B • TURBULENT FLOW 



The empirical laws of the skin friction coefficient for a compressible 

 fluid start from incompressible laws, and the compressibility effects are 

 incorporated by comparison with experiments. The power law 



c/. = ARe-- (13-5) 



is an example. Here Cf.^ is the skin friction coefficient for the incompressi- 

 ble boundary layer, Re is defined by Eq. 13-4, A and n are numbers 

 {A = 0.0262, n = Y, according to Falkner [61]). In order to estimate the 

 compressible skin friction coefficient (for example C/J, we assume that a 

 reference temperature Tr can be found so that the compressible skin fric- 

 tion coefficient C/^, defined by putting pr = p{Tr) into Eq. 13-1, satisfies 

 the incompressible formula (Eq. 13-5). Then we can write 



Cfr = C/XRe.) 



= Cf{Reef!lPi\ 



\ Mr Pe/ 



= "/.[««• (I;) """] (13-6) 



with pr/pe = Te/Tr aud jUe/Mr = (Te/Tr)". Slucc C/j foUows thc power law 

 (Eq. 13-5), Eq. 13-6 can be rewritten in the following form: 



Further, C/^ can be expressed in terms of c/^ by means of the definitions 

 (Eq. 13-1) which can be rewritten as follows: 



C/. = C/e B^ = Cf^ 



Pr 



SO that Eq. 13-6 becomes 



(sr 



®" 



CfSRee) /7^^\i-(i+")" 



Cf-iRSe) 



(13-7a) 



The right-hand side of Eq. 13-7a gives the effect of compressibility (or 

 Me). In a compressible boundary layer T varies between Te and T^. It 

 can be assumed that the compressibility effect is covered on the average, 

 if the average temperature 



Tr = UTe + T^) (13-8) 



is taken as the reference temperature. Then Te/2Tr or T^/iTe + T^) can 



be computed in terms of M^ on the basis of Eq. 9-9, so that finally Eq. 



13-7a becomes 



Cf(Ree) 

 ,p , = ^i-(i+«)" (13-7b) 



{ 114 > 



