ABSTRACT 



A procedure is presented for calculating the viscous drag o£ bodies of 

 revolution in axial motion from boundary-layer theory. Rapid approximate 

 methods are developed for computing the growth of the laminar and turbulent 

 boundary layers. A new empirical criterion is given for locating the position of 

 self-excited transition associated with low-turbulence flows. 



INTRODUCTION 



A completely immersed body moving rectilinearly with uniform velocity in an infinite 

 fluid at rest experiences a resisting force which may be termed viscous drag as it results pri- 

 marily from the viscous properties of the fluid. In designing low-drag bodies there often a- 

 rises the need for calculating the viscous drag of streamlined bodies of revolution in axial 

 motion when considering various proposed shapes. For bodies moving at high Reynolds num- 

 bers, which are of great technical importance, the viscous drag of streamlined bodies of revo- 

 lution is readily amenable to analytical treatment on the basis of the boundary-layer concept. 



Historically, the theoretical analysis of the drag of bodies in uniform motion by as- 

 suming an ideal (non-viscous) fluid gave the fruitless result of zero drag for all bodies, the 

 classical D'Alembert paradox. At the other extreme, the theoretical analysis of drag by apply- 

 ing the complete set of Navier-Stokes equations of motion for the flow of a viscous fluid led, 

 in general, to mathematical difficulties which were virtually unresolvable owing to the com- 

 plicated non-linear nature of these equations. For bodies moving at high Reynolds numbers, 

 however, the flow is virtually that of an ideal fluid except in a thin boundary layer next to the 

 body where substantial viscous forces are produced by the rapid drop in velocity to zero at the 

 body surface. Accordingly, by considering the viscous flow confined to the boundary layer, 

 Prandtl was able to derive the simpler boundary-layer equations of motion from the Navier- 

 Stolces equations. 



The principal purpose of this report is to describe methods of solving the boundary- 

 layer equations of motion to arrive at the. viscous drag of bodies of revolution of arbitrary 

 shape in uniform axial motion. The study is restricted to hydraulically or aerodynamically 

 smooth streamlined bodies in incompressible flow. A streamlined body may be defined as one 

 without appreciable separation of flow from its surface and consequently with small pressure 

 drag resulting from the generation of separation eddies. It is to be noted, however, that some 

 pressure drag is still present in the viscous drag of even perfectly streamlined shapes owing 

 to the effect of the boundary layer in displacing the main flow outward, especially near the 



tail. 



The calculation of the viscous drag of a body of revolution requires a detailed analysis 



of the development of each phase of the boundary layer from its origin on the nose of the body 



to its final phase as the frictional wake far downstream. In the downstream direction the 



