High Speed Vessel Rudders

Journal of Marine Science and Technology, Vol. 13, No. 1, pp. 61-72 (2005) 61

Short Paper

INVESTIGATION OF HYDRODYNAMICPERFORMANCE OF HIGH-SPEED CRAFT

RUDDERS VIA TURBULENT FLOWCOMPUTATIONS, PART I: NON-CAVITATING

CHARACTERISTICS

Shiu-Wu Chau*, Jen-Shiang Kouh**, Teck-Hou Wong***, and Yen-Jen Chen****

Paper Submitted 10/07/04, Accepted 05/05/05. Author for Correspondence:Shiu-Wu Chau. E-mail: [email protected].*Assistant Professor, Department of Mechanical Engineering, Chung Yuan

Christian University.**Professor, Institute of Engineering Science and Ocean Engineering,

National Taiwan University.***Graduate Student, Institute of Engineering Science and Ocean Engi-

neering, National Taiwan University.**** Assistant Professor, Department of Multimedia and Game Science,

Lunghwa University of Science and Technology.

Key words: high-speed craft rudder, non-cavitating hydrodynamiccharacteristics, turbulent flow computation, computationalfluid dynamics.

ABSTRACT

This paper focuses on the numerical analysis of non-cavitatinghydrodynamic characteristics of practical rudders used for high-speedcrafts, which is valid in the low-speed region. The force and momentacting on rudder are calculated by integrating the shear force and pres-sure force on the rudder surface, which are obtained from the turbulentflow calculation around rudder. The non-cavitating hydrodynamiccharacteristics of NACA rudders, as well as rudders using simple pro-files, are computed and then compared. Three non-dimensionalcoefficients, the lift, drag and stock moment coefficient of rudder, areevaluated to investigate the influences of profile shape and profilethickness. The influences of rudder’s geometrical parameters on thestall characteristics are also discussed. According to the predictedresults, the location of maximum thickness is the most importantfactor to influence the non-cavitating hydrodynamic characteristics.For all studied rudders, the thicker rudder generates larger stockmoment despite of the profile difference.

INTRODUCTION

For high-speed crafts, cavitation is an inevitable

phenomenon occurred at propeller and rudders. Thecavitating characteristics of rudder are valid for thehigh-speed region, while the non-cavitating character-istics of rudder are applicable in the low-speed region.The presented paper focuses on the non-cavitatinghydrodynamic characteristics, where cavitation isneglected. The cavitating characteristics of rudderare also very important for high-speed craft rudders,and are definitely different from the non-cavitatingcharacteristics. The cavitating performances of rudderswill be discussed in the second part of this study andpresented in a separate paper, where the turbulent cavi-tating rudder flow is computed through the coupling ofRayleigh-Plesset equation to take the influence of cavi-tation effect into account. Many experimental andnumerical analyses are conducted in the past to investi-gate the performance of low aspect ratio NACA foilsboth in fields of aerodynamics and hydrodynamics.Experimental measurements of a family of all-movable,low aspect ratio control surfaces in free-stream at vari-ous Reynolds number have been performed [10]. Arelevant experiment to investigate the force and mo-ment characteristics of six high-speed rudders withaspect ratio of 1.5 and different section shapes undercavitating conditions can be found in [2]. Furthermore,a series of rudders with a geometric aspect ratio of 1.5but widely varying section shapes was constructed todetermine the effect of section shape on the cavitatingperformances while operating in the propeller slip-stream [5]. However, few works for other profile shapesare observed. Recognizing the need for providing use-ful information in the rudder design and the properselection of rudder gears, further investigations on theperformance of high-speed rudders are essentially

Journal of Marine Science and Technology, Vol. 13, No. 1 (2005)62

necessary. As mentioned in [8], the model experimentsproduce accurate hydrodynamic forces at model-scaleReynolds numbers, but are poor to give maximum liftdue to the scale effect. One popular approach used toanalyze the rudder flow problems is the potential flowcalculation. However, it failed to give any stall condi-tion due to neglecting viscosity, which leads to turbu-lence and flow separation.

