appropriate "images" of the propeller and hull 

 singularity systems. 



Upon solving for the velocity potential, all 

 other quantities of interest can be determined. 

 The linearized, unsteady component of pressure is 

 given by* 



p(x,t) = 



—^ + V 

 3t s 



V (x) = iU + V4i^(x) 

 s ° 



(8) 



or from (1) and (2) 



p(x,t) = -p 



[inNco(t> + V^ • V(t)„] e 



inNut 



n=0 



p (x) e 

 n 



inNut 



(9) 



n=0 



where the Pn''^' ^^'^ amplitudes of harmonics of the 

 unsteady pressure. The periodic force, F(t), and 

 moment, M(t) acting on the hull surface (see Figure 

 1) may be written as 



F(t) = 



p n dS 



(10) 



and 



M(t) 



p X X n dS 



(11) 



Inserting the expression for p, one obtains the 

 amplitudes of the force and moment harmonics , as 



(inNMc))^ 



+ V, 



V<))n)n dS 



(12) 



281 



the propeller operates. As viewed in a coordinate 

 system rotating with the propeller, the flow 

 approaching the propeller consists of time-average 

 or circumferential mean component and an oscillatory 

 component. The oscillatory component gives rise to 

 unsteady loading on the blades in a manner analogous 

 to a hydrofoil encountering a sinusoidal gust. This 

 unsteady loading, summed over all the blades, yields 

 periodic shaft forces at blade frequency and integer 

 multiples. In contrast, the periodic pressure 

 forces acting on the hull surface arise from the 

 induced velocity and pressure fields from both the 

 mean and unsteady components of loading, as well 

 as the blade thickness, because of the varying 

 aspect of the rotating blades relative to the fixed 

 hull boundary. 



Propeller theory for unsteady flow has developed 

 as a logical extension of linearized lifting-surface 

 theory for hydrofoils. It is assumed that the 

 oscillatory components of the wake velocities are 

 small compared to the mean, and can be resolved by 

 Fourier analysis into "wake harmonics," the funda- 

 mental harmonic being the shaft rotation frequency. 

 Each of these harmonics , within the linear approxi- 

 mation, will produce a component of unsteady blade 

 loading with the same frequency. By virtue of the 

 propeller's symmetry, upon summing over all the 

 blades, only certain harmonics of the loading will 

 contribute to the net force on the shaft. However, 

 all the harmonics of loading contribute to the 

 forces on an individual blade, and, as will be seen, 

 to the radiated pressure field of the propeller. 



The propeller lifting-surface theory developed 

 by Tsakonas et al. (1973) is adopted in the present 

 work. This analysis and associated computer pro- 

 grams have been successfully applied in recent 

 propeller designs to minimize bearing forces, e.g., 

 Valentine and Dashnaw (1975) . In addition, the 

 analysis has been extended to compute field point 

 velocities and pressures, including the contributions 

 from the image of the propeller arising from the 

 presence of the free surface. As the details of 

 the development of these formulae have been largely 

 reported in the literature, we shall not burden 

 this paper by recounting them, being content to 

 outline the procedure. 



and 



M = p / / (inNu(j)j^ + Vg • V<})j^) x x n dS (13) 



Blade Loading Potential 



Until now, the propeller 'potential has been regarded 

 as a known function. Before proceeding with the 

 surface force analysis, it is appropriate to discuss 

 the analytical representation of the propeller and 

 the velocities and pressures induced at arbitrary 

 field points. 



3. REPRESENTATION OF THE PROPELLER 



The primary source of propeller exciting forces is 

 the spatially nonuniform wake of the hull in which 



The linearized equation of motion for unsteady flow, 

 referred to a non-rotating cylindrical coordinate 

 system (x,r,f) centered at the propeller axis 

 (Figure 2) , may be written 



-P' 



at 



+ u 



which has the solution 



*p(x,r,f,t) =- — 



3(|) 



E 



3x 



(14) 



x' ,r,f,t- 



T^>- 



(15) 



*To be strictly consistent with the high frequency approxi- 

 mation, the convective pressure term should be discarded. 

 However, this term adds no serious burden to the ensuing 

 analyses and by retaining it, numerical calculations can 

 be used to demonstrate that the contribution from this term 

 is, in fact, negligibly small. 



where p is the pressure induced by the loadings on 

 the blades due to camber and incidence and p ' , for 

 later convenience, denotes the fluid density. Here 

 the angles of attack are produced by each axial 

 and tangential spatial harmonic of the nominal hull 



