B,21 • LAW OF THE WAKE ACCORDING TO COLES 



The reader is referred to Clauser's paper [83] for a number of signifi- 

 cant facts brought to light in his investigation. One of these concerned 

 the downstream instability of a turbulent boundary layer with a large 

 adverse pressure gradient. When the pressure gradient was small, no diffi- 

 culty was experienced in adjusting the pressure distribution to obtain a 

 desired equihbrium profile; but when it was large, great difficulty was 

 experienced. He attributes the condition for large pressure gradients to a 

 downstream instabiUty, meaning that a change, say in the local gradient 

 or in 6, made at one point would produce further changes downstream 

 as the layer developed, rather than become damped out. This is an insta- 

 bility in X, not in time. 



B,21. Law of the Wake According to Coles. In the short space of 

 this article it is impossible to cover adequately the careful and extensive 

 study which led Coles [85] to propose the law of the wake as an extension 

 to the law of the wall. After having examined practically all available 

 experimental data on turbulent boundary layers in terms of the loga- 

 rithmic form of the law of the wall, expressed by Eq. 19-2, and noting 

 the universal agreement with the law near the wall and the characteristic 

 departure from it away from the wall, he concluded that the flow had a 

 wakelike character, modified in various degrees by wall constraints. He 

 concluded further that the wakelike form could be reduced to a second 

 universal similarity law which he called the "law of the wake." A Unear 

 combination with the law of the wall was then proposed as an over-all 

 similarity law representing the complete proffie for equilibrium and non- 

 equilibrium flows alike. 



Attempts to generalize the law of the wall and the defect law so as to 

 fit experimental results are not new. Millikan [73], for example, proposed 

 forms to fit the distribution in pipes and channels. Others have expressed 

 and employed ideas bearing certain similarities to the present one, those 

 known being Lees and Crocco [91], Ross and Robertson [93], and Rotta 

 [93], Coles, however, appears to have been the first to show evidence of a 

 universal wake law and to give it a rational physical explanation. 



In general form the mean velocity profile in turbulent shear flow may 

 be expressed as 



■^=/(-^)+/^(^,2/) (21-1) 



For equilibrium flows it is found experimentally that Eq. 21-1 may be 

 written 



Ur 



where tt is a parameter which is independent of x and y for a specific 



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