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Effects of wall compliance on the linear stability of Taylor–Couette flow

Published online by Cambridge University Press:  10 July 2009

ANAÏS GUAUS ,
CHRISTOPHE AIRIAU ,
ALESSANDRO BOTTARO  and
AZEDDINE KOURTA
ANAÏS GUAUS*
Affiliation:
Université de Toulouse, Institut de Mécanique des Fluides, UMR 5502 CNRS/INP-UPS, Allée du Professeur Camille Soula, 31400 Toulouse, France
CHRISTOPHE AIRIAU
Affiliation:
Université de Toulouse, Institut de Mécanique des Fluides, UMR 5502 CNRS/INP-UPS, Allée du Professeur Camille Soula, 31400 Toulouse, France
ALESSANDRO BOTTARO
Affiliation:
DICAT, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
AZEDDINE KOURTA
Affiliation:
Université de Toulouse, Institut de Mécanique des Fluides, UMR 5502 CNRS/INP-UPS, Allée du Professeur Camille Soula, 31400 Toulouse, France
*
†Email address for correspondence: anais.guaus@alumni.enseeiht.fr
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Abstract

The stability of the laminar flow in the narrow gap between infinitely long concentric cylinders, the inner of which rotates, is examined for the case of compliant bounding walls, modelled as thin cylindrical shells supported by rigid frames through arrays of springs and dampers. Sufficiently soft walls have a destabilizing influence on the axisymmetric Taylor vortices produced by the centrifugal force, although the effect is limited to modes with large axial wavelengths. Due to the walls flexibility, hydroelastic modes are generated. Complex modal exchanges are observed, as function of the wall properties and the Reynolds number. For axisymmetric modes an asymptotic analysis is conducted in the limit of small axial wavenumber, to show the correspondence between such exchanges and singularities in the analytical solutions. While the axisymmetric modes dominate the spectrum when the walls are rigid or very mildly compliant, a critical non-zero azimuthal wavenumber exists for which the hydroelastic modes become more unstable. Shorter azimuthal waves are favoured by increasing spring stiffness.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 630 , 10 July 2009 , pp. 331 - 365
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proceedings of the National Academy of Sciences of the United States of America, 102, 1829–1834.Google Scholar
Benjamin, T. B. 1960 Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9, 513–532.CrossRefGoogle Scholar
Benjamin, T. B. 1963 The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436–450.Google Scholar
Buckingham, A. C., Hall, M. S. & Chun, R. C. 1985 Numerical simulations of compliant material response to turbulent flow. AIAA J. 23, 1046–1052.CrossRefGoogle Scholar
Bushnell, D. M., Hefner, J. N. & Ash, R. L. 1977 Effect of compliant wall motion on turbulent boundary layers. Phys. Fluids A 20, S31.CrossRefGoogle Scholar
Carpenter, P. W. 1990 Status of transition delay using compliant walls. In Viscous Drag Reduction in Boundary Layers (ed. Bushnell, D. M. and Hefner, J. N.), pp. 79–113. AIAA.Google Scholar
Carpenter, P. W. 1993 Optimization of multiple-panel compliant walls for delay of laminar–turbulent transition. AIAA J. 31, 1187–1188.CrossRefGoogle Scholar
Carpenter, P. W., Davies, C. & Lucey, A. D. 2000 Hydrodynamics and compliant walls: does the dolphin have a secret? Curr. Sci. 79, 758–765.Google Scholar
Carpenter, P. W. & Garrad, A. D. 1985 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities. J. Fluid Mech. 155, 465–510.CrossRefGoogle Scholar
Carpenter, P. W. & Garrad, A. D. 1986 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199–232.Google Scholar
