F • CONVECTIVE HEAT TRANSFER IN GASES 



layer. However, the situation is complicated by the presence of the eddy 

 viscosity e^, which is a function of the eddying process itself, rather than 

 a function of temperature like the molecular viscosity n. As a result, the 

 simplest procedure to calculate the wall shear is to use the von Kdrmdn 

 integral method based upon a velocity profile obtained from the Prandtl 

 mixing-length theory. 



Application of the Prandtl mixing-length theory to compressible flow 

 leads to [21] 



(du\ 

 dy) 



r = PV (^^J (11-20) 



in which I is the mixing length analogous to the mean free path in kinetic 

 theory. Near the wall, one may assume that I = Ky where i^ is a con- 

 stant equal to about 0.40; however, away from the wall, one might assume 

 the von Kd,rman similarity law, viz. I = —K{du/dy)/{d}u/dy'^). For in- 

 compressible flow near the wall, both mixing-length assumptions lead to 

 the semilogarithmic velocity profile. For compressible flow, on the other 

 hand, the use of the different assumptions leads to sHghtly different 

 results. The shear stress t is usually taken as constant and equal to the 

 wall value Tw 



In order to account for the variation of the density of the fluid, it is 

 first remembered that, owing to the thinness of the boundary layer, the 

 pressure is constant across the layer. Therefore, from the perfect gas law, 

 when p'T' is neglected, 



^ = ^ (11-21) 



Pw y 



Next it is assumed that Pr — \ and Cp is constant. Eq. 10-11 then yields 

 or, upon rearrangement. 



7^ 



^ + [(i H- ^ ^') ^ - 1] S - '-^ ^' k ©' (11-^^) 



Hence, upon substitution of Eq. 11-23 in Eq. 11-21, the density relation 

 becomes 



P 1 



where 



Pw 1 +5(W/-Me)2 - A(M/We)2 



^^^ Ml 1 + ^^ Ml 



(11-24) 



< 382 ) 



