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(19) The free surface is modelled with the two phase volume of fluid technique (VOF). Further details about the code can be found in STAR-CCM+ user guide (2012). Finally, it must be noted that all computations are done in model scale with a model similar to the one used in the model test. The motions of the ship are described in terms of an earth-fixed inertial reference coordinate system and a ship fixed coordinate system. The motion of the ship, i.e. translations and rotations, are described with respect to the inertial frame. The linear and angular velocities plus the forces and moments are described with respect to the ship-fixed frame. In the present calculation, three degrees of freedom are considered and the ship motions in surge, sway and yaw are all prescribed. The motions are applied in the earth-fixed frame by mowing the complete computational domain and hereby introduce grid velocities according to the considered motion. This applies to both static conditions like static drift where the ship is moved with constant speed to obtain the desired drift angle and to dynamic tests like pure yaw motion, which are not covered in this paper. Heave and pitch motions in terms of dynamic sinkage and trim are taken from the model test, so they are not predicted in the CFD solver. The thrust and torque values applied in the prescribed body-force model are also taken from the model test. Computational domain The computational mesh is based on the trimmed mesh approach, which means that the mesh is dominated by hexahedral cells in most regions except close to the hull where the hull surface is used to trim the hexahedral cells leaving a set of polyhedral cells. Further, prism layer mesh is used near the hull surface to resolve the boundary layer flow. Concerning boundary conditions on the domain, an inlet condition 29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 prescribing the inlet velocity and the still water level for the volume fraction is given up-stream of the model and on the sides of the domain. On the outlet boundary a pressure condition is applied in order to have zero gradients for the velocity and volume fraction and prescribed hydrostatic pressure. On the top and bottom of the domain, slip walls are applied so there is no flow through the boundary. The inlet boundary is located 4.0 ship lengths from the ship, Outlet and sides are located 6.4 and 5.0 ship lengths from the ship, respectively. Figure 5. Illustration of domain boundaries. Figure 6 and Figure 7 show the computational mesh in the bow and stern regions, respectively. Refinement zones are applied in the free surface region and on the hull where the wave formation is expected to take place. Figure 6. Mesh in bow region. Figure 7. Mesh in bow region. Verification and validation Concerning verification focus is on mesh sensitivity. To study the sensitivity of the mesh size on the results one condition has been picked to be investigated. Three different grid sizes have been used and the corresponding forces have been evaluated for a static drift condition with 20 degrees drift angle at model speed of 1.318m/s, i.e. 0.775U0. Further, no running body-force propeller is included in this case.
With the unstructured mesh, it is not possible to make a completely systematic grid refinement, since it is not possible to directly control the size of the inpidual cells. However, all mesh quantities are given as a percentage of a base size, so in order to change the grid as systematically as possible the coarse (No. 3), medium (No. 2) and fine (No. 1) grids were made by changing the base size with a constant factor. Based on the three grids, the CFD simulations were conducted and the longitudinal X-force, X, the transverse Y-force, Y, and the yaw moments, N, taken around the Z-axis at mid-ship position were computed. It is difficult to estimate the grid uncertainty for unstructured meshes, so in lieu of an uncertainty method for unstructured meshes and since the present grid is hexahedral dominated, it has been attempted to estimate the grid uncertainty based on Richardson extrapolation. Solutions were obtained on 3 systematically refined grids by means of the refinement ratio, ݎீ ൌ √2. This gave mesh sizes ranging from approximately 1.5 to 7.0 million cells. Exact grid numbers and computed force and moment coefficients can be seen in Table 5. Forces and moments are non-dimensionalized with ሺ½ߩܷଶܮܶሻ and ሺ½ߩܷଶܮଶ ܶሻ respectively. 29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 Table 5: Non-dim forces and moments for different grids. No. of cells X’ Y’ N’ 1,491,946 -0.0208 0.1372 0.0451 3,068,474 -0.0214 0.1442 0.0446 6,969,575 -0.0211 0.1473 0.0451 The grid uncertainty ܷீ is estimated according to the procedure given in Stern et al. 2001 and Wilson et al. 2004. Briefly, the procedure is as follows. The changes in solutions between coarse and medium grids, 2 3 32S S G , and between medium and fine grids, 1 2 21S S G , are used to calculate the convergence ratio 32 21/ G G G R . Depending on the value of ܴீ three conditions can occur: i) 0 ൏ ܴீ ൏1, grid convergence, ii) ܴீ ൏0, oscillatory convergence and iii) 1൏ܴீ, grid pergence. In condition iii) no uncertainty can be estimated. In condition ii) the uncertainty is estimated by ) ½( L U G S S U (20) where U S and L S are the maximum and minimum of the solutions from the considered grids. In condition i) grid convergence occurs and generalized Richardson extrapolation (RE) is used to estimate the grid error ) 1 *(1 G RE and the order of accuracy G p , which are given as 1211) 1 ( *G G pGGREr (21) and ) ln(/ ln 21 32GG GGrp (22) 2 3 1 2 / / G G G G G x x x x r is the refinement factor, which defines the relation between the considered grids. When ) 1 *(1 G RE and G p are know it is possible to estimate the grid uncertainty. There are two ways to do this depending on whether the solutions are close to the asymptotic range or not. If the correction factor defined by 11GestGpGpGGrrC