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VjF - viscous excitation force. The indexes j and k indicate the direction of the fluid force and the mode of motion respectively. Cross - flow approach Cross-flow effects arise from three-dimensional boundary layer when the streamlines are curved. In the plane perpendicular to the main stream the secondary flow will produce lift and different pressure distribution along the body. In the aeronautic field this effect plays a significant role and is generally considered through the determination of two cross-flow coefficients: lift and drag as explained by Thwaites (1960). In Begovic (2002) the sectional cross-flow force for flow with small angle of incidence αj is defined as shown in Formula (2) () () () ( x v x v C x U AdxdFJ J D J Jj+ = α α ρ 221) (2) where AJ - projected plane area of the section in the j-th direction α - viscous lift coefficient αJ - angle of attack to uniform flow CD - viscous drag coefficient Vj - relative fluid velocity with respect to the section in the j-th direction, as shown in Fig. 2 U - body forward speed Figure 2. Kinematics scheme of cross-flow The coefficients α and CD depend on the geometrical characteristics of the body, the mode of motion and the frequency of oscillation; their values should be determined experimentally. During the past years, some authors considered the cross-flow for the vertical motion prediction of SWATH and catamaran vessels summing the additional damping coefficients to those obtained by potential theory using the linearity assumption. Different numerical potential flow methods were used: 2D strip theory (Centeno (2002), 3D Green function methods (Chan (1992, 1993, 1995), Fang et al (1996)), 3D Rankine source method (Schellin and Rathje (1997), 2 ½ D (Begovic et al (2002)), 2D-Time domain (Davis (2003)) in combination with the constant coefficients values along the ship length. All of the authors used the coefficient values derived from the experiments on airship models with circular or polygonal sections. According to Lee and Curphey (1977) α is about 0.07 and CD varies from 0.4 to 0.7. Fang (1996, 1997), Centeno (2002) have found that the combination α= 0.05 and CD = 0.0 best fit their experimental data. Hull forms and experimental results For the validation of the hypothesis of implementing the cross flow effect in the assessment of vertical motions, the results of two model tests performed at Towing Tank of University of Trieste were used as presented in Begovic, Boccadamo and Zotti (2002). Other results relative to four models with similar hull form covering a significant L/B ratio range from 4.5 to 14.1, were collected from the literature. The considered models are: 4797 from systematic Series 64, model 915, model 5 by Blok and Beukelman (1984), NPL models 4B and 5B by Molland et al (2000) and model NOVA II by Lahtiharju et al (1991). Each model was scaled to ship dimensions having the same displacement of 137 t. Main dimensions, principal characteristics and calculation speeds are summarized in Table 1, while in Figs 3-8 the relative body plans are presented. numerical code TRIM. It takes the results from the 2 ½D high speed theory based on the potential flowassumption, calculates forces due to the cross-flow andrecalculates motions. The reported works considered thesame coefficient values as in experiments with aeroprofiles in uniform flow; seakeeping is characterised bynon-uniform flow, so that a deeper investigation of theproblem seems quite necessary. For each model at each speed five calculations wereperformed as summarized in Table 2. Table 2. Calculation cases summary NFor the Model 4797, heave at FN= 0.496 is little bit overestimated by all methods, which are in this case very close to each other. It seems that the experimental results peak is higher and placed in the lower frequency range than the numerical results. Pitch motion at this speed is very well fitted by each one of the methods including cross flow coefficients. The calculation by DRAG underestimates experimental results so it could be considered as too much damping. At FN=0.798, heave is best predicted by: α=0.035 - CD=0.25, while Diagram 12. Pitch Motions of Model 915 at FN = 1.077 For the Model 915 at FN=0.498, heave motion is well predicted by all methods in the low frequency range while at the higher frequency all methods overestimate the experimental data. The experimental point relative to the highest ωe is unreasonably high and regarding the physical nature of phenomenon, it could be considered that some experimental error occurred. The pitch is perfectly predicted by LIFT. At FN=0.797, few experiment points do not show the real behaviour of heave motion. As the existing points are set it seems that there should be a peak between the 2nd and 3rd experimental points (ωE=2.2 &pide; 2.9). POT predicts this peak to be very high, while all the calculations are close to each other and perfectly fit the existing experimental data. Pitch is underestimated by POT and underestimated by LIFT. The other two methods are fair but not as good as LIFT. At FN=1.077 LIFT prediction is the best both for heave and for pitch motions. Model 5Diagram 16. Pitch Motions of Model 5 at FN=1.14 For the Model 5 at FN=0.57, the 2nd part of the heave experimental data are very well predicted by LIFT, by α=0.035 - CD=0.25 or by α=0.0 - CD=0.5. The 1st part is overestimated but also the peak position is significantly shifted. The pitch is best predicted by DRAG. At FN=1.14, heave is perfectly predicted by DRAG. Pitch is best predicted by α=0.0 - CD=0.5 even if there is no big differences among the various predictions. NPL Model 4B 2.0η3 /APOTDiagram 20. Pitch Motions of Model NPL 4B at FN = 0.8 For NPL model 4B at FN=0.5 heave is very well predicted by α=0.035 - CD=0.25.