Abstract: Based on recent and other available experimental results, a new mathematical model has been developed for prediction of bank effects and implemented in SSPA's general manoeuvring and seakeeping program SEAMAN II. The new model includes more influencing variables and can predict repulsion forces at extreme shallow water. The paper presents the model and its validation against experiments. It contains studies from different bank passages simulation with special emphasis on small bottom clearances and propeller loads. Comparison is also made between simulations by the new model, SSPA's old model and another known model. It shows that compared with the old model, the new model is in general more accurate and robust, especially the model for vertical and sloping banks. The model for flooded banks and propeller effect may need further improvement.

　

 As a ship travels in vicinity of the wall of a quay or a canal, a hydrodynamic interaction between the ship and the bank, known as "bank effects" occurs. As a result, the ship is subjected to a sway (side) force and a yaw (turning) moment that tend to alter its navigational course. If bank effects are sufficiently severe, the ship manoeuvrability can be deteriorated in shallow and restricted water. Thus it is important to use adequate bank effects in simulation studies of ship manoeuvring in fairways and ports as it may have a decisive influence on the design of the ship, fairway or port. For a given ship, fairway and port the handling of the ship may be studied, e.g., weather windows, feasible speed, and training etc.

　

 Numerous studies contributing to the knowledge of bank effects have been carried out in at least four aspects since 1950: (i) Experiments, (ii) Analytical methods, (iii) Numerical simulations, and (iv) Mathematical models. A more detailed review of the work in these aspects can be found in [l][2][3]. With regard to mathematical models, earlier works are mainly due to Norrbin [3]. Ch'ng and Renilson [9] recently applied a second order Taylor expansion and least square method to analyse their model test results and thereafter developed a model for the steady sway force and yaw moment. Vantorre [6] proposed a model for the sway force and yaw moment as a polynomial function of Froude number Fn and the ship-bank distance for each depth to draught ratio h/T. Both models have taken into account the contribution of an operating propeller.

　

 SSPA has been very active in both experiments and mathematical models since 1960. Most experiments carried out at earlier time were performed on large displacement ships with simplified bank forms, like vertical and flooded (dredged) banks. Based on these results, a mathematical model for the vertical and sloping bank was first developed by Norrbin [3], later extended to include the flooded bank and transient effects [4]. The model reflects the ship-bank interaction in a physically comprehensive but mathematically simple form, it only uses water depth and ship-bank distance as independent variables, assuming a dependence of U². This is the model implemented in SSPA's manoeuvring and seakeeping program PORTSIM. The model works well for vertical banks and low speeds, with the water depth to draught ratio varying in the range: 1.2< h/T <∞. However, it cannot predict the repulsion force at extreme shallow water (h/T < 1.2). The influence of hull form is not taken into account in that model. Since there has been an increasing need for a mathematical model to predict bank effects of more realistic topographies, at extreme shallow waters, as well as ships of various forms and speeds, more experiments have been carried out recently at SSPA with special concerns to these factors. Based on this recent and previous experimental results, a new mathematical model for prediction of the stationary sway force and yaw moment due to different bank configurations has been developed.

　

 The paper presents part of the experiments, the new mathematical model and its validation. Finally a simulation study with two ships comparing the effects of three different bank configurations and three different mathematical models is presented.

　

 The experiments were performed in SSPA's seakeeping and manoeuvring basin (MDL) with three different ships: a tanker, a Ro/Ro-passenger ship (Ferry) and a catamaran. The ship main dimensions are given in Table 1 and the test program, summarised in Table 2, comprised:

　

 The bank models include a vertical bank, a sloping bank with a 30°inclination, and three flooded banks, as illustrated in Fig.1.

　

　

　

　

 The measured forces and moments for the Catamaran are quite different from those measured for the two mono-hull ships. Therefore the data for Catamaran is not used in the development of present mathematical model. This means that the present model will not be valid for Catamarans. Only the experimental results that are used for validation are presented in δ4, while other results are referred to [10][11].

　

3.1 Methodology

　

 Some general rules have been followed during the construction of new models. Attempt is made so that the models contain as few terms as possible and the expression as simple as possible. But they should also be fine enough to reproduce all characteristics of the response. Excessive use of higher-order terms, which usually introduces oscillation in a model, has been avoided. Symmetry condition is taken into account within the formulation. Most of the influencing variables have been selected in such a manner that their values are finite under any circumstance and their boundary values represent a certain limit situation in reality.

　

 From [1] , the variables that have vital influence on bank effects include the hull geometry and fullness, ship speed, water depth, ship-bank distance y, flooded bank's width y1, and over-bank water depth h1, bank inclination angle α and propeller loading. Therefore it is assumed that the sway force and the yaw moment coefficients, CY and CN, are functions of the following non-dimensional variables: CB, Fn, ζ, η, σ1, α and CT. They are defined and explained in the end of the paper.

　

 Symbolically, coefficients CY and CN have been designed to take the following form,

　

CY = (CY_v +Cy_s)・CY_F +CY_P　(1)

CN = (CN_v +CN_s)・CN_F +CN_P

　

 Where the subscripts V, S and F mean the contribution due to vertical, sloping, flooded banks respectively. The subscript P indicates the contribution due to propeller action. Thus the total force and moment is the summation of contributions of the respective bank type. The model for each specific bank contribution will be described in the subsections below.

　

3.2 Model for vertical banks

　

 The mathematical model is constructed in the following form,

　

CY_v = CB^cl [a0 + a1ζ + a2ζ² + a3ζ³ + (a4 + a5ζ + a6ζ ² + a7ζ ³ - a8・sign(ζ-c2)(Fnζ)³)｜η｜]ηFn^c3　(2)

　

CN_v = CB^cl [b0 + b1ζ + b2ζ² + b3ζ³ + (b4 + b5ζ + b6ζ ² + b7ζ ³ ｜η｜]η　exp(c4・Fⁿζ　(3)

　

 In the above formulae a polynomial of ζup to the 3^rd order is employed to reflect the water depth effect. Coefficients a0,... a8, b0,...b7, cl and c4 are determined by regression analysis. The influence of speed on CY_v is represented by the power of Fn to a constant c3 while the influence on CN_V by the exponential of Fn. At higher speeds, the influence on CY_V is further enhanced by a high order term (Fnζ)³. In (2), the change of sign of CY_V is triggered by a sign function between the difference ζ and a critical value c2. The contribution from ship-bank distance is expressed by a linear part (η) and a non-linear part |η|η. Note that when there are banks on both sides of a ship, the sway force due to each bank on either side should be calculated separately and summed.

　

3.3 Model for flooded banks

　

 The effect of flooded banks is modelled by an influencing function, which will be multiplied to the model for vertical banks. The model has the form,

　

　

 Where η1 and σ1, are the non-dimensional width and water-depth variables of the flooded bank, whose definitions can be found at the end of the paper. Parameters k1, k2 and k3 are determined by the regression analysis.

　

3.4 Model for sloping banks

　

 The influence of bank inclination is considered as a modification to that of a vertical bank by means of an increment, CY_S and CN_S defined by,

　

CY_S = CB^C1(1-|α|/90)[d0 + d1ζ^C5 + d2((90-|α|)Fn)³]|η'|η'　(6)

CN_S = CB^C1(1-|α|/90)^C6[f0 + f1ζ² + f2((90-|α|)Fn)³]|η'|η'　(7)

　

 The parameters c1, c5, c6, d0, d1, d2, f0, f1, and f2 are obtained via regression analysis. CY_S and CN_S will approach zero as α reaches 90°. On the other hand, as α tends to 0°, CY_S and CN_S also become zero since the distance variable η' vanishes.

　

3.5 Model for propeller effects

　

 A general tendency observed in the previous experiments is that a working propeller contributes to a suction force and a bow-away moment (compared with the case without a propeller, CT = 0). Moreover, there seems to be a linear relation between the propeller-induced force (and moment) and the loading coefficient CT. Thus following model is devised for the contribution of propeller rotation,

　

CY_P = sign(η)・m0ζ^m1Fn^m2｜η｜^m3CT　(8)

CN_P = sign(η)・n0ζⁿ¹Fnⁿ²｜η｜ⁿ³CT　(9)

　

 The parameters m0, m1, m2, m3 and n0, n1, n2, and n3 are again determined by regression analysis.