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Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids

Published online by Cambridge University Press:  14 April 2009

NAZISH HODA ,
MIHAILO R. JOVANOVIĆ  and
SATISH KUMAR
NAZISH HODA
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
MIHAILO R. JOVANOVIĆ*
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
SATISH KUMAR*
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: [email protected], [email protected]
†Email address for correspondence: [email protected], [email protected]
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Abstract

Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input–output point of view by analysing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number (Re) scaling of frequency responses from different forcing to different velocity components is developed, showing the same Re dependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional time scales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds–Orr equation demonstrates that the energy-exchange term involving the streamwise/wall-normal polymer stress component τxy and the wall-normal gradient of the streamwise velocity ∂yu becomes increasingly important relative to the Reynolds-stress term as the elasticity number increases and is thus the main driving force for amplification in flows with strong viscoelastic effects.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 625 , 25 April 2009 , pp. 411 - 434
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitations. Phys. Fluids 13, 3258–3269.CrossRefGoogle Scholar
Bertola, V., Meulenbroek, B., Wagner, C., Storm, C., Morozov, A., van Saarloos, W. & Bonn, D. 2003 Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow. Phys. Rev. Lett. 90, 114502:1–4.CrossRefGoogle ScholarPubMed
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, vol. 2. Wiley.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Doering, C., Eckhardt, B. & Schumacher, J. 2006 Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newton. Fluid Mech. 135, 92–96.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5, 2600–2609.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618.CrossRefGoogle Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Hoda, N. 2008 Dynamics of complex fluids and complex flows near surfaces: continuum and molecular modeling. PhD thesis, University of Minnesota.Google Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid. Mech. 601, 407–424.CrossRefGoogle Scholar
Jovanović, M. R. 2004 Modeling, analysis, and control of spatially distributed systems. PhD thesis, University of California, Santa Barbara.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2006 A formula for frequency responses of distributed systems with one spatial variable. Syst. Control Lett. 55 (1), 27–37.CrossRefGoogle Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Math. 28, 735–756.CrossRefGoogle Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.CrossRefGoogle Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Larson, R. G. 2000 Turbulence without inertia. Nature 405, 27–28.CrossRefGoogle ScholarPubMed
Lin, B., Khomami, B. & Sureshkumar, R. 2004 Effect of non-normal interactions on the interfacial instability of multilayer viscoelastic channel flows. J. Non-Newton. Fluid Mech. 116, 407–429.CrossRefGoogle Scholar
McComb, W. D. 1991 The Physics of Fluid Turbulence. Oxford University Press.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Reynolds, W. C. & Kassinos, S. C. 1995 One-point modeling of rapidly deformed homogeneous turbulence. Proc. R. Soc. Lond. A 451 (1941), 87–104.Google Scholar
Sadanandan, B. & Sureshkumar, R. 2002 Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids 14, 41–48.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle 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.CrossRefGoogle 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-Newton. Fluid Mech. 82, 57–104.CrossRefGoogle Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoli, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Software 26 (4), 465–519.CrossRefGoogle Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control. Prentice Hall.Google Scholar