away from the boundary. The picture is then possible that the unstable oscillations, 

 when their amplitudes are large, might feed energy into the oscillations basically at the 

 scale of turbulent flow. The amplitude of the latter oscillations also show a change-over 

 from large to small scales like the intermittency phenomenon in fully developed turbu- 

 lence or the phenomena of turbulent spots in transition. 



The above discussions (and other experimental evidence at the Bureau of Stand- 

 ards) suggest the importance of a thorough study of the amplitude distribution of the 

 oscillations, especially that of the damped modes. We therefore turn to the discussion 

 of the improved mathematical theory of such oscillations. 



4. Improved mathematical theory 



The small two-dimensional disturbances in a parallel flow are assumed to be given 

 by a perturbation stream function of the wavy type 



1// = <p(y)e ia ^- ct) (2) 



such that 



v! = Rel — ) , v' = Re( ] (3) 



\ dy / \ dx 



The linearized Navier-Stokes equations then lead to the Orr-Sommerfeld equation 



4> iv ~ 2a 2 cf>" + a 4 = iaR [(w - c)(0" - a 2 cf>) - w"4>] (4) 



where w(y) is the basic velocity profile, R is the Reynolds number, and all the variables 

 involved are dimensionless. The conditions at the solid boundary are 



0(0) = 0'(O) = 0; (5) 



and those for the outer edge of the boundary layer are 



$,0'->O, as y-^co. (6) 



This leads to a characteristic value problem with a secular determinant 



F(a, R, c) = 0. (7) 



Instability of the flow obtains when c, > 0. 



As is well-known, the solution of the problem is usually based on the fact that 

 aR is large. However, the usual asymptotic solutions used for the calculation of the 

 stability characteristics are valid either in the immediate neighborhood of the critical 

 point where w — c, ox away from the critical point. Strictly speaking, it is not justifiable 

 to apply both types of solutions at the same time. It is due to a fortunate coincidence 

 of circumstances that the old process can be justified from a numerical point of view in 

 certain important special cases — including the Blasius flow and the plane Poiseuille flow. 

 Actual difficulties arise in other important cases; for example, in the extension of the 

 theory to supersonic boundary layers.* 



We are therefore led to seek asymptotic solutions which are uniformly valid in 

 a finite region containing the critical point. Such solutions are desired for at least two 

 other reasons mentioned in Section 3; namely: (1) In trying to study the physical nature 

 of such oscillations, the calculation of the amplitude distribution is essential. Such a 

 calculation cannot be conveniently and accurately made by using the old solutions. 

 (2) In an attempt to study oscillations of finite amplitude, the development of the non- 



* The most illustrative example is the calculation made by Bloom who followed the 

 older theory faithfully and arrived at incorrect results. Solutions adequate for a large number 

 of cases have been found recently by Dunn and Lin (See Ref. 18, p. 72) 



357 



