Skip to main content Accessibility help
×
Home

Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids

  • Mengqi Zhang (a1), Iman Lashgari (a1), Tamer A. Zaki (a2) and Luca Brandt (a1)

Abstract

We study the modal and non-modal linear instability of inertia-dominated channel flow of viscoelastic fluids modelled by the Oldroyd-B and FENE-P closures. The effects of polymer viscosity and relaxation time are considered for both fluids, with the additional parameter of the maximum possible extension for the FENE-P. We find that the parameter explaining the effect of the polymer on the instability is the ratio between the polymer relaxation time and the characteristic instability time scale (the frequency of a modal wave and the time over which the disturbance grows in the non-modal case). Destabilization of both modal and non-modal instability is observed when the polymer relaxation time is shorter than the instability time scale, whereas the flow is more stable in the opposite case. Analysis of the kinetic energy budget reveals that in both regimes the production of perturbation kinetic energy due to the work of the Reynolds stress against the mean shear is responsible for the observed effects where polymers act to alter the correlation between the streamwise and wall-normal velocity fluctuations. In the subcritical regime, the non-modal amplification of streamwise elongated structures is still the most dangerous disturbance-growth mechanism in the flow and this is slightly enhanced by the presence of polymers. However, viscoelastic effects are found to have a stabilizing effect on the amplification of oblique modes.

Copyright

Corresponding author

Email address for correspondence: luca@mech.kth.se

Footnotes

Hide All

Current address: Département Fluides, Thermique, Combustion, Institut PPrime, CNRS–Université de Poitiers–ENSMA, Poitiers, France.

Footnotes

References

Hide All
Arora, K. & Khomami, B. 2005 The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions. J. Non-Newtonian Fluid Mech. 129, 5660.
Arratia, P. E., Thomas, C. C., Diorio, J. & Gollub, J. P. 2006 Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett. 96, 144502.
Atalik, K. & Keunings, R. 2002 Non-linear temporal stability analysis of viscoelastic plane channel flows using a fully-spectral method. J. Non-Newtonian Fluid Mech. 102, 209319.
Berlin, S., Lundbladh, A. & Henningson, D. S. 1994 Spatial simulations of oblique transition. Phys. Fluids 6, 19491951.
Berti, S., Bistagnino, A., Boffetta, G., Celani, A. & Musacchio, S. 2008 Two-dimensional elastic turbulence. Phys. Rev. E 77, 055306.
Bird, R., Curtiss, C., Armstrong, R. & Hassager, O. 1987 Dynamics of Polymer Liquids. Vol. 2. Kinetic Theory. Wiley.
Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2007 Nonlinear dynamics of the viscoelastic Kolmogorov flows. J. Fluid Mech. 590, 6180.
Blonce, L. 1997 Linear stability of Giesekus fluids in Poiseuille flow. Mech. Res. Commun. 24, 223228.
Burghelea, T., Segre, E. & Steinberg, V. 2006 Role of elastic stress in statistical and scaling properties of elastic turbulence. Phys. Rev. Lett. 96, 214502.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 16371650.
Canuto, C. G., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2007 Spectral Methods: Evolution of Complex Geometries and Applications to Fluid Mechanics. Springer.
Cruz, D., Pinho, F. & Oliveira, P. 2005 Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution. J. Non-Newtonian Fluid Mech. 132, 2835.
De Angelis, E., Casciola, C. M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495507.
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.
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.
Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82, 026316.
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 20932102.
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional constant shear flow. Physics 5 (6), 13901400.
Govindarajan, R., Lõvov, V. S. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrasts. Phys. Rev. Lett. 87 (17).
Groisman, A. & Steinberg, V. 2001 Elastic turbulence in a polymer solution flow. Nature 405 (6782), 5355.
Groisman, A. & Steinberg, V. 2004 Elastic turbulence in curvilinear flows of polymer. New J. Phys. 6, 29.
Ho, T. C. & Denn, M. M. 1977 Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech. 3, 179195.
Hoda, N., Jovanović, 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.
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamic and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), pp. 81294. Cambridge University Press.
Joo, Y. L. & Shaqfeh, S. G. 1992 The effects of inertia on the viscoelastic Dean and Taylor–Couette flow instabilities with application to coating flows. Phys. Fluids 4, 24152431.
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22 (2)023101.
Jovanović, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166 (14/15), 755778.
Klinkenberg, J., De Lange, H. C. & Brandt, L. 2011 Modal and non-modal stability of particle-laden channel flow. Phys. Fluids 23 (6)064110.
Klinkenberg, J., Sardina, G., Lange, H. C. D. & Brandt, L. 2013 Numerical study of laminar–turbulent transition in particle-laden channel flow. Phys. Rev. E 86, 043011.
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31 (3), 213263.
Larson, R. G. 2000 Turbulence without inertia. Nature 405, 2728.
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely viscoelastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573600.
Lieu, B. K., Jovanovic, M. R. & Kumar, S. 2013 Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids. J. Fluid Mech. 723, 232263.
Meulenbroek, B., Storm, C., Morozov, A. N. & Saarloos, W. 2004 Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116, 235268.
Morozov, A. N. & Saarloos, W. 2005 Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett. 95, 024501.
Nouar, C., Bottaro, A. & Brancher, J. P. 2007 Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177194.
Porteous, K. C. & Denn, M. M. 1972 Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol. 16 (2), 295308.
Ranganathan, B. T. & Govindarajan, R. 2005 Stabilization and destabilization of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1).
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 1547.
Renardy, M. & Renardy, Y. 1986 Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Non-Newtonian Fluid Mech. 22, 2333.
Roy, A., Morozov, A., Saarloos, W. V. & Larson, R. G. 2006 Mechanism of polymer drag reduction using a low-dimensional model. Phys. Rev. Lett. 97, 234501.
Sadanandan, B. & Sureshkumar, R. 2002 Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids 14 (1), 4148.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129185.
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. J. Phys. Fluids 16 (9), 34703482.
Sureshkumar, R. & Beris, A. N. 1995 Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech. 56 (2), 151182.
Sureshkumar, R., Smith, M. D., Armstrong, R. C. & Brown, R. A. 1999 Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations. J. Non-Newtonian Fluid Mech. 82, 57104.
Toms, B. A. 1949 Some observations of the flow of linear polymer solution through straight tubes at large Reynolds numbers. In Proceedings of the First International Congress on Rheology (North-Holland, Amsterdam, 1949), vol. 2, pp. 135141.
Trefethen, N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.
Weideman, J. A. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465519.
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.
Zhao, L. H., Andersson, H. I. & Gillissen, J. J. J. 2010 Turbulence modulation and drag reduction by spherical particles. Phys. Fluids 22, 081702.
Zhu, L., Lauga, E. & Brandt, L. 2011 Self-propulsion in viscoelastic fluids: pushers vs. pullers. Phys. Fluids 24, 051902.
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