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. 横向流作用下瘦长船体的垂直运动预测英文文献和中文翻译(2):http://www.youerw.com/fanyi/lunwen_63319.html