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J. Ocean Eng. Technol. > Volume 38(6); 2024 > Article
Lee and Nam: Numerical Analysis of Wave Interference Effects on Ship Resistance in Parallel Arrangements

Abstract

Ships moving in formations experience complex hydrodynamic interactions, leading to changes in resistance and motion responses, which are important considerations in engineering applications. This study investigated the resistance of ships arranged in parallel formations using both theoretical and numerical methods. Thin ship theory and computational fluid dynamics (CFD) simulations were employed to analyze the wave interference effects among multiple advancing ships. First, variations in wave resistance as a function of ship speed were examined. It was observed that as ship speed increases, the wavelength becomes longer, resulting in periodic occurrences of destructive and constructive interference, which cause significant fluctuations in wave resistance. Next, the influence of separation distances on wave resistance was explored at a consistent ship speed. The results showed that smaller separation distances exhibit more pronounced variations in wave resistance. Finally, the overall characteristics of wave resistance changes with respect to the number of ships in parallel arrangements were reviewed. In conclusion, the study demonstrated that wave interference effects in parallel arrangements significantly impact wave resistance, depending on forward speed and separation distance.

1. Introduction

At sea, situations frequently arise where multiple ships move in formations or operate cooperatively. In such cases, complex hydrodynamic interactions occur, influenced by ship formation, separation distance, and speed conditions. These interactions can affect ship resistance, environmental load, and the motion characteristics of the ships. Notably, for ships moving in formations in still water, wave resistance components may increase or decrease due to constructive and destructive wave interference effects, which represent an important characteristic that can be utilized in engineering applications. To address representative challenges such as efficient catamaran and multi-hull ship design, optimal ship formation, and ship resistance evaluation in narrow waterways, research on wave interference effects and their evaluation methods is essential. In general, the wavelength of ship waves is determined as a function of ship speed, while wave elevation varies depending on factors such as ship speed and hull shape. Therefore, various experimental, theoretical, and numerical approaches are necessary to examine wave interference effects accurately and systematically.
Experimental studies have long been conducted to observe the resistance variation characteristics caused by wave interference effects. Since Herreshoff (1877) patented the first catamaran, various experiments have been conducted to explore optimal catamaran geometries in terms of energy efficiency, driven by growing interest in catamaran capable of stable navigation. Dubrovskiy (1968) demonstrated through experiments that the wave resistance of a catamaran is reduced by more than four times that of a single hull and revealed that wave resistance fluctuates significantly depending on the separation distances and the speed of the catamaran. Additionally, Soding (1997) experimentally analyzed changes in wave resistance due to increased longitudinal separation distances between two hulls in arrow arrangements, finding that the wave resistance of a catamaran in such arrangements was up to 50% lower than that of a single hull.
In addition to experimental studies, various theoretical approaches have been undertaken to evaluate changes in wave resistance caused by wave interference effects. In particular, many researchers have proposed theoretical approaches based on thin ship theory and validated their applicability to identify the trends in wave resistance with respect to variables such as ship arrangement, separation distance, and ship speed. Tuck and Lazauskas (1998) theoretically analyzed the magnitude of resistance for each ship using thin ship theory, which approximates a hull to a thin column, when Wigley parabolic hull forms were arranged in parallel, serial, and arrow configurations. Day et al. (2003) demonstrated that theoretical calculations based on thin ship theory showed good agreement with experimental results for parallel arrangements but highlighted relatively large discrepancies between theoretical and experimental results for arrow and serial configurations. Zhang and Maki (2023) analyzed the magnitude of resistance during ship movement in a narrow waterway using a theoretical approach presented by Tuck and Lazauskas (1998). They approximated ships in the narrow waterway as a parallel arrangement of multiple ships and calculated the resistance magnitude for the middle ship. The validity of their theoretical approach was confirmed by comparing these results with numerical analysis findings.
In recent years, studies have been conducted to evaluate wave interference effects through numerical analysis based on potential flow or computational fluid dynamics (CFD). Yu et al. (2017) utilized CFD analysis to study the flow field around two hulls in an arrow arrangement. They analyzed the effects of changes in separation distance and ship speed on wave resistance and examined the variation in wave resistance for the rear hull located on the divergent wave by comparing the results with experimental values. Yuan et al. (2021) performed a potential flow-based numerical analysis of wave resistance when multiple ducks moved in a serial arrangement and investigated the effects of destructive and constructive wave interferences on wave resistance. Specifically, they presented the effects of the movement patterns of a mother duck and multiple ducklings arranged in series at a fixed separation distance on wave resistance.
In this study, thin ship theory and CFD analysis were used to investigate wave resistance characteristics caused by wave interference effects under the parallel arrangement condition of multiple ships. The CFD analysis was performed to validate the results of theoretical calculations. Based on this, the wave resistance trend for the hull at the center of the parallel arrangement was examined, and the correlation between wave interference effects and wave resistance was analyzed. In section 2, the thin ship theory and CFD analysis methods are described. In section 3, the numerical analysis results are presented, detailing the wave resistance characteristics for the middle ship with respect to the number of ships, separation distance, and ship speed, in connection with wave interference effects. Finally, section 4 summarizes and presents the findings of this study.

