Comparison between the RANS Simulations of Double-Body Flow and Water–Air Flow around a Ship in Static Drift and Circle Motions

Abstract

Manoeuvrability is one of the important ship hydrodynamic performances. That is closely related to the safety and economy of navigation. The development of a high-accuracy and high-efficiency numerical method to compute the forces and moments on manoeuvring ships has been the main task for ship manoeuvring predictions. The numerical method by solving RANS (Reynolds-Averaged Navier–Stokes) equations may be the most used one nowadays for the computations of ship manoeuvring forces and moments. However, applying a RANS tool for ship manoeuvring prediction remains very low efficiency, especially considering the six-degrees-of-freedom ship motions on the water surface. Thus, it is very necessary to introduce a few assumptions to reduce the computational time when applying a RANS tool, e.g., the assumptions of double-body flow and body force propeller, and consequently improve the application efficiency. Generally speaking, the assumption of double-body flow, in which the free-surface effects are neglected, is more suitable for low-speed ships. Nevertheless, rare publications have been reported relating to how this assumption affects the accuracy of the computed manoeuvring forces and moments. To this end, this article presents a comparative study between the RANS simulations of double-body flow and water–air flow around a container ship performing static drift and static circle motions. Three ship speeds, corresponding to the Froude numbers 0.156, 0.201, and 0.260, respectively, are considered during the simulations. The computed side forces and yaw moments obtained by the water–air flow simulations are closer to the available experimental data than that obtained by the double-body flow simulations for all ship speeds. The computed surge forces obtained by water–air flow simulations also agree well with the experimental data, whereas the computed surge forces obtained by the double-body flow simulations are wrong. The reasons are analyzed by comparing the pressure distributions on the ship surface and the flow separations around the ship.

1. Introduction

With the increase in the number of ships, as well as ship dimensions and ship speed, ship control has become more difficult than before, which increases the risk of ship overwater accidents, probably leading to not only the loss of property and life but also environmental pollution. The IMO (International Maritime Organization) released ship manoeuvrability standards in 2002, which recommended that it should check the manoeuvrability of a newly designed ship at the early design stage. Moreover, the large amount of carbon dioxide emitted by the shipping industry will reinforce the greenhouse effects and worsen the global climate. To reduce carbon dioxide emissions, the ship EEDI (Energy Efficiency Design Index), also recommended by IMO [1,2], proposed high requirements for ship performance. The shipboard equipment should be of lower power as far as possible to reduce emissions but inversely requires that the ship must have enough performance to ensure navigation safety. Hence, the high-accuracy evaluation of ship manoeuvrability is even more necessary than before.

Generally, there are two kinds of experimental methods to evaluate ship manoeuvrability. One is to perform free-running tests for a ship model, e.g., turning test, zigzag test, etc. In this way, the manoeuvring characteristics of the ship can be obtained directly, such as advance, static turning circle diameter, and overshoot angles. The other is to conduct captive model tests to determine hydrodynamic derivatives or coefficients in advance, then to simulate ship free-running tests by solving ship motion equations in which the hydrodynamic forces and moments are approximated by using the hydrodynamic coefficients.

With the development of numerical methods, the technique of the virtual ship manoeuvring test has matured at present and is widely used for ship manoeuvring prediction. The numerical method by solving RANS (Reynolds-Averaged Navier–Stokes) equations may be the most used one now.

The RANS method can be used to simulate the captive model test and then predict ship manoeuvrability. Cura Hochbaum et al. [3,4] simulated the captive model test of KVLCC1 and KVLCC2 by using the RANS method and simulated the 10/10 zigzag manoeuvring motion by an Abkowitz-type model further. The first and second overshoot angles simulated are in good agreement with the experiment. Kim et al. [5] compared the virtual captive model test results based on a RANS solver with the experimental data. The results are in good agreement in the case of static drift and pure sway, but there are some differences in the case of a pure surge. Yao et al. [6] used a RANS solver to simulate the captive motions of the KVLCC2 bare ship model, such as static oblique motion, static turn, and pure sway. The calculated hydrodynamic forces and moments are highly consistent with the test results. Liu et al. [7,8] simulated the pure sway test of the DTC container ship model in shallow water towing tank based on an unsteady RANS (URANS) solver. The research shows that the results of the virtual dynamic PMM test are still not accurate enough to reliably predict the manoeuvrability of the ship.

The application of the RANS method to simulate a free-running model test to predict ship manoeuvrability has also made some progress. The direct simulations of zigzag and turn manoeuvring tests by Carrica et al. [9,10] and Mofidi [11] indicate that it is feasible to directly simulate the manoeuvring motion of ships with a rotating rudder and a real propeller by the RANS method (CFD-Iowa, RANS), at the cost of huge computational power. Stern et al. [12] and Moctar et al. [13] carried out a full time-domain numerical simulation of ship zigzag and turn manoeuvring motion based on the self-developed RANS/DES solver. Broglia et al. [14,15] simulated the manoeuvering motion of a twin-propeller ship based on the RANS solver in the commercial software star CCM+. In addition, the research of Sadat-Hosseini et al. [16] shows that the CFD method also has good applicability for the direct simulation of ship manoeuvering motion in waves.

The RANS method has been widely applied for the simulations of free-running model tests and captive model tests, as reported in the above works. However, the simulation will be very time-consuming if considering the free surface or a real rotating propeller (Carrica et al. [10]). It may take more than one month to complete a case simulation of a free-running model test. Thus, the assumptions of double-body flow and body force propeller are usually introduced in RANS applications, e.g., in the work of Yao et al. [6,17]. Due to the double-body assumption, the transient flow around a ship in static drift motions and/or static circle motions can be simplified as the steady flow. If a body force propeller is used instead of a real rotating propeller, larger time steps are allowed during RANS simulations, and the transient flow around the ship can also be simplified to the steady flow when the ship is in static motion. However, if a real rotating propeller is considered, typically, more than 200 time steps per propeller revolution are required to guarantee sufficient numerical accuracy, increasing the computation amount significantly.

The double-body assumption seems reasonable in the work of Yao et al. [17]) because the ship’s speed is low (the Froude number is 0.142), and the final predicted results of ship manoeuvrability seem good compared to the experimental data. Nevertheless, rare publications have been reported on how the double-body assumption affects the accuracy of the computed hydrodynamic forces and moments on a manoeuvring ship.

The purpose of this study is to investigate the differences between the RANS simulations of double-body flow and water–air flow around a container ship performing static drift motions and static circle motions. The KCS (MOERI Container Ship) is the worldwide benchmark ship for validating computational methods on ship hydrodynamics and ship manoeuvring prediction methods. The KCS was designed by KRISO (Korean Research Institute of Ships and Ocean Engineering), and the full-scale ship has not been built. To analyze the influence of ship speed on the applicability of the double-body flow assumption, three ship speeds were taken into account, corresponding to the Froude numbers 0.156, 0.201, and 0.260, respectively. The RANS simulations were performed on the open-source CFD platform OpenFOAM. A very simple body-force model was used to approximate the effects of a real rotating propeller on the flow. The computed surge forces, sway forces, and yaw moments obtained by the simulations of double-body flow and water–air flow were compared, and the differences were analyzed by comparing the pressure distributions on the ship surface and the flow separations around the ship.

2. RANS Method

2.1. Continuity Equation and RANS Equations

A ship-fixed coordinate system, o-xyz, was used to describe the flow around the ship. The origin o is located at the junction of the mid-ship section, the centre-line plane, and the still water plane. The x-axis points to the ship’s bow, the y-axis points to the starboard, and the z-axis points to the keel. Under the assumption of an incompressible Newtonian fluid, the continuity equation (mass conservation equation) and the RANS equations (momentum conservation equations) can be written as:

$$\frac{\partial U_i}{\partial x_i}=0$$

(1)

$$\begin{array}{ll}\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}& =f_i-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left[\nu \left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)-\overline{{U^{\prime }}_i{U^{\prime }}_j}\right]\\ & -\left(\frac{\partial u_i}{\partial t}+{\epsilon }_{ijk}{\mathrm{\Omega }}_j{u}_k\right)-{\epsilon }_{ijk}{\omega }_j{\epsilon }_{kmn}{\mathrm{\Omega }}_m{x}_n-2{\epsilon }_{ijk}{\mathrm{\Omega }}_j{U}_k-{\epsilon }_{ijk}\frac{\partial {\mathrm{\Omega }}_j}{\partial t}{x}_k\end{array}$$

(2)

where the subscripts i, j, and k run from 1 to 3, $x_i=(x,y,z)$ and $U_i=(U,V,W)$ are independent Cartesian coordinates and flow velocity, $u_i=(u,v,w)$ and ${\mathrm{\Omega }}_i=(p,q,r)$ are the ship linear velocity and angular velocity, $f_i$ is the body force per unit mass, $\nu$ is the kinematic viscosity, and ${\epsilon }_{ijk}$ is permutation symbol.

Note that there are four additional terms on the right side of the RANS equations, compared with the RANS equations in the earth-fixed inertial frame of reference. They are translational acceleration force per unit mass, centrifugal force per unit mass, Coriolis force per unit mass, and angular acceleration per unit mass from left to right. These inertial terms have been programmed into OpenFOAM in our previous work (Yao et al. [17], 2021).

The term, $-\rho \overline{{U^{\prime }}_i{U^{\prime }}_j}$, in Equation (2), refers to Reynolds stress. Based on the Boussinesq assumption, the specific Reynolds stress is expressed as:

$$-\overline{U_i^{\prime }{U}_j^{\prime }}={\nu }_t(\frac{\partial \overline{U_i}}{\partial x_j}+\frac{\partial \overline{U_j}}{\partial x_i})-\frac{2}{3}k{\delta }_{ij}$$

(3)

where ${\nu }_t$ is eddy viscosity, $k$ is turbulent kinetic energy, and ${\delta }_{ij}$ is the Kronecker symbol.

2.2. Turbulence Model

The k−ω SST model with wall functions (Menter et al. [18], 2003) was used to close the RANS equations in this study. The transport equations of $k$ and the dissipation rate $\omega$ are:

$$\frac{\partial k}{\partial t}+\overline{u_j}\frac{\partial k}{\partial x_j}=\stackrel{\sim }{P_k}-{\beta }^{\ast }k\omega +\frac{\partial }{\partial x_j}\left[\left(\nu +{\sigma }_k{\nu }_t\right)\frac{\partial k}{\partial x_j}\right]$$

(4)

$$\frac{\partial \omega }{\partial t}+\overline{u_j}\frac{\partial \omega }{\partial x_j}=\frac{\gamma }{{\nu }_t}{P}_k-\beta {\omega }^2+\frac{\partial }{\partial x_j}\left[\left(\nu +{\sigma }_{\omega }{\nu }_t\right)\frac{\partial \omega }{\partial x_j}\right]+2(1-F_1){\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$$

(5)

where the function $F_1$ is given by

$$F_1=\tanh \left\{{\left\{\min \left[\max \left(\frac{\sqrt{k}}{{\beta }^{\ast }\omega y_w},\frac{500\nu }{y_w^2\omega }\right),\frac{4\rho {\sigma }_{\omega 2}k}{C{D}_{k\omega }{y}_w^2}\right]\right\}}^4\right\}$$

(6)

with

$$C{D}_{k\omega }=\max \left\{2\rho {\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j},{10}^{-10}\right\}$$

(7)

and $y_w$ the distance to wall. The production of turbulent kinetic energy is given by:

$$P_k={\tau }_{ij}\frac{\partial \overline{u_i}}{\partial x_j}$$

(8)

$$\stackrel{\sim }{P_k}=\min \left(P_k,10{\beta }^{\ast }k\omega \right)$$

(9)

the turbulent eddy viscosity ${\nu }_t$ is given by:

$${\nu }_t=\frac{a_{1}k}{\max \left(a_1\omega ,S{F}_2\right)}$$

(10)

where $a_1=0.31$ is a constant and $S=2\sqrt{S_{ij}{S}_{ij}}$. The strain rate tensor is given by:

$$S_{ij}=\frac{1}{2}\left(\frac{\partial \overline{u_i}}{\partial x_j}+\frac{\partial \overline{u_j}}{\partial x_i}\right)$$

(11)

The function $F_2$ is given by:

$$F_2=\tanh \left\{{\left[\max \left(\frac{2\sqrt{k}}{{\beta }^{\ast }\omega y_w},\frac{500\nu }{y_w^2\omega }\right)\right]}^2\right\}$$

(12)

All of the above coefficients were calculated by the corresponding constant mixing function, $c$, which is expressed by:

$$c=c_1{F}_1+c_2(1-F_1)$$

(13)

The coefficients in the above model are listed in

Table 1. Constants for the k−ω SST model.

Symbol	${\mathit{a}}_1$	${\mathit{\beta }}^{\ast }$	${\mathit{\gamma }}_1$	${\mathit{\beta }}_1$	${\mathit{\gamma }}_2$	${\mathit{\beta }}_2$	${\mathit{\sigma }}_{\mathit{k}1}$	${\mathit{\sigma }}_{\mathit{\omega }1}$	${\mathit{\sigma }}_{\mathit{k}2}$	${\mathit{\sigma }}_{\mathit{\omega }2}$
Value	0.31	0.09	0.555	0.075	0.44	0.0828	0.85	0.5	1.0	0.856

.

