Calculation of hydrodynamic manoeuvring coefficients using viscous-flow calculations Serge Toxopeus Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands Delft University of Technology, Delft, The Netherlands ABSTRACT: In the present paper, the work conducted by the author regarding implementation and improvement of efficient calculation of hydrodynamic coefficients within the manoeuvring work package of VIRTUE is presented. The improvements are mainly realised using variation of grid topology and density. In the paper, a mathematical model for the bare hull forces and moments based on the viscous-flow calculation will be given. Comparisons with erimental data obtained within the project shows that using accurate viscous-flow calculations, a considerable improvement in the prediction of the forces and moments on the ship compared to conventional empiric methods can be obtained. 1 INTRODUCTION Ship-owners and shipyards increasingly require accurate predictions of the manoeuvrability of ships in order to verify compliance with manoeuvring criteria. To improve computer predictions, one of the aims in the European integrated project VIRTUE is therefore to derive hydrodynamic manoeuvring coefficients from viscous-flow calculations in order to be able to predict the manoeuvrability of ships more accurately than conventional empiric manoeuvring predictions. The derived coefficients are implemented in fast-time manoeuvring simulation programs and used to predict selected manoeuvres. This method provides an alternative to conducting fast-time simulations using empirical mathematical manoeuvring models on the one hand and conducting direct simulation of manoeuvres using viscous-flow solvers coupled with body motion equations in the time domain on the other. Although the first method provides results very quickly, the accuracy and resolution of design details are often insufficient for designers. The second method using direct simulation is ected to provide accurate results but at impractically long computation times. Therefore the method using hydrodynamic coefficients derived from viscous-flow calculations in fast-time simulations is at the moment an attractive solution to the designer. In the present paper, the work conducted by MARIN regarding implementation and improvement of efficient calculation of hydrodynamic coefficients within the manoeuvring work package of VIRTUE is presented. Based on earlier and present work, see e.g. Eça, Hoekstra and Toxopeus (2005), the improvements were mainly realised using variation of grid topology and density. Furthermore, improvements were obtained by comparing results of calculations for corresponding test cases from different partners within the VIRTUE project. As a first step, hydrodynamic coefficients for the bare hull are calculated. In future work within VIRTUE, also coefficients for the appended hull and rudder force coefficients will be derived. For each obtained coefficient, a sensitivity study is conducted in order to determine its relative importance on the manoeuvring behaviour of the ship. The work will show that using accurate viscous-flow calculations, a considerable improvement in the prediction of the forces and moments on the ship compared to conventional empiric methods is obtained.
2 PARTICULARS OF THE SHIP AND TEST CONDITIONS The hull form under consideration is the Hamburg Test Case (HTC) which is one of the test cases in the VIRTUE project. The erimental results for the HTC used in this paper were provided by HSVA. The particulars of this hull form are presented below, Table 1: Non-dimensional main particulars, HTC Description Symbol Magnitude Description Symbol Magnitude Block coefficient C b 50 Length/beam ratio L pp /B 5.582 Midship section coefficient C m 83 Length/draught ratio L pp /T 14.922 Prismatic coefficient C p 62 Beam/draught ratio B/T 2.673 Waterplane coefficient C wp 22 The measurements were carried out with the model restrained from moving in any direction relative to the carriage. Bilge keels, rudder and propeller were not present during the model tests and were therefore not modelled in the calculations. The calculations were conducted with an undisturbed water surface, i.e. neglecting the generation of waves. Unless otherwise indicated, the Reynolds number in the calculations was 6.29 10 6, corresponding to a full scale ship speed of 10 knots. 