The measurements of flow angles show that the flow was essentially parallel to the 

 direction of motion of the ship in the regions investigated. 



ESTIMATING THE THICKNESS OF THE BOUNDARY LAYER 



The Prandtl or Von Karman formula is often used to estimate the growth of the 

 turbulent boundary layer along a smooth flat plate: 



Spp = 0.37 xR^- 



[5] 



However, Equation [5] was derived on the basis of ti = 7 in Equation [1] and hence is 

 valid for only a limited range of Reynolds numbers. To determine whether Spp would provide 

 a useful engineering estimate for the magnitude of S on a ship a general expression was 

 derived for Spp as a function of n. (See Appendix.) 



n + 1 2n 2 



Spp {x,n) = 



(2 + n)(3+n) 



1 

 C{n) 



X fl. 



[6] 



C{n) is a dimensionless empirical coefficient related to the friction velocity. The 

 values of C{n), shown in Table 3, are given in Reference 6, as obtained from an analysis of 

 experimental data by Wieghardt. 



TABLE 3 



Values of C{n) 



n 



7 



8 



9 



10 



C{n) 



8.74 



9.71 



10.6 



11.5 



Equation [5] then becomes a special case of Equation [6] for ti = 7. Values of n were 

 obtained as a function of Reynolds number for Figure 7 from Landweber's data. ^ Values of 

 Spp computed from Equation [6], using values of n from Figure 7 and C{n) from Figure 8, are 

 also given in Table 2. 



The variation of C{n) and n with Reynolds number, together with Equation [6], explains 

 the difficulty of detecting a variation in 5 with ship speed, as experienced by Baker-^ and 

 Allan. ^ At high Reynolds numbers the simultaneous changes in C{n) and n occur in different 

 directions, giving an apparent relationship 



8~Kx [7] 



where /visa constant. 



11 



