B • TURBULENT FLOW 



With regard to the theories in question, three main facts stand out: 

 (1) only the smaller eddies of this double-structure picture can take part 

 in the gradient diffusion on which the theories are based, (2) the smaller 

 eddies are mixed to a state of near uniformity, and (3) the scale and 

 intensity of all eddies responsible for transfer are determined by general 

 conditions rather than local conditions. 



Fact 1 means that we cannot predict to what extent the theories will 

 apply. Fact 2 means that we can make a good case for Eq. 29-1 for that 

 part of the transport which is of the gradient type. Fact 3 means that we 

 must be skeptical of the kind of local dependence on which mixing length 

 theory rests. This refers specifically to Eq. 12-2b and 12-13 of Art. 12 

 which expresses v and I in terms of local mean flow parameters. Some 

 lessening of local dependence is achieved when I is taken to be constant 

 over a section of the flow and proportional to the width. This is commonly 

 done in free turbulent flows. We see that even with this compromise, 

 mixing length theory is scarcely tenable in free turbulent flows. 



Turning to comparisons with measured distributions, we find that 

 mixing length theory cannot be shown to be definitely wrong, although 

 the agreement with observations is rather casual, with vorticity transfer 

 turning out to be better in some cases and momentum transfer being 

 better in others. The vorticity transfer version of the theory when com- 

 bined with the heat transfer version does at least yield a broader tem- 

 perature distribution than velocity distribution [121]. 



Hinze and van der Hegge Zijnen [122] conducted an exhaustive series 

 of experiments in which they measured distributions of velocity, tem- 

 perature, and concentration of small amounts of added gas in a round 

 air jet. After comparing their results with mixing length theories they 

 concluded that these theories were unsatisfactory, and so set out to ex- 

 plore the possibilities of constant turbulent exchange coefficient. From 

 their measured velocity distributions and the equations of motion and 

 continuity, Z)„ was determined as a function of radius and axial distance. 

 It was found to remain nearly constant with increasing r from the center 

 outward, and then to decrease in the intermittent zone. They concluded, 

 however, that a constant Du was a sufficiently good assumption to 

 justify the adoption of the well-known laminar solution. The resulting 

 velocity distribution formula and the expression for Du are given in Art. 

 30. 



Hinze and van der Hegge Zijnen found that temperature and concen- 

 tration profiles indicated practically identical exchange coefficients. We 

 shall denote these by the common symbol Dh and refer to the ratio 

 Du/Dh = Pvt as the turbulent Prandtl number. (This ratio is known as 

 the Schmidt number when referring to matter in place of temperature.) 

 The value of Pft on the axis of the jet was found to be 0.685. How- 

 ever, Prt increased steadily with r and became greater than unity for 



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