3 NUMERICAL PROCEDURES 3.1 Flow solver, turbulence model and computational domain All calculations were performed with the MARIN in-house flow solver PARNASSOS, which is based on a finite-difference discretisation of the Reynolds-averaged continuity and momentum equations, using fully-collocated variables and discretisation. The equations are solved with a coupled procedure, retaining the continuity equation in its original form. The governing equations are integrated down to the wall, i.e. no wall-functions are used. More detailed information about the solver can be found in Hoekstra (1999) or Raven, Van der Ploeg and Eça (2006). For the calculations, the one-equation turbulence model, proposed by Menter (1997). The Spalart correction (see Dacles-Mariani et al. (1995)) of the stream-wise vorticity is included. The results presented in this paper were all obtained on structured grids with H-O topology, with grid clustering near the bow and propeller plane. Appendages and free surface deformation were not modelled. More details regarding the computational domain, the implementation of a drift angle in the calculations and the applied boundary conditions can be found in Toxopeus (2005). 3.2 Coordinate system and non-dimensionalisation The origin of the right-handed system of axes used in this study is located at the intersection of the waterplane, midship and centre-plane, with x directed aft, y to starboard and z vertically upward. The forces and moments presented in this paper are given relative to the origin of the coordinate axes, but in a right-handed system with the longitudinal force directed forward positive and the transverse force positive when directed to starboard. A positive drift angle β corresponds to the flow coming from port side (i.e. β=arctan(-v/u)). All forces and moments are presented non-dimensionally. The longitudinal force X and transverse force Y are made non-dimensional using 1 ρ VL 2 s ppt, the vertical force using 1 1 2 1 2 ρ VL 2 s ppb, the heeling moment K by ρ VL 2 s ppt, the pitch moment M by ρ VL 2 s pp B and the 1 2 yaw moment N by ρ VL 2 s pp T. 3.3 Uncertainty analysis For the uncertainty analysis, the procedure earlier applied to the KVLCC2M is used, see Toxopeus (2005). The background of this procedure is given in Eça and Hoekstra (2004).
In all calculations a reduction of the maximum difference in non-dimensional pressure between consecutive iterations to 5 10-5 was adopted as the convergence criterion. It is assumed that this is sufficiently small compared to the discretisation error and therefore the iteration error is ignored in the uncertainty analysis. In general, the adopted convergence criterion results in a reduction of the difference in the (total) force and moment components between consecutive iterations of well below 1 10-5. 4 HYDRODYNAMIC COEFFICIENTS FOR STEADY DRIFT MOTION 4.1 Influence of discretisation error Using the HTC hull form, a series of geometrically similar grids has been generated for a drift angle of 10, in order to investigate the discretisation error. The grid coarsening has been conducted in all three directions. For each grid, the variation in the number of grid nodes in the stream-wise, normal and girth-wise (n ξ, n η and n ζ ) directions is presented in Table 2, which includes also the maximum y + value for the cells adjacent to the hull, designated y + 2, that was obtained during the calculations. For grid 5, the results were not converged until the adopted convergence criterion and therefore the results for this grid are dropped from further analysis. Table 2: Properties of grids for uncertainty analysis, HTC, β=10. id β n ξ n η n ζ h i Nodes y 2 + Comment 1 10 377 95 51 2 1.00 3653130 0 2 10 361 91 49 2 1.04 3219398 6 3 10 297 77 41 2 1.25 1875258 7 4 10 257 65 35 2 1.47 1169350 1.19 5 10 185 48 26 2 2.00 461760 1.48 based on grid 1, coarsened by 2 2 2 6 10 177 46 25 2 