B • TURBULENT FLOW 



these motions, then it is to be expected, by analogy with Eq. 10-5 and 

 10-6, that a turbulent flux will result in the form 



3$ 



J = — p^v = D -r- 



dy 



where J is turbulent flux, v is the turbulent velocity in the direction y, 

 and D is turbulent coefficient of transport. In general the transport coef- 

 ficient D may depend upon a number of unknown factors among which 

 are the property to be transferred, and the intensity and scale of the tur- 

 bulent motion. For example, its value may vary according to whether 

 we have a transport of momentum, heat, or matter. A knowledge of its 

 structure necessitates a detailed investigation of the basis of the tur- 

 bulent exchange term by a procedure analogous to that which yielded 

 the exchange coefficients n and k of Eq. 10-4 from the solution of the 

 complete Boltzmann equation including the collision term. It is hoped 

 that some insight into the essential structure of the turbulent transport 

 can be gained by proceeding in this way on a somewhat simplified basis 

 made possible by adopting an approximate form of the Boltzmann equa- 

 tion. When applied to turbulent motion, the right-hand side of Eq. 10-1 

 represents the effects of the turbulent exchanges on the distribution func- 

 tion. It can be regarded as a forcing term which distorts the distribution 

 from its equilibrium. Therefore we can write Eq. 10-1 approximately as 

 follows : 



^=-k(/-/J (10-7) 



where / is the nonequilibrium distribution, /eq is the equilibrium distribu- 

 tion, and K depends on the efficiency of the turbulent mixing. The idea of 

 writing such a simple relaxation type of exchange term in Eq. 10-7 in the 

 place of the complete collision integral in Eq. 10-1 is not new. Lorentz [22], 

 Van Vleck and Weisskoff [23] had initiated such a simplification in their 

 study of microwave line shapes. Later Bhatnagar, Gross, and Krook [24] 

 applied an essentially similar simplification for studying the collision proc- 

 esses in gases. According to Eq. 10-2 and 10-7, we can write 



^ = _«($_$) (10-8) 



Here ^ — $ is the fluctuation of the transferable property. Eq. 10-8 can 

 be used to find the evolution of the property $ carried by a lump of fluid 

 when the latter moves and mixes with its surroundings. 



Being given $, the value of <l> at any instant t is given by the integral 

 of Eq. 10-8 as follows: 



$(f) = K f " dre-^'^t - r) (10-9a) 



< 100) 



