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On the stability of a laminar incompressible boundary layer over a flexible surface

Published online by Cambridge University Press:  28 March 2006

Marten T. Landahl
Marten T. Landahl
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
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Abstract

The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.

What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.

The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 13 , Issue 4 , August 1962 , pp. 609 - 632
Copyright
© 1962 Cambridge University Press

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References

Benjamin, T. Brooke 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161.Google Scholar
Benjamin, T. Brooke 1960 Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9, 513.Google Scholar
Betchov, R. 1959 Simplified analysis of boundary-layer oscillations. Douglas Aircraft Company Report, no. ES-29174.Google Scholar
Boggs, F. W. & Tokita, N. 1960 A theory of the stability of laminar flow along compliant plates. Paper presented at the Third Symp. on Naval Hydrodynamics, Scheveningen, Holland, September, 1960.
Hains, F. D. & Price, J. F. 1961 Effects of flexible walls on the stability of plane Poiseulle flow. Paper presented at the 1961 Annual Meeting of the Amer. Phys. Soc. February 1961.
Krämer, M. O. 1960a Reader's Forum. J. Aero/Space Sci. 27, 68.Google Scholar
Krämer, M. O. 1960b Boundary-layer stabilization by distributed damping. J. Amer. Soc. Nav. Engrs, 72, 25.Google Scholar
Less, L. 1947 The stability of the laminar boundary layer in a compressible fluid. NACA Rep. no. 876.Google Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Parts I, II and III. Quart. Appl. Math. 3, 117, 218, 277.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Linebarger, J. H. 1961 On the stability of a laminar boundary layer over a flexible surface in a compressible fluid, S.M. Thesis, Dept. Aeronautics and Astronautics, M.I.T.
Miles, J. W. 1956 On the aerodynamic instability of thin panels. J. Aero. Sci. 23, 77.Google Scholar
Miles, J. W. 1960 The hydrodynamic stability of a thin film of liquid in uniform shearing motion. J. Fluid Mech. 8, 593.Google Scholar
Nonweiler, T. 1961 Qualitative solutions of the stability equation for a boundary layer in contact with various forms of flexible surface. A.R.C. Rep. no. 22, 670.Google Scholar
Schlichting, H. 1960 Boundary Layer Theory, 4th ed. New York: McGraw-Hill.
Squire, H. B. 1933 On the stability of the three-dimensional disturbances of viscous flow between parallel walls. Proc. Roy. Soc. A, 142, 621.Google Scholar
Tollmien, W. 1947 Asymptotische Integration der Störungsdifferentialgleichung ebener laminarer Strömungen bei hohen Reynoldschen Zahlen. Z. angew. Math. Mech. 25/27, 33, 70.Google Scholar