Impact of the Longitudinal Center of Buoyancy on the Total Resistance of a Passenger Ship

Abstract

A numerical investigation into the impact of the longitudinal position of the center of buoyancy on the total resistance of a passenger ship is conducted using the computational fluid dynamics software package STAR-CCM+. The modification of the hull form is performed using the CAESES software package, respecting the limitations on the longitudinal position of the center of buoyancy set by Flow Ship Design d.o.o. The total numerical uncertainty for the total resistance, sinkage, and trim angle of the original hull form is assessed within the verification study. The flow around the ship hull is analyzed in detail, including the determination of the wave pattern and free surface elevation as well as the hydrodynamic pressure and tangential stress distributions. The obtained values for total resistance, sinkage, and trim angle for all modified hull forms are compared. The study indicated that shifting the longitudinal position of the center of buoyancy by 0.4% can lead to a 2.11% reduction in total resistance compared to the original hull form. Resistance tests are conducted at two additional speeds to determine the resistance curve for the hull form with the most favorable total resistance characteristics. The results indicate that simple modifications of the hull form can lead to a reduction in the total resistance without necessitating complex optimization algorithms.

1. Introduction

According to the regulations proposed by the International Maritime Organization (IMO), all newly built ships must meet the requirements of the minimum level of energy efficiency following the Energy Efficiency Design Index (EEDI) [1]. To improve the energy efficiency of ships, one of the technical measures proposed by the IMO is the optimization of the ship’s hull form [2]. To identify the optimal hull form from the hydrodynamic point of view, it is crucial to employ suitable optimization methods, typically involving computational fluid dynamics (CFD), as it offers a detailed representation of the flow around the hull. Combining CFD with optimization techniques provides an efficient approach to hull form optimization, improving the design quality of energy-efficient ships [3].

Feng et al. [4] outlined the procedure for generating the optimal parametric hull form for three containerships of different sizes using a fully parametric modeling method. A multi-objective optimization algorithm, coupled with the boundary element method based on nonlinear potential flow theory, was used to reduce the wave resistance and, consequently, the total resistance. The effect of the bow shape on the total resistance of a yacht was evaluated for three models through towing tank tests and numerical simulations. The results indicated that a bulbous bow can decrease total resistance by up to 7% [5]. Employing a multi-objective shape optimization approach to enhance the bow region of a trimaran, along with a CFD-based design, can reduce total resistance in calm water by more than 5% [6]. Miao et al. [7] proposed a CFD-based multi-objective optimization procedure aimed at reducing the resistance of Series 60 catamarans by addressing the separation and shape of the demihulls. By comparing the results for the initial and optimized catamaran forms, the authors confirmed the effectiveness of the proposed optimization method. By combining computer-aided design with CFD, Zhao et al. [8] performed an optimization procedure for an ocean-going trawler using the STAR-CCM+ 2306 and CAESES v.5.2. software packages. To validate the results, the authors also conducted an experimental study on both the initial and optimized hull forms. Recognizing that conducting CFD simulations during an optimization procedure is highly time-consuming and computationally demanding, Zhang et al. [9] developed an optimization framework to optimize hull forms with considerably reduced computational effort. The IPSO–Elman (Improved Particle Swarm Optimization) algorithm was introduced for the approximate calculation of total resistance, while the hull form optimization was carried out using the IPSO algorithm. In order to minimize total resistance and maximize the volume of a small underwater vehicle, Hou et al. [10] employed a Kirgin-based Response Surface Method (RSM) model. The obtained results offer technical support for the design and optimization of similar small underwater vehicles. To assess the accuracy and efficiency of the optimization algorithm, a sampling method was employed to gather information about the optimization space in the initial stage [11]. Jambak and Bayezit [12] developed a rapid optimization algorithm to identify a robust optimal controller for a ship heading angle control, accounting for uncertainties in hydrodynamic coefficients that are challenging to solve deterministically.

