Numerically Investigating the Effect of Trim on the Resistance of a Container Ship in Confined and Shallow Water

Abstract

One of the promising strategies for the improvement of the resistance characteristics and energy efficiency of a ship is trim optimization. Most of the studies conducted so far regarding trim optimization deal with unrestricted water. The effect of trim on the total resistance and its components for the KRISO Container Ship model in confined and shallow water is investigated using computational fluid dynamics. Numerical simulations of resistance tests with and without the free surface effects using two turbulence models are carried out for even keel and four trims in restricted water. A verification study is conducted for the total resistance, sinkage, and trim in terms of grid resolutions and time steps to assess the numerical uncertainty. The numerical results are validated against the experimental ones available in the literature. Performing the double body simulations enabled the analysis of the effect of trim on the resistance components. The numerical results pointed out that by adjusting the trim a reduction in the total resistance in confined and shallow water can be obtained.

1. Introduction

Due to global climate changes, one of the main requirements in the shipping industry is the reduction in fuel consumption and harmful gas emissions, aiming to decrease navigation costs and meet the requirements imposed by the International Maritime Organization (IMO) [1]. Most ships during their service enter restricted waterways, necessitating consideration of the effect of confined and shallow water. To meet the requirements for reduced energy consumption, IMO has proposed trim optimization as a solution, which does not require altering the hull form. Trim optimization can be achieved by ballasting or redistributing the load [2]. When optimizing trim, it is important to understand that the ship will have different total resistances at the same draft and speed for different trims during navigation. Trim optimization can be assessed by towing tank testing or using computational fluid dynamics (CFD).

Islam and Soares [3] conducted a study on the total resistance of the container ship for three values of the Froude number and draughts using Reynolds Averaged Navier–Stokes (RANS) simulations. The authors concluded that the optimal trim varies at different draughts and Froude numbers. Initially, they predicted the ship resistance on an even keel and then conducted an optimization procedure, demonstrating that by varying the trim, reduced total resistance of the ship and improved energy efficiency can be achieved. Sogihara et al. [4] investigated the trim optimization effect on the propulsion characteristics of the ship using model testing. Model tests were conducted for various trims, and the required power was evaluated to investigate the effect of trim on the propulsion characteristics in calm water. The alterations in ship propulsion characteristics concerning trim and draft have been studied, and the trim for which the required power can be reduced was identified. Sherbaz and Duan [5] investigated the influence of trim on the total resistance of the KRISO Container Ship (KCS) model and found that with a slight trim adjustment at the stern, the total resistance of the ship decreased by 2.29%. Sun et al. [6] applied a combination of experimental and numerical methods to investigate the total resistance of a 4250 TEU container ship. The authors conducted trim optimization with respect to the total resistance of the ship and concluded that there was a different optimal trim for each draught and speed. Shivachev et al. [7] predicted that the relative deviations of the total resistance of the KCS ship trimmed by bow at the slow steaming speed are larger for the ballast draught in comparison to the design draught. Le et al. [2] investigated the effect of trim and draught on the total resistance of the DTMB 5415 ship. The research results indicated that it is possible to achieve a reduction in total resistance of 1.5%, while a reduction of up to 8% can be achieved for pressure resistance. Tsugane et al. [8] investigated the possibility of fuel savings by employing model testing and field measurements and concluded that in certain cases, trim optimization can reduce fuel consumption. Tu et al. [9] employed experimental fluid dynamics (EFD) and CFD to assess the effect of trim on the total resistance of the ship in the model scale and demonstrated good agreement between the experimental and numerical results.

When a ship enters shallow and confined water, the total resistance of the ship increases, which is caused by the smaller under-keel clearance and limited width of a waterway. The flow is accelerated around the hull, which, according to Bernoulli’s principle, results in reduced pressure below and around the hull. These changes lead to additional sinkage of the ship, a decrease in the water level around the ship, and increased wave resistance [10]. Martić et al. [11] investigated the effect of shallow water on the total resistance of the solar catamaran by performing CFD simulations in deep and shallow water. The authors conducted a detailed analysis of the flow around the catamaran. The results showed that the total resistance of the catamaran increases due to shallow water and the wave pattern is altered. Du et al. [12] investigated numerically the total resistance and wave characteristics of a ship for inland navigation in a restricted waterway. The results of numerical simulations were validated against experimental data while considering all constraints of the restricted waterways.

