B,5 • EQUATION OF KINETIC ENERGY 



B,5. Equation of Kinetic Energy. The equation of kinetic energy 

 for the mean motion is obtained by multiplying Eq. 4-5 by C/,-. We ob- 

 tain, after some transformation, 



(5^^') + 4(^^'^') + ^' 



I (,'„<) + A (,-„,c/,.) 



= — ^ [Uiiphij -\- pUiUj) -{- ^p'ujW,] 



+ -T— {p8ij + pU^^j) — <po (5-1) 



where 



<Po 



-Ui 



dXj 



d 



2 .- 

 3 



dXj 





3Mf5.. 



dXj 



(5-2a) 



and f>o is defined by 



$0 = — 



2-2,1 (dU, ,dU\^ ,_„, 



It is remarked that f>o is the Rayleigh dissipation function [5, p. 580]. 

 In deriving Eq. 5-1, use has been made of Eq, 4-6. On the left-hand side 

 of Eq. 5-1, we have the rate of change of kinetic energy, 



l(5^^-)+46^^'^') 



and the convection by density fluctuations, 



iip'u.)+l^ip'um 



On the right-hand side of Eq. 5-1, the term 



— [Uiipdij + pUiUj) + IpUjUj] 



accounts for the diffusion of energy by turbulence and pressure; the term 

 p'^ represents the rate of change of energy due to expansion; the term 

 pUiUjd Ui/dXj is the rate of production of turbulent energy from the energy 

 of the mean flow, as a result of Reynolds stresses pUiUj] and finally the 

 term <po, as given by Eq. 5-2a, is the action of viscosity, which takes the 

 form of a dissipation $0, and a spatial transfer 



-i-\Ui 

 dXj 



<83 > 



<Po — $0 (5-3a) 



