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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.

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Abbreviations

a i :

Regression coefficient (–)

AEP:

Expanded area ratio of propeller (–)

A R :

Rudder area (m2)

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/s2)

G*:

Function (–)

h :

Depth/thickness (m)

h*:

Hydrodynamically equivalent depth (m)

I zz :

Moment of inertia about z-axis (kg m2)

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/s2)

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/s2)

UKC:

Under keel clearance

v :

Lateral velocity (m/s)

V R :

Velocity near rudder (m/s)

\( \dot{\text{v}} \) :

Lateral acceleration (m/s2)

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/m3)

ρ*:

Non-dimensional density, Eq. 47 (–)

Φ:

Fluidization parameter (–)

Φ ij :

Regression coefficient (–)

χ:

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

1:

Water layer

2:

Mud layer

H:

Hull

P:

Propeller

Q:

Torque

R:

Rudder

T:

Thrust

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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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Authors and Affiliations

  1. Flanders Hydraulics Research, Berchemlei 115, 2140, Antwerp, Belgium

    Guillaume Delefortrie

  2. Maritime Technology Division, Ghent University, Technologiepark 904, 9052, Ghent, Belgium

    Marc Vantorre

Authors
  1. Guillaume Delefortrie
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  2. Marc Vantorre
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Corresponding author

Correspondence to Guillaume Delefortrie.

Appendix: Influence of sway and yaw acceleration on the longitudinal force

Appendix: Influence of sway and yaw acceleration on the longitudinal force

Here, the reason for extending the model of the longitudinal force to encompass the influence of sway and yaw acceleration is explained. The yaw acceleration is taken as an example. Consider Fig. 31: when building a model for X′(β) + X′(γ) with a constant drift angle, e.g. 0°, and small yaw angles, the data can normally be modelled using a parabolic expression G* as a function of the yaw angle. However, when navigating in contact with highly viscous mud layers, some measurements have shown a large deviation from the proposed model G*. These deviations are presented in Fig. 32 as a function of the yaw acceleration. It seems that a counteracting force effect occurs that depends on the square of the yaw acceleration.

Fig. 31
figure 31

Measured drift–yaw function for the longitudinal force versus the modelled values for mud g2 and a UKC of −1.1%

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Fig. 32
figure 32

Model rest fraction as a function of the yaw acceleration for mud g2 and a UKC of −1.1%

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The deviations are probably related to the behaviour of the mud–water interface. It is very clear that it would be more appropriate to represent these effects by non-stationary terms, but this was not possible within the scope of the project. Moreover, the sway and yaw acceleration-dependent terms appear to have only a small effect compared to the other terms.

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Delefortrie, G., Vantorre, M. Prediction of the forces acting on container carriers in muddy navigation areas using a fluidization parameter. J Mar Sci Technol 14, 51–68 (2009). https://doi.org/10.1007/s00773-009-0041-x

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