Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T06:37:56.017Z Has data issue: false hasContentIssue false

The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

Published online by Cambridge University Press:  20 July 2015

Jacob Page
Affiliation:
Mechanical Engineering, Imperial College London, LondonSW7 2AZ, UK
Tamer A. Zaki*
Affiliation:
Mechanical Engineering, Imperial College London, LondonSW7 2AZ, UK Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
*
Email address for correspondence: [email protected]

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.

Type
Papers
Copyright
© 2015 Cambridge University Press 

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

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.Google Scholar
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205213.Google Scholar
Azaiez, J. & Homsy, G. M. 1994 Linear stability of free shear flow of viscoelastic liquids. J. Fluid Mech. 268, 3769.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 21602163.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers, 1st edn. McGraw-Hill.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn. vol. 1. Wiley.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Optimal perturbations in viscous shear flow. Phys. Fluids 4, 16371650.Google Scholar
Cadot, O. & Kumar, S. 2000 Experimental characterization of viscoelastic effects on two- and three-dimensional shear instabilities. J. Fluid Mech. 416, 151172.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Drazin, P. & Reid, W. H. 1995 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 110817.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 (4), 311329.Google 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, 271280.Google Scholar
Farrell, B. 1987 Developing disturbances in shear. J. Atmos. Sci. 44, 21912199.Google Scholar
Graham, M. D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26, 101301.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 5355.CrossRefGoogle Scholar
Haj-Hariri, H. & Homsy, G. M. 1997 Three-dimensional instability of viscoelastic elliptic vortices. J. Fluid Mech. 353, 357381.Google Scholar
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.Google Scholar
Hinch, E. J. 1977 Mechanical models of dilute polymer solutions in strong flows. Phys. Fluids 20 (10), S22.CrossRefGoogle Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407424.CrossRefGoogle Scholar
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.Google Scholar
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25 (11), 110814.CrossRefGoogle Scholar
Jin, S. & Collins, L. R. 2007 Dynamics of dissolved polymer chains in isotropic turbulence. New J. Phys. 9, 360.CrossRefGoogle Scholar
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22, 023101.Google Scholar
Jovanović, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755778.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, 281299.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Luchik, T. S. & Tiederman, W. G. 1988 Turbulent structure in low-concentration drag-reducing channel flows. J. Fluid Mech. 190, 241263.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Page, J. & Zaki, T. A. 2014 Streak evolution in viscoelastic Couette flow. J. Fluid Mech. 742, 520551.CrossRefGoogle Scholar
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.Google 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, 251291.CrossRefGoogle Scholar
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311324.Google Scholar
Ray, P. K. & Zaki, T. A. 2014 Absolute instability in viscoelastic mixing layers. Phys. Fluids 26 (1), 014103.Google Scholar
Ray, P. K. & Zaki, T. A. 2015 Absolute/convective instability of planar viscoelastic jets. Phys. Fluids 27, 014110.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601639.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Terrapon, V. E., Dubief, Y. & Soria, J. 2014 On the role of pressure in elasto-inertial turbulence. J. Turbul. 16 (1), 2643.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Walker, D. T. & Tiederman, W. G. 1990 Turbulent structure in a channel flow with polymer injection at the wall. J. Fluid Mech. 218, 377403.Google Scholar
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.Google Scholar
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235256.Google Scholar
Xi, L. & Graham, M. D. 2010 Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units. J. Fluid Mech. 647, 421452.Google Scholar
Xi, L. & Graham, M. D. 2012 Intermittent dynamics of turbulence hibernation in Newtonian and viscoelastic minimal channel flows. J. Fluid Mech. 693, 433472.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