B • TURBULENT FLOW 



Finally, by introducing the mean and fluctuating value of the total 

 energy content per unit mass E'^, E, E, and E', such that 



E' = c^T + ^C/f 



E = c^{T + T') + UUi + UiY 



E = c^T -\- UU', + ^) 



E' = E -E 



and adding Eq. 5-1 to Eq. 6-2, we obtain 



^ ipE') + U, ^, („%) + ■^, (c,p'T') - (f + «, ^) 



§t(pE') + u4,iyu,) + §i 



= — — ip'ujE'^ + UipUiUj + CppT'uj — ^ ^ ] + pUiUj — * + ^0 — <po 



- I p'ujE^ + UipUiUj + CppT'uj — ^ ^ ) 



(6-3a) 

 where the operator D/Dt on any function / denotes 



The left-hand side of Eq. 6-3a expresses the rate of change of quanti- 

 ties pE'^, Vui, p'T', and p. These rates are the result of diffusion and tur- 

 bulent energy production expressed by the terms on the right-hand side. 

 Here we find the production term —pUiUjdUi/dxj which decreases the 

 energy of mean motion, and diffusion terms within the brackets which 

 include Reynolds stress terms of transport of mass and temperature along 

 with the better-known term pUiUj. Here also is the molecular contribution 

 expressed by $o — (po, which we have already noted in Eq. 5-3 as a spatial 

 transfer and not an energy dissipation. No molecular dissipation appears 

 in Eq. 6-3a because the kinetic energy dissipated appears in the form of 

 heat. 



The corresponding equation for E may be obtained by adding Eq. 

 5-4 and 6-2. Thus 



= - -^ (plW + 7^^ + 7W + V^ - i ^ ) + § - 9 (6-4a) 



oXj \ dXj/ 



As would be expected, Eq. 6-4a, which accounts for both the mean and 

 the turbulent energy, is of simpler form and does not contain the pro- 

 duction term pUiUjdUi/dXj. 



The diffusion terms in the energy equations (Eq. 6-3a and 6-4a) con- 

 tain the thermal diffusion with flux kdT/dXj. This may be expressed as an 



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