F,ll • HEAT TRANSFER 



laminar Prandtl number, turbulent Prandtl number, and Reynolds num- 

 ber can be derived. 



The general expression for recovery factor is [22] 



r = 2R{1) =2 Pr„ exp 



/: 



dr 

 (1 - Pr J - 



(0) T . 



exp 



' (1-PrJ^ 



(0) T . 



du^ \ du^ (11-4) 



with variable Prandtl number. However, before the integral can be evalu- 

 ated, it is necessary to have information on the shear distribution across 

 the layer, except for the case of Pr = 1 when r = 1. Now for turbulent 

 boundary layers, there is no analytical expression for r as there is in the 

 case of laminar boundary layers. Therefore, with turbulent flow, it is 

 necessary to assume a shear distribution which conforms with experi- 

 ment. In the first place, owing to the experimental fact that the sublayer 

 and transition regions are very small compared to the turbulent part of 

 the boundary layer, it is certainly sufficient to assume that the total 

 shear is constant across those subregions. Hence, Eq. 11-4 becomes, for 

 averaged (constant) Pria^ and Pn, 



r = 2Priam / "* ^"^ u^du^ + 2 / ** Pr^u^du^ 



Jo yu^lam 



+ 2Pn [' r^^^-'du^ r*\i-p^du^ + 2Pn r T^t-^ r* ti-p"^^ du^du^ 



(11-5) 



where w^um indicates the outer edge of the laminar sublayer, %t the inner 

 edge of the turbulent zone, and r^ = t/t{0). 



Experimentally [25], the shear distribution may be well approximated 

 by a straight line over the turbulent region of the boundary layer, whence 



r, = 1 - I (11-6) 



Therefore t^('u^.) is represented by the ordinate above the y^{= yu^/h) 

 profile. Typical velocity and shear profiles for a turbulent boundary layer 

 are plotted in Fig. F,lla. The Blasius velocity and shear distribution for 

 laminar flow are also plotted in the figure for comparison. It is seen that 

 the turbulent shear t^ as a function of relative velocity u^ is much fuller 

 than the laminar shear, and therefore tends to justify the assumption 

 usually made in turbulent flow analysis that the shear is constant across 

 the layer. Also, the assumption that the turbulence Prandtl number 

 Prt is near unity tends further to diminish the shear effect. If the semi- 

 logarithmic law for velocity is used, then the shear distribution as a 



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