280 



favorably with the theoretical predictions. It is 

 recommended that this experimental technique be 

 extended to study the effects of nonuniform flow and 

 intermittent blade surface cavitation. 



2. FORMULATION OF THE PROBLEM 



Vcfi^ -* X ^ -"> 



(4) 



and at large downstream distances, x ->■ + <»,(|i 

 satisfies a suitable radiation condition. 



The boundary condition on the hull surface, 

 denoted by S, requires that the fluid velocity must 

 be tangent to the surface , or 



Consider a ship moving at constant speed U through 

 otherwise undisturbed water. We seek to determine 

 the periodic forces and moments exerted on the 

 ship hull surface arising from the unsteady propeller 

 velocity and pressure fields. The fluid is con- 

 sidered to be incompressible and inviscid and within 

 the domain bounded by the free surface, the hull 

 boundary, and the propeller blades (and trailing 

 vortex wakes), the flow is assumed to be irrotational. 

 Under these circumstances, a fluid velocity potential 

 exists which can be expressed in terms of steady 

 and unsteady components as 



>(x,t) = Ux + 



;(X) 



*p(x. 



+ <t>j^{x,t) 







X on S 



(5) 



n being the outward unit normal vector to the sur- 

 face (see Figure 1) . Here we have assumed the 

 hull to be rigid and stationary with respect to the 

 translating coordinate system (i.e. hull motion 

 and deformation due to propeller excitation is 

 ignored) . 



The linearized free surface boundary condition 

 may be written in the form 



3*n 



•(nNu)^(J)„ + (2inNuU) + U^ 



n Sx 



3(t> 



+ g 



3x^ 



3z 



Here, x = (x,y,z) is a cartesian coordinate system 

 fixed to the ship with the x and y axes in the 



is the steady disturbance flow about the bare hull 

 in the presence of the free surface, <l)p(x,t) is the 

 propeller potential, and i)>Q(x,t) is the potential 

 of the flow arising from the propeller-hull inter- 

 action, often termed the scattering or diffraction 

 potential. It should be noted that the presence 

 of the viscous, rotational wake of the ship is 

 ignored in the diffraction problem, i.e. , it is 

 assumed that the unsteady pressure fo'rces on the 

 hull can be derived from potential flow considera- 

 tions alone . 



The propeller potential is periodic in time and, 

 by virtue of the symmetry of identical, equally 

 spaced blades, may be expressed as a Fourier series 

 with harmonics in blade passage frequency as 



,(x,t) 



n=0 



''Pn 



(x)e 



inNut 



(1) 



with ((ip being the complex amplitude of nth harmonic. 

 (In this and all subsequent expressions involving 

 einNojt the real part is understood to be taken.) 

 Similarly, the diffraction potential will be of the 

 form 



♦o'^'t) 



(x)e 



inNiot 



(2) 



n=0 



We now consider the boundary value problem for 

 the potential (^ = (\> + <ii^, assuming the fluid 

 disturbance velocities to be small compared to the 

 ship speed, i.e., | Vcf | and | V(j)g | <^U. Within the 

 fluid domain, the potential must satisfy Laplace's 

 equation 



V'^<J>n(x) = 



(3) 



At large depth and distances upstream of the hull 

 and propeller the disturbance must vanish 



on z = 



(6) 



In order to establish the relative magnitude of 

 terms the equation is recast in nondimensional form 

 using the ship speed U and propeller radius Rg for 

 reference length and time scales, obtaining 



3(|) 



32(j) 



^n + 2iE — 



+ z^ 



gR 3()> 



,02 '' 



02 "^ 3z 



= on z = 



where e = J/irnN, J being the propeller advance 

 coefficient. It may now be observed that typical 

 propeller applications, e '^^^^l and the first term 

 will dominate. Thus, as a first approximation the 

 free surface boundary condition (6) reduces to 



.(x) 



(7) 



This completes the statement of the boundary value 

 problem for the diffraction potential as summarized 

 in Figure 1. It should be noted that by virtue of 

 (7), the function (^^(x) can be analytically continued 

 into the upper half plane, z > , in a straight- 

 forward manner. As will be shown in subsequent 

 sections a solution can be constructed in terms of 



VELOCITY POTENTIAL 



^l iJ>(XJl)'UX +ci^(X) + 0(x,-t) 



4>Mi <i)(X,t) = ? (^(Xje'"""" 



K*n|— °,|x| — 

 KO 



FIGURE 1. Coordinate system and boundary value 

 problem in propeller-hull interaction analysis. 



