On the Effect of a Rudder on Propulsive Performance

(昭和 56 年 11 月日本造船学会秋季講演会において講演)

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 estimation method9) on the performance of the propeller-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 substitute 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 obtain as

( 4 ),

Photo. I Visualization of propeller and rudder 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 rudder 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).

In the case that the propeller and the rudder

are in the wake, the pressure on the surface of the rudder and the viscous head loss due to the

rudder are defined by pi and ƒÏgƒÂHR, where g (23). represents the acceleration of gravity, and then The rudder surface normal vector and the inflow pi is expressed as velocity vector on the rudder surface (SR) are denoted by nR=(nx, ny, nz) and VR =( [V x*](SR), [V y*](SR) [Vz*](SR)) respectively, and then the surface condition is expressed as

(24). (29).

Kutta condition of the rudder can be considered Denoting the surface of the rudder as the differ- as the boundary condition on the free vortex ence of the pressure ‡™p, ‡™p is defined as sheet shedding from the trailing edge at a only short distance. Denoting the normal vector (30). on the sheet and the inflow velocity vector on Considering the viscous head loss due to the the point by nF= (0, 0, 1) and VR*=([Vx*](TE), rudder, the velocity on the rudder surface is [Vy*](TE),[Vz*](TE)), Kutta condition is expressed expressed as as (31). (25). 2.6 Boundary layer on a rudder The constant dipole distribution ƒÊ(ƒÌ,ƒÅ) has to The boundary layer on the rudder behind the be taken on the free vortex sheet along the rotating propeller can be estimated by applying constant y-line, but the arbitrary dipole dis- the 2-D boundary layer theory13)14) to the flow tribution on the rudder surface can be adopted. field in each section (xz-plane) of the rudder. when the distribution is continuous at the trail- The distance from the stagnation point to the ing edge. Therefore, ,ƒÊ(ƒÌ,ƒÅ) is devided into rudder surface point O' (xR, yR, zR) along a con- due to the distribution pattern and μ0(ξ,η) stant y-line is defined by S. The component on B(ƒÅ) due to the strength, and is expressed as the xz-plane of the velocity at O' is adopted as (26). the velocity at the outer edge of the boundary layer Ue. Futher, the momentum thickness, the Solving the equations (18), (20), (24), (25) on shape factor and skin frictional coefficient are the unknown quantities ƒ¡ (ƒÁ, ƒÆ), h(ƒÁ), ƒÐ(ƒÌ, ƒÅ), denoted by ƒ¦, H, Cf, respectively, and v re- B(ƒÅ), the disturbed velocity potentials ƒÓP, ƒÓR presents the kinematic viscousity of water. The can be obtained. Denoting the velocity vector symbols l, t indicate the quantities in the laminar of the propeller-rudder interactive flow field by and turbulent regions respectively.

VI, VI is expressed as The laminar boundary layer on the rudder

can be estimated by applying Thwaits' method as

(27). 2.5 Pressure and velocity field on a rudder The pressure in the non-uniform wake (VA+ uc, uy, uz) without a propeller and a rudder is (32), On the Effect of a Rudder on Propulsive Performance 67 where where CPD is estimated experimentally based on the results of the propeller open test and is given as16)

(42),

where (33).

At the laminar separation point, the point S indicating the K-value as

(34)

is adopted. And then, at the transition point,

by using Cebeci-Smith's formula, S indicating

the Rƒ¦-value as (35) is adopted, where (43). (36). The rudder drag is denoted by FRx. And

When the laminar separation or the transition denoting the pressure component due to the occurs, it is assumed that the laminar flow potential interaction, the viscous pressure com- translates into the turbulent flow at that point. ponent and the frictional component by FL,

The symbol tƒÁ indicates the quantities at this F(VP)Rx,F(F)Rx, respectively, FRx is expressed as point. And the turbulent region is connected (44), with the laminar region15) as where

(37).

The turbulent boundary layer on the rudder (45). can be estimated by applying Head's method as

(38)

(39) The viscous head loss pgƒÂHR can be estimated by the experimental data and Cf is estimated by

the results of the rudder boundary layer calculation. The propeller advance ratio J, the pro-

peller thrust coefficient KT, the torque coefficient KQ and the rudder drag coefficient KFX are ex-

(40). pressed as

2.7 Interactive forces on a propeller and a (46). rudder Denoting the sectional drag coefficient of the 3. Procedures for numerical calculation propeller blade by CPD, the thrust T and the torque Q are expressed as The propeller disk and the rudder surface are devided into the elements in the numerical calculation (shown in Fig. 3). The performance of propeller is estimated by the infinitely bladed propeller theory and the performance of the rudder is estimated by applying the thick wing theory by J. L. Hess in the non-uniform propeller slip stream. It is assumed that the slip stream does not contract itself and expand by the existence of the rudder. As the quantity of the term due to the rudder dipole distribution

μ0 (ξ, η), we adopt the value in the proportion (41), to the distance from the trailing edge on the 68 日本造船学会論文集第150号

Fig. 3 Element division on propeller and rudder surfaces

Fig. 4 Dipole distribution on rudder and Fig. 5 Calculation flow of propeller and free vortex sheet rudder interactive flow field

rudder surface and the constant value on the

free vortex sheet (shown in Fig. 4).