Carpenter, P. W. & Morris, P. J. 1990 The effect of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. 218, 171–223.Google Scholar
Carpenter, P. W. & Pedley, T. J. 2003 Editors of IUTAM Symposium on Flow Past Highly Compliant Boundaries and in Collapsible Tubes. Kluwer Academic Publishers.Google Scholar
Carpenter, P. W. & Thomas, P. J. 2007 Flow over compliant rotating disks. J. Engng Math. 57, 303–315.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Colley, A. J., Carpenter, P. W., Thomas, P. J., Ali, R. & Zoueshtiagh, F. 2006 Experimental verification of Type-II-eigenmode destabilization in the boundary layer over a compliant rotating disk. Phys. Fluids 18, 054107.CrossRefGoogle Scholar
Colley, A., Thomas, P. J., Carpenter, P. W. & Cooper, A. J. 1999 An experimental study of boundary-layer transition over a rotating disk. Phys. Fluids 11, 3340–3352.CrossRefGoogle Scholar
Cooper, A. J. & Carpenter, P. W. 1997 a The effect of wall compliance on inflexion point instability in boundary layers. Phys. Fluids 9, 468–470.Google Scholar
Cooper, A. J. & Carpenter, P. W. 1997 b The stability of rotating-disk boundary-layer flow over a compliant wall. Part 2. Absolute instability. J. Fluid Mech. 350, 261–270.CrossRefGoogle Scholar
Couette, M. 1890 Etudes sur le frottement des liquides. Ann. Chim. Phys. 21, 433–510.Google Scholar
Davies, C. & Carpenter, P. W. 1997 a Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997 b Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361–392.CrossRefGoogle Scholar
Denier, J. P. & Hall, P. 1991 The effect of wall compliance on the Görtler vortex instability. Phys. Fluids A 3, 2000–2002.CrossRefGoogle Scholar
Domaradzki, J. A. & Metcalfe, R. W. 1987 Stabilization of laminar boundary layers by compliant membranes. Phys. Fluids 30, 695–705.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duncan, J. H., Waxman, A. M. & Tulin, M. P. 1985 The dynamics of waves at the interface between a viscoelastic coating and a fluid flow. J. Fluid Mech. 158, 177–197.CrossRefGoogle Scholar
Ehrenstein, U. & Rossi, M. 1996 Nonlinear Tollmien–Schlichting waves for a Blasius flow over compliant coatings. Phys. Fluids 32, 256–267.Google Scholar
Gad-el-Hak, M. 1986 The response of elastic and viscoelastic surfaces to a turbulent boundary layer. J. Appl. Mech. 53, 206–212.CrossRefGoogle Scholar
Gad-el-Hak, M. 2002 Compliant coatings for drag reduction. Prog. Aerosp. Sci. 38, 77–99.Google Scholar
Gad-el-Hak, M., Blackwelder, R. F. & Riley, J. J. 1984 On the interaction of compliant coatings with boundary-layer flows. J. Fluid Mech. 140, 257–280.Google Scholar
Gaster, M. 1987 Is the dolphin a red herring? In IUTAM Symposium on Turbulence Management and Relaminarisation (ed. Liepmann, H. W. and Narasimha, R.), pp. 285. Springer.Google Scholar
Guaus, A. 2008 Analyse linéaire des instabilités dans les écoulements incompressibles à parois courbes compliantes. PhD thesis, Université Toulouse III – Paul Sabatier.Google Scholar
Guaus, A. & Bottaro, A. 2007 Instabilities of the flow in a curved channel with compliant walls. Proc. R. Soc. London A 463, 2201–2222.Google Scholar
Hoepffner, J., Bottaro, A. & Favier, J. Submitted. Mechanisms of non-modal energy amplification in channel flow between compliant walls. J. Fluid Mech.Google Scholar
Joslin, R. D. & Morris, P. J. 1992 Effect of compliant walls on secondary instabilities in boundary-layer transition. AIAA J. 30, 332–339.Google Scholar
Joslin, R. D., Morris, P. J. & Carpenter, P. W. 1991 Role of three-dimensional instabilities in compliant wall boundary-layer transition. AIAA J. 29, 1603–1610.Google Scholar