The contemporary approaches for predicting a ship’s total resistance include experimental testing and numerical methods. Experimental tests conducted in towing tanks are time-consuming and expensive, but when combined with numerical methods, they provide a comprehensive understanding of the ship’s hydrodynamic characteristics. Numerical methods have proven to be more practical for ship design and optimization. CFD is a numerical method that offers a detailed insight into the flow around a ship’s hull compared to experimental testing. Within CFD, there are several approaches to solving the mathematical model, the most common being the RANS equations, which allow for solutions of satisfactory accuracy within a reasonable time frame [13]. Farkas et al. [14] demonstrated the importance of employing CFD to enhance extrapolation methods. The authors highlighted the advantages and limitations of extrapolation methods and compared results obtained using four turbulence models. Elsherbiny et al. [15] investigated the squat of the KCS ship model by gauging the ship’s sinkage and trim. They varied the width of the channel, thereby studying the blockage effect. Conducted studies have shown that the channel width has no significant impact on the squat of the ship. Hadi et al. [16] examined not only the influence of shallow water but also the channel width on ship resistance. The authors conducted numerical simulations using RANS equations and investigated several channels of different shapes and widths. The results showed that the narrower the channel, the greater the increase in total resistance, and that the total resistance depends on the channel shape. Prakash and Chandra [17] examined how shallow water affects the total resistance of the ship at different speeds, employing a RANS solver within the Fluent software package. The authors concluded that by using CFD, satisfactory predictions of the total resistance and wave patterns in shallow water can be achieved.

In this paper, the numerical simulations with and without the free surface effects were carried out using two turbulence models to assess the effect of trim on the total resistance and its components in shallow and confined water. The flow around the hull in shallow and confined water is significantly altered in comparison to the flow in unrestricted water, and the adjustment of trim could lead to a significant reduction in the total resistance and thus improvement of the ship’s energy efficiency. Numerical simulations are performed for an even keel and four trims at one Froude number. Numerical uncertainty for the total resistance, sinkage, and trim is assessed using different grid resolutions and time steps. The obtained numerical results are validated against the experimental data available in the literature. A detailed analysis of the wave patterns, distribution of hydrodynamic pressure and wall shear stress is presented.

The remainder of the paper is structured as follows: the case study is introduced in Section 2. Section 3 provides the mathematical model, while Section 4 outlines the numerical setup of the CFD simulations. The depiction and analysis of the results are presented in Section 5, while in Section 6, the conclusions drawn from the conducted study are given.

2. Case Study

The numerical simulations of the resistance tests for the KCS ship model in shallow and confined water were conducted for an even keel and four trims.

Table 1. Main features of the KCS container ship model.

Parameters	Symbol	Value
Scale	λ	75
Length between perpendiculars	$L_{PP}$	3.067 m
Breadth	$B$	0.429 m
Depth	$D$	0.25 m
Design draught	$T$	0.144 m
Block coefficient	$C_B$	0.651

shows the main features of the KCS ship model.

Figure 1 depicts a 3D model of the ship with a defined coordinate system within the computational domain. The origin is set at the intersection of the design waterline and the aft perpendicular. The positive direction of the x-axis is from stern to bow, the y-axis to the port side, and the z-axis vertically upwards. Negative values of trim refer to a ship trimmed by stern, while positive values indicate a ship trimmed by bow.

3. Mathematical Model

To describe turbulent fluid flow, a mathematical model based on RANS equations and the averaged continuity equation was utilized:

$${\displaystyle \frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho {\overline{u}}_i{\overline{u}}_j+\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)=-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{\partial {\overline{\tau }}_{ij}}{\partial x_j}}$$

(1)

$${\displaystyle \frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial x_i}}=0$$

(2)

where $\rho$ represents the fluid density, ${\overline{u}}_i$ denotes the averaged Cartesian components of the velocity vector, $\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ represents the Reynolds stress tensor, and $\overline{p}$ denotes the mean pressure. The mean viscous stress tensor reads as follows:

$${\overline{\tau }}_{ij}=\mu \left({\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_i}}\right)$$

(3)

where $\mu$ denotes the dynamic viscosity.

