Sec. 74.16 



MOVABLE-APPENDAGE DESIGN 



731 



vertical transverse planes between the leading 

 and the trailing edges of the rudder. 



The steps listed in the foregoing require some 

 explanation, given here in the same order as 

 (a) through (i). 



The velocity diagrams of Fig. 37. K, for a contra- 

 rudder installation only, are supplemented by 

 those of Fig. 74. M, for an installation of both 

 contra-guide skeg ending and contra-rudder. In 

 the latter diagrams the incident flows on the pro- 

 peller are shown for two adjacent blade sections, 

 unrolled into the flat. This is an aid in visualizing 

 the flow leaving the skeg endmg, represented by 

 the vector U a , and that forming one of the 

 components of incident flow on a blade section at 

 the same level. 



It is customary to show the flow meeting a 

 moving-blade element with reference to axes in 

 that element. The vector Ua is therefore combined 

 with the rotational vector 2irrii2 and with the 

 induced-velocity vector fcj Uj to give the incident- 

 velocity vector Uri . When the flow leaves the 

 moving blade along Ur2 it is again referred back 

 to axes fixed in the ship by combining it with the 

 rotational vector 2imR, but in a reverse direction. 

 This is equivalent to combining the inflow vector 

 U A with the induced-velocity vector at the trailing 

 edge of the blade element, the latter equal to 

 k^Uj . The flow meeting the hydrofoil section of 

 the rudder is then represented nominally by the 

 vector ?7jj3 . 



It is now required to select a section shape for 

 the contra-rudder, and a nominal angle of in- 

 cidence with respect to the ship centerplane, 

 which will insure smooth flow around the rudder 

 with a reasonable lift force and forward-thrust 

 component. What is wanted first is the value of 

 the angle Br which the vector U r^ makes with 

 the ship axis. Second, the amount of offset of the 

 leading edge of the contra-rudder section is to be 

 determined. 



If the fraction fej is estimated (or determined in 

 some manner) it is possible to derive Br by a 

 graphic construction such as that in diagram 2 

 of Fig. 74. M. The velocity induced by the hydro- 

 foil section of the rudder, shown as /cjC// in the 

 small-scale diagram 1 of the figure, is omitted 

 from the large-scale diagram 1 to avoid confusion. 

 It is also assumed, for the sake of simplicity but 

 without appreciable error, that the induced- 

 velocity vectors fci Ui and fca Uj he normal to the 

 base chord of the blade section, at the geometric 



blade angles marked in the diagram. The 

 nominal value of the obliquity 6r of the velocity 

 vector U R3 , incident on the rudder section, may 

 be calculated by Eq. (74.i), set down in the upper 

 RH corner of the figure. 



For a contra-guide skeg ending designed as 

 in Sec. 67.22, the angle ds is known for any blade 

 radius or level. If the skeg ending is symmetrical, 

 as in Fig. 37. K, the expression for Br reduces to 

 that of Eq. (74. ii), alongside the large-scale 

 diagram 3 of Fig. 74.M. The value of Ui is always 

 small with respect to U a , and /cj is always less 

 than 1.00, because the full value of Ui is developed 

 only far astern. Eq. (74.ii) therefore reduces to 

 Br = tan~^ (C sin 0). Here C, representing 

 kiUi/U A , is a small fraction, say 0.15. For any 

 normal propeller the geometric blade angle 4> 

 seldom exceeds 65 deg at the hub surface, hence 

 C sin has a maxunum value of about 0.135. 

 The angle 9r then has a maximum value of about 

 8 deg, where Br = sin Br = tan Br , approximately. 



Actually, it is possible to derive the average 

 value and direction of U a only from model tests 

 or very special ship tests. Further, U a is almost 

 never constant along the upper or the lower 

 blade radii, either in magnitude or direction, nor 

 does the mean upper value of U a equal the mean 

 lower one. As a consequence, and because of the 

 varying circulation at different blade radii, the 

 maximum induced velocity Uj is almost never 

 the same for all radii, nor is it known precisely 

 at any radius. To cap all the foregoing, the factors 

 /ci and ^2 are not well known. While values reason- 

 ably close to the actual ones could undoubtedly 

 be substituted in the expressions for Br in Fig. 

 74. M, there are practical factors which require a 

 more realistic approach to this design problem. 



It is recalled that for the design of a contra- 

 guide skeg ending discussed in Sec. 67.22 the 

 value of Br derived analytically for points opposite 

 the outer propeller radii are small but they are 

 still too large for practical use. Employed here, 

 they would produce a rudder which had no straight 

 or symmetrical sections whatever. Whether this 

 much of a departure from the orthodox stream- 

 Imed rudder would be the best device for steering 

 a straight course for long periods could only be 

 determined by successive full-scale installations 

 on the same vessel. The doubt expressed in the 

 foregoing is heightened by the fact that it is also 

 rarely possible to use hydrofoil sections which 

 deliver the water directly astern mth the rudder 

 at zero angle. The large camber necessary to 