2.08 407100 1.51 based on grid 2, coarsened by 2 2 2 7 10 145 39 21 2 2.50 237510 1.76 based on grid 3, coarsened by 2 2 2 8 10 129 33 18 2 2.94 153252 2.28 based on grid 4, coarsened by 2 2 2 9 10 89 23 13 2 4.17 53222 3.08 based on grid 2, coarsened by 4 4 4 10 10 73 19 11 2 5.00 30514 4.06 based on grid 1, coarsened by 5 5 5 Table 3: Uncertainty analysis, HTC, β =10. Item φ 0 φ 1 U φ p Item φ 0 φ 1 U φ p X -1.40 10-2 -1.57 10-2 14.1% 4 K -1.89 10-2 -1.79 10-2 7.5% 1.42 X f - -1.22 10-2 5.0% K f 1.79 10-3 1.76 10-3 3.3% 1.97 X p -5.33 10-4 -3.40 10-3 111.1% 7 K p -2.07 10-2 -1.96 10-2 7.0% 1.46 Y 3.76 10-2 4.42 10-2 18.4% 5 M -1.92 10-3 -1.25 10-3 72.6% 6 Y f 1.18 10-3 1.12 10-3 7.2% 1.80 M f - 3.03 10-4 3.0% Y p 3.68 10-2 4.31 10-2 19.5% 9 M p -2.24 10-3 -1.55 10-3 59.7% 5 Z 7.86 10-2 8.60 10-2 2.7% 0.21 N 2.45 10-2 2.44 10-2 3.3% 3.48 Z f - 3.40 10-4 8.5% N f -2.78 10-5 -2.48 10-5 54.8% 1.98 Z p 7.86 10-2 8.56 10-2 2.8% 0.22 N p 2.45 10-2 2.44 10-2 3.5% 3.45 Monotonous divergence For a drift angle of 10, the predicted values φ 1 of the friction (subscript f) and pressure (subscript p) components as well as the total force and moment coefficients are presented in Table 3 with the estimated uncertainties U φ. Based on an analysis of the results for each grid, it was decided to use the eight finest grids for the uncertainty analysis. The number of grids n g used depended on the scatter in the results for the coarsest grids. As already found during an uncertainty study for the KVLCC2M hullform, see Toxopeus (2005), the absolute uncertainty in the pressure components is larger than in the friction
components. The uncertainty in the longitudinal friction component X f is about one-third of the uncertainty in the longitudinal pressure component X p. For the other forces and moments, the uncertainty in the friction component is at least one order of magnitude smaller than the uncertainty in the pressure component. Since most integral forces and moments are dominated by the pressure component, this results in relatively large uncertainties in the overall forces and moments. In Raven, Van der Ploeg and Eça (2006), an extensive study to improve the uncertainty and accuracy of the pressure resistance component is presented. In Figure 1 the longitudinal force X, transverse force Y and yawing moment N and the nondimensional de-stabilising arm N/Y are graphically presented for the different grids. The scatter in the results is much smaller than found for the KVLCC2M results. For a relative step size below 3, the results appear to converge. The convergence rate p, however, is found to be small for both X and Y (p= and respectively). Due to the slow convergence, the difference between the extrapolated value φ 0 for zero step-size and the value φ 1 is large and hence the uncertainty is relatively large. Noteworthy is the fact that based on the trends with the current grids, the estimations (indicated by ) for X, Y, N and N/Y for increasing numbers of grid nodes do not converge to the erimental values (indicated by ). This may be caused by either modelling errors or by uncertainties in the erimental values. 0.013 0.06 0.014 0.015 0.016 0.017 p = U = 14.1% 0.055 0.05 X 0.018 0.019 0.02 0.021 Y 0.045 0.04 p = U = 18.4% N 0.022 0.0255 0.025 0.0245 0.024 0.0235 0.023 p = 3.5, p* = 2.0 U = 3.3% NdivY 5 0.55 0.5 0.45 0.4 p = U = 14.6% 0.0225 relative step size Figure 1: Convergence with grid refinement, HTC, β=10 0.35 relative step size 4.2 Influence of Reynolds number A calculation for β=10 has been conducted for a full-scale Reynolds number of 7.4 10 8. The grid was geometrically similar to Grid 1, except for an increase in the number of grid nodes in wall-normal direction to capture the gradients in the thinner boundary layer at full scale. With n η =137, the total number of nodes in this grid was 5.3 10 6, with a y + 2 of 0.56. In Figure 2, the calculated axial velocity field at the aft perpendicular for model scale and full scale is compared. Due to the higher Reynolds number, a somewhat thinner boundary layer is present at full scale. However, the structure of the wake does not change drastically.
