As the axial component of the pressure integral over the body is characterized 

 by the velocity field and Bernouilli theorem, it is legitimate to apply Lagally's exterior 

 theorem [2] to calculate this force of resistance in the potential flow that comprises the 

 body. In this way, a general expression for the "thrust deduction fraction" is derived: 



fi + ( - ) *2 

 AT \ VI 



Q = — = (31) 



1*1 / c\ 



1 + % ( — ) - ( P cUA)- l AP 



where T is the thrust force over the propeller support. Lty l5 is the average of the nor- 

 mal velocity component over the propeller circle, for the body without propeller (t/^, 

 the "nominal wake"), and cif 2 is the average of the normal velocity component of the 

 interaction potential. The density is denoted by p , the propeller circle area by A, and 

 AP is the increment in the integral of the pressures over the intersection of the rotational 

 slipstream with the plane at infinity. 



AP is a consequence of the rotation of the water in the slipstream and intro- 

 duces a dependence of on the torque. Since in general the angular velocities 

 impressed on the particles of the slipstream are small compared with the angular velocity 

 of the propeller [7, 8, 9] it is assumed AP = 0. In the limiting case of very light 

 loadings (c — > 0), it is -> ^ 15 and the thrust deduction may be approximated by 

 = if/^ which is a generalization of Fresenius's expression [10]. The expression for 



c 



(generalization of Helmbold formula [6]) for the moored condition, with ( — ) — > oo, 



must be taken with reservation due to the asymptotic character of the first approxi- 

 mation to the solution of Oseen equations. 



lim 

 Therefore, the consideration that = © — 2xh is the thrust deduction 



17-0 



coefficient for U — > (moored condition) must be taken as a conjecture of empirical 

 validity. 



The hypothesis of a boundary layer with extreme backward separation corre- 

 sponding to a favorable distribution of pressures due to the propeller inflow, seems 

 to accord with bodies of reasonable shape such as torpedoes and streamlined submarines. 



The solution of the corresponding harmonic problem for the case of the ovoid 

 ellipsoidal hull was developed in Legendre functions and \j /l and ^, have been tabulated 

 for different configurations [3]. The results indicate that the thrust deduction fraction 

 exhibits a rapid diminution with the increase of the distance of the propeller to the 

 hull and with the increase of propeller diameter. (See Fig. 1 .) 



These theoretical results are slightly lower than the experimental values for the 

 ovoid hulls tested by Weitbrecht [11]. This confirms the assertions of Dickmann [12] 

 and van Manen [13] that for the case of radial symmetry the "frictional thrust deduc- 

 tion" is small. The theoretical values of the figure correspond to one of the cases 

 investigated experimentally by Weitbrecht and show a typical variation of with the 



/ c . 



d.stance ratio — of propeller to hull, and for different rations — . (a, b, ellipsoid axes, 



a U 



R radius of propeller.) 



Dr. Lerbs [1] has pointed out that "imperfections of the theory of interaction 

 exist relative to viscous flow and to time-dependent flow on the propeller which arises 

 with a finite number of blades in a non-homogeneous wake." A possible source of small 

 discrepancies with experiments [11] is that the hull is not exactly our prolate ellipsoid. 

 Experiments with such prolate hulls may be performed and it is very reasonable 

 to expect that the results would be also in good agreement. 



170 



