19 



turbulence. If the curve for transition for any boundary layer with a pressure gradient is as- 

 sumed geometrically similar to that for a flat plate as plotted in Figure 5, the Reynolds 

 number for transition Ra j^ for boundary layers on bodies in a stream with turbulence may be 

 stated as 



^ 6 tb ~ ^6' " ' '^ '*■ ~ ^ 



^e.ts ~ ^( 



yv ^ / ^e, tb - ^g, JV \ [54] 



where the subscript refers to flat-plate values without pressure gradient. For more precise 

 results, it is best to depend upon measured locations of the position of transition to obtain 

 values of the Reynolds numbers for transition characteristic of the particular wind tunnel or 

 flow facility being utilized. 



AXISYMMETRIC TURBULENT BOUNDARY LAYER 



GENERAL 



Owing to the incomplete state of present knowledge, concerning the mechanics of 

 turbulent flow processes, the analysis of turbulent boundary layers lacks the clearly defined 

 features of laminar boundary layers. In order to arrive at results of immediate utility, exten- 

 sive reliance has to be placed on empirical data to augment theoretically derived relations. 



Turbulent flows, in general, may be treated from the Reynolds viewpoint, this consid- 

 ers turbulent flow to consist of a mean flow upon which a fluctuation flow of much smaller 

 magnitude is superimposed. After the combined mean and fluctuation quantities are substi- 

 tuted into the Navier-Stokes equations of motion for viscous flow, appropriate time averages 

 of the resulting flow lead to the Reynolds equations of motion containing separate terms for 

 the mean quantities and the fluctuation quantities. The form of the Reynolds equations is 

 similar to the Navier-Stokes equations with the significant exception of the presence of addi- 

 tional terms which consist of averages of various products of the fluctuation velocities. 

 These additional terms may be shown to act as apparent stresses (Reynolds stresses) within 

 the flow. It is, however, tnt; present lack of analytical relations for the Reynolds stresses 

 which has made the theoretical treatment of turbulent flows so difficult. 



Turbulent boundary-layer equations are derived from the Reynolds equations in the 

 same way as laminar boundary-layer equations are derived from the Navier-Stokes equations, 

 i.e., by the process of eliminating terms of negligible magnitude. 



In the case of axisymmetric flow past a body of revolution with negligible longitudinal 

 curvature, the Reynolds equations of motion become* 



♦The Reynolds equations of motion for general curvilinear coordinates are written in tensor notation in Refer- 

 ence 30. 