0.02 0.00 0.02 z 0.04 0.06 0.08 Parnassos, HTC, x=0.50l pp, β=10 0.10 0.10 0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 y Figure 2: Comparison of axial velocity field for model scale and full scale Reynolds numbers HTC, β=10 (solid lines: model scale, dotted lines: full scale) Parnassos, HTCfs, x=0.50l pp, β=10 The values of the integral forces and moments for both model scale and full scale are presented in Table 4. In this table, also the relative change when going from model scale to full scale and the uncertainty in the model scale results are given. The changes in X, K and the ratio N/Y are clearly larger than the uncertainties in X, K and N/Y on model scale and therefore it is concluded that the trends in these components are caused by the change in Reynolds number. For the other components however, the change is smaller than the uncertainty and therefore conclusions can only be drawn based on physical interpretation. Theoretically, results for increasing Reynolds numbers are ected to be between results for model-scale Reynolds numbers and potential flow solutions. For a ship at steady drift motion, this means that both the X and Y force components will decrease (paradox of D'Alembert) and the N moment component will approach the so-called Munk moment (which is much larger than the N moment in viscous flow). Consequently, the de-stabilising arm N/Y will increase. Considering the values of X, Y, N and N/Y in Table 4, the trends comply with these statements and therefore it is concluded that the full scale values of X, Y, K, N and N/Y are realistic values for the full scale situation. Table 4: Comparison of integral forces and moments for model scale and full scale, HTC, β=10 Condition X Y Z K M N N/Y model scale (ms) -0.0157 0.0442 0.0860-0.0179-0.00125 0.0244 0.551 full scale (fs) -0.0091 0.0387 0.0871-0.0201-0.00152 0.0251 48 (fs-ms)/ms -41.7% -12.5% 1.3% 12.6% 21.7% 2.9% 17.6% U φ,ms 14.1% 18.4% 2.7% 7.5% 72.6% 3.3% 14.6% 5 HYDRODYNAMIC COEFFICIENTS FOR STEADY YAW MOTION 5.1 Implementation of rotational motion The approach to incorporate rotational motion adopted in PARNASSOS is using a non-inertial reference system. This approach has been used by several authors, see for example section 3.2 in Batchelor (1967) or section 1.15 in Wesseling (2000) and the applications to ships of e.g. Alessandrini & Delhommeau (1998) or Cura Hochbaum (1998). Using this system, the grid is attached to the hull form and rotates with the ship. However, each water particle now should erience centrifugal and coriolis forces due to the rotation of the coordinate system. These forces have to be added to the momentum equation as source terms. Originally, the momentum equation in PARNASSOS read in a cartesian coordinate system (see equation (2.1) in Hoekstra (1999)): ( ) ρ uu + p µ u +ρ uu = f i i,j,i i,jj i j i,j
with f i the force per unit volume that is exerted on a discrete flow volume. Assuming a steady flow, the rotational motion is simulated by implementing the centrifugal and coriolis force as additional force terms, such that the modified momentum equation reads: ( ) ( ) ( ( )) ρ uu + p µ u +ρ uu = f ρ 2Ω u ρ Ω Ω r i i,j,i i,jj i j i,j i i with f i a remaining force term per unit volume (e.g. propeller forces), Ω the vector of rotation, u = (u 1, u 2, u 3 ) = (u, v, w) the velocity vector and r = (x x R ) the radius of rotation with x R the position of the centre of rotation. In the equation above, the coriolis force is represented by 2ρΩ u while the centrifugal force is ρω (Ω r). At the outer boundaries, it is assumed that the solution corresponds to a solution for potential flow. Therefore, potential flow calculations are used to calculate the velocities at the outer boundaries and the pressure is obtained from the velocities. Using section 3.5 of Batchelor (1967), it can be shown that the pressure follows from: 2 2 (( r) u ) p p 1 = Ω ρ 2 (3) 5.2 Computational grids In order to maintain the usual definitions of inflow plane, outflow plane, no-slip/symmetry/jj=1 and off-body plane, the grid needs to adopt a shape facilitating these definitions. This means that the outer boundary (i.e. the off-body plane) should have a torus shape. Also the in and outflow planes have to be rotated to allow for perpendicular in and outflow. Figure 3 provides an example of a grid generated for rotational motion. In this figure, it is seen that the same base grid can be used for all rotation rates or combinations of drift and rotation and the outer blocks are deformed to suit the computational condition. Figure 3: Inner and outer blocks (coarsened) for γ=-0.4 5.3 Influence of discretisation error For a non-dimensional rotation rate of γ=-0.2, a series of geometrically similar grids has been generated in order to investigate the discretisation error. The grid coarsening has been conducted in all three directions. Table 5 presents the number of nodes and y + 2 values for these grids.
