Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T05:12:44.509Z Has data issue: false hasContentIssue false
  • English
  • Français

On the maximum drag reduction due to added polymers in Poiseuille flow

Published online by Cambridge University Press:  27 July 2010

JAMES D. WOODCOCK ,
JOHN E. SADER  and
IVAN MARUSIC
JAMES D. WOODCOCK*
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia Department of Mechanical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia
JOHN E. SADER
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
IVAN MARUSIC
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The addition of elastic polymers to turbulent liquids is known to produce significant drag reduction. In this study, we prove that the drag in pipe and channel flows of an unforced laminar fluid constitutes a lower bound for the drag of a fluid containing dilute elastic polymers. Further, the addition of elastic polymers to laminar fluids invariably increases drag. This proof does not rely on the adoption of a particular constitutive equation for the polymer force, and would also be applicable to other similar methods of drag reduction, which are also achieved by the addition of certain particles to a flow. Examples of such methods include the addition of surfactants to a flowing liquid and the presence of sand particles in sandstorms and water droplets in cyclones.

JFM classification

Flow Control: Drag reduction
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 473 - 483
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

Barenblatt, G. I., Chorin, A. J. & Prostokishin, V. M. 2005 A note concerning the Lighthill ‘sandwich model’ of tropical cyclones. Proc. Natl. Acad. Sci. 102, 11148–11150.CrossRefGoogle ScholarPubMed
Bewley, T. R. 2009 A fundamental limit on the balance of power in a transpiration-controlled channel flow. J. Fluid Mech. 632, 443–446.CrossRefGoogle Scholar
Bewley, T. R. & Aamo, O. M. 2004 A ‘win–win’ mechanism for low drag transients in controlled two-dimensional channel flow and its implications for sustained drag reduction. J. Fluid Mech. 499, 183–196.CrossRefGoogle Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41 (1), 219–240.CrossRefGoogle Scholar
Collis, S. S., Joslin, R. D., Seifert, A. & Theofilis, V. 2004 Issues in active flow control: theory, control, simulation and experiment. Prog. Aerosp. Sci. 40, 237–289.CrossRefGoogle Scholar
Gore, R. & Crowe, C. 1989 Effect of particle size on modulating turbulent intensity. Intl J. Multiphase Flow 15, 279–285.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4, 473–494.CrossRefGoogle Scholar
James, D. F. 2009 Boger fluids. Annu. Rev. Fluid Mech. 41, 129–142.CrossRefGoogle Scholar
Karniadakis, G. E. & Choi, K.-S. 2003 Mechanisms on transverse motions in turbulent wall flows. Annu. Rev. Fluid Mech. 35, 45–62.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, 281–299.CrossRefGoogle Scholar
Luchik, T. S. & Tiederman, W. G. 1988 Turbulent structure in low-concentration drag-reducing channel flows. J. Fluid Mech. 190, 241–263.CrossRefGoogle Scholar
Marusic, I., Joseph, D. D. & Mahesh, K. 2007 Laminar and turbulent comparisons for channel flow and flow control. J. Fluid Mech. 570, 467–477.CrossRefGoogle Scholar
Min, T., Kang, S., Speyer, J. L. & Kim, J. 2006 Sustained sub-laminar drag in a fully developed channel flow. J. Fluid Mech. 558, 81–100.CrossRefGoogle Scholar
Min, T., Yoo, J. Y., Choi, H. & Joseph, D. D. 2003 Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech. 486, 213–238.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.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
Ptasinski, P. K., Nieuwstadt, F. T. M., Van den Brule, B. H. A. A. & Hulsen, M. A. 2001 Experiments in turbulent pipe flow with polymer additives at maximum drag reduction. Flow Turbulence Combust. 66, 159–182.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.CrossRefGoogle 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.CrossRefGoogle Scholar
Thomas, T. Y. 1942 Qualitative analysis of the flow of fluids in pipes. Am. J. Maths. 64, 754–767.CrossRefGoogle Scholar
Toms, B. A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proc. 1st Intl Congr. Rheol. 2, 135–141.Google Scholar
Virk, P. S. 1971 An elastic sublayer model for drag reduction by dilute solutions of linear macromolecules. J. Fluid Mech. 45 (3), 417–440.CrossRefGoogle Scholar
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21, 625–656.CrossRefGoogle Scholar
Virk, P. S., Merrill, E. W., Mickley, H. S. & Smith, K. A. 1967 The Toms phenomenon: turbulent pipe flow of dilute polymer solutions. J. Fluid Mech 30 (2), 305–328.CrossRefGoogle Scholar
Warholic, M. D., Schmidt, G. M. & Hanratty, T. J. 1999 The influence of a drag-reducing surfactant on a turbulent velocity field. J. Fluid Mech. 388, 1–20.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