32 



layer (which occurs for low frequencies Q and large 

 amplitudes A) , then the Blasius boundary layer can 

 be stabilized by these oscillations. However, the 

 Stokes layer stability is very sensitive to extra- 

 neous effects such as streamline curvature. For 

 instance, experiments show that transition of plane 

 Stokes layers occurs at Stokes layer Reynolds 

 numbers, Rgf (where R . = ARr^/B) on the order 

 of 500 [see Li (1954)]. However, if a slight amount 

 of streamline curvature exists, as would occur in 

 Stokes layers induced on the bottom of a water chan- 

 nel supporting free-surface gravity waves [see 

 Collins (1963)], the transition Reynolds number is 

 reduced to about 160. Thus, the effect on the in- 

 stability of the Blasius boundary layer of free 

 stream oscillations with a spatial structure such 

 as Uq + Ujcos (kx-iot) can be expected to be different 

 from the parallel flow oscillations considered above. 



It is well known that ambient turbulence tends 

 to promote laminar to turbulent transition of the 

 boundary layer. Thus, if some oscillatory boundary 

 layer does in fact properly model certain features 

 of the interaction of the ambient turbulence with 

 the underlying steady boundary layer then it is to 

 be expected that such a oscillatory boundary layer 

 is less stable than the underlying steady boundary 

 layer. Although the present numerical results show 

 only a stabilizing effect for the type of oscilla- 

 tion considered, as inferred above there is reason 

 to believe that a more complex form of oscillation 

 of the boundary layer can be destabilizing. The 

 theory of the instability of forced oscillatory 

 boundary layers provides an alternative point of 

 view from that of Rogler and Reshotko (1974) and 

 Mack (1975) on the role of the interaction of free- 

 stream disturbances with Tollmien-Schlichting waves. 



ACKNOWLEDGEMENT 



This work was supported by the Naval Sea Systems 

 Command . 



REFERENCES 



Ackerberg, R. C, and J. H. Phillips, (1972). The 

 Unsteady Laminar Boundary Layer on a Semi- 

 Infinite Flat Plate Due to Small Fluctuations in 

 the Magnitude of the Free Stream Velocity. J. 

 Fluid Mechanics, 51, 137-157. 



Coddington, E. A., and N. Levinson, (1955). The 

 Theory of Ordinary Differential Equations , 

 McGraw-Hill, New York. 



Collins, J. I. (1963). Inception of Turbulence at 

 the Bed under Periodic Gravity Waves. J. Geo- 

 physical Res., 68, 6007-6014. 



Craik, A. D. D. , (1971). Nonlinear Resonant In- 

 stability in Boundary Layers. J. Fluid Mechanics, 

 50, pp. 393-413. 



Davis, S. H., (1976). The Stability of Time-Periodic 

 Flows. Annual Review of Fluid Mechanics , 8, 

 57-74. 



Gaster, M. , (1963). A Note on the Relation Between 

 Temporarily-Increasing and Spatially- Increasing 

 Disturbances in Hydrodynamic Stability. J. Fluid 

 Mechanics, 14, 222-224. 



Hayashi , C. , (1964). Nonlinear Oscillations in 

 Physical Systems, McGraw-Hill Book Co., New 

 York. 



Kerczek, C. von, and S. H. Davis, (1974). Linear 

 Stability Theory of Oscillatory Stokes Layers, 

 J. Fluid Mechanics, 62, 753-773. 



Li, H., (1954). Tech. Mem. 47, Beach Erosion Board, 

 U.S. Army Corps of Engineers, Washington, D.C. 



Loehrke, R. I., M. V. Morkovin, and A. A. Fejer 



(1975). Transition in Nonreversing Oscillating 

 Boundary Layers, Transactions ASMS, J. Fluid 

 Engineering, 97, 534-549. 



Mack, L. M. (1975) . Linear Stability Theory and 

 the Problem of Supersonic Boundary-Layer Trans- 

 ition, AIAA J., 13, 278-289. 



Mack, L. M. , (1976). A Numerical Study of the 

 Temporal Eigenvalue Spectrum of the Blasius 

 Boundary Layer, J. Fluid Mechanics, 73, 497-520. 



Obremski, H. J. and M. V. Morkovin, (1969) . Appli- 

 cation of Quasi-Steady Stability Model to Periodic 

 Boundary-Layer Flows, AIAA J., 7, 1298-1301. 



Orszag, S. A., (1971). Accurate Solution of the 

 Orr-Sommerfeld Stability Equation, J. Fluid 

 Mechanics, 50, 689-70 3. 



Rogler, H. L. and E. Reshotko, (1975). Disturbances 

 in a Boundary Layer Introduced by a Low Intensity 

 Array of Vortices, SIAM J. Appl . Math., 28, 431- 

 462. 



Rosenhead, L. , (1963). Laminar Boundary Layers, 

 Oxford. 



Sarpkaya, T. , (1966). Experimental Determination 

 of the Critical Reynolds Number for Pulsating 

 Poiseuille Flow, Trans. ASME , J. Basic Engineer- 

 ing, 88, 589-598. 



Yakubovich, V. A. and V. M. Starzhinskii , (1975). 

 Linear Differential Equations with Periodic Co- 

 efficients, Translated from the Russian by D. 

 Lauvish, John Wiley S Sons. 



