Lumley 



of ten percent of the free stream velocity. It must be anticipated that a laminar 

 boundary layer can be successfully stabilized only in the absence of large dis- 

 turbances, since once transition has occurred, few stabilization techniques could 

 be expected to have the capacity to reestablish laminar flow downstream of a dis- 

 turbance." Disturbances appear either at the boundary, or in the free stream; 

 consequently, great care must usually be exercised to make both as free of dis- 

 turbances as possible. The only kind of stabilization that appears to be possible, 

 then, is stabilization to small disturbances; that is, the preventing of small dis- 

 turbances from growing to be big ones. This is what is customarily meant by 

 stabilization. 



There are, generally, two types of small disturbances to which laminar 

 boundary layers are unstable. One consists of progressive waves; these are 

 known as ToUmien-Schlichting waves (Lin (1955)). The other consists of stream- 

 wise standing vortices; these are known as Taylor-Goertler vortices (see Lin 

 (1955), p. 96). The Taylor-Goertler type of instability only appears where there 

 is concave curvature in the streamwise direction or where a surface is heated 

 with a liquid flow above it in a gravitational field (Goertler (1959)). However, 

 the condition on the curvature in order to assure the appearance of ToUmien- 

 Schlichting waves before Taylor-Goertler ones is quite stringent, and probably 

 few supposedly flat plates satisfy it. Practically without exception, the analyses 

 which indicate a possibility of stabilization have reference only to ToUmien- 

 Schlichting waves. In addition, most of these analyses have reference only to 

 progressive waves in the streamwise direction. While for the ordinary boundary 

 layer Squire's theorem (Lin 1955)) assures us that such waves become unstable 

 first, in some of the situations under discussion, we may not have such assur- 

 ance. While nearly all of the suggested techniques attempt to control the growth 

 rate of the streamwise progressive waves, at least one (Kramer (1962a)) attempts 

 to prevent the development of three -dimensionality, which appears to be (Kleb- 

 anoff, Tidstrom & Sargent (1962)) a necessary prelude to transition, while another 

 (Kramer (1962b)) attempts to control what appears to be a secondary instability 

 associated with the developing three -dimensionality (Klebanoff, Tidstrom & Sar- 

 gent (1962)). 



There are distinct differences between discussions of stability on two di- 

 mensional bodies and on bodies of revolution. If the diameter of the body is in- 

 creasing, two conflicting effects are felt. In the first, an increase in diameter 

 means that the boundary layer must be spread over an ever widening area, pro- 

 moting thinning and altering the profile (much as suction does). It might be ex- 

 pected that this would delay instability beyond the point to which it is already 

 delayed by the favorable pressure gradient usually present on the forward part 

 of a body of revolution. In the second, cross-stream vorticity is being stretched, 

 which, due to the associated increase in intensity, should result in an earlier 

 occurrence of ToUmien-Schlichting instability. There is evidence (Groth (1957)) 

 to indicate that the stretching dominates. The picture is complicated further, how- 

 ever, by the possibility of Taylor-Goertler instabilities in the concave flow near 

 the stagnation point (Goertler (1955), Goertler-Witting(1958)), and by the stretch- 

 ing (and intensification) of vorticity which may be present in the free stream. 



'^Although probably most will reestablish laminar flow i£ the disturbance is 

 removed. 



916 



