Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-09T22:14:36.009Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of boundary-layer flow over single-and multi-layer viscoelastic walls

Published online by Cambridge University Press:  21 April 2006

K. S. Yeo
K. S. Yeo
Affiliation:
Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we are concerned with the linear stability of zero pressure-gradient laminar boundary-layer flow over compliant walls which are composed of one or more layers of isotropic viscoelastic materials and backed by a rigid base. Wall compliance supports a whole host of new instabilities in addition to the Tollmien-Schlichting mode of instability, which originally exists even when the wall is rigid. The perturbations in the flow and the compliant wall are coupled at their common interface through the kinematic condition of velocity continuity and the dynamical condition of stress continuity. The disturbance modes in the flow are governed by the Orr-Sommerfeld equation using the locally-parallel flow assumption, and the response of the compliant layers is described using a displacement-stress formalism. The theoretical treatment provides a unified formulation of the stability eigenvalue problem that is applicable to compliant walls having any finite number of uniform layers; inclusive of viscous sublayer. The formulation is well suited to systematic numerical implementation. Results for single- and multi-layer walls are presented. Analyses of the eigenfunctions give an insight into some of the physics involved. Multi-layering gives a measure of control over the stability characteristics of compliant walls not available to single-layer walls. The present study provides evidence which suggests that substantial suppression of disturbance growth may be possible for suitably tailored compliant walls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 196 , November 1988 , pp. 359 - 408
Copyright
© 1988 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.)

References

Benjamin, T. B. 1960 Effects of a flexible boundary on hydrodynamic 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
Benjamin, T. B. 1964 Fluid flow with flexible boundaries. In Proc. 11th Intl Congr. Appl. Mech., Munich, Germany, p. 109.
Bers, A. & Briggs, R. J. 1964 Bull. Am. Phys. Soc. 9, 304.
Bland, D. R. 1960 Linear Viscoelasticity. Pergamon.
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., Gaster, M. & Willis, G. J. K. 1983 A numerical investigation into boundary layer stability on compliant surfaces. In Numer. Meth. Lam. Turb. Flows, Swansea, p. 165.
Chung, K. H. & Merrill, E. W. 1984 Drag reduction by compliant surfaces measured on rotating discs. Abstract submitted to the Compliant Coating Drag Reduction Prog. Rev., Off. Naval Res.
Craik, A. D. D. 1966 Wind-generated waves in thin liquid films. J. Fluid Mech. 26, 369.Google Scholar
Davey, A. 1981 A difficult calculation concerning the stability of Blasius boundary layer. Proc. IUTAM Symp. on Stability in Mech. of continua, Numbrecht, Germany, p. 365.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fraser, L. A. 1984 The hydrodynamic stability of flows over homogeneous elastic layers. Presented at Euromech Colloquium 188, 23–26 Sept. 84, University of Leeds, England.
Fraser, L. A. & Carpenter, P. W. 1985 A numerical investigation of hydroelastic and hydrodynamic instabilities in laminar flows over compliant surfaces comprising of one or two layers of viscoelastic materials. In Numerical Methods in Laminar and Turbulent Flows, p. 1171. Pineridge.
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433.Google Scholar
Gaster, M. 1974 On the effects of boundary layer growth on flow stability. J. Fluid Mech. 66, 465.Google Scholar
Gaster, M. 1984 Stability of laminar boundary layers on compliant boundaries: Theory and experiments at NMI Ltd. Abstract submitted to the Compliant Coating Drag Reduction Prog. Rev., Off. Naval Res.
Gaster, M. 1985 Growth of instability waves over compliant coatings. Bull. Am. Phys. Soc. 30, 1708.Google Scholar
Gaster, M. 1987 Is the dolphin a red herring. Proc. IUTAM Symp. on Turbulence Management and Laminarization, Bangalore, India, p. 285.
Gilbert, F. & Backus, G. E. 1966 Propagation matrices in elastic wave and vibration problems. Geophysics 31, 326.Google Scholar
Gyorgyfalvy, D. 1967 Possibilities of drag reduction by the use of flexible skin. J. Aircraft 4, 186.Google Scholar
Jaffe, N. A., Okamura, T. T. & Smith, A. M. O. 1970 Determination of spatial amplification factors and their application to predicting transition. AIAA J. 8, 301.Google Scholar
Jordinson, R. 1970 The flat plate boundary layer. Part 1. Numerical integration of the Orr-Sommerfeld equation. J. Fluid Mech. 43, 801.Google Scholar
Kaplan, R. E. 1964 The stability of laminar boundary layers in the presence of compliant boundaries. ScD thesis, MIT.
Kramer, M. O. 1957 Boundary-layer stabilization by distributed damping. J. Aero Sci. 24, 459.Google Scholar
Kramer, M. O. 1960 Boundary layer stabilization by distributed damping. J. Am. Soc. Naval Engng 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
Landahl, M. T. & Kaplan, R. E. 1965 Effect of compliant walls on boundary layer stability and transition. AGARDograph 97, 363.Google Scholar
Lighthill, M. J. 1963 Boundary layer theory. In Laminar Boundary Layer (ed. L. Rosenhead). Oxford University Press.
Ng, B. S. & Reid, W. H. 1979 An initial-value method for eigenvalue problems using compound matrices. J. Comp. Phys. 30, 125.Google Scholar
Nonweiler, T. 1963 Qualitative solutions of the stability equation for a boundary layer in contact with various forms of flexible surface. Aeronaut. Res. Council Rep. CP622.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar boundary-layer oscillations and transition on a flat plate. NACA Rep. 909.Google Scholar
Smith, A. M. O. & Gamberoni, H. 1956 Transition, pressure gradient and stability theory. Douglas Aircraft Co., Long Beach, Calif., Rep. ES26388.Google Scholar
Thomson, W. T. 1950 Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21, 89.Google Scholar
Wells, C. S. 1967 Effects of freestream turbulence on boundary-layer transition. AIAA J. 5, 172.Google Scholar
Willis, G. J. K. 1987 Hydrodynamic stability of boundary layers over compliant surfaces. PhD thesis, University of Exeter.
Yeo, K. S. 1986 The stability of flow over flexible surfaces. PhD thesis, University of Cambridge.
Yeo, K. S. & Dowling, A. P. 1987 The stability of inviscid flows over passive compliant walls. J. Fluid Mech. 183, 265.Google Scholar