# Manoeuvring model of an estuary container vessel with two interacting Z-drives

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## Abstract

A 6 degrees of freedom manoeuvring model of an estuary container vessel was implemented in the inland navigation simulator Lara at Flanders Hydraulics Research (FHR). The container vessel, with dimensions 110 m × 17.5 m × 4.5 m, is equipped with two Z-drives, each one consisting of two contra rotating propellers. Both Z-drives have a 360° azimuth angle of operation during manoeuvring. The mathematical manoeuvring model was built to cover all degrees of operation. This required the execution of a significant number of captive model tests in the shallow water towing tank of FHR (in co-operation with Ghent University). Based on these tests a new mathematical manoeuvring model was built to cover all effects of operation, including the interaction effects between the two Z-drives. The new mathematical model was implemented in the simulator and both fast time and real time manoeuvring simulations have been carried out. A paper discussing the validation of the manoeuvring model has been presented during MARSIM 2015 (Vos, Delefortrie, and Van Hoydonck in Validation of the manoeuvring behaviour on an estuary vessel. MARSIM, Newcastle, 2015). The present paper discusses the details of the manoeuvring model, with emphasis on the interaction effects between the Z-drives and the influence on the hydrodynamic forces.

## Keywords

Manoeuvring Shallow water Thruster interaction## List of symbols

*a*Flow acceleration factor (−)

*a*_{H}Additional steering induction for

*Y*(−)*A*_{R}Rudder area (m

^{2})- AEP
Expanded area ratio of propeller (−)

*A*_{W}Water plane area (m

^{2})*B*Ship breadth (m)

*C*_{B}Block coefficient (−)

*C*_{D}Drag coefficient (−)

*C*_{L}Lift coefficient (−)

*C*_{T}Thrust coefficient (−)

*D*Ship depth; propeller diameter (m)

*D*Drag in drive bound axis system (N)

*D*_{DEN}Duct entry diameter (m)

*D*_{DEX}Duct exit diameter (m)

*F*Force (N)

- GM
_{(T)} Transverse metacentric height above centre of gravity (m)

- GM
_{L} Longitudinal metacentric height above centre of gravity (m)

*I*Moment of inertia (kg m

^{2})*J*Propeller advance (−)

*K*_{(*)}Roll moment [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (Nm)*K*_{T}Thrust coefficient (−)

- KM
Transverse metacentric height above keel (m)

*L*Lift in drive bound axis system (N)

*L*_{D}Duct length (m)

*L*_{OA}Ship length over all (m)

*L*_{(PP)}Ship length between perpendiculars (m)

*m*Ship’s mass (kg)

*M*_{(*)}Pitch moment [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (Nm)*N*Propeller rate (rpm, 1/s)

*N*_{(*)}Yawing moment [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (Nm)*O*Origin of coordinate system (−)

*p*Roll velocity (rad/s)

*P*Propeller pitch (m)

- PS
Port side (−)

*q*Pitch velocity (rad/s)

*r*Yaw velocity (rad/s)

- SS
Starboard side (−)

*t*Time (s)

*T*Ship draft (m)

*T*_{P}Thrust in drive bound axis system (N)

*T*_{xp}Longitudinal force in drive bound axis system (N)

*T*_{yp}Lateral force in drive bound axis system (N)

*u*Longitudinal velocity (m/s)

*v*Lateral velocity (m/s)

*w*Vertical velocity (m/s)

*w*_{R}Wake factor (steering) (−)

*w*_{T}Wake factor (thrust) (−)

*V*Global velocity in the horizontal plane (m/s)

*x*Longitudinal position (m)

*X*_{(*)}Longitudinal force [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (N)*x*_{H}Additional steering induction for

*N*(−)*y*Lateral position (m)

*Y*_{(*)}Lateral force [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (N)*z*Vertical position (m)

*Z*_{(*)}Vertical force [derivative of * (* represents any combination of kinematical parameter(s)

*u, v, w, p, q, r*or their derivatives)] (N)*z*_{H}Additional steering induction for

*K*(−)*z*_{HX}Steering induction for

*M*(m)*z*_{HZ}Steering induction for

*Z*(−)- \(\alpha\)
Effective inflow angle (°, rad)

- \(\beta\)
Drift angle (°, rad)

- \(\gamma\)
Yaw angle (°, rad)

- \(\delta\)
Azimuth angle (°, rad)

- \(\varepsilon\)
Hydrodynamic advance angle (°, rad)

- \(\Delta\)
Displacement (N)

- \(\Delta *\)
Differential (acting on *) (−)

- \(\vartheta\)
Pitch angle (°, rad)

- \(\lambda\)
Regression coefficient (−)

- \(\rho\)
Water density (kg/m

^{3})- \(\sigma\)
Sign (± 1) (−)

- \(\tau\)
Regression coefficient (−)

- \(\varphi\)
Heel angle; phase angle (°, rad)

- \(\psi\)
Course angle (°, rad)

- \(\omega\)
Pulsation (°, rad)

## Subscripts

*D*Drag

*G*Regarding centre of gravity

*H*Regarding the hull

*i*Interaction; summation

*I*Inertia

*L*Lift

*o*Fixed coordinate system

*P*Regarding the propeller

- PS
Port side

- PT
Regarding the propeller thrust

- PTA
Oscillation amplitude regarding the propeller thrust

*R*Regarding the rudder

- SS
Starboard side

*X*In longitudinal direction

- xx
About

*x*-axis*y*In lateral direction

- yy
About

*y*-axis- zz
About

*z*-axis

## Superscripts

- (eff)
Effective

- (int)
Interaction

- .
Derivative

- \('\)
Dimensionless

- *
Apparent

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