The propeller-rudder interactive flow field is computed according to the flow chart (shown in

Fig. 5). When the geometric forms of a propeller and a rudder are given, at first the performance

of the propeller without a rudder and the inflow

velocity to a rudder are computed as the initial values. And the performance of the rudder and

the induced velocity on the propeller are com-

puted. The interative operation of the procedure is executed by adopting h(ƒÁ) as the itera-

tive parameter and the solution to be satisfied the boundary conditions can be obtained after

the convergence. Each components of the drag acting on the

rudder behind a propeller is computed, accord-

ing to the procedure shown in Fig. 6. At first, the pressure distribution on the rudder (‡™p+ Fig. 6 Calculation flow of the drag acting ρgδHR) is computed by using the numerical on rudder behind propeller result") of the interactive flow and F(P)Rx, is

estimated. The viscous pressure correction is

done experimentally,11) considering the experi- 4. Numerical examples and comparison mental results of the rudder surface pressure , and with experimental results F(VP)RX is estimated. Then the modified ve- 4.1 Propeller-rudder interaction in uniform locity on the rudder surface V1 to the modified flow pressure ‡™p can be obtained, and the boundary The measurement of the forces (propeller layer on the rudder is estimated by applying the thrust, torque and rudder drag) and the rudder 2-D theory and F(F)Rx is computed by using Cf, surface pressure and the visualization of the the result of the boundary layer calculation. The flow in the propeller-rudder interaction test were total rudder drag FRx can be obtained by sum- carried out in the cavitation tank of the Ship ming up the three components. Research Institute. Tables 1, 2 show the princi- On the Effect of a Rudder on Propulsive Performance 69

Table 1 Principal dimension of propeller model

Table 2 Principal dimension of rudder models

Fig. 8 Effect of rudder thickness on propeller and rudder interactive forces (J=0.405, l/D= 0.291)

Fig. 9 Effect of propeller-rudder gap on propeller and rudder interactive flow (J=0.405, MR-2)

factors on the propeller-rudder interaction. Figs. Fig. 7 The measurement points of rudder 11-1 to 11-5 show the pressure distributions on surface pressure the rudder behind the propeller. Fig. 12 shows the numerical result of the velocities on the pal dimensions of the propeller model and the rudder behind the propeller and Fig. 13 shows rudder model. Fig. 7 shows the pressure meas- the flow direction on the rudder visualized by uring holes on the rudder. In the experiments, surface tufts. The propeller slip stream expands the number of propeller revolution was con- on the rudder surface. trolled so as to correspond to the Reynold's 4.2 Propeller-rudder interaction behind a hull number Rnk(Kempf) =0.48 •~ 106. Figs. 8, 9, 10 The propeller loading test with the measure-

show the effects of the propeller-rudder gap, ment of the rudder drag in the non-uniform wake advance ratio and the rudder thickness on the behind a hull was executed in the towing tank. interactive forces. The gap and the rudder we adopted a moderate speed cargo ship model thickness are understood to be the important and the length Lpp is 6 meters and the block 70 日本造船学会論文集第150号

Fig. 10 Effect of propeller load on propeller and rudder interactive forces (l/D= 0.291, MR-2)

Fig. 11-1 Pressure distribution Fig. 11-2 Pressure distribution Fig. 11-3 Pressure distribution on rudder behind propeller on rudder behind propeller on rudder behind propeller (CHORD-A) (CHORD-B) (CHORD-C) (MP-B/MR-2, J=0.501, (MP-B/MR-2, 1=0.501, (MP-B/MR-2, T= 0.501, l/D = 0.500) l/D = 0.500) l/D = 0.500)

Fig. 11-4 Pressure distribution Fig. 11-5 Pressure distribution on rudder behind propeller on rudder behind propeller (CHORD-D) (CHORD-E) (MP-B/MR-2, J=0.501, (MP-B/MR-2, J=0.501, l/D ＝ 0.500) ｌ/D = 0.500) On the Effect of a Rudder on Propulsive Performance 71

Table 4 Principal dimension of rudder models

Fig. 12 Numerical results of the velocities on rudder surface ( 0.405, l/D = 0.291,MR-2)