Equations (1) and (2) do not constitute a closed system of equations; thus, a turbulence model must be employed. In this study, the $k-\omega$ SST (SSTKO) and the realizable $k-\epsilon$ (RKE) turbulence models were selected. The RKE turbulence model consists of two transport equations, one for the specific turbulent kinetic energy and one for the specific dissipation rate. The RKE turbulence model belongs to the group of eddy viscosity models, which are based on the similarity between the molecular gradient-diffusion process and turbulent motion [18]. It is a relatively recent development and it provides superior performance for flows including rotation, boundary layers with strong adverse pressure gradients, separation, and recirculation [19].

4. Numerical Setup

Numerical simulations of resistance tests were carried out utilizing the RANS equations. The finite volume method (FVM) is the integral method by which the governing equations were discretized. The volume of fluid (VOF) method combined with the high-resolution interface capturing (HRIC) scheme was utilized to ascertain the location of the free surface. The VOF method is a multiphase model that resolves the interface between the phases of the mixture of immiscible fluids, i.e., water and air. The physical properties of water in the numerical simulations corresponded to the ones during the towing tank tests [15].

4.1. Computational Domain and Boundary Conditions

Half of the ship model was simulated considering lateral symmetry condition, with the boundaries of the computational domain defined as follows: the inlet boundary was positioned 1.5$L_{PP}$ away from the hull, while the top boundary was 2$L_{PP}$ above the mean waterline. The position of the outlet boundary is 3$L_{PP}$ behind the hull, the bottom boundary was set 0.1$L_{PP}$ below the hull, and the side boundary was located 0.75$L_{PP}$ from the symmetry plane of the ship model.

The velocity inlet boundary condition was applied at both the inlet and top boundaries of the computational domain. The pressure outlet boundary condition was specified at the outlet boundary. A symmetry boundary condition was set at the symmetry plane of the ship model. Additionally, the no-slip wall boundary condition was assigned to the hull, bottom, and side boundaries of the ship model [20]. Figure 2 provides an overview of the boundary conditions used for numerical simulations in confined water. During the numerical simulations, the ship model was fixed while the speed of the incoming flow was equal to the negative value of the model speed.

4.2. Discretization of the Computational Domain

The computational domain was discretized by hexahedral cells. The number of cells greatly influences the results of numerical simulations, necessitating a balance between the time required to conduct numerical simulations and solution accuracy [21,22].

Volumetric controls in specific regions with abrupt changes in flow or where the flow needs to be described in more detail are generated. The mesh was additionally refined at the anticipated position of the free surface to accurately capture the Kelvin wake. Mesh at the symmetry plane is given in Figure 3, and at the position of the free surface in Figure 4. Considering that the computational domain represents the water of limited depth and width, boundary layers develop both at the bottom and the side boundaries. Therefore, the areas around the hull, as well as along the side and bottom boundaries, were refined with prism cells. Figure 5 depicts the discretization of the boundary layer around the hull in the stern region, while Figure 6 shows the discretized boundary layer along the bottom and side boundaries. The value of the $y^{+}$ parameter around the hull, at the bottom and side boundaries, was kept below 1 [23] within numerical simulations performed using both turbulence models. That enabled the comparison of the numerical results obtained using RKE and SSTKO turbulence models. Within the software package STAR-CCM+ both SSTKO and RKE can be used for the all $y^{+}$ wall treatment. RKE is available with the option of using a two-layer approach, which enables it to be used with fine mesh that resolves the viscous sublayer. The two-layer all $y^{+}$ wall treatment uses an approach that is identical to the all $y^{+}$ wall treatment. Figure 7 illustrates the distribution of the dimensionless parameter $y^{+}$ along the hull.

Table 2. The number of cells in the numerical simulations.

Grid	Number of Cells
Coarse mesh	1,191,294
Medium mesh	2,537,981
Fine mesh	5,513,192

displays the number of cells of the mesh utilized in the numerical simulations conducted for shallow and confined water.

4.3. Verification and Validation

Verification is the process of assessing the numerical uncertainty of results and verifying the convergence of results obtained by iterative methods [24]. The total numerical uncertainty $U_{SN}$ was determined based on the number of iterations ${\delta }_I$, grid size ${\delta }_G$, time step ${\delta }_T$, and other parameters ${\delta }_P$, and it is calculated as follows:

$$U_{SN}^2=U_I^2+U_G^2+U_T^2+U_P^2$$

(4)

where $U_I$ represents the numerical uncertainty due to the number of iterations, $U_G$ is the numerical uncertainty due to the grid size, $U_T$ denotes the numerical uncertainty due to the time step, and $U_P$ stands for the numerical uncertainty due to other parameters.

This research assesses the numerical uncertainty stemming from grid size and time step variations in the determination of the total resistance, trim, and sinkage of a ship model navigating in shallow and confined water. The numerical uncertainty due to the number of iterations and other parameters can be disregarded. In the verification process, the mesh and time step were refined by a constant factor, referred to as the refinement factor $r$ equal to $\sqrt{2}$ for the grid size and 2 for the time step [25].

To establish the convergence ratio based on the differences in solutions obtained with different mesh densities and time steps, a minimum of three solutions is necessary:

$${\epsilon }_{i,21}={\varphi }_i-{\varphi }_j$$

(5)

$$R={\epsilon }_{i,21}/{\epsilon }_{i,32}$$

(6)

where $\varphi$ represents the solution, ${\epsilon }_{ij}$ is the difference between the obtained results, and $R$ stands for the convergence ratio.

Table 3. Types of convergence.

Convergence Ratio	Convergence Type
$-1<R_i<0$	Oscillatory
$0<R_i<1$	Monotonic
$\left|R_i\right|>1$	Divergence

provides the types of convergence categorized according to convergence ratios.

For the case of monotonic convergence, generalized Richardson extrapolation was used to estimate numerical uncertainty, with the expressions for calculating the error ${\delta }_{RE}^{*}$ and the order of accuracy $p_i$ given as follows:

$${\delta }_{RE}^{*}={\displaystyle \frac{{\epsilon }_{i,21}}{r_i^{pi}-1}}$$

(7)

$$p_i={\displaystyle \frac{\ln \left(\frac{{\epsilon }_{i,32}}{{\epsilon }_{i,21}}\right)}{\ln \left(r_i\right)}}$$

(8)

Validation is the process of determining the deviation of obtained numerical results from available experimental data. Validation enables an assessment of the accuracy of numerical simulations and provides insight into the capability of a mathematical model to describe real phenomena [26]. The comparison of the numerically obtained total resistance with the one obtained experimentally is given based on the relative deviation determined by the following expression:

$$RD={\displaystyle \frac{R_{\mathrm{T},\mathrm{CFD}}-R_{\mathrm{T},\mathrm{EFD}}}{R_{\mathrm{T},\mathrm{EFD}}}}\cdot 100\%$$

(9)

where $R_{T,CFD}$ is total resistance obtained from numerical simulation, and $R_{T,EFD}$ represents total resistance determined experimentally. The relative deviations of sinkage and trim were determined analogously to the relative deviation of total resistance, as shown in Equation (9).

4.4. Physics Modelling and Solver Parameters

The Eulerian Multiphase model was used to define two fluid phases, with the initial free surface position and fluid velocity determined using the VOF approach. To reduce the impact of wave reflections against the computational domain boundaries, a damping layer approach was employed. The damping length was chosen equal to the length of the ship model. The damping layer was introduced at the inlet and outlet boundaries. The dynamic fluid body interaction (DFBI) model was used to calculate sinkage and trim, with heave and pitch motions enabled within the simulations at an even keel for the validation study. For simulations at different trims, DFBI was disabled. To accommodate potentially larger sinkage and trim values in numerical simulations at an even keel for the validation study, mesh morphing was utilized. The time step was determined using the following equation:

$$\Delta t={\displaystyle \frac{T}{c}}$$

(10)

where $T$ represents the period, defined as:

$$T={\displaystyle \frac{L_{PP}}{v}}$$

(11)

where $v$ represents the ship speed in m/s. The beforementioned coefficient c for the fine, medium, and coarse time steps is equal to 200, 100, and 50, respectively. The fine, medium, and coarse time steps were 0.024 s, 0.048 s, and 0.096 s, respectively. Five inner iterations were set within each time step.

