The rectangular domain covering the catamaran is generated to simulate the virtual captive model test. The computational domain is selected to be sufficiently large to avoid backflow and side flow at a high drift angle. In the static test, the dimension of the rectangular is 7L in length, 4L in width, and 2.5L in height. The rectangular is also covered for the inner domain. In the dynamic test, the dimensions are 6L, 4L, and 2.5L for the length, width, and height of the rectangular, respectively. The inner domain is covered a cylinder with a radius of 2.5 m and a height of 1 m to avoid mesh failure at sharp corners. In addition, a physical condition is applied to the boundaries domain. The inlet boundary condition upstream is taken as a pressure-inlet with an open channel; the outlet condition downstream is taken as a pressure-outlet with an open channel. A no-slip wall is set for the hull face and symmetry is assigned for the top face, bottom face, and side faces. The Volume of Fluid is applied to model the two-phase volume of the fluid technique. The fluid domain is discretized into a tetrahedral unstructured mesh in the inner domain of the static test and the whole domain of dynamic test. Furthermore, a hexahedron structured with multi-block is generated in the outer domain of the static test in order to reduce the number of elements. Because the wall function is applied to improve the accuracy of the flow resolution, prism layer mesh is used near the hull surface to resolve the boundary layer flow. The y+ value of 30 is estimated for Reynolds number of 9.7E+5. The mesh generation for the static test and the dynamic test is shown in Fig. 2.

According to Practical Guidelines for ship CFD Application (ITTC, 2011), a two-equation turbulence model is used to enable an accurate prediction of the ship dynamic. The turbulence models of k-ε Realizable and k-ω SST (Shear Stress Transport) are employed in the calculation. In addition, the SIMPLE (Semi-implicit Method for Pressure Link Equations) algorithm is applied to obtain the pressure and velocity field. The quantities at the cell faces are calculated from the cell centered values using the second order upwind method.

Before proceeding with the computation, it is necessary to first perform a verification study. Thus, the turbulence study and mesh independence study are conducted to select the most appropriate method for this study. Experimental results obtained from the BSHC (Bulgarian Ship Hydrodynamic Center) are used for verification in the present study.

In the turbulence study, two turbulence models of k-ε Realizable and k-ω SST are evaluated initially. A drift test case of 6 degrees is carried out at the Froude number of 0.3 for two turbulence model to obtain the hydrodynamic force.

For the mesh independence study, three different grid size are used and the corresponding forces are evaluated in a drift test with a drift angle of 6 degrees at Froude number 0.3. The grid refinement is achieved by applying a refinement factor rG=2 to the base size. The Grid Convergence Index (GCI) method is applied to evaluate the discretization error of the calculation method. Using this method, the grid densities of fine mesh, medium mesh, and coarse mesh is prepared by changing the base size with a refinement factor. The grid parameter is expressed in Eq. (6) and the GCI is defined as Eq. (7).

where ∈ij=ϕi−ϕj, and ϕ indicate the solutions with fine, medium, and coarse input parameters. eij=|(ϕj−ϕi)/ϕj| is the approximate relative error.

The ship is towed at a constant speed with a constant drift angle ranging from zero to 180 degrees. Fig. 3 shows a schematic of this motion.

Fourier approximation is used in deriving the hydrodynamic coefficients. The following expression using Fourier series is expressed as follows,

The test is conducted with a ship speed of zero while rotating around a vertical axis through the center of gravity of the ship with a constant yaw rate, as shown in Fig. 4. The hydrodynamic force in the pure rotating test is given in Eq. (9) as:

As shown in Fig. 5, the ship is simulated to rotate around the vertical axis through the center of gravity at a constant speed and a constant yaw rate. The motion given to the ship during the yaw rotating test with a large drift angle can be described as follows,

The hydrodynamic force for the drift angle measured between zero degrees and 360 degrees is described as the sine and cosine functions. Thus, the hydrodynamic force can be expressed in terms of Fourier approximation with respect to the drift angle in Eq. (11). On the other hand, the variations of these coefficients are supposed as shown in Eq. (12).

By substituting Eq. (12) into Eq. (11), X’, Y’ and N’ can be expressed as shown in Eqs. (13)-(15), respectively.

On the other hand, the terms of the Fourier coefficients can be transformed into conventional expressions with hydrodynamic coefficients according to the relationship between β and u, and v based on trigonometry. According to the mathematical model proposed by Takashina (1986), the hydrodynamic equations for X’, Y’, and N’ are respectively expressed as follows,

Finally, the hydrodynamic coefficients for X’, Y’, and N’ can be respectively obtained from the following relationships shown in Eqs. (19)∼(21).

It should be noted that X′uu is given as a function of velocity based on the resistance results.

