Abstract
An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.
The theory is applied to various cases including two- and three-dimensional disturbances, developing in boundary layers over isotropic and anisotropic compliant walls. The disturbances can be treated as either temporally or spatially growing. Eigenvalues are very accurately predicted by means of the theory, especially near points of neutral stability. The computational requirements are trivial compared with those required for full numerical solution of the Orr-Sommerfeld equation. For isotropic compliant walls the theory confirms the earlier result of Miles and Benjamin that the phase shift in the disturbance velocity across the critical layer plays a dominant role in destabilization of the Class B travelling-wave flutter through making irreversible energy transfer possible due to the work done by the fluctuating pressure at the wall. The theory elucidates the secondary role played by the phase shift occurring across the wall layer. Viscous effects are much more important for anisotropic compliant walls which admit substantial horizontal, as well as vertical, displacement. For these walls an important mechanism for irreversible energy transfer is the work done by fluctuating shear stress. This almost invariably has a stabilizing effect on the travelling-wave flutter. In addition there is a weaker effect arising from the effect of anisotropic wall compliance on the phase shift across the wall layer. This may be stabilizing or destabilizing.
Similar content being viewed by others
References
Benjamin, T.B. (1959) Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.
Benjamin, T.B. (1960) Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9, 513–532.
Benjamin, T.B. (1963) The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436–450.
Bodonyi, R.J., and Smith, F.T. (1981) The upper branch stability of the Blasius boundary layer including nonparallel flow effects. Proc. Roy. Soc. London Ser. A 375, 65–92.
Bridges, T.J., and Morris, P.J. (1984) Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460.
Carpenter, P.W. (1984a) The effect of a boundary layer on the hydroelastic instability of infinitely long plates. J. Sound Vibration 93, 461–464.
Carpenter, P.W. (1984b) A note on the hydroelastic instability of orthotropic panels. J. Sound Vibration 94, 553–554.
Carpenter, P.W. (1984c) The hydrodynamic stability of flows over non-isotropic compliant surfaces. Bull. Amer. Phys. Soc. 29, 1534.
Carpenter, P.W. (1985) Hydrodynamic and hydroelastic stability of flows over non-isotropic compliant surfaces. Bull. Amer. Phys. Soc. 30, 1708.
Carpenter, P.W. (1987a) The optimization of compliant surfaces for transition delay. In Proc. IUTAM on Turbulence Management and Relaminarization, Bangalore (ed. H.W. Liepmann and R. Narasimha), pp. 305–313. Springer-Verlag, New York.
Carpenter, P.W. (1987b) The hydrodynamic stability of flows over simple non-isotropic compliant surfaces. In Proc. Int. Conf. on Fluid Mechanics, Beijing, pp. 196–201. Peking University Press, Beijing.
Carpenter, P.W. (1989) Status of transition delay using compliant walls. In Viscous Drag Reduction (eds. D.M. Bushnell and J.N. Heffner), pp. 79–113. AIAA, New York.
Carpenter, P.W., and Garrad, A.D. (1985) The hydrodynamic stability of flows over Kramer-type compliant surfaces. Part 1. Tollmien-Schlichting instabilities. J. Fluid Mech. 115, 465–510.
Carpenter, P.W., and Garrad, A.D. (1986) The hydrodynamic stability of flows over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199–232.
Carpenter, P.W., and Morris, P.J. (1985) The hydrodynamic stability of flows over non-isotropic compliant surface. Numerical solution of the differential eigenvalue problem. In Numerical Methods in Laminar and Turbulent Flow (eds. C. Taylor, M.D. Olson, P.M. Gresho, and W.G. Habashi), pp. 1613–1620. Pineridge, Swansea.
Carpenter, P.W., and Morris, P.J. (1989a) Growth of three-dimensional instabilities in flow over compliant walls. In Proc. 4th. Asian Congr. of Fluid Mechanics, Hong Kong, pp. A206–209.
Carpenter, P.W., and Morris, P.J. (1990) The effects of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. (to appear).
