B,23 • EFFECT OF ROUGHNESS 



law (Eq. 23-4) for the fully rough condition. This means that the linear 

 portion of the velocity distribution curve for a rough wall parallels that 

 for a smooth wall but is stepped down by an amount AC7/(7t. With experi- 

 mentally determined values of bJJ IVr, the velocity distribution for a 

 fully rough wall may be expressed by the aid of the smooth wall formula. 

 For this we use Eq. 19-1 containing the constants given by Clauser. The 

 rough wall formula is then 



= 5.6 log (^) - (^) + 4.9 (23-5) 



_f7 



A skin friction formula results at once by subtracting Eq. 23-5 from 

 the logarithmic form of the velocity-defect law. Clauser [55] has obtained 

 a universal law applicable to equihbrium flows including the effect of the 

 pressure gradient by noting, on the basis of Fig. B,20b, that a pressure 

 gradient also has the effect of producing a step-down in the velocity, 

 bXIilVj. Accordingly he writes the generalized defect law for equilibrium 

 flows 



ILrJ^.5.61o.(|)-(f-^) + 0.6 (23-6) 



Since Eq. 23-5 is unaffected by the pressure gradient, and Eq. 23-6 takes 

 the effect of the pressure gradient into account, a universal skin friction 

 law results by subtraction of Eq. 23-6 from Eq. 23-5. The end result may 

 be written 



I = 5.6 log Re,* -^\ne^^[-l]J^^ (G) + 4.3 (23-7) 



where \/cj2 = Ur/U., d* = \/cj2 A, Reu = U^k/v, Re,* = Ue8*/p, and 

 iAU/Ur){Rek \/cf/2) and {AU2/Ur)(G) denote functions of the argu- 

 ments. The integral shape parameter G is defined in Art. 20. 



In order to put Eq. 23-7 into a more convenient form for engineering 

 applications, Clauser proposes the introduction of two auxiliary factors 



which permit Eq. 23-7 to be written 



^~ = 5.6 log (^Re,* p^ + 4.3 (23-8) 



Factors Fi and F2 have been determined by Clauser using Prandtl- 

 Schlichting data for sand-grain roughness for the calculation of Fi and 

 his own data for equilibrium profiles for the calculation of F2. These are 

 presented in Fig. B,23b and B,23c. A plot of Eq. 23-8 for Fi and F2 equal 

 to unity is given in Fig. B,23d. If a fictitious Reynolds number, ReCFi/Fy, 

 is first obtained, C/ may be found from this figure. Since values of F2 are 

 based on only two equihbrium pressure distributions, more data are to 



< 151 ) 