Table 5: Properties of grids for uncertainty analysis, HTC, γ=-0.2. id γ n ξ n η n ζ h i Nodes y 2 + Comment 3-0.2 297 77 41 2 1.25 1875258 7 4-0.2 257 65 35 2 1.47 1169350 1.19 6-0.2 177 46 25 2 2.08 407100 1.51 based on grid 2, coarsened by 2 2 2 8-0.2 129 33 18 2 2.94 153252 2.28 based on grid 4, coarsened by 2 2 2 9-0.2 89 23 13 2 4.17 53222 3.08 based on grid 2, coarsened by 4 4 4 10-0.2 73 19 11 2 5.00 30514 4.06 based on grid 1, coarsened by 5 5 5 Table 6: Uncertainty analysis, HTC, γ =-0.2. Item φ 0 φ 1 U φ p Item φ 0 φ 1 U φ p X -1.35 10-2 -1.38 10-2 7.2% 6.05 K -2.49 10-3 -2.33 10-3 16.3% 5.60 X f - -1.17 10-2 4.5% K f -1.77 10-4 -1.94 10-4 43.9% 2.68 X p -1.83 10-3 -2.07 10-3 37.6% 5.46 K p -2.31 10-3 -2.14 10-3 21.6% 5.00 Y - -5.98 10-3 8.3% M - -9.50 10-4 11.8% Y f -2.09 10-4 -2.21 10-4 48.5% 2.54 M f - 2.93 10-4 3.6% Y p - -5.76 10-3 5.7% M p - -1.24 10-3 8.4% Z 6.16 10-2 6.06 10-2 1.8% 9 N - 7.22 10-3 14.8% Z f - 1.62 10-4 6.4% N f -1.06 10-4 -1.11 10-4 14.9% 2.24 Z p 6.14 10-2 6.04 10-2 1.8% 1 N p - 7.34 10-3 14.4% Oscillatory convergence Monotonous divergence 0.0125 0.013 0.0135 2 x 10 3 2.5 3 3.5 U = 8.3% X N 0.014 0.0145 0.015 0.0155 p = 6.1, p* = 2.0 U = 7.2% 0.016 8.5 7.5 7 6.5 6 5.5 9 x 10 3 8 U = 14.8% 5 relative step size Figure 4: Convergence with grid refinement, HTC, γ=-0.2 Y NdivY 4 4.5 5 5.5 6 6.5 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 U = 14.5% 0.12 relative step size The predicted values of the friction (index f) and pressure (index p) components as well as the total force and moment coefficients are presented in Table 6 with the estimated uncertainties. Based on an analysis of the results for each grid, it was decided to use the four (grids 3, 4, 6 and 8) finest grids for the uncertainty analysis.
