B,6 • EQUATION OF ENERGY AND ENTHALPY 



Then we obtain 



'( 



dUidUj 



dXj dXi 



dUidU 

 dXi dx 



?) 



+|".(g+S-«.) 



(5-3c) 



The last term vanishes if n is taken as constant. The third term is not 

 important, as can be shown by special examples, for instance, a boundary 

 layer flow. The other terms are also not important if the effect of the 

 compressibility is small. Hence $o — (po has predominantly the function 

 of a spatial transfer. Similar formulas for $ — ^, $ — ^, or <!>' — (p' can 

 readily be obtained, and so also can formulas for x, x, and %', for example, 



3 ^ ^ S""' dxj ^ 



(dUi dUj dUi dUj\ 

 \dXj dXi dXi dxjj 



(5-3d) 



B,6. Equation of Energy and Enthalpy. The production and 

 transfer of heat in a turbulent flow is now considered. The derivation of 

 the energy equation for the total motion is well known (see [6, p. 603] 

 and also [7, p. 57]) and need not be repeated here. It is written as follows 

 for constant specific heat and for variable thermal conductivity and 

 viscosity: 



(P + P') [i + (Uj + uj) 4z 



dt 



dXj_ 



c^T + T') 



-I 



i + <"' + "'>S,j 



(p + pO = $ + 



dXj 



^"l^iT + T') 



(6-1) 



where Cp is the specific heat at constant pressure; k is the thermal con- 

 ductivity; p, p' are the mean and fluctuating pressures; T, T' are the 

 mean and fluctuating temperatures; $ is the dissipation function defined 

 by Eq. 5-2d; and CpT is the mean enthalpy. The energy equation (Eq. 6-1) 

 can be separated into an equation for the mean motion and an equation 

 for the fluctuating motion. Since the need for the latter equation is pres- 

 ently not apparent, only the energy equation for mean motion is devel- 

 oped and written as follows: 



I (P^.^) + A (,,^TU,) - 



+ ^ (c,,'T'V,) 



ft^l^^-^ 



+ ^ (c,,'T') 



ki' 





D 



( - dUk , , dUk\ , s ,r, r,\ 



(85 ) 