Liu et al. [13] optimized a high-speed slender hull both with and without the bulbous bow and proposed a method for designing the bulbous bow to reduce wave resistance. Tran et al. [14] proposed a strategy for optimizing a planning hull to achieve minimal resistance by integrating the Kriging surrogate model with the Nelder–Mead optimization algorithm. A tool for multi-objective ship hydrodynamic optimization was introduced and used for the design optimization of the DTMB 5512, aiming to reduce total resistance in calm water and improve vertical motion performance [15]. The optimization tool employed several NURBS-based (non-uniform rational B-spline) hull surface modification methods to achieve both global and local deformations of the hull form. An automated and integrated design procedure was developed in [16] to establish feasible ship dimensions during the conceptual design stage. By systematically varying the main dimensions, this procedure enables the evaluation of a design space to identify the most effective solution. In [17], a joint trim and engine power optimization method was developed to minimize ship fuel consumption and enhance the energy efficiency of ships.

Pak et al. [18] evaluated the optimal position of the longitudinal center of buoyancy (LCB) for an LNG-fueled ship and a small-scale LNG bunkering vessel using three methods: a statistical approach, partially parametric hull design, and conventional non-parametric hull design. Additionally, the authors optimized the bulb shape and stern profile, achieving a 9.5% reduction in effective power for the optimized hull form compared to the initial one. In [19], a methodology was proposed to assess the impact of the LCB position for a case study of a fishing vessel. Based on the performed calculations using the Holtrop–Mennen method, the authors obtained a minimum value of the calm water resistance for the LCB position of −2% from midship. Yu et al. [20] investigated the possibility of preventing the surf-riding of a high-speed vessel in quartering seas by adjusting the LCB position. Although the results indicated that adjusting the LCB position cannot prevent surf-riding, it was observed that shifting the LCB towards the bow allows surf-riding to occur at higher speeds. Adjusting the LCB position had a more pronounced impact on pressure resistance compared to frictional resistance, and even a minor shift in its position can result in a substantial reduction in the ship’s total resistance [21].

In this study, the impact of the longitudinal center of buoyancy on the total resistance, sinkage, and trim of a passenger ship is analyzed by performing numerical simulations for the hull forms, which are modified by shifting the cross sections. The ship used as a case study is currently in the design phase at the Flow Ship Design d.o.o. company [22]. Nine LCB positions, including the initial one, are analyzed within the specified range determined by the ship’s mass distribution. The results highlighted the potential for reducing a ship’s total resistance through simple hull modifications, offering an alternative to complex optimization procedures. This approach could be particularly valuable during the early stages of hull design. The remainder of this paper is organized as follows: the overview of the ship selected as a case study is given in Section 2, while the description of the hull form modification is given in Section 3. A mathematical model is given in Section 4 and a numerical model in Section 5. The results of the performed verification study are provided in Section 6. Section 7 presents and discusses the numerically obtained results for the initial and modified hull forms, while the conclusions are summarized in Section 8.

2. Case Study

The impact of the LCB position on calm water resistance, sinkage, and trim is examined for a passenger ship developed by Flow Ship Design d.o.o. (Figure 1). The passenger ship under analysis serves a dual purpose. It can operate as an excursion vessel for up to 150 passengers on one-day voyages and also offers luxury cabins to accommodate 8 passengers for multi-day cruises. Designed for coastal waters of the Adriatic Sea, the hull form was developed to house all necessary facilities for passengers and crew while maintaining optimal hydrodynamic performance. The hull will be built from steel, while the superstructure will be made of aluminum. The hull form is not fully finalized, leaving space for minor adjustments and potential improvements. Similarly, the general arrangement plan, including mass distribution, is not entirely defined, allowing for slight shifts in the longitudinal position of the center of mass. The main particulars of a ship are outlined in

Table 1. Main particulars of the passenger ship.

Parameters	Symbol	Value
Length between perpendiculars	$L_{PP}$	23.73 m
Breadth	$B$	7.072 m
Draught (summer load line)	$T$	1.9 m
Displacement mass	$\Delta$	168 t
Longitudinal center of gravity	$LCG$	10.111 m
Vertical center of gravity	$\overline{KG}$	3.69 m
Longitudinal center of buoyancy	$LCB$	10.111 m
Vertical center of buoyancy	$Z_{CB}$	1.174 m
Prismatic coefficient	$C_P$	0.57
Froude number	$Fn$	0.388

. The origin of the coordinate system is placed at the intersection of the design waterline and the aft perpendicular. The x-axis points from the stern to the bow, the y-axis extends to the port side, and the z-axis points vertically upward.

