Sec. 45.16 



FRICTION-RESISTANCE CALCULATIONS 



113 



Assume for the 20-ft towing model of this 

 destroyer a point 10 ft abaft the stem and a 

 speed of 10 kt, equivalent to 16.89 fps. The 

 kinematic viscosity v for the fresh water of the 

 basin is 1.2285(10"'). The value of p/2 is 0.969 

 slugs per ft'' for fresh water and Ul is 285.3 fps^. 



From Table 45.a, R^ is about 13.75 miUion, 

 whence E°' = 26.77. Then 



To = [0.059(0.969)285.3]/26.77 



= 0.6093 lb per ft' 



fcA. = [1.2285(10"')5]/0.6093 = 10.08(10'') ft, 



whence 



/cav = 1.209(10"') in. 



This value for the model, although not for 

 exactly similar conditions, is about twice the 

 permissible value for the ship. 



For the ABC ship, at an a;-distance of 500 ft 

 from the FP, and a speed of 20.5 kt, equivalent 

 to 34.625 ft per sec, i2, for standard salt water is, 

 from Table 45.b, about 1,350(10'). The shear 

 stress To at the ship hull is 



To = 0.059 I UlR-J-^ 



1 QQfl5 

 = 0.059 ^^^^ (34.625)'[1,350(10'')]- 



= 1.051 lb per ft'. 

 Hence the shear velocity is 



(5.iii) 



U. = 



/ 1.051 

 \ 1.9905 



= 0.726 ft per sec. 



whence the laminar-sublayer thickness is 



, _ lofiJl- igr l-2817(10-°) 

 h, - 12.6 ^^ - 12.6 ^^26 



= 22.24(10"') ft or 2.67(10"') in, 



and the permissible average roughness height for 

 a hydrodynamically smooth surface is 



fcA,< 5^ <5i^?||^< 8.83(10-') ft 



or < 1.06(10"') in. 



For the Lucy Ashton [Conn, J. F. C, Lackenby, 

 H., and Walker, W. P., INA, Oct 1953, p. 350ff] 

 the waterhne length is 190.5 ft and the speed is 

 12 kt, or 20.27 fps. The value of U^ or F' is then 

 410.87 ftVsec'. From Table 45.b, R„ is about 301 

 million and Rl'' = 49.63. The estimated kine- 



matic viscosity v (the water temperature is 

 unknown) is 1.42(10"') ft' per sec and the mass 

 density p is 1.985 slugs per ft'. Then, as a local 

 value at the stern. 



To = 0.059[^^)U^{RS 



^ 0.059(0.9925)(411) 

 49.63 



= 0.485 lb per ft'. 



'-(?r=(^r=°--"-- 



Then 





5(1.42)10" 

 0.494 



= 14.37(10"') ft or 1.725(10"') in. 



45.16 Equivalent Sand Roughness. Although 

 the procedure is admittedly empirical, and is not 

 based upon good roughness criteria, some practical 

 use has been made of a quantitative measure of 

 roughness based upon numerous tests of flat and 

 curved surfaces covered with sand grains of 

 different mean diameters. When a rough surface, 

 of whatever configuration, has a specific friction 

 resistance, either average Cp or local Clf , equal 

 to that of a similar surface completely covered 

 with sand of uniform size, it is said to have an 

 equivalent sand roughness equal to the mean 

 diameter of the sand grains on the reference sur- 

 face. This diameter corresponds to the height of 

 the grains as individual protuberances and is 

 represented by the symbol Ks . 



With grains completely covering the reference 

 plate surface, the sand-grain density is assumed 

 to be 100 per cent. The actual mean roughness 

 height is then less than the mean grain diameter 

 Ks , indicated in diagram 3 of Fig. 5.0. However, 

 this detail is overlooked when establishing equiv- 

 alent sand roughness as a practical comparator 

 for surfaces whose irregular roughness can not be 

 measured by any available method. 



M. L. Acevedo at the Madrid Model Basin and 

 W. P. A. van Lammeren at the Netherlands Model 

 Basin in Wageningen have pubUshed tables 

 [TABLAS, Madrid, 1943, pp. 98-117; van Lam- 

 meren, W. P. A., RPSS, 1948, pp. 66-69; 3rd 

 ICSTS, Berhn, 1937, pp. 42-47, 100-101] of 

 roughness allowances based upon the R. E. 

 Froude formulation and the roughness effects 

 developed by H. M. Weitbrecht from the Prandtl 