2.3. VOF Equation

For the water–air flow simulations, the method of VOF (Volume of Fluid) is applied to capture the free-surface. The transport equation is:

$$\frac{\partial \alpha }{\partial t}+\frac{\partial (\alpha U_i)}{\partial x_i}=0$$

(14)

where the fraction function $\alpha$ represents that a computational cell with $\alpha =0$ is full of air, if $\alpha =1$ it is full of water, and if $0<\alpha <1$, the cell locates at the interface between water and air (Hirt and Nichols [19], 1981). The fluid density and viscosity are computed by:

$$\rho ={\rho }_{water}\alpha +{\rho }_{air}\left(1-\alpha \right)$$

(15)

$$\nu ={\nu }_{water}\alpha +{\nu }_{air}\left(1-\alpha \right)$$

(16)

2.4. Body-Force Propeller

In this study, a very simple body-force model was used to approximate the effects of a real rotating propeller on the flow. The term $f_i$ in the RANS equations is not zero inside the region of body force. The distributions of the axial and tangential components of body force on the propeller disk are prescribed by (e.g., Stern et al. [20], 1988).

$$f_b{}_x=A_x{R}^{\ast }\sqrt{1-R^{\ast }}$$

(17)

$$f_b{}_{\theta }=A_{\theta }\frac{R^{\ast }\sqrt{1-R^{\ast }}}{R^{\ast }(1-R_h^{\prime })+R_h^{\prime }}$$

(18)

where

$$R^{\ast }=\frac{R^{\prime }-R_h^{\prime }}{1-R_h^{\prime }}$$

(19)

$$R^{\prime }=\frac{R}{R_P}$$

(20)

$$R_h^{\prime }=\frac{R_H}{R_P}$$

(21)

$$A_x=\frac{105}{8}\frac{T}{\pi {\delta }_P(3R_H+4R_P)(R_P-R_H)}$$

(22)

$$A_{\theta }=\frac{105}{8}\frac{Q}{\pi {\delta }_P{R}_P(3R_H+4R_P)(R_P-R_H)}$$

(23)

with $T$ thrust and $Q$ torque of the propeller, $R_P$ propeller radius, $R_H$ hub radius, ${\delta }_P$ thickness, and $R$ radius of propeller disk. The body force propeller is limited inside a cylinder with the volume defined by $\pi (R_P^2-R_H^2){\delta }_P$. Once $T$ and $Q$ are known, the body force distribution at the propeller disk can be calculated by Equations (17) and (18). In the present application, the $T$ and $Q$ were estimated by making use of the open water curves of propeller. A program module was also inserted into the OpenFOAM to compute the body force distributions in our previous work (Yao et al. [17], 2021).

2.5. Numerical Discretization

The RANS solver in OpenFOAM, based on a finite volume method, provides many numerical discretization schemes. In the present applications, steady-state computations were performed for double-body flow, and the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm was used to correct the pressure. The gradient term was calculated by Gaussian integral, and the CDS (central difference scheme) was used to calculate the interpolation from the body centre to the face centre of the CV (control volume). The convection scheme was also based on the Gaussian integral, and the second-order UDS (upwind difference scheme) was adopted. For the velocity gradient, the same limiter was applied in different directions, which is based on the direction with the largest change of gradient. Therefore, the effect of the limiter will be very strong and stable, but at the expense of accuracy. For the convection of $k$ and $\omega$, in which the second-order UDS, based on the Gaussian integral, was adopted, the spatial derivative term was included to improve the convergence and stability, and the gradient was completely limited by the body centre gradient. When calculating the surface normal gradient, a sub relaxation factor was applied to the orthogonal calculation part of implicit discrete and the non-orthogonal part of explicit discrete to increase the diagonal dominance. For the discretization of the diffusion term, the CDS was adopted, and the non-orthogonal correction was also added due to its call of the surface normal phase gradient.

For the water–air flow simulation, PIMPLE was used to calculate the coupling function of velocity and pressure, which combines PISO (Pressure Implicit with Splitting of Operators) and SIMPLE. The implicit Euler method was used for the discretization of the time term to ensure numerical stability. For the discretization of the gradient term, the same scheme as the double-body flow simulation was adopted. The convection term was also calculated based on the Gaussian integral. For the approximation of the velocity convection, the second-order UDS was adopted, but the limiter mentioned above was removed when specifying the gradient direction. The convection scheme of the scalar field was similar to that of the velocity field. However, for the scalar fields, boundedness is more demanding than accuracy. So, the VanLeer scheme was adopted to approximate the convection term of the phase volume fraction.

2.6. Computational Domain and Boundary Conditions

There are two computational domains in the present consideration, one for the double-body flow simulation and one for the water–air flow simulation. Each domain is limited by a box. The range of the former is $-2.5\le x/L_{pp}\le 2.0$, $-1.5\le y/L_{pp}\le 1.5$, $0\le z/L_{pp}\le 1.0$, while it is $-2.5\le x/L_{pp}\le 2.0$, $-1.5\le y/L_{pp}\le 1.5$, $-1.0\le z/L_{pp}\le 1.0$ for the latter, as shown in Figure 1a,b. Note that the underwater part of the domain for the water–air flow simulation is identical to the domain for the double-body flow simulation.

For both simulations of double-body flow and water–air flow, on the four side boundaries, i.e., the boundaries at $x=2L_{pp}$, $x=-2.5L_{pp}$, and $y=\pm 1.5L_{pp}$, if the fluid enters the domain, the Dirichlet boundary condition was imposed, while if the fluid exits the domain, the Newman boundary condition was imposed.

For the Dirichlet boundary condition, the free-stream flow velocity relative to the manoeuvring ship was specified by:

$$U_f=-u_0\cos \beta +ry$$

(24)

$$V_f=u_0\sin \beta -ry$$

(25)

$$W_f=0$$

(26)

The free-stream values for $k$ and $\omega$ were estimated by:

$$k_{in}=\frac{3}{2}{(I_t{U}_{in})}^2$$

(27)

$${\omega }_{in}=\frac{k_{in}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{C_{\mu }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{L}_s}$$

(28)

where $I_t$ (here given to 5%) is the turbulent intensity as a fraction of mean flow velocity $U_{in}$, $C_{\mu }=0.09$ is a constant and $L_s$ is the turbulent length scale often close to cell size.

For the Newman boundary condition, the pressure was set to zero, and the constant gradients of the flow velocity, $k$ and $\omega$, were given as $k_{in}$ and ${\omega }_{in}$, separately. The specific calculated values of them were determined by the relevant parameters in the specific case.

For both of the simulations of the double-body flow and water–air flow, the boundary at water bottom, i.e., the boundary at $z=1.0L_{pp}$, was considered as a slip boundary. For the water–air flow simulation, the air boundary at $z=-1.0L_{pp}$ was also considered as a slip boundary. However, for the double-body flow simulation, because the free-surface was assumed to be a rigid plane at $z=0$, a symmetric boundary condition was applied.

The no-slip boundary condition was enforced on walls, i.e., hull and rudder. For this condition, the flow velocity was set as zero, and the normal gradient for the pressure was set as zero too (because the wall flux is zero). Since there is no turbulence on walls, $k$ is set as zero. Because wall functions are used in present simulations, ${\omega }_{wall}$ was estimated by ${\mu }_{\tau }/C_{\mu }^{\frac{1}{4}}\kappa y_w$, where ${\mu }_{\tau }=\sqrt{{\tau }_w/\rho }$ is the fraction velocity, ${\tau }_w$ is the wall shear stress, $\kappa =0.41$ is the Kármán constant and $y_w$ is the distance to walls.

3. Ship Data and Computational Cases

In the present consideration, all RANS simulations for both double-body flow and water–air flow were performed for a KCS model. Although the rudder geometry was covered during the simulations, it was not deflected. In order to compare the computed results with the available experimental data from SIMMAN, the KCS model with a scale ratio of 52.667 was selected. The principal particulars are listed in

Table 2. Principal particulars of KCS.

Items	Full Scale	Model for CFD
Hull
Lpp (m)	230	4.3671
Lwl (m)	232.5	4.4141
Bwl (m)	32.2	0.6114
D (m)	19	0.4500
T (m)	10.8	0.2051
Displacement (m3)	52.030	0.3562
CB	0.651	0.651
CM	0.985	0.984
LCB (%), fwd+	−1.48	−1.48
Rudder
S of rudder (m2)	115	0.0415
Lat. area (m2)	54.45	0.0196
Propeller
Type	FP	FP
No. of blades	5	5
D (m)	7.9	0.150
P/D (0.7R)	0.997	1.000
Ae/A0	0.8	0.700
Rotation	Right hand	Right hand
Hub ratio	0.18	0.227

. The numerical setups were in accordance with the experimental conditions reported in SIMMAN. Three ship speeds and eleven drift angles were considered for the static drift motion, where the drift angle was defined by $\beta ={\tan }^{-1}(-v/u)$. For the static circle motion, one ship speed and seven dimensionless turning rates, defined by $r\prime =r(L_{pp}/U)$, were taken into account. The computational cases are summarized in

Table 3. Computational cases.

Motions	$\mathit{F}\mathit{r}$	$\mathit{\beta }$ [deg]	${\mathit{r}}^{\prime }$
Static drift	0.156, 0.201, 0.260	0, ±4, ±8, ±12, ±16, ±20	0
Static circle	0.201	0	0, ±0.2, ±0.4, ±0.6

.

4. Results and Discussion

4.1. Grid Generation

The software HEXPRESS was used to generate the grids for the present simulations. Figure 2 illustrates two grids for the body-force flow simulation and water–air flow simulation. Although the grid for the water–air flow simulation is refined near the water surface (see Figure 2b compared with the grid for double-body flow simulation, the cell distributions around the hull and rudder are similar (see Figure 2c,d). The surface meshes on the hull and rudder are very similar as well. This is beneficial for a comparison between the computed results obtained by the body-force flow simulation and water–air flow simulation.

Note that wall functions are applied to model the flow in the boundary layer. Thus, the cells adjacent to walls have to locate in the log layer, i.e., the dimensionless distance between the hull and rudder, $y^{+}$, should range for each cell from 30 to 100, as recommended by the Resistance and Propulsion Committee of ITTC (International Towing Tank Conference). So that, when generating the grids in the present application, the sizes of the cells adjacent to the hull and rudder were adjusted to satisfy the use condition of the wall functions.

4.2. Grid Independence Analysis

To ensure numerical accuracy, grid independence analysis was conducted first, according to the procedure proposed by Stern et al. [21] and Wilson et al. [22]. Three grids for both the double-body flow simulation and the water–air flow simulation were generated by systematically decreasing the cell size. The decrease ratio is around $\sqrt{2}$. The cell numbers and mean values of $y^{+}$ for both are listed in

Table 4. Cell numbers (million).

Grid	Double-Body Flow Simulation	Mean ${\mathit{y}}^{+}$	Water–Air Flow Simulation	Mean ${\mathit{y}}^{+}$
Coarse	0.34	70.21	0.62	68.48
Medium	0.66	58.32	1.39	52.68
Fine	1.37	41.67	2.82	40.52

. It can be seen that the cell number roughly becomes twice that of the decreasing cell size.

The refinement ratio $r_G$, which equals to $\sqrt{2}$ here, is expressed by:

$$r_G=\frac{\Delta x_1}{\Delta x_2}=\frac{\Delta x_2}{\Delta x_3}$$

(29)

where $\Delta x_1$, $\Delta x_2$, and $\Delta x_3$ represent the cell sizes of the coarse, medium, and fine grids, respectively. The convergence ratio $R_G$ is defined by:

$$R_G=\frac{{\epsilon }_{23}}{{\epsilon }_{12}}=\frac{S_2-S_3}{S_1-S_2}=\left\{\begin{array}{l}{R}_G<0 :Oscillatory convergence\\ 0<R_G<1:Monotoni cconvergence\\ R_G>1 :Divergence\end{array}\right.$$

(30)

In the case of divergence, i.e., $R_G>1$, the grid uncertainty cannot be estimated.

In the case of oscillatory convergence, i.e., $R_G<0$, the grid uncertainty $U_G$ is estimated by:

$$U_G=\left|\frac{1}{2}(S_U-S_L)\right|$$

(31)

where $S_U$ and $S_L$ are the computed maximum and minimum values, respectively.

In the case of monotonic convergence, i.e., $0<R_G<1$, the generalized RE (Richardson extrapolation) is used to estimate the order of accuracy $P_G$ and the grid error ${\delta }_{R{E}_G}^{\ast (1)}$, which are expressed by:

$$P_G=\frac{\ln {\epsilon }_{12}-\ln {\epsilon }_{23}}{\ln r_G}$$

(32)

$${\delta }_{R{E}_G}^{\ast (1)}=\frac{{\epsilon }_{23}}{r_G^{P_G}-1}$$

(33)

Once $P_G$ and ${\delta }_{R{E}_G}^{\ast (1)}$ are known, it is possible to estimate the grid uncertainty. There are two ways to estimate the grid uncertainty depending on whether the computed values are close to the asymptotic range or not. The correction factor $C_G$ is defined by:

$$C_G=\frac{r_G^{P_G}-1}{r_G^{P_{Gest}}-1}$$

(34)

where $P_{Gest}$ is the limiting or theoretical accuracy of the applied numerical method. If $C_G$ is close to unity and has confidence, the computed values are close to the asymptotic range. Then, the sign of the grid error is known, so the numerical error ${\delta }_{SN}^{\ast }$, benchmark $S_C$, and uncertainty $U_{GC}$ can be calculated by:

$${\delta }_{SN}^{\ast }=C_G{\delta }_{R{E}_G}^{\ast (1)}$$

(35)

$$S_C=S-{\delta }_{SN}^{\ast }$$

(36)

$$U_{GC}=\left\{\begin{array}{l}\left(2.4{\left(1-C_G\right)}^2+0.1\right)\left|{\delta }_{R{E}_G}^{\ast }\right|,\left|1-C_G\right|<0.125\\ \left|1-C_G\right|\left|{\delta }_{R{E}_G}^{\ast }\right| ,\left|1-C_G\right|\ge 0.125\end{array}\right.$$

(37)

If $C_G$ is far from unity and lacking confidence, only the numerical uncertainty $U_G$ can be calculated by:

$$U_G=\left\{\begin{array}{l}9.6{\left(1-C_G\right)}^2+1.1\left|{\delta }_{R{E}_G}^{\ast }\right|,\left|1-C_G\right|<0.125\\ \left(2\left|1-C_G\right|+1.1\right)+\left|{\delta }_{R{E}_G}^{\ast }\right|,\left|1-C_G\right|\ge 0.125\end{array}\right.$$

(38)

Grid independence analysis is first carried out for the case of straight-ahead motion. The Froude number is 0.260. During the simulation for each grid, the propeller revolution is adjusted to produce a thrust that can balance ship resistance. The computed longitudinal forces acting on the hull and rudder $X^{\prime }{}_{H+R}$ on the hull and rudder are listed in

Table 5. Grid independence analysis for straight-ahead motion.

Grid	Double-Body Flow Simulation	Diff. (%)	Water–Air Flow Simulation	Diff. (%)
Coarse	−0.0163	-	−0.0168	-
Medium	−0.0150	−7.98	−0.0172	2.38
Fine	−0.0151	0.67	−0.0173	0.58

.

Grid independence analysis was also carried out for the case of static drift motion with a drift angle of −12 deg. The Froude number is 0.201. For the static drift motion simulation, the propeller revolution for each grid was equal to the revolution at the self-propulsion point, which was determined by adjusting the revolution to achieve a balance between the ship resistance and propeller thrust for the ship in straight-ahead motion. The computed longitudinal forces, sway forces, and yaw moments acting on the hull and rudder are listed in

Table 6. Grid independence analysis for static drift motion by double-body flow simulation.

Grid	${\mathit{X}}^{\prime }{}_{\mathit{H}+\mathit{R}}$	Diff. (%)	${\mathit{Y}}^{\prime }$	Diff. (%)	${\mathit{N}}^{\prime }$	Diff. (%)
Coarse	−0.0188	-	−0.0736	-	−0.0211	-
Medium	−0.0173	−7.98	−0.0699	−5.03	−0.0206	−2.37
Fine	−0.0175	1.16	−0.0690	1.29	−0.0205	−0.49

and

Table 7. Grid independence analysis for static drift motion by water–air flow simulation.

Grid	${\mathit{X}}^{\prime }{}_{\mathit{H}+\mathit{R}}$	Diff. (%)	${\mathit{Y}}^{\prime }$	Diff. (%)	${\mathit{N}}^{\prime }$	Diff. (%)
Coarse	−0.0195	-	−0.0702	-	−0.0215	-
Medium	−0.0219	12.3	−0.0721	2.71	−0.0222	3.26
Fine	−0.0223	1.83	−0.0725	0.55	−0.0225	1.57

, respectively. In the independent study, the theoretical order of accuracy $P_{Gest}=2$ was adopted according to the spatial discretization scheme.

The force or moment shown in Table 5, Table 6 and Table 7 have been made non-dimensional by:

$${Force}^{\prime }=\frac{Force}{\frac{1}{2}\rho U^2{L}_{PP}T}$$

(39)

$${Moment}^{\prime }=\frac{Moment}{\frac{1}{2}\rho U^2{L}_{PP}^{2}T}$$

(40)

It can be seen from Table 5, Table 6 and Table 7 that the computed forces or moments become closer as increasing grid resolution. The grid uncertainty was also analyzed for the case of drift motion, as shown in

Table 8. Grid uncertainty analysis for static drift motion by double-body flow simulation. ${\delta }_{R{E}_G}^{\ast (1)}$, $U_{GC}$ and $U_G$ are % $S_3$.

Quantity	${\mathit{R}}_{\mathit{G}}$	${\mathit{P}}_{\mathit{G}}$	${\mathit{C}}_{\mathit{G}}$	${\mathit{\delta }}_{\mathit{R}{\mathit{E}}_{\mathit{G}}}^{\ast (1)}$	${\mathit{U}}_{\mathit{G}\mathit{C}}$	${\mathit{S}}_{\mathit{C}}$	${\mathit{U}}_{\mathit{G}}$	Convergence Type
$X^{\prime }{}_{H+R}$	−0.13	-	-	-	-	-	−4.29	Oscillatory
$Y^{\prime }$	0.24	4.08	3.11	0.42	0.89	−0.0681	−3.33	Monotonic
$N^{\prime }$	0.20	4.60	3.93	0.12	0.36	−0.0204	−1.45	Monotonic

and

Table 9. Grid uncertainty analysis for static drift motion by water–air flow simulation. ${\delta }_{R{E}_G}^{\ast (1)}$, $U_{GC}$ and $U_G$ are % $S_3$.

Quantity	${\mathit{R}}_{\mathit{G}}$	${\mathit{P}}_{\mathit{G}}$	${\mathit{C}}_{\mathit{G}}$	${\mathit{\delta }}_{\mathit{R}{\mathit{E}}_{\mathit{G}}}^{\ast (1)}$	${\mathit{U}}_{\mathit{G}\mathit{C}}$	${\mathit{S}}_{\mathit{C}}$	${\mathit{U}}_{\mathit{G}}$	Convergence Type
$X^{\prime }{}_{H+R}$	0.16	5.22	5.11	−0.35	−1.44	−0.0227	−6.37	Monotonic
$Y^{\prime }$	0.21	4.50	3.75	−0.14	−0.40	−0.0729	−1.59	Monotonic
$N^{\prime }$	0.50	2.02	1.01	−1.53	−0.02	−0.0228	−2.32	Monotonic

. For the simulation of double-body flow, the grid study shows monotonic convergence for the sway force on hull and rudder with $R_G$ of 0.24 and yaw moment on hull and rudder with $R_G$ of 0.20, respectively. The longitudinal force shows oscillatory convergence with $R_G$ of −0.13, and the order of accuracy for them cannot be estimated. For the simulation of double-body flow, the grid study shows monotonic convergence for longitudinal forces, sway forces, and yaw moments acting on the hull with $R_G$ of 0.16, 0.21, and 0.50, respectively.

As shown in Table 8 and Table 9, the $U_G$ of side force and yaw moment acting on the hull and rudder for both simulations of the double-body flow and water–air flow are below 5% $S_3$. The $U_G$ of the longitudinal force for double-body flow and water–air flow is −4.29% $S_3$ and −6.37% $S_3$, respectively. The ${\delta }_{R{E}_G}^{\ast (1)}$ for all variables is between −0.4% $S_3$ and 0.5% $S_3$. In addition, for the grid independence analysis for static drift motion by double-body flow simulation, the $P_G$ of sway force and the yaw moment are 4.08 and 4.60, respectively, suggesting that the solutions for them are far from the asymptotic range. Furthermore, the $P_G$ of sway force for static drift motion by water–air flow simulation, is 4.50, deviating from the theoretical order of accuracy $P_{Gest}$ obviously. Only the solution of the yaw moment for static drift motion by water–air flow simulation is close to the asymptotic range with $P_G=2.02$. Overall, the sway force and yaw moment show better grid convergence than the longitudinal force both in double-body flow simulation and water–air flow simulation. It indicates that the longitudinal force is more affected by the grid uncertainty.