3. Hull Form Modification

The modification of the hull form is carried out using the CAESES software package [23] employing the Lackenby method to shift cross sections, thereby altering the prismatic coefficient of the fore and aft sections of the hull. The transformation begins by taking the initial sectional area curve and incorporating inputs for changes in displacement and the center of buoyancy. A smooth shift function is then applied within a specified range to perform the volume shift. The function curve is computed automatically by the Lackenby method. It is important to note that while adjusting the position of the LCB, the prismatic coefficient value remains constant. The limits of the LCB position are defined by the mass distribution of the passenger ship, allowing for a variation of $\Delta LCB=\pm 1.6\%$ relative to the initial LCB position. Eight variants of modified hull form are analyzed within the defined limits of the LCB position with a uniform step of 0.4%, Figure 2. For each hull modification, the center of mass is aligned with the new LCB, ensuring that each modified hull remains on an even keel, just like the initial one.

After performing the numerical simulations for both the initial and modified hull forms, the results are compared by calculating the relative deviation (RD) as follows:

$$RD={\displaystyle \frac{x_M-x_I}{x_I}}\cdot 100\%$$

(1)

where $x_M$ denotes the value obtained for the modified hull form, while $x_I$ corresponds to the value obtained for the initial hull form. Relative deviations for the total resistance, trim, and sinkage are calculated based on Equation (1).

4. Mathematical Model

A mathematical model for describing turbulent fluid flow is based on the Reynolds Averaged Navier–Stokes (RANS) equations and the averaged continuity equation as follows:

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

(2)

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

(3)

where $\rho$ is the fluid density, ${\overline{u}}_i$ is the averaged Cartesian components of the velocity vector, $\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}$ is the Reynolds stress tensor, while $\overline{p}$ is the mean pressure. The mean viscous stress tensor is given as:

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

(4)

where $\mu$ is the dynamic viscosity.

To close the system of Equations (2) and (3) the $k-\omega$ SST (SSTKO) turbulence model is employed, which solves two additional transport equations: one for turbulent kinetic energy $k$ and another for specific dissipation $\omega$:

$${\displaystyle \frac{\partial }{\partial t}}(\rho k)+{\displaystyle \frac{\partial }{\partial x_i}}(\rho k{\overline{u}}_i)={\displaystyle \frac{{\partial }^{2}k}{\partial x_i^2}}(\mu +{\sigma }_k{\mu }_t)+G_k+G_{nl}+G_b-\rho {\beta }^{*}{f}_{{\beta }^{*}}(\omega k-{\omega }_0{k}_0)+S_k$$

(5)

$${\displaystyle \frac{\partial }{\partial t}}(\rho \omega )+{\displaystyle \frac{\partial }{\partial x_i}}(\rho \omega {\overline{u}}_i)={\displaystyle \frac{{\partial }^2\omega }{\partial x_i^2}}(\mu +{\sigma }_{\omega }{\mu }_t)+G_{\omega }+D_{\omega }-\rho \beta f_{\beta }({\omega }^2-{\omega }_0^2)+S_{\omega }$$

(6)

where ${\sigma }_k$, ${\sigma }_{\omega }$, ${\beta }^{*}$ and $\beta$ are model-depending coefficients, $S_k$ and $S_{\omega }$ are the user specified terms, $G_{nl}$ is the nonlinear production term, $G_k$ and $G_{\omega }$ are turbulent production terms, $D_{\omega }$ is the cross-derivative term, $f_{\beta }$ is the free-shear modification factor, $f_{{\beta }^{*}}$ is the vortex-stretching modification factor, ${\mu }_t$ is the eddy viscosity, and $k_0$ and ${\omega }_0$ are the ambient values that counteract turbulence decay [24].

5. Numerical Setup

Numerical simulations of the resistance test are conducted using the RANS equations. The governing equations are discretized utilizing the Finite Volume Method (FVM). To determine the location of the free surface, the Volume of Fluid (VOF) method is used [25,26].

