E,4 • ANALYSIS FOR CONSTANT FLUID PROPERTIES 



Fig. E,4a shows, on semilogarithmic coordinates, velocity profile data 

 from [20,22], together with the analytical curves obtained. The data are 

 for fully developed adiabatic flow in tubes. The constants k and n in Eq. 

 3-1 and 3-2 were found, from the data, to have values of 0.36 and 0.124 

 respectively. The relation for €„ from Eq. 3-2 applies for ^* < 26, and 

 that from Eq. 3-1, for ?/* > 26. In matching the two solutions it was 

 assumed that the velocity is continuous at the junction of the two regions. 



The velocity distribution is often divided into three regions rather 

 than two: the laminar layer, where turbulence is supposed to be absent, 

 the buffer layer, and the turbulent core [6]. The use of Eq. 3-2' for €„ close 



30 



25 





20 



15 



10 



1 10 100 1000 10,000 



*_Vtw/p 



Fig. E,4a. Generalized velocity distribution for adiabatic turbulent flow 

 (vertical line is dividing line between equations at y-^ = 26). 



to the wall eliminates the need for a laminar layer and reduces the num- 

 ber of regions to two. For values of ?/* < 5, Eq. 3-2 indicates that e„/ 

 (/i/p) <$C 1, so that the flow is nearly laminar in that region. This can be 

 seen from the plot of m* against ?/* in Fig. E,4a, where, for small values of 

 ?/*, M* = y*. 



Several other analyses were made recently and might be mentioned 

 at this point. Van Driest [23] used a modified form of Prandtl's mixing- 

 length theory which assumed that the presence of the wall reduces the 

 universal constant k or the mixing length. By introducing a damping 

 factor into the expression for shear stress, he was able to obtain a velocity 

 profile w^hich agreed with the data for the regions both close to and away 

 from the wall. 



< 293 ) 