Kempf, M. & McHugh, J. 1996 Stability of Taylor–Couette flow with an elastic layer on the outer cylinder. APS Division of Fluids Dynamics Meeting. Oral communication.Google Scholar
Kempf, M. & McHugh, J. 1998 Fluid-structure interaction in Taylor–Couette flow. APS Division of Fluids Dynamics Meeting. Oral communication.Google Scholar
Kramer, M. O. 1957 Boundary-layer stabilization by distributed damping. J. Aero. Sci. 24, 459–460.Google Scholar
Kramer, M. O. 1960 a Boundary-layer stabilization by distributed damping. J. Am. Soc. Nav. Engrs 72, 25–33.Google Scholar
Kramer, M. O. 1960 b The dolphin's secret. New Scientist 7, 1118–1120.Google Scholar
Kramer, M. O. 1965 Hydrodynamics of the dolphin. Adv. Hydrosci. 2, 111–130.CrossRefGoogle Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609–632.CrossRefGoogle Scholar
Levinski, V., Levy, D.-E. & Cohen, J. 2001 Effects of wall compliance on the hydrodynamic stability and transition delay of a wall-jet flow. Fluid Dyn. Res. 29, 115–134.CrossRefGoogle Scholar
Lucey, A. D., Cafolla, G. J. & Carpenter, P. W. 1997 a Numerical simulation of a boundary-layer flow interacting with a passive compliant boundary. Lecture Notes Phys. 490, 406–411.CrossRefGoogle Scholar
Lucey, A. D., Cafolla, G. J. & Carpenter, P. W. 1998 The effect of a boundary layer on the hydroelastic stability of a flexible wall. In Proceedings of the Third International Conference on Engineering Aero-Hydroelasticity, Prague. pp. 268–273.Google Scholar
Lucey, A. D., Cafolla, G. J., Carpenter, P. W. & Yang, M. 1997 b The nonlinear hydroelastic behaviour of flexible walls. J. Fluid. Struct. 11, 717–744.CrossRefGoogle Scholar
Lucey, A. D. & Carpenter, P. W. 1992 A numerical simulation of the interaction of a compliant wall and an inviscid flow. J. Fluid Mech. 234, 121–146.CrossRefGoogle Scholar
Mallock, A. 1888 Determination of the viscosity of water. Proc. R. Soc. Lond. A 45, 126–132.Google Scholar
Mallock, A. 1896 Experiments on fluid viscosity. Phil. Trans. R. Soc. Lond. A 187, 41–56.Google Scholar
Prigent, A. & Dauchot, O. 2000 “Barber pole turbulence” in large aspect ratio Taylor-Couette flow. arXiv:cond-mat/0009241 v1 15 Sept. 2000.Google Scholar
Riley, J. J., Gad-el-Hak, M. & Metcalfe, R. W. 1988 Compliant coatings. Annu. Rev. Fluid Mech. 20, 393–420.CrossRefGoogle Scholar
Sen, P. K. & Arora, D. S. 1988 On the stability of laminar boundary-layer flow over a flat plate with a compliant surface. J. Fluid Mech. 197, 201–240.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotationg cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Timoshenko, S. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells. McGraw–Hill.Google Scholar
Wiplier, O. & Ehrenstein, U. 2000 Numerical simulation of linear and nonlinear disturbance evolution in a boundary layer with compliant walls. J. Fluid. Struct. 14, 157–182.Google Scholar
Yeo, K. S. 1988 The stability of boundary-layer over single- and multi-layer viscoelastic walls. J. Fluid Mech. 196, 359–408.Google Scholar
Yeo, K. S., Khoo, B. C. & Chong, W. K. 1994 The linear stability of boundary-layer over compliant walls: effects of boundary-layer growth. J. Fluid Mech. 280, 199–225.Google Scholar
Yeo, K. S., Khoo, B. C. & Zhao, H. Z. 1996 The absolute instability of boundary-layer flow over viscoelastic walls. Theor. Comp. Fluid Dyn. 8, 237–252.Google Scholar
Yurchenko, N. F. & Babenko, V. V. 1987 Stability criterion of three-dimensional perturbations on concave elastic surfaces. J. Engr. Phys. Thermophys. 52, 568–573.Google Scholar