B,16 • WALL LAW AND VELOCITY-DEFECT LAW 



in terms of the characteristic friction velocity Ur and the characteristic 

 length v/Ur. The functional equation (Eq. 16-1) is the law of the wall. 

 In the laminar sublayer it takes the special form 



which arises from the circumstance that the sublayer is so thin that r 

 therein is constant and equal to r^. In Eq. 16-2 the density included in 

 the terms automatically cancels out. 



The range of y over which Eq. 16-1 is valid must be established by 

 experiment. It might be supposed that the range would be severely Umited 

 by pressure gradient effects when these are present, since, as we have seen, 

 the pressure acting across an area of unit width and height y has been 

 neglected. Recent data, to be discussed in Art. 19, show that there re- 

 mains a considerable range over which the law is valid for both rising 

 and falling pressures and that the law is not so restricted as to be useless 

 until conditions of near-separated flow are reached. Thus there is a range, 

 even though possibly short, beyond the laminar sublayer, where the func- 

 tional relation (Eq. 16-1) is universally of the same form. This is true 

 only when there is a laminar sublayer, and therefore true only when the 

 wall is aerodynamically smooth. 



The argument leading to the velocity-defect law is that the reduction 

 in velocity {U^ — U) at distance y is the result of a tangential stress at 

 the wall, independent of how this stress arises but dependent on the dis- 

 tance 5 to which the effect has diffused from the wall. We may then write 



U.-U = G{Ur, y, 8} 



and in terms of dimensionless ratios 



This is the velocity-defect law. 



The law (Eq. 16-3), unlike Eq. 16-1, holds true for rough as well as 

 smooth walls, provided the roughness elements are not so large that y 

 becomes indeterminate. Data for boundary layers with constant pressure 

 are found to fall on a single curve within the precision fixed by the 

 experimentalscatter.-; Thisiis shown by Fig. B,16 which presents various 

 data collected by Clauser [7S] for different Reynolds numbers and for 

 smooth and rough walls. Aside from the fact that the law cannot apply 

 in the vicinity of the laminar sublayer nor at distances comparable to 

 the height of roughness elements, it appears to exhibit a universality for 

 constant pressure boundary layers. Clauser has shown, however, by a 

 formal argument that the law is fundamentally not universal when Ur 

 varies from one set of data to another, but that the dispersion will gener- 



< 123 ) 