According to the grid independence analysis above, the medium grids seem accurate enough. Thus, they were selected for case simulations, as listed in Table 3, i.e., there are about 0.62 million cells for double-body flow simulations and 1.39 million cells for water–air flow simulations. All these simulations are run on DELL workstations with 8 or 16 processors. Each double-body flow simulation case takes about 0.5 h of wall clock time, while each water–air flow simulation case takes about 10 h of wall clock time.

4.3. Simulation of Static Drift Motion

Figure 3 shows the comparisons of the computed longitudinal forces on the hull and rudder, side forces, and yaw moments with the available EFD (experimental fluid dynamics) data for the case of static drift motions.

Table 10. The error between the CFD computed results and EFD data for double-body flow simulation, $Fr$ = 0.156.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−16	−0.0193	−0.0214	9.74	−0.1105	−0.1107	0.21	−0.0271	−0.0300	9.66
−12	−0.0198	−0.0217	9.04	−0.0743	−0.0724	2.61	−0.0185	−0.0210	11.87
−8	−0.0200	−0.0226	11.22	−0.0422	−0.0411	2.61	−0.0112	−0.0129	12.75
−4	−0.0203	−0.0223	8.87	−0.0176	−0.0159	10.18	−0.0055	−0.0071	23.27
0	−0.0202	−0.0227	10.68	-	-	-	-	-	-
4	−0.0207	−0.0239	13.27	0.0150	0.0188	20.16	0.0063	0.0048	30.73
8	−0.0208	−0.0244	15.05	0.0378	0.0416	9.22	0.0134	0.0115	16.01
12	−0.0206	−0.0245	15.78	0.0697	0.0741	5.98	0.0208	0.0187	11.19
16	−0.0204	−0.0252	19.00	0.1076	0.1143	5.80	0.0286	0.0265	7.88

,

Table 11. The error between the CFD computed results and EFD data for water–air flow simulation, $Fr$ = 0.156.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−16	−0.0247	−0.0214	15.10	−0.1080	−0.1107	2.41	−0.0318	−0.0300	5.75
−12	−0.0257	−0.0217	18.42	−0.0714	−0.0724	1.39	−0.0237	−0.0210	9.47
−8	−0.0243	−0.0226	7.44	−0.0399	−0.0411	3.02	−0.0145	−0.0129	12.98
−4	−0.0225	−0.0223	0.79	−0.0141	−0.0159	11.54	−0.0082	−0.0071	16.08
0	−0.0221	−0.0227	2.28	-	-	-	-	-	-
4	−0.0224	−0.0239	6.37	0.0152	0.0188	18.78	0.0062	0.0048	28.77
8	−0.0242	−0.0244	1.00	0.0402	0.0416	3.41	0.0130	0.0115	12.34
12	−0.0257	−0.0245	4.73	0.0762	0.0741	2.79	0.0204	0.0187	9.51
16	−0.0254	−0.0252	0.94	0.1165	0.1143	1.99	0.0277	0.0265	4.63

,

Table 12. The error between the CFD computed results and EFD data for double-body flow simulation, $Fr$ = 0.201.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−12	−0.0173	−0.0205	15.46	−0.0699	−0.0726	3.71	−0.0212	−0.0241	12.31
−8	−0.0175	−0.0202	13.40	−0.0394	−0.0401	1.99	−0.0134	−0.0144	6.72
−4	−0.0173	−0.0193	10.39	−0.0163	−0.0155	5.25	−0.0060	−0.0073	17.66
0	−0.0172	−0.0190	9.34	-	-	-	-	-	-
4	−0.0175	−0.0202	13.15	0.0137	0.0170	19.32	0.0067	0.0056	18.94
8	−0.0180	−0.0213	15.39	0.0362	0.0399	9.27	0.0140	0.0126	11.59
12	−0.0178	−0.0227	21.43	0.0679	0.0741	7.97	0.0216	0.0187	2.69

,

Table 13. The error between the CFD computed results and EFD data for water–air flow simulation, $Fr$ = 0.201.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−12	−0.0206	−0.0205	0.59	−0.0747	−0.0726	2.91	−0.0226	−0.0241	6.50
−8	−0.0191	−0.0202	5.43	−0.0428	−0.0401	6.70	−0.0146	−0.0144	1.83
−4	−0.0184	−0.0193	4.50	−0.0171	−0.0155	9.95	−0.0069	−0.0073	5.81
0	−0.0180	−0.0190	5.22	-	-	-	-	-	-
4	−0.0189	−0.0202	6.37	0.0188	0.0170	10.61	0.0058	0.0056	3.73
8	−0.0200	−0.0213	5.88	0.0427	0.0399	6.82	0.0140	0.0126	10.93
12	−0.0220	−0.0227	2.87	0.0791	0.0741	7.17	0.0204	0.0187	2.73

,

Table 14. The error between the CFD computed results and EFD data for double-body flow simulation, $Fr$ = 0.260.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−8	−0.0153	−0.0193	20.57	−0.0362	−0.0413	12.39	−0.0140	−0.0169	17.67
−4	−0.0151	−0.0182	17.37	−0.0147	−0.0177	16.92	−0.0068	−0.0083	18.11
0	−0.0149	−0.0179	16.83	-	-	-	-	-	-
4	−0.0153	−0.0190	19.89	0.0128	0.0170	24.51	0.0077	0.0068	12.26
8	−0.0156	−0.0206	24.29	0.0349	0.0406	14.21	0.0146	0.0150	2.73

and

Table 15. The error between the CFD computed results and EFD data for water–air flow simulation, $Fr$ = 0.260.

$\mathit{\beta }$ (°)	${\mathit{X}}_{\mathit{H}+\mathit{R}}^{\prime }$			${\mathit{Y}}^{\prime }$			${\mathit{N}}^{\prime }$
	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$	CFD	EFD	$\mathit{E}$
−8	−0.0191	−0.0193	1.23	−0.0409	−0.0413	1.00	−0.0161	−0.0169	5.19
−4	−0.0173	−0.0182	4.94	−0.0160	−0.0177	9.16	−0.0082	−0.0083	1.15
0	−0.0167	−0.0179	6.53	-	-	-	-	-	-
4	−0.0171	−0.0190	10.39	0.0162	0.0170	4.45	0.0081	0.0068	18.57
8	−0.0189	−0.0206	8.12	0.0409	0.0406	0.58	0.0160	0.0150	6.82

show the error between the CFD computed results and EFD data for double-body flow simulation and water–air flow simulation, relatively. The EFD data are published by FORCE Technology in the workshop SIMMAN 2014 [23].

