B • TURBULENT FLOW 

 Again taking B = 75/72 and using Eq. 15-9, 



(^1 



Cf = 0.074 1^1 (15-11) 



where Cf is the mean friction coefficient from the leading edge to the 

 point X. Eq. 15-11 checks the tests on smooth plates for Uex/v up to 

 about 3 X 10«. 



Power formulas should be regarded as interpolation formulas, useful 

 over a limited range of Reynolds number. For Ue8/v over 100,000, Eq. 

 15-6a agrees better with measurements when the exponent y is replaced 

 by I", and even -g- when the Reynolds number is sufficiently high. Skin 

 friction formulas may likewise be improved for agreement with measure- 

 ment over a greater range of Reynolds number by adjusting the exponent. 

 For example, as we have seen in Art. 13, Falkner [61] uses an exponent of 

 — Y instead of — ■§■ and gives 



Cf = 0.0262 (^) ' (15-12) 



Cf = 0.0306 (J~] " (15-13) 



It must be remembered that the foregoing considerations apply only 

 to smooth walls. Except for Art. 23, where the effect of roughness is con- 

 sidered, and elsewhere where roughness is mentioned, the smooth-wall 

 condition is implied throughout this chapter. 



B,16. Wall Law and Velocity-Defect Law. Two laws that have 

 gone far toward giving order and meaning to the seemingly confusing 

 and confficting data on flows bounded or partially bounded by walls are 

 the "law of the wall" attributed to Prandtl (for example [70]) and the 

 ''velocity-defect law" introduced by von Karman [71]. The first pertains 

 to the region close to the wall where the effect of viscosity is directly felt 

 and the second pertains to the bulk of the shear layer where viscous forces 

 become negligible. 



The law of the wall is based on the logical premise that the tangential 

 stress at the wall r^ must depend on the velocity U at the distance y 

 from the wall and on the viscosity /x and density p. Assuming that the 

 stress at the wall is the only constraint on the flow, we may write 



F{t^, U, y, tJL,p)=0 



This may be expressed in dimensionless form by 



< 122 ) 