5. Results and Discussion

This section presents an overview of the results obtained from the numerical simulations of the resistance tests conducted for two turbulence models, the SSTKO and RKE for the KCS ship model. The results include the total resistance, sinkage, and trim obtained from the numerical simulations using three grid densities with the smallest time step, as well as three different time steps with a fine grid. The verification procedure was conducted, and the numerical uncertainty of the total resistance, sinkage, and trim was determined. The results of the numerical simulations obtained using the fine mesh and the smallest time step were validated against experimental results from the literature [15]. Additionally, a depiction of the wave patterns, hydrodynamic pressure, and tangential stresses along the hull obtained using the RKE turbulence model is provided. A comparison of the total resistance of the ship model on an even keel and at different trims is presented. Double body simulations were conducted for the aforementioned two turbulence models, and the decomposition of total resistance was performed.

5.1. Double Body Simulations

The double-body approach asserts that the flow around the submerged body is symmetrical relative to the undisturbed waterplane [27]. The boundary conditions of the computational domain remained the same as in the free-surface simulations, with the exception of the top boundary for which a symmetry plane boundary condition was applied. Figure 8 shows the computational domain with the specified boundary conditions for double-body simulations.

The double-body simulation approach is based on neglecting the influence of the free surface, so the total resistance of the ship model equals the viscous resistance, which is divided into frictional resistance and viscous pressure resistance:

$$R_T=R_F+R_{VP}=R_V$$

(12)

The form factor can be calculated as the ratio of viscous pressure resistance and frictional resistance:

$$k={\displaystyle \frac{R_{VP}}{R_F}}$$

(13)

5.2. Free-Surface Simulations

Free-surface simulations were also conducted for two turbulence models. The total resistance obtained by the free surface simulations includes the wave component as well:

$$R_T=R_F+R_R=R_F+R_{VP}+R_W$$

(14)

$$R_R=R_{VP}+R_W$$

(15)

Viscous resistance obtained in the double-body simulations was used to calculate the wave resistance component as follows:

$$R_W=R_T-R_V$$

(16)

5.3. The Results of the Verification Study

The verification study was conducted for the ship model at a Froude number calculated based on the water depth equal to $F{n}_h=0.469$, which corresponds to the ratio of the water depth to the ship draft of 1.78. The results of the verification study are presented in

Table 4. The numerical uncertainty due to the grid size for the total resistance, sinkage, and trim.

Turbulence Model		${\mathit{\epsilon }}_{\mathit{i},21}$	${\mathit{\epsilon }}_{\mathit{i},32}$	$\mathit{R}$	${\mathit{U}}_{\mathit{G}},$%
RKE	$R_{\mathrm{T}}$, N	−0.001	0	0.424	0.023
	Sinkage, m	0.0008	−0.0008	−0.994	7.94
	Trim, °	0.0017	−0.0009	−1.9	2.84
SSTKO	$R_{\mathrm{T}}$, N	0.004	0.006	0.72	0.53
	Sinkage, m	−0.00001	−0.000006	1.8	0.33
	Trim, °	−0.00004	0.0002	−0.17	0.18

and

Table 5. The numerical uncertainty due to the time step for the total resistance, sinkage, and trim.

Turbulence Model		${\mathit{\epsilon }}_{\mathit{i},21}$	${\mathit{\epsilon }}_{\mathit{i},32}$	$\mathit{R}$	${\mathit{U}}_{\mathit{T}},$%
RKE	$R_{\mathrm{T}}$, N	0.025	0.056	0.436	0.845
	Sinkage, m	0.000015	0.00003	0.49	0.34
	Trim, °	−0.00059	−0.00071	0.826	5.68
SSTKO	$R_{\mathrm{T}}$, N	−0.005	−0.012	0.43	0.18
	Sinkage, m	−0.00005	−0.00013	0.34	0.56
	Trim, °	0.00015	0.002	0.09	0.03

. The total numerical uncertainty of the results obtained using both turbulence models is given in

Table 6. Total numerical uncertainty for the total resistance, sinkage, and trim.

Turbulence Model	${\mathit{U}}_{\mathit{S}\mathit{N}},$%
	RKE	SSTKO
$R_{\mathrm{T}}$, N	0.85	0.56
Sinkage, m	7.94	0.65
Trim, °	6.35	0.18

.

