Sec. 41.5 



GENERAL LIOUID-FLOW FORMULAS 



15 



41.5 Calculation of the Reynolds Numbers. 

 Although many rudders lie partly or wholly 

 within the boundary layer of the main hull, it may 

 be assumed in Fig. 41. A that the point P is 

 outside (below) that layer. At the orifice position 

 the boundary layer is assumed to be that due to 

 flow over the rudder alone. 



For a ship speed of 29 kt, the velocity U in the 

 Reynolds-number expression UL/v{rm) for the 

 rudder only is again 48.98 ft per sec and the 

 significant length L of the rudder section, at 8 ft 

 below the at-rest WL, is 10.2 ft. From Table 

 X3.h the kinematic viscosity v of fresh water at 

 59 deg F is 1.2285(10"') ft' per sec. Hence, for 

 the rudder section as a whole, sketched in diagram 

 1 of Fig. 41.B, 



Rn = 



48.98(10.2) 

 1.2285(10"') 



= 40.66(10") 



As the Reynolds number rarely needs to be 

 expressed in exact terms, its value for a speed of 

 29 kt, or about 49 ft per sec, and for a length of 

 10.2 ft, may be taken by inspection from Table 

 45. a. This gives about 40(10**), almost exactly 

 the same as by computation. 



If the flow at the orifice position P is to be 

 studied, the significant Reynolds number is the 

 x-Reynolds number R^ at that point, indicated in 

 Fig. 41. B. It is customary, in asymmetrical as 

 well as symmetrical shapes of this kind, to measure 

 the x-distance from the leading edge along the 

 base chord or other convenient dimension parallel 

 to the direction of flow. In this case it is measured 

 along the meanline. It is customary, also, when 

 the velocity of the hquid along the boundary is 

 not accurately known, to consider it equal to the 

 undisturbed stream velocity. Here Ua, = V. 



If the orifice at P lies opposite a point 2.84 ft 

 downstream from the leading edge, then the 

 a;-Reynolds number is 



R. = 



Vix) ^ 48.98(2.84) 

 V ~ 1.2285(10"') 



= 11.32(10''') 



by calculation or 11.2(10') from inspection of 

 Table 45.a. For the 1/25 scale model, run at a 

 speed equal to 48.98/5 = 9.79 ft per sec with the 

 test point 2.82/25 = 0.113 ft downstream from 

 the leading edge, the a;-Reynolds number is 



R. = 



9.79(0.113) 



1.2285(10"') 



0.09(10') 



In the case of flow around a body of short 

 length, broadside to the stream, the body beam 



B or body diameter D rather than its length L is 

 the determining factor in the type and nature of 

 viscous flow encountered. For this reason, a 

 dimension transverse to the flow rather than one 

 parallel to the flow is used as the space dimension 

 in the numerator of the Reynolds number. For 

 the case of the underwater sound head of Fig. 

 41. D of Sec. 41.6 it is the diameter D of the head, 

 say 1.72 ft. The relationship so formed is called 

 the rf-Reynolds number, represented by 



Rd = 



UD 



(2.xxii) 



For a ship speed of 14 kt, or 23.64 ft per sec, 

 this is 



Rd = 



UD ^ 23.64(1.72) 

 V 1.2285(10"') 



= 3.3(10') 



There are several other Reynolds numbers in 

 use by hydrodynamicists, such as the 5-Reynolds 

 number Rs . In this case the thickness 8 of the 

 boundary layer replaces L as the space dimension 

 in the expression UL/v. 



One of great importance is the blade-Reynolds 

 number R^ for the blade sections of screw and 

 rotating-blade propellers. This expression is 

 built up in the manner shown by Fig. 41. C. It 



Blode 

 Section 

 Qt 0.7 Rodiusl 



Tonqential 

 Velocitij 

 2-rrn(0.7R) 



Lonqitudmal 



Chord Length at 0.7 R= c q.yr 



Blade Rev/nolds 

 \- Number Reiode 



^/.[2rrn(0.7R)]t-^(co.,,) 

 V 



Nominal Blode Velocity 



{v/t[a^n(o.7R)]2)0-5 



Velocity VA Effect of Small 



^ Angle of Attack 



is Nei^lected 



Fig. 41. C Definition Sketch and Formula fob 

 Blade Reynolds Number 



utihzes as the length dimension the chord length c 

 of a typical blade section, at 0.7i2 on a screw pro- 

 peller, and a nominal velocity generally parallel to 

 the base chord of the blade. For TMB model pro- 

 peller 2294 shown in Fig. 78. L, where the chord 

 length c at 0.7R is 2.682 inches or 0.2235 ft, the 

 value of 0.7R is 3.378 inches or 0.2815 ft, Va is 