The purpose of this paper is to analyze the non-cavitating hydrodynamic performances of NACArudders, as well as rudders using simple profiles. Anumerical method based on computational fluid dynam-ics (CFD) is applied to predict the non-cavitating hydro-dynamic characteristics of rudder. The force and mo-ment acting on a rudder are calculated by integrating theshear force and pressure force on the rudder surface,which are obtained from the turbulent flow calculationaround rudder. Therefore, viscous effects, such as flowseparation, can be taken into account. Therefore, thestall phenomenon can be reasonably predicted, whichcannot be predicted by the potential methods. Throughthe calculated results, valuable information can be con-cluded for the rudder design, which compromises be-tween the hydrodynamic performance and the manufac-turing cost.

GOVERNING EQUATION AND TURBULENCEMODEL

The steady and imcompressible Reynolds-aver-aged Navier-Stokes equation and continuity equationare used as the governing equation to describe the flowaround rudder:

∂(U i U j )∂x j

= – 1ρ

∂P∂x i

+ ∂∂x j

v∂U i

∂x j+

∂U j

∂x i–

∂ u i' u j

'

∂x j

(1)

∂U i

∂x i= 0 (2)

u i' u j

' = – v t∂U i

∂x j+

∂U j

∂x i+ 2

3δij k (3)

where Ui, P denotes the mean velocity and the mean

pressure, respectively, u i' the fluctuating velocity

component, ρ‚ the density, v the fluid viscosity, vt theturbulent viscosity, and xi the Cartesian coordinate. Atwo-equation turbulence model, i.e. the k − ε model [4]is used in the numerical computations:

∂(k U j )∂x j

= ∂∂x j

v +v tσ k

∂k∂x j

+ P k – ε (4)

∂(εU j )∂x j

= ∂∂x j

v +v tσε

∂ε∂x j

+ c ε1P kεk

– c ε2ε2

k(5)

P k = – u i' u j

'∂U i

∂x i(6)

v t = C µk 2

ε (7)

where k denotes the turbulent kinetic energy, and ε thedissipation rate of k. The constants in the above equa-tions are given in Table 1. Because this paper focuseson the non-cavitating performance of rudders for high-speed crafts, the computations are performed for turbu-lent flow at Reynolds number (Re) equal to 106. Theeffect of transition from laminar to turbulence isneglected, which means that the flow is fully turbulentalong the wall. The wall function approach is adoptedto bridge the viscous sublayer near wall, where theequations (4) and (5) are no more valid. The shear stresson solid wall τw is computed as follows, where up

denotes the local velocity component parallel to thewall of the first grid point from wall, y the distance towall and µe the effective viscosity (≡ µ + µt):

τ w = µe

u p

y , if y+ ≤ 20 (8)

τ w = µeρkC µ

, if y+ > 20 (9)

y + =y

τ wρ

v(10)

NUMERICAL METHOD

All the vector quantities such as position vector,velocity, and moment of momentum, are expressed inCartesian coordinates system. A second order finitevolume method is utilized to discretize the governingequation into a system of algebraic equations. Thegoverning equations are first expressed in an integralform over a control volume V bounded by the controlsurface A with outward normal vector n . The integral

Table 1. Coefficients of the k − ε model

Cµ σk σε cε1 cε2

0.09 1.00 1.30 1.44 1.92

S.W. Chau et al.: Investigation of Hydrodynamic Performance of High-Speed Craft Rudders Via Turbulent Flow Computations, Part I 63

form of generic function φ (= 1, Ui, k and ε) is given asfollows:

φA

( u ⋅ n ) dA = Γφ

A( ∇ φ ⋅ n ) dA + q φ

VdV

(11)

where u denotes the velocity vector at cell interface,the diffusion constant Γφ and the source term of qφ. Thevalue of φ is then expressed in terms of the neighboringnodal values. The number of neighboring points andmathematical form depend on the order of the approxi-mation scheme. Here a deferred correction approach [3]is employed to approximate the convection term, wherea central difference scheme is blended with an upwinddifference. Then the governing equation is discretizedinto a system of algebraic equations as follows:

a cφφc = Σ

ia i

φφi + S cφ

(12)

where the subscript c denotes the considered cell node,the subscript i the neighboring cells, a the linearizedcoefficient and S the linearized source term. The re-sulted algebraic equations are solved by the SIP method[9]. The SIMPLE algorithm [6] is applied to computevelocity and pressure updates. The pressure interpola-tion at cell faces follows a special treatment [7] isadopted to obtain a non-oscillating pressure field. Pleasereference [1] for more details.

