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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 14
LIFTING-SURFACE FLOW: PROPELLER-RUDDER INTERACTIONS, AND OTHERS

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Computation of Viscous Flow Around a Rudder Behind a Propeller: Laminar Flow Around a Flat Plate Rudder in Propeller Slipstream
H.Suzuki, (NKK Corporation, Japan)
Y.Toda (University of Mercantile Marine, Japan)
T.Suzuki (Osaka University, Japan)
ABSTRACT
The viscous flow computation of propeller-rudder interaction is presented through comparisons with experimental data including flow visualization and mean-flow measurements. The steady flow field is calculated by a viscous flow code coupled with a body-force distribution which represents the propeller. The transport equations are discretized using a staggered grid and the exponential scheme. The velocity-pressure coupling is accomplished based on the SIMPLER algorithm. Qualitative agreement is obtained between the calculations and the mean-flow data. Although the details of the flow field is different because of the laminar flow computation and numerical treatment, the computational results show the essential feature such as upward movement of propeller slipstream in port side vise versa in starboard side. The streaklines from one blade position are traced and compared with the flow visualization using dye and air bubbles. The results show very similar trends. Those comparisons show the conclusion that the present approach can simulate qualitatively the steady part of the flow field around a rudder in propeller slipstream.
NOMENCLATURE
CT
=thrust coefficient
DP
=propeller diameter
fbx
=x wise body force per unit volume
fby
=y wise body force per unit volume
fbz
=z wise body force per unit volume
fbθ
=θ wise body force per unit volume
J
=advance coefficient (=VA/nDp)
KT
=thrust coefficient
KQ
=torque coefficient
p
=pressure
Q
=propeller torque
Rn
=Reynolds number (=VADP/ν)
Rh
=hub radius
Rp
=propeller radius (=Dp/2)
T
=propeller thrust
u,v,w
=velocity components in cartesian coordinates
x,y,z
=cartesian coordinates
x,r,θ
=cylindolical coordinates
n
=number of propeller revolution
VA
=propeller advance speed
Greek symbols
Г
=circulation distribution
ν
=kinematic viscosity
ρ
=fluid density
1.
INTRODUCTION
The interaction between a propeller and a rudder is one of the major problems from the viewpoints of not only maneuverability but also propulsive performance, so numerous studies have been performed for flow field around rudder behind a propeller without rudder angle.
(Nakatake's review of this topic1) Among those studies, the flow field is calculated mainly by invicid-flow method under the assumption that the interaction is invicid, and in theoretical works the shapes of trailing vortex are assumed comparatively simple such as only consideration of propeller-slipstream contraction (Tamashima et al.2, Ishida et al.3) . On the contrary, in experimental studies, it is reported that propeller slipstream is dramatically deformed by the rudder effect from the data of mean-flow measurement by Baba et al.4 and Ishida et al.3 and from the result of flow visualization of the propeller tip-vortex by Tanaka et al.5 and Tamashima et al.2.
Recently, a lot of fin type energy-saving devices which is installed on a rudder are proposed

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
and put to practical use. (for example, NKK-SURF6, IHI A.T.Fin7). For the better understanding and improvement of the performance of such devices, it is advisable to develop the calculation model which can express the phenomena which is observed in experiments.
On the other hand, with respect to the time-averaged flow of the propeller slipstream, it is reported that the computed propeller-hull interaction flow field by the method that propeller effect is represented by the body force distribution in the computation code of Navier-Stokes equation shows good agreement with the experimental result in propeller slipstream8,9.
In this paper, the flow field with a flat plate rudder is computed by the Navier-Stokes solver coupled with analytical prescribed body-force distribution as the simplest model of propeller-rudder interaction problem. And the result are compared with experimental data for similar condition carried out in circulating water channel. Although the details of the flow field is different because of the laminar flow computation and numerical treatment, it is appeared that the method can express phenomenon which appeared in experimental data with respect to time-averaged flow. A simulation of the streakline from a rotating point on the propeller plane is similar to the flow visualization result. Moreover half domain computation and full domain computation are carried out and compared. These results are almost same for present laminar and steady flow computation.
In the presentation of the results and the discussion to follow, a Cartesian coordinate system is adopted in which x-, y- and z- axes are in the direction of the uniform flow, starboard side of the rudder and upward respectively. The origin is at the intersection of the shaft center line and the propeller plane. The mean velocity components in the direction of the coordinate axes are denoted by u,v,w. Unless otherwise indicated, all variables are nondimentionalized using the propeller diameter Dp, propeller advance speed VA and fluid density ρ. In some part, the cylindolical (x,r,θ) coordinates in which x=x,y=rcosθ and z=rsinθ are used.
2.
EXPERIMENTS
Mean-flow measurements and flow visualization were carried out in NKK Tsu Ship Model Basin, circulating water channel to compare with computational results. Experimental results are shown first to explain the phenomena of propeller-rudder interaction.
2.1
Model Rudder, Propeller and Their Arrangements
The principal dimensions of the propeller and the flat plate rudder are given in table 1. The leading edge and trailing edge of the 8mm flat plate rudder are tapered as shown in Fig. 1.
The arrangement of the flat plate rudder and propeller open boat with the propeller is shown in Fig. 2. The propeller open boat was attached front-side back so that the propeller shaft did not pass through the rudder. But this arrangement had a point that wake of the open boat was generated. So an extension shaft was attached to the propeller open boat so that the open boat did not have a large disturbance on the flow field. Length between propeller plane and the rudder leading edge was 0.30DP (66mm) and the propeller shaft center depth from the free surface is 1.50DP (330 mm) in order to minimize the free surface effect.
2.2
Mean-Flow Measurement
The device for measuring the propeller slipstream, which is unsteady flow field, should be preferably be carried out by a non-contact type Laser Doppler velocimetory or the equivalent; however, instruments of this type cannot be widely used now. Therefore, two spherical-type 5 hole pitot probes, one for the port and the other for starboard side of the center plane, were used in mean-flow measurements because it is easy to handle and able to measure time averaged velocities. Velocities were measured for the with and without rudder condition. The number of propeller revolution and the corresponding thrust and torque coefficients for both conditions are shown in table 2. Note that the thrust and torque for the with rudder condition are 4% larger and 2% smaller than those for the without rudder condition, respectively. It is similar as the other experimental data and might be due to the displacement effect of the rudder and the distortion of the trailing vortex geometry by rudder discussed later.
The mean-velocity field measurements were performed for both conditions and for three axial stations shown in Fig. 3. These locations were just behind propeller plane (x=0.125, 0.125DP downstream of propeller plane) , x=1.23 (in case of with rudder, the trailing edge of the flat plate rudder) and the position at x=2.0 (2DP downstream of the propeller plane.
The results of the mean-flow measurement of the propeller slipstream are shown in Fig. 4 and Fig. 5 for the with and without rudder condition, respectively. The mean-velocity field without

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
propeller and rudder at first measurement station is shown in Fig. 6 in order to check the effect of the open boat. Although a small region where the velocity defect is observed and it is asymmetric, the flow field is seems to be almost uniform in the present experimental region. In Fig. 4, the time averaged propeller slipstream similar to a swirling jet is observed for the without rudder condition. From the cross-plane vectors, the swirl velocity is maximum just downstream of the propeller and decays gradually with downstream distance. The crossplane vectors outside the propeller slipstream at x=0.125 show the flow direction towards the shaft center. It shows the flow contraction by the propeller. The axial velocity is increased from x=0.125 to x=1.23 and decay very gradually. The concentration of the axial velocity contours near the propeller tip is the trace of the tip vortices and its associated vortex sheet, which diffuses with downstream distance.
For the with rudder condition shown in Fig. 5, very similar velocity distribution as that for the without rudder condition is observed at x=0.125. The diffusion of tip vortex sheet and so on are similar. However, the slipstream is clearly altered due to the rudder effect at the latter two stations. The slipstream moves upward (toward the free surface) in the port side and moves downward in the starboard side. The outer shape of the slipstream in one side altered from the half circle at x=1.23. It shows that the movement seems to be larger near the rudder surface. Tanaka et al.10 explained the phenomena by the mirror image vortex due to the rudder. The propeller slipstream shows complicated shape at x=2.0 and the outer shape is enlongated in vertical direction and almost same in horizontal direction as compared with that for the without rudder condition. The cross plane velocity is smaller and the axial velocity is a little bit larger for the with rudder condition. The overall results of mean-flow measurement show similar results of Baba et al4. and Ishida3 although the one measurement was carried out behind the hull.
2.3
Flow Visualization
Flow visualization was carried out for both with and without rudder condition. Streakline from one blade position was visualized by dye method according to Nagamatsu et al.11. The device is the tank which is attached on the boss part of the propeller and dye in the tank. When the propeller is rotating, water enters from boss part inlet and colored water go out from the small diameter tube which attached propeller trailing edge by their head difference. But this method can not be used for fast flow because dye defuses immediately. So, the number of propeller revolution n was selected as 7.16 (r.p.s.) and propeller advance speed VA was 0.63 (m/s) to keep the advance coefficient J same as in mean flow measurements.
The results are shown in Fig. 7 for both conditions. The streaklines from r=0.5RP and r=0.9 RP are shown in figure; where RP is the propeller radius. For the without rudder condition (a) and c)), the helical streaklines are seen as usual and the streaklines are deformed drastically for the with rudder condition. It moves upward in port side and downward in starboard side. Note the transparent rudder enables to see streaklines in both side. The movement of streaklines is very similar to the tip vortex visualization by Tanaka et al5. shown in Fig. 8. In this figure, the air bubble method was used for visualization. The results for similar condition are shown in figure. The deformation of those streaklines is corresponding to the enlargement of slipstream in vertical direction.
In the experiment, the phenomena observed in the previous studies is reproduced for the flat plate rudder. It is explained by Tanaka et al.10 by images. So, if the nonlinear trailing vortex geometry including rudder effect is used for the calculation using iterative procedure, it can be expressed by invicid method. But, it seems difficult to treat the induced velocity at the vortex segment near the rudder. So, in this paper, the computation of Navier-Stokes equations for the time averaged flow field has been investigated if it can express the before mentioned phenomena or not.
3
COMPUTATION
3.1
Governing Equations and Computational Method
In this paper, the computation was carried out for the zero thickness flat plate rudder which had same profile as the rudder used in experiment and the time averaged flow using time averaged body force distribution following Stern et al.9. The computation was carried out for steady laminar flow case because the turbulence model in the slipstream was not clear, the present approach can not treat the complicated unsteady phenomena in the slipstream such as blade wake and so on and the grid number was limited due to the memory size of the computer. So, three-dimensional steady Navier-Stokes equations and the continuity equation are used for the governing equations. The equations are written in cartesian coordinates discussed in section 1 in the physical domain as follows;

