# The Prediction of the Performance of a Twisted Rudder
## 3 Numerical Approaches
### Numerical Solution Procedure  
Figure 2 shows the computational domain of the full appended ship model which has the same test section of the LCT. The velocity inlet boundary was located at the starting point of the LCT test section, and the pressure outlet boundary was placed 1.5 model lengths downstream of the hull stern. The wall boundary condition was applied on the hull and other boundaries. Figure 2b shows the local computational domain rotating with the propeller. The governing equations for two-phase incompressible flows were solved in the commercial program Star-CCM+ by using the finite volume method and a volume of fluid approach [42]. The segregated flow approach was used to solve the equations in which the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm was adopted to resolve the pressure-velocity coupling. A second-order accurate upwind scheme was used for the spatial differencing of the convective terms. A second-order central differencing was used for the viscous terms. The solution procedure was separated into two steps in the  
simulation of non-cavitating flows. In the first step, the steady computation of the propeller rotation was carried out using the moving reference frame (MRF) method. The second step involved the unsteady computation of the propeller rotation using a rigid body motion (RBM), the so-called sliding mesh method until a periodic convergence was achieved. Here, the computational time step corresponded to the time taken to rotate \(2^{\circ}\) at a given propeller rotation speed shown in Table 2. The cavitating flows were solved by employing a second-order time implicit scheme and 10 inner iterations to reach convergence at a given physical time, where the converged unsteady-state solutions for non-cavitating flows were utilized as the initial conditions. The time step for the cavitating flow simulations corresponded to \(1^{\circ}\) rotation time at a given propeller rotation rate.  
Figure 3 shows the surface and field grids around the full appended hull. An unstructured grid based on hexahedral and polyhedral meshes topology was used in the present simulations. To see the grid convergence on rudder forces, three grids called coarse, medium, and fine with 5.2 M, 7.5 M, and 11 M cells, respectively, were generated by a refinement ratio of \(\sqrt{2}\). A prism layer was used to generate orthogonal prismatic cells next to the wall surfaces, in which the height of the first cells from the wall was determined so that the dimensionless wall distance \(y^{+}\) was set to be 70 or less. Here, all grids had 11 prism grid layers to have a proper resolution of the boundary layer around the hull, propeller, and rudder. Especially, an overset grid was adopted to consider the rotations of each rudder in the starboard and port side at a given rudder angle. The present computations were carried out on a Linux-based PC cluster system whose each node has 20 Intel(r)_Xeon(r) 2.40 GHz processors. All runs used a total of 280 processors.  
Figure 2: Computational domain: (**a**) computational domain of the ship model and boundary conditions; (**b**) starboard and port rotational regions containing the propellers.