DESCRIPTION OF RUDDERS

The geometric definition of rudder used in thispaper is shown in Figure 1. It consists with the one usedin [10]. The mean chord length (cm), taper ratio (Tr), andaspect ratio (Ar) of rudder are defined in Eq. (1) to (3):

c m = c t + c r

2(13)

T r = c tc r

(14)

A r = hc m

(15)

where ct and cr denote the chord length at the tip and theroot of the rudder, respectively. The height of therudder is represented by h. The angle intersects be-tween the quarter chord axis and Z-axis is known as thesweep angle β. The geometric aspect ratio and taperratio of the rudders are 1.5 and 1.0, respectively. Allrudders are with zero sweep angles.

Six rudder shapes: NACA 00 series, NACA 16series, NACA 66 series, wedge shapes, ellipse shapesand rectangle shapes, with different geometric param-eters were evaluated. The first two profile series are

widely used for rudders due to their smooth pressuredistributions along the rudder surface, which couldsignificant reduce the vibration of rudder caused by anunsmoothed pressure distribution. The third one isoften used as the basic profile for propeller blades. It isinvestigated in order to validate the differences betweenthe rudder profile and propeller profile. For smallcrafts, such as boats, in order to minimize the manufac-turing cost of rudder, rudders of simple shapes are oftenemployed, which fall into the category of the latterthree. The name of studied shapes is expressed byalphabetic characters and numeric digits, where thealphabetic characters denote the profile shape, and thenumeric value after the alphabetic characters gives themaximum thickness of the profile, e.g. Rectangle08means the rectangle rudder with a maximum thicknessof 8% chord length. All rudders are geometricallysymmetrical about their nose-tail line. For rectangleand wedge type rudders, the trailing edge doesn’t con-verge to a single point and have a finite end thickness.Because the slope nears the trailing edge changesdrastically, a significant pressure change near the trail-ing edge is expected, which deteriorates the conver-gence of the turbulence model with wall functionemployed. Hence, small geometrical simplificationsnear the leading edge and trailing edge are made, asshown in Figure 2, in order to obtain a better numericalconvergence in the turbulence model. This geometrical

quarter chord axis

h/2

h/2

Z

β

Cr/4

Ct/4

Cr

Cm

Ct

Fig. 1. The geometric definition of a rudder.

WedgeXX

RectangleXX

L.E.

L.E.

T.E.

T.E.

Original GeometryModified Geometry

Fig. 2. Simplification of rudder geometry.


T

R

R

RR

YJ

I

I h/2

JJ

K

K

K

I

Z

X

Fig. 3. Computational domain for rudder flow. Fig. 4. Grid arrangement near the rudder surface.

Fig. 5. Profile shape of NACA series.

simplification could influence the pressure distributionalong the profile, especially near the trailing edge.Because the pressure near trailing edge has only littlecontribution to the total force calculation, the pressurechange due to the geometry change near the trailingedge has quit limited influence on the lift and drag,which are calculated through the integral of stressesalong the profile. This approach has been justified in[1], in which a similar approach was adopted.

GRID TOPOLOGY

The arrangement of the computational domain,Figure 3, is created with the strategy described in [1]. Theupstream part of the computational domain is a halfcylinder with radius R and height T, whereas the down-stream part is a rectangular block with length R, width2R and height T. Since the rudders are assumed to besymmetrical about their root section in the numericalcomputation, only one half of rudder is modeled. Inorder to avoid unfavorable effects of the artificial outletboundary, R and T are specified as 5 cm. The computa-tional domain is divided into 400,000 cells based on a C-H topology, Figure 4. The local grid quality can influencethe discretization error significantly. Therefore, thegrid lines in the horizontal planes near rudder surfacewere clustered with elliptic smoothing to keep the gridlines almost orthogonal in order to reduce the interpola-tion errors due to grid skewness.

