B • TURBULENT FLOW 



The other approach is to regard the statistical effect of turbulence 

 on the mean flow as being similar to that of molecular viscosity or heat 

 conduction, so that the turbulent transport terms can be treated by the 

 same statistical methods as those applied to transport processes in non- 

 turbulent motions. To this end, and as a basis of the statistical theory of 

 transport processes by molecular motions in a gas, we use the Boltzmann 

 equation 



df , ^ dj _hl 

 dt 





(10-1) 



where df/dt is a symbol representing the collision integral, f{t, x; i^dxd^ is 

 the number of particles in the space and velocity elements dxdiE, at the 

 instant t; x and ^ are the vectors of position and velocity, X is the external 

 force per unit mass. The left-hand side of Eq. 10-1 represents the rate of 

 increase in time of the number of particles in the phase element dxd^ 

 when we move together with the particles in the phase space x, ?. The 

 right-hand side represents the effect of restoring and direct collisions 

 which throw the particles respectively in and out of the phase element. 

 A consequence of the Boltzmann equation is the equation of evolution 

 of a transferable property ^{t, x) defined by 



Ht, x) = 



/ dkct>W 

 f dkf 



(10-2) 



where 0(^) is a function of the random velocity ^. As special cases it is 

 interesting to put (/> = 1, ^i, |^?, thus obtaining from Eq. 10-1 and 10-2 

 the general hydrodynamic equations 



i+4<^^'-^ 



DUi 

 Dt 



. (dUi J J dUi\ _„ daij 



_ BI dqj 

 ^ Dt dXj 



^ij^ij P^ j-^j 



(10-3) 



Here the density p, the speed Ui, and the internal thermal energy per 

 unit volume I are defined by the mean values 



p = m j fd^ = 



mn 



I = ^C? 



-^^i 



<98) 



