-dimensional elastoinertial turbulence (EIT). These results suggest that, in the parameter range considered here, the bypass transition leading to EIT is mediated by nonlinear amplification and self-sustenance of perturbations that excite the TS mode.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.

References

Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2019 Elastoinertial wall mode instabilities in viscoelastic plane Poiseuille flow. J. Fluid Mech. 881, 119163.CrossRefGoogle Scholar
Chaudhary, I., Garg, P., Subramanian, G. & Shankar, V.2020 Linear instability of viscoelastic pipe flow. Preprint, arXiv:2003.09369v1.Google Scholar
Choueiri, G. H., Lopez, J. M. & Hof, B. 2018 Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120 (12), 124501.CrossRefGoogle ScholarPubMed
Dallas, V., Vassilicos, J. & Hewitt, G. 2010 Strong polymer-turbulence interactions in viscoelastic turbulent channel flow. Phys. Rev. E 82 (6), 066303.Google ScholarPubMed
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge Mathematical Libraries. Cambridge University Press.CrossRefGoogle Scholar
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elastoinertial turbulence. Phys. Fluids 25 (11), 110817.CrossRefGoogle Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008a Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.CrossRefGoogle Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008b Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.CrossRefGoogle Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366 (1868), 12971315.CrossRefGoogle ScholarPubMed
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.CrossRefGoogle Scholar
Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121 (2), 024502.CrossRefGoogle ScholarPubMed
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.CrossRefGoogle Scholar
Graham, M. D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26, 101301.CrossRefGoogle Scholar
Hameduddin, I., Gayme, D. F. & Zaki, T. A. 2019 Perturbative expansions of the conformation tensor in viscoelastic flows. J. Fluid Mech. 858, 377406.CrossRefGoogle Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 15941598.CrossRefGoogle ScholarPubMed
Jiménez, J. 1990 Transition to turbulence in two-dimensional Poiseuille flow. J. Fluid Mech. 218, 265297.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.CrossRefGoogle Scholar
Kim, K., Li, C.-F., Sureshkumar, R., Balachandar, S. & Adrian, R. J. 2007 Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow. J. Fluid Mech. 584, 281299.CrossRefGoogle Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160 (1), 241282.CrossRefGoogle Scholar
Lee, S. J. & Zaki, T. A. 2017 Simulations of natural transition in viscoelastic channel flow. J. Fluid Mech. 820, 232262.CrossRefGoogle Scholar
Li, W. & Graham, M. D. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19 (8), 083101.CrossRefGoogle Scholar
Li, W., Stone, P. A. & Graham, M. D. 2005 Viscoelastic nonlinear traveling waves and drag reduction in plane Poiseuille flow. Fluid Mech. Appl.: Proceedings of the IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitute Solutions 77, 285308.Google Scholar
Li, W., Xi, L. & Graham, M. D. 2006 Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. Fluid Mech. 565, 353362.CrossRefGoogle Scholar
Lopez, J. M., Choueiri, G. H. & Hof, B. 2019 Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. J. Fluid Mech. 874, 699719.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.CrossRefGoogle Scholar
Min, T., Choi, H. & Yoo, J. Y. 2003 Maximum drag reduction in a turbulent channel flow by polymer additives. J. Fluid Mech. 492, 91100.CrossRefGoogle Scholar
Page, J. & Zaki, T. A. 2015 The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow. J. Fluid Mech. 777, 327363.CrossRefGoogle Scholar
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430454.CrossRefGoogle Scholar
Patera, A. T. & Orszag, S. A. 1981 Finite-amplitude stability of axisymmetric pipe flow. J. Fluid Mech. 112, 467474.CrossRefGoogle Scholar
Pereira, A., Thompson, R. L. & Mompean, G.2019a Beyond the maximum drag reduction asymptote: the pseudo-laminar state. Preprint, arXiv:1911.00439.Google Scholar
Pereira, A. S., Thompson, R. L. & Mompean, G. 2019b Common features between the Newtonian laminar–turbulent transition and the viscoelastic drag-reducing turbulence. J. Fluid Mech. 877, 405428.CrossRefGoogle Scholar
Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B. 2013 Elastoinertial turbulence. Proc. Natl Acad. Sci. USA 110 (26), 1055710562.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.CrossRefGoogle Scholar
Shekar, A. & Graham, M. D. 2018 Exact coherent states with hairpin-like vortex structure in channel flow. J. Fluid Mech. 849, 7689.CrossRefGoogle 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 (12), 124503.CrossRefGoogle ScholarPubMed
Sid, S., Terrapon, V. E. & Dubief, Y. 2018 Two-dimensional dynamics of elastoinertial turbulence and its role in polymer drag reduction. Phys. Rev. F 3 (1), 011301.Google Scholar
Stone, P. A. & Graham, M. D. 2003 Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids 15 (5), 12471256.CrossRefGoogle Scholar
Stone, P. A., Waleffe, F. & Graham, M. D. 2002 Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89 (20), 208301.CrossRefGoogle Scholar
Stone, P. A., Roy, A., Larson, R. G., Waleffe, F. & Graham, M. D. 2004 Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16 (9), 34703482.CrossRefGoogle 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 (1–3), 322.CrossRefGoogle Scholar
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625656.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81 (19), 41404143.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 15171534.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.CrossRefGoogle Scholar
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235256.CrossRefGoogle Scholar
Zammert, S. & Eckhardt, B. 2014 Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348359.CrossRefGoogle Scholar
Zhang, M., Lashgari, I., Zaki, T. A. & Brandt, L. 2013 Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737, 249279.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 21
Total number of PDF views: 200 *
View data table for this chart

* Views captured on Cambridge Core between 09th June 2020 - 17th January 2021. This data will be updated every 24 hours.

Hostname: page-component-77fc7d77f9-cdnjt Total loading time: 0.283 Render date: 2021-01-17T19:53:16.022Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags last update: Sun Jan 17 2021 18:54:59 GMT+0000 (Coordinated Universal Time) Feature Flags: { "metrics": true, "metricsAbstractViews": false, "peerReview": true, "crossMark": true, "comments": true, "relatedCommentaries": true, "subject": true, "clr": true, "languageSwitch": true, "figures": false, "newCiteModal": false, "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Self-sustained elastoinertial Tollmien–Schlichting waves
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Self-sustained elastoinertial Tollmien–Schlichting waves
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Self-sustained elastoinertial Tollmien–Schlichting waves
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *