Numerical investigation on the influence of Froude number on the maneuvering characteristics of a container ship

Abstract

Captive ship model tests are conducted to determine the hydrodynamic derivatives appearing in the maneuvering equations of motions of a ship. These hydrodynamic derivatives have an important role in maneuvering prediction of a ship at early design stages. Practically, surface ships operate at different speed conditions. Variation in vessel speed will affect the hydrodynamic derivatives and subsequently maneuvering characteristics of the ship. This paper investigates the effect of vessel speed on the derivatives and maneuvering characteristics of a ship. Captive model tests are numerically simulated in a CFD environment for a container ship at different Froude numbers to estimate the influence of Froude number on hydrodynamic derivatives and on the turning characteristics of the ship.

Keywords

Ship maneuvering planar motion mechanism turning circle test computational fluid dynamics

1. Introduction

Investigation of maneuvering abilities of a ship is important for both ship designers and operators. Poor controllability of a ship is considered as a major cause for most of the marine accidents, which may result in loss of life and property. Hence maneuvering prediction of a ship has to be done in its early design stages to ensure its safety and efficiency. Predicted maneuvering qualities of the vessel must be verified with the mandatory “Standards of ship maneuverability” approved by International Maritime Organisation (IMO) in 2002. Maneuvering behaviour of a ship can be predicted using free running model tests and system based methods. Free running model tests are more direct method for the prediction of maneuvering trajectories as well as maneuvering qualities. In system based methods, maneuvering prediction starts with the selection of suitable mathematical model representing the motion of the vessel under consideration acknowledges the maneuvering derivatives appearing in the model are required to solve the equations of motion. Once the mathematical model is chosen, the forces and moments acting on the ship hull can be estimated numerically and from this the hydrodynamic derivatives in the mathematical model are estimated. Usually, these hydrodynamic derivatives are determined from experimental, theoretical or numerical methods. A series of model experiments were carried out by many researchers [1,3,7,9,10,23,25,28] for the determination of hydrodynamic derivatives. Experimental methods are found to be more accurate and reliable than empirical and numerical methods used for maneuvering predictions, but it is time-consuming and requires expensive hydrodynamic test facilities to carry out the model tests. Free running model tests are conducted by several researchers to get the maneuvering trajectories. Empirical expressions for hydrodynamic derivatives derived from theoretical or regression analysis give an only approximate assessment. The limitations of experimental and theoretical methods leave room for the application of numerical methods in the maneuvering investigation. Among the numerical methods, researchers are mainly focusing on the application of RANSE based solvers for the solution of maneuvering problems. They have introduced different approaches on the numerical simulation of surface ship maneuvering, which include direct maneuvering simulation and simulation of captive model tests in a numerical environment. Compared to other maneuvering simulation methods, free running model simulations are more time consuming and require high computational resources. Verification and validation studies were carried out by several authors for establishing benchmarks for ship hydrodynamics. The SIMMAN (2008) workshops were conducted to benchmark the maneuvering prediction capabilities of simulation methods like system based and CFD based through systematic comparison and validation against EFD data for 3 types of selected ships such as Tanker (KVLCC), container ship (KCS) and surface combatant (5415) [21]. Carrica [2] carried out numerical simulations of zig-zag maneuvers for a containership model in a self-propulsion model. Dynamic overset grid is used for modelling the moving semi-balanced horn rudder and rotating propeller. Initial attempt for simulating captive model tests in a numerical wave tank was carried out by Ohmori [13] to estimate velocity dependent linear derivatives. Finite volume method is used for estimating viscous forces around the ship for captive model tests and showed good agreement by comparing the numerical results with the experiment. Kim [8] predicted the maneuvering of KRISO Container Ship with four degrees of freedom including roll motion by conducting captive model tests in a computerized planar motion carriage. It was concluded that twin propeller ship had better directional stability and worse turning ability than single propeller ship. Simonsen [19] validated CFD simulations of PMM model tests with experimental data by performing standard deep water maneuvering simulations on KCS container ship. Sakamoto [15] carried out URANS computations of static and dynamic test using viscous CFD solver. Verification of numerical convergence and validation of forces and moment coefficients were performed. The author concluded that compared to single run method, multiple run curve fitting method gives better results for the estimation of non-linear derivatives in maneuvering equations of motion. Toxopeus [22] made an attempt to improve the efficient calculation of hydrodynamic coefficients in maneuvering simulation. The improvements mainly focused on the variation of grid topology and density. Application of CFD promises an efficient and economical alternative to experimental captive model tests and it is gradually attracting the ship designers and operators in estimating the maneuvering qualities of a ship.

Muhammad [12] studied the effect of speed on a ferry ship maneuvering performance during turning circle and zig-zag manuevers of the ship. Maneuvering mathematical model presented in this paper did not consider the influence of Froude number on the hydrodynamic derivatives that make the prediction less accurate. Yoon [29] has performed captive model tests experiments on a surface combatant for different speeds and estimated variation of surge, sway and yaw derivatives with speed. This study has not extended to the simulation of maneuvering trajectory which would have given information about the speed effect on the turning abilities of the ship. Ship operates at different speed conditions and the influence of the speed on the hydrodynamic derivatives and controllability characteristics need to be studied for the accuracy of maneuvering predictions. Present study deals with numerical simulation of captive model tests in different forward speeds for a container ship to estimate the speed effect on the hydrodynamic derivatives. These derivatives are used in the maneuvering equations of motion to simulate the ship turning trajectory and to get knowledge about the influence of Froude number on the turning characteristics of a ship. Lower to medium Froude number range from 0.144 to 0.289 are selected for the present study.

1.1. Mathematical model

Mathematical model representing the maneuvering of surface ship in still water are formed based on the fact that the hydrodynamic forces are the functions of the velocities and acceleration involved in a motion [24]. The selection of suitable mathematical model forms the initial step in the maneuvering prediction of a surface ship. A nonlinear mathematical model for high speed container ship proposed by Son and Nomoto [20] is used for the present study, presented as MMG (Mathematical Modelling Group) [27] which describes each hull forces such as surge force, sway force and yaw moment as separate modules. Maneuvering motions of a ship in calm water are represented in three degrees of freedom, describing the surge, sway and yaw motions and are expressed as $\begin{array}{l} (1) & (m - X_{\dot{u}}) \dot{u} - (m - Y_{\dot{v}}) v r = X \\ (2) & (m - Y_{\dot{v}}) \dot{v} + (m - X_{\dot{u}}) u r - Y_{\dot{r}} \dot{r} = Y \\ (3) & (I_{Z} - N_{\dot{r}}) \dot{r} + N_{\dot{v}} \dot{v} = N \end{array}$ where X, Y, N are expressed as sum of hull, propeller and rudder forces as shown below $\begin{array}{l} (4) & X = X (u) + (1 - t_{r}) T_{P} + X_{v r} v r + X_{v v} v^{2} + X_{r r} r^{2} + X_{δ} sin δ \\ (5) & Y = Y_{v} v + Y_{r} r + Y_{v v v} v^{3} + Y_{r r r} r^{3} + Y_{v v r} v^{2} r + Y_{v r r} v r^{2} + Y_{δ} cos δ \\ (6) & N = N_{v} v + N_{r} r + N_{v v v} v^{3} + N_{r r r} r^{3} + N_{v v r} v^{2} r + N_{v r r} v r^{2} + N_{δ} cos δ \end{array}$

In this mathematical model, hull, rudder and propeller forces are represented by separate modules. Rudder and propeller derivatives are found out using empirical expressions, coupling between rudder angle and hull velocities are not considered. Hull derivatives are of prime importance, hence modelling of rudder and propeller is not considered and a bare hull form of ship without rudder and appendages are selected. $X (u) = X_{| u | u} | u | u$ is velocity dependent damping function. Ship roll motion become significant when it takes a strong turn and the effect is more if the turn is taken at high speeds. To consider the roll effects in ship maneuvering one has to include the roll motion and the coupled effects on the other these modes such as surge, sway and yaw. Present mathematical model does not consider the roll degree of freedom. Inclusion of roll in modelling and analysis is more involving and computationally complicated, hence the original model is simplified by neglecting roll effects.

Propeller thrust $T_{P}$ is given by, $\begin{matrix} (7) & T_{p} = ρ n^{2} D^{4} K_{T} \end{matrix}$

Where ρ is the density of the fluid, n is the propeller revolutions per second. D is the propeller diameter and $K_{T}$ is the thrust coefficient. $\begin{matrix} (8) & K_{T} = 0.527 - 0.455 J \end{matrix}$

Where J is the advance ratio, given by, $\begin{matrix} (9) & J = \frac{u_{p}^{'} U_{T}}{n D} \end{matrix}$

$U_{T}$ is the service speed of the ship, $u_{p}^{'}$ is the effective wake fraction given by, $\begin{matrix} (10) & u_{P}^{'} = cos v^{'} [(1 - w_{p}) + τ_{P} {(v^{'} + x_{P}^{'} r^{'})}^{2} \end{matrix}$

Where $w_{p}$ is the wake fraction in the straight ahead condition, $v^{'}$ and $r^{'}$ are the non-dimensionalised sway velocity and yaw rate respectively, $x_{P}^{'}$ is the x-location of the propeller plane and $τ_{P}$ is the parameter to account for the lateral velocity components.

The non-dimensionalised rudder derivatives are given by the following empirical expressions. $\begin{array}{l} (11) & X_{δ}^{'} = c_{R X} F_{N}^{'} \\ (12) & Y_{δ}^{'} = (1 + a_{H}) F_{N}^{'} \\ (13) & N_{δ}^{'} = - (x_{R}^{'} + a_{H} x_{H}^{'}) F_{N}^{'} \end{array}$

Where $c_{R X}$ , $a_{H}$ , $x_{H}^{'}$ represents the rudder-hull interaction coefficients and $x_{R}^{'}$ represent the non-dimensionalised rudder position in longitudinal directhe tion. $F_{N}^{'}$ represents the rudder normal force which is given by the following expression, $\begin{matrix} (14) & F_{N}^{'} = - \frac{6.13 Λ}{Λ + 2.25} \frac{A_{R}}{L^{2}} (u_{R}^{^{'} 2} + v_{R}^{^{'} 2}) sin α_{R} \end{matrix}$

Where Λ, $A_{R}$ , $α_{R}$ is the aspect ratio, mid-ship area and effective flow angle of the rudder, respectively. $u_{R}^{'}$ , $v_{R}^{'}$ are the non-dimensional axial and lateral component of effective rudder inflow velocity.

Fig. 1.

Earth fixed and ship fixed co-ordinate system.

The two right-handed coordinate systems used for describing the ship motions in the present maneuvering study are earth fixed and ship fixed ones (Fig. 1). Axes fixed related to earth are known as earth fixed coordinate system ( $o x_{0} y_{0} z_{0}$ ), and axes fixed to centre of gravity of the ship and move with the motion of ship are known as ship fixed coordinate system ( $G x y z$ ), where $x_{O G}, y_{O G}$ represent the position of centre of gravity of ship with respect to earth fixed co-ordinate system and ψ is the heading or yaw angle which refers to the direction of the ship and β is the drift angle defined as the angle between resultant velocity in the ship fixed co-ordinate system and the longitudinal axis of the ship. The container ship S175 is selected for the present study and principal particulars of the ship are given in Table 1.

Table 1

Main particulars of the ship S175

Particulars	Prototype	Model (1:36)
Length, L_BP	175 m	4.86
Beam, B	25.4 m	0.705
Draft at FP, $T_{f}$	8.0 m	0.22
Draft at AP, $T_{a}$	9.0 m	0.25
Draft at midship, $T_{m}$	8.5 m	0.236
Depth, D	11.0 m	0.305
Displacement, ∇	21,222 m³	0.4548
Block coefficient, $C_{B}$	0.559	0.559

2. Simulation of captive model tests

Captive ship model tests are numerically simulated to determine the coefficients appearing in the maneuvering mathematical model. These tests are of two types namely static or straight line test and dynamic or Planar Motion Mechanism (PMM) test, respectively. For the present study, static drift test and planar motion mechanism tests are carried out numerically for different speeds in a numerical towing tank. Circular motion tests are not included in the present study and these tests also give only velocity dependant derivatives, whereas the PMM tests provide with both velocity and acceleration dependant derivatives.

2.1. Simulation of straight line test

Straight line test is carried out to determine sway velocity dependent derivatives by towing the model of the ship at a constant velocity Um at various angles of attack β (Fig. 2). For the present study static test is simulated in a CFD environment at different Froude numbers ( $Fn = 0.14, 0.18, 0.22, 0.25$ and 0.29) for determining the vessel speed effect on velocity dependent hydrodynamic derivatives.

Fig. 2.

Model orientation in straight line test.

2.2. Planar motion mechanism

Planar motion mechanism is used for conducting dynamic ship model tests in a conventional towing tank to determine the maneuvering hydrodynamic derivatives of surface ships, PMM consists of two oscillators placed on the ship model on bow part (B) and stern part (S), which are oscillated in-phase or out of phase to get the required oscillations modes, in which the model is towed along the towing tank at the presented speed, $U_{m}$ (Fig. 3).

Fig. 3.

Model set up in PMM test.

2.2.1. Pure sway mode

In this mode of PMM operation, phase angle between the oscillators at bow and stern of the model are made to zero. So that model experiences harmonic translational motion in y-direction with prescribed amplitude and frequency where the model axis always remain parallel to the towing tank axis (Fig. 4). Along with this motion, the model is towed down the tank at a constant velocity. This test is conducted to determine both velocity and acceleration derivatives in pure sway. Model kinematic parameters are represented as $\begin{array}{l} (15) & Transverse displacement, y_{0} = - y_{0 a} sin Ω_{0} t \\ (16) & Transverse velocity, {\dot{y}}_{0} = - y_{0 a} Ω_{0} cos Ω_{0} t \\ (17) & Transverse acceleration, \ddot{y_{0}} = y_{0 a} Ω_{0}^{2} sin Ω_{0} t \end{array}$ t is the time taken for simulation, $Ω_{0}$ is the frequency of transverse oscillation of PMM, $y_{0 a}$ is the sway amplitude and $y_{0}$ is the instantaneous sway motion. For numerical simulation of the PMM test, User Defined Function (UDF) available in CFD tool box assign the translational velocity to the model and mesh morphing technique in which mesh deforms close to hull to account for hull motion. The grid oscillates in transverse direction in the body co-ordinate system and the mesh positions are updated in each time step.

Fig. 4.

Orientation of ship model in pure sway model.

Fig. 5.

Orientation of ship model in pure yaw mode.

2.2.2. Pure yaw mode

In this mode of PMM operation, phase angle 2ε is given between the two oscillators, where ε is given by ${tan}^{- 1} {d Ω / 2 U_{m}}$ , where d is given by two times distance between oscillators and $Ω_{0}$ is the frequency of oscillation. This phase angle ensures that the model axis is always tangential to the oscillation path (Fig. 5) while the model is towed down the tank at a constant velocity ( $U_{m}$ ) with zero drift angle. This results in a harmonic yaw motion with prescribed amplitude and frequency. This test is conducted to determine yaw dependant acceleration and velocity derivatives. $\begin{array}{l} (18) & Yaw angle, Ψ = - Ψ_{a} cos Ω_{0} t \\ (19) & Yaw rate, \dot{Ψ} = - Ψ_{a} Ω_{0} sin Ω_{0} t \\ (20) & Yaw acceleration, \ddot{Ψ} = - Ψ_{a} Ω_{0}^{2} cos Ω_{0} t \end{array}$ In the numerical simulation of this mode of PMM, a rotational velocity is assigned to the mesh on the hull using morphing technique available in the CFD software. The mesh position on the hull are updated at each time step while the outer domain remain stationary.

2.2.3. Combined sway and yaw mode

In combined sway and yaw test, the model is towed with a harmonic pure yaw motion mode with a constant drift angle. The phase angle ( $2 ε$ ) between the oscillators to give a yaw motion along with a sway motion is given by the relation, $ε \neq {tan}^{- 1} \frac{d Ω}{2 U_{m}}$ , This phase angle ensures that the centreline of the model is not tangent to its path (Fig. 6), but with the prescribed drift angle. This test is used to determine sway and yaw coupled velocity and acceleration dependent derivatives. In the numerical simulation of the combined mode of PMM, a translational velocity in y-direction superimposed with rotational velocity in z-direction are assigned to the ship model using morphing technique.

Fig. 6.

Orientation of ship model in combined sway and yaw mode.

3. Numerical CFD simulation

In the present study, static and dynamic tests are simulated in a CFD environment at different Froude numbers. Hull form of the ship is modelled and imported in the RANSE based CFD solver, STARCCM+. Suitable computational domain for static and dynamic test are selected separately according to the International Towing Tank Conference (ITTC) guidelines [6], by creating a rectangular block around the ship model. According to ITTC guidelines on the use of RANSE tools for maneuvering predictions, typical domain dimensions are 3–5 ship lengths in the longitudinal direction and 2–3 ship lengths in transverse direction. Domain within the range of ITTC is selected for static test simulation. Domain larger than ITTC standards is selected for the numerical simulation of dynamic tests as the hydrodynamic disturbances are expected to be high in the case of dynamic tests compared to static one. The fluid domain for static simulation extends 1 L_PP from bow, 2 L_PP from stern and 1.5 L_PP from each of port and starboard sides, 0.5 L_PP from the deck to top boundary, 1 L_PP from keel to bottom sides (Fig. 7). The fluid domain for dynamic simulation extends 4 L_PP from the aft, 2 L_PP from the bow, 3 L_PP from each of port and starboard sides and 1 L_PP each from keel to bottom side, where L_PP is the length between perpendiculars (Fig. 8). Boundaries of a domain are composed of velocity inlet for inlet boundary, downstream pressure at outlet boundary, wall with no slip for the hull and wall with slip for the tank.

Fig. 7.

Computational domain for static test.

Fig. 8.

Computational domain for dynamic test.

An unstructured trimmed hexahedral cell with prismatic near wall layers is selected to capture the flow properties around the hull surface (Fig. 9). Rectangular volumetric control block with high resolution is selected to resolve the free surface (Fig. 10). Anisotropic refinement is chosen for the refinement of water surface. Mesh density is created at the free surface with dimensions sufficient enough to resolve the generated waves due to ship hull. It can be seen in the scalar scene of free surface updated at each time step. Volume mesh is further refined based on the knowledge of flow gradients (Fig. 11). Mesh refinement in volumetric control is done separately for each speed. Contour plot of wall Y+ for the hull at 1 m/s, 1.5 m/s and 2 m/s are shown in Figs 12, 13 and 14. Surface averaged wall Y+ for the submerged part of hull is included in Table 2. As the speed of ship increases, hydrodynamic forces on the hull become over predicted due to the reflection of wave from the outlet boundary of the computational domain, Hence wave damping option in the software is activated to damp the waves in order to avoid the wave reflection. Here, wave damping started at a location towards the outlet boundary where the volumetric blocks of fine mesh ends, otherwise, the Kelvin wave pattern produced by the ship system, which has a non-negligible contribution in the hull hydrodynamic forces, got interrupted by the damper. To investigate the Kelvin wave pattern due to steady speed forward motion of ship, the free surface waves are captured at different speeds (Figs 15–17) and analysed. The Kelvin wave pattern showed its shape clearly as the speed increases and it is decayed away from the ship.

Fig. 9.

Structure of mesh generated for hull.

Fig. 10.

Generated mesh in sectional view.

Fig. 11.

Plan view of generated grid in pure yaw mode.

Fig. 12.

Wall Y+ for the hull at 1 m/s.

Fig. 13.

Wall Y+ for the hull at 1.5 m/s.

Fig. 14.

Wall Y+ for the hull at 2 m/s.

Table 2

Surface averaged wall Y+ at different speeds

Speed (m/s)	Surface averaged wall Y+
1	33
1.25	36
1.5	37
1.75	38
2	40

Fig. 15.

Plan view of water surface captured for 1 m/s of ship model speed in pure yaw mode.

Fig. 16.

Plan view of water surface captured for ship 1.5 m/s of ship model speed in pure yaw mode.

Fig. 17.

