Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-26T00:15:47.532Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  25 November 2013

Mengqi Zhang ,
Iman Lashgari ,
Tamer A. Zaki  and
Luca Brandt
Mengqi Zhang
Affiliation:
Linné Flow Centre and SeRC (Swedish e-Science Research Centre), KTH Mechanics, S-100 44 Stockholm, Sweden
Iman Lashgari
Affiliation:
Linné Flow Centre and SeRC (Swedish e-Science Research Centre), KTH Mechanics, S-100 44 Stockholm, Sweden
Tamer A. Zaki
Affiliation:
Mechanical Engineering, Imperial College, London SW7 2AZ, UK
Luca Brandt*
Affiliation:
Linné Flow Centre and SeRC (Swedish e-Science Research Centre), KTH Mechanics, S-100 44 Stockholm, Sweden
*
†Email address for correspondence: luca@mech.kth.se
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Instability: Instability Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 249 - 279
Copyright
©2013 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.)

Footnotes

‡

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

References

Arora, K. & Khomami, B. 2005 The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions. J. Non-Newtonian Fluid Mech. 129, 56–60.Google Scholar
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.Google Scholar
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, 209–319.Google Scholar
Berlin, S., Lundbladh, A. & Henningson, D. S. 1994 Spatial simulations of oblique transition. Phys. Fluids 6, 1949–1951.Google Scholar
Berti, S., Bistagnino, A., Boffetta, G., Celani, A. & Musacchio, S. 2008 Two-dimensional elastic turbulence. Phys. Rev. E 77, 055306.Google Scholar
Bird, R., Curtiss, C., Armstrong, R. & Hassager, O. 1987 Dynamics of Polymer Liquids. Vol. 2. Kinetic Theory. Wiley.Google Scholar
Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2007 Nonlinear dynamics of the viscoelastic Kolmogorov flows. J. Fluid Mech. 590, 61–80.Google Scholar
Blonce, L. 1997 Linear stability of Giesekus fluids in Poiseuille flow. Mech. Res. Commun. 24, 223–228.Google Scholar
Burghelea, T., Segre, E. & Steinberg, V. 2006 Role of elastic stress in statistical and scaling properties of elastic turbulence. Phys. Rev. Lett. 96, 214502.CrossRefGoogle ScholarPubMed
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.CrossRefGoogle Scholar
Canuto, C. G., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2007 Spectral Methods: Evolution of Complex Geometries and Applications to Fluid Mechanics. Springer.Google Scholar
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, 28–35.Google Scholar
De Angelis, E., Casciola, C. M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495–507.Google 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, 92–96.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, 271–280.CrossRefGoogle Scholar
Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82, 026316.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 2093–2102.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional constant shear flow. Physics 5 (6), 1390–1400.Google Scholar
Govindarajan, R., Lõvov, V. S. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrasts. Phys. Rev. Lett. 87 (17).Google Scholar
Groisman, A. & Steinberg, V. 2001 Elastic turbulence in a polymer solution flow. Nature 405 (6782), 53–55.Google Scholar
Groisman, A. & Steinberg, V. 2004 Elastic turbulence in curvilinear flows of polymer. New J. Phys. 6, 29.Google Scholar
Ho, T. C. & Denn, M. M. 1977 Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech. 3, 179–195.CrossRefGoogle Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407–424.Google Scholar
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, 411–434.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamic and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.Google Scholar
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, 2415–2431.Google Scholar
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22 (2)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 (14/15), 755–778.CrossRefGoogle Scholar
Klinkenberg, J., De Lange, H. C. & Brandt, L. 2011 Modal and non-modal stability of particle-laden channel flow. Phys. Fluids 23 (6)064110.Google Scholar
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.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31 (3), 213–263.Google Scholar
Larson, R. G. 2000 Turbulence without inertia. Nature 405, 27–28.Google Scholar
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely viscoelastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573–600.Google Scholar
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, 232–263.Google Scholar
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, 235–268.CrossRefGoogle Scholar
Morozov, A. N. & Saarloos, W. 2005 Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett. 95, 024501.CrossRefGoogle ScholarPubMed
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, 177–194.CrossRefGoogle Scholar
Porteous, K. C. & Denn, M. M. 1972 Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol. 16 (2), 295–308.Google Scholar
Ranganathan, B. T. & Govindarajan, R. 2005 Stabilization and destabilization of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1).Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 15–47.Google Scholar
Renardy, M. & Renardy, Y. 1986 Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Non-Newtonian Fluid Mech. 22, 23–33.Google Scholar
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.Google Scholar
Sadanandan, B. & Sureshkumar, R. 2002 Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids 14 (1), 41–48.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.Google Scholar
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), 3470–3482.Google Scholar
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), 151–182.Google Scholar
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, 57–104.Google Scholar
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. 135–141.Google Scholar
Trefethen, N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Weideman, J. A. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519.Google 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.Google Scholar
Xi, L. & Graham, M. D. 2010 Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units. J. Fluid Mech. 647, 421–452.CrossRefGoogle Scholar
Zhao, L. H., Andersson, H. I. & Gillissen, J. J. J. 2010 Turbulence modulation and drag reduction by spherical particles. Phys. Fluids 22, 081702.CrossRefGoogle Scholar
Zhu, L., Lauga, E. & Brandt, L. 2011 Self-propulsion in viscoelastic fluids: pushers vs. pullers. Phys. Fluids 24, 051902.Google Scholar