J Mar Sci Technol (2010) 15:131–142 DOI 10.1007/s00773-010-0084-z

ORIGINAL ARTICLE

Prediction of ship steering capabilities with a fully nonlinear ship motion model. Part 1: Maneuvering in calm water Ray-Qing Lin • Michael Hughes • Tim Smith

Received: 25 July 2008 / Accepted: 11 January 2010 / Published online: 25 February 2010 Ó JASNAOE 2010

Abstract This paper introduces a new method for the prediction of ship maneuvering capabilities. The new method is added to a nonlinear six-degrees-of-freedom ship motion model named the digital, self-consistent ship experimental laboratory (DiSSEL). Based on the first principles of physics, when the ship is steered, the additional surge and sway forces and the yaw moment from the deflected rudder are computed. The rudder forces and moments are computed using rudder parameters such as the rudder area and the local flow velocity at the rudder, which includes contributions from the ship velocity and the propeller slipstream. The rudder forces and moments are added to the forces and moments on the hull, which are used to predict the motion of the ship in DiSSEL. The resulting motions of the ship influence the inflow into the rudder and thereby influence the force and moment on the rudder at each time step. The roll moment and resulting heel angle on the ship as it maneuvers are also predicted. Calm water turning circle predictions are presented and correlated with model test data for NSWCCD model 5514, a pre-contract DDG-51 hull form. Good correlations are shown for both the turning circle track and the heel angle of the model during the turn. The prediction for a ship maneuvering in incident waves will be presented in Part 2. DiSSEL can be applied for any arbitrary hull geometry. No

R.-Q. Lin (&)  M. Hughes  T. Smith Hydromechanics Department, David Taylor Model Basin, NSWCCD, 9500 MacArthur Boulevard, West Bethesda, MD 20817-5700, USA e-mail: [email protected] M. Hughes e-mail: [email protected] T. Smith e-mail: [email protected]

empirical parameterization is used, except for the influence of the propeller slipstream on the rudder, which is included using a flow acceleration factor. Keywords Ship steering capabilities  Maneuvering  Seakeeping  Rudder angle  Rudder force  Roll motion  Comparison of the numerical solution and experimental data List of symbols xr, yr, zr Location of the center of pressure of the rudder xp The location of a point, p, on the ship surface I Moment vector of inertia about the ship I(i,i) Moments of inertia about the ship’s center of mass in the ith direction F Force vector acting at the ship’s center of mass in the ship-fixed frame C Moment vector acting at the ship’s center of mass in the ship-fixed reference frame Fp, Fq Exciting force and restoring force acting at the ship’s center of mass in the ship-fixed frame xc, yc, zc Center of mass in the ship-fixed coordinate system x Unit vector in the ship-fixed coordinate system xs, ys, zs Three components of unit vectors in the shipfixed coordinate x^; y^; ^z Unit vectors in the Earth-fixed coordinate system X Translational motion of the ship Xi Translational motion of the ship in the i direction us Ship speed vector ur Effective inflow velocity into the rudder

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ut vs vh n n x, n y, n z b d q P h hi X Xh Xi h* X* g ge gs  ge

u ue mship v A0 R3 R1 R i step Fship Fl Frudder Fx Fy Cship Crudder Crudder(3) Cc(1) Dtrans

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Total velocity at a point on the ship hull including all ship motions Translational velocity of the ship’s center of mass Horizontal component vector of the translational velocity Normal vector of the ship’s surface Three components of the normal vector of the ship’s surface Drift angle of the ship Effective rudder deflection angle Density of the water in which the rudder is operating Pressure Rotational angle of the ship Rotational angle of the ship in the ith direction Angular velocity vector of the ship Horizontal angular vector (roll and pitch) of the ship Angular velocity of the ship in the i direction Total heading angle, equal to h3 ? b  Total heading angular velocity, equal to dh dt Free surface elevation Environmental free surface elevation Free surface deformation due to the ship’s motion The over bar indicates the average (this is the average of the environmental free surface elevation) Velocity potential Velocity potential of the environment, including incident waves Total ship mass Dissipation coefficient due wave breaking Surface area of the rudder planform Moment arm for the rudder force yaw moment Moment arm for the roll moment Ship maneuvering radius vector Number of the time step Force on the ship’s surface, except on the rudder Lift force in the zs coordinate Force on the rudder Force on the rudder in the xs direction Force on the rudder in the ys direction Moment on the ship, except on the rudder Moment on the rudder Yaw moment on the rudder Roll moment due to the direction of ship speed change Dissipation of translational motion due to wave breaking [1]

Drotat H V dV xtrack xtrack ytrack

Dissipation of rotational motion due to wave breaking and bilge keels Water depth Volume Element of the volume Track in Earth coordinates Track in the x direction in Earth coordinates Track in the y direction in Earth coordinates

