B • TURBULENT FLOW 



hulk convective type, and will be discussed in Art. 29 in connection with 

 the coarse eddies of free turbulent flow. 



The structure of the transport coefficients can be determined by 

 means of kinetic equations more general than the Boltzmann equation. 

 This attempt has been made in an article by Tchen published in the 

 Proceedings of the International Symposium on Atmospheric Diffusion 

 (1958). 



B,ll. Reynolds Analogy between Heat Transfer and Skin Fric- 

 tion. As an application of the transport processes treated in Art. 10, 

 let us study the Reynolds analogy between heat transfer and momentum 

 transfer. Let q = —pCpvT' be the rate of turbulent heat transfer in the 

 y direction across the unit area normal to this axis, and r = — puv be the 

 rate of momentum transfer or turbulent shear stress. According to Eq. 



10-6 we have _ 



dT 



q = —pCpvT' = pCpDk-— 



T = —pUV = pUu-TT- 



dy 



The following expressions written in nondimensional form may be 

 compared : 



" T 



and -TTTT rr^i (H-l) 



pCp{U - t/e)(r - n) p{U - Ue)' 



Here Ue and Te are the velocity and temperature at a reference plane, 

 which, in the present discussion, is taken at the edge of the boundary 

 layer. The Reynolds analogy is a statement of equality of the two ex- 

 pressions of Eq. 11-1. Let us investigate this analogy in some detail. Of 

 special interest are the heat and momentum transfers at the wall, where 

 the two expressions of Eq. 11-1 become 



^^ ^ -p^CpU.iT^ - Te) ' 2^' = '^l ^^^"^^ 



where the subscript ^ denotes the value at the wall, St is the coefficient 

 of heat transfer or Stanton number, and c/ the coefficient of skin friction. 

 Then the Reynolds analogy leads to 



St = ^Cf (11-3) 



This result was first obtained by Reynolds [28] and is also given by Squire 

 [7, p. 819] and Goldstein [6, p. 654] in their study of heat transfer. 



It is easy to see that Eq. 11-3 cannot be valid in general because, 

 in the compressible case of an insulated boundary layer, we must have 

 St = and C/ ^ 0, which obviously violate Eq. 11-3. Therefore it is 

 worthwhile to derive a more general relationship between the heat trans- 

 fer and skin friction. For this purpose we make use of the relation (Eq. 



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