boundary layer may consist successively of: a laminar boundary layer, a transition zone from 

 laminar to turbulent flow, a turbulent boundary layer, and a frictional wake. 



In the laminar boundary layer an approximate method involving a simple quadrature is 

 derived for the rapid calculation of the changes in momentum. The derivation consists of an 

 extension to axisymmetric flows past bodies of revolution of a method of successive approx- 

 imation introduced by Shvets^ for two-dimensional laminar boundary layers. 



New empirical criteria are presented for locating the position of transition for either 

 low-turbulence or turbulent free-streams from the position of neutral stability. The position 

 of so-called self-excited transition occurring in low-turbulence streams or under flight con- 

 ditixjns is based on the average pressure gradient from the position of neutral stability to that 

 of transition. Although the test data are for two-dimensional flows, the criterion is extended 

 to axisymmetric flows past bodies of revolution by means of Mangier' s transformations. An 

 approximate criterion for estimating the position of transition on a body in a turbulent free- 

 stream is established on the basis of measured positions of transition for flat plates in free 

 streams with various degrees of turbulence. 



The analysis of the axisymmetric turbulent boundary layer on a body of revolution is 

 divided into that for the main portion of the body, where the boundary layer is relatively thin 

 compared to the radius of the body, and that for the tail portion, where the boundary layer is 

 relatively thick. The momentum changes in the thin boundary layer may be calculated by a 

 rapid method involving a simple quadrature wherein a power-law relation for the skin friction 

 of flat plates is incorporated. Flat-plate values for skin friction are deemed reliable even 

 where the local skin friction is diminishing in an adverse pressure gradient owing to the com- 

 pensating effect of the Reynolds normal-stress term. The turbulent boundary layer on the tail 

 is analyzed by means of appropriate linear simplifications which give a rapid method for cal- 

 culating the momentum changes. An expression is derived for the change in momentum pro- 

 duced by the pressure difference in the wake at the tail and in the wake far downstream in ac- 

 cordance with a method presented by Young^ wherein, however, a more general relationship is 

 employed for the variation of the shape parameter of the velocity profile. 



For quick reference the various steps involved in calculating the development of the 

 boundary layer on a body of revolution are summarized at the end of this report. 



GENERAL CONSIDERATIONS 



AXISYMMETRIC BOUNDARY-LAYER FLOW 



The main elements comprising axisymmetric flow past a body of revolution at high 

 Reynolds numbers are shown in Figure 1 for the meridian plane. Two principal regions of 

 flow are indicated: the boundary layer next to the body with viscous flow and the region 



References are listed on page 29. 