In the case of the numerical uncertainty due to the grid size, it can be seen from Table 4 that the numerical uncertainty for the total resistance is sufficiently low for both analyzed turbulence models. The numerical uncertainties for the sinkage and trim are significantly lower in the case of SSTKO in comparison to the RKE turbulence model. It should be noted that for the total resistance, monotonic convergence was obtained for both turbulence models, while for sinkage and trim oscillatory convergence and divergence were obtained, respectively. Regarding the numerical uncertainty due to the time step, as can be seen from Table 5 the monotonic convergence was obtained for the total resistance, sinkage, and trim for both turbulence models. The numerical uncertainties are below 1%, except for the trim obtained using the RKE turbulence model.

Based on the obtained verification results, it is evident that the total numerical uncertainty for the total resistance is very small for both analyzed turbulence models. In contrast, the total numerical uncertainties for trim and sinkage obtained using the RKE turbulence model are significantly higher.

Table 7. The relative deviations of the total resistance, sinkage, and trim of the ship model for the RKE turbulence model.

	${\mathit{R}}_{\mathbf{T}}$, N	Sinkage, m	Trim, °
Experimental results	3.001	−0.0056	0.069
Numerical results	2.8	−0.0051	0.061
Relative Deviation, %	−6.67	−8.5	−10.55

provides the values of the relative deviations for the total resistance, sinkage, and trim of the ship model calculated according to Equation (9) for the RKE turbulence model, while

Table 8. The relative deviations of the total resistance, sinkage, and trim of the ship model for the SSTKO turbulence model.

	${\mathit{R}}_{\mathbf{T}}$, N	Sinkage, m	Trim, °
Experimental results	3.001	−0.0056	0.069
Numerical results	2.64	−0.0051	0.062
Relative Deviation, %	−12.13	−8.68	−9.54

shows the relative deviations of the results obtained using the SSTKO turbulence model. It can be seen that the relative deviation for the total resistance obtained using the RKE turbulence model is significantly lower in comparison to the one obtained using SSTKO. The relative deviations for the sinkage and trim are quite pronounced for both turbulence models ranging from 8.5% to 10.55%. However, it should be noted that the experimental uncertainties for both trim and sinkage are higher than 90% at low speeds [15].

5.4. The Effect of Trim on the Total Resistance and Its Components

The effect of trim on the total, frictional, and pressure resistance obtained with free surface simulations using fine mesh and fine time step is presented in

Table 9. Total, frictional, and pressure resistance for the ship model at even keel and various trims.

Trim, °	${\mathit{R}}_{\mathbf{T}}$, N	RD, %	${\mathit{R}}_{\mathbf{F}}$, N	RD, %	${\mathit{R}}_{\mathbf{P}}$, N	RD, %
−0.4	2.89	4.51	2.32	1.66	0.57	17.98
−0.2	2.81	1.51	2.30	0.56	0.51	6.18
0	2.77	/	2.28	/	0.48	/
0.2	2.71	−2.00	2.27	−0.53	0.45	−6.86
0.4	2.68	−3.05	2.25	−1.41	0.43	−11.00

and Figure 9. According to the results of numerical simulations of resistance tests at various trims, it is evident that for a trim by bow, the total resistance decreases in comparison to an even keel. The largest decrease in the total resistance is obtained for a trim of 0.4°, and it amounts to −3.05%. Conversely, for a trim by stern, the total resistance increases, even by 4.51% for a trim of −0.4° in comparison to an even keel.

Based on the obtained relative deviations, it is evident that the pressure resistance changes the most with trim variation. For a trim by bow, there is a reduction in pressure resistance compared to an even keel, which contributed to an overall decrease in the total resistance of the ship model at those trims. Conversely, for a trim by stern, there is a significant increase in pressure resistance, ultimately resulting in higher total resistance. Regarding frictional resistance, it decreases at trims by bow and increases for trims by stern. The wetted surface area for trims by bow are smaller in comparison to the one at an even keel. For trim by stern, the wetted surface area is 1.04% and 0.62% lower for trims of −0.4° and −0.2°, respectively, in comparison to an even keel. The reduction in the wetted surface area for the trim by bow amounts to −0.84% and −1.61% for trims of 0.2° and 0.4°, respectively, in comparison to an even keel. In general, the effect of trim on the frictional resistance is much smaller in comparison to pressure resistance. It can be concluded that the trim has a significant impact on altering the pressure resistance, while its effect on frictional resistance is also present but to a lesser extent.

