Sec. 45.11 



FRICTION-RESISTANCE CALCULATIONS 



105 



1.0 50 10.0 50.0 100 500 1000 



Reynolds Number in million5 



Fig. 45.F Variation of Laminar Sublayer Thick- 

 ness Si WITH X-DlSTANCB (L) AND SPEED 



ically smooth surface, given by the exiDression 

 described at greater length in Sec. 45.15 [Gold- 

 stein, S., ARC, R and M, 1763, Jul 1936, p. 113], 

 namely 



fc.v < 5 



Ur 



(45.iii) 



indicates that the roughnesses should average 

 somewhat less than half the laminar-sublayer 

 thickness S^ , in the ratio of say 5 to 12.6, if 

 they are to produce no roughness effects. 



Granting that the relationship between the 

 height of the roughness on a solid surface and the 

 laminar-sublayer thickness has an important 

 influence on the roughness effects, it is useful 

 to know what factors influence this thickness 6i . 

 To understand this, Eq. (5.vi) may be written 



6i = 12.6Kp)°'(r„)- 



(45. iv) 



From Eq. (S.iii) of Fig. 5.R or the corresponding 

 equation from the turbulent-flow column in Fig. 

 45. A, one may also write 



To = 0.059 I UlRZ 



= 0.059 I Ul 



U„x 



(5.iii) 



Combining these two equations, there is obtained 

 (^0.059^ ~°' 



8l = 12.6Kp) 



/ \-0.57-7— 1 t-tO.I 0.1 

 (P) U„ U„ X V 



= 12.6(0.0295)-°- V"f/;"'(a;)''-' 



Dropping out the numerical values for the 

 moment, and assuming that the kinematic 

 viscosity v is constant for this study, 



Ul 



(45 .v) 



This means that Sl increases very slowly as x 

 increases and that it decreases rapidly as U«, 

 increases. For example, if C/„ increases from 10 

 kt to 20 kt and then to 30 kt, 6x, decreases very 

 nearly to | and then to | of its 10-kt value. If x 

 increases from 5 ft to 15 ft, the increase in 8l 

 occasioned by it is only in the ratio of 1 to (3)°' 

 or 1.116. 



At the same time, an increase in either U„ or 

 x to three times its original value multiplies the 

 original Reynolds number by three. It can not 

 be said, therefore, that the change in 5i, , and 

 hence the change in the effect of a given roughness 

 of a certain absolute size and shape, is a function 

 solely of the Reynolds number of the flow, as 

 determined by the product of the relative velocity 

 [/„ and the distance x from the leading edge 

 (neglecting the kinematic viscosity). 



A given barnacle at the bow of a long ship 

 might project through the thin laminar sublayer 

 there but lie just within this layer at the stern. 

 However, as the ship speed increases, the value 

 of 5i diminishes. A barnacle of the same size at 

 the stern projects more and more through the 

 laminar sublayer and becomes more and more 

 effective as an item of roughness. This is the 

 reason why, in aeronautical circles, it is customary 

 to use a special Revnolds number with the rough- 

 ness height as a space dimension, instead of the 

 distance x from the nose or leading edge of the 

 body or plate. In fact, if the Goldstein relationship 

 mentioned earlier in this section is rearranged in 

 the manner 



5 > 



fcAv?7. 



(45 .vi) 



it forms a Reynolds number of this kind, where 

 /cav is the space dimension. 



This relationship is discussed further in Sec. 

 45.15. 



45. 11 Friction Data for Water Flow in Internal 

 Passages. This is not the place for an extended 

 discussion of the friction resistance offered by the 

 internal surfaces, both smooth and rough, of 

 water ducts and passages which are separate 

 from those in the maciiinery plant. Some reference 

 data on this subject are given by J. K. SaUsbury 

 [ME, 1944, Vol. II, Fig. 51 on p. 61 and Art. 12 

 on pp. 62-63], but it must be noted that the 

 weight density (lb per ft^) of the liquid is repre- 



