F • CONVECTIVE HEAT TRANSFER IN GASES 



and constant wall temperature, exact solutions can be obtained, and these 

 have practical application for supersonic flow over cones and slender 

 ogives, and over wedges and thin airfoils, especially when covered with 

 thin skins. 



The analytical and numerical results of Crocco [1] for the case of con- 

 stant free stream velocity, wall temperature, and Prandtl number are 

 singular because of their extensiveness and apphcabihty. Crocco not only 

 developed an accurate method of numerical solution of the momentum 

 equation (Eq. 3-2a), but also gave a practical solution of the energy 

 equation (Eq. 3-4) for Prandtl number near unity. (For a review of the 

 Crocco analysis and detailed calculations, the reader is referred to [2].) 

 van Driest [3] in turn has extended the Crocco analysis to include variable 

 Prandtl number in the solution of the energy equation. 



Owing to the importance of the numerical results derivable therefrom, 

 the extension of the Crocco analysis to include variable Prandtl number 

 will be outlined here with pertinent formulas. Following the procedure of 

 Crocco, the independent variables x and y are first transformed to x and u 

 by w = u{x, y) and x = x. Eq. 3-1, 3-2a, and 3-4 then become, upon 

 eUmination of v, 



(3-5) 



)! = « 



(3-6) 



where shear stress r = n{du/dy). These equations are still in general form. 

 However, when dh/dx = and dp/dx = 0, they simplify considerably. 

 The enthalpy h is accordingly a function of u only, and for a perfect gas 

 the density varies inversely with the temperature. Since ix = ni{T) = 

 iJLiiu), Crocco next showed, upon satisfaction of the boundary condition 

 T -^ CO as a; -^ 0, that Eq. 3-5 becomes 



^|| + PMW = (3-7) 



where g{u) = t {x, y) ^/2x. In dimensionless form, Eq. 3-5 and 3-6 are 

 then, for dh/dx = and dp/dx = 0, 



g*g* + 2m^p^m* = (3-8) 



(|)'H.a-POf^(|) = -f (3-9) 



<342 ) 