Based on the results of the conducted free surface and double body numerical simulations the total resistance is decomposed, and the effect of trim on the resistance components is analyzed using both RKE and SSTKO turbulence models. The portion of resistance components in the total resistance is shown in Figure 10 and Figure 11.

Based on the displayed results, it can be seen that the frictional resistance forms the largest part of the total resistance, followed by viscous pressure resistance, while the part of the wave resistance is the smallest. The part of the frictional resistance in the total resistance increases for trim by bow, as well as for a trim of −0.2°. On the other hand, the viscous pressure resistance increases for trim by bow and decreases for trim by stern in comparison to an even keel. The most significant differences can be observed for wave resistance, whose part in the total resistance decreases for trim by bow. It can be seen that the increase in the total resistance for a trim of −0.4° is mainly caused by an increase in the wave resistance. Regarding the influence of the turbulence model on the obtained results, it should be noted that the parts of the frictional and viscous pressure resistance obtained by the SSTKO turbulence model are smaller than the ones obtained by RKE. On the contrary, the part of the wave resistance obtained by the SSTKO turbulence model is larger in comparison to the one obtained by the RKE for all analyzed trims. For example, the part of the wave resistance in the total resistance for a trim of 0.4° obtained by RKE and SSTKO is equal to 3.32% and 9.36%, respectively. The effect of trim on the form factor is almost negligible, while the form factor calculated based on the results obtained using the SSTKO turbulence model is approximately 3% larger than the one calculated using RKE.

5.5. Wave Patterns

Figure 12 depicts wave patterns around the ship model at an even keel and different trims. When a ship enters shallow water, several changes occur due to the interaction between the ship and the bottom. Namely, the fluid accelerates, and the pressure under the hull decreases, which may result in significant changes in sinkage and trim. This leads to an increase in frictional and wave resistance along with the changes in the wave pattern generated around the hull. The effects of side boundaries, i.e., banks are similar to those experienced in shallow water. The Froude number relevant for the ships in restricted waterways is the one calculated based on the water depth ($F{n}_h$). Speeds corresponding to the $F{n}_h<1$ are denoted as subcritical speeds, while the ones corresponding to $F{n}_h>1$ as supercritical speeds [28]. Around the critical speed solitary waves may be generated that move ahead of the ship. At speeds well below $F{n}_h=1$, the wave system consists of a transverse wave system and a divergent wave system propagating away from the ship. Within this study, the Froude number based on the water depth is $F{n}_h=0.469$, and the wave system of transverse and divergent waves along with the Kelvin wave pattern can be observed similarly to the one in unrestricted water. As the speed approaches the critical one a significant amplification of wave resistance occurs, while at speeds greater than critical, the resistance again decreases. In confined water, the wave system around the ship hull consisting of transverse and divergent waves interferes with the waves reflected against the side walls of the channel, which can be noticed in Figure 12. Under such conditions complex wave patterns are generated considering that waves propagate from different directions, which is consistent with findings available in the literature [12,29]. In the case of a trim by stern, an increase in wave elevations behind the stern can be noticed.

The wave elevations along the model hull at an even keel and various trims are given in Figure 13. It can be noticed that the wave elevations in the bow region are the highest for trims by stern, while in the stern region, the highest wave elevations are obtained for trims by bow. In comparison to an even keel wave systems obtained for all analyzed trims are shifted towards the stern. Changes in the wave elevations for trims by stern in comparison to an even keel are more significant in the bow region than in the stern region. Higher wave elevations are linked to an increase in the pressure resistance (Table 9). The lowest wave elevations are obtained for the largest analyzed trim by bow, which corresponds to the larger decrease in the pressure resistance of −11%. By comparing the wave elevations at the centerline in front of the hull it can be noticed that the lowest wave elevations are obtained for the trim of 0.4°, while the highest ones correspond to trims by stern (Figure 14). The underwater shape of the bulbous bow is very important, especially for the wave-making resistance. As it changes with the change in trim it affects the formation of the bow wave system.

