The Instability of 

 Oscillatory Boundary Layers 



Christian von Kerczek 



David W. Taylor Naval Ship Research and Development 



Center, Bethesda, Maryland 



ABSTRACT 



The instability of the two-dimensional flat plate 

 oscillatory boundary layer induced by a stream with 

 velocity Ug + Uj cos ut is considered. The velocity 

 amplitudes, Uq and Uj , are constants and Ui/Uq is 

 assumed to be small. The instability of this oscil- 

 latory boundary layer is analyzed by a time-dependent 

 linear parallel flow instability theory. The change 

 of the Tollmien-Schlichting growth rates due to the 

 imposed oscillations are computed to second order in 

 Uj/Uq. It is found that for imposed oscillation 

 frequencies in the range of the Tollmien-Schlichting 

 frequencies of the underlying Blasius flow, the 

 boundary layer is stabilized by the oscillations of 

 the external flow. 



1. INTRODUCTION 



In this paper, we study the instability of the two- 

 dimensional oscillatory laminar boundary layer which 

 forms on a flat plate that is exposed to a stream 

 with a velocity, Uq + Uj cos ut, perpendicular to the 

 plate's leading edge. The velocity amplitudes, U 

 and Uj, are constants, id is the angular frequency of 

 the oscillation, and t denotes time. The considera- 

 tions of the instability of oscillatory flows has 

 become an important field of research in recent years 

 and has been reviewed by Davis (1976) . The partic- 

 ular class of problems concerned with the instability 

 and laminar-turbulent transition of oscillatory bound- 

 ary layers has been reviewed by Loehrke, Morkovin, 

 and Fejer (1975) . The latter review indicates that 

 very few studies of instability and transition have 

 focused directly on the subject of oscillatory bound- 

 ary layers. Such studies that have concentrated on 

 oscillatory boundary layers have been mainly experi- 

 mental investigations which were restricted to low 

 frequency oscillations compared to the oscillation 

 frequency of unstable Tollmien-Schlichting waves. 

 The only analytical work concerning the instability 



of oscillatory boundary layers has been the quasi- 

 steady analysis of Obremski and Morkovin (1969) which 

 was aimed at these low frequency cases. 



The study of the instability of oscillatory 

 boundary layers has technological as well as funda- 

 mental importance. Examples of a fundamental nature 

 for which the study of the instability of oscillatory 

 flows may have relevance are the problems of how 

 ambient disturbances affect the instability of the 

 underlying steady boundary layer. Specific examples 

 might be the effects of ambient acoustic waves or 

 ambient turbulence on steady boundary layer insta- 

 bility. The problem of the effects of ambient tur- 

 bulence on the instability of a steady boundary layer 

 probably is not completely accessible by the theory 

 of the instability of oscillatory boundary layers. 

 However, a sufficiently complex, but organized, am- 

 bient oscillation may be adequate for duplicating 

 some aspects of the effects of ambient turbulence 

 on steady boundary layer instability. We are hope- 

 ful that this may be the case because of similar 

 phenomena in the field of nonlinear ordinary dif- 

 ferential equations. The study of the instability 

 of forced periodic solutions of nonlinear ordinary 

 differential equations has furnished a much richer 

 class of phenomena than the corresponding study of 

 the instability of only the steady solutions of these 

 equations [see, for example, Hayashi (1964); in 

 particular, the results for the forced van der Pol 

 equation, pp. 286-300]. 



In the present study, we focus on the very simple 

 oscillatory boundary layer that was described ear- 

 lier. The purely oscillatory part of this boundary 

 layer is approximated by the oscillatory Stokes layer 

 which has no spatial structure in the plane of the 

 plate, i.e., it is an exactly parallel flow. Thus, 

 this model problem may be too simple to reveal any 

 particularly important features of realistic ambient 

 disturbances. However, the model problem is a good 

 starting point and serves as a basis on which to 

 develop the appropriate methods of analysis for the 



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