Although there are no objections against such a methodology in order to simulate ship-bank interaction effects for one particular ship in given loading conditions, the results can hardly be used for developing generic formulae covering a range of ships in varying operational and environmental conditions.

　

　

 For this reason, the number of terms in (6-7) was first reduced from 18 to 5, considering the most significant terms only. The results were discussed by Vantorre et al [11]; a clear relationship between the regression coefficients (α, β) and hull form parameters could not be found.

　

 Taking account of the observations described in section 3.4, it was investigated whether the dependency on the Froude number in (6-7) could be avoided by replacing the non-dimensional inverse gross under keel clearance T/(h-T) by a non-dimensional representation of the inverse net under keel clearance, T/(heff-T), with

　

　

 Zm being the average sinkage of the ship due to squat effects. In this way, the number of regression coefficients was reduced to 6:

　

　

 The regression coefficients α and β were determined for the three ship models; the agreement appeared to be satisfactory. As a following step, it was checked whether a linear relationship between these coefficients and combinations of the non-dimensional hull form parameters CB, B/L, B/T and T/L.

　

 The following regression coefficients were obtained:

　

α^(H)10 = -4.61E-01 + 3.68E-01 CB^-1

α^(H)11 = -1.73E-01 + 1.14E-02 CBLT^-1

α^(H)12 = 6.34E-03 - 4.60E-04 CBLT^-1

α^(H)20 = 1.36E+00 - 6.58E-02 LT^-1

α^(H)21 = 2.57E-01 - 8.96E-02 BT^-1

α^(H)22 = -1.9 IE-02 + 6.34E-03 BT^-1

β^(H)10 = 1.19E-01 + 2.11E+00 TL^-1

β^(H)11 = -3.54E-02 + 1.95E-01 BL^-1

β^(H)12 = 2.29E-04 - 6.92E-05 BT^-1

β^(H)20 = 1.3 4E-01 - 2.35E+00 TL^-1

β^(H)21 = -4.23E-02 + 2.93E-03 CBLT^-1

β^(H)21 = 1.23E-03 - 8.53E-05 CBLT^-1　(11)

　

5.3 Effect of propulsion: bollard pull conditions

　

 In bollard pull conditions, non-dimensional formulations containing the square of the forward speed have to be avoided. For this reason, another reference velocity VT is introduced:

　

　

 The effect of propeller action on the shi -bank inter-action force and moment, denoted Y^(P) en N^(P), is modelled as follows:

　

　

with following numerical values for the regression coefficients:

　

α^(P)10 = 5.24E-02 - 1.5 1E-01 TB^-1

α^(P)11 = 8.70E-03 - 3.86E-02 BL^-1

α^(P)12 = -1.05E-04 - 9.23E-07 LB^-1

α^(P)20 = 5.92E-02 - 2.11E-02 BT^-1

α^(P)21 = -2.82E-04 - 2.42E-04 LB^-1

α^(P)22 = 2.32E-04 - 1.98E-05 LB^-1

β^(P)10 = -4.95 E-03 + 1.25E-01 CBTL^-1

β^(P)11 = 1.29E-03 - 5.37E-03 TB^-1

β^(P)12 = -2.68E-04 - 8.78E-04 TB^-1

β^(P)20 = 3.81E-03 - 1.29E-02 CBTB^-1

β^(P)21 = 3.44E-03 - 1.10E-03 BT^-1

β^(P)22 = 4.88E-04 - 1.52E-03 TB^-1　(15)

　

5.4 Effect of propulsion combined with forward speed

　

 As the effect of the propeller action on the ship-bank interaction force and moment appear to be amplified if the ship has a nonzero forward speed, additional terms have to be added:

　

　