5.6. Distribution of Hydrodynamic Pressure Along the Hull

The distribution of the hydrodynamic pressure along the hull of the ship model is given in Figure 15. It can be seen that there are no significant changes in the pressure distribution for various trims. In comparison to an even keel, an area of underpressure can be noticed in the bow region for trims by stern. This is in accordance with wave elevations given in Figure 13 where it can be seen that the largest wave troughs are obtained for trims by stern. It increases the pressure resistance, and consequently the total resistance for trims by stern. As was already shown, wave resistance is more sensitive to trim changes in comparison to frictional resistance. Although, in restricted water, pressure resistance forms a small part of the total resistance, it becomes more important than in deep water [20]. As the waterline shape changes with the change in trim, the pressure distribution changes as well affecting the wave elevations around the hull.

5.7. Distribution of Tangential Stresses Along the Hull

In Figure 16, the distribution of tangential stresses along the hull is provided at an even keel and various trims. Generally speaking, the pressure drop along the ship hull induces changes in the flow velocity around the hull and the distribution of tangential stresses. Tangential stresses increase as the clearance between the ship’s bottom and the bottom of the waterway decreases. An increase in tangential stresses leads to an increase in frictional resistance. Since in numerical simulations conducted within this study, the under−keel clearance is not altered, changes in tangential stresses are not significant. As can be seen from Table 9 changes in trim mostly affect pressure resistance, while frictional resistance remains almost the same as already shown in other studies [3,30].

6. Conclusions

In this study, numerical simulations of resistance tests were conducted for the KCS container ship model at an even keel and four trims to determine the effect of trim on the total resistance in confined and shallow water using two turbulence models, i.e., RKE and SSTKO. The mathematical model was based on the RANS equations discretized by the FVM method, while the VOF method was used to determine the free surface position. Verification and validation studies were carried out for the ship model at an even keel to assess the numerical uncertainty and relative deviations from the experimental data. A detailed analysis of the flow around the hull is given, including wave patterns and wave elevations along the hull, the distribution of the hydrodynamic pressure, and tangential stresses for all analyzed trims. Based on the conducted study, the following conclusions are drawn:

• The obtained results have shown the importance of the turbulence model selection, as in this study, the total resistance obtained using the RKE turbulence model is closer to the experimental value than the one obtained using SSTKO.
• By assessing the effect of trim on the total resistance it was found that it is possible to achieve a reduction in total resistance at trims by bow in confined and shallow water. The largest decrease in the total resistance was obtained for a trim of 0.4°, and it amounts to −3.05%.
• It was shown that pressure resistance is more sensitive to trim variations in comparison to frictional resistance. For a trim by bow, there is a reduction in pressure resistance compared to one at an even keel, which contributed to an overall decrease in the total resistance of the ship model at those trims. On the other hand, there is a significant increase in pressure resistance for a trim by stern, resulting in higher total resistance. It can be concluded that the trim has a significant impact on altering the pressure resistance, while its effect on frictional resistance is also present but to a lesser extent.
• To analyze the effect of trim on the total resistance components, double−body simulations were also conducted using the aforementioned turbulence models. The results showed that by using the SSTKO turbulence model the part of the wave resistance in the total resistance is notably larger in comparison to results obtained using RKE.
• The frictional resistance forms the largest part of the total resistance, followed by viscous pressure resistance, while the part of the wave resistance is the smallest. The part of the frictional resistance in the total resistance increases for trims by bow.
• The viscous pressure resistance increases for trims by bow and decreases for trims by stern in comparison to one at an even keel.
• The most significant differences can be observed for wave resistance, whose part in the total resistance decreases for trims by bow. The increase in the total resistance for the largest analyzed trim by stern of −0.4° is mainly caused by an increase in the wave resistance.
• The effect of trim on the form factor is almost negligible.

Although this study showed a decrease in the total resistance at trims by bow and an increase at trims by stern, the trend by which a certain trim will affect the total resistance of the ship depends on many parameters such as the draught, speed, type of the ship, etc.

This study has shown that CFD can be successfully applied to investigate the flow around the hull of a ship model when navigating in confined and shallow water. This enables the trim optimization with the aim of increasing the energy efficiency of the ship in confined and shallow water. As a part of future research, the squat effect on the total resistance and its components in shallow and confined water will be taken into account as well.