B • TURBULENT FLOW 



are respectively the instantaneous local density and the instantaneous 

 local temperature. The bars denote mean parts, and the primes denote 

 turbulent parts. Other physical quantities like viscosity, coefficient of 

 heat condition, and specific heat are considered to have negligible fluctu- 

 ating parts compared to their mean parts. 



The Navier-Stokes equation for the total motion is written as follows: 



Tt + (^'- + «') k 



(p + p') 



Here a-ji is a stress tensor [5, p. 574] defined by 



iUi + w.) = ^ fe + <.;,) (4-1) 



..= -(^P + |.f)a« + .gf + f; (4-2) 



where f = 6(f7jt + Uk)/dXk; f = dUk/dXk; f' = duk/dxk; n is the viscosity 

 supposed to be variable, but with a negligible fluctuating part; and bji is 

 the Kronecker delta having the value 1 for i = j and for i 7^ j. In 

 those equations a summation is understood for repeated indices.^ A similar 

 expression for o-^ can be written but this is omitted here. 



The variables Ui and Ui must moreover satisfy the equation of 

 continuity 



I (P + P') + ^. [(P + p'){Uj + uj)] = (4-3) 



With the aid of Eq. 4-3, the equation of motion (Eq. 4-1) may also be 

 written in the following form : 



^ [(P + p')(Ui + ui)] = -^ [{aj. + <r;,) - (p + p'){U, + ui){Uj + u,)] 



(4-4) 



One way of obtaining the momentum equations for the mean and 

 fluctuating motions is to start from Eq. 4-4 instead of Eq. 4-1. By aver- 

 aging we obtain the following momentum equation for the mean motion: 



— (pC/i + Vui) + Q- (pUiUj) = J^ ~ Q~- ^P'^''^^ "^ Ujp'ui + Uip'uj] 



(4-5) 



Similarly by averaging Eq. 4-3, the continuity equation for the mean 

 motion is 



^ + ^(pf/i + A.) =0 (4-6) 



01 OXj 



Corresponding equations could be written for the fluctuating motion, but 

 this will not be done. 



2 The indicial notations are advantageous in the general discussion of the equations 

 of motion. However, in the following articles when dealing with properties in two 

 dimensions the indicial notations will usually be abandoned in favor of x, y, U, and V. 



(82) 