1 Introduction The prediction of a ship’s steering capabilities is an important part of the design process. Traditional maneuvering prediction methods are based almost entirely on empirical coefficients [2–6]. Such methods are not appropriate for new and unconventional ships, for which empirical data are not available. With the increase in computer capabilities and the development of advanced computational flow prediction methods, the use of numerical methods to predict the steering capabilities of ships has become feasible. For example, Chau [7] and El Moctar [8] used viscous flow methods to predict the rudder flow, and Lee [9], Tamashima et al. [10], Han et al. [11], Kinnas et al. [12], and Hacket et al. [13] used the panel method in a potential flow to compute the rudder force and moment. In particular, Kinnas et al. [12] used a finite element boundary condition to improve the panel method. These works have significantly improved our physical understanding of rudder forces and moments. However, as So¨ding [14] pointed out, those potential flow methods did not take into account the viscosity, turbulence, and flow separation. Viscous flow methods for predicting a ship’s steering capabilities still remain technically difficult and computationally expensive. Therefore, in spite of the limitations of potential flow, most practical flow problems are still solved either by obtaining experimental data, or by potential flow calculations [14]. In the current paper, a physics-based method for predicting ship maneuvering is described. This new method, which is added to a nonlinear six-degrees-of-freedom ship motion model named the digital, self-consistent ship experimental laboratory (DiSSEL), is dynamically consistent and based on the dynamic equations derived from first principles. No empirical parameterization is introduced in the model ([15–16]; Lin R-Q, Kuang W, A fully nonlinear, dynamically consistent numerical model for solid-body motion. Part I: ship motion with fixed heading, submitted to Proc R Soc). Even though DiSSEL is based on potential flow, it includes kinematic viscosity in wave breaking effects ([15–16]; Lin R-Q, Kuang W, A fully nonlinear,

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dynamically consistent numerical model for solid-body motion. Part I: ship motion with fixed heading, submitted to Proc R Soc) and flow separation, which is calculated by ‘‘blocking theory’’ [17]. Therefore, DiSSEL contains advantages of both potential and viscous methods. The new method is not only much more accurate than the previous methods, but it is also computationally efficient. However, as a model for propellers has not yet been implemented in DiSSEL, the influence of the propeller slipstream on the rudder is included empirically by modifying the effective inflow velocity into the rudder to account for the acceleration of the flow by the propeller. In the initial part of this effort, a method is presented for maneuvering in calm water. Future efforts will extend the method in order to predict the motions of a vessel controlled by an autopilot and operating in a seaway. Such a method will give the naval architect the capability to compute the nonlinear motions of a steered ship in waves. This will assist naval architects by providing improved seakeeping performance predictions in higher sea states as well as predicting the dynamic stability performance of a ship during the early stages of the design. In the following section, an overview of DiSSEL is provided. In Sect. 3, the use of the method to calculate the additional forces and moments due to the deflected rudder is described. Section 4 shows the correlation of the predicted steering capabilities with experimental data for Model 5514, which was tested at the Naval Surface Warfare Center Carderock Division (NSWCCD). Conclusions are then provided.

2 Nonlinear ship motion model DiSSEL includes two components. The first component is the ship–wave interaction component, which computes the flow around the ship including the influence of the water free surface, the pressure distribution on the ship hull and the hydrodynamic forces on the ship. The second component is the solid body motion component, which uses the forces computed by the ship–wave interaction component to predict the six-degrees-of-freedom motions of the ship by solving the equations of motion in the time domain. The rudder is included as a separate force in the rigid body motion component of the code. To calculate the forces and moments on the ship hull and all appendages, as well as on the rudders, DiSSEL performs a body exact integration by a finite element/ finite difference method. At each time step, the water line, the underwater volume and the wetted surfaces, as well as the normal vectors that are based on the wetted surfaces’ positions and coordinates are redefined. Two coordinate

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systems are used; except for the ship track and incident waves, all of the simulations are in the ship coordinate system. The ship reference frame of the model is defined with its origin on the mean free surface, and it moves with the total horizontal translational velocity plus the yaw motion crossed with the turning radius vector (us þ vh þ X3  R). Since the ship is moving with a total heading angle of h*, the normal vector is nistepþ1 ¼ nistep þ ðXh Þ  nistep Dt: For the far field, DiSSEL employs the pseudo-spectrum method. At each time step, the forces and moments in the solid body motion model are obtained by integrating the pressure over the wetted ship surface, where the pressures are obtained from the ship wave interaction model. An FFT is used to transfer information between the spectrum modes and the collocation points in the pseudo-spectrum method. A quasilinear method is used to transfer information between the collocation points of the pseudo-spectrum method and the grids of the finite element/finite difference method. This method is very accurate and efficient; the convergent speed is NlogN instead of N2, where N is the number of unknown variables; the details are described in Lin et al. [15]. In the current work, us is constant and prescribed by the user, while vs is computed by the program at each time step. 2.1 Ship–wave interaction model The ship–wave interaction component is computed in a fluid domain of depth H, width D and length L. Except for the free surface and ocean bottom boundaries, all other boundaries are open, which allows fluid to move in and out through these faces. Such boundaries not only allow the use of a small computational domain, but also increase accuracy. The fluid is incompressible: r2 u  r2h u þ

o2 u ¼ 0 for  H  z  g: oz2

ð1Þ

The dynamic and kinematic boundary conditions at the free surface, z = g are   ou 1 P þ ru þ us þ vh  ru þ gg þ ot 2 q oðus þ vh Þ þ  x  mr2h u ¼ 0; ð2Þ ot og ou þ ðrh gÞ  ðrh u þ us þ vh Þ ¼ ; ð3Þ ot oz where P is the pressure, q is the fluid density, m is the kinematic viscosity, here representing the wave breaking effects, and x is the position vector of a point in space. For a purely potential flow, the viscous dissipation vanishes in

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(2). The above equations are solved with the impenetrable boundary conditions at the bottom, z = -H ou þ ðrh HÞ  ðrh u þ us þ vh Þ ¼ 0; oz

ð4Þ

and at the ship hull: n  ðru þ us Þ ¼ 0;

ð5Þ

where, at a point xp on the ship hull surface, ut ¼ us þ vs þ X  ðxp  xc Þ:

ð6Þ

At the side and aft boundaries (x = c), and the forward boundary (x = b) of the domain, the radiation boundary conditions are used to ensure that the waves generated by the ship are left behind the ship and do not radiate ahead of the ship. They are implemented as open boundary conditions away from the ship (the far field): e ¼ ou x ¼b ox  ðus þvh Þ; g ¼ ge  r  u ¼ rue þ rus  rus  ðus þ vh Þ; g ¼ ge þ ðgs  gs Þ; x¼c

ou ox

ð7Þ

The over bar indicates the spatial average of the quantity. The boundary conditions are consistent with mass conservation. In calm water, ue and ge vanish. 2.2 Solid body motion model In this section, the basic equations describing the motions of the rigid ship hull are presented. Both the motions and the forces are defined in the ship reference frame, with the origin located at the ship’s center of mass. Therefore, in the solid body motion model of DiSSEL [17], the ship’s translational motion is expressed by: mship

dvs þ Dtrans vs ¼ Fship þ Frudder þ Fl z þ FCoriolisð3Þs ; dt

ð8Þ

where Fship includes the restoring forces and the pressure forces on the ship hull, Flzs is the lift force of the bare hull, and FCoriolis(3) is the Coriolis force due to the movement of the ship coordinates with the yaw motion, which is equal to  Z  dX     x dv: FCoriolisð3Þ ¼ q 2X  vs þ X  ðX  xÞ þ dt V

The solid body rotation is governed by the Liouville equation defined in the ship reference frame: I

dX þ X  ðI  XÞ þ Drotat X dt ¼ Cship þ Crudder þ Ccð1Þ þ CCoriolisð3Þ :

ð9Þ

Positive xs is in the forward direction, positive ys is to port, and positive zs is upwards. The origin is at the center of mass of the ship.

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The ship forces Fship and moment Cship are calculated by integrating the pressure and the pressure multiplied by the torque arm over the wetted ship surface in normal vector coordinates, respectively. The details are given in ([15–16]; Lin R-Q, Kuang W, A fully nonlinear, dynamically consistent numerical model for solid-body motion. Part I: ship motion with fixed heading, submitted to Proc R Soc). CCoriolis(3) is the moment due to the movement of the ship coordinates with yaw, and it is equal to CCoriolisð3Þ   Z dX     x dv: ¼ qR  2X  vs þ X  ðX  xÞ þ dt V

R is the maneuvering radius vector. Flzs is the lift force on the ship hull in Eq. 8. It is computed by integrating the fluid pressure over the wetted surface of the ship hull in the z direction of the ship coordinates at each time step [18]. The lift force, Flzs, on the ship hull is obtained in a similar way to the lift force on the rudder in DiSSEL. 2.3 Blocking theory In order to show how to calculate the lift force accurately in a potential flow ship motion model, this section summarizes ‘‘blocking theory’’ [17]. Blocking theory was developed to estimate the separation effects in a potential flow model. The lift force from the ship hull is simulated in a similar way to that of the rudder, as described in Sect. 3 after the blocking theory has been described. The blocking theory is based on the physical concept that when a flow passes an object, its total pressure changes, as defined by Landau and Lifshitz [19]. Lin and Kuang [17] proposed a nonlinear method for calculating the total pressure on the object in a potential flow. In this method, they suggested that the total pressure lost due to separation on the object can be obtained by integrating the pressure over the ‘‘effect area.’’ The effect area can also be referred to as the blocking area or separated area. There is a long history of studying blocking mechanisms in a wide variety of fluid systems; see for example Freund and Meyer [20], Garner [21], Lin and Chubb [22], and Holton [23]. However, in this study, we use the term for the net pressure force on the surface. When a flow passes an object in the absence of viscosity and turbulence, the pressure is P(s). When viscosity and turbulence are present, the pressure P(s) will change to P0ðsÞ on the object’s surface, and P0ðsÞ \PðsÞ . This is the separation effect. If Ar is the surface area of the object, then the total pressure on the object is

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Z

nP0ðsÞ ds:

135

ðBK1Þ

Ar

For a potential flow, P(s) is calculated instead of P0ðsÞ : In order to obtain the total pressure accurately, we can reduce the object surface area from Ar to Ar–Ae. Then the total pressure on the object will be Z Z nPðsÞ ds  nP0ðsÞ ds; ðBK2Þ Ar Ae

Ar

ðBK3Þ Ae ¼ Lobject Wblock j k Wblock ¼c1 ðhm xÞ 1þa1 ðhm xÞ2 þa2 ðhm xÞ4 þa3 ðhm xÞ6 þ ¼

K X

c2kþ1 ðhm xÞ2kþ1

ðBK4Þ

k¼0

where ci is a constant and k is the truncation order defined in [17]. Here hm is the angle of attack of the rudder or the roll or pitch angle for a ship hull. For roll and pitch motions, x is the natural frequency associated with the roll or pitch motion in calm water. The frequency of the angle of attack can be calculated as xturn ¼ T2p ; where the ship’s  turn   turning period, Tturn ¼ uship þv2pR , and its turning h þX3 R _

