293 



-30p- 



s 



-35A- 



-40L 

 0.06r 



0.04 



0.02 



,% 



a 



o 



D 



J_ 



JL 



I 



I 



o 



^ 



0.0 0.2 0.4 



0.6 ITS" 

 J ■= U/nO 



iTO o r4 



FIGURE 10. Blade frequency induced pressure on body 

 with propeller 4118 located at H = 16 in. and a tip 

 clearance C = 3.0 in. 



Application of the Theory 



only 1.0 percent of the force arising from the 

 time rate of change of the potential. This surely 

 justifies the order of magnitude argument given 

 earlier. 



Alternative Approach 

 (Varus, 1974) 



Oscillatory Body Potential 



In order to apply Eq. (47) to the experimental 

 configuration, it is convenient to consider the 

 velocity potential H-^n °f the body travelling back- 

 wards and executing simple vertical oscillations, 

 so that o. =03 = k in Eq. (53). The free surface 

 condition ^i^i = on z = , Eq. (54), can be satis- 

 fied by reflecting the body surface into the upper 

 half space and satisfying the body boundary condition 

 additionally on the image surface, S-j^. In Appendix 

 B it is shown that the vertical force induced by 

 the propeller on a ship in the free surface is 

 equal to the force on the "doable hull" deeply 

 submerged. If we make the further assumption that 

 the force due to the convective pressure can be 

 omitted, the problem for Hj_j^ now reduces to 



H. = 

 in 



in V 



V K. = inNu(n'k) 

 m 



on S + S . 



V H. 



0, 



(63) 



(64) 



(65) 



Direct Approach - Extended Lagally Theorem 

 (Breslin and Eng , 1965) 



The test body surface was divided into 154 elements 

 as shown in Figure 13 with finer subdivisions made 

 in way of the nearest approach of the propeller 

 blades. Panels 93 through 100 were used to close 

 the body. The geometry of these elements, together 

 with the normal velocity induced by the propeller 

 due to loading and blade thickness formed the input 

 to the generalized Hess-Smith program which inverts 

 Eq. (33) to yield the source densities on each of 

 the panels. 



A typical velocity variation, as given in Figure 

 14, shows that, downstream of the propeller, the 

 loading contribution is oscillatory, requiring 

 great care as the body sections are becoming larger. 

 This test case presents a somewhat difficult appli- 

 cation of this technique for this reason. In the 

 ship case, there is only a small portion of the 

 hull downstream of the propeller, and the sections 

 are generally becoming smaller. As a result of 

 this non-ship arrangement, difficulty was encountered 

 in securing an accurate answer, requiring several 

 adjustments of the size and location of the source 

 panels. 



A calculation for a single set of conditions, 

 specified by the geometry of DTNSRDC Propeller 4113 

 set at a tip clearance of 3.0 inches (1.18 cm) at 

 an axial distance of 16.0 inches (6.3 cm) downstream 

 of the nose of the body gives a blade-frequency 

 force coefficient Cp = 3.4 x 10"^ and a phase angle 

 6p = -2.0°. These results are quite close to the 

 measured values shown in Figure 9. It should be 

 remarked that the evaluation included the Lagally 

 force corresponding to the integral of the convective 

 pressures, i.e., the action of the transverse pro- 

 peller velocity component on the sources which 

 generate the body in the uniform axial flow. This 

 contribution, as expected, is indeed small yielding 



where V is the whole space outside the "double-hull" 

 surface, S + S^. 



This method is particularly convenient in the 

 present application because the velocity potential 

 of an oscillating spheroid is well known, e.g. Lamb 

 (1932) . With slightly modified notation 



20 



-20 - 



DESIGN ZERO THRUST 

 (J = 0.83) (J = 1.16) 



-40 



,<L 



§ 4119 [S- 4118 & 4119 

 C^4118 



r 



4.0 



,^ 2.0 



a 



s 

 e 



-4118 - Open Symbols 

 4119 - Solid Symbols 



1.0 



EXP. 



I 



CALC. 



-= Vorus{1974) 



4119 



4118 



I 



4119 



4118' 



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 

 J = U/nD 



FIGURE 11. Calculated and measured blade frequency 

 force for propellers 4118 and 4119 located at i = 10.0 

 in. with tip clearance C = 4.5 in. 



