simple manner, quantitative relations between the Reynolds number of the basic flow 

 and other characteristic parameters of the oscillations. These relations could be obtained 

 otherwise only by following very complicated mathematical analysis of the secular 

 determinant of the characteristic value problem. 



3. Oscillations of finite amplitudes 



The unstable oscillations predicted by the instability theory are still far from 

 turbulent flow. In fact, experimental observations and theoretical considerations suggest 

 the following five stages of the phenomenon of transition from laminar to turbulent flow: 



1 . instability of laminar flow 



2. amplification of small disturbances 



3. amplification of finite disturbances 



4. breakdown of finite disturbances 



5. formation of turbulence 



(a) formation of turbulent spots 



(b) spread of turbulent spots 



(c) fully developed turbulence. 



One of the current issues is whether step 4 is basically a two-dimensional or 

 three-dimensional phenomena. I shall not elaborate on this point, as Mr. Klebanoff will 

 no doubt tell you something more about it in his discussion. I shall now turn to a 

 discussion of the space structure of these oscillations. 



The space structure of the oscillations indicates clearly that they are far from 

 turbulent motions, since their scales are far too large. In the direction of the flow, they 

 are of the order of the boundary layer thickness 8; even its scale across the boundary 

 layer is only of the order of SRd~ y3 close to the boundary. On the other hand, turbulent 

 motions have a vorticity scale of the order of 8Rs~% which is still smaller by a factor 

 Rs 1 '' 6 - Thus, we must have a further breakdown of the motion into smaller eddies. 

 This is in line with our general concepts of turbulent motion where energy is fed into 

 motions of smaller and smaller scales, and is eventually dissipated into heat. Either the 

 small scale motions are produced as harmonics of the oscillation when its amplitude 

 becomes large, or they are made possible by distortion of the basic flow. In either case, 

 it is necessary to study oscillations of finite amplitudes. By considering the distortion of 

 the basic flow Meksyn and Stuart [14] have made some interesting calculations of such 

 oscillations. More recently, Stuart [15] calculated the development of such motions in 

 the course of time. 



The study of higher harmonics, on the other hand, requires a more complete 

 knowledge of the theory of small oscillations. It is known that a perturbation theory 

 based on non-linear oscillations must be based on a complete set of normal modes in 

 the linear theory. In the usual calculations, only one of the modes is calculated. To 

 obtain the other modes, * an improved mathematical theory (such as that outlined in 

 Section 4) is desirable. 



I shall therefore refrain from discussing in detail the possible mechanisms leading 

 from regular oscillations to the threshold of transition,** except to make the following 

 two observations. First, there is a general concept*** originally suggested by Landau 

 [17] who regards the final state of turbulent motion as reached through the successive 

 instability of the flow system with respect to more and more modes as the basic flow 

 is being distorted by the occurrence of the lower modes. Secondly, I wish to point out 

 that a more extensive study of the linear theory, particularly of the amplitude distribu- 

 tion of these oscillations, does show that damped oscillations have a scale 8Rq~^ even 



Some calculations of higher modes were recently made by Grohne. 

 *Cf. Dryden [22] p. 57 for some discussions of this point. 

 * * Also independently set forth by J. T. Stuart. 



356 