_

DC radius, R ¼ Dh  (C is the turning curl). The separation effect RRcausing the total pressure reduction is Fseparationeffect ¼  Ae nPðsÞ ds: The influence of the separation effect on the moment is given by ZZ Cseparationeffect ¼  ðxp  xc Þ  nPðsÞ ds: Ae

The roll damping effects obtained by blocking theory agree well with the experimental data [17]. 3 Ship maneuvering predictions

moments generated by the deflected rudder are described by the following expressions: The effective force on the rudders in the ship coordinate system can be expressed as ZZ nPrudder ðsÞds ¼ Fx xs þ Fy ys þ Flr zs : ð10Þ Frudder ¼ A0 A0e

If the rudder surface is a planar surface, Eq. 10 can be written as 1 Fx ¼  q u2r ðA0  A0e Þ sin2 d; and 2 1 2 1 Fy ¼ q ur ðA0  A0e Þ sin d cos d ¼ q u2r ðA0  A0e Þ sin 2d; 2 4 where A0 is the total surface of the rudder and A0e is the effect area of the rudder. The effect area can be obtained from Eqs. BK3–BK4. Flrzs is the vertical lift force on the rudders. To calculate the lateral force and yaw moment acting on the turning hull, it is important to accurately predict the lift forces on the rudders and on the hull. The geometries of the rudders and the ship’s hull are often symmetric. Angles of attack or roll and pitch angles are required to create lift forces. For a pure potential flow, the Kutta condition, that there are velocity stagnation points at the trailing edges (TE) of the rudders and ship hull (i.e., VTE = 0), is not satisfied. Therefore, the lift forces will be underestimated, even with an angle of attack or roll and pitch angles. However, for a viscous flow, the flow separates at the trailing edge, and lift forces occur with angles of attack or roll and pitch angles. The lift force can be calculated with reasonable accuracy. DiSSEL satisfies the Kutta condition at trailing edges, with velocity stagnation points VTE = 0, and the lift force can be obtained from DiSSEL with reasonable accuracy by integrating the total pressure on the non-effect wetted surface. The force along the z-coordinate of the ship is the lift force: ZZ Fl z s ¼ nz zs Pship ðsÞds: ð11aÞ Ar Ae

3.1 Force and moment on the rudder There are a number of papers that describe low-aspect-ratio lifting surfaces [2, 3]. One can use a well-known empirical function derived from experimental data for low-aspectratio lifting surfaces at moderate angles of attack (\12°): p

cl ¼ 5:7  hattack ; 180 1 Fl ¼ cl A0 qu2r ; 2 where hattack is the angle of attack of the rudder [2, 3, 24, 25]. Instead of these empirical methods, DiSSEL integrates the pressure over the hull and rudders. The forces and

Similarly, the lift force on the rudder is ZZ nz zs Prudder ðsÞds; Flr zs ¼

ð11bÞ

A0 A0e

where Pship(s) and Prudder(s) are the pressures on the wetted surfaces of the ship and rudder for a potential flow, respectively; Ae is the effect area of the ship’s surface, Ar is the total wetted surface of the ship, and the non-effect area on the rudder is A00 ¼ A0  A0e . The effective inflow into the rudder is derived from both the ship’s forward speed and the accelerated flow in the propeller slipstream. DiSSEL does not have a propulsion

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model, so the velocity into the rudder, including the influence of the propeller, is obtained empirically as ur = 1.6us. The 1.6us approximation is a reasonable estimate for naval vessels with Froude numbers of between 0.2 and 0.4 and propellers designed for that speed (based on a suggestion of Dr. Young Shen, 2008). For other types of vessels, the flow acceleration at the propeller may differ from this. In these cases, the effective inflow velocity into the rudder should be computed from the conservation of momentum based on a user-specified propeller diameter and thrust coefficient. The yaw moment produced by the rudder about the ship’s center of gravity is obtained by multiplying the lift on the rudder by the appropriate moment arm: ZZ Crudderð3Þ ¼ ðR3 xs Þxðny ys ÞPrudder ðsÞds: ð12Þ A0 A0e

If the rudder surface is a planar surface, Eq. 12 can be written as 1 0 Crudderð3Þ ¼ q u2r A0 ðsin 2dÞR3 zs ; 4 where R3 ¼ x r  x c : After the forces and moments on the rudder have been obtained, these forces and moments on the rudder are added to the translational and rotational motion equations in the solid body ship motion of DiSSEL. The calculation of the translational and rotational motions of the ship is discussed in the following section.

mship

d2 X 1 0 ¼ F ¼ Fship þ Fl zs þ qw u2r A0 sin dððsin dÞxs dt2 2 þ ðcos dÞys Þ þ Flr zs þ FCoriolisð3Þ : 2

For surge x1, mship ddXt21 ¼ Fx ¼ FðxÞship  12qw u2r A00 sin2 d þ 2 FðxÞCoriolisð3Þ : For sway x2, mship ddXt22 ¼ Fy ¼ FðyÞship þ 1 2 0 2qw ur A0 sin d cos d þ FðyÞCoriolisð3Þ : The drift angle is computed from the velocities in the ship-fixed frame: b ¼ tan1

dX2=dt : us þ dX1=dt

ð17Þ

The total moment acting about the ship’s center of mass is C ¼ Cship þ Crudderð3Þ þ Ccð1Þ þ CCoriolisð3Þ :

