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On the stability of an inviscid shear layer which is periodic in space and time

Published online by Cambridge University Press:  28 March 2006

R. E. Kelly
R. E. Kelly
Affiliation:
National Physical Laboratory, Teddington
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Abstract

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.

At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.

Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 27 , Issue 4 , 20 March 1967 , pp. 657 - 689
Copyright
© 1967 Cambridge University Press

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References

Ball, F. K. 1964 Energy transfer between external and internal gravity waves J. Fluid Mech. 19, 465–478.Google Scholar
Benney, D. J. & Niell, A. M. 1962 Apparent resonances of weakly nonlinear standing waves J. Math. Phys. 41, 254–263.Google Scholar
Betchov, R. & Criminale, W. O. 1966 Spatial instability of the inviscid jet and wake J. Phys. Fluids, 9, 359–362.Google Scholar
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams Phys. Fluids 6, 1391–1396.Google Scholar
Bradshaw, P. 1966 The effect of initial conditions on the development of a free shear layer J. Fluid Mech. 26, 225–236.Google Scholar
Browand, F. K. 1966 An experimental investigation of the instability of an incompressible, separated shear layer J. Fluid Mech. 26, 281–307.Google Scholar
Cole, J. D. & Kevorkian, J. 1963 Uniformly valid asymptotic approximation for certain non-linear differential equations. Nonlinear Differential Equations and Nonlinear Mechanics (J. P. LaSalle & S. Lefschetz, ed., pp. 113–120). New York: Academic Press.
Domm, U. 1956 Über eine Hypothese, die den Mechanismus der Turbulenz-Entstehung betrifft. Deutsche Versuchsanstalt fÜr Luftfahrt (DVL) Rep. 23.Google Scholar
Drazin, P. G. & Howard, L. N. 1962 The instability to long waves of unbounded parallel inviscid flow J. Fluid Mech. 14, 257–283.Google Scholar
Gallagher, A. P. & Mercer, A. McD. 1962 On the behaviour of small disturbances in plane Couette flow J. Fluid Mech. 13, 91–100.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability J. Fluid Mech. 14, 222–224.Google Scholar
Greenspan, H. P. & Benney, D. 1963 On shear-layer instability, breakdown and transition J. Fluid Mech. 15, 133–153.Google Scholar
Ince, E. L. 1956 Ordinary Differential Equations. New York: Dover.
Kelly, R. E. 1965 The stability of an unsteady Kelvin-Helmholtz flow J. Fluid Mech. 22, 547–560.Google Scholar
Lamb, H. 1959 Hydrodynamics, 6th ed. Cambridge University Press.
Lessen, M. & Fox, J. A. 1955 The stability of boundary layer type flows with infinite boundary conditions. 50 Jahre Grenzschichtforschung (pp. 122–126). Braunschweig: Friedr. Vieweg und Sohn.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent profile J. Fluid Mech. 19, 543–556.Google Scholar
Michalke, A. 1965a Vortex formation in a free boundary layer according to stability theory J. Fluid Mech. 22, 371–383.Google Scholar
Michalke, A. 1965b On spatially growing disturbances in an inviscid shear layer J. Fluid Mech. 23, 521–544.Google Scholar
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Norair Rep. NOR-59-383. Hawthorne, California.Google Scholar
Sato, H. 1956 Experimental investigation on the transition of laminar separated layer. J. Phys. Soc. Japan 11, 702–709, 1128.Google Scholar
Sato, H. 1959 Further investigation on the transition of two-dimensional separated layer at subsonic speeds J. Phys. Soc. Japan 14, 1797–1810.Google Scholar
Sato, H. & Kuriki, K. 1961 The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow J. Fluid Mech. 11, 321–352.Google Scholar
Schade, H. 1964 Contribution to the nonlinear stability theory of inviscid shear layers Phys. Fluids 7, 623–628.Google Scholar
Segel, L. A. 1962 The non-linear interaction of two disturbances in the thermal convection problem J. Fluid Mech. 14, 97–114.Google Scholar
Stuart, J. T. 1962 Nonlinear effects in hydrodynamic stability. Proc. Xth Int. Congr. Appl. Mech., Stresa, 1960. Amsterdam: Elsevier.
Stuart, J. T. 1966 On finite-amplitude oscillations in laminar mixing layers. II, Liouville's partial differential equation (to be published as an NPL report).
Stoker, J. J. 1950 Nonlinear vibrations. New York: Interscience.
Tatsumi, T. & Gotoh, K. 1960 The stability of free boundary layers between two uniform streams J. Fluid Mech. 7, 433–441.Google Scholar
Tatsumi, T., Gotoh, K. & Ayukawa, K. 1964 The stability of a free boundary layer at large Reynolds numbers J. Phys. Soc. Japan 19, 1966–1980.Google Scholar
Wehrmann, O. & Wille, R. 1958 Beitrag zur Phänomenologie sep laminar-turbulenten Übergangs im Freistrahl bei kleinen Reynoldszahlen. Boundary Layer Research (H. Görtler, ed., pp. 387–404). Berlin: Springer-Verlag.
Wille, R. 1963 Growth of velocity fluctuations leading to turbulence in free shear flow. AFOSR-TRAF 61(052)-412.Google Scholar