Only half of the computational domain is discretized, taking into account lateral symmetry conditions. The boundaries of the domain are positioned far enough from the hull in accordance with the recommendations of the International Towing Tank Conference (ITTC) [27]. The inlet and top boundaries are placed 1.5$L_{PP}$ from the hull, the side boundary is placed 2$L_{PP}$ from the symmetry plane, and the bottom and outlet boundaries are placed 2.5$L_{PP}$ and 4$L_{PP}$ from the hull, respectively. The boundary conditions applied at the domain boundaries are given in Figure 3. It should be noted that in the numerical simulations, the ship remains stationary, with the incoming flow speed set to the negative value of the ship’s speed. The computational domain was discretized using hexahedral cells, with volumetric controls applied in particular areas with sudden flow variations or where more detailed flow descriptions are needed. The mesh is further refined at the expected free surface position in order to accurately capture the Kelvin wake. To conduct a verification study and assess the numerical uncertainty of the obtained results, three mesh densities are analyzed. The fine, medium, and coarse mesh contains approximately 4.22, 2.77, and 1.55 million cells, respectively, with base sizes of 0.38 m, 0.54 m, and 0.76 m. The cross sections of the fine mesh are shown in Figure 4. To properly capture the flow in the vicinity of the hull, the boundary layer is discretized using prism layers, and the wall functions are applied. The total thickness of the prism layer is 0.0749 m at the design speed, consisting of 11 cells, with a stretching factor of 1.4. The value of the parameter $y^{+}$ on the hull is kept above 30. The prism layer generated on the bulb can be seen in Figure 5.

The Eulerian Multiphase flow model is used to describe the two fluid phases, while the fluid velocity and the initial position of the free surface are defined by the VOF approach. In order to mitigate the effects of wave reflections at the boundaries of the computational domain, a damping layer approach is applied at the inlet, outlet, and side boundaries, with the damping length set equal to the ship length. Ship translation in the z-axis direction and rotation around the y-axis is enabled to assess the impact of LCB position on the sinkage and trim of the passenger ship. The time step is determined based on the ship length and speed and amounts to 0.023 s, 0.046 s, and 0.091 s for the fine, medium, and coarse time steps, respectively. Within each time step, five inner iterations are set.

The verification study was conducted using three different grid densities and three time steps, allowing for an assessment of numerical uncertainty for both the fine mesh and fine time step. Since the numerical and physical setups were kept consistent across simulations for the modified hull forms, the same verified grid and time step were utilized for all simulations, except for two additional simulations performed at speeds both lower and higher than the design speed. This adjustment was necessary due to the change in Reynolds number that occurs with varying speeds. Since the prism layer, discretized to describe the flow within the boundary layer, is dependent on the Reynolds number, the mesh parameters for the prism layer were modified to ensure an acceptable y⁺ value.

6. Verification Study

To calculate the numerical uncertainty, a verification study following the ITTC guidelines is conducted [28]. The total numerical uncertainty $U_{SN}$ is calculated based on the numerical uncertainty arising from the grid size $U_G$, and numerical uncertainty due to the time step $U_T$. The verification study is based on the obtained results using three mesh densities, with the refinement ratio ($r_i$) of $\sqrt{2}$ and three time steps, with the refinement ratio equal to 2. To determine the convergence ratio by comparing solutions obtained using various mesh densities and time steps, at least three solutions are required.

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

(7)

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

(8)

where $\varphi$ is the solution, ${\epsilon }_{ij}$ is the difference between the obtained results, and $R$ is the convergence ratio. In case the convergence ratio is $-1<R_i<0$, the oscillatory convergence is obtained; if $0<R_i<1$, the monotonic convergence is achieved; and in case $\left|R_i\right|>1$, the divergence is obtained. When monotonic convergence is attained, the generalized Richardson extrapolation method is applied to estimate the numerical uncertainty. The error ${\delta }_{RE}^{*}$ and the order of accuracy $p_i$ can be determined as follows:

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

(9)

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

(10)

7. Results and Discussion

The results of the performed numerical simulations for the initial and modified hull forms in terms of LCB position are presented within this section. The numerical uncertainty of the calm water resistance, sinkage, and trim is calculated for the design speed of 11.5 knots. The comparison of the results obtained for the initial and modified hull forms is given and discussed, along with the analysis of the wave pattern, hydrodynamic pressure, and tangential stress distributions along the hull. In addition, the impact of the LCB position on the frictional and pressure resistances is assessed.

