Taking a very simplified two-dimensional viewpoint, we may define local displace- 

 ment and momentum thickness by 



A*^ / ( 1 )dy, (58) 



<8>= / — ( 1 )dy. (59) 



Jo U, 



u 



Introducing U — U + u, we get mean thicknesses 



A* = 



U 



1 )dy, 



e = 



and mean square fluctuations 



U 



U 



) dy - / dy, 



a- 



- II u(yi)u(y 2 ) dy 2 dyi, 



x~ J o Jo 



and 



oo J J 



U 2(y0 

 1 



2U(y 2 ) 



(60) 

 (61) 



(62) 



u(yi)u(y 2 ) + 0(w 3 )> dy 2 dy u 



(63) 



where terms of order u s and higher may be neglected. 



Using very rough approximations for mean velocity profile and lateral correlation 

 function from reference 6, we estimate the following values for 8*' and •&': 



8*' 8*' 



— « .027 ; == tt 0.23 (64) 



5 A* 



— ^.014; —^0.15 

 5 © 



(65) 



8 is a "geometrical boundary layer thickness," the distance at which U zz U x 



It would, of course, be most desirable to be able to predict these values from a 

 rough estimate of skin friction fluctuations. The proper approach must involve a two- 

 dimensional random field of skin friction and the proper three-dimensional generaliza- 

 tion of the von Karman integral relation. 



A rough estimate was attempted, however, by using an over-simplified form of 

 two-dimensional Karman integral relation: 



d© T 



— = , (66) 



dx 



>l\ 



397 



