B ■ TURBULENT FLOW 



mean quantities in accordance with the relations (Eq. 10-11 and 10-12) 

 as follows: 



r = -pvi^ (12-2a) 



dy 



where, following Prandtl, vl is expressed by 



7l = P^ (12-2b) 



dy 



The mixing length I is now to be expressed in terms of the local mean 

 flow parameters. This can be done [6, Chap. 8] either by means of the 

 Kdrmdn similarity hypothesis 



dU/dy 



or by means of the Prandtl hypothesis 



I = K,y (12-4) 



where ki and k2 are numerical constants. With the Karmdn hypothesis, 

 Eq. 12-2a can be written as follows: 



,_ {dU/dyY 

 ^ = ''' {d^U/dyr ^ ^^ 



In boundary layer theories the equations are usually rendered di- 

 mensionless by introducing a reference velocity 



C/, = . P (12-6a) 



and a reference length 



(12-6b) 



where the subscript w denotes the value at the wall. 



Using the reference velocity and length, as defined by Eq. 12-6, we 

 can write the following dimensionless quantities: 



(12-7a) 



(12-7b) 

 (12-7c) 



and rewrite Eq. 12-5 in the dimensionless form as follows: 



{ 108 ) 