Plan view of water surface captured for 2 m/s of ship model speed in pure yaw mode.

Transient finite volume simulations employed an implicit unsteady with a segregated (predictor-corrector) flow solver and SIMPLE (Semi Implicit Method for Pressure Linked Equations) solution algorithm. Second order upwind spatial and first order temporal discretisation scheme are used. Based on prior studies [16,17], SST k-Ω turbulence model [11], which is a two-equation eddy-viscosity model is used in the present study. This model is more stable than k-ε model because it doesn’t involve any damping function. Problems involving high pressure gradients, it is advisable to use k-Ω model to capture the flow [18] and this model can accurately capture the near wall physics, that is necessary for the estimation of hydrodynamic forces acting on the hull. Different turbulent models are compared by Chandran [14] for studying the hydrodynamic forces for an underwater body. Among different models, k-Ω model gives closer prediction for hydrodynamic forces and moments to experimental results. Study conducted by Shenoi [18] on the influence of free surface modelling in the simulations of static drift tests shows that the derivatives obtained from the simulations including free surface modelling are more comparable with the experimental results than the simulations without including the free surface. Hence for the present study free surface modelling is done using VOF method, which captures movement of the interface between the fluid phases. The High Resolution Interface Capturing (HRIC) convective discretization scheme is used to track the air-water interface. The transient simulation is initialised at $t = 0$ and the whole domain of fluid flow will have initial values. The solution advances through each time step assuming new values in the fluid flow dimensions. Appropriate time step $Δ t$ of order $10^{- 2}$ is chosen based on the velocity of flow and also on grid size in the direction of flow. Grid independence study is carried out for the proper selection of grid size for the static drift test with drift angle 5 degrees. Convergence of surge force, sway force and yaw moment are observed. Grid independence study is carried out to obtain reliable values based on a grid refinement ratio of $\sqrt{2}$ [4] where the minimum cell size near the hull surface is multiplied by $\sqrt{2}$ to get next grid (Table 3). Change of minimum size on the hull surface are not proportionately reflecting in the total number of faces due to the fact that grid generation is carried out using unstructured grids. Also, change of minimum cell size is a driving factor for an unstructured grid to refine the curvature parts which can directly effect in the change of total number of faces in a disproportionate way. From the grid independence study, grid no. 3 is selected for the numerical simulation, which will give a total cell count of 1.32 million. Convergence ratio, $R_{k} = \frac{ε_{k 21}}{ε_{k 32}}$ ,where changes between medium-fine, $ε_{k 21} = S_{k 2} - S_{k 1}$ and changes between coarse to medium $ε_{k 32} = S_{k 3} - S_{k 2}$ , $S_{k 1}, S_{k 2}, S_{k 3}$ correspond to solution with fine, medium and coarse input parameter respectively. Estimated convergence ratio for surge force, sway force and yaw moment are $- 0.085$ , $- 0.02$ and $- 0.025$ respectively. Present convergence ratio less than zero are called solutions with oscillatory convergence [5]. For oscillatory convergence, uncertainties are estimated based on oscillation maximums $S_{u}$ and minimum $S_{L}$ , i.e. uncertainty $U_{k} = 1 / 2 (S_{U} - S_{L})$ . Estimated $U_{k}$ for surge force, sway force and yaw moment are 0.235, 0.23 and 0.18 respectively. Mesh finalized from grid independent study is chosen for domain dependence study (Table 4). A small computational domain of 0.75 times the dimension in all direction of ITTC standard fluid domain and a bigger domain of size 2 times the dimension in all direction of ITTC standard fluid domain are chosen. Hydrodynamic disturbance is more in smaller computational domain due to wave reflection from outer boundary which results in the inaccurate estimation of forces and moments. For domain extends greater than the ITTC standard domain, the results remain unchanging and thus ITTC domain which extends 1 L_PP from bow, 2 L_PP from stern and 1.5 L_PP from each of port and starboard sides, 0.5 L_PP from the deck to top boundary, 1 L_PP from keel to bottom sides is selected for static simulation. Total cell count used for different speeds are presented in Table 5.

Table 3

Grid independent study for static drift test with drift angle 5 degree

Grid No.	Minimum cell size on hull surface (cm)	Total number of faces on the hull	Cell count in free surface refinement zone (million)			Total Cell count (million)	Solver elapsed time/time step	Surge force (N)	Sway force (N)	Yaw moment (N)

			Water phase	Air phase	Total
1	25	44292	0.458	0.374	0.832	0.95	0.257	−8.92	8.73	22.64
2	17.6	55178	0.585	0.478	1.063	1.22	0.29	−8.52	8.42	22.12
3	12.5	67213	0.63	0.51	1.15	1.32	0.32	−8.45	8.27	21.92
4	8.875	81244	1.25	1.02	2.273	2.702	0.59	−8.456	8.273	21.925

Table 4

Domain independent study for static drift test with drift angle 5 degree

Grid No.	Domain	Cell count (million)	Solver elapsed time/time step	Surge force (N)	Sway force (N)	Yaw moment (N)
1	0.75 ITTC	0.61	0.16	−8.21	8.35	22.1
2	ITTC	1.32	0.32	−8.45	8.27	21.92
3	2 ITTC	2.42	0.53	−8.451	8.278	21.929

Table 5

Cell count for different speeds

Speed (m/s)	No. of grid points per representative wave height	Number of grid points per representative wave length	Cell count in free surface refinement zone (million)			Total cell count (million)

			Air phase	Water phase	Total
1	18	36	0.95	1.166	2.12	2.42
1.25	19	37	1.0305	1.259	2.29	2.6
1.5	20	38	1.089	1.331	2.42	2.76
1.75	21	40	1.19	1.47	2.66	2.89
2	23	41	1.33	1.633	2.97	3.39

In the numerical dynamic simulation, the prescribed motion of the mesh around the hull are imposed using mesh morphing technique, where the motion parameters are expressed in terms of field functions available within the solver and the model is given freedom in heave and pitch. The solution process runs for a total physical time of 80 seconds with a number of time step of 1256 per period of oscillation for pure sway mode and 661-time steps per period of oscillation for pure yaw and combined sway and yaw mode of simulation are used. Time step chosen is enough to give a Courant Friedrichs Lewy (CFL) number less than unity. Solver executes 6 number of inner iterations for each unsteady time step and Root Mean Squared residuals are showed to be error of order $10^{- 4}$ for every simulation. The mesh generated around the hull are updated for each time step as the solution advances while the outer domain remains stationary. Selection of oscillation frequency for different modes of PMM is based on the guidelines given by Vantorre [5]. Based on the formula frequency selected for pure sway mode is taken as 0.5 rad/s and for pure yaw and combined mode is 0.95 rad/s. Motion parameters of PMM simulation are given in Table 6.