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(1)
(2)
(3)
(4)
where p is the pressure normalized by , Rn=VADp/ν is the Reynolds number defined in terms of VA,DP and molecular kinematic viscosity ν. The terms fbx, fby and fbz in the momentum equations are the components of the body force, normalized by and represent the influence of the propeller. These will be discussed subsequently.
The equations are transformed into irregular orthogonal coordinates shown in Fig. 9. The transformed equations are discretized by exponential scheme 12 using the staggered grid in which pressure is defined at the grid point and the velocity components are defined at half grid shifted points in x,y and z direction for u,ν and w, respectively. The pressure-velocity coupling is accomplished based on the SIMPLER algorithm. The matrix is solved using tri-diagonal matrix solver and line-by-line iteration method. The steady converged solution was obtained by iterative procedure from the guessed initial condition (uniform flow except for the plate surface). After about 500 iterations, the converged solution was obtained.
103 was used for Reynolds number in the computation from the grid size discussed later. Note that the Reynolds numbers in the experiment are 2.4×105 and 1.2×105 for the mean-flow measurements and flow visualization, respectively.
3.2
Analytically-prescribed body force distribution
To represent the propeller effect in the numerical method, the body force fbx in axial direction and fbθ in circumferential direction are used corresponding to thrust and torque. Following the Stern et al.9, the body force fbx and fbθ are prescribed using the loading condition in the experiment shown in table 2. Although the computation was carried out for both with and without rudder condition and the loading conditions were a little bit different in experiment, the loading condition for without-rudder condition was used for both condition because the zero thickness plate rudder which had a small displacement effect was used in the computation. Of course, the interactive method using a invicid propeller theory is preferred and the body force should be the function of θ for the with rudder condition. But, because the computer program which can treat the nonlinear training vortex geometry for with rudder condition like the program of Ishii13 for the without propeller condition was not available, the same distribution as for the without rudder condition was used.
Following the noniterative calculation of Stern et al.9, the circulation distribution on the propeller blade of Hough and Ordway14 was used to determine body force.
The body force fbx and fbθ are written using the loading coefficient , torque coefficient and advance coefficient J (=VA/nDp) as follows;
(5)
(6)
(7)
(8)
where r*=(Y−Yh) / (1−Yh), Yh=Rh/RP and Y=r/RP; Rh is boss radius; RP is propeller radius; T is thrust; Q is torque, Δx means x direction grid size at propeller plane. And CT is easily calculated from KT .
Body force distributions from KT ,KQ and

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J for without propeller condition in table 2 are shown in Fig. 10. In the present computation, propeller is assumed without boss. fbθ is non-zero from the equation (6) at r=0. But fbθ is thought to be zero at the shaft center as shown in figure. fby and fbz are decomposed from fbθ as follows.
fby=−fbθsinθ, fbz=fbθcosθ (9)
In the computation, the fbx,fby and fbz are given at the points where u,ν and w are defined, respectively, corresponding to the pressure points at the propeller plane. So, r and θ are calculated from y and z at those points and fbx and fbθ are calculated from eq.(5) through (8).
3.3
Solution domain and Computational Grid
In the computation, the detail propeller geometry, the thickness of rudder and the rudder stock were ignored. The arrangements of the propeller and the rudder is defined as follows; The propeller plane is x=0 and the propeller disk in which the body forces are non-zero is the circle whose radius is 0.5. With respect to the flat plate rudder, leading edge is x=0.3; trailing edge is x=1.23; upper edge is z=0.65; lower is z=−0.65, and it exists on y=0 in z-x plane. Those arrangement is same as that in experiment.
Two Solution domains are used for the computation. One is the half (starboard side) domain using symmetric condition with respect to the shaft center line for the present geometry written as follows;
u(x, y, z)=u(x,−y,−z)
ν(x, y, z)=−ν(x,−y,−z)
w(x,y,z)=−w(x,−y,−z)
p(x,y,z)=p(x,−y,−z) (10)
This computation was carried out first to save the memory size. In this case, The solution domain is [−3, 3.63], [−0.05, 3.0] and [−3.0, 3.0] in x,y and z directions, respectively. The other is the full (port and starboard sides) domain. The solution domain is [−3.0, 3.0] in y direction and the same for the other direction as the half domain. Computational grid of the former case is shown in Fig. 9. The grid system of latter case extended to negative y direction as same as positive y direction. The grid numbers are (63, 32, 61) in (x, y, z) direction, respectively for half domain computation and (63, 61, 61) for full domain computation. The minimum grid spacing is 0.01 at propeller plane, rudder leading edge and trailing edge in x directions, and 0.05 in y and z directions in the region where y or z is less than 0.7. This uniform grid size near the propeller circle in cross plane is chosen from experimental results. Axial body force (fbx) is embedded at x=0.005 and y and z direction body force (fby,fbz) at x=0 in x direction due to the staggered grid system. Note fby is given at ν point and fbz at w point at x=0.0.
3.4
Boundary Conditions
The boundary conditions are as follows;
on inlet plane x=−3.0, uniform flow condition is given. u=1.0, ν=w=p=0
on exit plane x=3.63, zero-normal(axial)-gradient condition is applied.
on the outer boundary in y direction y=±3.0, ∂(u,w,p)/∂y=0, ν=0
on the outer boundary in z direction z=±3.0, ∂(u, ν, p)/∂z=0, w=0
on the rudder surface, no slip condition and zero-normal-pressure-gradient condition are imposed. u=v=w=0.0 and ∂u/∂y=0
For the half domain computation, above symmetric condition eq(10) is used for u,w,p at y=−0.05 and ν at y=−0.025 by using u,w,p at y=0.05 and ν at y=0.025 of previous iteration except for on the flat plate rudder.
4.
COMPUTATIONAL RESULTS AND DISCUSSION
Axial velocity contours and cross flow vectors of half domain computation for the with and without rudder conditions are shown in Fig. 11 and Fig. 12, respectively. The results of full domain computation are shown in Fig. 13 and Fig. 14 for the with and without rudder conditions, respectively. The velocity distributions are drawn at three axial stations; a) at x=0.01 (just downstream of the propeller plane), b) x=1.23 (at rudder trailing edge), and c) x=2.13 (grid point near the station of experiment). From those figures, two computations show almost same results although the full domain computation show a little bit asymmetry with respect to the center line. So, for the present steady laminar flow computation, half domain computation can be used to get the very similar results by small computer as those of full domain computation. In the following, the results of full domain computation are discussed.
For the without rudder condition shown in

