E,5 ■ ANALYSIS FOR VARIABLE FLUID PROPERTIES 



apply to variable properties; that is, if e^ = €u{du/dy, d^u/dy^) away from 

 the wall and e„ = eu{u, y, ju/p) close to the wall. 



The relations between u* and y* and T* and y* are thus obtained for 

 various values of /3. The result for u* against y* for a Prandtl number of 

 0.73 is shown in Fig. E,5a. The positive value of /3 corresponds to heat 

 addition to the gas, the negative value to heat extraction. Similar curves 

 were obtained for T* against y* [14]- 



For calculating the curves for various values of /3 in Fig. E,5a it was 

 assumed, somewhat arbitrarily, that the value of y*, which is the point 

 of intersection of the curves for the regions close to and away from the 



35 

 30 

 25 



20 



15 



10 



00 

 * 



1000 



10,000 



y 



Fig. E,5a. Predicted velocity distributions for air with 

 variable properties (Prandtl number = 0.73). 



wall, remains constant at y\ = 26. A somewhat more reasonable assump- 

 tion for y\ might be made by including the molecular shear stress and 

 heat transfer terms in the region away from the wall, and using the con- 

 dition that yX occurs at a constant ratio of turbulent to molecular shear 

 stress tu/{iJ./p). That is, the turbulence changes over from that described 

 by Eq. 3-2 to that described by Eq. 3-1 when the ratio of turbulent to 

 molecular shear stress reaches a certain value. This assumption was made 

 in [Uf, Fig. 13], where it was shown to give essentially the same results 

 as the assumption of constant yX- 



With the relations among w*, T*, and y* known, the relations be- 

 tween Nusselt number and Reynolds number can be calculated from Eq. 

 3-4, 3-5, 3-6, and 3-7. For variable properties, Eq. 3-4 and 3-5 give 

 Nusselt and Reynolds numbers with properties evaluated at the wall 

 temperature. The integrands in the numerator and denominator of Eq. 

 3-6 must be multiplied by p/pw when the density is variable. Nusselt and 



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