Abstract: An accurate prediction of hydrodynamic forces and moments acting on the hull of a ship during basic steady motions is a prior requirement for a reliable simulation of ship manoeuvres based on viscous flow computations. In the course of our developments towards such simulations, a detailed study has been performed for a twin screw, twin rudder RoRo ship model. In this work, the flow and the hydrodynamic forces on the hull have been predicted using our own RANS code for many combinations of yaw rate and drift angle. Since free surface computations are highly time consuming the present computations were performed neglecting the free surface. The predicted forces and moments and the velocity distribution in selected cross sections are compared with experimental data obtained in the towing tank. The agreement between computational and experimental results is in general good.

　

 The long-term objective of our current CFD developments is to achieve reliable and afford-able simulations of rudder manoeuvres of a vessel based on the computation of the free surface viscous flow around it. However, despite recent encouraging advances it will take some more years to reach this goal. The main reason is the extremely high computational time of 3D time simulations on relatively fine grids (e.g. 1 million cells) especially when the free surface is taken into account. In addition, sliding grid and/or overlapping grid techniques have to be used in order to turn the part(s) of the grid surrounding the rudder(s) respect to the rest of the grid fixed to the hull in the course of the simulation. This increases both, the computational time and numerical difficulties like stability and convergence. An intermediate step will be to use the RANS code to determine usual hydrodynamic coefficients for conventional manoeuvring simulations numerically. For this purpose, the flow around the ship performing prescribed (mostly harmonic) motions is computed and the resulting time histories of the hydrodynamic forces are analysed in the same way as when performing a captive model test with a Planar Motion Mechanism (PMM). This procedure seems much easier than the direct manoeuvring simulation, even if the free surface is taken into account. Once the hydrodynamic coefficient set has been determined, say from a dozen of 'numerical runs', all desired manoeuvres can easily be simulated in a conventional way with negligible computational effort. An open issue when performing RANS based manoeuvring simulations remains the proper modelization of the propeller forces and their effect on the flow, which is usually implemented applying body forces in those cells of the grid which take the space of the non-present propeller. In the present work, a simple empirical body force model depending on drift angle and yaw rate has been implemented derived from the thrusts measured during the model tests.

　

 In order to get useful simulation results, the prediction of the forces and moments acting on the hull of the ship has obviously to be 'sufficiently' accurate. Because the used numerical method can take almost all relevant hydrodynamic aspects involved into account (like viscosity, free surface, dynamic sinkage and trim) , this is expected to be the case. However, there is still a lack of validation of the predicted forces and moments on a ship when turning and drifting.

　

 This paper briefly outlines the computational technique used and compares selected results for some steady basic ship motions with model test measurements. Results are presented for a modern RoRo ship of the German shipyard FSG for different combinations of drift angle and yaw rate at zero rudder angle. The computed transverse velocity field in selected cross sections of the ship in steady turning motion is compared with preliminary PIV mea-surements performed at the Hamburg Ship Model Basin (HSVA).

 The governing equations of the fluid motion are the RANS equations and the continuity equation. Reynolds stresses are approximated with the k-ω turbulence model, k being the turbulent kinetic energy and ω the specific dissipation rate of k. For k and ω two additional transport equations are solved, see [8] . The governing equations are written in non-dimensional form in a Cartesian ship fixed coordinate system. Inertial forces stemming from the acceleration and rotation of this coordinate system, e.g. centrifugal and Coriolis forces, are added to the right hand side of the RANS equations. In the present case, the inertial forces are calculated in the course of the simulation with the known prescribed velocities of the model and their time derivatives. By integrating the pressure and shear stresses on the hull and appendages we get the hydro-dynamic forces and moments acting on the ship.

　

 If the free surface is taken into account, the fluid flow around the ship is considered to be the flow of a single incompressible fluid with two immiscible phases (water and air). The free surface is then taken as the interface between the two phases. This is captured with the Level Set method, see for instance [6] or [7] . The numerical method, including the free surface technique and the prediction of motions, has been described in detail in [2].

　

 Because of the large amount of computations performed, wall functions were used and the free surface was neglected in all computations presented in this paper in order to reduce the computational time and numerical difficulties.

　

　

 A block-structured grid is used to approximate the differential expressions of the governing equations of the fluid with algebraic expressions by means of a Finite-Volume technique. The quality of this grid (e.g. smoothness and orthogonality at boundaries) is crucial for the accuracy of the results and for the convergence behaviour of the method. While the loss of accuracy due to a poor grid can (in principle) be reduced increasing the grid resolution, convergence problems mostly remain.

　

 The finest grid used in the present study had about 2 million cells and more than 200 blocks for both ship sides when including the brackets of the propeller shafts. In order to calculate the asymmetric flow around the ship the starboard side of the grid was mirrored to the port side. Fig. 1 shows a partial view of the surface grid on the stern of the ship including rudders, without and with V-shaped brackets on the top and bottom of the figure respectively. As can be seen, the inclusion of appendages makes the grid more complicated and affects the smoothness of the grid.

　

 Computations for many combinations of the non-dimensional yaw rate r'=rL/U0 and the drift angle β at the main section were per-formed for the twin screw, twin rudder RoRo ship mentioned above at scale 21.4. The main dimensions of the full scale ship are:

 L=182.4 m, B=26.0 m, T=5.7 m and CB=0.569. The speed of the model was Uo=1.80 m/s yielding a Reynolds number of Rn=1.5310⁷. During the computations the ship motions were prescribed.

　

 The predicted forces and moments on the hull are compared with experimental data obtained with the Computarized Planar Motion Carriage (CPMC) in the towing tank of the HSVA. Because of the relatively low Froude number Fn = 0.20, and especially because roll, sinkage and trim of the model were suppressed during the model tests, we do not expect the free surface to have a significant influence on the measured forces for moderate drift angles.

　

 The computations were performed on grids with roughly 0.7 and 2 million cells. Between 1500 and 4000 SIMPLE iterations were necessary to achieve a good convergence, demanding a CPU time from 0.5 up to 4 days on a normal PC. The higher values correspond to the larger grid and the largest drift angles.

　

 The computed non-dimensional side force

　

　

is compared with measurements in Fig. 2 for all computed combinations of drift angle and rate of turn. The agreement is in general very satisfactory.

　

 The comparison of the non-dimensional yaw moment about the vertical axis amidships

　

　

with experiments is shown in Fig. 3. Although the overall trends are captured well, larger discrepancies are evident for larger drift angles. The conjecture that they could stem from the omission of the V-brackets which support the propeller shafts have been discarded after repeating the computations including the brackets in the computational grid, see Fig. 1. The new results do not differ much from those without brackets for large drift angles. Note that both the computations and the measurements were performed suppressing dynamic heel, trim and sinkage. We are now going to repeat the computations