α^(HP) 10 = 2.37E-0.1 + 4.22E-01 TB^-1

α^(HP) 11 = 2.89E-02 - 2.20E-03 LB^-1

α^(HP) 12 = -6.57E-03 + 1.18E-03 LB^-1

α^(HP) 20 = 4.19E-01 - 4.79E-02 LB^-1

α^(HP) 21 = -4.49E-02 + 9.98E-03 LB^-1

α^(HP) 22 = 1.06E-02 - 4.24E-02 TB^-1

β^(HP) 10 = -2.89E-01 + 5.71E+00 TL^-1

β^(HP) 11 = -1.22E-01 + 3.53E-01 TB^-1

β^(HP) 12 = 7.87E-03 - 2.33E-02 TB^-1

β^(HP) 20 = -9.33E-02 + 1.57E-02 LB^-1

β^(HP) 21 = 2.60E-01- 7.95E-01 TB^-1

β^(HP) 22 = -1.99E-02 + 5.98E-02 TB^-1　(18)

　

 Note: in expressions (6), (7), (9), (10), (13), (14), (16) and (17) yB3 should be written as yB3｜y^B3｜.

　

 By means of equations (9-18), a reasonable estimation of lateral forces and yawing moments acting on a ship due to interaction with a vertical bank can be made based on geometrical ship characteristics, water depth, ship-bank distance and propeller loading. Reasonable agreement is obtained for the three tested ship models, but it should be emphasised that the validity of these formulae has not been tested yet for other ships and bank geometries such as sloping and submerged banks.

　

 Following limitations must be respected when applying the formulae: 0.56 < C <B < 0.84; 6.0 < L/B < 7.3; 2.6<B/T<3.0; 17.8 < L/T < 21.5; h/T> 1.07.

　

 It must be noted that an estimation of the average sinkage of the ship due to squat and of the propeller thrust has to be available as well.

　

 The set of formulae in section 5 is certainly no definitive solution for simulation problems in which bank effects have a dominant effect. The phenomenon is determined by a large number of parameters, so that it is hard to cover the full range of conditions by an experimental program. Exchange of data between institutions is therefore highly recommended.

　

[1] M. Fujino "Experimental studies on ship manoeuvrability in restricted waters - Part I", International Shipbuilding Progress, Vol. 15, No. 168, pp. 279-301, 1968.

[2] H. Eda "Directional stability and control of ships in restricted channels", Transactions SNAME, pp. 71-116, 1971.

[3] N. Norrbin "Bank effects on a ship moving through a short dredged channel", 10th ONR Symposium on Naval Hydrodynamics, Cambridge, Mass., 1974.

[4] N. Norrbin "Bank clearance and optimal section shape for ship canals", 26th PIANG International Navigation Congress, Brussels, Section 1, Subject 1, pp.167-178, 1985.

[5] M. Fuehrer "The results of systematic investigations into lateral forces for determining the effects of hydraulic asymmetry and eccentricity on the navigation of sea-going ships in canals". Symposium on aspects of navigability, Delft, 1978.

[6] M. Fuehrer, K. Romisch "Effects of modem ship traffic inland and ocean-waterways and their structures", 24th International Navigation Congress, PIANC, Leningrad, pp. 236-244, 1974.

[7] K. Römisch "Recommendations for dimensioning of harbour entrances" (in German), Wasserbauliche Mitteilungen der Technischen Universitat Dresden, Heft 1, pp. 39-63, 1989.

[8] I. Dand "Some measurements in interaction between ship models passing on parallel courses", NMIR108, 1981.

[9] P.W. Ch'ng, L.J. Doctors, M.R. Renilson "A method of calculating the ship-bank interaction forces and moments in restricted water", International Shipbuilding Progress, Vol.40, No.412, pp. 7-23, 1993.

[10] D.-Q. Li, M. Leer-Andersen, P. Ottoson, P., Trägårdh "Experimental investigation of bank effects under extreme conditions", PRADS 2001 - Practical Design of Ships and Other Floating Structures Shanghai, 2001.

[11] M. Vantorre, G. Delefortrie, K. Eloot "Modelling of ship-bank interaction forces", Colloquium "The ship in interaction with the waterway", Duisburg, 2002.