Table 6

PMM text matrix for different speeds

Tests	Speed, U (m/s)	Sway amplitude (m)	Sway velocity (m/s)	Yaw amplitude (rad)	Yaw velocity (rad/s)
Pure sway	1	0.3	0.1424	–	–
	1.25
	1.5
	1.75
	2.0
Pure yaw	1.0
	1.25			0.142	0.134
	1.5			0.1312	0.1239
	1.75			0.121	0.1143
	2.0			0.11187	0.1122
Combined mode	1	0.2785	0.263	0.06765	0.0639
	1.25
	1.5
	1.75
	2.0

3.1. Hydrodynamic forces and derivative determination

For the static simulation, the estimated surge and sway forces and yaw moment are plotted against sway velocity for each speed of model. The magnitude of estimated forces are found to increase with the model speed. Hydrodynamic derivatives are derived using curve-fitting method on the above data. Third order polynomial is used for fitting curve and the coefficient of curve fit are calculated. These terms are equated to the coefficient of the sway velocity dependant terms of the third order Taylor series in the nonlinear mathematical model proposed by Abkowitz [26]. The dynamic simulation runs for a total time of 80 seconds. First and second cycle of the force/moment time history showed fluctuations due to unsteady nature of flow, resulting in over-prediction of the forces/moments. After second cycle steady trends are noticed with the absence of transients, and hence the third cycle is chosen for the Fourier series analysis of the time series force data. Hydrodynamic derivatives are derived from the estimated force-time series using Fourier-series expansion method in which the resulting force/moment is reconstructed as a Fourier series. The computed forces/moments are numerically integrated over the full cycle using trapezoidal integration rule to get Fourier coefficients and these coefficients are compared with the coefficients of the mathematical model for deriving the expression for hydrodynamic derivatives. Hydrodynamic forces and moments on the ship hull can be expressed as $\begin{matrix} (21) & f (t) = \sum_{m = 0}^{\infty} (a_{m} cos m Ω_{o} t + b_{m} sin m Ω_{0} t) \end{matrix}$

Fourier series for the forces and moments can be expressed as $\begin{array}{l} (22) & X_{H Y} = \sum_{m = 0}^{3} {a_{x A m} cos m Ω_{o} t + b_{x A m} sin m Ω_{o} t} \\ (23) & Y_{H Y} = \sum_{m = 0}^{3} {a_{y A m} (cos m Ω_{o} t + b_{y A m} sin m Ω_{o} t} \\ (24) & N_{H Y} = \sum_{m = 0}^{3} {a_{n A m} cos m Ω_{o} t + b_{n A m} sin m Ω_{o} t} \end{array}$

Fourier constants are given by, $\begin{array}{l} (25) & a_{0} = \frac{1}{T} \int_{0}^{T} f (t) d t \\ (26) & a_{m} = \frac{2}{T} \int_{0}^{T} f (t) cos m Ω_{0} t d t \\ (27) & b_{m} = \frac{2}{T} \int_{0}^{T} f (t) sin m Ω_{0} t d t \end{array}$

Detail description of mathematical formulation of the PMM for estimating hydrodynamic derivatives are given in the reference [18].

Table 7
Hydrodynamic derivatives obtained from pure sway test for different Froude numbers

Hydro dynamic derivative $Fn = 0.14$ $Fn = 0.18$ $Fn = 0.22$ $Fn = 0.25$ $Fn = 0.29$ % change from 0.14 to 0.29

$Y_{v}^{'}$ −0.00978 −0.0099 −0.009 −0.0084 −0.0087 11.037

$Y_{\dot{v}}^{'}$ −0.00726 −0.007 −0.0074 −0.0076 −0.0081 −11.59

$Y_{v v v}^{'}$ −0.1432 −0.13 −0.1276 −0.0965 −0.059 58.78

$N_{v}^{'}$ −0.00389 −0.004 −0.00456 −0.0046 −0.00495 −27.16

$N_{\dot{v}}^{'}$ −0.0002 −0.00006 −0.00006 −0.0001 −0.00002 87.87

$N_{v v v}^{'}$ −0.0191 −0.0005 0.01573 0.0039 0.0443 332.03

$X_{v v}^{'}$ −0.0014 −0.0029 −0.0041 −0.0042 −0.0039 −178.5

Hydro dynamic derivative	$Fn = 0.14$	$Fn = 0.18$	$Fn = 0.22$	$Fn = 0.25$	$Fn = 0.29$	% change from 0.14 to 0.29
$Y_{v}^{'}$	−0.00978	−0.0099	−0.009	−0.0084	−0.0087	11.037
$Y_{\dot{v}}^{'}$	−0.00726	−0.007	−0.0074	−0.0076	−0.0081	−11.59
$Y_{v v v}^{'}$	−0.1432	−0.13	−0.1276	−0.0965	−0.059	58.78
$N_{v}^{'}$	−0.00389	−0.004	−0.00456	−0.0046	−0.00495	−27.16
$N_{\dot{v}}^{'}$	−0.0002	−0.00006	−0.00006	−0.0001	−0.00002	87.87
$N_{v v v}^{'}$	−0.0191	−0.0005	0.01573	0.0039	0.0443	332.03
$X_{v v}^{'}$	−0.0014	−0.0029	−0.0041	−0.0042	−0.0039	−178.5

Table 8

Hydrodynamic derivatives obtained from pure yaw test for different Froude numbers

Hydro dynamic derivative	$Fn = 0.14$	$Fn = 0.18$	$Fn = 0.22$	$Fn = 0.25$	$Fn = 0.29$	% change from 0.14 to 0.29
$Y_{r}^{'}$	0.003	0.0033	0.00361	0.00410	0.00413	−37.86
$N_{r}^{'}$	−0.0022	−0.0021	−0.0016	−0.0014	−0.0015	31.82
$X_{u u}^{'}$	−0.00062	−0.00059	−0.00058	−0.00037	−0.00026	58.14
$X_{r r}^{'}$	−0.00065	−0.00068	−0.00069	−0.00075	−0.00079	−21.54
$N_{\dot{r}}^{'}$	−0.00052	−0.00049	−0.00046	−0.00046	−0.00057	−9.61
$Y_{\dot{r}}^{'}$	−0.0001	−0.00012	−0.00012	−0.0001	−0.00011	−10
$N_{r r r}^{'}$	−0.00393	−0.00409	−0.00494	−0.0058	−0.00682	−73.53
$Y_{r r r}^{'}$	−0.00644	−0.01256	−0.01593	−0.00989	−0.00227	64.51
$X_{\dot{u}}^{'}$	−0.00063	−0.00015	−0.00023	−0.00022	−0.00093	−47.22

Table 9

Hydrodynamic derivatives obtained from combined sway and yaw test for different Froude numbers