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Fig. 13 a), b), c), the computational results show general feature of the propeller slipstream. The axial velocity is accelerated and the swirl velocity is produced by the propeller at x=0.01. The swirl velocity is maximum just downstream of the propeller and decays with downstream distance. At x=0.01, the flow toward the shaft center exists outside the propeller slipstream corresponding to the contraction. The axial velocity is increased from x=0.01 to x=1.23 and decay very gradually. The axial velocity contours also show the diffusion of the tip vortex sheet. So, the computation can capture the flow field qualitatively. In comparison with experimental results shown in Fig. 4, the acceleration of the axial velocity and swirl velocity are both underpredicted at just behind the propeller. It might be due to the coarse grid size at propeller plane and the effect of the boss. The body force distribution applied for this computation which is made based on the force measurement and Hough and Ordway circulation distribution also might be different from load distribution of mean-flow measurement condition. The acceleration and decay of the axial velocity is predicted fairly well if the underprediction at x=0.01 is considered. But the diffusion of the concentration of axial velocity contours near the propeller radius is faster than experiment. The reason is considered as the low Reynolds number computation and the numerical diffusion due to coarse grid in cross plane. The decay of the swirl velocity is much faster than that in experiment especially near the shaft center line. In experiment, the swirl velocity is larger as the position is closer to the shaft center line at all three stations. It suggests that the strong hub vortex exists, the diameter of its core is very small and its decay is slow. But in the computation, The large core of the hub vortex is observed at x=0.01 and decays very fast. It is due to the difference of Reynolds number and the coarse grid.
For the with rudder condition shown in Fig. 15 a), b), c), the computation also show the general qualitative feature of flow field. At x=0.01, the velocity distribution is almost same as that for the without rudder condition. The high velocity region moves upward in the port side and downward in the starboard side at the latter two stations. The cross plane vectors are smaller than those for the without propeller condition. In comparison with experimental results shown in Fig. 5, both axial velocity acceleration and swirl velocity are underpredicted similar as for the without rudder condition. At the rudder trailing edge, x=1.23, high velocity region has similar shape as the experimental high velocity region except near the flat plate. Near the flat plate, the cross plane velocity is small due to the laminar boundary layer and the present computation resolution. It might be the reason why the upper edge of the slipstream has different shape from the experimental results. The difference of the decay of the axial velocity contour concentration and the cross plane velocity between in experiment and in computation is almost same as for the without rudder condition. The change of the direction of cross plane vector from the without rudder condition to the with rudder condition is predicted fairly well, although the difference is observed in detail due to the thickness effect and so on. At x=2.13, comparing with the experimental results at x=2.0, the position of high velocity region is predicted fairly well, but the outer shape of the slipstream is different. The movement of the slipstream near the center plane is small due to the laminar boundary layer and wake and small cross plane velocity. So, the gap between port side and starboard looks small, but the gap of the high velocity region show similar trend. Note that the vortices whose turning direction is unti-clockwise are observed in the upper part of the port side and in the lower part of the starboard side. It is also observed in experimental result of Tanaka et al.10. It is also noted that the lower velocity region near the center plane at x=1.23 is observed in lower part of the port side and upper side of the starboard side in experiment. It might be the movement of divided hub vortex. This phenomena is also seen in the computational results, but the movement is small and not clear due to the laminar boundary layer and the diffusion of hub vortex.
To compare the computation and experiment more precisely, the velocity profiles at nine heights at x=1.23 are shown in Fig. 15 and Fig. 16 for the without and the with rudder conditions, respectively. For the without rudder condition, the axial velocity and cross flow velocity are both underpredicted as discussed previously. For the axial velocity, the shape of the acceleration is well predicted. For ν and w, overall shapes are predicted, but the w for z=0 show clearly the much larger diffusion of the hub vortex. For the with rudder condition, the three velocity components are also underpredicted. The velocity outside the boundary layer at z=0.545 is not affected by propeller in the computation although the influence of the propeller is observed in experiment. It might be due to the underprediction of the cross flow velocity. At z=0.409, the high axial velocity in the port side is predicted although the extent and the magnitude are underpredicted. At z=0.273 and 0.136, the larger region of high axial velocity and higher

