63 (昭 和 56 年 11 月 日本 造 船 学 会 秋 季 講 演 会 にお い て 講 演) On the Effect of a Rudder on Propulsive Performance by Fumio Moriyama, Member* Summary In designing a ship, it is important to estimate the effects of the propeller-rudder interaction, as the rudder behind a propeller has a great effect on the propulsive performance of a ship. This interaction problem has been treated theoretically by applying the thin wing theory to the rudder, and the hydrodynamical forces have been discussed. In this paper, the author develops the estimation method on the propeller- rudder interaction by applying thick wing theory and boundary layer theory to a rudder with thickness. And then the forces, the velocity field, and the pressure on the rudder are discussed. pressure and the effect of the boundary layer in 1. Introduction addition to the forces. Furthermore the author Since the effect of a rudder on the propulsive formulizes the pressure on the rudder behind a performance was investigated experimentally by rotating propeller, and then develops the meth- Yamagata-Kikuchi1), a lot of experimental od11)to estimate the potential and viscous com- works2) have been done in order to improve the ponents of the rudder drag by using the rudder propusive efficiency due to the rudder behind a pressure formula and the boundary layer theory. propeller. And he carried out some measurements of the As developing the theory and the numerical forces and the pressure on the rudder in the procedure to estimate the performance of marine propeller-rudder interaction test in the cavita- propellers, this problem was treated theoretically tion tank and the loading test with the measure- by Takagi3), Isay4), Yamazaki5), as the propeller- ment of the rudder drag12) in the towing tank, rudder interaction problem, by applying the and compares the numerical results with the rudder theory based on the lifting line or lifting experimental results. surface wing theory, and discussed qualitatively on the hydrodynamical forces. Nakanishi-Ueda- 2. Fundamental theory Yamazaki6) solved the interaction problem by We consider a thick rudder with zero helm using the theoretical rudder model based on the angle behind a operating propeller (Fig. 1). The thin wing theory and Michell approximation, and propeller and rudder are in the non-uniform suggested the effect of the rudder thickness on wake and the propeller rotates at a constant the interaction. Further, Nakatake et al.7)8) number of revolution. reformed the numerical model based on the thin 2.1 Coordinate systems wing theory to obtain the estimation on the We substitute the infinitely bladed propeller effects of the rudder load and displacement with highly accuracy, and compared the numerical results with the experimental results of the basic experiments and the self propulsion tests, on the hydrodynamical forces. In this paper, the author develops the estima- tion method9) on the performance of the pro- peller-rudder interaction by applying the thick wing theory10) as a rudder theory so as to discuss the flow field near the rudder, the rudder surface * Ship Research Institute, Ship Propulsion Division Fig. I Coordinate systems 64 日本 造 船 学 会 論 文 集 第150号 for the ordinary propeller with a finite number of where Cd represents the correction factor of the blades, from the view point of the estimation cascade effect and is nearly equal to 0.96 in the on the propulsive performance. And then the case of ordinary marine propellers. We sub- stitute the propeller neglected the rake, the propeller-rudder interaction problem can be dealt with as a steady problem. skew and the wing thickness which has a infinite As the main coordinate system, a rectangular number of blades and maintains the radial dis- coordinate system O-xyz fixed to the center tribution of Nc(ƒÁ) and 2ƒÎƒ¿(ƒÁ) for the ordinary point of propeller O is defined. The x-axis is propeller. And then the propeller plane (SP) taken on the propeller shaft center line and is expressed as directed from the propeller to the rudder. The ( 5 ), y-axis is taken upward positive, and the z-axis is directed as O-xyz is a right hand coordinate where the propeller plane (SP) is defined by the system. As the auxiliary coordinate system to yz-plane expressed as ƒÁ=0.7ƒÁ0 on the generator represent the propeller, a cylindrical coordinate line. system O-xyƒÆ is defined. The relationship be- The rudder has the chord length 2xR, the span tween O-xyz and O-xrƒÆ is expressed as length 2yR and the maximum thickness tR, and the distance between the leading edge of the ( 1 ). rudder and the propeller disk is denoted by 1. As