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The stability of an incompressible two-dimensional wake

Published online by Cambridge University Press:  29 March 2006

G. E. Mattingly  and
W. O. Criminale
G. E. Mattingly
Affiliation:
Department of Civil and Geological Engineering, Princeton University Present address: Department of Oceanography and Geophysics Group, University of Washington, Seattle.
W. O. Criminale
Affiliation:
Department of Civil and Geological Engineering, Princeton University
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Abstract

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.

Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.

In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 51 , Issue 2 , 25 January 1972 , pp. 233 - 272
Copyright
© 1972 Cambridge University Press

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References

Betchov, R. & Criminale, W. O. 1966 Spatial instability of the inviscid jet and wake. Phys. Fluids, 9, 359.Google Scholar
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Plows. Academic.
Betchov, R. & Szewczyk, A. A. 1963 Stability of a shear layer between parallel streams. Phys. Fluids, 6, 1391.Google Scholar
Curle, N. 1957 On hydrodynamic stability in unlimited fields of viscous flow. Proc. Roy. Soc. A 238, 489.Google Scholar
Freymuth, P. 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 683.Google Scholar
Gaster, M. 1963 A note on a relation between temporally increasing and spatially increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222.Google Scholar
Gaster, M. 1965 The role of spatially growing waves in the theory of hydrodynamic stability. Prog. Aeron. Sci. 6, 251.Google Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. FEuids, 11, 723.Google Scholar
Gaster, M. & Davey, A. 1968 The development of three-dimensional wave-packets in unbounded parallel flows. J. Fluid Mech. 32, 801.Google Scholar
Goldstein, S. 1929 Concerning some solutions of the boundary layer equations in hydrodynamics. Proc. Camb. Phil. Soc. 26, 1.Google Scholar
Mattingly, G. E. 1962 An experimental study of the three-dimensionality of the flow around a circular cylinder. Inst. Fluid Dyn. Appl. Math., University of Maryland. Tech. Note, BN-295.
Mattingly, G. E. 1966 The hydrogen-bubble visualization technique. David Taylor Model Basin Rep. no. 2146.
Mattingly, G. E. 1968 The stability of a two-dimensional incompressible wake. Dept. Aerospace Mech. Sci., Princeton University, Rep. no. 858.
Sato, H. 1960 The stability and transition of a two-dimensional jet. J. Fluid Mech. 7, 53.Google Scholar
Sato, H. & Kuriki, K. 1961 Mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321.Google Scholar
Taneda, S. 1963 The stability of two-dimensional laminar wakes at low Reynolds numbers. J. Phys. Soc. Japan, 18, 288.Google Scholar