Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T21:22:07.930Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units

Published online by Cambridge University Press:  18 March 2010

LI XI  and
MICHAEL D. GRAHAM
LI XI
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA
MICHAEL D. GRAHAM*
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA
*
†Email address for correspondence: graham@engr.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The observation that addition of a minute amount of flexible polymers to fluid reduces turbulent friction drag is well known. However, many aspects of this drag reduction phenomenon are not well understood; in particular, the origin of the maximum drag reduction (MDR) asymptote, a universal upper limit on drag reduction by polymers, remains an open question. This study focuses on the drag reduction phenomenon in the plane Poiseuille geometry in a parameter regime close to the laminar–turbulent transition. By minimizing the size of the periodic simulation box to the lower limit for which turbulence persists, the essential self-sustaining turbulent motions are isolated. In these ‘minimal flow unit’ (MFU) solutions, a series of qualitatively different stages consistent with previous experiments is observed, including an MDR stage where the mean flow rate is found to be invariant with respect to changing polymer-related parameters. Before the MDR stage, an additional transition exists between a relatively low degree (LDR) and a high degree (HDR) of drag reduction. This transition occurs at about 13%–15% of drag reduction and is characterized by a sudden increase in the minimal box size, as well as many qualitative changes in flow statistics. The observation of LDR–HDR transition at less than 15% drag reduction shows for the first time that it is a qualitative transition instead of a quantitative effect of the amount of drag reduction. Spatio-temporal flow structures change substantially upon this transition, suggesting that two distinct types of self-sustaining turbulent dynamics are observed. In LDR, as in Newtonian turbulence, the self-sustaining process involves one low-speed streak and its surrounding streamwise vortices; after the LDR–HDR transition, multiple streaks are present in the self-sustaining structure and complex intermittent behaviour of the streaks is observed. This multistage scenario of LDR–HDR–MDR recovers all key transitions commonly observed and studied at much higher Reynolds numbers.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 421 - 452
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Benzi, R., De Angelis, E., L'vov, V. S., Procaccia, I. & Tiberkevich, V. 2006 Maximum drag reduction asymptotes and the cross-over to the Newtonian plug. J. Fluid Mech. 551, 185–195.CrossRefGoogle Scholar
Bird, R. B., Curtis, C. F., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn., vol. 2. John Wiley & Sons.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carlson, D. R., Widnall, S. E. & Peeters, M. F. 1982 A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121, 487–505.Google Scholar
De Angelis, E., Casciola, C. M., L'vov, V. S., Piva, R. & Procaccia, I. 2003 Drag reduction by polymers in turbulent channel flows: energy redistribution between invariant empirical modes. Phys. Rev. E 67, 056312.CrossRefGoogle ScholarPubMed
De Angelis, E., Casciola, C. M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495–507.Google Scholar
Dimitropoulos, C. D., Sureshkumar, R. & Beris, A. N. 1998 Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters. J. Non-Newton. Fluid Mech. 79, 433–468.Google Scholar
Dubief, Y. & Delcayre, F. 2000 On coherent-vortex identification in turbulence. J. Turbul. 1, 1–22.Google Scholar
Dubief, Y., Terrapon, V. E., White, C. M., Shaqfeh, E. S. G., Moin, P. & Lele, S. K. 2005 New answers on the interaction between polymers and vortices in turbulent flows. Flow Turbul. Combust. 74, 311–329.CrossRefGoogle Scholar
Dubief, Y., White, C. M., Terrapon, V. E., Shaqfeh, E. S. G., Moin, P. & Lele, S. K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Flyvbjerg, H. & Petersen, H. G. 1989 Error-estimates on averages of correlated data. J. Chem. Phys. 91, 461–466.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanotić, P. 2008 Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Graham, M. D. 2004 Drag reduction in turbulent flow of polymer solutions. In Rheology Reviews (ed. Binding, D. M. & Walters, K.), pp. 143–170. British Society of Rheology.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google 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 Recurrence of travelling waves in transitional pipe flow. Science 305, 1594–1597.Google Scholar
