B • TURBULENT FLOW 



shear stress at the wall r^. This means, of course, that the effect of a 

 pressure gradient is neglected, and we must limit ourselves to cases where 

 the pressure is constant or changing so slowly in the stream direction 

 that its effect is minor compared to the effect of r^. The coefficients in- 

 volved are accordingly: 



The local friction coefficient 1-772 ~ ^f (15-1) 



The friction velocity -vp = ^r (15-2) 



Assuming that the local friction coefficient depends on the Reynolds 

 number and may be expressed in powers of the Reynolds number, the 

 relation may be written 



const 

 ^/ = /.r .\m (15-3) 



("0" 



where 11^8/ v is a Reynolds number based on the maximum velocity and 

 the distance from the wall to the point of maximum velocity. It follows 

 from Eq. 15-3 and the definitions (Eq. 15-1 and 15-2) that 



^ = const (^J^ (15-4) 



where Ur8/v is a Reynolds number based on the friction velocity and 8. 

 Eq. 15-3 and 15-4 are both expressions for conditions near the wall. How- 

 ever, it may be argued that a formula similar to Eq. 15-4 may be used 

 to express the velocity at any distance from the wall without appreciable 

 error because the main increase in velocity takes place near the wall. 

 Assuming this, the velocity distribution is written 



—- = const (-^)^ (15-5) 



where U is the mean velocity at the distance y from the wall. 



We assume now that all mean velocity profiles are similar, and accord- 

 ingly that U/Ue is a function of y/d. While this assumption is exactly true 

 for laminar flow, it is only an approximation for turbulent flow. The 

 appropriate power-law form of the function is indicated by Eq. 15-5 and 

 is written 



y,-,., (15-6) 



By taking to = ^ it is found that Eq. 15-3 expresses the variation of 

 friction coefficients in pipes over the range 3000 < JJ^b/v < 70,000. With 



< 120 ) 



