A General Theory for Marine Propellers 



N- 1 M- 1 



■ V(r^,^.)= ^ ^ ^n. •Mrk,^j)nn, , ^ . .:,:^ : (86b) 



n= m= 



in which u{T\^,(pj)^^ and v(t^^,4>- )„„ are induced velocities as obtained from Eqs. 

 (19a) and (19b) for the various modes of the load functions, with the load coeffi- 

 cient a^^ equal to unity. 



Combining Eqs. (85) and (86) we can establish a set of J times K linear 

 equations as follow: 



N- 1 M- 1 " . - ^■< ■ 



2] E -anrntuC^k-'^j )nm - vC'-k'^^j )nm ta" ^('"k'^j )1 

 n= m=0 



= r^n tan '/'(r^,^. ) - V , for j = 1, J and k = 1, K . (87) 



Consequently, the unknown load distribution coefficients a^^ can be obtained by 

 solving the J times K equations with a^^n, as the corresponding number of un- 

 knowns. ^ , 



As an example of a performance prediction, the simple case of the inverse 

 calculation of the propeller design example is considered. It is felt that this 

 example provides a check on the numerical accuracy of the method. 



Five points on each of nine radial sections were used as control points. 

 Table 1 shows the output from the computer program. Figure 7 gives the radial 

 load distribution compared with the load distribution used in the design. Figure 

 8 shows chordwise load distributions for four of the nine radial sections. 



The results from the computer program obtained so far have confirmed that 

 an efficient computer program can be developed on the basis of the theory and 

 the numerical technique. Less than 10 minutes of computer time was required 

 on the IBM 7090 at the Naval Ship Research and Development Center for both the 

 design example and the inverse calculation. 



CONCLUDING REMARKS 



1. The general theory for marine propellers outlined in the paper is based 

 on an exact acceleration potential. There is no linearization of the equations in- 

 volved; it is a higher order theory. It imposes no limit on loading of a propeller 

 under normal practical operating conditions. 



2. The theory is derived from the equations of motion and the equation of 

 continuity. An irrotational fluid motion has not been assumed. Therefore, the 

 theory can be applied to a propeller operating behind another propeller or another 

 lifting surface where free vortex distribution exists. 



3. The theory uses information about the moving propeller blades and their 

 load distributions and positions, whereas it is not required that the fluid flow 

 induced by the propeller be established beforehand. 



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