A. S. Iberall 



1 - (p = 1 - ax2M + bx2P 



is not satisfactory. It can satisfy the qp boundary conditions, but it is not a 

 completely accurate enough description of the cp function. 



The salient characteristics of the ip function beyond its boundary conditions - 

 to be noted experimentally — are the magnitude, location, and width of the very 

 high impulse in cp", and possibly even its slope at x = ±1. For this there is no 

 theory, none at least in a self-consistent sense. This is no trivial observation. 



Kraichnan, in private criticism of this work, called attention to the possible 

 pertinence of the work of Landahl. Landahl pointed out the differences in our 

 motivation and purpose, and kindly supplied material that he considered relevant 

 from his work [14]. He makes therein an important point. He points up, validly, 

 that the Orr-Sommerfeld theory, essentially obtained by eliminating pressure 

 between the incompressible set of the Navier-Stokes equation and continuity, 

 leading (in his terminology) to a result like 



(U- C) ((p"-K2(p) - U"<p + i/aR ((p"" -2K2(p" + K2(p) = [his Eq. (30] 



cannot or has not been correctly applied to turbulent fields. The essence of the 

 matter, he states, is that in stability theory u" (our R'J) is assumed to be of 

 order unity. This is fine for the laminar flow field, but far from true in turbu- 

 lence. He points up that the results depend on u", and that values of u" run up 

 to the thousands, (on the basis of boundary -layer thickness). Thus, Orr- 

 Sommerfeld stability theory is only valid for the transition from laminar flow. 

 In the turbulent field, it is not correct. In order to be applicable, it must deal 

 with the form and boundary conditions for u" as well as u. However, this diffi- 

 culty is intensified in Orr-Sommerfeld theory. Actually the Orr-Sommerfeld 

 theory is embedded in the theory herein developed as part of the equation set. 

 However, we do not "eliminate" any variables, such as pressure. It is this 

 elimination, by differential operations, that introduced u" (or our cp"). The 

 original inhomogeneous linear equation set does not contain terms higher than 

 first order in cp'. It is here that the theory of linear equations is not complete. 

 A mathematical colleague pointed out that the results in Poole and other books 

 on linear differential equations are not complete for inhomogeneous linear sets; 

 that it is moot whether derivatives higher than the coefficients that appear in 

 the original equation set appear in the solution. The standard theoretical course 

 is the discussion of the transformed set of first order equations. To avoid re- 

 lated difficulties, the earlier treatment [10] stumbled on a valid path, purely by 

 necessity. 



The elimination of all of the variables but one (i.e., the reduction to one 

 higher- order equation), in addition to possible ambiguities, leads to thousands 

 of coefficients for the much higher ordered equation set for compressible flow. 

 It was only in desperation that Frobinius-type series solutions were elected for 

 exploration. It was quickly realized that the solution of the five-equation set is 

 arrived at by a "relaxation" of terms in the power series one at a time, by 

 cycling through the equation set, with quite a few being developed before a cycle 

 of repetition could be obtained. The second method, having then found these 

 series summable essentially into modalities, was to then derive solutions by 



740 