7.1. Verification Study Results

Table 2. Verification study for grid size.

	${\widehat{\mathit{S}}}_{\mathit{i},1}$	${\widehat{\mathit{S}}}_{\mathit{i},2}$	${\widehat{\mathit{S}}}_{\mathit{i},3}$	${\mathit{\epsilon }}_{\mathit{i},21}$	${\mathit{\epsilon }}_{\mathit{i},32}$	$\mathit{R}$	${\mathit{U}}_{\mathit{G}}, \%$
$R_{\mathrm{T}}$, N	24,739	24,736	24,736	−2.66	436.30	−0.0061	0.8818
Sinkage, m	−0.1990	−0.1989	−0.1995	0.0001	−0.0006	−0.1771	−0.1539
Trim, °	0.6267	0.6305	0.6238	0.0038	−0.0066	−0.5733	0.5284

shows that varying mesh densities result in monotonic convergence for total resistance and trim, while sinkage exhibits oscillatory convergence. The numerical simulations conducted to evaluate the numerical uncertainty related to grid size are carried out using a fine time step, while the ones performed to assess the numerical uncertainty arising from the time step are performed with fine mesh. It can be noticed that the numerical uncertainty due to grid size is well below 1% for all physical quantities.

Table 3. Verification study for time step.

	${\widehat{\mathit{S}}}_{\mathit{i},1}$	${\widehat{\mathit{S}}}_{\mathit{i},2}$	${\widehat{\mathit{S}}}_{\mathit{i},3}$	${\mathit{\epsilon }}_{\mathit{i},21}$	${\mathit{\epsilon }}_{\mathit{i},32}$	$\mathit{R}$	${\mathit{U}}_{\mathit{T}}, \%$
$R_{\mathrm{T}}$, N	24,739	24,575	24,995	−163.67	419.83	−0.3899	0.8485
Sinkage, m	−0.1990	−0.1991	−0.2074	−0.0002	−0.0082	0.0197	−0.0020
Trim, °	0.6267	0.6328	0.6017	0.0061	−0.0311	−0.1975	2.4824

shows that numerical uncertainty due to the time step results in monotonic convergence for total resistance, oscillatory convergence for sinkage, and divergence for trim. The largest numerical uncertainty is approximately 2.5%, which remains within acceptable limits.

The total resistance of the initial hull form obtained using a fine grid and fine time step amounts to 24,436 N. The sinkage is approximately 20 cm, while the trim angle is larger than 0.62°.

The total numerical uncertainty is reasonably low, amounting to 1.22% for the total resistance, 0.15% for the sinkage, and 2.5% for the trim (

Table 4. Total numerical uncertainties of the total resistance, sinkage, and trim.

	${\mathit{U}}_{\mathit{T}},\%$	${\mathit{U}}_{\mathit{T}},\%$	${\mathit{U}}_{\mathit{S}\mathit{N}},\%$
$R_{\mathrm{T}}$, N	0.8818	0.8485	1.2238
Sinkage, m	−0.1539	−0.0020	0.1539
Trim, °	0.5284	2.4824	2.5380

).

7.2. Impact of LCB Position on Total Resistance, Sinkage, and Trim

The impact of the LCB position on the total resistance ($R_T$) of the passenger ship is illustrated in

Table 5. Impact of the LCB position on the total resistance.

