filfi 



HYDRODYNAMICS IN SHIP DESIGN 



Ser. 7n.2S 



The calculation of ffi from Eci. (70.i) is shown 

 in Table 70.d. 



The next step is to calculate a thrust-load co- 

 efficient d{CTL)s , at each radius, based on the 

 ship's speed, using the formulas to be given 

 presently. These values, when integrated over 

 the whole radius, should give a thrust-load co- 

 efficient which is close to the desired coefficient, 

 {Ctl)s , calculated earlier in Sec. 70.26. If the 

 values are not close, within 1 or 2 per cent, then 

 it is necessary to make additional approximations 

 by modifying the hydrodynamic pitch angle Pi 

 until the required accuracy is obtained. 



The formulas necessary for executing this step 

 are: 



tan ^ = ^ = ^ (1 - w,.) (70.ii) 



UiT _ 2 sin )3, sin {I3j 

 Va ~ sin^ 



(70.iii) 



(70 .va) 



^ = ^(1-h;..) (70.iv) 



(7 



(70. vb) 



where 13 is the advance angle 



UiT is the tangential component of the induced 

 velocity 



K (kappa; capital) represents what is known as 

 the Goldstein factor. 



Most of the relationships in the preceding 

 paragraphs and several which appear in sub- 



of Aheod Thrust 



it Selected Radius R 

 where x - p~ 



-Lin Line 



Effective Anqle 

 / of Attock 



rUj ; Induced 

 Velocity 



y^ 2;Ujy, Tangential 



Speed of Advance at Radius R Component 



Fig. 70.F Definition Diagram fob Velocity 

 Vectors at a Blade Element 



seciuent sections can be determined directly from 

 the basic propeller-blade-section velocity diagram 

 which is shown in Fig. 70.F. 



The calculations for the first approximation 

 are given in Table 70.e. In Col. of this table the 

 Goldstein factor K is introduced to take into 

 account the effect of a finite number of blades, 

 since the fundamental theory is based upon an 

 infinite number. Fig. 70.G shows curves of this 

 factor for 4 blades and for various 0-diml radii, 

 plotted on a base of 1/X/ , where Xj = x' tan /?/ . 

 Somewhat similar curves are given by J. G. Hill 

 for 3, 4, and 8 blades [SNAME, 1949, Figs. 19, 

 20, 21, pp. 159-160]. For 5 or 7 blades it is neces- 

 sary to interpolate between the adjacent even- 

 blade values. This introduces a slight error but 

 it is the best that can be done at present. The 

 Goldstein factor K, as shown by the curves of 

 Fig. 70.G, is calculated with certain simplifying 

 approximations, which introduce small inaccu- 

 racies near the hub and tip sections. For this 

 reason, new and more accurate Goldstein factors 

 have been calculated on the TMB Univac for 

 3, 4, 5, and 6 blades. They are tabulated and 

 published as graphs in TMB Report 1034 of March 

 1956; the graphs also appear as Figs. 1-3 on pages 

 326-329 of SNAME, 1955. The resultant changes 

 in K entail slight changes of rj,^ in Fig. 70.D, not 

 taken care of here. 



When calculating (C'tl)s in Col. V of Table 

 70.e, the small thrust developed between the 

 0.2R and the hub at 0.18/2 is neglected. At the 

 most this would introduce only a very small 

 error. Actually, it is negligible because this area 

 of the blade contributes little or nothing to the 

 thrust when the root fillets are added. 



The {Ctl)s calculated by using Eqs. (70.ii), 

 (70.iii), (70.iv), and (70.v), as shown in Table 70.e, 

 is 0.452 as compared to the desired value of 0.473. 

 This is 4.4 per cent low, which is too large a dis- 

 crepancy. It means that a second approximation 

 must be made by modifying the hydrodynamic 

 pitch angle ffi . 



70.28 Second Approximation of /3/ and the 

 Radial Thrust Distribution. When modifying 

 Pi , it is found that a 1 per cent increase in tan Pt 

 causes approximately a 5 per cent increase in 

 {Ctl)s , and vice versa. The new tan Pi is thus 

 given by the formula 



2nd approx. tan (3/ = 



1st approx. tan fii 



1 [ MCrL 

 '^5 1 (CrL) 



(70. vi) 