2. Methodology

2.1 Thin Ship Theory

Ship waves refer to the waveform of the Kelvin wake pattern generated at the stern of an advancing ship. These waves consist of divergent waves, which propagate diagonally, and transverse waves, which propagate horizontally from the stern to the rear. The wave elevation (ζ(x, y)) at a certain location (x,y) within three-dimensional (3D) ship waves can be expressed as the real part of the overlap of unit component waves propagating at various wave angles (θ) relative to the ship’s direction, as shown in Eq. (1) (Newman, 1977). In this instance, A (θ), a complex-valued function representing the wave elevation of unit component waves, is referred to as the complex wave amplitude function. If the wave resistance for the hull is calculated using this wave amplitude function and the energy conservation relationship, the wave resistance value (Rw) can be obtained using Eq. (2) (Newman, 1977).
(1)
ζ(x,y)=Rπ/2π/2A(θ)eik(θ)[xcosθ+ysinθ]]dθ
(2)
Rw=π2ρU2π/2π/2|A(θ)|2cos3θdθ
(3)
k(θ)=k0sec2θ
where U is the forward speed of the ship, and ρ is the density of the fluid. k(θ) is the wave number of the unit component wave that propagates at each advancing angle. It is expressed as Eq. (3), using the wave number ( k02=g/U2) for the ship direction based on the dispersion relation and taking into account the advancing angle. Here, g represents the gravitational acceleration. The wave amplitude function (A (θ)) can be theoretically estimated by considering the ship’s geometry, speed, and frequency. Many previous studies have proposed various methods for calculating A (θ), often using approximate hull shapes. In this study, Michell’s (1898) thin ship theory was applied to analyze ship waves and wave resistance. Thin ship theory assumes that the hull width is very small relative to the ship’s length. Based on this assumption, the wave amplitude function is expressed, as shown in Eq. (4), as an integral of the product of the micro-ship waves formed on the hull surface and the x-direction slope (ηx (x, z)) of the hull surface (Michell, 1898).
(4)
A(θ)=2π(gU2)sec3θηx(x,z)exp[(gU2)sec2θ(zixcosθ)]dxdz
The wave elevation and wave resistance around a ship moving at a constant speed can be calculated by substituting Eqs. (1) and (2) into Eq. (4). Since the analytical calculation of the hull surface slope required for determining the wave amplitude function is challenging for hulls with complex geometry, the ship geometry in this study was simplified into a hexagonal column, as shown in Fig. 1, by referring to the study by Zhang and Maki (2023). When the dimensions of the hexagonal column in Fig. 1 are set equal to the ship’s dimensions, the hull surface slope is expressed as different constant values depending on the location of the bow, midship, and stern, as shown in Eq. (5). Here, C is the hull surface slope of the bow and stern, LP is the length where the hull surface slope is zero, and L is the length between perpendiculars (LBP, LWL).
(5)
ηx(x,z)={BLLP=Cforx=[L2,LP2)0forx=[LP2,LP2)BLPL=Cforx=[LP2,L2]
The wave amplitude function can be calculated using Eq. (6) by substituting the hull surface slope from Eq. (5) into Eq. (4). Substituting Eq. (6) into Eq. (2) yields the wave resistance calculation formula for the simplified hexagonal column hull form, as shown in Eq. (7):