CONSTANT MAXIMUM THICKNESS

The influence of the rudder shape is examined inthis section. Six rudder profiles, including Ellipse08,Rectangle08, Wedge08, NACA 0008, NACA 1608 andNACA 6608 are considered with a constant maximumthickness of 8% chord length, Figure 5. The drag coeffi-cient (CD), the lift coefficient (CL), and the stock mo-ment coefficient about the axis at 25% chord lengthfrom the leading edge (CM, 0.25) of these rudders areshown in Figures 6-8, respectively. Generally speaking,the CD of the Rectangle08 rudder is much greater thanthe other rudders. Besides, the drag produced by theWedge08 rudder is also larger than those of NACAseries. The CD of the Ellipse08, NACA 0008, NACA1608 and NACA 6608 rudders are almost in the sameorder of magnitude. This is due to that the streamliningshape has a considerable effect on the drag reduction.The differences of CD between the NACA series ruddersare quite small. Obvious differences are only observedat large angles of attack (before the stall angle). Amongthese rudders, the ones with convex end seem to producelower drag for large angles of attack. The resultsdemonstrate that the streamlining of section shape is animportant factor that affects the drag coefficient.

In Figure 7, the CL curve of the Rectangle08rudder deviates significantly from the other rudderswith the increasing angle of attack. With the angle ofattack α = 10° for instance, the CL of the Rectangle08


rudder is 29.3% smaller than that of the NACA 0008rudder. The Wedge08 rudder with the maximum thick-ness at the trailing edge seems to stall earlier among allrudders except the Rectangle08. The Ellipse08 rudder,which has a convex end and the maximum thicknesslocated at the chord middle point, generates smaller liftwhen compared to the NACA rudders. Rudder is a kindof hydrofoil that should operate over a wide range ofangle of attack, and the location of maximum thicknessof the rudder section should be designed close to theleading edge in order to deliver large lift. The maximumthickness of NACA 1608 and NACA 6608 section aresituated near the middle chord. This is why they yieldlower lift than the NACA 0008 rudder. Therefore, they

are not favorable for applications with large variation inangle of attack, and this explains why they are preferredin the propeller blade design, where small operationrange of angle of attack are expected. The NACA 0008rudder provides the largest maximum lift coefficient(CL, max) and stall angle (αstall), whereas the Rectangle08rudder offers the smallest ones. The CL, max and the αstall

for all rudders are summarized in Table 2, where xmax

denotes the location of maximum thickness. Figure 8demonstrates the required turning moment for eachrudder. The Wedge08 rudder with an unusual locationof maximum thickness at the trailing edge shows themost unfavorable performance (the largest moment) inCM, 0.25 for all angles of attack. This is due to that thehigh-pressure region of this rudder, which is generallycorresponding to the location of maximum thickness, isfound most distant from the stock location than otherrudders. The same mechanism leads to that the NACA0008 rudder produces the smallest CM, 0.25, because itsmaximum thickness is located very close to the rudderstock. From the predicted results, both the location ofmaximum thickness and the shape at trailing edge playan important role on the rudder performance.

Fig. 6. CD of rudders with different profile shapes.

Fig. 7. CL of rudders with different profile shapes. Fig. 8. CM, 0.25 of rudders with different profile shapes.

Table 2. CL, max and αstall of rudders

Rudder CL, max αstall (°) xmax

Ellipse08 0.53 15.6 0.5Rectangle08 0.32 11.2 −

Wedge08 0.45 11.8 1.0NACA 0008 0.88 21.0 0.3NACA 1608 0.56 15.0 0.5NACA 6608 0.54 14.4 0.45


Figures 9 and 10 depict the drag-lift ratio (C D/CL)and moment-lift ratio (CM, 0.25/CL) curves for the rudders.Again, the NACA 0008 rudder shows its superior per-formance on CD/CL and CM, 0.25/CL compared with theother rudders. The Rectangle08 rudder, which has azero slope on the profile shape along the downstreamdirection, produces the largest CD/CL and CM, 0.25/CL.The Wedge08 rudder gives lower CD/CL and CM, 0.25/CL

than that of the Rectangle08 one but obviously greaterthan the other rudders. Therefore, a foil-shaped ruddernot only produces lower CD but also offers greater CL,higher CL, max and larger α stall. The calculated resultsindicate that the location of maximum thickness is themost important factor that influences the rudder

Fig. 9. CD vs. CL of rudders with different profile shapes.

Fig. 10. CM, 0.25 vs. CL of rudders with different profile shapes.

Fig. 11. CD of NACA rudders with different profile thicknesses.

Fig. 12. CL of NACA rudders with different profile thicknesses.

performance.

