Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-29T17:39:21.041Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of inviscid flows over passive compliant walls

Published online by Cambridge University Press:  21 April 2006

K. S. Yeo  and
A. P. Dowling
K. S. Yeo
Affiliation:
Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge 0511, Singapore
A. P. Dowling
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear temporal stability of incompressible semi-bounded inviscid parallel flows over passive compliant walls is studied. It is shown that some of the well-known classical results for inviscid parallel flows with rigid boundaries can, in fact, be extended in modified form to passive compliant walls. These include a result of Rayleigh (1880) which shows that the real part of the phase velocity of a non-neutral disturbance must lie within the range of the velocity distribution; the semi-circle theorem of Howard (1961) and a result of Høiland (1953) which places a bound on the temporal amplification rates of unstable disturbances. The bounds on the phase velocity and the temporal amplification rates of unstable two-dimensional disturbances provide useful guides for numerical studies.

The results are valid for a large class of passive compliant walls. This generality is achieved through a variational-Lagrangian formulation of the essential dynamics of wall motion. A general treatment of the marginal stability of thin shear flows over general passive compliant walls is given. It represents a generalization of the analysis given by Benjamin (1963) for membrane and plate surfaces. Sufficient conditions for the stability of thin shear flows over passive compliant walls are deduced. The applications of the stability criteria to simple cases of compliant wall are described to illustrate the use and the effectiveness of these criteria.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 265 - 292
Copyright
© 1987 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

Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangian-Mean flow. J. Fluid Mech. 89, 609.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647.Google Scholar
Biot, M. A. 1984 New variational-Lagrangian irreversible thermodynamics with applications to viscous flow, reaction-diffusion and solid mechanics. In Adv. Appl. Mech. 24, 1.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161.Google Scholar
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
Bland, D. R. 1960 Linear Viscoelasticity. Pergamon.
Callan, K. & Case, K. 1981 Drag reduction and adaptive boundary conditions. SRI International, Tech. Rep. JSR-81–17.
Carpenter, P. W. & Garrad, A. D. 1982 Effects of a fluid substrate on the flow-induced vibrations of a compliant coating. Proc. Intl Conf. Flow-induced Vib. in Fluid Engng, Reading, England, paper H3.
Carpenter, P. W. & Garrad, A. D. 1985 The hydrodynamics 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 hydrodynamics of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199.Google Scholar
Chin, W. C. 1979 Stability of inviscid shear flow over flexible membranes. AIAA J. 17, 665.Google Scholar
Drazin, P. G. & Howard, L. N. 1962 The instability to long waves of unbounded parallel inviscid flow. J. Fluid Mech. 14, 257.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. In Adv. Appl. Mech. 7, 1.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Duncan, J. H., Waxman, A. M. & Tulin, M. P. 1985 The dynamics of waves at the interface a viscoelastic coating and a fluid flow. J. Fluid Mech. 158, 177.Google Scholar
Eringen, A. C. & Suhubi, E. S. 1975 Elastodynamics, Vols 1, 2. Academic.
Evrensel, C. A. & Kalnin, A. 1985 Response of a compliant slab to inviscid incompressible fluid flow. J. Acoust. Soc. Am. 78, 2034.Google Scholar
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofys. Publ. 17, 1.Google Scholar
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.
Fung, Y. C. 1965 Foundations of Solid Mechanics. Prentice-Hall.
Gad-El-Hak, M. 1986 The response of elastic and viscoelastic surfaces to a turbulent boundary layer. J. Appl. Mech. 53, 1.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.Google Scholar
Hansen, R. J. & Hunston, D. L. 1974 An experimental study of turbulent flows over compliant surfaces. J. Sound Vib. 34, 297.Google Scholar
Hansen, R. J., Hunston, D. L., Ni, C. C. & Reischmann, M. M. 1980 An experimental study of flow-generated waves on a flexible surface. J. Sound Vib. 68, 317.Google Scholar
Høiland, E. 1953 On two-dimensional perturbation of linear flow. Geofys. Publ. 18, 1.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509.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
Landau, L. D. & Lifshitz, E. M. 1970 The Theory of Elasticity. Pergamon.
Lin, C. C. 1955 The Theory of Hydrodynamic Instability. Cambridge University Press.
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185.Google Scholar
Nonweiler, T. 1961 Qualitative solutions of the stability equation for a boundary layer in contact with various forms of flexible surface. Aeronaut. Res. Council Rept CP 622.
Rayleigh, Lord 1880 On the stability, or the instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 57.Google Scholar
Washizu, K. 1982 Variational Methods in Elasticity and Plasticity, 3rd edn. Pergamon.
Yeo, K. S. 1986 The stability of flow over flexible surfaces. Ph. D. thesis, University of Cambridge.