(6)
A(θ)=4Cπik0(1ek0sec2θD)[cos(0.5k0secθLP)cos(0.5k0secθL)]
(7)
R0=π2ρU2-π/2π/2(i4Cπik0(1-e-k0sec2θD)[cos(0.5k0secθLp)-cos(0.5k0secθL)])2cos3θdθ
The aforementioned ship wave and wave resistance formulas can be extended to conditions in which multiple ships navigate in formations. When multiple ships are navigating, the wave field and wave resistance are altered due to the interference of ship waves. This phenomenon can be explained using wave interference effects by linearly superposition of the ship waves generated by each hull, following the thin ship theory method presented by Tuck and Lazauskas (1998). Specifically, when the j-th ship is located at (xj, yj) and the wave amplitude function of the ship wave at the (x, y) point is defined as Aj(θ), the total wave amplitude function for N ships can be expressed as Eq. (8). If each ship in the formation has identical geometry, the wave amplitude functions become equal (Aj(θ)=A0 (θ)). If the exponential term representing the interference effect due to ship positions is replaced with F (θ), as shown in Eq. (9), the wave amplitude function can then be expressed as Eq. (10). Based on this, the formulas for the wave elevation in the wave field and wave resistance for multiple ships in formation are given as Eqs. (11) and (12), respectively.
(8)
A(θ)=j=1NAj(θ)eik(θ)[xjcosθ+yjsinθ]
(9)
F(θ)=j=1Neik(θ)[xjcosθ+yjsinθ]
(10)
A(θ)=A0(θ)F(θ)
(11)
ζ(x,y)=Rπ2π2A0(θ)F(θ)dθ
(12)
Rw=π2ρU2π/2π/2|A0(θ)|2|F(θ)|2cos3θdθ
Assuming that the separation distance between ships is identical (yj = w) in a parallel arrangement (xj = 0) where a total of N=2M+1 ships (M ships to the left and right, respectively) are arranged transversely, the exponential term (G (θ)=F (θ)2) based on the ship position can be expressed as Eq. (13). By substituting this into Eq. (12), the wave resistance for the middle ship in a parallel arrangement can be calculated using Eq. (14) (Zhang and Maki, 2023):
(13)
G(θ)=m=1M2cos(mw/LFr2cos2θsinθ)+1
(14)
R=π2ρU2π/2π/2|A0(θ)|2G(θ)cos3θdθ
Fig. 2 shows the results of calculating the ship waves of a single ship and those of three ships moving in a parallel arrangement with a separation distance of w/B = 3 based on the thin ship theory at a Froude number of 0.3. For the single ship, it can be observed that transverse waves are formed behind the stern and divergent waves are generated at an advancing angle of 19.47°. Meanwhile, in a parallel arrangement, the wave field geometry is altered by the interference effects between the waves generated by adjacent ships. This interference increases or decreases the transverse wave elevation, resulting in a more complex shape for the divergent waves. To verify the calculation results of the thin ship theory applied in this study, Fig. 3 compares the wave resistance results of the middle ship in the parallel arrangement of 15 ships (seven ships to the left and right, respectively) with the calculation results of Zhang and Maki (2023). The comparison shows that the wave resistance change characteristics with respect to separation distance and ship speed are in good agreement with the findings of Zhang and Maki (2023)