Carpenter, P.W., Gaster, M., and Willis, G.J.K. (1983) A numerical investigation into boundary layer stability on compliant surfaces. In Numerical Methods in Laminar and Turbulent Flow (eds. C. Taylor, J.A. Johnson, and W.R. Smith), pp. 166–172. Pineridge, Swansea.
Cebeci, T., and Stewartson, K. (1980) On stability and transition in three-dimensional flows. AIAA J. 18, 398–405.
Cebeci, T., and Stewartson, K. (1981) Asymptotic properties of the Zarf. AIAA J. 19, 806–807.
Domaradzki, J.A., and Metcalfe, R.W. (1987) Stabilization of laminar boundary layers by compliant membrances. Phys. Fluids 30, 695–705.
Drazin, P.G., and Reid, W.H. (1981) Hydrodynamic Stability. Cambridge University Press, Cambridge.
Gad-El-Hak, M. (1986) The response of elastic and viscoelastic surfaces to a turbulent boundary layer. J. Appl. Mech. 53, 206–212.
Grosskreutz, R. (1971) Wechselwirkungen zwischen turbulenten Grenzschichten und weichen Wänden. MPI für Strömungsforschung und der AVA, Göttingen, Mitt. No. 53.
Grosskreutz, R. (1975) An attempt to control boundary-layer turbulence with nonisotropic compliant walls. Univ. Sci. J. (Dar es Salaam) 1, 67–73.
Heisenberg, W. (1924) Über Stabilität und Turbulenz von Flussigkeitsströmen. Ann. Physsik (4) 74, 577–627.
Landahl, M.T. (1962) On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609–632.
Mackerrell, S.O. (1988) Hydrodynamic instabilities of boundary layer flows. Ph.D. thesis, University of Exeter.
Miles, J.W. (1957) On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–199.
Miles, J.W. (1959a) On the generation of surface waves by shear flows. Part 2, J. Fluid Mech. 6, 568–582.
Miles, J.W. (1959b) On the generation of surface waves by shear flows. Part 3, J. Fluid Mech. 6, 583–598.
Miles, J.W. (1962) On the generation of surface waves by shear flows. Part 4. J. Fluid Mech. 13, 433–448.
Rothmayer, A.P., and Hiemcke, C. (1988) The stability of a Blasius boundary-flowing over an elastic solid. Unpublished report, Department of Aerospace Engineering, Iowa State University.
Sen, P.K., and Arora, D.S. (1988) On the stability of laminar bounday-layer flow over a flat-plate with a compliant surface. J. Fluid Mech. 197, 201–240.
Smith, F.T. (1979a) On the non-parallel stability of the Blasius boundary layer. Proc. Roy. Soc. London Ser. A 366, 573–589.
Smith, F.T. (1979b) Nonlinear stability of boundary layers for disturbances of various sizes. Proc. Roy. Soc. London Ser. A 368, 573–589 (correction (1980), 371, 439–440).
Smith, F.T., Papageorgiou, D., and Elliott, J.W. (1984) An alternative approach to linear and non-linear stability calculations at finite Reynolds numbers. J. Fluid Mech. 146, 313–330.
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. (1989) The hydrodynamic stability of boundary-layer flow over a class of anisotropic compliant walls. Submitted to J. Fluid Mech.
Yeo, K.S., and Dowling, A.P. (1987) The stability of inviscid flows over passive compliant walls. J. Fluid Mech. 183, 265–292.
Author information
Authors and Affiliations
Additional information
Communicated by Philip Hall
This work was carried out with the support of the Ministry of Defence (Procurement Executive) and the Office of Naval Research and was completed while P.W.C. and J.S.B.G. were on study leave at the Department of Aerospace Engineering, The Pennsylvania State University, and the Department of Mathematics, Iowa State University, Ames, respectively. They would like to express their gratitude to those institutions and the Office of Naval Research for financial support during their study leaves.
Rights and permissions
About this article
Cite this article
Carpenter, P.W., Gajjar, J.S.B. A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theoret. Comput. Fluid Dynamics 1, 349–378 (1990). https://doi.org/10.1007/BF00271796
Issue Date:
DOI: https://doi.org/10.1007/BF00271796