Difficulties with slow convergence which arise with Goldstein's series expansions for 

 great values of Xi are avoided in numerical computations when applying the integral 

 equation. 



The relations mentioned so far hold both for an airplane and for a marine 

 propeller. For the latter, however, the replacement of the blade by a line vortex does 

 not suffice because its aspect ratio is usually small. This arises from the requirement 

 to avoid the onset of cavitation from which condition an upper limit for the lift 

 coefficient follows. Replacing a blade by a lifting surface instead of by a lifting line 

 introduces a boundary value problem, viz., to determine the shape and the geometric 

 angle of attack of an infinitely thin section such that a prescribed pressure distribution 

 is realized within the propeller flow. There are two ways known in literature to treat 

 this problem. In a direct way, the velocity components induced from the free and 

 bound vortex sheets are calculated at several stations of the chord length and the 

 boundary condition is satisfied at these stations [4, 5]. These calculations are fairly 

 lengthy also when using precalculated functions. In a faster working approximate 

 method one starts from a skeleton which generates the prescribed pressure distribution 

 in 2-dimensional flow. The propeller flow requires corrections on both the shape and 

 the angle of attack of this skeleton. These corrections follow from the curvature of 

 the propeller flow at the half-way point, which is known from papers by Ludwieg 

 and Ginzel [9], and from the boundary condition at the % -point. This latter means 

 an application of Weissinger's approximate lifting surface theory to the propeller 

 flow [10]. 



Marine propellers also differ from airplane propellers relative to the ratio of 

 the hub diameter to the propeller diameter which is greater for marine propellers. 

 The question arises whether or not the boundary condition on the hub requires cor- 

 rections on the components of the induced velocity in the case of marine propellers 

 since this boundary condition is not taken into account within equations (4) to (9) 

 for the induction factors. Attempts to determine the order of magnitude of these 

 corrections are made for the free vortex sheets in a paper by Lerbs [6] and for the 

 bound vortices in a paper by Ackeret [11]. Numerical calculations have shown that 

 this correction may still be neglected for a hub-diameter ratio of 0.2. 



II. Theory of Interaction Between Hull and Propeller 



The older method to explore the interaction is based on the laws of conserva- 

 tion of energy and of momentum. The outstanding papers are those by Fresenius [12] 

 and by Horn [13]. From these investigations it became apparent that there is a 

 marked difference in the effects of a viscous and of a non-viscous fluid on the pro- 

 pulsive power. It is shown that a positive viscous wake has a favorable effect on the 

 propulsive coefficient and that a positive displacement wake is unfavorable. The reason 

 for the opposite behaviour of these two components of the wake lies in the energy of 

 the obsolute motion which is left far behind the propeller body. In the case of a viscous 

 fluid the absolute motion consists of 2 parts of opposite direction, viz., the viscous 

 wake of the body and the induced velocity of the propeller race, whereas in the case 

 of potential flow only the propeller field is present. From these considerations it appears 

 that a wake adapted propeller should be designed such that the remaining absolute 

 motion becomes as small as possible. This requires a large axial component of the 

 induced velocity and, therefore, a large thrust element to be generated on those radii 

 on which a large viscous wake is found. 



These general statements on the interaction have been considerably extended 

 and refined by Dickmann [14]. In addition to the afore mentioned effects on the pro- 

 pulsive coefficient an influence of the free surface is also considered. From the inter- 

 ference of the wave systems of the hull and of the propeller it is shown that an improve- 

 ment of the propulsive coefficient takes place if the propeller is placed below a hump 

 of the hull's wave system. 



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