2.2 CFD Analysis

In this study, CFD simulation was performed to numerically analyze ship waves and ship resistance, considering the accurate ship geometry and viscosity effects. STAR-CCM+ (Version 2020.2 Build 15.04.008), a commercial CFD software program, was used. A computational mesh was constructed to analyze the flow field around the ship, and solutions to the mass, momentum, and energy conservation equations were numerically calculated by applying the finite volume method.
For the governing equations of the incompressible viscous flow field, the Navier-Stokes equation (Eq. (15)), representing the momentum conservation law, and the continuity equation (Eq. (16)), representing mass conservation, were applied. The flow field solution was derived by imposing appropriate boundary conditions.
(15)
uit+uiujxj=1ρpxi+v2uixjxj+1ρFi
(16)
ρt+(ρui)xi=0
The Reynolds-averaged Navier-Stokes (RANS) equation was used to analyze the wave resistance of the ship moving at a constant speed, and the kϵ model was applied as the turbulence model. To examine the ship waves generated by a ship moving over a free water surface, the volume of fluid (VOF) technique, capable of simulating ideal fluids for water and air, was utilized. The ONR-Tumblehome (ONRT) hull form was selected as the target ship, as shown in Fig. 4. Table 1 summarizes the specifications of the full-scale and model-scale ships.
Considering the calculation efficiency of numerical analysis, CFD simulation was performed for the fluid domain that included only half of the hull, based on the xz-plane passing through the fixed midship (half-domain). The size of the entire domain was set to 5 LWL in the flow direction, 2.5 LWL in the width direction, and 3 LWL in the vertical direction. The trim cell mesher was used for the hull surface mesh, and 10 prism layer meshers were generated from the hull surface to ensure that y+ remained at 30 or less. To clearly observe the ship waves, a detailed fan-shaped mesh system was formed within the ship wave advancing angle of 19.47° at the stern. An additional precise mesh domain was also created in the vertical direction near the free water surface to form a dense mesh. Fig. 5 shows the mesh system in the fluid domain. The total number of mesh cells ranged from 2.5 to 3 million, depending on the number of ships in the parallel arrangement. A time interval of △t = 0.01 s was applied, and calculations were performed for up to 60 s.

3. Numerical Analysis Results and Analysis

3.1 Effect of the Ship Speed

In this section, the wave resistance characteristics of three ships moving in a parallel arrangement were analyzed using thin ship theory-based calculations and CFD simulations, with a focus on variations in ship speed. Fig. 6 presents the wave resistance changes for the middle ship as a function of ship speed, with the separation distance fixed at w/B = 3.0. The x-axis represents the Froude number (Fn), which nondimensionalizes the ship speed, while the y-axis represents the wave resistance of the middle ship (R) in the parallel arrangement, normalized by the wave resistance of the ship under single operation conditions. In the figure, the solid line denotes the results from thin ship theory-based calculations, and the circular symbols correspond to the CFD simulation results. Regarding the wave resistance results based on thin ship theory, it was observed that the wavelength of ship waves increased as the ship speed increased, leading to periodic occurrences of destructive and constructive wave interferences. Within the Froude number range of 0.2 to 0.7, where calculations were performed, three maximum and minimum wave resistance values were identified. Specifically, wave resistance reductions of 23.3%, 38.4%, and 37.0% were observed at Froude numbers of 0.206, 0.239, and 0.329, respectively, due to destructive interference between ship waves. Conversely, wave resistance increased by 23.7%, 80.2%, and 64.4% at Froude numbers of 0.217, 0.258, and 0.410, respectively, due to constructive interference. The CFD simulation results revealed a similar overall trend of wave resistance increases and decreases compared to thin ship theory for Froude numbers up to 0.5. However, the maximum changes in wave resistance caused by wave interference were slightly less pronounced in the CFD simulations. For example, at Fn = 0.283, the CFD simulation indicated a maximum wave resistance increase of approximately 58%, which was 18.4% lower than the value predicted by thin ship theory. Similarly, at Fn = 0.325 the CFD simulation showed a resistance reduction of approximately 23.1%, compared to the 37.0% reduction predicted by thin ship theory. These discrepancies likely arise from the more complex wave patterns modeled in the CFD simulations, which accounted for the precise hull geometry and viscosity effects, resulting in weaker wave interference effects compared to thin ship theory. Furthermore, CFD simulations provided a more detailed analysis of wave interference by incorporating coupled effects of flow velocity, pressure, and free surface in a unified flow field. In contrast, thin ship theory approximates interference effects by linearly overlapping the waves generated by each ship, potentially oversimplifying the interactions.
Fig. 7 illustrates the ship wave field at different Froude numbers in a parallel arrangement, based on the CFD simulation results. In the graph, the wave elevation is nondimensionalized using the LBP of ONRT. High waves are represented by red contour plots, while low waves are depicted in blue. As the Froude number increased, both the wavelength and wave elevation of the ship waves increased. The wave field exhibited changes due to wave interference effects with adjacent ships under certain speed conditions. Fig. 8(a) compares the wave field at a Froude number of 0.283, where constructive interference between ship waves occurred, with the results for a single ship. While the wave field geometry at the bow was similar, more complex wave patterns were observed around the stern due to interference effects. Additionally, the parallel arrangement generated more complex and higher wave elevations in the wake region compared to the wave field of a single ship. Conversely, Fig. 8(b) shows the wave field geometry at a Froude number of 0.325, where destructive interference occurred. Compared to the constructive interference case, the waveform at this Froude number was relatively simple, and a low wave field was formed. To further examine wave field changes around the hull due to constructive and destructive interference, Fig. 9 compares wave elevations around the ship hull for the parallel arrangement and the single ship. The x-axis represents the distance along the ship direction, nondimensionalized by the ship length, while the y-axis represents the nondimensionalized wave elevation. At a Froude number of 0.283, corresponding to constructive interference, a clear increase in wave elevation toward the stern was observed, correlating with an increase in resistance. At a Froude number of 0.325, associated with destructive interference, a decrease in wave elevation toward the stern was evident.