Here, several statistics are introduced to evaluate the error of the calculated value. Once the computed and experimental values are known, the relative error $E$ between CFD and EFD can be calculated by the following formula:

$$E=\left|\frac{EFD-CFD}{EFD}\right|\times 100\%$$

(41)

The average relative error $\overline{E}$ and the standard deviation $\sigma$ can be calculated by:

$$\overline{E}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{E}_i}$$

(42)

$$\sigma =\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(E_i-\overline{E})}^2}}$$

(43)

where $n$ is the number of samples.

In the prediction of $X^{\prime }{}_{H+R}$, it can be seen from Figure 3a,d,g that the longitudinal force by water–air flow simulation shows the same trend as the experimental data, while the longitudinal force by the double-body flow simulation does not display the same trend. It can be seen from Figure 3a that even if the longitudinal forces by the water–air flow simulations show differences from EFD at negative drift angles when $Fr$ = 0.156, the computed longitudinal forces are much closer to the experimental value at the positive drift angles. Moreover, the $\overline{E}$ for the longitudinal force by the water–air flow simulation is about 6.34%, and the $\sigma$ is about 0.06. For $Fr=0.201$ and $Fr=0.260$, the $\overline{E}$ is about 4.41% and 6.24%, the $\sigma$ is about 0.02 and 0.03, respectively.

In the prediction of $Y^{\prime }$, it can be seen from Figure 3b,e,h that, in general, the side forces by the double-body flow simulations and water–air flow simulations are both acceptable; however, the side forces by the water–air flow simulations are closer to the experimental values, especially at a higher ship speed. As shown in Table 10 and Table 11, when $Fr=0.156$, the $\overline{E}$ for side force by double-body flow simulation is about 6.31%, and it is about 5.03% by the water–air flow simulation, while the difference is not obvious. As shown in Table 12 and Table 13, when $Fr=0.201$, the $\overline{E}$ for sway force by double-body flow simulation is about 6.79%, and it is about 6.31% by the water–air flow simulation, while the $\sigma$ for them is 0.06 and 0.04, respectively.

The advantage in the accuracy of the water–air flow simulation becomes more and more obvious with the increase in speed and drift angle. As shown in Figure 3h and Table 14, when $Fr=0.260$, the $\overline{E}$ for side force by double-body flow simulation is about 13.61%, and the $\sigma$ is about 0.09. However, the $\overline{E}$ for the side sway force by the water–air flow simulation is about only 3.04%, and the $\sigma$ is about 0.04, as can be seen from Table 15.

In the prediction of $N^{\prime }$, it can be seen from Figure 3c,f,i that, similarly to the side force, yaw moments by double-body flow simulations and water–air flow simulations both show good agreement with the experimental data. Table 14 and Table 15 indicate that when $Fr=0.260$ the $\overline{E}$ for the yaw moment by double-body flow simulation is 10.15%, and the $\sigma$ is about 0.08. The $\overline{E}$ for the yaw moment by the water–air flow simulation is only about 6.35%, and the $\sigma$ is about 0.07. It indicates that the simulation based on water–air flow is more accurate than that based on double-body flow in the prediction of yaw moment.

Figure 4 shows the comparison of static drift results by the water–air flow simulation at different speeds. It can be seen from Figure 4a that the dimensionless longitudinal forces are quite different at different speeds, and the difference decreases with the increase in speed and drift angle. However, the three curves of $Y^{\prime }$ and $N^{\prime }$ are quite close. This means that the side force and yaw moment are almost proportional to the square of ship speed, but the longitudinal force is not.

Figure 5 shows the comparison of the dynamic pressure distribution on the hull and rudder by the simulations of double-body flow and water–air flow under several typical cases. As can be seen from Figure 5a, when $Fr=0.156$ and $\beta =16$ deg, the dynamic pressure distribution on the hull surfaces between the double-body flow and water–air flow simulations are similar.

As shown in Figure 5b, when $Fr=0.201$ and $\beta =16$ deg, there is a larger positive pressure zone at the fore shoulder of the leeward (left) surface for the result of the water–air flow simulation than that of double-body flow simulation. At the bow of the windward (right) surface, a small part of the positive pressure zone above the waterline can be observed in the result of the water–air flow simulation. In other parts, there is little difference between the pressure distributions. Perhaps it is this “offset” phenomenon of the pressure on the front of the ship’s windward and leeward surfaces that makes the predictions of sway force and yaw moments very close.

Comparing Figure 5a–d, the following conclusions can be drawn: (1) For the results of double-body flow simulations, with the increase in ship speed and drift angle, the distribution form of positive and negative pressure on the windward and leeward sides of the hull changes little, but increases in the area and magnitude, and this increase is more obvious with the increase in ship speed. (2) For the results of the water–air flow simulations, with the increase in ship speed and drift angle, the distribution form of positive and negative pressure on the windward, and the leeward surfaces of the hull in the underwater part change little but increase in the area and magnitude, and this increase is more obvious with the increase in ship speed. (3) For the results of the water–air flow simulations, with the increase in the ship speed and drift angle, the area and magnitude of the positive pressure above the waterline on the windward side of the bow increase. (4) At the same ship speed and drift angle, the simulation results of the double-body flow and water–air flow are almost the same, except that the pressures on both sides of the bow are different, which leads to the similarities in the prediction of sway forces and yaw moments. In general, the predictions of hull surface dynamic pressure distribution are consistent with the results of force and moment.

Figure 6 shows the cross-sections of the flow field for the case corresponding to Figure 5. The cross-sections are coloured by vorticity magnitude, and the hull is coloured by dynamic pressure. As is shown in Figure 6, in general, for the results of both simulations based on double-body flow and water–air flow, the section vortices due to flow separation are already evident at drift angles of 16 and 20 degrees. Moreover, with an increasing drift angle, this separation moves towards the bow and central longitudinal section of the ship hull, which is consistent with the study by Zhang et al. [24].

Figure 7 shows the streamlines around the hull for $Fr$ = 0.201 and $\beta$ = 16 deg. It can be seen in Figure 7b that, for the simulation of water–air flow, the streamlines near the ship bow of the windward rise in the vertical direction, which means that there is an area of positive pressure due to the rise of the free-surface. The streamlines swirl near the ship bow of the leeward and then rise in the vertical direction near the fore shoulder of the leeward, as observed in Figure 7b. However, for the simulation based on the double-body flow, as observed in Figure 7a, the streamlines near the windward bow do not rise in the vertical direction. Furthermore, the vortex of the streamlines near the leeward bow is less than that of the simulation based on water–air flow. Figure 7c,d show similar streamlines distributions near the side of the leeward. In addition, the streamlines swirl near the stern of the leeward in Figure 7d, which indicates that there is a negative pressure area. Those phenomena in Figure 7 confirm the results shown in Figure 5 and Figure 6.