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443, 59–62.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Housiadas, K. D. & Beris, A. N. 2003 Polymer-induced drag reduction: effects of variations in elasticity and inertia in turbulent viscoelastic channel flow. Phys. Fluids 15, 2369–2384.CrossRefGoogle Scholar
Housiadas, K. D., Beris, A. N. & Handler, R. A. 2005 Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow. Phys. Fluids 17, 035106.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M. & Shiba, M. 2005 Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids 17, 015105.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.Google Scholar
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle Scholar
Kim, K., Adrian, R. J., Balachandar, S. & Sureshkumar, R. 2008 Dynamics of hairpin vortices and polymer-induced turbulent drag reduction. Phys. Rev. Lett. 100, 134504.Google 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, 281–299.CrossRefGoogle Scholar
Li, C.-F., Sureshkumar, R. & Khomami, B. 2006 a Influence of rheological parameters on polymer-induced turbulent drag reduction. J. Non-Newton. Fluid Mech. 140, 23–40.Google Scholar
Li, W. & Graham, M. D. 2007 Polymer-induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19, 083101.Google Scholar
Li, W., Stone, P. A. & Graham, M. D. 2005 Viscoelastic nonlinear travelling waves and drag reduction in plane Poiseuille flow. In IUTAM Symposium on Laminar–Turbulent Transition and Finite Amplitude Solutions (ed. Mullin, T. & Kerswell, R. R.), pp. 289–312. Springer.Google Scholar
Li, W., Xi, L. & Graham, M. D. 2006 b Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. Fluid Mech. 565, 353–362.Google Scholar
Min, T., Choi, H. & Yoo, J. Y. 2003 a Maximum drag reduction in a turbulent channel flow by polymer additives. J. Fluid Mech. 492, 91–100.Google Scholar
Min, T., Yoo, J. Y., Choi, H. & Joseph, D. D. 2003 b Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech. 486, 213–238.CrossRefGoogle Scholar
Oldaker, D. K. & Tiederman, W. G. 1977 Spatial structure of viscous sublayer in drag-reducing channel flows. Phys. Fluids 20, S133–S144.CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.Google Scholar
Procaccia, I., L'vov, V. S. & Benzi, R. 2008 Colloquium: theory of drag reduction by polymers in wall-bounded turbulence. Rev. Mod. Phys. 80, 225–247.CrossRefGoogle Scholar
Ptasinski, P. K., Boersma, B. J., Nieuwstadt, F. T. M., Hulsen, M. A., van den Brule, B. H. A. A. & Hunt, J. C. R. 2003 Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms. J. Fluid Mech. 490, 251–291.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.CrossRefGoogle Scholar
Sankaran, R., Sokolov, M. & Antonia, R. A. 1988 Substructures in a turbulent spot. J. Fluid Mech. 197, 389–414.Google Scholar
Smith, C. R. & Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region of a turbulent boundary-layer. J. Fluid Mech. 129, 27–54.Google Scholar
Sreenivasan, K. R. & White, C. M. 2000 The onset of drag reduction by dilute polymer additives, and the maximum drag reduction asymptote. J. Fluid Mech. 409, 149–164.Google Scholar
Stone, P. A. & Graham, M. D. 2003 Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids 15, 1247–1256.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, 3470–3482.CrossRefGoogle Scholar
Stone, P. A., Waleffe, W. & Graham, M. D. 2002 Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89, 208301.Google Scholar
Sureshkumar, R. & Beris, A. N. 1997 Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9, 743–755.Google Scholar
Toms, B. A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In Proceedings of the First International Congress on Rheology, vol. 2, pp. 135–141. Amsterdam.Google Scholar
Toms, B. A. 1977 Early experiments on drag reduction by polymers. Phys. Fluids 20, S3–S5.Google Scholar
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21, 625–656.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Warholic, M. D., Massah, H. & Hanratty, T. J. 1999 Influence of drag-reducing polymers on turbulence: effects of Reynolds number, concentration and mixing. Exp. Fluids 27, 461–472.CrossRefGoogle Scholar
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–256.CrossRefGoogle Scholar
White, C. M., Somandepalli, V. S. R. & Mungal, M. G. 2004 The turbulence structure of drag-reduced boundary layer flow. Exp. Fluids 36, 62–69.Google Scholar
Willis, A. P. & Kerswell, R. R. 2007 Critical behaviour in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98, 014501.CrossRefGoogle ScholarPubMed
Wu, J. Z., Xiong, A. K. & Yang, Y. T. 2005 Axial stretching and vortex definition. Phys. Fluids 17, 038108.Google Scholar
Zang, T. A. 1991 On the rotation and skew-symmetrical forms for incompressible-flow simulations. Appl. Numer. Math. 7, 27–40.Google Scholar