ð18Þ

If the rudder surface is a planar surface, Eq. 18 can be written as C ¼ Cship þ Crudderð3Þ þ Ccð1Þ 1 ¼ Cship þ q u2r A00 ðsin 2dÞR3 þ Ccð1Þ þ CCoriolisð3Þ : 4 Cc(1) is an additional roll moment resulting from the turning of the ship that will be discussed separately in Sect. 3.4. Combining Eq. 18 with Eq. 9 allows us to calculate the yaw motion as follows: Ið3;3Þ

3.2 Translational and rotational motions of the ship

d2 h3 þ ðI22  I11 ÞX1 X2 ¼ Cð3Þship þ Crudderð3Þ dt2 1 ¼ Cð3Þship þ q u2r A00 ðsin 2dÞR3 ; 4 ð19Þ

and the roll moment can be calculated as The total force acting on the ship in the ship-fixed reference frame, including the force contribution from the rudder, is F ¼ Frudder þ Fship þ Fl zs þ FCoriolisð3Þ :

ð13Þ

From Eq. 8, Fship ¼ Fp þ Fq  Dtrans vs ; so

Ið1;1Þ

d2 h1 þ ðI33  I22 ÞX2 X3 ¼ Cð1Þship þ Ccð1Þ dt2 þ CðxÞCoriolisð3Þ :

ð20Þ

3.3 Heading and effective rudder angle

2

mship

dX ¼ F ¼ Fship þ Fl zs þ Fx xs þ Fy ys dt2 þ Flr zs þ FCoriolisð3Þ :

ð14Þ

For surge x1, mship

d 2 X1 ¼ Fx ¼ FðxÞship þ Fx þ FðxÞCoriolisð3Þ : dt2

ð15Þ

For sway x2, mship

d 2 X2 ¼ Fy ¼ FðyÞship þ Fy þ FðyÞCoriolisð3Þ : dt2

ð16Þ

If the rudder surface is a planar surface, Eqs. 14, 15, and 16 can be written as

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The heading angle of the ship, h3, is the angle between the ship’s longitudinal axis and the Earth-fixed x-axis. The drift angle, b, is the angle between the ship’s velocity vector in the horizontal plane and the ship’s longitudinal axis. The total heading angle, h*, is h ¼ h3 þ b:

ð21Þ

The total heading angle represents the angle between the ship’s total velocity vector and the Earth-fixed x-axis in the horizontal plane. The track of the ship in the Earth-fixed coordinate system is defined as xtrack ¼ xtrack ^x þ ytrack y^; and the track is computed at each time step using the following equation:

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137

xtrackðistepÞ ¼ xtrackðistep1Þ þ X1ðistepÞ      dX2   dX1 cos h3ðistepÞ  sin h3ðistepÞ dtðistepÞ þ usðistepÞ þ dt dt ytrackðistepÞ ¼ ytrackðistep1Þ þ X2ðistepÞ   

dX   dX1 2 cos h3ðistepÞ dtðistepÞ : þ usðistepÞ þ sin hðistepÞ þ dt dt

R1 ¼ ½ðxp  xc Þ  n  x: The roll motion can be calculated by substituting Eq. 25 into Eq. 20. This roll motion often results in a ship heel angle when the ship is performing a steady turning maneuver, as shown in Fig. 1.

ð22Þ

2

Ið3;3Þ ddth23  Cð3Þship d ¼ 0:5 sin1 : 1 2 0 4qw ur A0 R3

ð24Þ

3.4 Roll moment An additional roll moment acts on the ship as a result of its turning motion. This additional roll moment about the ship’s center of gravity can be computed as  2 dX 2 ZZ 1 us þ dX þ dt2 dt Ccð1Þ ¼ ds mship R1 r s  2 dX 2 ZZ 1 us þ dX þ dt2 dt  mship R1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds; 2 2 s ðus þdXdt1 Þ þðdXdt2 Þ dh dt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ZZ dX1 2 dX2 2 dh us þ ds ¼ mship R1 þ dt dt dt s

ð25Þ

where r is the turning radius, and the roll moment arm is

(a)

(b) Surge Force (X)

After obtaining the total heading angle h from Eq. 23, the time history of the rudder angle required to produce the specified trajectory can be obtained from Eqs. 15, 16, 17, 18, 19, and 21 using an iterative method. If h3

b; then h3  h when R3 (moment arm for the rudder force yaw moment) is large or the yaw moment of inertia I(3,3) is small, the rudder angle can be found by applying Eqs. 23 and 19:

Accurately calculated values for sway force and yaw moment are very important for predicting ship maneuvering. To benchmark the sway forces and yaw moments, this

-20

Surge Forces of 5514 Bare Hull in Pure Drift 0 Measured -10 -0.005 0 10 20 (Fr=0.41) -0.01 DiSSEL (Fr=0.41) -0.015

Measured (Fr=0.138)

-0.02

DiSSEL (FR=0.138)

-0.025 -0.03 (deg)

(c)

Sway Forces of 5514 Bare Hull in Pure Drift 0.15

Swary Force (Y)

ð23Þ *

4 Correlation with model test data

Measured (Fr=0.41) DiSSEL (Fr=0.41)

0.1 0.05 -15

-10

0 -5 0 -0.05

5

10

15

-0.1 -0.15 (deg)

(d) Yaw Moment (N)

