Propeller Pressure Field in a Nonuniform Flow 



P(x«,^„r.) 



y 



Xo,X 



Fig. 1 - Coordinate systems and notations 



arising from blade thickness is negligibly small. Therefore, only the influence 

 of flow nonuniformity on the pressure due to loading will be treated. 



Disregarding the physical effects which contribute to the origin of the non- 

 uniform flow, we assume the fluid to be ideal. In addition we assume that the 

 propeller-induced flow is potential. The vortex system of a propeller can be 

 represented by: z radial bound vortices whose strength r( r ' , t) depends on ra- 

 dial coordinate r ' and time t ; z helicoidal free vortex surfaces formed by heli- 

 coidal and radial vortices arising due to radial and temporal variation of r re- 

 spectively. Propeller-induced velocities are assumed to be small compared to 

 Vp and the pitch of helicoidal surface is taken to be constant and equal to 27rVp/aj. 



We define a cylindrical coordinate system x^.r^.y^ fixed in space and a 

 system x,r,y advancing along the x axis with a speed Vp (Fig. 1). The angular 

 coordinate / is measured in the direction of the propeller rotation. 



The velocity potential of a single propeller blade can be written in terms of 

 a distribution of doublets whose axes are perpendicular to the helicoidal surface 

 swept out by the advancing lifting line. The strength of the doublets is equal to 

 the discontinuity of the potential [<t>^] between the upper and lower sides of the 

 helical surface, so that one is led to 



-^0 



where , 



[(X, 



V„t + V„T)- 



2r^T' cos (y^- cxjt + cot)]^^'^ 



