# Prediction of the forces acting on container carriers in muddy navigation areas using a fluidization parameter

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## Abstract

To assess the safety of navigation in muddy areas, a comprehensive captive manoeuvring model test program was executed. Based on the results of this experimental program, a four-quadrant manoeuvring model was built with a separate set of coefficients for each combination of under-keel clearance and mud layer characteristics. The disadvantage of this model is that only conditions corresponding with the experimental ones can be simulated. A more consolidated mathematical model was needed. This was achieved with the introduction of a fluidization parameter that determines the corresponding hydrodynamically equivalent depth above a solid bottom. As a result, the under-keel clearance dependency of a given mathematical manoeuvring model can be reformulated in such way that the effect of any realistic muddy condition is included. In this article, the modelling of the hull forces, the propeller thrust and torque, and the forces acting on the rudder will be discussed.

## Keywords

Nautical bottom Mud Manoeuvring Container Fluidization Captive model testing## List of symbols

*a*_{i}Regression coefficient (–)

- AEP
Expanded area ratio of propeller (–)

*A*_{R}Rudder area (m

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

*C*_{B}Block coefficient (–)

*C*_{D}Drag coefficient (–)

*C*_{L}Lift coefficient (–)

*C*_{T(Q)}Thrust (torque) coefficient (–)

*D*_{P}Propeller diameter (m)

*F*Force component (–)

*F*_{X}Longitudinal rudder force (N)

*F*_{Y}Lateral rudder force (N)

*f*Function (–)

*g*Function (–)

*g*Gravity constant (m/s

^{2})*G**Function (–)

*h*Depth/thickness (m)

*h**Hydrodynamically equivalent depth (m)

*I*_{zz}Moment of inertia about

*z*-axis (kg m^{2})*J*Advance (–)

*J*′Apparent advance (–)

*K*_{T(Q)}Thrust (torque) coefficient (–)

*k*Sway velocity or yaw rate (m/s or rad/s)

*L*,*L*_{PP}Ship length (m)

*m*Ship mass (kg)

*N*Yawing moment (Nm)

*n*Propeller rate (1/s)

*N*_{i}Hydrodynamic derivative \( \left({{i}} = \dot{\text{v}}, \; \dot{\text{r}}, \; \ldots \right) \)

*N*′(β)Non-dimensional yawing moment due to drift: \( N^{\prime}\left( \beta \right) = \frac{N\left( \beta \right)}{{\frac{1}{2}\rho L^{2} T\left( {u^{2} + v^{2} } \right)}}\left( {\text{--}} \right) \)

*N*′(γ)Non-dimensional yawing moment due to yaw: \( N^{\prime}\left( \gamma \right) = \frac{N\left( \gamma \right)}{{\frac{1}{2}\rho L^{2} T\left( {u^{2} + \left( {\tfrac{1}{2}rL} \right)^{ 2} } \right)}}\left({\text{--}}\right) \)

*N*′(χ)Non-dimensional yawing moment due to yaw–drift correlation: \( N^{\prime}\left( \chi \right) = \frac{N\left( \chi \right)}{{\frac{1}{2}\rho L^{2} T\left( {v^{2} + \left( {\tfrac{1}{2}rL} \right)^{2} } \right)}}\left({\text{--}}\right) \)

*P*Ship’s length or draught (m)

*P*Propeller pitch (m)

*Q*_{P}Propeller torque (Nm)

*r*Yaw rate (rad/s)

- \( \dot{r} \)
Yaw acceleration (rad/s

^{2})- sgn
Sign function (–)

*T*Ship draught (m)

*T*_{P}Propeller thrust (N)

- TEU
Twenty feet equivalent unit (–)

*u*Longitudinal velocity (m/s)

*u*_{P}Longitudinal velocity at propeller (m/s)

- \( \dot{{u}} \)
Longitudinal acceleration (m/s

^{2})- UKC
Under keel clearance

*v*Lateral velocity (m/s)

*V*_{R}Velocity near rudder (m/s)

- \( \dot{\text{v}} \)
Lateral acceleration (m/s

^{2})*w*Wake factor (–)

*X*Longitudinal force (N)

*X*_{i}Hydrodynamic derivative \( \left({{{i}} = \dot{{u}},\; \ldots} \right) \)

*X*′(β)Non-dimensional longitudinal force due to drift: \( X^{\prime}\left( \beta \right) = \frac{X\left( \beta \right)}{{\frac{1}{2}\rho LT\left( {u^{2} + v^{2} } \right)}}\left( {\text{--}} \right) \)

*X*′(γ)Non-dimensional longitudinal force due to yaw: \( X^{\prime}\left( \gamma \right) = \frac{X\left( \gamma \right)}{{\frac{1}{2}\rho LT\left( {u^{2} + \left( {\tfrac{1}{2}rL} \right)^{2} } \right)}}\left( {\text{--}} \right) \)

*X*′(χ)Non-dimensional longitudinal force due to yaw–drift correlation: \( X^{\prime}\left( \chi \right) = \frac{X\left( \chi \right)}{{\frac{1}{2}\rho LT\left( {v^{2} + \left( {\tfrac{1}{2}rL} \right)^{2} } \right)}}\left( {\text{--}} \right) \)

*x*_{G}Longitudinal position of centre of gravity (m)

*Y*Sway force (N)

*Y*_{i}Hydrodynamic derivative \( \left({{\text{i}} = \dot{\text{v}},\;\dot{\text{r}},\ldots} \right) \)

- Y′(β)
Non-dimensional lateral force due to drift: \( Y^{\prime}\left( \beta \right) = \frac{Y\left( \beta \right)}{{\frac{1}{2}\rho LT\left( {u^{2} + v^{2} } \right)}}\left( {\text{--}} \right) \)

- Y′(γ)
Non-dimensional lateral force due to yaw: \( Y^{\prime}\left( \gamma \right) = \frac{Y\left( \gamma \right)}{{\frac{1}{2}\rho LT\left( {u^{2} + \left( {\tfrac{1}{2}rL} \right)^{2} } \right)}}\left( {\text{--}} \right) \)

- Y′(χ)
Non-dimensional lateral force due to yaw–drift correlation: \( Y^{\prime}\left( \chi \right) = \frac{Y\left( \chi \right)}{{\frac{1}{2}\rho LT\left( {v^{2} + \left( {\tfrac{1}{2}rL} \right)^{2} } \right)}}\left( {\text{--}} \right) \)

- α
Angle of attack of flow (°)

- β
Drift angle, Eq. 14 (°)

- γ
Yaw rate angle, Eq. 15 (°)

- δ
Rudder angle (°)

- ε
Hydrodynamic angle, Eq. 42 (°)

- ε*
Apparent hydrodynamic angle, Eq. 50 (°)

*μ*Dynamic viscosity (Pa s)

*μ*′Non-dimensional dynamic viscosity:

*μ*′ =*μ*/(1 Pa s) (–)- \(\Uppi\)
_{T} Keel penetration parameter, Eq. 17 (–)

- \(\Uppi\)
Alternative keel penetration parameter, Eq. 20 (–)

- ξ
_{i} Regression coefficient or function (–)

- ρ
Density (kg/m

^{3})- ρ*
Non-dimensional density, Eq. 47 (–)

- Φ
Fluidization parameter (–)

- Φ
_{ij} Regression coefficient (–)

- χ
Drift–yaw correlation angle, Eq. 16 (°)

## Subscripts

- 1
Water layer

- 2
Mud layer

- H
Hull

- P
Propeller

- Q
Torque

- R
Rudder

- T
Thrust

## Notes

### Acknowledgments

The generalised mathematical model is based on experimental results obtained at and in close co-operation with Flanders Hydraulics Research (Antwerp, Belgium) and has been developed in the framework of the research project Validation of the Nautical Bottom Concept by order of the Flemish Government (ref 16 EB/0501), Belgium.

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