$\Delta \mathit{L}\mathit{C}\mathit{B}, \%$	$\mathit{L}\mathit{C}\mathit{B}, \mathbf{m}$	${\mathit{R}}_{\mathbf{T}}, \mathbf{N}$	$\mathit{R}\mathit{D},\%$
1.6	10.487	24,879	0.57
1.2	10.393	24,227	−2.07
0.8	10.299	24,441	−1.20
0.4	10.205	24,216	−2.11
0 (initial)	10.111	24,739	/
−0.4	10.017	24,787	0.20
−0.8	9.923	25,301	2.27
−1.2	9.829	25,523	3.17
−1.6	9.735	25,847	4.48

. Shifting the LCB towards the bow results in a decrease in the total resistance compared to the initial hull form, except for the $\Delta LCB=1.6\%$, where a slight increase in the total resistance is observed. However, the obtained value falls within the range of numerical uncertainty for total resistance. The most favorable LCB position with respect to the total resistance corresponds to $\Delta LCB=0.4\%$, for which the decrease in the total resistance amounts to −2.11%. On the other hand, by shifting the LCB towards the stern, the total resistance of the passenger ship increases. As can be seen from

Table 6. Impact of the LCB position on pressure and frictional resistance.

$\Delta \mathit{L}\mathit{C}\mathit{B}, \%$	${\mathit{R}}_{\mathbf{P}}, \mathbf{N}$	$\mathit{R}\mathit{D},\%$	${\mathit{R}}_{\mathbf{F}}, \mathbf{N}$	$\mathit{R}\mathit{D},\%$
1.6	17,463	1.12	7418	−0.60
1.2	16,776	−2.86	7451	−0.17
0.8	16,974	−1.71	7467	0.04
0.4	16,762	−2.94	7456	−0.10
0 (initial)	17,270	/	7463	/
−0.4	17,306	0.20	7483	0.26
−0.8	17,819	3.18	7484	0.27
−1.2	18,032	4.41	7489	0.34
−1.6	18,396	6.52	7452	−0.16

, the aforementioned changes in the total resistance are mainly due to the change in the pressure resistance ($R_P$). Altering the LCB position has a more significant impact on the pressure resistance in comparison to the frictional resistance ($R_F$). For $\Delta LCB=0.4\%$, the pressure resistance is decreased by almost 3%, resulting in a decrease in the total resistance for that particular LCB position. A notable increase in the pressure resistance can be observed when the LCB position is shifted towards the bow. It is worth mentioning that the displacement volume and draught for modified hull forms are kept constant and equal to the ones for the initial hull form. The afterbody of the hull typically generates less wave-making resistance than the forebody, largely due to the boundary layer suppressing the waves at the stern. When the LCB is moved aft, the reduction in wave-making resistance from the forebody generally outweighs the increase in resistance from the afterbody, despite a rise in pressure resistance in the stern region. For faster, finer vessels, the LCB is typically positioned aft of the midship. Although there is a general trend showing that shifting the LCB towards the stern increases total resistance, while shifting it towards the bow decreases it, optimal LCB positions are often associated with specific speed ranges, which relate the center of buoyancy to the Froude number. It is also worth noting that the optimal LCB position will vary with different hull parameters, such as the inclusion of a bulbous bow. For the passenger ship, the initial LCB position is aft of the midship. Shifting the LCB forward resulted in a decrease in total resistance, with the most favorable shift being $\Delta LCB=0.4\%$ forward from the initial position, though it remained aft of midship.

Altering the LCB position affects the sinkage ($z$) of the passenger ship. For shifts towards the bow, the sinkage decreases in comparison to the sinkage of the initial hull form and vice versa. It appears that the LCB position has a more significant impact on the trim ($t$) than on the sinkage. From

Table 7. Impact of the LCB position on sinkage and trim.

$\Delta \mathit{L}\mathit{C}\mathit{B}, \%$	$\mathit{z}, \mathbf{m}$	$\mathit{R}\mathit{D},\%$	$\mathit{t}{, }^{\circ }$	$\mathit{R}\mathit{D},\%$
1.6	−0.1937	−2.65	0.77	22.71
1.2	−0.1955	−1.73	0.75	20.42
0.8	−0.1965	−1.24	0.70	11.86
0.4	−0.1958	−1.61	0.67	7.68
0 (initial)	−0.1990	/	0.63	/
−0.4	−0.2004	0.72	0.59	−6.45
−0.8	−0.2014	1.23	0.53	−15.04
−1.2	−0.2036	2.31	0.50	−20.98
−1.6	−0.2053	3.17	0.44	−30.36

, it can be seen that for $\Delta LCB=1.6\%$, the trim increases by 22.71%. Conversely, shifting the LCB position towards the stern results in a decrease in trim, with the largest decrease of approximately 30% obtained for $\Delta LCB=-1.6\%$. The trim of the initial hull form amounts to 0.63°. For the optimal LCB position from the total resistance point of view, the trim increases by 7.68%. It should be noted that the negative trim values denote trim by bow, and positive values trim by stern.