The ship is simulated to oscillate in a sinusoidal motion at a constant speed and a constant frequency. The pure sway test and the pure yaw test are achieved in this study. The main reason these tests are performed is to estimate the added mass coefficients with v oscillating harmonically in the pure sway test and the coefficients relative to the yaw rate with harmonic oscillating yaw motion in the pure yaw test. The motions of the pure sway test and the pure yaw test is described in Eqs. (22) and (23), respectively. Figs. 6~7 show the definitions of the pure sway test and the pure yaw test, respectively.

The mathematical models for the harmonic test (Hajivand et. al., 2015 and Liu et. al., 2018) are shown in Eqs. (24) and (25) for pure sway test and pure yaw test, respectively.

Fourier approximation for the harmonic test is adopted to derive the hydrodynamic coefficients. The harmonic forms of the pure sway test and the pure yaw test are achieved by substituting the motion into the mathematical model as presented in Eqs. (26) and (27) for pure sway and pure yaw, respectively.

The Fourier sine and cosine coefficients in these expressions are given in Table 1. The functions of the N component are not listed in the table since they are the same as those of the Y component, and it can be obtained by replacing "Y" with "N".

Fig. 9 shows the comparison of the turbulence study results. It can be seen that both of the turbulence models predict the hydrodynamic force of the catamaran with good accuracy. However, the results of the k-ω SST turbulence model are slightly better than those of the other two.

The three densities of the fine grid, the medium grid, and the coarse grid are counted to be approximately 6.77M, 4.90M, and 3.55M, respectively. The mesh independence study is performed with the k-ω SST turbulence model. Table 5 shows the solution of the mesh convergence study. In addition, the computation results obtained from the different mesh sizes are shown in Fig. 10. It indicates that the error of the solution is reduced as the generated grid becomes finer.

The verification is simulated for a fine grid with the k-ω SST turbulence model at a Froude number of 0.3 in the case of the drift test. In Fig. 11, the calculated hydrodynamic force is compared with the experimental results from BSHC. It can be seen that the hydrodynamic force of computation results are consistent with the experimental results. Therefore, fine grid and k-ω SST are simulated in further study.

The hydrodynamic forces of the static drift test for drift angles ranging from zero to 180 degrees at low speed are shown in Fig. 12. The comparison with other ship types shows that the trend of the hydrodynamic force in the present study is consistent with the trends of the experimental results of the other ship types. However, the difference among results due to the different ship types is observed, especially tugboats. Furthermore, slender shape of a catamaran and slender body ship gives a similar results.

By approximating the measured data, the Fourier coefficients are obtained by using Eq. (8) and presented in Table 6.

Furthermore, the below figure shows the velocity contour of static drift test at 90 degrees. It can be seen that the velocity between two side faces where symmetry condition is set. It is nearly same value with inlet velocity despite of low speed condition. Since velocity changes occur around the ship, applying symmetry conditions seems appropriate.

Fig. 14 shows the yaw moment acting on the catamaran against the yaw rate squared r|r|. The hydrodynamic coefficient N’r|r| can be obtained based on the ratio of yaw moment to yaw rate squared.

The Fourier coefficients obtained from yaw rotating test are plotted against the non-dimensional yaw rates in Figs. 15~17 for X’, Y’, and N’, respectively. In addition, the trends of the Fourier coefficients of catamaran are compared with those of the Fourier coefficients of tugboats obtained by Takashina(1986). It can be observed that Fourier coefficients of the tugboats are smaller than the catamaran results due to the smaller values of hydrodynamic forces of the tugboats as shown in Fig. 12. Nevertheless, the similar trends between tugboats and catamaran can illustrate that the obtained coefficients of catamaran are appropriate results.

The Fourier coefficients obtained from the harmonic test for in-phase components are drawn versus lateral acceleration in Fig. 18 for the pure sway test and versus yaw rate in Fig. 19 for the pure yaw test. By fitting the in-phase components, the added mass coefficients can be estimated.

Finally, the hydrodynamic coefficients of a Delft 372 catamaran operating at low speed are estimated using the virtual captive model test and the results are presented in Table 7.

The water-jet propulsion is used as propulsor device and steering device. Thus, the right-hand side of Eq. (1) is expressed as

subscript H and WJ indicate hull and water-jet, respectively.

A experiment for Delft 372 catamaran with water-jet propulsion was performed by Milanov, et al., (2012) in order to obtained mathematical model and hydrodynamic coefficients of water-jet as shown in Table 8.

Fig. 20 shows the simulation result of turning circle maneuvers at low speed when the water-jet impeller RPM is kept to be constant and water-jet nozzle deflection is 35 degrees. The trajectory is described for the turning path. The speed and yaw rate are changing throughout turning circle simulation.