Another more important source may be the separation of the flow, due to viscos- 

 ity, in the hull after-body. Further, the simplifying assumption ±P — gives lower 

 values for the theoretical thrust deduction fraction. 



Besides the proof of the hydrodynamic equivalence of the infinite blades pro- 

 peller to steady flow, the approximate steadiness of the flow, depends essentially on 

 the angular velocity O of the propeller, since the periodicity of the flow field is 

 ex (2 7 rfiz)" 1 , where z is the number of blades [8]. The thrust of the propeller and the 

 thrust deduction should be quasi-non-periodic, if fi is great enough. 



Because our results are in substantial agreement with the observed values of 

 time average thrust deduction [10], it is very plausible to expect that a more involved 

 time-dependent theory, that is, for a finite number of blades, would not introduce 

 differences significant for practical purposes. 



REFERENCES 



1. Lerbs, H. W., Symposium on Naval Hydrodynamics, (pp 155-165, this volume). 



2. Lagally, M., Berechnung der Krafte und Momente, die stromenden Flussigkeiten auf ihre 



Begrenzung ausben. ZAMM. December 1922. 



3. Hunziker, R. R.. Final Report HI on Effective Wake and Thrust Deduction for Deeply 



Submerged Bodies, "The Influence of Body Shape Variation on the Thrust Deduction 

 Coefficient". Reed Research, Inc. to David Taylor Model Basin, August 30, 1956. 



4. Oseen, C. W.. Neure Methoden und Ergenbnisse in der Hydrodynamik. Leipzig. 1927. 



5. Burgers, J. M.. On the Application of Oseen's Hydrodynamical Equations to the Problem 



of the Slipstream from an Ideal Propeller. Kon. Ak. Wetenshapen Amsterdam 32, 

 p. 1278, 1929. 



6. Helmbold, H. B., Schraubensog und Nachstrom. Werft-Reedereihafen Bd. 19 (1938) 



p. 354. 



7. Joukowski, N. E., Theorie Tourbillonaire de L'Helice Propulsive. Soc. Math. Moscou 



1912. 



8. Pistolesi, E., Aerodinamica, p. 455, Torino 1932. 



9. Glauert, H., in Durand's "Aerodynamic Theory" Vol. IV, p. 193. 



10. Fresenius, R., Das grundsatzliche Wesen der Wechselwirkung zwischen Schiffskorper und 



Propeller. Schiffbau 1921, p. 257. 



11. Weitbrecht, H. M., Vom Sog, ein Versuch seiner Berechnung. 



12. Dickmann, H., Wechselwirkung zwischen Propeller und Schiff unter besonderer Beruck- 



sichtigung des Welleneinflusses. Jahr.d. Schiff. Ges. 1939, p. 234. 



13. van Manen, J. D. Thrust Deduction and a Proposed Formula for its Radial Distribution. 



International Shipbuilding Progress Vo. 2 No. 8. 1955. 



/. S. Florio 



The purpose of my comment is to emphasize some critical aspects of the theory 

 of the "thrust deduction" set forth by Dr. Lerbs [I] and commented on by Dr. 

 Hunziker [2], as essentially a theory of hydrodynamical interaction, and particularly 

 the generalized formula of the Helmbold type [3], which is a definitive conclusion, on 

 some established hypotheses, following the line of thought mentioned by Dr. Lerbs. 



The primitive Helmbold formula was derived directly from the principle of con- 

 servation of energy [4], and was the most important contribution, without the application 

 of the Lagally theorem. In the other development of the theory (Dickmann) [5, 3] the 

 starting point is the fundamental "exterior theorem" of Lagally [6], based on the Ber- 

 noulli equation which, as is well known, expresses the same Conservation Principle; this 

 theorem, which goes beyond the earlier work by Cisotti on the "D'Alembert Paradox," 

 taking account of continuous distributions of singularities of a general field (ideal fluid) 

 expresses the corresponding resistance force and torque. Betz and Dickmann [5] applied 



172 