Hydro dynamic derivative	$Fn = 0.14$	$Fn = 0.18$	$Fn = 0.22$	$Fn = 0.25$	$Fn = 0.29$	% change from 0.14 to 0.29
$Y_{v r r}^{'}$	−0.03917	−0.02598	−0.02187	−0.01895	−0.01169	70.16
$Y_{v v r}^{'}$	0.01112	0.032616	0.040556	0.041228	0.048	−331.655
$N_{v v r}^{'}$	−0.0177	−0.02952	−0.04739	−0.03245	−0.03064	−73.07
$N_{v r r}^{'}$	0.003221	0.001054	0.002148	0.002541	0.003884	−20.59
$X_{v r}^{'}$	−0.00981	−0.00856	−0.00358	−0.00124	−0.00191	80.48

Table 10

Comparison of hydrodynamic derivative results

Hydro dynamic derivative	Experimental (Son and Nomoto, 1981)	Numerical (Present CFD)
$Y_{v}^{'}$	−0.0116	−0.00978
$Y_{\dot{v}}^{'}$	−0.00705	−0.00726
$Y_{v v v}^{'}$	−0.109	−0.1432
$N_{v}^{'}$	−0.00386	−0.00389
$N_{\dot{v}}^{'}$	−0.00035	−0.0002
$N_{v v v}^{'}$	0.001492	−0.0191
$X_{v v}^{'}$	−0.00386	−0.0014
$Y_{r}^{'}$	0.00242	0.003
$N_{r}^{'}$	−0.0022	−0.0022
$X_{u u}^{'}$	−0.00042	−0.00062
$X_{r r}^{'}$	0.0002	−0.00065
$N_{\dot{r}}^{'}$	−0.000419	−0.00052
$Y_{\dot{r}}^{'}$	−0.0003525	−0.0001
$N_{r r r}^{'}$	−0.00229	−0.00393
$Y_{r r r}^{'}$	0.00177	−0.00644
$X_{\dot{u}}^{'}$	−0.000238	−0.00063
$Y_{v r r}^{'}$	−0.0405	−0.03917
$Y_{v v r}^{'}$	0.0214	0.01112
$N_{v v r}^{'}$	−0.0424	−0.0177
$N_{v r r}^{'}$	0.00156	0.003221
$X_{v r}^{'}$	−0.00311	−0.00981

3.2. Results and discussion of CFD simulations

Static and dynamic tests are conducted for the container ship for different Froude numbers in a numerical towing tank. Estimated forces and moments from the numerical tests are analysed using curve fitting method and Fourier series expansion method to get the hydrodynamic derivatives (Table 7, 8, 9). Numerically obtained hydrodynamic derivatives estimated at 1 m/s speed are compared with the derivatives obtained from captive model experiments done by Son and Nomoto [20] (Table 10). Results of hydrodynamic derivatives values from present CFD study matches well with the experimental results. Force-time series obtained from static test are plotted against sway velocity in non-dimensional form. (Figs 18–20). It is observed that the magnitude of estimated forces increase with the model speed except for surge force. Forces and moments are made dimensionless by the factors $0.5 ρ L_{m}^{2} U_{m}^{2}$ for force and $0.5 ρ L_{m}^{3} U_{m}^{2}$ for moment. Hydrodynamic derivatives are derived from this resulted force-time series using curve fitting method. Curves are fitted separately for each speed and the hydrodynamic derivatives are estimated to know about the dependency of the derivatives on ship model speed.

Fig. 18.

Variation of non-dimensional surge force against drift angle at different speeds in static test.

Fig. 19.

Variation of non-dimensional sway force against drift angle at different speeds in static test.

Fig. 20.

Variation of non-dimensional yaw moment against drift angle at different speeds in static test.

Non dimensional forces/moment times series obtained from pure sway test for different Froude numbers are plotted (Figs 21–23). Influence of model forward speed on the hydrodynamic derivatives is visible from the estimated hydrodynamic derivatives from captive PMM tests. Comparison is made with estimated hydrodynamic derivatives from both static and dynamic test (Figs 24–28). First order derivatives like $Y_{v}^{'}$ , $N_{v}^{'}$ obtained from static tests are matching reasonably well with the dynamic test results whereas higher order derivatives like $Y_{v v v}^{'}$ , $N_{v v v}^{'}, X_{v v}^{'}$ show high deviation. These deviation could be due to the effect of PMM frequency. Shenoi et al. [18] reported from their study that sway dependant derivatives obtained from dynamic pure sway test are better when compared with the static test results. Hence for the present study, results of sway dependant derivatives from PMM pure sway mode are taken for the simulation of trajectory. First order derivatives obtained from pure sway, pure yaw and combined sway and yaw tests showed less influence on Froude number (Figs 24, 26, 29–31, 34, 35, 39) while the higher order derivatives showed more dependency on Froude number (Figs 25, 27, 32, 33 and 37–43), but sensitivity studies reveal that many of the higher order derivatives have less dependency on maneuvering characteristics of a ship compared to first order derivatives [18]. First order velocity dependant derivatives like $Y_{v}^{'}$ , $N_{v}^{'}$ show a variation of 11%, 27% and acceleration derivative like $Y_{\dot{v}}^{'}$ shows a variation of 12% when Froude number changes from 0.14 to 0.29. Third order velocity derivatives like $Y_{v v v}^{'}$ , $N_{v v v}^{'}$ showed a variation up to 58%, 332% with change in Froude number from 0.14 to 0.29. The derivatives obtained for pure yaw test such as $Y_{r}^{'}, N_{r}^{'}$ , $Y_{\dot{r}}^{'}$ , $N_{\dot{r}}^{'}$ showed less variation with speed. (38%, 32%, 10%, 9.6% respectively) and the third order yaw dependant derivatives such as $Y_{r r r}^{'}$ , $N_{r r r}^{'}$ showed higher variation (65%, 75% respectively). The derivatives like $Y_{v}^{'}$ , $N_{v}^{'}$ , $Y_{v v v}^{'}$ , $N_{v v v}^{'}$ , $Y_{r}^{'}$ , $N_{r}^{'}$ , $Y_{r r r}^{'}$ , $N_{r r r}^{'}$ obtained from pure sway and pure yaw model tests follow trends shown by Yoon [29] who conducted pure sway and pure yaw tests on a surface combatant at different speeds. Surge derivatives like $X_{u u}^{'}$ , $X_{v v}^{'}, X_{r r}^{'}$ , $X_{v r}^{'}$ , are highly influenced by model forward speed shows variation of 58%, 178%, 22%, 80% respectively when Froude number changes from 0.14 to 0.29. Sway and yaw coupled derivatives estimated from both pure yaw and combined sway and yaw tests are also influenced by Froude number. Hydrodynamic derivatives like $Y_{v r r}^{'}$ , $N_{v v r}^{'}, N_{v v r}^{'}$ show a variation up to 75% with increase of Froude number and $Y_{v v r}^{'}$ is highly influenced by Froude number with increase of 331%.

Fig. 21.

Variation of non-dimensional surge force time series at different speeds in pure sway test.

Fig. 22.

Variation of non-dimensional sway force time series at different speeds in pure sway test.

Fig. 23.

Variation of non-dimensional yaw moment time series at different speeds in pure sway test.

Fig. 24.

Variation of $Y_{v}^{'}$ with Froude number.

Fig. 25.

Variation of $Y_{v v v}^{'}$ with Froude number.

Fig. 26.

Variation of $N_{v}^{'}$ with Froude number.

Fig. 27.

Variation of $N_{v v v}^{'}$ with Froude number.

Fig. 28.

Variation of $X_{v v}^{'}$ with Froude number.

Fig. 29.

Variation of $Y_{\dot{v}}^{'}$ with Froude number.

Fig. 30.

Variation of $Y_{r}^{'}$ with Froude number.

Fig. 31.

Variation of $N_{r}^{'}$ with Froude number.

Fig. 32.

Variation of $X_{u u}^{'}$ with Froude number.

Fig. 33.

Variation of $X_{r r}^{'}$ with Froude number.

Fig. 34.

Variation of $N_{\dot{r}}^{'}$ with Froude number.

Fig. 35.

Variation of $Y_{\dot{r}}^{'}$ with Froude number.

Fig. 36.

Variation of $N_{r r r}^{'}$ with Froude number.

Fig. 37.

Variation of $Y_{r r r}^{'}$ with Froude number.

Fig. 38.

Variation of $X_{\dot{u}}^{'}$ with Froude number.

Fig. 39.

Variation of $Y_{v r r}^{'}$ with Froude number.

Fig. 40.

Variation of $Y_{v v r}^{'}$ with Froude number.

Fig. 41.

Variation of $N_{v v r}^{'}$ with Froude number.

Fig. 42.

Variation of $N_{v r r}^{'}$ with Froude number.

Fig. 43.

Variation of $X_{v r}^{'}$ with Froude number.

The results of sensitivity study conducted by Shenoi [18] show that the yaw derivatives $N_{r}^{'}$ , $N_{v v r}^{'}$ , $N_{v}^{'}$ and $N_{r r r}^{'}$ have high impact on steady turning radius, advance, transfer and tactical diameter. $N_{v v v}^{'}$ has insignificant impact on steady turning radius and advance and low impact on transfer and tactical diameter. But for the present case, variation of around 300% from lower to higher speed for the derivative effected the turning circle parameters significantly. $Y_{v}^{'}$ has high impact on transfer and medium impact on steady turning radius and advance, but low influence on tactical diameter. $Y_{r}^{'}$ has low impact on steady turning radius, advance and tactical diameter and moderate impact on transfer. Impact of sway acceleration derivatives like $Y_{\dot{v}}^{'}$ , $Y_{\dot{r}}^{'}$ are less significant on turning parameters for the present case. Yaw acceleration derivative $N_{\dot{r}}^{'}$ has a moderate influence on advance of the turning circle and low impact on steady turning radius, transfer and tactical diameter. Higher order sway derivatives like $Y_{v v v}^{'}$ , $Y_{r r r}^{'}$ , $Y_{v v r}^{'}$ have low impact on steady turning radius, advance and tactical diameter of the turning circle. Surge derivative like $X_{u u}^{'}$ has moderate impact on steady turning radius and high impact on advance, transfer and tactical diameter. $X_{v v}^{'}$ has moderate impact on steady turning radius and low impact on advance, transfer and tactical diameter. $X_{r r}^{'}$ has low impact on steady turning radius, transfer and tactical diameter and insignificant impact on advance of the turning circle. $X_{v r}^{'}$ has medium impact on steady turning radius, transfer and tactical diameter and low impact on advance of the turning circle. Sway-yaw coupled derivative like $Y_{v r r}^{'}$ has moderate impact on steady turning radius, transfer and advance and low impact on tactical diameter. $N_{v v r}^{'}$ has high impact on all the turning circle parameters and $N_{v r r}^{'}$ has moderate impact on steady turning radius, transfer and tactical diameter and low influence on advance.

3.3. Simulation of turning circle

IMO has prescribed standard maneuvers like turning circle test, zig-zag test and stopping test to identify the maneuvering characteristics of conventional surface ships. Turning circle trajectory for the model at an approach speed of 1.0 m/s and rudder angle 35 degrees is compared with the turning circle simulated using the experimentally determined hydrodynamic derivatives presented by Son and Nomoto [20] (Fig. 44). These trajectories match well and hence validate the reliability of the current numerically predicted values. Speed of ship during turning test for simulation and experiments are also well compared (Fig. 45). Turning circle maneuver at rudder angle, $δ = 35^{\circ}$ is simulated for different speeds to understand the influence of Froude number on the turning ability of a ship. Hydrodynamic derivatives obtained from numerical simulation of different forward speeds are directly substituted in the equations of motion and are solved using Euler’s integration method to simulate turning circle trajectory (Fig. 46). Coefficients are not interpolating for different speeds in the present case. Yaw rate and drift angle changes during the simulation of turning circle are also presented (Figs. 47, 48). Change of speed during turning also plotted against time (Fig. 49). Turning parameters like steady turning radius, transfer, advance and tactical diameter are estimated from the numerically simulated turning circle test and variation of turning parameters with respect to ship speed is analysed (Table 11). Results show that variation in forward speed affects the maneuvering characteristics of a ship considerably. Turning parameters are found to be increased when Froude number changes from 0.14 to 0.29. The obtained trends of turning circle trajectory match with the simulation results obtained by Muhammad [12] who studied the effect of forward speed on turning trajectory of a Ferry ship.

Fig. 44.

Comparison of turning circle results of simulation using present study and coefficients obtained by experiments by Son and Nomoto.

Fig. 45.

Speed of ship during turning test.

Fig. 46.

Turning circle results at different Froude numbers.

Fig. 47.

Change of yaw rate during turning circle simulation for different speeds.

Fig. 48.

Change of drift angle during turning circle simulation for different speeds.

Fig. 49.

Change of speed during turning circle simulation for different speeds.

Table 11

Turning parameters obtained from turning circle test at different Froude numbers

Turning parameters (m)	Froude Number, Fn					% variation

	0.14	0.18	0.22	0.25	0.29
Steady turning radius	304	316	351	359	373	−22.69
Transfer	292	301	317	329	346	−18.49
Advance	546	567	590	599	621	−13.73
Tactical Diameter	693	722	778	806	845	−21.93
Steady Turning rate	0.57	0.625	0.63	0.64	0.65	12.3
Drift angle	14.07	15.7	17.4	18.3	19.1	−35.74

3.4. Summary and conclusion

In the present study, static and dynamic simulation are carried out for a container ship for different speeds and the estimated hydrodynamic derivatives are fitted in the mathematical model of Son and Nomoto [20] to simulate the maneuvering trajectory in order to understand the dependency of Froude number on the turning ability of ship. Captive model tests are simulated in a numerical towing tank for Froude numbers from 0.14 to 0.29 using RANS based CFD approach and the hydrodynamic derivatives are determined from estimated static and dynamic test results using curve fitting method and Fourier series expansion method, respectively. Numerically obtained hydrodynamic derivatives are fitted in the equation of motion to simulate turning circle trajectory.

First order derivatives like $Y_{v}^{'}$ , $N_{v}^{'}, Y_{r}^{'}, N_{r}^{'}$ estimated from both pure sway and pure yaw test show a variation up to 40% when Fn changes from 0.14 to 0.29. Sensitivity study conducted by Shenoi [11] on the same model shows that these derivatives have a high influence on the maneuvering parameters. The higher order derivatives like $Y_{v v v}^{'}$ , $N_{v v v}^{'}, Y_{r r r}^{'}$ , which have a low impact on the maneuvering parameters, show an increase of 58.78%, 332.03%, 64.75% respectively with respect to changes of Fn from 0.14 to 0.29. Acceleration derivatives like $Y_{\dot{v}}^{'}$ , $Y_{\dot{r}}^{'}$ , $N_{\dot{r}}^{'}$ show less dependency on Froude number, which are decreased by 11.59%, 10%, 9.6% respectively with increase of Fn from 0.14 to 0.29. These derivatives also have moderately less influence on standard maneuvering parameters. Surge derivatives like $X_{u u}^{'}$ , $X_{r r}^{'}, X_{v v}^{'}$ show a variation up to 200% when Fn changes from 0.14 to 0.29, in which $X_{u u}^{'}$ has moderate to high impact on all the test parameters. Sway and yaw coupled derivatives estimated from combined sway and yaw test also influenced by Froude number. Hydrodynamic derivatives like $Y_{v r r}^{'}$ , $N_{v r r}^{'}, N_{v v r}^{'}$ show a variation up to 75% with increase of Froude number and $Y_{v v r}^{'}$ is highly influenced by Froude number with increase of 331%. Sensitivity study [18] shows that these sway and yaw coupled derivatives have low impact on turning parameters.

Hydrodynamic derivatives estimated from PMM simulation for different speeds are substituted in the equations of motion to simulate the turning trajectory and turning parameters like steady turning radius, advance, transfer, tactical diameter are estimated. The test parameters of the definitive maneuvers are found to increase as Froude number changes from 0.14 to 0.29. Steady turning radius and tactical diameter are increased by 22.69%, 21.93% respectively, and transfer and advance are increased by 18.79%, 13.73%, respectively. Influence of roll motion is not taken into account in the present study, this may effects the accuracy of results. The results of simulation indicated that vessel speed has an important role in vessel maneuverability. Hence for the conscious evaluation of the derivatives, the effect of speed need to be considered which will give accurate prediction of trajectory and subsequent contribution in the process of design of a vessel.

Nomenclature

Beam (m)

C_{B}

Block coefficient

Depth (m)

Propeller diameter

I_{Z}

Moment of inertia in Z direction

Advance ratio

K_{T}

Thrust coefficient

L_BP

Length between perpendiculars (m)

L_{m}

Length of model ship (m)

Mass of the ship (kg)

m_{z}

Yaw moment amplitude

N_{δ}

Rudder deflection derivative in z-direction

N_{\dot{r}}

Hydrodynamic coupled derivative of yaw moment with respect to yaw acceleration

N_{r}

Hydrodynamic linear coupled derivative of yaw moment with respect to yaw rate

N_{r r r}

Hydrodynamic third order coupled derivative of yaw moment with respect to yaw rate

N_{\dot{v}}

Hydrodynamic coupled derivative of yaw moment with respect to sway acceleration

N_{v}

Hydrodynamic linear coupled derivative of yaw moment with respect to sway acceleration

N_{v v v}

Hydrodynamic third order coupled derivative of yaw moment with respect to sway velocity

N_{v v r}

Hydrodynamic third order cross coupled derivative of yaw moment with respect to sway velocity and yaw rate

N_{v r r}

Hydrodynamic third order cross coupled derivative of yaw moment with respect to sway velocity and yaw rate

Propeller revolutions per second

Yaw rate (rad/s)

\dot{r}

Yaw acceleration (rad/s²)

t_{r}

Thrust deduction factor

T_{p}

Propeller Thrust

T_{a}

Draft at the aft end (m)

T_{m}

Mean draft (m)

T_{f}

Draft at the fore end (m)

Linear velocity in surge direction (m/s)

\dot{u}

Linear acceleration in surge direction (m/s²)

Linear velocity in sway direction (m/s)

\dot{v}

Linear acceleration in sway direction (m/s²)

∇

Displaced volume (m³)

Rudder angle (rad)

X_{δ}

Rudder deflection derivative in x-direction

X_{r r}

Hydrodynamic second order coupled derivative of surge force with respect to yaw rate

X_{\dot{u}}

Hydrodynamic uncoupled derivative in surge with respect to surge acceleration

X_{u u}

Hydrodynamic second order uncoupled derivative of surge with respect to surge velocity

X_{v v}

Hydrodynamic second order coupled derivative of surge force with respect to sway velocity

X_{v r}

Hydrodynamic second order cross coupled derivative of surge force with respect to sway velocity and yaw rate

Y_{δ}

Rudder deflection derivative in y-direction

Y_{\dot{r}}

Hydrodynamic coupled derivative of sway force with respect to yaw acceleration

Y_{r}

Hydrodynamic linear coupled derivative of sway force with respect to yaw rate

Y_{r r r}

Hydrodynamic third order coupled derivative of sway force with respect to yaw rate

Y_{\dot{v}}

Hydrodynamic coupled derivative of sway force with respect to sway acceleration

Y_{v}

Hydrodynamic linear coupled derivative of sway force with respect to sway velocity

Y_{v v v}

Hydrodynamic third order coupled derivative of sway force with respect to sway velocity

Y_{v v r}

Hydrodynamic third order cross coupled derivative of sway force with respect to sway velocity and yaw rate

Distance between two oscillators (m)

Ω_{0}

Frequency of oscillation (rad/sec)

U_{m}

Model forward speed (m/s)

u_{p}^{'}

Effective wave fraction

U_{T}

Service speed of the vessel (m/s)

w_{p}

Wake fraction in the straight ahead condition

x_{P}^{'}

X-location of the propeller plane

Y_{v r r}

Hydrodynamic third order cross coupled derivative of sway force with respect to sway velocity and yaw rate

y_{0 a}

Amplitude of transverse oscillation (m)

Aspect ratio of the rudder

α_{R}

Effective flow angle of the rudder

2 ε

Phase difference between the oscillators of PMM

Yaw angle

Drift angle

Density of water

τ_{P}

Parameter to account for the lateral velocity components

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