Fr  ¼ KrAHr  — brIrxr; ð6Þ

ar  ¼ Fr=Ir; ð7Þ

xr  ¼ xr  þ arDt; ð8Þ

hr   ¼ xrDt; ð9Þ

Fig。 5  The weights of grid points is related to their Zs  coordinates。

Height fields in the right side have positive weights while the weights

of the height field in the other side are   negative Fig。 6  Roll motion of  ship

When a ship rests on sea surface without  any  internal force or external force acting on it, its forward velocity  is

0。 If its propeller is turned on, the ship is accelerated and driven forward。 As the ship moves, the drag force of water will slow down the ship。 Gradually, the force of propeller is canceled by the drag force, and the ship reaches  a steady speed。 Theoretically, drag force is proportional to the square of speed and the area of ship hull surface submerged in water。 It can be formulated as (Batchelor 1967):

Rs  ¼ bsAskvsk2;

where Rs is the resistance (drag force), vs is the velocity of surge, bs is the drag force coefficient, and As is the area of ship hull submerged in water。 However, As is difficult to decide。 Instead, we replace it with ship mass M。 The resistance is computed by:

Rs  ¼ bsMkvsk2:

Let the force produced by the propeller be T, and assume

the cross-section of ship is narrow enough such that currents and winds make no contribution for surge, then the net force for surge is T-Rs。 The force of the propeller is related to its revolution speed n and diameter D (Rawson and Tupper 2001)。 The surge motion magnitude is deduced as follows:

T ¼ KT n2D4; ð10Þ

as ¼ ðT — RsÞ=M; ð11Þ

vs  ¼ vs  þ asDt; ð12Þ

surge ¼ vsDt; ð13Þ

where KT is the thrust coefficient of propeller。 In the first equation, the thrust force of the propeller is computed。 In the second equation, the acceleration is obtained by piding the net force with the ship mass。 Then the velocity is updated by using the acceleration。 The magnitude of surge is computed by multiplying the velocity with the time step size。

According to this model, bs plays a key role in the surge motion。 To demonstrate its effects, two examples are shown in Figs。 7 and 8。 In these examples, the ship mass is 570 kg and the propeller force is 1,200 newtons。 The drag force coefficient is set to 0。05 and 0。08。 Initially, the speed of ship is zero。 Then the engine is turned on and the ship is sped up until it reaches a threshold speed。 The speed of the ship is shown in Fig。 7。 As revealed in the figure, if bs is smaller, then the ship can be accelerated longer before the drag force cancels the propeller thrust。 Therefore its final speed is quicker。 In the second test, the propeller is   turned

Fig。 7  The ship speed becomes steady as the drag force cancels the

thrust of propeller

Fig。 8 As the propeller is shut down, the drag force causes the ship to stop

off when the ship moves at the threshold speed。 The drag force slows down the ship。 The ship speed is reduced more quickly, if the drag force coefficient is higher。 The results are displayed in Fig。  8。

We assume that the rudder of the ship is the major force of yaw。 The rudder force is related to the area of the rudder, the angle of the rudder, and the square of the velocity of water passing the rudder surface。 Many simplified formulae have been suggested for calculating rudder force (Rawson and Tupper 2001)。 We adopt the following one for approximating the rudder  force: