F,ll • HEAT TRANSFER 



to the wall value in both Eq. 11-20 (with I = Ky) and 11-27, i.e. through- 

 out the entire analysis. One obtains, then, for the heat transfer case, 



0.242 



c%T^/T.)^ 



= 0.41 -|- log (Re ■ Cf) — n log 



(11-32) 



which differs from Eq. 11-31 by the factor log (T^/Te). Von Karman also 

 originally derived his equation for the insulated-plate case, because he 

 (like Cope later) desired to find the effect of speed, only, upon drag. 



1.0 



0.8 



Cf 



Cf; 



0.6 



0.4 



0.2 



10 



M. 



Fig. F,lli. Effect of heat transfer and Mach number on local skin friction 

 coefficient according to Eq. 11-29 for a Reynolds number of 10^. 



The general question that must now be answered is: Which of the 

 above formulas is the most valid for engineering purposes? Although the 

 question can best be answered by experimental data, a preliminary check 

 on the form of the equations can be made upon observation of the effect 

 of heat transfer (T^/Te) on the local skin friction coefficient. Eq. 11-28, 

 11-29, 11-31, and 11-32 are plotted in Fig. F,llh, F,lli, F,llj, and F,llk 

 where the ratio of the compressible to the incompressible flow coefficient 

 for one Reynolds number and n = 0.76 is shown as a function of Mach 

 number for various wall-to-free stream temperature ratios. It is immedi- 

 ately seen from Fig. F,llh and F,lli that, regardless of mixing-length 

 theory assumed, Eq. 11-28 and 11-29 yield friction coefficients which are 

 definite functions of Mach number for a constant wall temperature. On 



( 385 ) 



