E • CONVECTIVE HEAT TRANSFER AND FRICTION 



terms of the series result in 



-^ = n'y*^ (3-8) 



M/P 



In connection with this equation it is of interest that Reichardt [3] as- 

 sumed €„ proportional to y^ for moderate Prandtl numbers and to y^ for 

 higher Prandtl numbers. Also, an analysis by Hama [4.I] assumed an ex- 

 pression for e„/(M/p) similar to Eq. 3-8, except that the right-hand side 

 was multiplied by du*/dy*. For small values of y*, where du*/dy* = 1, 

 his expression therefore reduces to Eq. 3-8. Eq. 3-8 is also consistent with 

 exact information obtainable from the continuity relation and the con- 

 dition that the fluctuating velocity components are zero at the wall. It is 

 shown by Elrod [42] that u'v' at the wall cannot be proportional to less 

 than the fourth power of y. Inasmuch as du/dy approaches a constant 

 at the wall, the same result holds for e. 



Substituting Eq. 3-8 in 2-6 and integrating gives [15]: 



St = ?^^ (3-9) 



where the friction factor / can be calculated from 



and n has the value 0.124, as determined in Fig. E,4a. Eq. 3-9 is indi- 

 cated by the dotted line in Fig. E,4d and is seen to be in good agreement 

 with the predicted hne obtained previously for Prandtl numbers greater 

 than 200. 



A comparison of analyses by various investigators is given in Fig. 

 E,4e. It can be seen that they all more or less converge at the lower 

 Prandtl numbers. At the high Prandtl numbers, the analysis described 

 herein [15] and the analyses from [7,12] are in fair agreement, whereas 

 those from [6,11] diverge considerably. The present analysis and that 

 from [12] represent the experimental data about equally well. The analy- 

 sis in [12] modifies that of von Karman, which utilized a laminar layer, 

 a buffer layer, and a turbulent core. A small amount of turbulence was 

 introduced into the laminar layer in order to give better agreement with 

 experiment at high Prandtl and Schmidt numbers. 



The entrance region. Most of the results given so far are for fully 

 developed flow or heat transfer, that is, for the region at a large distance 

 from an entrance, or from the point where heat transfer begins. In the 

 region close to an entrance, the heat transfer coefficients and shear stresses 

 are higher than those for fully developed flow, because of the thin bound- 

 ary layers and consequently severe temperature and velocity gradients 

 at the wall near the entrance. One of the first studies of turbulent heat 



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