F • CONVECTIVE HEAT TRANSFER IN GASES 



which, when substituted back into Eq. 3-13, gives 



Kiu^) = K{Q) - [Km - 1] ^ + 



K 



S{1) 



R{1) - 72K) 



(3-17) 



S and R will be found to depend upon the free stream Mach number, the 



free stream temperature, and the plate temperature (i.e. heat transfer). 



When the Prandtl number is constant, *S = PrI and R = PrJ, where 



/(w*) = 



^*(0)J 



duj 



and 



/( 





_9^ 

 L^*(0)J 



Hence there results Crocco's original formula: 



du^du^ 



(3-18) 

 (3-19) 



K{u^) = K{0) - [KiO) - 1] 



J(l) "^^"^Ae 



/(I) 



J{1) - J{u^) 



(3-20) 



Crocco tabulated / and / for various fixed Pr and the Blasius shear dis- 

 tribution. Extensive calculations by Crocco had shown that I and J were 

 approximately independent of Mach number and heat transfer for moder- 

 ate supersonic speeds, regardless of viscosity-temperature law (i.e. p^/x^ 

 variation) when the Prandtl number was not too far from unity. (Indeed, 

 this is exactly true for Pr = 1.) Hence it was concluded that the Blasius 

 (incompressible flow) shear profile, which resulted from the assumption 

 that^ p^H^ equaled unity, was appropriate for the calculation of I and J 

 and consequently the approximate enthalpy distribution from Eq. 3-19, 

 given an average constant Prandtl number. The Blasius shear distribu- 

 tion is tabulated in Table F,3a. The /'s and J's, calculated by Crocco, 

 are tabulated in Table F,3b. For moderate Mach numbers, the specific 



Table F,Sa. Shear function gf*, when p*ju* = 1. 



< 344 ) 