PROFILE THICKNESS

In this section, rudders of four common profileshapes: NACA 00 series, ellipse, rectangle and wedgewith various profile thickness are analyzed. Figure 11,Figures 12 and 13 describe the computed C D, CL andCM, 0.25 curves of the NACA00 rudders (NACA 0008,NACA 0012, NACA 0015 and NACA 0018). As shownin the figures, the differences of drag and lift coefficientbetween the NACA 00XX rudders are quite small formost angles of attack. Hence, no notable difference onCD/CL curves is observed in Figure 14. The thicker


Fig. 13. CM, 0.25 of NACA rudders with different thicknesses.

Fig. 14. CD vs. CL of rudders with different thicknesses.

Fig. 15. CM, 0.25 vs. CL of rudders with different thicknesses.

Fig. 16. CD of ellipse rudders with different profile thicknesses.

NACA00 rudder gives larger CL, max and greater α stall

without much increase in drag force. This is because theeffective angle of attack of thicker rudder is smallerthan that of thinner one, so that the thinner rudderproduces larger CL at a fixed angle of attack and stalls ata smaller angle of attack than that of thicker one.However, the thicker rudder required a larger moment.The CM, 0.25/CL curves of the rudders deviate from eachother with the increase of angle of attack, Figure 15.

The CD, CL and CM, 0.25 curves of three ellipserudders (Ellipse08, Ellipse09 and Ellipse11) were shownin Figures 16-18, respectively. For α < 10°C, the C D

curves are quite similar among the ellipse rudders. Butafter α = 10°C, Ellipse08 rudder experiences larger

drag than the other ellipse rudders. Figure 17 depicts thatthe thinner ellipse rudder produces larger lift for afixed angle of attack. Nevertheless, the thinner oneseems to still stall at a smaller angle of attack than thethicker one. The CL,max of the Ellipse11 rudder is 28.3%larger than that of the Ellipse08 rudder, whereas theα stall is 31.4% greater than that of the Ellipse08 rudder.However, the Ellipse11 rudder requires larger turningmoment for all attack angles. The predicted resultsshow that only 3% increase in profile thickness forellipse rudder will result in about 28% to 31% increasein CL,max and αstall, respectively, but at the same time themoment coefficient |CM, 0.25| increases by 36.9%. Formost angles of attack, the difference in CD/CL curves,


Figure 19, is somewhat small. Besides, the C M, 0.25/CL

curves, Figure 20, significantly differ from each other withthe increasing angle of attack. The CD, CL and CM, 0.25

curves of the rectangle rudders are illustrated in Fig-ures 21-23. The thicker rectangle rudder (Rectangle08)produces a drag increase by 200% when compared to thedrag generated by the thinner one (Rectangle04). Be-fore α = 5°C, the lift gradient of the two rudders arealmost identical. After α = 5°C, the slope of lift curvefor the thicker rectangle rudder decreases severely. TheRectangle08 rudder generates a lower CL, max and smallerα stall than that of the Rectan-gle04 one, which is differ-ent from the trend given by the NACA 00 rudders. Theeffective angle of attack of rectangle rudders does not

decrease with the increasing profile thickness, becausethe profile slope significantly influences the effectiveangle of attack of rudders. When compared with theRectangle08 rudder, the Rectangle04 rudder gives aincrease of 71.9% in CL, max and a increase of 31.2% inα stall. As illustrated in Figure 23, the Rectangle08 rudderhas a worse characteristic on turning moment than thethinner one. The CD/CL and CM, 0.25/CL curves of therectangle rudders are given in Figures 24 and 25,respectively. Although the performance of the Rect-angle04 rudder is more favorable than that of the Rect-angle08 rudder, the thinner one may fail to providesufficient structure strength.

Three wedge rudders (Wedge08, Wedge12 and

Fig. 17. CL of ellipse rudders with different profile thicknesses.

Fig. 18. CM, 0.25 of ellipse rudders with different thicknesses.

Fig. 19. CD vs. CL of ellipse rudders with different thicknesses.



Wedge14) are also computed to compare their non-cavitating hydrodynamic performances on lift, drag andmoment, Figures 26-28. The Wedge08 rudder experi-ences larger CD than two other thicker ones, while thedrag coefficient of the Wedge14 rudder is almost thesame as that of the Wedge12 rudder. Because thegeometric difference near the leading edge betweenthese two rudders is quite small, it explains the smalldeviation in the drag coefficient. The Wedge08 ruddergives a larger lift gradient but a smaller CL, max and αstall

than that of the Wedge12 rudder, as shown in Figure 26.The Wedge12 rudder yields larger CL than the Wedge14rudder for the most angles of attack, while the latter oneproduces a lower CL, max and a smaller α stall. As ob-

Fig. 21. CD of rectangle rudders with different thicknesses.

