B,10 • TRANSPORT OF PROPERTIES IN A TURBULENT FLUID 



(Eq. 10-11), which imphes Z)„ = Dn, predicts a turbulent Prandtl num- 

 ber of unity. However, experiments show that Pvi is about 0.7, a value 

 very close, incidently, to the laminar Prandtl number for air. 



The fact that the turbulent Prandtl number, as given by the ratio 

 Du/Dn, is different from unity is interesting and indicates that Eq. 

 10-lOb, rather than the mixing length formula (Eq. 10-11), should be 

 more correct. However, due to the simplification introduced in the trans- 

 port equation (Eq. 10-7), the parameter k is not determined in terms of 

 the transferable property, so that the numerical value of the ratio of the 

 two exchange coefficients cannot be computed from Eq. 10-lOb alone. 

 An auxiliary equation is needed to determine the transfer of property, 

 for example, heat or particles, under the action of a turbulent fluid. In 

 the case of the transfer of particles, such an equation may govern the 

 motion of a small spherical particle suspended in a turbulent fluid. On 

 the basis of it, the velocity correlation for the particles can be computed 

 in terms of the velocity correlation of the ambient fluid or vice versa, 

 and hence the ratio of the two exchange coefficients can be obtained. 

 This has been done by Tchen [25], and, for the case of k = 0, it has been 

 found that the exchange coefficient of particles is equal to that of the 

 fluid (Eq. 10-11). This case is not surprising, since consistently the relax- 

 ation is neither involved in the motion of the fluid nor in the motion of 

 the particles, and no difference in exchange coefficients should exist, as 

 already revealed by the simple theory of Eq. 10-lOb. The ratio of the 

 two exchange coefficients for the case of k 5^ has not yet been studied 

 on this basis. Several authors are concerned with such difficulties of dif- 

 fusion phenomena, see e.g. [25] and the Burgers lecture on the turbulent 

 fluid motion [26]. 



In the integrand of Eq. 10-lOb, the exponential term can be con- 

 sidered as a retarding effect of the relaxation between the equilibrium 

 and nonequilibrium distribution in the transport phenomena (Eq. 10-8). 

 Hence the complete integrand of Eq. 10-lOb can be considered as a corre- 

 lation corrected for the relaxation by means of the exponential factor. 

 In the diffusion problem based on the model of a random walk, such an 

 effect has been considered by Tchen [27] in the form of a more general 

 memory, which could be either negative or positive, so that the corrected 

 correlation will contain a factor respectively smaller or larger than unity. 



Before leaving the discussion, it is important to remark that the dif- 

 fusion phenomena, described by the above transport phenomena, are only 

 valid for irregular movements of small scales, since we have used in Eq. 

 10-9b a series expansion in terms of a length and some gradient of the 

 transferable property. Such a diffusion can be called diffusion of the 

 gradient type. On the other hand, when the irregular movements are of 

 coarse scales, the bulk property rather than its local gradient must be 

 the governing factor. The latter diffusion can be called diffusion of the 



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