F,7 • STAGNATION POINT SOLUTION 



on cones, provided that the Mach number is the local value just outside 

 the boundary layer. In the case of the local heat transfer coefficient, Fig. 

 F,5p can be used also for cones except that the ordinate must be multi- 

 pUed by \/3 in accordance with the above discussion. Fig. F,6d repre- 

 sents a corrected (by \/3) plot for moderate heat transfer to or from 

 cones in heated wind tunnels. 



F,7. Stagnation Point Solution. Because of its importance in 

 general missile design, heat transfer at stagnation points of cylindrical 

 and spherical surfaces should be given a few words at this time. It may be 

 desirable to round the leading edges of airfoils and the noses of bodies of 

 revolution at high speeds in order to diminish the local heat transfer rates 

 at those locations and to allow easier internal cooling, if necessary. 



The heat transfer problem for incompressible flow lends itseK readily 

 to analysis. For the cyhndrical surface, Squire [7, p. 631] used Homann's 

 solution [8] of the momentum equation near the stagnation point and 

 found, upon simultaneous solution with the energy equation, that the 

 local heat transfer coefficient St, defined by 



9w = -Stc^pU{T„ - n) (7-1) 



may be expressed by the relation 



where D is the diameter of curvature of the cylindrical surface and 

 /3 = {duJdx)x=o where x is measured along the body from the stagnation 

 point. Subscript « refers to the undisturbed flow. From incompressible 

 perfect fluid theory, it is found that ^D/u^ = 4. This relationship has 

 been verified experimentally [7, p. 631]. 



For the spherical surface, Sibulkin [9] also used Homann's results, and 

 following the method of Squire [7, p. 631], obtained the formula 



in which D is the diameter of curvature of the spherical surface. Also 

 from incompressible perfect fluid theory, ^D/U = 3. 



Eq. 7-1, 7-2, and 7-3 can be used for approximate heat transfer 

 calculation with supersonic flow about a body when it is remembered that 

 the problem is strictly a local one and therefore the fluid properties Cp, 

 k, n, and p in all three equations must now be taken at the stagnation 

 temperature T" at the outer edge of the boundary layer. Thus, Eq. 7-1 

 becomes for gases 



q^ = St'p'Uih' - K) (7-la) 



< 365 > 



