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The three-dimensional stability of boundary-layer flow over compliant walls

Published online by Cambridge University Press:  26 April 2006

K. S. Yeo
K. S. Yeo
Affiliation:
Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Republic of Singapore
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Abstract

This paper examines the linear stability of the Blasius boundary layer over compliant walls to three-dimensional (oblique) disturbance wave modes. The formulation of the eigenvalue problem is applicable to compliant walls possessing general material anisotropy. Isotropic-material walls and selected classes of anisotropic-material walls are studied. When the properties of the wall are identical with respect to all oblique wave directions, the stability eigenvalue problem for unstable three-dimensional wave modes may be reduced to an equivalent problem for two-dimensional modes. The results for isotropic-material walls show that three-dimensional Tollmien–Schlichting instability modes are more dominant than their two-dimensional counterparts when the walls are sufficiently compliant. The critical Reynolds number for Tollmien-Schlichting instability may be given by three-dimensional modes. Furthermore, for highly compliant walls, calculations based solely on two-dimensional modes are likely to underestimate the maximum disturbance growth factor needed for transition prediction and correlation. However, because the disturbance growth rates on highly compliant walls are much lower than those on a rigid wall, significant delay of transition may still be possible provided compliance-induced instabilities are properly suppressed. Walls featuring material anisotropy which have reduced stiffness to shear deformation in the transverse and oblique planes are also investigated. Such anisotropy is found to be effective in reducing the growth rates of the three-dimensional modes relative to those of the two-dimensional modes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 537 - 577
Copyright
© 1992 Cambridge University Press

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References

Benjamin, T. B. 1960 Effects of a flexible boundary on hydrodvnamic stability. J. Fluid Mech. 9, 513.Google Scholar
Benjamin, T. B. 1963 The three-fold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436.Google Scholar
Briggs, R. J. 1964 Electron-stream Interaction with Plasmas. Monograph 29. MIT Press.
Carpenter, P. W. 1984 A note on the hydroelastic instability of orthotropic plates. J. Sound Vib. 95, 553.Google Scholar
Carpenter, P. W. 1985 The optimization of compliant surfaces for transition delay. University of Exeter, School of Engineering, Tech. Note 85/2.Google Scholar
Carpenter, P. W. 1990 Status of transition delay using compliant walls. In Viscous Drag Reduction in Boundary Layers (ed. D. M. Bushnell & J. N. Heffner), p. 79. AIAA.
Carpenter, P. W. 1991 The optimization of multiple-panel compliant walls for delay of laminarturbulent transition. AIAA Paper 91–1772.Google Scholar
Carpenter, P. W. & Gajjar, J. S. B. 1990 A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theor. Comp. Fluid Dyn. 1, 349.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.Google 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.Google Scholar
Carpenter, P. W. & Morris, P. J. 1989 Growth of three-dimensional instabilities in flow over compliant walls. In Proc. Fourth Asian Congr. Fluid Mech., Hong Kong (ed. N.W.M. Ko & S.C. Kot), Vol. 2, p. A206.
Carpenter, P. W. & Morris, P. J. 1990 The effects of anisotropic wall compliance on boundarylayer stability and transition. J. Fluid Mech. 218, 171.Google Scholar
Cebeci, T. & Stewartson, P. J. 1980 On stability and transition in three-dimensional flows. AIAA J. 18, 398.Google Scholar
Christensen, R. M. 1979 Mechanics of Composite Materials. John Wiley & Sons.
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393.Google Scholar
Davey, A. & Reid, W. H. 1977 On the stability of stratified viscous plane Couette flow. Part 1. Constant buoyancy frequency. J. Fluid Mech. 80, 509.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fung, Y. C. 1965 Foundations of Solid Mechanics. Prentice-Hall.
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.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 271.Google Scholar
Gaster, M. 1987 Is the dolphin a red herring? In Proc. IUTAM Conf. on Turbulence Management and Relaminarization. Bangalore, India (ed. H. W. Liepmann & R. Narisimha), p. 285. Springer.
Gray, J. 1936 Studies in animal locomotion VI: The propulsive power of the dolphin. J. Exp. Biol. 13, 192.Google Scholar
Green, A. E. & Zerna, W. 1968 Theoretical Elasticity. Oxford University Press.
Gustavsson, L. H. & Hultgren, L. S. 1980 A resonant mechanism in plane Couette flow. J. Fluid Mech. 98, 149.Google Scholar
Joslin, R. D., Morris, P. J. & Carpenter, P. W. 1991 The role of three-dimensional instabilities in compliant wall boundary layer transition. AIAA J. (to appear).Google Scholar
Kachanov, Yu. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech. 138, 209.Google Scholar
Kaplan, R. E. 1964 The stability of laminar boundary layers in the presence of compliant boundaries. Sc.D. thesis, MIT.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1.Google Scholar
Kramer, M. O. 1960 Boundary layer stabilization by distributed damping. J. Am. Soc. Naval Engrs 73, 25.Google Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609.Google Scholar
Lekhnitskii, S. G. 1963 Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day.
Mack, L. M. 1978 Three-dimensional effects in boundary-layer stability. In Proc. 12th Symp. Naval Hydrodynamics, p. 63. National Academy of Science, Washington.
Nayfeh, A. H. 1980 Stability of three-dimensional boundary layers. AIAA J. 18, 406.Google Scholar
Nonweiler, T. 1963 Qualitative solution of the stability equation for a boundary layer in contact with various forms of flexible surface. Aero Res. Coun. Rep. CP 622.Google Scholar
Riley, J. J., Gad-el-Hak, M. & Metcalfe, R. W. 1988 Compliant coatings. Ann. Rev. Fluid Mech. 20, 393.Google Scholar
Saric, W. S. & Thomas, A. S. W. 1984 Experiments on subharmonic route to turbulence in boundary layers. In Turbulence and Chaotic Phenomena in Fluids, Proc. IUTAM Symp., Kyoto (ed. T. Tatsumi). North-Holland.
Square, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. A 142, 621.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layer (ed. L. Rosenhead), p. 492. Clarendon.
Willis, G. J. K. 1986 Hydrodynamic stability of boundary layers over compliant surfaces. Ph.D. thesis, University of Exeter.
Yeo, K. S. 1986 The stability of flow over flexible surfaces. Ph.D. thesis, University of Cambridge.
Yeo, K. S. 1988 The stability of boundary-layer flow over single- and multi-layer viscoelastic walls. J. Fluid Mech. 196, 359.Google Scholar
Yeo, K. S. 1990 The hydrodynamic stability of boundary-layer flow over a class of anisotropic compliant walls. J. Fluid Mech. 220, 125.Google Scholar
Yeo, K. S. & Dowling, A. P. 1987 The stability of inviscid flows over passive compliant walls. J. Fluid Mech. 183, 265.Google Scholar