Additional numerical simulations were performed for the most favorable hull form regarding the total resistance for 10 and 12 knots. The obtained results are presented in

Table 8. Impact of the LCB position on the total resistance, sinkage, and trim.

$\mathit{V}, \mathbf{kn}$	${\mathit{R}}_{\mathbf{T}}, \mathbf{N}$	$\mathit{z}, \mathbf{m}$	$\mathit{t}{, }^{\circ }$
10	11,425	−0.1416	0.6093
11.5	24,216	−0.1958	0.6748
12	32,765	−0.2173	0.5465

. Decreasing the design speed by 1.5 knots results in a decrease of more than 52% in the total resistance. The trim and sinkage at the speed of 10 knots are also notably smaller in comparison to the ones at the design speed. The smallest trim is obtained for a speed of 12 knots, with a decrease in the total resistance of approximately 35%.

7.3. Wave Patterns, Hydrodynamic Pressure, and Tangential Stress Distributions

The wave patterns around the initial and modified hull forms can be seen in Figure 6. The system of transverse and divergent waves can be noticed, and shifting the LCB position does not have a notable effect on the wave pattern around the hull. From the distribution of the hydrodynamic pressure on the initial and modified hull forms given in Figure 7, it is evident that shifting the LCB position toward the stern increases the overpressure area on the bow compared to that of the initial hull form, leading to an overall increase in pressure resistance. In comparison to the initial hull form, a slight decrease in the overpressure area can be noticed in the stern region for modified hull forms with the LCB position shifted towards the stern.

Figure 8 illustrates the distribution of tangential stresses on the initial and modified hull forms. Shifting the LCB position towards the bow results in a slight reduction in tangential stress in the midship area. However, since shifting the LCB position causes no significant changes in tangential stress, the frictional resistance of the modified hull forms remains unchanged compared to that of the initial hull form, which can be seen in Table 6 as well.

8. Conclusions

Numerical simulations were performed to evaluate the effect of the longitudinal position of the center of buoyancy on the total resistance, sinkage, and trim angle of the passenger ship at design speed. The most favorable hull form in terms of total resistance was identified based on the CFD results, while hull form modifications were executed using the CAESES software package. The numerical results were verified, and the total numerical uncertainty for resistance, sinkage, and trim angle was calculated.

The comparison of the results revealed that shifting the longitudinal position of the center of buoyancy by 0.4%, 0.8%, and 1.2% towards the bow reduces the total resistance compared to the initial hull form. The most significant reduction of 2.11% in total resistance was observed with a 0.4% shift towards the bow from the initial position. In contrast, shifting the longitudinal center of buoyancy towards the stern resulted in increased total resistance and sinkage while reducing the trim angle. The largest reduction in trim angle of approximately 30% compared to the initial hull form was obtained for the position of the longitudinal center of buoyancy closest to the stern. The results indicate that shifting the longitudinal center of buoyancy has a minimal impact on frictional resistance, but significantly affects pressure resistance, leading to an overall increase or decrease in total resistance.

The results of this study indicate that even with simple modifications to the hull form, a reduction in the ship total resistance can be achieved without the need for complex optimization algorithms. However, future research could benefit from employing optimization algorithms for multi-objective optimization to further reduce both total resistance and trim. To ensure reliable and accurate results, the analysis should be performed across a range of operating conditions of interest, encompassing various loading conditions, draughts, and sailing speeds. Such a comprehensive approach would provide a thorough understanding of the ship’s performance under different scenarios, ensuring that the findings are applicable across a variety of operational conditions.