F • CONVECTIVE HEAT TRANSFER IN GASES 



with 



for the cylindrical face, and 



« = o..e3(f)-"Xf)"'(^T 



for the spherical nose, where the superscript " indicates stagnation con- 

 ditions. (Note that U in the equations is arbitrary, because the effect of 

 speed is already accounted for in Cp, k, n, p, and jS.) However, in terms of 

 undisturbed conditions, Eq. 7-la, 7-2a, and 7-3a become 



(7-2a) 



(7-3a) 



-St^p^Uih' - K) 



(7-lb) 



«-=- =«-(¥y(=iF)-('^r©"(M)" ™ 



for cylinders, and 



— •(vT'(=^)""(fr(£r(£)" "* 



for spheres. 



With supersonic flow, jS can be approximated upon the assumption 

 that Newtonian flow prevails between the bow wave and the body. The 

 derivation goes as follows: In Newtonian flow [10], the pressure p over 

 the surface of a circular arc of diameter D is given by 



P - Poo 



But, at the stagnation point. 



2 cos^ jY 



J23, = _ 1 ^ 

 p dx 



Hence, from Eq. 7-4 



= # 



p« D 



(7-4) 



(7-5) 

 (7-6) 



in which p^ is the density at the stagnation (local ambient) point. 

 Across a normal shock, 



p^ iy + l)Mi \ T - 1 (7 - l)Ml + 2 " 



p„ (7 - l)Mi + 2 L "^ 2 2yMi - (t - 1). 



so that, from Eq. 7-6, the nondimensional quantity I3D/U is 



7 - 1 (t - l)Ml + 2 



1 

 7-1 



^ _ [ 8[(7-l)M^ + 2] 

 f/ - ( (7 + 1)M^ 



1 + 



2 27M^ - (t - 1) 

 < 366 ) 



1 H 



7-1 



(7-7) 



(7-8) 



