B,23 • EFFECT OF ROUGHNESS 



The foregoing considerations regarding a constant eddy viscosity are 

 given more for the physical ideas that they embody than for any possible 

 expediency in methods of computation. 



B,23. Effect of Roughness. The treatment of roughness and its 

 effects is rendered difficult and somewhat inexact by the varied geometri- 

 cal forms of roughness and the variety of ways in which it may be dis- 

 tributed. Again we are confronted with a subject that cannot be treated 

 adequately in a short space, and the reader can profit by consulting addi- 

 tional sources of information, such as [95,78,6,96,97]. 



The pattern of roughness studies was set largely by the extensive work 

 of Nikuradse [95] on sand-grain roughness in tubes. Sand-grain roughness 

 has been adopted as a standard in skin friction studies, and is taken to 

 mean roughness elements consisting of grains, either being sand or like 

 grains of sand, of nearly uniform size but generally of irregular shape 

 spread with maximum density on a plain surface. The significant dimen- 

 sions then reduce to one, this one being the mean height of the roughness 

 element, denoted by k. It is customary to express the effect of an arbi- 

 trary type of roughness in terms of an equivalent sand-grain roughness. 

 For example, the effect of a given distribution of rivets of height k^ is re- 

 duced to the effect of equivalent sand roughness of height k. A number of 

 such equivalents are given by Schlichting [96]. 



It has been found that the onset of an effect of sand-grain roughness 

 on skin friction and on the flow near the wall depends on k relative to 

 the thickness of the laminar sublayer. A more precise length, avoiding the 

 arbitrariness of the sublayer thickness, is v/Ur. Using this, the criterion 

 becomes a roughness Reynolds number 



Urk 

 V 



It has been found that below some value of this number roughness has no 

 effect. The surface is then said to be aerodynamically smooth. Above 

 this value an effect sets in, at first as a mixture of smooth-wall and rough- 

 wall behaviors, involving both the roughness and viscous effects. When 

 Vrk/v reaches a sufficiently large value, the behavior is characteristic of 

 the roughness only, becoming independent of viscosity. The final con- 

 dition is termed "fully rough." When the final condition is reached, the 

 laminar sublayer no longer exists since the particles themselves induce 

 turbulent mixing by the flow about them. Broadly speaking, the foregoing 

 is true of all types of roughness but the limits are different for different 

 types. 



We shall shortly return to these limits and the importance of the 

 parameter JJrk/v, but first we turn our attention to the fully rough con- 

 dition where viscosity no longer enters explicitly into the picture. Here 



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