B ■ TURBULEXT FLOW 



B.29. Transport Proce??e? in Free Turbulent F1oa\-. In order to 

 solve tiie equations of moLion and heat iransfer given in Art. 27 and thus 

 obtain velocity and temperature distributions in i/ or r, it is necessary to 

 expre^ the quantities on tiie right-hand side of the equations in terms 

 that can be related to the derivatives of velocity and temperature with 

 respect to y or r. The auxiliary expressions for this purpose have been 

 discu^ed in Art. 10. Specifically. Eq. 10-12 or 10-13 are used with the 

 coefl&cients D^. Di. e^^. or ej specified either by general conditions of the 

 problem or expressed in terms of local conditions. 



The former usually takes the form of an assumption that the coef- 

 ficients are constant over a given cross section of the flow but vary from 

 one section to the next. In recent years the following expression proposed 

 by Prandtl [120] has been extensively used: 



D. or D, = K^r^, - r^l (29-1) 



where h is the width of the region at a given cross section, V ^^^ and 

 tJaia are the extremes of mean velocity across the section, and K is an 

 experimentally determined constant of proportionality whose value de- 

 pends on the quantity D« or D-,,. 



Specification of transi)ort in terms of local conditions takes the form 

 of mixing length theory. This theory has already been discussed in Art. 12. 

 Its apphcation to free turbulent flows has been so widely discussed in the 

 literature, for example [6,111], that only a few remarks are called for here. 

 Much of the discussion has had to do with the relative merits of mo- 

 mentum transfer theory on the one hand and Taylor's vorticity transfer 

 theory on the other. Yorticity transfer theory is generally favored on the 

 grounds that it is consistent with a wider distribution of temperature 

 than of velocity, but which of the two theories agrees the better with 

 observed velocity distributions depends on cases. 



We shall here concern ourselves with the broader question regarding 

 the foundation of the foregoing procedures rather than with the details 

 of their apphcation. The basis for judgment rests largely on the work of 

 Townsend with the plane wake and that of Corrsin with the round jet. 

 As mentioned in Art. 28. there is evidence that large eddies operate in 

 free turbulent flows to contort the whole flow field and thus transport 

 fluid with smaller scales of turbulence over much of the width occupied 

 by the flow. The next idea to be introduced is that mixing of aU proper- 

 ties by large and small scale motions has gone on for a considerable time 

 over the previous course of the flow. In this connection it is ad^*isable to 

 restrict the discussion to jets and wakes, for in these cases aU of the 

 properties in question have been put in at the beginning and through 

 mixing have covered much of the cross section during their previous his- 

 tory. Eddies of any scale significant in diffusion will have existed for a 

 considerable time, and their size and intensity found at a particular lo- 



( 168 )' 



