G • COOLING BY PROTECTIVE FLUID FILMS 



layer on an impermeable surface was developed by Tollmien, Schlichting, 

 and Lin for an incompressible fluid. Lees and Lin [^4] extended the theory 

 to include the effect of compressibility. By application of the stability 

 theory it is possible to determine, from the velocity distribution in the 

 boundary layer, the local Reynolds number at which a flow with such a 

 velocity distribution becomes unstable. Transition to a turbulent bound- 

 ary layer may be expected to occur somewhere downstream of the point 

 of stability. The works of the above investigators indicate that an adverse 

 pressure gradient in the flow direction destabilizes the boundary layer and 

 a favorable pressure gradient increases the stability. 



In the analyses of the stability of compressible laminar boundary 

 layers [24,25], the results indicate that stability is greatly influenced by 

 the heat transfer from the wall to the gas. In accordance with Eq. 4-14 

 at the wall condition of the fiat plate it follows that the curvature of the 

 velocity profile at the wall is proportional to a negative product of the 

 temperature gradient and the velocity gradient at the wall. Then if the 

 wall is hotter than the free stream fluid the temperature gradient at 

 the wall will be negative, and in turn, the curvature of the velocity profile 

 at the wall will be positive. It follows that in the boundary layer on a 

 heated wall the velocity profile has a point of inflection which is a neces- 

 sary and sufficient condition for the existence of amplified disturbance, 

 hence, its instability. On the other hand, in the boundary layer on a 

 cooled wall, the curvature of the velocity profile at the wall is negative 

 and consequently the limit of complete stability increases. 



It is known that the effect of fluid injection is to destabilize the 

 boundary layer in a way similar to the effect of an adverse pressure 

 gradient. It can be seen that fluid injection (1) increases the boundary 

 layer thickness (a growing boundary layer is more prone to become tur- 

 bulent) and (2) fluid injection creates a velocity profile which is less stable 

 than one without injection. Since cooling of the wall and fluid injection 

 at the wall have opposite effects on the stability of the laminar boundary 

 layer with coolant injection, it is desirable to determine the simultaneous 

 effects on transition. 



Based on the improved viscous solutions of the stability equations 

 [26], calculations of the stability of the compressible laminar boundary 

 layer with coolant injection are made in [23] for the reduction of aero- 

 dynamic heating in high speed flight. The results apply at moderate 

 supersonic speeds and indicate the complete stability limits for two- 

 dimensional disturbances. In Fig. G,4f, the complete stability curves for 

 several rates of coolant injection are shown. For a given rate of coolant 

 injection, each of the curves depicts the region of complete stabilit3^ For 

 any given Mach number, the wall temperature must be below the curve 

 in order to attain a completely stable laminar boundary layer. 



Sublayer Theory in Turbulent Flow. The heat transfer in the 



(452 > 