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axial velocity in the port side than in the starboard side observed in experiment are predicted similar fairly well. Horizontal velocity is smaller for the with rudder condition compared with for the without rudder condition. It is similar with experiment. The profiles of the axial velocity are very similar with experiment. At z=0.0, the velocity defect due to flat plate rudder is over-predicted because of the laminar flow computation and the coarse grid. The velocity distribution of both experiment and computation has the symmetric feature written in eq.(10).
From Fig. 11 through Fig. 16, the present computation show the large diffusion and the underprediction of velocity. It might be improved by turbulence model for at least plate boundary layer and wake and the finer grid. It also might be effective to use the Euler code.
Fig. 17 and Fig. 18 show the result of streakline tracing from the rotating point at propeller plane which is rotating at the same speed as the propeller blade. It has been conducted to investigate the model of training vortices in invicid theory for the with rudder condition. The procedure is as follows; (1) from the 72 points at every 5 degree θ at prescribed r, the streamlines are traced with time step J/72 (n=1/J, where n is the number of propeller revolution in the computation DP=1.0, VA=1.0). (2) The points at propeller plane were numbered in the unti-clockwise (positive θ) direction from 0 to M from θ=0 and the 72nd point is same as the 0-th point. (3) For the streakline from 0-th point, the point at K-th time step on the streamline from K-th point (K=0,M) were drawn. Although the flow field is steady, the present streakline show the helical streaklines which are observed usually. In comparison with the flow visualization, the pitch of the helical streaklines is similar for both r=0.5RP and r=0.9RP for without rudder condition. It shows the pitch of the helical streakline is larger at r=0.5RP than r=0.9RP. For the with rudder condition shown in Fig. 18, the distortion of the helical streaklines are clear. It shows the helical streakline moves upward in the port side and downward in the starboard side. It also shows the low velocity region where the simulated air bubbles do not go further compared with outer part of the boundary layer. These streaklines show very similar phenomena as experimental results shown in Fig. 7 and Fig. 8. The streaklines which is observed from downstream show the enlargement of the slipstream in the vertical direction clearly.
Limiting streamlines on the flat plate rudder surface at the port side are shown in Fig. 19. The uniform flow direction is from left to right in this figure. Although computational grid is so coarse, it is suggestive that upward flow exists on the rudder surface at the port side. The direction of limiting streamlines is similar as the direction of tufts on the surface of flow visualization by Tanaka et al.10.
5.
CONCLUDING REMARKS
This work presents mean flow measurement data and flow visualization of the flow field around the rudder in the propeller slipstream. The numerical method and the computational results for the flat plate rudder behind the propeller which represented by body force distribution are also presented. Although the Reynolds number of the computation is small and the number of grid is limited, the salient feature of the flow field for the time-averaged flow field has been predicted by present approach. In detail, there is some discrepancy due to the laminar flow computation and the numerical treatment. The present approach can be extended easily for the flat plate rudder with angle of attack in propeller slipstream and the rudder with zero thickness fins.
Finally, some of the issues that must be addressed while further developing the present approach are as follows: improvement of accuracy in calculating the propeller flow field; introduction of appropriate turbulence model for at least the rudder boundary layer; including the rudder shape (thickness and so on) effects by using body-fitted coordinate system; using finer and appropriate grid for propeller flow field. Also of interest is to use present approach for the improvement of the invicid theory which treat the propeller-rudder interaction.
ACKNOWLEDGEMENTS
The authors wishes to thank Dr. Y.Kasahara and Mr. Y.Okamoto at NKK Tsu Laboratories for their valuable discussion and encouragement. It is noted that the numerical work in this study has been carried out on the CONVEX C-120 at NKK Tsu Laboratories and on the AV6220 at Kobe University of Mercantile Marine.

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13. Ishii, N.: “The Influence of Tip Vortex on Propeller Performance”, Journal of the Society of Naval Architects Japan, Vol.168, 1991 (in Japanese)
14. Hough, G.R., and Ordway, D.E.: “The Generalized Actuator Disk”, Developments in Theoretical and Applied Mechanics, Vol.2, Pergamon Ga., 1965, pp 317–336
15. Suzuki, H., Toda, Y., and Suzuki, T.: “Numerical Simulation of a Flow Field Around a Flat Plate Rudder in Propeller Slipstream”, Journal of the Kansai Society of Naval Architects, Japan, No.219, 1993 (in Japanese)
Table 1. Principal dimensions propeller and rudder
Propeller
Number of Blades
5
Diameter (mm)
220.0
Pitch Ratio
0.700
E.A.R.
0.600
Boss Ratio
0.170
Blade section
MAU-M
Direction of Rotation
Right
Rudder
Thickness (mm)
8.0
Chord (mm)
205.0
Span (mm)
286.0
Table 2. Propeller condition
Propeller condition
VA (m/s)
1.26
n (r.p.s.)
14.32
J
0.40
WITHOUT RUDDER
KT
1.91×10−1
KQ
2.54×10−2
WITH RUDDER
KT
1.98×10−1
KQ
2.50×10−2

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Fig.1 Profile of the rudder
Fig.2 Experimental set
Fig.3 Measuring sections

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Fig 7: Streamlines in a moving frame ( =0.6 k=0.52)
Figure 8: Surface pressure coefficient –CP ( =0.6, k=0.52)
Fig. 9: Drag (CX), lift (CY) and moment (CM) coefficients (=0.6, k=0.52)

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Fig.10: Streamlines in a moving frame (=1.2, k=0.26; small acceleration)
(Fig. 11 c)). The leading edge vortex formed during the downstroke is finally shed and improves the circulation around the foil to cause a better propulsion.
Performance
Fig. 12 and Fig. 13 present the drag, lift and moment coefficients for the discussed cases. The thrust force for the k=1.04 (Fig. 13) is larger than that of the k=0.26 (Fig. 12) and slightly larger than the ”standard” one (Fig. 9). The realized efficiency, η=0.16, is comparable with that of the standard case (η=0.10).
Small increase of the reduced frequency leads to slight improvement in the efficiency. In the cases of very high reduced frequency with small amplitude of oscillations, the efficiency drops rapidly as the thrust/lift ratio
Fig. 11: Streamlines in a moving frame (= 0.3, k=1.04; large acceleration)
Fig.12: History of drag (CX), lift (CY) and moment (CM) coefficient (=1.2, k= 0.26)

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Fig. 13: Drag (CX), lift (CY) and moment (CM) coefficient (=0.3 k=1.04)
Fig. 14: Drag (CX), lift (CY) and moment (CM) coefficient (=0.018, k=7.78)
decreases. The coefficients of the hydrodynamic loads for the case of k=7.78 and =0.018 are presented in Fig. 14. Because the inertial terms are large in this case, the viscous effect is reduced and there is no evidence of separation. The lift is completely out of phase which corresponds qualitatively to the result of Choi and Landweber [15]. However, the results differ quantitatively from those obtained by the potential flow methods. This is because the assumption of the Kutta condition is strongly violated.
Potential methods predict that the efficiency will decrease with the increase of reduced frequency (see for example Katz[17]). The results shown in Fig. 12 and Fig. 13 lead to the just opposite conclusion. The main reason is that the higher frequency decrease leading edge separation which is not included in potential schemes. When the reduced frequency is low and amplitude large, the flow is characterized by a strong leading edge separation at the early stages of heaving. Such result is caused by weak unsteady effects. This separation has strong impact on the wake and the resulted circulation. It leads to a large drag and low efficiency.
The principal result is that pure having motion may produce a propulsive force but the efficiency is low.
PITCHING MOTION
The pitching motion is performed symmetrically according to the eq.3. The reduced frequency k, angular amplitude αA and normalized pivot point location can be the parameters of motion.
Cases when the pivot point is located far ahead or far behind the foil were studied in [16]. In these cases, the foil produces thrust but small in mean value. Fig. 15 shows a typical example of the time history of the force coefficients; the pivot point is located between the leading edge and the middle of the foil chord. When the foil is pitching around an axis located near the middle of the chord, it can not produce a propulsive force.
Fig. 16 shows one of the flow patterns, when the pivot point is located at the leading edge of the foil. In this case, leading edge separation is prevented almost completely. However, large trailing edge separation is generated which induces an increase of the drag.
Comparing with the heaving motion, it is revealed that acceleration of the foil is responsible for the unsteady effects. In the

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pitching case, the acceleration is expressed by k2αA, so that the flow near the trailing edge is highly dependent on the reduced frequency and the distance to the pivot point location. It is found out that the flow changes dramatically when the pivot point is located relatively near and ahead of the leading edge.
Fig. 17: Streamlines in a moving frame (= 0.0, k=3.0, αA=15º t*=1.05)
When the oscillations are of relatively low frequency, the trailing edge separation is large to lead to a large drag force. As the frequency of oscillations increases, the inertial terms play a more significant role. Trailing edge separation decreases rapidly when the leading edge separation is negligible (Fig. 17). Wake development assists production of a propulsive force and relatively large forward thrust may be obtained. Time variations of the coefficients for this case are presented in Fig. 18. The shift in phase of the lift improves additionally the efficiency (η=0.23).
Further improvement of propulsive abilities was reached when the pivot point location was moved ahead to the leading edge by a half chord. The flow pattern is illustrated in Fig. 19. Although small but intensified separation vortices are observed around the leading edge, quite high efficiency of 0.62 is realized for this case. The coefficients of
Fig. 15: Drug (CL), lift (CD) and moment (CM) coefficients ( = 0.1, k = 0.5, αA = 15º)
Fig. 16: Streamlines in a moving frame ( = 0.0 k = 1.5, αA = 15º, t* = 1.05)

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Fig.18: Drag (CX), lift (CY) and moment (CM) coefficients (=0.0, k=3.0, αA= 15º)
induced forces are shown in Fig. 20. The combined effects of lift phasing, large suction at the leading edge, wake interaction and energy extraction from the near wake results in such higher efficiency. The present result may suggest how the pure pitching motion could be efficiently propulsive.
COMBINED MOTION
The case of combined motion is that heaving, pitching and surging motions are combined.
The ”standard case” has the same reduced frequency and heaving amplitude as the standard heaving case (k=0.52, hA= 0.6). Pitching is performed around a pivot point located at 0.25c from leading edge with the amplitude of 20º. The effective incidence angle at the pivot point location is ranging in [−15º, 15º] during one cycle. Such motion may lead to an efficient propulsive force as strong separation can not be expected.
Fig. 21 shows the flow pattern at the selected stages, visualized by the stream function contours in a moving frame fixed to the foil. Soon after the motion has begun, very small separation bubble formed earlier near the leading edge and a weak trailing edge sep
Fig. 19: Streamlines in a moving frame ( = –0.5 k = 3.0 αA = 15º, t* = 1.45)
Fig. 20: Drag (CX), lift (CY) and moment (CM) coefficients (=−0.5, k=3.0, αA= 15º)

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aration are observed (Fig. 21 a)). Stronger separations have been prevented by the foil acceleration at the initial stage. At the moment shown in Fig. 21 a) the foil already decelerates and separation of the boundary layer grows. The trailing edge vortex is shed when the bubble separates in two smaller structures which travel into downstream over the foil.
Fig. 21 b) illustrates the flow, when the foil is at the maximum amplitude and starts its motion downward. Separation is stronger on the lower side. At the following stage, the separation vortices are washed into downstream (Fig. 21 c)). A developed wake leads to delay of the flow evolution in comparison with the initial stage (Fig. 21 a)) but does not prevent the flow separation when the foil is altering the direction of the heaving motion again (Fig. 21 d)).
Such flow behavior is common and is repeated in the following cycles. Comparing with the result for the pure heaving and the pure pitching cases, the unsteady effects and wake interaction are less.
Fig. 22 shows the pressure distribution on the foil. The wavy peak of pressure is caused by the leading edge vortex and its propagation in time. Fig. 23 shows the time histories of drag, lift and moment coefficients. Qualitatively and in average, the behavior of the induced forces is comparable with the predictions made by the potential model (see for example [17]).
However, they differ in details. Separation of the boundary layer at the stages when the foil is about turning position is causing larger drag and loss of efficiency. It seems that viscous effects are considerable for the flows whose Reynolds number is around 0.5 ×104.
Parametric Study and Performance
Parametric studies are carried out by changing the frequency k, the amplitude heaving and pitching hA, αA, and the position of pitching axis (Table1). C0 is the standard case.
Fig. 21: Streamlines in a moving frame (=0.6, k=0.52, αA=20º)

