B • TURBULENT FLOW 



It is remarked that the Reynolds stresses puiUj which characterize the 

 important nonhnear mechanism in the turbulent transfer, occur in two 

 places, namely in the diffusion and in the production. 



It is much easier to derive the equation of total kinetic energy, 

 K = ^{U1 + ul). It suffices simply to multiply Eq. 4-4 by (C7i + Wi) and 

 obtain, after some transformation, 



m (^^') + a^ (^^^'-) = - It ''^' - ai". ('''''■^' + ^«'-^' + ^'^^'^' 



+ Kp'«,)-y,|.-%^.-. (5-4) 



where 



The quantities <p and $ can be easily obtained by taking the mean values 

 of Eq. 5-2c and 5-2d. Reynolds stress terms are included in Eq. 5-4 but 

 only as diffusion terms. There is no energy production term associated 

 with these stresses such as we find in Eq. 5-1 for the kinetic energy of 

 mean motion. This is not surprising since the Reynolds stresses, which 

 transfer energy from the mean motion into turbulent motion, must have 

 a vanishing balance in the production of the total kinetic energy by reason 

 of conservation. The molecular motion contributes a pure dissipation ^ 

 and a spatial transfer ^ — $ which plays the role of viscous diffusion of 

 energy. Its structure can be clarified by transforming either Eq. 5-2c or 

 5-3a. For the sake of abbreviation, let us take Eq. 5-3a and rewrite it as 

 follows : 



2 - 1 a/xf , M fdUj dU^Y 



3'^' ' 3 ' dXj 



+ u4 



ax 



■j \ dXj / dXj \dXi / 





3^^ +3^^a^ 



dXj 



Let us introduce 



Xo = 





<84> 