Fig. 22. CL of rectangle rudders with different thicknesses.

Fig. 23. CM, 0.25 of rectangle rudders with different thicknesses.

Fig. 24. CD vs. CL of rudders with different thicknesses.

served among the performance of NACA 00 rudders, thethicker rudder should provide a larger CL, max and also agreater α stall, which is not found in the wedge rudders.Through the geometry simplification of the Wedge12and Wedge14 rudders, the thickness distribution onprofile is slightly different from the original shape. Themaximum thickness between the Wedge12 and Wedge14rudders is only differ by 2% of the chord length. Forsmall variation in the maximum thickness, the simpli-fied geometry may not reflect the correct influence ofthe maximum thickness. Hence, the comparisons be-tween the Wedge08 and Wedge12 rudders or the com-parison between the Wedge08 and Wedge14 rudders aremore appropriate to illustrate the influence of profile



Fig. 26. CD of wedge rudders with different profile thicknesses.

Fig. 27. CL of wedge rudders with different profile thicknesses.

Fig. 28. CM, 0.25 of wedge rudders with different thicknesses.

Table 3. CL, max and αstall of rudders

Rudder CL, max αstall (°)

NACA 0008 0.88 21.0NACA 0012 1.13 27.1NACA 0015 1.21 29.3NACA 0018 1.23 30.2

Ellipse08 0.53 15.6Ellipse09 0.61 18.0Ellipse11 0.68 20.5

Rectangle04 0.55 14.7Rectangle08 0.32 11.2

Wedge08 0.45 11.8Wedge12 0.47 14.1Wedge14 0.44 13.6

thickness for wedge rudders. The thicker wedge rudderdemands a larger CM, 0.25 than those thinner ones. Thistendency is also found in other rudder shapes mentionedabove. For instance, |CM, 0.25|max of the Wedge12 rudderis 39.9% greater than that of the Wedge08 rudder. TheCD/CL curves are depicted in Figure 29, and the thinnerwedge rudder offers smaller CD/CL values. The CM, 0.25/CL values , Figure 30, of the Wedge12 and Wedge14rudders are much larger than that of the Wedge08rudder. Therefore, the thinner wedge rudder shows abetter hydrodynamic performance compared with thatof thicker one. The CL, max and α stall of the abovediscussed rudders are gives in Table 3.


Fig. 29. CD vs. CL of wedge rudders with different thicknesses. Fig. 30. CM, 0.25 vs. CL of rudders with different thicknesses.

CONCLUSION

In the presented work, non-cavitating hydrody-namic performances of different rudders used for high-speed crafts are analyzed using a CFD approach. Thethree-dimensional, turbulent flow around rudders hasbeen computed by solving the Reynolds-averagedNavier-Stokes equations combined with the k− ε turbu-lent model.

For a given maximum thickness, the foil-shapedrudder not only produces lower drag and greater lift ata specified angle of attack before stall but also offershigher maximum lift and larger stall angle. The NACA00 rudder generates the largest maximum lift coeffi-cient and stall angle among other rudders with the samemaximum thickness. The rectangle rudders show themost unfavorable performance, i.e. the smallest liftcoefficient gradient and the largest drag coefficient forall angles of attack. The stock moment coefficient ofthe NACA 00 rudder is also much smaller than otherrudder profiles, which is favorable from the maneuver-ing point of view. The predicted results reveal that boththe location of maximum thickness and the shape attrailing edge play important roles on the rudderperformance. The profile with a maximum thicknesslocation near to the leading edge is favorable in therudder design. For the NACA and ellipse rudders, aslight growth in thickness increase the stall angle andgives larger lift without significant increase in the dragforce. However, the thicker rectangle rudder produceslarger drag and seems to stall at smaller angle of attackthan the thinner one. For wedge rudders, the thicker onegives larger stall angle, larger maximum lift and also

larger drag. For all the investigated rudder profiles, thethicker rudder yields larger stock moment.

REFERENCES

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High Speed Vessel Rudders

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