Skip to main content Accessibility help
×
Home

The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

  • Jacob Page (a1) and Tamer A. Zaki (a1) (a2)

Abstract

The viscoelastic analogue to the Newtonian Orr amplification mechanism is examined using linear theory. A weak, two-dimensional Gaussian vortex is superposed onto a uniform viscoelastic shear flow. Whilst in the Newtonian solution the spanwise vorticity perturbations are simply advected, the viscoelastic behaviour is markedly different. When the polymer relaxation rate is much slower than the rate of deformation by the shear, the vortex splits into a new pair of co-rotating but counter-propagating vortices. Furthermore, the disturbance exhibits a significant amplification in its spanwise vorticity as it is tilted forward by the shear. Asymptotic solutions for an Oldroyd-B fluid in the limits of high and low elasticity isolate and explain these two effects. The splitting of the vortex is a manifestation of vorticity wave propagation along the tensioned mean-flow streamlines, while the spanwise vorticity growth is driven by the amplification of a polymer torque perturbation. The analysis explicitly demonstrates that the polymer torque amplifies as the disturbance becomes aligned with the shear. This behaviour is opposite to the Orr mechanism for energy amplification in Newtonian flows, and is therefore labelled a ‘reverse-Orr’ mechanism. Numerical evaluations of vortex evolutions using the more realistic FENE-P model, which takes into account the finite extensibility of the polymer chains, show the same qualitative behaviour. However, a new form of stress perturbation is established in regions where the polymer is significantly stretched, and results in an earlier onset of decay.

Copyright

Corresponding author

Email address for correspondence: t.zaki@jhu.edu

References

Hide All
Agarwal, A., Brandt, L. & Zaki, T. A. 2014 Linear and nonlinear evolution of a localized disturbance in polymeric channel flow. J. Fluid Mech. 760, 278303.
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205213.
Azaiez, J. & Homsy, G. M. 1994 Linear stability of free shear flow of viscoelastic liquids. J. Fluid Mech. 268, 3769.
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 21602163.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers, 1st edn. McGraw-Hill.
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn. vol. 1. Wiley.
Butler, K. M. & Farrell, B. F. 1992 Optimal perturbations in viscous shear flow. Phys. Fluids 4, 16371650.
Cadot, O. & Kumar, S. 2000 Experimental characterization of viscoelastic effects on two- and three-dimensional shear instabilities. J. Fluid Mech. 416, 151172.
Dimitropoulos, C. D., Sureshkumar, R., Beris, A. N. & Handler, R. A. 2001 Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow. Phys. Fluids 13, 1016.
Doering, C. R., Eckhardt, B. & Schumacher, J. 2006 Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newtonian Fluid Mech. 135, 9296.
Drazin, P. & Reid, W. H. 1995 Hydrodynamic Stability. Cambridge University Press.
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 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 (4), 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.
Farrell, B. 1987 Developing disturbances in shear. J. Atmos. Sci. 44, 21912199.
Graham, M. D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26, 101301.
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 5355.
Haj-Hariri, H. & Homsy, G. M. 1997 Three-dimensional instability of viscoelastic elliptic vortices. J. Fluid Mech. 353, 357381.
Harder, K. J. & Tiederman, W. G. 1991 Drag reduction and turbulent structure in two-dimensional channel flows. Phil. Trans. R. Soc. Lond. A 336 (1640), 1934.
Hinch, E. J. 1977 Mechanical models of dilute polymer solutions in strong flows. Phys. Fluids 20 (10), S22.
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407424.
Hoda, N., Jovanović, M. R. & Kumar, S. 2009 Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech. 625, 411434.
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25 (11), 110814.
Jin, S. & Collins, L. R. 2007 Dynamics of dissolved polymer chains in isotropic turbulence. New J. Phys. 9, 360.
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22, 023101.
Jovanović, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755778.
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.
Kumar, S. & Homsy, G. M. 1999 Direct numerical simulation of hydrodynamic instabilities in two- and three-dimensional viscoelastic free shear layers. J. Non-Newtonian Fluid Mech. 83, 249276.
Kupferman, R. 2005 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation. J. Non-Newtonian Fluid Mech. 127, 169190.
Lagnado, R. R. & Simmen, J. A. 1993 The three-dimensional instability of elliptical vortices in a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 50, 2944.
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.
Lieu, B. K., Jovanović, M. R. & Kumar, S. 2013 Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids. J. Fluid Mech. 723, 232263.
Luchik, T. S. & Tiederman, W. G. 1988 Turbulent structure in low-concentration drag-reducing channel flows. J. Fluid Mech. 190, 241263.
Min, T., Yul Yoo, J., Choi, H. & Joseph, D. D. 2003 Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech. 486, 213238.
Moffatt, H. K.1967 The interaction of turbulence with strong wind shear. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom & V. I. Tatarski), pp. 139–156. Moscow.
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.
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, 251291.
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311324.
Ray, P. K. & Zaki, T. A. 2014 Absolute instability in viscoelastic mixing layers. Phys. Fluids 26 (1), 014103.
Ray, P. K. & Zaki, T. A. 2015 Absolute/convective instability of planar viscoelastic jets. Phys. Fluids 27, 014110.
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601639.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Samanta, D. S., Dubief, Y., Holzner, H., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110, 1055710562.
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.
Terrapon, V. E., Dubief, Y. & Soria, J. 2014 On the role of pressure in elasto-inertial turbulence. J. Turbul. 16 (1), 2643.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
Walker, D. T. & Tiederman, W. G. 1990 Turbulent structure in a channel flow with polymer injection at the wall. J. Fluid Mech. 218, 377403.
Wang, S.-N., Graham, M. D., Hahn, F. J. & Xi, L. 2014 Time-series and extended Karhunen–Lo\`eve analysis of turbulent drag reduction in polymer solutions. AIChE J. 60 (4), 14601475.
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.
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.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed