Fig. 17 
Nondimensional yawing moment N'^{(γ)}as function of yaw rate angle. 
Fig. 18 
Four quadrants tabular models for forces and yawing moment (E at 20% UKC). 
For the ship moving astern the scatter on the nondimensional longitudinal force X'^{(γ)} is important.
5 .3 Combination of sway and yaw: additional forces
Cross flow effects are partly included in the lateral forces and yawing moments due to pure swaying and pure yawing. Additional forces and moment measured during harmonic yaw tests with constant drift angle are subject to errors as the values are small. In figure 1 9 the following force is shown for the tanker at 50% UKC as a function of Arctan (rl/v):
with YTotal the measured total hull force.
A comparable figure as figure 1 9 can be drawn up for the yawing moment.
The scatter occurring at angles Arc tan (rl/v) around 0°and 180° can be reduced by executing tests at higher drift angles (β>5), possibly together with a small yaw amplitude and a moderate test frequency.
Fig. 19 
Nondimensional lateral force as a sum of an additional force due to a combination of yawing and drifting and the force due to pure drifting (E at 50% UKC, see table 5 for test parameters). 
Table 5 Test parameters for tanker at 50% UKC
Run 
Fn 
β 
ΨA 
ω’ 
EHGC06 
0.065 
2.5° 
15° 
2.2 
EHGC08 
0.065 
2.5° 
15° 
2.2 
EHGC12 
0.065 
5.0° 
15° 
2.2 
EHGC09 
0.065 
2.5° 
25° 
2.2 
EHGC13 
0.065 
5.0° 
25° 
2.2 
EHGF02 
0.065 
8.0° 
10° 
2.9 
EHGF03 
0.065 
8.0° 
10° 
2.9 
EHGF04 
0.065 
8.0° 
20° 
1.4 
EHGF05 
0.065 
8.0° 
20° 
1.4 

The relationship between the crosscoupling force and moment and the angle Arc tan(rl/v) is displayed in figure 20 for the tanker at an under keel clearance of 20%.
Fig. 20 
Additional lateral force and yawing moment due to a combination of say and yaw (tanker E at 20% UKC). 
6. CONCLUSIONS AND PERLIMINARY GUIDELINES
The test results discussed in this paper belong to an extensive research program on mathematical modelling of ship manoeuvring in shallow water conditions based on captive model testing.
The introduction of full guidelines for physical model testing in shallow water, however, requires a further step in this research:
□ the development of mathematical models describing the influence of propeller and rudder action
□ the validation of the mathematical models obtained from captive model test results, based on full scale trials.
Nevertheless, preliminary guidelines can be formulated:
□ The horizontal forces and yawing moment measured during captive model tests in shallow and very shallow water conditions (h/d<1.5) are clearly influenced by the selected test parameters. In deep water conditions, this influence is negligible.
□ The choice of the parameters determining the harmonic captive tests highly affects the most important acceleration derivatives, the added mass due to sway and the added moment of inertia due to yaw. The added mass due to sway is subject to nonstationary effects both for slender and full ships. The execution of alternative sway tests does not solve the problem. For the added moment of inertia, especially the results for slender ships are influenced by the test parameters.
□ At equal water depth to draft ratio, the nondimensional velocity dependent lateral force due to pure drift and the nondimensional velocity dependent yawing moment due to pure yaw, respectively, are almost equal for the tanker and the container carrier.
□ Yaw amplitude and frequency are test parameters with important influence on the velocity dependent lateral force due to pure yaw. This influence is partly, but not completely, caused by an increasing sinkage occurring at maximum yaw velocity during harmonic yaw tests.
□ In order to obtain tabular hull models for four quadrants of operation, oscillating sway or yaw tests with zero forward velocity are required to determine the cross flow effects at 90°drift angle or yaw rate angle. Measured values are restricted to the first oscillation cycle as memory effects are generated due to the motion of the ship model through its own wake.
Harbour manoeuvres assisted by tugs can be very diverse, so that mathematical manoeuvring models covering four quadrants of operation and simulating the hydrodynamic hull forces and moments in a realistic way are required. The development of generally accepted guidelines for physical model testing techniques in shallow and very shallow water conditions is recommended, as numerical methods based on CFD calculations are still not satisfactory.
REFERENCES
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[5] Martinussen K., Ringen E. "Manoeuvring prediction during design stage", International Workshop on Ship Manoeuvrability at the Hamburg Ship Model Basin, pp, 121, 2000
[6] Eloot K., Vantorre M. "Nonconventional captive manoeuvring tests", International Workshop on Ship Manoeuvrability at the Hamburg Ship Model Basin, Paper No. 3 20 pp., 2000
[7] The Research Committee of Dynamic Performance, Manoeuvring and Control Section "Prediction of manoeuvrability of a ship", Bulletin of the Society of Naval Architects of Japan No. 668, 1985
[8] Eloot K., Vantorre M. "Alternative captive model tests, possibilities and limitations", International Symposium and Workshop on Forces Acting on a Maneuvering Vessel, France, pp. 110J, 1998
AUTHOR'S BIOGRAPHY
Katrien Eloot obtained a Master's degree in Naval Architecture from Ghent University Belgium in 1995.
She worked as an academic assistant at the university from 1995 until 2000. There she started her principal research concerning mathematical modelling of ship manoeuvres in shallow water and captive model testing. In 2001 she entered her present post at Flanders Hydraulics Research, a laboratory of the Flemish Community. As studyengineer she carries out research contracts on realtime and fasttime manoeuvring simulation for the public and the private sector.