If we assume that the time history of the rudder angle is input data, then we can use Eq. 19 to obtain the yaw angle h3 and Eqs. 15–17 to obtain the drift angle b at each time step. The track (xtrack(istep), ytrack(istep)) can be calculated using Eq. 22. However, if one assumes that the time history of the navigation positions, xtrack and ytrack, in Earth-fixed coordinates is the input dataset provided, the time history of the total heading angle h* can easily be obtained using 8 track > tan1 dx if dxtrack \0 and dytrack \0 > dytrack   > > >   dx 1 track > <  tan dytrack  þ p if dxtrack \0 and dytrack \0   h ¼ 1 dxtrack  > tan if dxtrack \0 and dytrack \0  þp > dy >  track  > >   > dx 1 3p track : tan  if dxtrack [ 0 and dytrack \0 dytrack  þ 2

Yaw Moments of 5514 Bare Hull in Pure Drift 0.06

Measured (Fr=0.138) DiSSEL (FR=0.138)

0.04

Mesured (FR=0.41)

0.02

DiSSEL (Fr=0.41)

0 -15

-10

-5-0.02 0 -0.04 -0.06 (deg)

5

10

15

Mesured (Fr=0.138) DiSSEL (Fr=0.138)

Fig. 1 a Ship maneuvering in a steady circle. b The surge forces in pure static drift at two different Froude numbers for a bare hull. c The sway forces in pure static drift at two different Froude numbers for a bare hull. d The yaw moments in pure static drift at two different Froude numbers for a bare hull

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Table 1 Comparison of experimental data and DiSSEL predictions for the pure drift and fully appended model Run no.

Measured conditions b (°)

d (°)

Measured

Heel (°)

Speed (kts)

Surge (KN)

DiSSEL Sway (KN)

Yaw (KN*M)

Roll (KN*M)

Surge (KN)

Sway (KN)

Yaw (KN*M)

Roll (KN*M)

1022

-4

-0.1

-0.1

18.01

-318.4

-797.5

-44746.7

4716.9

-321.2

-805.5

-45194.5

1023

-2

-0.1

-0.1

18.01

-306.3

-366.1

-17249.6

2240.1

-305.1

-362.4

-16991.3

4764 2217.7

1026 1029

0 2

-0.1 -0.1

-0.2 -0.2

18.02 18

-263.1 -284.3

91.4 518.1

-834.7 18268

-886.7 -3919.6

-265.7 -282.8

92.5 512.4

-844.7 18067

-897.3 -3876.5

1030

4

-0.1

-0.2

18

-210.2

983

42413.1

-6434.4

-212.3

992.3

42815.9

-6525.1

1100

0

39.7

-0.1

7.22

266.8

417.5

-25338.2

-3067.1

269.7

422.9

-25667.4

-3106.9

1153

16

-39.8

0

10.81

1.8

1378.6

108973.7

-7548.4

1.8

1397.3

110500

-7638.5

1187

-16

34

-0.1

10.77

62.9

-1266.9

-100509.2

6754.2

62.4

-1255.2

-99503.9

6686.7

1278

12

0

-8

10.8

360.3

1393.4

37675.5

-8551.7

363.9

1408.7

38089.9

-8645.8

1279

16

0

-8

10.8

341.9

1945.3

60696.8

-11231.2

338.5

1923.9

60028.3

-11107.5

(a)

Ship Scale Track of 5514 in Calm Water (Fr=0.2734, Rudder Angle=19.5deg) 100

(a)

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(Fr=0.2949, Rudder Angle=11.3deg) 200 -200 0

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Experimental Data

(b) (b)

Roll Motion of 5514 in Calm Water (Fr=0.2949, Rudder Angle=11.3deg)

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Fig. 2 a Track of model 5514 in calm water; Fr = 0.2949, rudder angle = 11.3°. b Roll angle time series with the same conditions as in a

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Roll Angle (deg)

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2

-2

Roll Motion of 5514 in Calm Water (Fr=0.2734, Rudder Angle=19.5deg)

Fig. 3 a Track of model 5514 in calm water; Fr = 0.2734, rudder angle = 19.5°. b Roll angle of model 5514 with the same conditions as in a

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study compared results with the planar motion mechanism (PMM) model tests performed on DTMB model 5415, which were used as a test case for the SIMMAN 2008 workshop. Model tests for a fully appended model are described by Moelgaard [20] and for the bare hull by Simonsen [27] and Benedetti et al. [28]. DTMB models 5514 and 5415 are geosyms, with model 5415 being a largerscale model used for resistance testing and model 5514 being used for free-running maneuvering and dynamic stability model tests. The geometry is unclassified and it is a commonly used test case that is employed to validate computational methods. The DiSSEL simulations were performed at full scale with a length, beam, and depth of 142, 19, and 16 m, respectively. The draft was 7 m and the weight was 9.549 tonnes. The mass gyradii were: k_roll = 0.383, k_pitch = 0.25, and k_yaw = 0.25. The

(a)

Ship Scale Track of 5514 in Calm Water (Fr=0.2332, Rudder Angle=28.1deg)

metacentric height was (GM) 0.91 m. It had two rudders and two propellers, and the total rudder area was 14.2 m2, with a root chord, cr, of 1.36 m and a tip chord, ct, of 0.23 m. Comparisons are made for the ‘‘pure drift’’ case, where the model is towed at a constant yaw angle. Figure 1b–d shows good correlations between the measured data [28] and DiSSEL in surge, sway forces and yaw moment due to the pure drift angle for a bare hull, respectively. Table 1 shows the run conditions associated with steady surge, sway forces and yaw moments in the experimental data [26] and in DiSSEL simulations for a fully appended model. In the following section, the predictions obtained using the proposed method are compared with the results of freerunning maneuvering model tests of model 5514 in calm water.

(a)

Ship Scale Track of 5514 in Calm Water (Fr=0.1957, Rudder Angle=10.0deg)

700

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Roll Motion of 5514 in Calm Water (Fr=0.2332, Rudder Angle=28.1deg)

Roll Motion of 5514 in Calm Water (Fr=0.1957, Rudder Angle=10.0deg)

12

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(b)

Experimental Data

8 6 4 2 0 -2 0

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Time (sec) DiSSEL Simulation Experimental Data

Fig. 4 a Track of model 5514 in calm water; Fr = 0.2332, rudder angle = 28.1°. b Roll angle of model 5514 with the same conditions as in a

Time (sec) DiSSEL Simulation Experimental Data

Fig. 5 a Track of model 5514 in calm water; Fr = 0.1957, rudder angle = 10.0°. b Roll angle of model 5514 with the same conditions as in a

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(a)

(a)

Ship Scale Track of 5514 in Calm Water (Fr=0.1635, Rudder Angle=19.5deg)

Ship Scale Track of 5514 in Calm Water (Fr=0.1367, Rudder Angle=29.3deg) 100

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Roll Motion of 5514 in Calm Water (Fr=0.1635, Rudder Angle=19.5deg)

(b)

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Roll Motion of 5514 in Calm Water (Fr=0.1367, Rudder Angle=29.3deg) 3 2 1 0 -1 0 -2 -3 -4 -5 -6

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Time (sec) DiSSEL Simulation Experimental Data

Time (sec) DiSSEL Simulation Experimental Data

Fig. 6 a Track of model 5514 in calm water; Fr = 0.1635, rudder angle = 19.5°. b Roll angle of model 5514 with the same conditions as in a

4.1 Ship track This section compares DiSSEL numerical simulations of the ship track for model 5514 with experimental data over a wide range of Froude numbers and rudder angles. The ship speed and rudder angle time series are the input data. Equations 15–19, 21, and 22 are used in DiSSEL (Eqs. 1– 11b) to calculate the ship track. Equations 20 and 25 are used to calculate the heel angle. The following DiSSEL simulations only study the steady turning circles; the initial transient processes are not included. In Figs. 2, 3, 4, 5, 6 and 7, the ship track is shown in panel (a) while the roll motion is shown in panel (b). The results are presented for six cases. Additional cases were also compared and showed very similar correlations.

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Fig. 7 a Track of model 5514 in calm water; Fr = 0.1367, rudder angle = 29.3°. b Roll angle of model 5514 with the same conditions as in a

The above six cases demonstrate that the differences in measured steady turning diameter and steady heel angle between the numerical simulation results obtained by DiSSEL and experimental data are generally less than 15%. There is only one case in which the difference between the DiSSEL predictions and the experimental data is about 20%. 4.2 Rudder angle In contrast with the previous section, the time histories of the ship speed and track are provided as input data in this section. DiSSEL is used to compute the required time history of the rudder angle. Equation 23 is first used to obtain the total heading angle, h*, and then Eqs. 15–19, and 21 are used with DiSSEL (Eqs. 1–11a) to calculate the rudder angle by an iterative method. However, if h* & h3,

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we can simply use Eqs. 19 and 23 to obtain the rudder angle; see Eq. 24. The following six cases correspond to the six cases shown in the previous section. The results are shown in Figs. 8, 9, 10, 11, 12 and 13.

The differences between the rudder angle values obtained in the numerical simulations by DiSSEL and from the experimental data, as shown in Figs. 8, 9, 10, 11, 12 and 13, are again less than 15%, which is smaller than the

Rudder Deflection of 5514 for Run 150 (Fr=0.1957)

Rudder Deflection of 5514 for RUN 157 (Fr=0.2949)

2

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-4 -6 DiSSEL Simulation Experimental Data

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Experimental Data -8 -10

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Fig. 8 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.2949

Fig. 11 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.1957

Rudder Deflection of 5514 for Run 159b (Fr=0.2734) 0 100

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Rudder Deflection of 5514 for Run 152 (Fr=0.1635)

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Fig. 9 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.2734

Fig. 12 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.1635

Rudder Deflection of 5514 for Run 155 (Fr=0.1367)

Rudder Deflection of 5514 for 162 Run (Fr=0.2332)

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Fig. 10 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.2332

-35

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Fig. 13 Comparison between the rudder angles predicted by DiSSEL and experimental data; Fr = 0.1367

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measurement error. The roll angles in all of the above cases are not shown because they are very similar to those shown in Figs. 2b, 3b, 4b, 5b, 6b and 7b.

5 Conclusion A method has been presented for computing the forces and moments generated by the rudder on a maneuvering ship. This rudder force model was implemented in the nonlinear ship motion model known as DiSSEL. The numerical predictions of ship steering capabilities as well as the associated roll motions obtained with the new method implemented in DiSSEL agree reasonably well with the experimental data. Acknowledgments The authors wish to thank Mr. Terry Applebee, Head of the Seakeeping Division, NSWCCD, for his many useful comments and suggestions. We also want to thank the Seakeeping Division at NSWCCD for providing the experimental data used for the correlation shown in this paper. Finally, we wish to thank Dr. Young T. Shen and Mr. Mike Davis for their many useful suggestions, and Scott Carpenter, who drew Fig. 1. This work is supported by the Naval Surface Warfare Center Independent Laboratory In-House Research (ILIR) program administered by Dr. John Barkyoumb.