Specifically, when $Fr=0.156$ and $\beta =16$ deg, there is little difference between the section vortex results for the double-body flow simulation and the two-phase flow simulation. As the ship speed increases, i.e., when $Fr=0.201$ and $\beta =16$ deg, the difference between the section vortex results of the two flow models in the front half of the hull begins to show, but the difference is not significant. Moreover, the section vortices generated in the alt half of the hull are still similar.

When $Fr=0.201$ and $\beta =20$ deg, the difference between the simulation results based on double-body flow and water–air flow does not further increase compared to the results for $Fr=0.201$ and $\beta =16$ deg. With the further increase in the Froude number, i.e., when $Fr=0.260$ and $\beta =20$ deg, the difference between the section vortex results of the double-body flow simulation and the water–air flow simulation in the front half of the hull is already obvious; that is, the magnitude value and area of the former are smaller than those of the latter. Furthermore, the difference in the section vortices generated in the alt half of the hull cannot be ignored.

The phenomena shown in Figure 6 are also a further explanation of the relevant conclusions mentioned above. That is to say, the similarity in section vortex at $Fr=0.156$ and $Fr=0.201$ for simulations based on double-body flow and water–air flow can explain the similarity in their lateral force and yaw moment results. The difference in the section vortex at $Fr=0.260$ can also explain the difference in their lateral force and yaw moment results.

Figure 8 shows the free-surface for the simulations of water–air flow, which is coloured by the value of the z-coordinate. The Kelvin wave patterns are well captured after the hull. The bow and stern waves are both very steep, and gird refinements are recommended in these regions to resolve possible breaking. Compared with the case of $Fr$ = 0.156, the wave patterns on the free-surface are larger when $Fr$ = 0.260. With the increase in drift angle, the asymmetry of the wave pattern around the hull is more obvious, but its range is not expanded. It can be concluded that the ship’s speed has a great influence on the free-surface, while the drift angle has a smaller influence.

4.4. Static Circle Motion

Figure 9 shows the computed longitudinal forces, side forces, and yaw moments on the hull and rudder for the static circle motion in comparison with the available EFD data. Unfortunately, there is no publicly available data for the KCS model with a scale ratio of 52.667 in the static circle motion. There are experimental data with a scale ratio of 75.5 from SIMMAN2014, NMRI (National Maritime Research Institute, Tokyo, Japan). Thus, the experimental data are used for the present validation.

As can be seen from Figure 9a, when $Fr$ = 0.201, the trend of the longitudinal force by the water–air flow simulation is the same as that displayed by the EFD data. The $\overline{E}$ is 8.21% and $\sigma$ is 0.03. Regrettably, similar to the static drift motion, the longitudinal force by the double-body flow simulation also fails to reflect the trend of the EFD data of a static circular motion. As can be seen from Figure 9b,c, when $Fr$ = 0.201, the computed side forces and yaw moments for the simulations based on double-body flow and water–air flow are both very close to EFD, except for the sway force at the maximum positive $r^{\prime }$.

It should be noted that the side force of EFD shows obvious asymmetry for the maximum positive and negative turning rate. It is worth mentioning that when $Fr$ = 0.201 and $r^{\prime }$ = −0.6, the $E$ for the computed side force of the simulation based on water–air flow is only 1.19%, and that of the simulation based on double-body flow is about 10%.

In the prediction of $Y^{\prime }$, the $\overline{E}$ for the computed sway force by the simulations based on double-body flow is 12.67% with a $\sigma$ of 0.11. Furthermore, the $\overline{E}$ for the computed sway force of simulations based on water–air flow is −7.58%, with a standard deviation of 0.07. In the prediction of $N^{\prime }$, they are 5.41%, 0.06, 5.40%, and 0.05, relatively.

Figure 10 shows the dynamic pressure distribution on the hull and rudder surface for the simulations of the two flow models in static circle motions. Figure 11 shows the free-surface for the water–air flow simulations, which is coloured by the z-coordinate value. As is shown in Figure 10, there is a positive pressure region on the bow of the windward surface and a negative pressure region on the stern. At the leeward surface, there is a negative pressure area on the bow and a positive pressure region on the front shoulder. In Figure 10a,b, a positive area above the water line can be observed on the windward surface for the water–air simulation. It also can be observed that there is a more obvious positive region in the fore shoulder of the leeward surface of the water–air simulation results compared to the double-body flow simulation. It can be seen from Figure 11 that with the increase in turning rate, there is a more obvious form of circular motion and corresponding dynamic pressure distribution.

Figure 12 shows that the streamlines are arc-shaped around the hull for a static circular motion. For the simulations of water–air flow, the streamlines near the ship bow of the windward rise in the vertical direction. Furthermore, the streamlines swirl near the ship bow of the leeward, then rise in the vertical direction near the fore shoulder of the leeward. However, it is ignored in the double-body flow simulation. For both simulations, compared with the streamlines at $r^{\prime }$ = −0.2, there is an obvious vortex near the bow of the leeward at $r^{\prime }$ = −0.6. With the increases in $r^{\prime }$, the swirls of the streamlines are more complex around the bow and the bilge both for the simulations of the double-body flow and the water–air flow. It should be noted that for the simulations of water–air flow, there is also a more obvious streamline rising at the bow of the windward and the fore shoulder of leeward with the increase in $r^{\prime }$.

5. Conclusions

The differences between the RANS simulations of double-body flow and water–air flow around a container ship performing static drift motions and static circle motions were investigated in this paper. The results show that, as predicted, the calculation accuracy of a water–air flow simulation is higher than that of a double-body flow simulation in general. For double-body flow, although the prediction of longitudinal force is not accurate under the two motion forms, the predictions of side force and yaw moment are still satisfactory. Moreover, when the Froude number is 0.156 and 0.201, the predictions of side force and yaw moment based on a double-body flow simulation are close to those based on a water–air flow simulation. However, as the range of ship motion (ship speed, drift angle and ship rotation angular velocity) increases, the difference between the predicted results based on double-body flow simulation and water–air flow simulation becomes larger. Moreover, this difference is most sensitive to ship velocity, followed by drift angle, and finally rotation angular velocity. Fortunately, the predicted results from the water–air flow simulation indicate that the side force and yaw moment are both proportional to the square of the ship velocity in static drift motion. Perhaps, the predictions of side force and yaw moment with low Froude number based on double-body flow can be used to yield relative hydrodynamic derivatives.

To sum up, in this study, the average computational cost of double-body flow simulation is much less than that of a water–air flow simulation in each case. In terms of the reduced computing time, this paper may be helpful for the balance of efficiency and accuracy in the application of the RANS method. However, for static circle motion, only one ship speed is considered in this paper, and the verification of hydrodynamic derivatives will make it have a wider application space. Future work can be carried out to expand on these aspects.