B • TURBULENT FLOW 



served temperature and velocity distributions when I is assumed constant 

 over the cross section. Momentum transfer theory and vorticity transfer 

 theory give the same results for velocity distribution, but vorticity trans- 

 fer must be used in connection with the heat transfer equation to get the 

 proper result for temperature distribution. The results are 







r - Te ^ ^ 



T„- r, 



where TJe and T^ are respectively the velocity and temperature of the free 

 stream, U^ and T^ are respectively the velocity and temperature at the 

 center, and y^ is the extreme limit in each case, y^ being the greater for 

 temperature distribution. 



We may conclude this discussion by noting that recent findings have 

 given us a clearer physical picture but little by way of a fundamental 

 theory. It has not been possible to clarify the question as to why turbu- 

 lent motions act differently toward heat and matter than toward mo- 

 mentum. Some discussion of this question is given by Townsend; and 

 since this cannot readily be taken out of context, the reader is referred to 

 [1, pp. 164, 165]. 



B,30. Velocity Distribution Formulas for Jets and Wakes. The 



advantage of a constant exchange coefficient is not so much in any marked 

 improvement in accuracy over mixing length theory, but rather that it 

 permits the adoption of laminar-type solutions. When similarity exists, 

 the form of the dependence of the exchange coefficient on x is known, 

 but the absolute magnitude must be found from experiment. The purpose 

 here is to give examples of final results based on this method. For the 

 purpose of comparison a mixing length formula will be shown for one case. 

 It is assumed that mixing length theory and the resulting formulas have 

 been given sufficient attention in other literature, notably in [6,111]. 



A comparison of formulas for the plane wake, made by Townsend 

 [126], is shown in Fig. B,30. Compared with an observed velocity dis- 

 tribution curve are 



1. Mixing length theory, I constant over the width: 



/i = 1.835 1 - 

 2. Constant exchange coefficient: 

 /i = 1.835 exp 



< 172 > 



(Ay 



\0.48/ 

 VO.253/ 



(30-1) 



(30-2) 



