15 



where the tildes over the quantities refer to equivalent two-dimensional conditions and C is 



an arbitrary constant. 



A simple expression will now be derived for Z_ 5L^ which can be used to analyze two- 



dimensional data and which can be extended into an equivalent axisymmetric system. First 

 the following two-dimensional momentum equation is obtained from the axisymmetric momentum 

 equation, [30], by utilizing Mangier' s relations [42] 



Since in general 



it 



dl 



6 dU_ 



V dx 



:1-K. 



[43] 



then from [43] 



P. dU_^ 1 d(UP) _ U dp 

 V dx V dx V dS 



[44] 



V dS i5 5 u dx 



[45] 



Averaging — i^-^ over the distance from neutral stability to transition in accordance with 

 V dx 



[41] gives simply 



P dU 4 

 1/ dx 45 



5u 



{UeX -(Ud^)^ 



[46] 



In order to use the two-dimensional data of Figure 4 to determine the self-excited tran- 

 sition point for axisymmetric flow, it is necessary to express — ^-^ in terms of an equiva- 



V dx^ 



lent axisymmetric system. Applying Mangler's transformations to [45] and integrating in ac- 



cordance with [41] gives -2- =^ in equivalent axisymmetric quantities as 

 V dx 



H AM. = -£ 



V dx 45 



5v 



(i^ "A. - [^ "'\ 



J r 2 



dx 



[47] 



Furthermore, in order to use the two-dimensional data of Figure 4 to determine the self- 

 excited transition points for axisymmetric flows, it is necessary to convert the Reynolds 

 number for self-excited transition in two-dimensional flows Rq ^^ to that for axisymmetric 

 flows on bodies of revolution Rq ^^. Now Mangler's transformations [42] give for all Reynolds 

 numbers of laminar flows 