the auxiliary coordinate system to repre- The surface of the rudder is expressed as sent the rudder, a rectangular coordinate system O'-ξ ηζ fixed to the rudder surface 0'(xR, yR, zR) is defined. The a-axis is directed to the outer- where ( 6 ). ward normal, and the a-axis is taken on the 2.3 Velocity potential and propeller slip stream tangential plane of the rudder surface. By using We deal with the case that the propeller and the transfer matrix (E), the relationship between the rudder are in the non-uniform wake and the O-xyz and O '- ξηζ is expressed as propeller rotates at a constant anguler velocity Ω (=2πn) in the direction of negative ƒÆ. The ( 2 ). non-uniform inflow to the propeller can be devided 2.2 Expression of propeller and rudder into the main uniform flow having the constant The propeller has the number of blades N, velocity VA and the disturbed flow dependent radius ro(=D/2), boss radius ƒÁB. The geometrical on space. The x, ƒÁ, ƒÆ-directional components of pitch and the effective pitch are denoted by the disturbed flow are denoted by ux(ƒÁ, ƒÆ), 2ƒÎp(ƒÁ) and 2ƒÎƒ¿(ƒÁ). The chord length, the length uγ(γ,θ), uθ(γ,θ), respectively. between the leading edge and the point of the The propeller can be represented hydrody- maximum wing thickness, the height from the namically by bound vortices distributing on the base line to the nose-tail line at the leading edge propeller disk (SP) and having the axis in the and the trailing edge, and the maximum wing direction of the propeller radius, and free vortices thickness are denoted by c(ƒÁ), c'(ƒÁ), yi(ƒÁ), yt(ƒÁ) shedding from them rearward. The strength and t(ƒÁ), respectively. The zero-lift angle ƒ¿g1(ƒÁ) of the bound vortices is denoted by ƒ¡(ƒÁ, ƒÆ). It from the base line is expressed as is assumed that the free vortices are shedding rearward on the herical and non-contractive plane having a constant pitch 2ƒÎ-h(ƒÁ). Photo. 1 ( 3 ). The effective pitch 2ƒÎƒ¿(ƒÁ) is approximately ob- tain as ( 4 ), Photo. I Visualization of propeller and rud- der interactive flow field by pro- Fig. 2 Propeller blade section peller tip vortex cavitation On the Effect of a Rudder on Propulsive Performance 65 shows the visualization of the propeller and direction of normal to the rudder surface (SR) rudder interactive flow by using the propeller or the rudder free vortex sheet (SF). The free tip cavitation. The propeller tip vortices, visu- vortex sheet (SF), smoothly shedding from the alized by the tip vortex cavitation, are not in- trailing edge are expressed as fluenced greatly by the rudder and shed rearward (SF) z=0 in maintaining a constant herical pitch. The where disturbed velocity potential due to the infinitely (12). bladed propeller ƒÓp is expressed as The disturbed velocity potential due to the rud- der OR is expressed as (13) ( 7 ), where (14) G p(x , y, z; ƒÁ ', ƒÆ') (15), where ( 8 ). (16). In the flow field on and behind the propeller disk (SP), there exists a non-potential flow due to the vortices. The x,ƒÁ,ƒÆ-directional com- 2.4 Boundary condition ponents of this flow wix, Wir, wie are expressed Denoting the x, ƒÆ-directional components of the as inflow velocity on the propeller by [Vx*](sp), [V ƒÆ*](sp), we have (17). The boundary condition on the propeller disk (SP) is expected as ( 9 ). The correction of the induced velocity on (SP) for a finite number of blades has to be done. Therefore, for the region x=0, ƒÁB •… ƒÁ •… ƒÁ0, the (18), non-potential velocity components on the pro- where k1 represents the lifting surface correc- peller blade are modified by Prandtl's tip cor- tion factor of the lift slope influenced by the rection factor ƒÈ(ƒÁ, h) as chord length, and is expressed as (10), (19). The condition on shedding of the propeller free vortices is expressed as (11). (20) In the propeller slip stream, the rudder with zero helm angle acts as a lifting body. Regard- ing the rudder as a thick wing, both of the effects of the displacement and the lift have to be (21), considered. The rudder can be represented hy- drodynamically by the source distributed on (SR) and by the dipole having the axis in the 66 日本 造 船 学 会 論 文 集 第150号 where A indicates the correction factor on the defined by po, and the density of water is defined herical pitch of the free vortices and is estimated by ƒÏ. And in the case that the propeller rotates experimentally as itself in the non-uniform wake, the pressure is defined by pi*. And then pi* is expressed as (22). Denoting the x, y, z-directional components of the inflow velocity on the rudder surface (SR) by [-Vx*]SR),[Vy*](SR), [Vz*](SR), we have (28).