B,17 • LOGARITHMIC FORMULAS 



course to a physical model, the authors feel that this deduction must be 

 ranked among the major contributions to the subject. A simple way of 

 arriving at this result is to reexamine Eq. 16-1 and 16-3, written in the 

 following forms : 





= f 



U f/e Jy 



m 



Ur Ur ^ V5 



(17-1) 

 (17-2) 



Since these are two expressions for the same quantity, and since a multi- 

 plying factor inside a function must have the same effect as an additive 

 factor outside a function, the functions / and g must be logarithms. 

 The two formulas are usually written in the form 



U 2.3, fUry\ , ,,^o^ 



tt; = X i°s (^j + -^^ (17-3) 



U, - U 2.3, M ,,, ,. 



where K, Ci, and C2 are experimentally determined constants. It follows 

 from Eq. 17-1 and 17-2, when / and g are expressed as logarithms, that 

 K must be common to both Eq. 17-3 and 17-4. The constant K is uni- 

 versal and the logarithmic form of the functions do fit the observations, 

 but only over a limited range of the variables. More specifically they 

 have the logarithmic form where they overlap, but not necessarily much 

 beyond this region. This may be taken as evidence that the empirically 

 established overlap is not a basic condition and therefore not a sufficiently 

 strong one to impel a long range validity for the laws deduced from it. 

 The extent to which these laws fit the data and are influenced by various 

 conditions will be taken up in Art. 19, 20, and 21. 



For the present we direct our attention to Eq. 17-4 in order to call 

 attention to the fact that the constant C2 is found to be the same for 

 pipes and channels, but that it has a different value for boundary layers 

 of flat plates. This is shown in Fig. B,17a in which pipe data have been 

 omitted. The data are taken from [67,74,75,76,77]. The Reynolds number 

 Res is in all cases U^b/v. The constant 5.75, corresponding to i^ = 0.40, 

 is common to both, provided the curves are fitted near the wall. It is 

 seen that the log law does not fit well for the full range of y/b. This 

 means only that the logarithmic form of the defect law is at fault, not 

 the functional form of the law itself. More significant is the fact that the 

 function g in Eq. 16-3 is different in boundary layers from that in chan- 

 nels. This difference is evidently due mostly to a sensitivity to conditions 

 at the outer limit y = b rather than to the presence of a small falling 



< 125) 



