Linear Instability of the Supersonic Boundary Layer on a Compliant Surface


In the paper the influence of flexible covering properties on the linear development of disturbances in a supersonic boundary layer is investigated for Mach numbers M = 1.0, 2.0, 5.3, 6.0. As a model of a covering the porous plate closed by a flexible film is used. In the absence of gas in pores it is established that the flexible covering stabilizes boundary layer in the area of large Reynolds numbers and destabilizes it at small Reynolds numbers. Joint influence of the thickness and tension of a film leads to an appearance of additional unstable waves. For filled with gas pores the researches are conducted as taking into account losses of energy of disturbances in pores and in their absence. Calculations without power losses indicate possibility of existence of an absolute instability of the boundary layer on the flexible surface. The damping properties of a flexible covering connected with power losses in pores reduce their stabilizing role.

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Gaponov, S. (2014) Linear Instability of the Supersonic Boundary Layer on a Compliant Surface. Journal of Applied Mathematics and Physics, 2, 253-263. doi: 10.4236/jamp.2014.26030.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kramer, M.O. (1957) Boundary Layer Stabilization by Distributed Damping. Journal of the Aeronautical Sciences, 24, 459-460.
[2] Kramer, M.O. (1960) Boundary Layer Stabilization by Distributed Damping. Journal of the American Society for Naval Engineers, 72, 25-33.
[3] Benjamin, T.B. (1960) Effects of a Flexible Boundary on Hydrodynamic Stability. Journal of Fluid Mechanics, 9, 513-532.
[4] Benjamin, T.B. (1963) The Threefold Classification of Unstable Disturbances in Flexible Surfaces Bounding Inviscid Flows. Journal of Fluid Mechanics, 16, 436-450.
[5] Landahl, M.T. (1962) On the Stability of a Laminar Incompressible Boundary Layer over a Flexible Surface. Journal of Fluid Mechanics, 13, 609-632.
[6] Carpenter, P.W. and Garrad, A.D. (1985) Kramer-Type Compliant Surfaces. Part 1. Tollmien-Schlichting Instabilities. Journal of Fluid Mechanics, 155, 465-510.
[7] Babenko, V.V. and Kozlov, L.F. (1973) Experimental Study of Hydrodynamical Stability on Rigid and Elastically Damping Surfaces. Fluid Dynamics, 8, 109-114.
[8] Benjamin, T.B. (1964) Fluid Flow with Flexible Boundaries. In: Gortler, H., Ed., Proc. II-th Int. Congr. Appl. Math, Springer-Verlag, Berlin, 109-128.
[9] Bushnell, D.M., Hefner, J.N. and Ash, R.L. (1977) Effect of Compliant Wall Motion on Turbulent Boundary Layers. Physics of Fluids, 20, 31-48.
[10] Riley, J.J., Gad-el-Hak. and Metcalfe, R.W. (1988) Compliant Coatings. Annual Review of Fluid Mechanics, 20, 393- 420.
[11] Carpenter, P.W. (2008) Recent Progress in the Use of Compliant Walls for Laminar Flow Control. Progress in Industrial Mathematics at ECMI 2006, Mathematics in Industry, 12, 178-187.
[12] Manuilovich, S.V. (2003) Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance. Fluid Dynamics, 38, 529-544.
[13] Walker, J.D.A., Fletcher, A. and Ruban, A.I. Instabilities of a Flexible Surface in Supersonic Flow. The Quarterly Journal of Mechanics and Applied Mathematics, 59, 253-276.
[14] Gaponov, S.A. and Terekhova, N.M. (2012) Influence of a Compliant Surface on a Supersonic Boundary Layer Stability. In: Arslan, O. and Oprisan, S, Eds., Resent Advances in Mechanical Engineering and Automatic Control. Proceedings of the 3rd European Conference of MECHANICAL ENGINEERING (ECME '12), Paris, WSEAS, 87-92.
[15] Gaponov, S.A. and Maslov A.A. (1980) Development of Disturbances in Compressible Flows. Science, Novosibirsk, 144 p. (In Russian).
[16] Petrov, G.V. (2000) New Parabolized System of Equations of Stability of a Compressible Boundary Layer. Journal of Applied Mechanics and Technical Physics, 41, 55-61.
[17] Gaponov, S.А. and Yudin, A.V. (2002) Interaction of Hydrodynamic External Disturbances with the Boundary Layer. Journal of Applied Mechanics and Technical Physics, 43, 83-89.
[18] Gaponov, S.A. and Smorodsky, B.V. (2012) Acoustics and Instability of High-Speed Boundary Layers. International Journal of Mechanics, 6, 9-16
[19] Gaponov, S.A. and Smorodsky, B.V. (2009) Linear Stability of Supersonic Boundary Layer on Porous Surface. Proceedings of the 7th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics (FMA'09), Moscow, WSEAS, 68-73.
[20] Gaponov, S.A. (1977) Stability of Supersonic Boundary Layer on Permeable Surface with Heat Exchange. Fluid Dynamics, 41-46
[21] Landau, L.D. and Lifshitz, E.M. (1970) Theory of Elasticity. 2nd Engl. Edition, Revised end Enlarged, Pergamon Press, 165 pp (Vol. 7 of Course of Theoretical Physics).

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