Similar to what was found for steady drift, the absolute uncertainty in the pressure components is larger than in the friction components. Compared to the calculations for steady drift, the relative uncertainties for the rotational motion results are in most cases lower. In Figure 4 the longitudinal force X, transverse force Y, yawing moment N and stabilising arm N/(Y-m u r) are graphically presented for the different grids. It is seen that the results do not differ much between the individual results, but convergence is not always found due to scatter. For a relative step size below 3, quite consistent results are however found. 6 MATHEMATICAL MODEL Based on the viscous-flow calculations discussed above, hydrodynamic coefficients for the forces on the bare hull were derived. First, the linear manoeuvring coefficients for drift and rotation were obtained by determination of the slope for zero drift or yaw rate. More information about deriving linear coefficients is found in Toxopeus (2006), in which also the predicted relation between the forces and moments and the drift angle or yaw rate for the HTC and other ships is given. Subsequently, non-linear terms were determined to describe the forces for large drift angles or yaw rates. To determine the coefficients Y v r and N v r use was made of calculations for combined drift and yaw motion. For illustration purposes, the derived coefficients for the transverse force Y and yaw moment N are given below: Table 7: Estimated manoeuvring coefficients for HTC bare hull Coefficient Value Coefficient Value Y uv 0.183 N uv 0.140 Y ur 0.017 N ur -0.0239 Y vv 1.118 N rr -0.0378 Y uuuvv -57 N uvv 0.034 Y v r 0.302 N v r -0.0555 With these coefficients, the mathematical model for the bare hull forces amounts to: Y ' = Y ' cosβ sinβ+ Y ' cosβ γ+ Y ' sin β sinβ + Y ' sinβ γ uv ur vv vr 3 2 uuuvv ( ) + Y cos β sin β sign sinβ N ' = N ' cosβ sinβ+ N ' cosβ γ+ N ' γ γ + N ' sinβ γ uv ur rr vr 2 uvv ( ) + N ' cosβ sin β sign sin β In Figure 5 a comparison is given of the relation between the transverse force and yawing moment as a function of the drift angle or non-dimensional yaw rate for the eriments (), viscous-flow calculations (), semi-empiric method of SurSim (sb) (see e.g. Toxopeus (2006)) and based on the mathematical model presented in equation (4) using the coefficients in Table 7 ( fit). The comparison shows that for drift motion the derived mathematical model resembles the eriments much better than the semi-empiric method. Unfortunately, no erimental data for rotational motion was available at the time of writing of this paper and therefore no conclusions can be drawn regarding the accuracy of the mathematical model for steady yaw rate. (4)
Y 0.000-0.100-0.200-0.300 sb fit HTC -0.400 0 5 10 15 20 25 30 β N 0.000-0.010-0.020-0.030-0.040-0.050 sb fit -0.060-0.070 HTC 0 5 10 15 20 25 30 β Y 0.060 0.040 0.020 sb fit 0.000 HTC -0.020 0 0.2 0.4 1 γ N 0.000-0.010 sb -0.020 fit -0.030-0.040-0.050-0.060-0.070 HTC -0.080 0 0.2 0.4 1 γ Figure 5: Comparison between eriments and predicted forces and moments, HTC 7 SENSITIVITY STUDY In order to determine the influence of estimation errors in each linear hydrodynamic manoeuvring derivative on the results for standard manoeuvres, a sensitivity study was conducted. Similar studies have been conducted in the past, see e.g. Lee and Shin (1998). In the present study, a set of fast-time manoeuvres using the mathematical model above was conducted during which one of the coefficients was individually multiplied by a factor of 1.1. Zig-zag manoeuvres were conducted to obtain the first and second overshoot angles (osa) and the initial turning ability (ITA). From turning-circle manoeuvres with 35 steering angle, the advance (AD) and tactical diameter (TD) were obtained. Based on the sensitivity study, the results as collected in Figure 6 were obtained. It is clearly seen that for the HTC inaccuracies in N uv have the largest impact on the accuracy of the prediction. N ur is also an important coefficient. Y ur is the least important linear coefficient for accurate predictions. Similar conclusions were found by Lee and Shin (1998). This means that for accurate predictions of the manoeuvrability using coefficients derived from CFD calculations, accurate predictions of the especially the yawing moment must be made. Yuv*1.1 30% Nuv*1.1 Yur*1.1 20% Nur*1.1 10% Yvv*1.1 Yuuuvv*1.1 0% Y v r*1.1 Nrr*1.1-10% Nuvv*1.1 1st osa 10/10 2nd osa 10/10 ITA 1st osa 20/20 AD TD N v r*1.1 Figure 6: Change in manoeuvring performance due to 10% change in input variable, HTC, 10 knots Change