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Fig. 22: Surface pressure coefficient −CP (=0.6, k=0.52, αA= 20º)
Fig. 23: Drag (CX), lift (CY) and moment (CM) coefficient (=0.6, k=0.52, αA= 20º)
Table 1: Cases for the parametric studies of combined motion.
Case
k
hA
αA
η
C0
0.52
0.6
20º
0.25
0.47
C1
0.26
1.2
20º
0.25
0.51
C2
1.04
0.3
20º
0.25
0.41
C3
0.52
0.6
20º
0.00
0.55
C4
0.52
0.6
20º
0.50
0.48
C5
0.52
1.2
40º
0.25
0.61
C6
0.52
2.4
60º
0.25
0.55
The effects of the reduced frequency and the heaving amplitude are studied by the cases C1 and C2. The effective incidence angle for these two cases is the same as the standard case. In case C1, the leading edge separation appears earlier due to the smaller dynamic effects. It covers a larger domain (Fig.24 a)) comparing with the standard case. The magnitude of induced forces is bigger as no stall effect is observed. When the foil is altering its direction of heaving, the energy of the shedding vortices is partially extracted back as shown in Fig.24 a). All these facts result in an improvement of efficiency (refer Table.1).
In the case C2 where the acceleration and deceleration is high, the dynamic effects are stronger. The phase of acceleration delays leading edge separation much longer Fig.24 b). At the same time the trailing edge separation is more intensive to cause a worse performance.
The dependency on the pivot point location is investigated by the cases C3 and C4. In the case C3 where the pivot point is at the leading edge, the leading edge separation is prevented (Fig.24 c)). In the case C4 an opposite tendency is observed as shown in Fig.24 d). The leading edge separation appears earlier due to the larger angular deceleration of the leading edge. Its reattachment leads to a trailing edge vortex. When the foil approaches the position of the maximum amplitude, the case of C4 has a worse efficiency.

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After the turn, the flow near the leading edge is accelerated, so that the disturbances are washed easier and the foil starts to produce thrust earlier than the other cases. Such a compensation results in a similar efficiency compared with the standard case.
The effect of pitching and heaving amplitude is studied in the case C5. Larger amplitude of heaving combined with larger pitching amplitude may be more efficient. The main reason is that the foil inclination compensates the increase of the effective incidence angle to make the thrust component of the reaction force larger. If the energy losses are not so large during the turn, a better efficiency could be obtained. Leading edge separation appears earlier and forms a series of vortices (Fig. 24 e)). The mean value of the ratio CX/CY is very high. When the foil approaches its heaving amplitude, the separation grows considerably. The shedding of the edge vortices leads to an energy extraction. However, much larger amplitude fails to increase the efficiency as seen in the case C6. The separation is much more intensive and remain longer after the foil is altering the direction of heaving. It causes a large drag force. This fact leads to a worse efficiency.
GENERAL DISCUSSION
Parametric studies described in the previous sections indicate that flow phenomena depend much more on the parameters of the motion than on the type of motion itself such as heaving, pitching and surging. Driving parameters have a complicated and interrelated influence on the flow pattern and induced forces. In the light of obtained results, conclusions like ”as the reduced frequency increases, the mean thrust increases while the efficiency decreases.” seem to be ambiguous. It could be true or could not be as it depends on the other parameters and all complexity of dynamic effects. For example, the resultant effective angle of attack, strong trailing and leading edge separation and stall depend much on the parameters. When driv-
Fig. 24: Parametric study of combined motion. Streamlines for the middle of the second cycle

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ing parameters postulate a foil motion with significant foil acceleration and deceleration, strong unsteady dynamic effects govern the flow evolution. When the foil is accelerating, leading edge separation is prevented or postponed. If a separation appears at this stage, it produces intensive vortical structures which evolve rather by diffusion than by convection. On the other hand, the deceleration assists separation process. Quite different flow phenomena are observed for the cases characterized by small amplitude oscillations with extremely high reduced frequency Strong inertial forces govern the flow pattern overwhelming viscous effects. They will dump transition and turbulent effects as well.
From this point of view, the results obtained by a potential method may be questioned even qualitatively. Obviously, such methods will provide unrealistic results and misleading conclusions for strong unsteady viscous effects as studied.
The Reynolds number may affect the results not only quantitatively but qualitatively as well. Its effect is studied for the combined motion (case C0). Computations were carried out for Re=1.0×103 and Re=1.0×104, including Re=0.5×104 for the case C0. Its influence on the flow pattern is illustrated in Fig. 25 as in Fig. 21 d) for the standard case. It is well seen that the effect is not negligible even within this small variation. In the case of Re=1.0×103, the leading edge separation vanishes but the trailing edge separation is much stronger. For Re=1.0×104 the leading edge separation is stronger, but the scale is smaller and more intensive. The flow reat-taches faster. Such qualitative differences are valid all the way during the motion. As a result a better efficiency (η=0.54) is realized at higher Reynolds number. In the case of low Reynolds number the efficiency dropped to η=0.30.
The time histories of force coefficients are shown in Fig. 26 a) and b) for Re= 1.0×103 and Re=1.0×104. the low and high Re respectively. Comparing efficiency
Fig. 25: Effect of Reynolds number on the flow pattern (=0.6, k=0.52, αA=20º)
coefficients for the three Reynolds numbers we may suggest that at really high Reynolds numbers the propulsion may be of considerable for the practice efficiency.
Except the cases of pitching with high frequency around a pivot point in front of the leading edge, the only motion with considerable high efficiency is the combined one. The efficiency is very sensitive to the variations of parameters. An extensive research is needed to specify in details the best combination of parameters for a more efficient propulsion system.
CONCLUSION
Numerical simulation of viscous unsteady flow around an oscillating 2D hydrofoil is performed. An implicit finite difference method is implemented. Main flow features and its dependency to the foil motion are investigated. Obtained results lead to the following main conclusions:
Pilot computations and comparison of the numerical results with available experimental data concluded that the ba

