F • CONVECTIVE HEAT TRANSFER IN GASES 



heat can be considered constant, whence ul/h^ may be expressed by 

 (7 — l)Ml, where 7 is the ratio of the specific heat at constant pressure 

 to that at constant volume and M^ is the free stream Mach number. 



It is seen above that the energy equation can be written as an inte- 

 gral, dependent, however, upon the momentum equation. On the other 

 hand, the momentum equation can be written as an integral equation, 

 dependent upon the energy equation. Thus 



g^{u^) = du^2 ±^^*^*^cZw,x (3-21) 



./«* Jo y* 



which can be integrated by the method of successive approximations, 

 starting with the Blasius solution (Table F,3a), if none better can be 

 assumed as a first approximation. However, as Crocco pointed out, the 

 iterative process does not converge upon a single solution in general but 

 rather yields values of Q:^, which oscillate about the exact value. For, if 

 an initial value of the shear function gr^i, equal to Ag^^^ in which A is a 

 constant and g^^ex is the exact value, is substituted into the right-hand 

 side of Eq. 3-21, the new g^2, obtained upon integration, will be equal to 

 g^eJA. Resubstitution of g^2 yields gr^s = Afif^ex = fif^i. Therefore it is 

 seen that the next substitution should be -s/g^ig^z = g^^^. In the iter- 

 ative process of solving Eq. 3-8 (Eq. 3-21), it follows that successive 

 values of g^ for resubstitution into Eq. 3-21 should be the geometric mean 

 of the two previous substitutions. 



It is next observed that a singularity exists at u^ = 1, owing to the 

 boundary condition g^ = at the outer edge of the boundary layer. 

 Therefore, in the case of Eq. 3-21, numerical integration cannot be carried 

 all the way across the layer but must stop at some point u^ = 1 — i just 

 short of w^ = 1, where i is arbitrarily small. Hence Eq. 3-21 becomes 



g^iu^) = gM -i)+ du^2 / "l";^* du^r (3-22) 



Ju^ Jo g*\u^) 



or, in order to avoid double integration, 



/""* f(u ) f^~^ f(u ) 



g^{u^) = g^O- - i) -1- (1 — u^) \ *\ du^ — i / ■ \*\ du^ 



^*^ * * * Jo g^u*) * Jo gM*) 



+ f \l-u^) /%^ du^ (3-23) 

 J "* y* \ "^* / 



where /(m^) = 2m^p^m*- 



Since the first term, g^{\ — i), on the right-hand side of Eq. 3-23 is 

 unchanged by successive iteration, a method to adjust g^il — i) in each 

 iteration is necessary so that the boundary condition g^ (1) = is more 

 nearly approached. Following Crocco, one notes that for w^ — > 1, Eq. 3-8 



< 346 ) 