Fig. 14 Wake distribution (Fn=0.22)

Fig. 13 Visualization of propeller-rudder interactive flow by surface tuft (MP-B/MR-2, J=0.405, l/D= 0.291)

Table 3 Principal dimension of propeller Fig. 15 The arrangement of propeller and model rudder behind hull

coefficient CB is 0.742. Fig. 14 shows the wake distribution at the propeller disk without a rudder. Tables 3, 4 show the principal dimensions of the propeller model and the rudder model. Fig. 15 shows the arrangement behind the hull. Figs. 16, 17 show the effective wake fraction 72 日本造船学会論文集第150号

mental results. Furthermore the rudder effects on manoeuvrability should be investigated and the method estimating overall performance of rudders should be developed. The author would like to express his deep ap- preciation to Prof. Ryusuke Yamazaki, Kyushu University, for his theoretical guidance and steady encouragement, and then feels very grateful to Dr. Hajime Takahashi and the staffs of the Propulsion Division in the Ship Research Institute for their help. Numerical calculation was carried out by FACOM M180IIAD of the Ship Research Institute.

References 1) Yamagata, M., Kikuchi, Y.: Experiments on the Mutual Action between Propeller Fig. 16 Effective wake fraction by rudder and Rudder, Journal of the Society of (Results of propeller loading tests Naval Architects of Japan, Vol. 52 (1933). with and without rudder, Fn= 0.22) 2) Okada, S.: On the Results of Experi- ments on Model Rudders in the Propeller Race, Journal of the Society of Naval Architects of Japan, Vol. 104 (1958). 3) Takagi, M.: A Theoretical Study Con- cerning Mutual Interference of Propeller and Rudder, Journal of the Society of Naval Architects of Japan, Vol. 109 (1961). 4) Isay, W. H.: Uber die Wechselwirkung zwischen Schiffsruder andSchraubenpro- peller, Schifftechnik, Bd.12 (1965). 5) Yamazaki, R.: On the Propulsion Theory of Ships on Still Water-Introduction-, Memoirs of the Faculty of Engineering, Kyushu University, Vol. 27 (1968). 6) Nakanishi, M., Ueda, K., Yamazaki, R.: On the Interaction between a Propeller and a Rudder, Transactions of the West- Japan Society of Naval Architects, No. 36 (1968). 7) Nakatake, K., Arimura, F., Yamazaki, R.: Fig. 17 Rudder drag (Results of propeller Effect of a Rudder on Propulsive Per- loading tests, F=0.22) formance of a Ship, Transactions of the West-Japan Society of Naval Architects, No. 55 (1978). 1 - wT,and the rudder drag FRx corresponding to 8) Nakatake, K., Koga, T., Yamazaki, R.: the variation of the propeller thrust. In this On the Interaction between Screw Pro- computation, it is assumed that [Vx*](SR) is peller, Transactions of the West-Japan equal to [Vx*](sR) and [Vy*] and [Vz*] are neg- Society of Naval Architects, No. 61 (1981). lected. 9) Moriyama, F., Yamazaki, R.: On the Effect of Propeller on Rudder, Transac- 5. Concluding remarks and tions of the West-Japan Society of Naval acknowledgement Architects, No. 61 (1981). In the preceding section, the method to calcu- 10) Hess, J. L.: Calculation of Potential Flow late the performance of the propeller-rudder about Arbitrary Three Dimensional Lifting interaction by applying the thick wing (rudder) Bodies, Douglas Report MDC-J5679 (1972). theory was developed and the numerical results 11) Moriyama, F., Yamazaki, R.: On the were compared with the several kinds of experi- Forces Acting on Rudder in Propeller Slip On the Effect of a Rudder on Propulsive Performance 73

Stream, Transactions of the West-Japan 14) Ohji, M.: Calculation of Turbulent Flow Society of Naval Architects, No. 62 (1981). - The Progress of Hydrodynamics, edited 12) Moriyama, F., Sugai, N.: On the pro- by Tani, I.-, Maruzen Corporation, Tokyo pulsive Performance of a Ship with a (1980). Rudder - Propeller Loading Effect-, Re- 15) Schlichting, H.: Boundary Layer Theory, port of Ship Research Institute, Vol. 18 6-Edition, McGraw-Hill Book Company, (1981). New York (1968). 13) Cebeci, T., Bradshaw, P.: Momentum 16) Moriyama, F.: On an Approximate Nu- Transfer in Boundary Layers, Hemisphere merical Method for Estimating the Per- Publishing Corporation, Washington formance of Marine Propellers, Report of (1977). Ship Research Institute, Vol. 16 (1979).