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Fig. 26: Effect of the Reynolds number on the indiced forces ( =0.6, k=0.52, αA= 20º)
sic flow features are modeled with sufficient accuracy to merit physical analysis. However, the accuracy simulations may be limited with respect to the used grid resolution.
Basic flow features depend strongly on the main parameters governing the foil motion. The parametric study indicated their interrelated influence. The foil acceleration is revealed as a major physical parameter for the unsteady and viscous effects.
The ability of an oscillating foil to produce forward thrust is in direct connection with the foil motion and resulted flow. Pure heaving motion can produce thrust but of limited efficiency. This conclusion is valid for the pitching motion, except the cases of very high frequency and the pivot point located slightly ahead of the leading edge of the foil.
Combined foil motion can produce thrust force with high efficiency. Simulated results show the complicated nature of thrust production and the heavy dependence of the efficiency on the parameters of the motion. Further investigations may be needed for tuning these parameters if efficiency has to be improved.
Complicated flow phenomena lead to the conclusion that 3D effects as well as turbulent effects may play an important role.
References
[1] McCroskey, W.J., ”Unsteady Airfoils,” Annual Review of Fluid Mechanics, 1982, pp. 285–311.
[2] Carr, L.W., ”Dynamic Stall Progress in Analysies and Prediction,” AIAA Paper 85–1769CP, 1985.

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[3] Lighthill, M.J., ”Note on the swimming of slender fish,” Journal Fluid Mechanics, Vol. 9, 1960, pp. 305–317.
[4] Wu, T.Y., ”Swimming of waving plate,” Journal Fluid Mechanics, Vol. 10, 1961, pp. 321–344.
[5] Lighthill, M.J., ”Biofluiddynamics of balistiform and gymnotiform locomotion. Part 2. The pressure distribution arising in two-dimensional irrotational flow from a general symmetrical motion of a flexible flat plate normal to itself,” Journal Fluid Mechanics, Vol. 213, 1990, pp. 1–10.
[6] Karpouzian, G., Spedding, G. and Cheng, H.K., ”Lunate—tail swimming propulsion. Part 2. Performance analysis,” Journal Fluid Mechanics, Vol. 210, 1990, pp. 329–351.
[7] Chopra, M.G., ”Hydromechanics of lunate—tail swimming propulsion,” Journal Fluid Mechanics, Vol. 64, 1974, pp. 375–391.
[8] Chopra, M.G., ”Large amplitude lunate— tail theory of fish locomotion,” Journal Fluid Mechanics, Vol. 74, 1976, pp. 161– 182.
[9] Chopra, M.G., ”Hydromechanics of lunate—tail swimming propulsion. Part 2,” Journal Fluid Mechanics, Vol. 79, 1977, pp. 49–69.
[10] Lan, C.E., ”The unsteady quasi—vortex—lattice method with applications to animal propulsion,” Journal Fluid Mechanics, Vol. 93, Part 4, 1979, pp. 749–765.
[11] Cheng, J.Y., Zhuang, L.X., Tong, B.G., ”Analysis of swimming three—dimensional waving plates,” Journal Fluid Mechanics, Vol. 232, 1991, pp. 341–355.
[12] Kudo, T., Kubota, A., Kato, H., Yamaguchi, H., ”Study on Propulsion by Partially Elastic Oscillatinf Foil. 1st Report. Analysis by Linearized Theory,” J. Soc. Naval Arch. of Japan, Vol. 156, Nov. 1984, pp. 82–91. (in Japanese).
[13] Kubota, A., Kudo, T., Kato, H., Yamaguchi, H., ”Study on Propulsion by Partially Elastic Oscillatinf Foil. 2nd Report. Numerical simulation by singularity distribution method and evaluation of scope for application to ship propulsion,” J. Soc. Naval Arch. of Japan, Vol. 156, Nov. 1984, pp. 82–91. (in Japanese)
[14] Mehta, U.B., ”Dynamic Stall of an Oscillating Airfoil,” AGARD Paper 23, Unsteady Aerodynamics, AGARD CP-227, Sept. 1977.
[15] Choi, D.H., Landweber, L., ”Inviscid Analysis of Two-Dimensional Airfoils in Unsteady Motion Using Conformal Mapping,” AIAA Journal, Vol. 28, No. 12, Dec.1990, pp. 2025–2033.
[16] Videv, T.A. and Doi, Y., ”Numerical Study of the Flow and Thrust Produced by a Pitching 2D Hydrofoil,” J. Soc. Naval Arch. of Japan, Vol. 172, Nov. 1992, pp. 165–174.
[17] Katz, J. and Weihs, D., ”Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility,” Journal Fluid Mechanics, Vol. 88, Part 3, 1978, pp. 485–497.