i-V(v«V) = (V-V)V + V X (V X V) • (15) 



and 



-0.71 (fi X r ) = yV|Jl X r|^, 



one can express Equation (13) as follows: 



vUv. 



(16) 



1+ -^ - -T 1^ X r|^ + gy \ = - -_ + V X (o) + 2f^) (17) 



3V 



Here, (o is the vorticity in the fluid measured in the moving frame of reference. 

 Now, consider a propeller rotating at a constant angular velocity, Q = -0± 

 (a right-hand rotation propeller, see Figure 1), in an axisymmetric wake of a ship, 

 where the flow is steady in the rotating frame of reference. The ship wake velo- 

 city, V , can be expressed in polar components as: 



V =Vi+Ve + V^e. (18) 

 -w X— r— r 0— e ^ 



where V^, V and V_ are radially varying axial, radial, and tangential components, 

 respectively. It is assumed that the variation of the ship wake velocity in the 

 radial direction is small. 



In addition to the ship wake velocity we assume that there exists another 

 axlsyflnnetric disturbance velocity component, V , that is introduced locally by 

 nearby appendages or other lifting surfaces: 



V =V i+V e +V^p (19) 



^o ox— oi^r 00^ 



Then the total velocity, V, in a cylindrical coordinate system rotating with 

 the propeller can be expressed as follows: 



V = V + r$7e„ + V + V (20) 

 — — w ^ — p — o 



where V is the perturbation velocity due to the presence of the propeller. 