8 CONCLUSIONS The uncertainty studies presented in this paper provide clear insight into the relation between the predicted values and the number of grid nodes used for a calculation. Even with geometrically varied grids, scatter between individual solutions is found, resulting in relatively large uncertainties of the estimated values. It is concluded that in order to obtain consistent results, grids with a number of nodes of at least 10 5 should be used. However, due to slow convergence upon grid refinement, considerable grid dependency is even found with 4 10 6 nodes. Using calculations for various drift angles, rotation rates and combined motion, a mathematical model was derived. The predicted forces and moments have been compared to predictions using empirical methods. It is demonstrated that using a mathematical model derived from viscous-flow calculations, better agreement with the eriments is obtained. Based on a sensitivity study in which the linear manoeuvring coefficients were individually varied, it is found that the simulations are most sensitive to changes in the N uv derivative. Therefore, accurate prediction of especially the yaw moment as a function of the drift angle is required. 9 ACKNOWLEDGEMENTS Part of the work conducted for this paper has been funded by the Commission of the European Communities for the Integrated Project VIRTUE under grand 516201 in the 6 th Research and Technological Development Framework Programme (Surface Transport Call). REFERENCES Alessandrini, B. and Delhommeau, G. 1998. Viscous free surface flow past a ship in drift and in rotating motion. 22 nd Symposium on Naval Hydrodynamics, pages 491 507, August 1998. Batchelor, G.K. 1967. An Introduction to Fluid Mechanics. Cambridge University Press. ISBN 0 521 66396 2. Cura Hochbaum, A. 1998. Computation of the turbulent flow around a ship model in steady turn and in steady oblique motion. 22nd Symposium on Naval Hydrodynamics, pages 550 567, August 1998. Dacles-Mariani, J., Zilliac, G.G., Chow, J.S. and Bradshaw, P. 1995. "Numerical/erimental Study of a Wing Tip Vortex in the Near Field", AIAA Journal, Vol. 33, September 1995, pp. 1561-1568. Eça, L. and Hoekstra, M., editors. 2004. Workshop on CFD Uncertainty Analysis, October 2004. Eça, L., Hoekstra, M. and Toxopeus, S.L. 2005. "Calculation of the flow around the KVLCC2M tanker". CFD Workshop Tokyo, March 2005. Hoekstra, M. 1999. Numerical Simulation of Ship Stern Flows with a Space-Marching Navier-Stokes Method. PhD thesis, Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, October 1999. Lee, H.-Y. and Shin, S.-S. 1998. "The Prediction of Ship's Manoeuvring Performance in Initial Design Stage", Practical Design of Ships and Mobile Units (PRADS), September, 1998. Menter, F.R. 1997. "Eddy Viscosity Transport Equations and Their Relation to the k-ε Model", Journal of Fluids Engineering, Vol. 119, December 1997, pp. 876-884. Raven, H.C., Ploeg, A. van der, and Eça, L. 2006. "Extending the benefit of CFD tools in ship design and performance prediction". 7 th International Conference on Hydrodynamics, October 2006. Toxopeus, S.L. 2005. "Verification And Validation Of Calculations Of The Viscous Flow Around KVLCC2M In Oblique Motion". 5 th Osaka Colloquium, March 2005. Toxopeus, S.L. 2006. "Validation of slender-body method for prediction of linear manoeuvring coefficients using eriments and viscous-flow calculations". 7 th International Conference on Hydrodynamics, October 2006. Wesseling, P. 2000. Principles of Computational Fluid Dynamics. Springer-Verlag. ISBN 3-540-67853-0.