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Linear and nonlinear evolution of a localized disturbance in polymeric channel flow

  • Akshat Agarwal (a1), Luca Brandt (a2) and Tamer A. Zaki (a1)


The evolution of an initially localized disturbance in polymeric channel flow is investigated, with the FENE-P model used to characterize the viscoelastic behaviour of the flow. In the linear growth regime, the flow response is stabilized by viscoelasticity, and the maximum attainable disturbance-energy amplification is reduced with increasing polymer concentration. The reduction in the energy growth rate is attributed to the polymer work, which plays a dual role. First, a spanwise polymer-work term develops, and is explained by the tilting action of the wall-normal vorticity on the mean streamwise conformation tensor. This resistive term weakens the spanwise velocity perturbation thus reducing the energy of the localized disturbance. The second action of the polymer is analogous, with a wall-normal polymer work term that weakens the vertical velocity perturbation. Its indirect effect on energy growth is substantial since it reduces the production of Reynolds shear stress and in turn of the streamwise velocity perturbation, or streaks. During the early stages of nonlinear growth, the dominant effect of the polymer is to suppress the large-scale streaky structures which are strongly amplified in Newtonian flows. As a result, the process of transition to turbulence is prolonged and, after transition, a drag-reduced turbulent state is attained.


Corresponding author

Present address: Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. Email address for correspondence:


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Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Annu. Rev. Fluid Mech. 20, 359391.
Breuer, K. S. & Landahl, M. T. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances. J. Fluid Mech. 220, 595621.
Dallas, V., Vassilicos, J. C. & Hewitt, G. F. 2010 Strong polymer–turbulence interactions in viscoelastic turbulent channel flow. Phys. Rev. E 82, 066303.
De Angelis, E., Casciola, C. M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31 (4–7), 495507.
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-Newtonian Fluid Mech. 79, 433468.
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25, 110817.
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, 311329.
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, 271280.
El-Kareh, A. W. & Leal, L. G. 1989 Existance of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion. J. Non-Newtonian Fluid Mech. 33, 257287.
Henningson, D. S. & Andersson, P. H. 1987 The wave structure of turbulent spots in plane Poiseuille flow. J. Fluid Mech. 178, 405421.
Henningson, D. S. & Kim, J. 1991 On turbulent spots in plane Poiseuille flow. J. Fluid Mech. 228, 183205.
Henningson, D. S., Lundbladh, A. & Johansson, A. V. 1993 A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. J. Fluid Mech. 250, 169207.
Hoda, N., Jovanovic, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407424.
Hoda, N., Jovanovic, M. R. & Kumar, S. 2009 Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech. 625, 411434.
Jovanovic, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755778.
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28, 735756.
Li, C.-F., Sureshkumar, R. & Khomami, B. 2006 Influence of rheological parameters on polymer induced turbulent drag reduction. J. Non-Newtonian Fluid Mech. 140, 2340.
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, 2747.
Min, T., Yoo, J. Y. & Choi, H. 2003a Maximum drag reduction in a turbulent channel flow by polymer additives. J. Fluid Mech. 492, 91100.
Min, T., Yoo, J. Y., Choi, H. & Joseph, D. D. 2003b Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech. 486, 213238.
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689703.
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159205.
Page, J. & Zaki, T. A. 2014 Streak evolution in viscoelastic Couette flow. J. Fluid Mech. 742, 520551.
Pan, L., Morozov, A., Wagner, C. & Arratia, P. E. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110, 174502.
Ray, P. K. & Zaki, T. A. 2014 Absolute instability in viscoelastic mixing layers. Phys. Fluids 26, 014103.
Richter, D., Iaccarino, G. & Shaqfeh, E. S. G. 2010 Simulations of three-dimensional viscoelastic flows past a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 651, 415442.
Samanta, D., Dubief, Y., Holzner, M., Schafer, C. & Morozov, A. N. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110, 1055710562.
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129185.
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.
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, 5380.
Sureshkumar, R., Beris, A. N. & Handler, R. A. 1997 Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9, 743755.
Toms, B.1948 Observation on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In Proceedings of the International Rheological Congress, vol. 2 pp. 135–141.
Tsukahara, T., Ishigami, T., Yu, B. & Kawaguchi, Y. 2011 DNS study on viscoelastic effect in drag-reduced turbulent channel flow. J. Turbul. 12, 125.
Vaithianathan, T. & Collins, L. R. 2003 Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. J. Comput. Phys. 187, 121.
Virk, P. S. & Mickley, H. S. 1970 The ultimate asymptote and mean flow structures in Tom’s phenomenon. Trans. ASME E: J. Appl. Mech. 37, 488493.
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235256.
Xi, L. & Graham, M. D. 2010 Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units. J. Fluid Mech. 647, 421452.
Xi, L. & Graham, M. D. 2012 Intermittent dynamics of turbulence hibernation in Newtonian and viscoelastic minimal channel flows. J. Fluid Mech. 693, 433472.
Yu, B., Li, F. & Kawaguchi, Y. 2004 Numerical and experimental investigation of turbulent characteristics in a drag-reducing flow with surfactant additives. Intl J. Heat Fluid Flow 25, 961974.
Zhang, M., Lashgari, I., Zaki, T. A. & Brandt, L. 2013 Linear stablity analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737, 249279.
Zhou, Q. & Akhavan, R. 2003 A comparison of FENE and FENE-P dumbbell and chain models in turbulent flow. J. Non-Newtonian Fluid Mech. 109, 115155.
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