References 1. Lin L, Lin R-Q (2004) Wave breaking function. In: 8th Int Workshop on Wave Hindcasting and Forecasting, North Shore, Hawaii, USA, 14–19 Nov 2004 2. Whicker LF, Fehlner LF (1958) Free-stream characteristics of a family of low-aspect-ratio, all-movable control surfaces for application of ship design (David Taylor Model Basin Report 933). David Taylor Model Basin, Navy Dept, Washington, DC, pp 121 3. Chaplin HR (1954) Wind-tunnel tests of low-aspect-ratio control surfaces with trailing edge to wind (David Taylor Model Basin Report 944). David Taylor Model Basin, Navy Dept, Washington, DC 4. Hacket JP, Burgh COE, Brewer WH (2005) Manufacturing tolerance effects on ship rudder force/cavitation performance. In: Proceedings of the 2005 SNAME Marine Technol Conf and Expo and Ship Production Symp, Houston, TX, USA, 19–21 Oct 2005 5. Abkowitz MA (1980) Measurement of hydrodynamic characteristics from ship maneuvering trials by system identification. Trans SNAME 30:2379–2404 6. Abkowitz MA (1964) Lectures on ship hydrodynamics—steering and maneuverability (Tech Rep Hy-5). Hydro- and Aerodynamics Laboratory, Lyngby 7. Chau S-W (1998) Computation of rudder force and moment in uniform flow. Ship Technol Res 45(1):3–13 8. El Moctar OM (1998) Numerical determination of rudder forces. In: Euromech 374, Futurescope, Poitiers, 27–29 April 1998

123

J Mar Sci Technol (2010) 15:131–142 9. Lee J-T (1988) A potential based panel method for the analysis of marine propellers in steady flow (Rep 87–13). Dept Ocean Eng, MIT, Cambridge, MA 10. Tamashima M, Matsui S, Yang J, Mori K, Yamazaki R (1993) The method for predicting the performance of propeller-rudder system with rudder angles and its application to rudder design. Trans West Jpn Soc Naval Architects 86:53–76 11. Han JM, Kong DS, Song IH, Lee CS (2001) Analysis of the cavitating flow around the horn-type rudder in the race of propeller. In: Fourth International Symposium on Cavitation, California Institute of Technology, Pasadena, CA, USA, 20–23 June 2001 12. Kinnas SA, Lee H, Gu H, Natarajan S (2007) Prediction of sheet cavitation on a rudder subject to propeller flow. J Ship Res 51:1–65 13. Hacket JP, Burgh COE, Brewer WH (2005) Manufacturing tolerance effects on ship rudder force/cavitation performance. In: Proceedings of the 2005 SNAME Marine Technol Conf and Expo and Ship Production Symp, Houston, TX, USA, 19–21 Oct 2005 14. So¨ding H (1999) Limits of potential theory in rudder flow predictions (1999 Weinblum Lecture). In: Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, DC, USA, 9–14 Aug 1998, pp 622–637 15. Lin R-Q, Kuang W, Reed AM (2005) Numerical modeling of nonlinear interactions between ships and surface gravity waves I: ship waves in calm water. J Ship Res 49(1):1–11 16. Lin R-Q, Kuang W (2006) Numerical modeling of nonlinear interactions between ships and surface gravity waves II: ship boundary condition. J Ship Res 50(2):181–186 17. Lin R-Q, Kuang W (2008) Modeling nonlinear roll damping with a self-consistent, strongly nonlinear ship motion model. J Mar Sci Technol 13:127–137 18. Lin R-Q, Hoyt J (2007) Fast ship motion in coastal region. In: 7th International Conference on Fast Ship, Shanghai, China, 23–27 Sept 2007 19. Landau LD, Lifshitz EM (1987) Fluid mechanics. Pergamon, Oxford, p 539 20. Freund DD, Meyer RE (1972) On the mechanism of blocking in a stratified fluid. J Fluid Mech 54:719–744 21. Garner ST (1998) Blocking and frontogenesis by two-dimensional terrain in baroclinic flow. Part II: analysis of flow stagnation mechanisms. J Atmos Sci 56:1509–1523 22. Lin R-Q, Chubb SR (2003) A comparison between radar imagery and coupled wave-current model results for a study of northwest Pacific seamount trapped waves. J Phys Ocean 108(C2):3032 23. Holton JR (1979) An introduction to dynamic meteorology. Academic, New York, p 391 24. Anderson JD (2005) Introduction to flight, 5th edn. McGraw-Hill, Boston, pp 763 25. Anderson JD (2007) Fundamentals of aerodynamics, 5th edn. McGraw-Hill, Boston, pp 500 26. Moelgaard A (2000) PMM tests with a model of a frigate (DMI 2000071, Rep 1). Danish Maritime Institute, Lyngby 27. Simonsen CC (2004) PMM model test with DDG51 including uncertainty assessment. FORCE Technology, Department of Maritime Industry, Lyngby 28. Benedetti L, Bouscasse B, Broglia R, Fabbri L, La Gala F, Lugni C (2006) PMM model test with DDG51 including uncertainly assessment (INSEAN Rep 14/06). INSEAN, Rome