249 



by (()g(x,y,z), the velocity potential for the over- 

 all flow field can be expressed by U-x+<t>s, and the 

 following equation must be satisfied for i/^: 



3-^(1) 



3x2 



3'^d) 3'^d> 

 s , s s „ 

 + —IT- + . ^ = 



3y2 



3z2 



(4) 



Boundary conditions are given as follows. On the 

 hull surface, Sg, the following equation must be 



satisfied: 



FIGURE 1. Coordinate systems. 



o^-xiyiziand 0i-xir6 related with the propeller 

 as indicated in Figure 1 . In the coordinate system 

 Oj-xiyizi, the origin o\ coincides with the propeller 

 center and we assume that the xj-axis coincides 

 with the propeller axis and is parallel to the x- 

 axis. Further, the coordinate system oi-xjyizi has 

 the following relationship with the coordinate 

 system o-xyz : 



Xp + XI, 



-f + yi, z 



zi. 



(2) 



where (Xp, -f, o) are the coordinates of the pro- 

 peller center on o-xyz. Moreover, the following 

 relationship is satisfied between Oj-xjyizi and 

 Oj-xirG : 



XI 



XI' Yl 



zi 



(3) 



U + 



3* 



£ 



8y 



n + -5— 

 y dz 



= 



(5) 



on S 



where Hj,, ny, and n^ represent x-, y- , and z- com- 

 ponents of the outward normal unit vector on Sg . 

 On the free surface, we have two boundary conditions. 

 One of them can be obtained from the Bernoulli ' s 

 law and the condition of constant pressure there, 

 as follows: 



3*3X2 



3*3X2 

 3x / ^ \^^) ^ 



3*s\2 



3z" 



(6) 



where f;s(x,z) represents the vertical displacement 

 of the free surface, namely, wave height. Another 

 boundary condition on the free surface is the 

 kinematical condition as indicated below: 



U + 



3x 



3(()^ 



3y" 



3z 



3z 



= 0. 



(7) 



Y=Ce 



3. PRESSURE ON A HULL SURFACE AND ACCELERATION 

 POTENTIAL 



At infinity, the following boundary conditions 

 might be given: 



Pressure generated on the hull surface in the towed 

 condition differs from that in the self-propulsion 

 condition because of the influence of propeller 

 action. The time-independent part of this change 

 corresponds to the pressure component of the thrust 

 deduction and the time-dependent part corresponds 

 to the propeller-induced surface force. Now, with 

 conventional methods devised to calculate these 

 forces, numerical procedures tend to be extremely 

 troublesome. Consequently, a great deal of calcu- 

 lation time is required, especially in calculating 

 propeller induced velocity, and it is hard to apply 

 to a practical hull of a complicated form. Hence, 

 an easy method with which the calculations of pro- 

 peller influences can be reduced is needed. 



In this chapter, the method which can calculate 

 change of pressure induced by a propeller on the 

 hull surface is explained. This method can be 

 obtained by using acceleration potential. 



Fundamental Equation 



In this section, we assume that the flow field 

 around the hull is inviscid. This assumption may 

 be considered reasonable in solving the problem of 

 pressure on the hull surface when the boundary layer 

 on the hull surface is thin. 



At first, let us examine the flow field around 

 the hull in the towed condition. Denoting the 

 velocity potential of disturbance due to the hull 



Derivatives of <j) ^ when /x^ + y^ 



when /x" 



(8) 



Next, let us examine the flow field around the 

 hull in the self-propulsion condition. We assume, 

 similarly to the towed condition, that the velocity 

 potential of disturbance exists. Then, we can 

 express the velocity potential of the overall flow 

 field by U*x + (fig + i)* . Here, the (|)* (x,y ,z ; t) rep- 

 resents the change of the velocity potential due 

 to the propeller action when the moving condition 

 is changed from the towed condition to the self- 

 propulsion condition, and (})* must satisfy the 

 following equation: 



3V: 



3x2 



3^(1)* 



3y^ 



3^(j)* 

 3z^ 



= 



C9) 



We can also obtain the boundary conditions under 

 the self-propulsion condition in the same manner 

 as under the towed condition. In this case, however, 

 time derivatives appear in some conditons by the 

 influence of propeller rotation. On the hull sur- 

 face, the following boundary condition is given: 



3()> 



U + 



341 



3<})< 



3y By 



3(fi 

 3F 



3T 



on So 



(10) 



