G • COOLING BY PROTECTIVE FLUID FILMS 



Eq. 4-2 can be further simplified if the stagnation enthalpy h^ = {u/2y + 

 CpT is used instead of the absolute temperature T. If, with the above 

 assumptions, both the velocity and stagnation enthalpy profiles are 

 assumed as fourth degree polynomials in t, satisfying appropriate con- 

 ditions at the outer edge of the boundary layer and at the wall, and these 

 profiles are substituted into the modified equations (Eq. 4-1 and 4-2), 

 then two ordinary differential equations in the nondimensional hydro- 

 dynamic and thermal thicknesses are obtained. On the basis of a uniform 

 wall temperature, general approximate solutions of these differential equa- 

 tions for the boundary layer thicknesses are derived. These solutions are 

 valid for a prescribed external flow as given by u^/U and M^, and for a 

 given wall temperature and mass flow injection distribution. By this 

 means the boundary layer characteristics can then be calculated with 

 comparative ease. The following general conclusions are drawn from the 

 above analysis: (1) in the region of an adverse pressure gradient, the cool- 

 ing of the wall tends to delay the separation of the flow, (2) for a fixed 

 wall temperature, normal mass flow injection tends to promote separation, 

 although in the absence of an adverse pressure gradient, injection alone 

 cannot cause separation; and (3) the effect of the wall temperature on 

 the boundary layer characteristics depends on whether the axial pressure 

 gradient is adverse or favorable. The skin friction tends to be diminished 

 by a decrease in the wall temperature (for fixed injection) in a favorable 

 pressure gradient, but tends to be increased in an adverse pressure gradi- 

 ent. Similar conclusions hold for the Nusselt number but it is less sensitive 

 to change in the wall temperature than the skin friction. 



In the preceding analysis, the wall temperature and the injection mass 

 flow have been treated as independent quantities. Actually, however, a 

 consideration of the heat balance at the wall indicates that the wall tem- 

 perature and the amount of injection mass flow are related to each other 

 through the temperature of the coolant. Thus, by considering such a heat 

 balance, a new parameter involving the coolant temperature is intro- 

 duced. The details of this analysis can be found in [13,lli]. 



Exact Solution of Heat Transfer in the Laminar Boundary 

 Layer. In the preceding articles, approximate methods for the solution 

 of heat transfer in laminar boundary layers on a transpiration-cooled wall 

 have been discussed. The solutions obtained by approximate methods 

 have explained most of the physical phenomena in the transpiration- 

 cooling problems, even though they satisfy the differential equations of 

 boundary layer flow only on the average. 



The present article considers some exact solutions of the equations of 

 boundary layer flow. The essential restrictions of the exact solutions are 

 that they are based on the case in which the velocity outside of the bound- 

 ary layer is proportional to a power of the distance along the main flow 

 (wedge flow) and that the velocity of fluid injection is proportional to 



< 446 ) 