3.2 Effect of the Separation Distance

Even under the same ship speed condition, the change in separation distance between ships (w/B) causes changes in ship wave interference characteristics. In this section, wave resistance change characteristics as a function of the separation distance were examined. Fig. 10 shows the changes in wave resistance with respect to the separation distance (w/B) under four ship speed conditions (F n = 0.1, 0.2, 0.3, 0.5) based on the calculation results of the thin ship theory. The x-axis represents the separation distance, and the y-axis indicates the nondimensionalized wave resistance value of the middle ship. Overall, wave resistance changes were clearly observed due to wave interference effects as the separation distance decreased. However, as the separation distance increased, the results converged to the wave resistance value of a single ship (R0) because wave interference effects from adjacent ships diminished. Since the wavelength of ship waves is proportional to the square of the ship speed (λ(θ)= 2πU2cos2 θ/g), numerous constructive and destructive interferences were observed with varying separation distances at low ship speeds due to the reduced wavelength of ship waves. At higher ship speeds, however, the number of constructive and destructive interferences within the same separation distance tended to decrease due to the increased wavelength of ship waves. As seen in the figure, the peak and trough values in wave resistance were observed at two and three separation distances (w/B), respectively, when the Froude number was low (F n = 0.1). In particular, wave resistance decreased by up to 80.6% at a separation distance of 2.137 due to destructive interference between ship waves. At a separation distance of 1.689, wave resistance increased by up to 60.3% due to constructive interference between ship waves. At Fn = 0.2, wave resistance increased by up to 69.1% (maximum value) at a separation distance of 2.418, and it decreased by 73.3% (minimum value) at a separation distance of 1.762. At Fn = 0.3, on the other hand, a relatively small wave resistance reduction of 11.5% occurred only at a separation distance of 3.322. At Fn = 0.5, there was no separation distance at which wave resistance decreased.
Fig. 11 compares the CFD analysis results with the thin ship theory results for wave resistance as a function of the separation distance at a Froude number of 0.3. Overall, the trends of increasing and decreasing wave resistance show good agreement between the two calculation methods within the separation distance (w/B) range of 2 to 5. Specifically, wave resistance decreased by approximately 8.0% at a separation distance of 3 in the CFD simulation results, which closely aligns with the prediction from the thin ship theory. However, at a lower separation distance (w/B = 2), wave resistance increased by 18.0% in the CFD simulation results compared to an increase of 33.7% in the thin ship theory results. This discrepancy appears to arise due to increased effects of the ship geometry and wake field as the separation distance decreases. Fig. 12 illustrates the wave field obtained from the CFD simulation under four separation distance conditions. Overall, the wave field approaches that of a single ship as the separation distance increases, owing to the reduction in wave interference effects. Conversely, as the separation distance decreases, an increase in wave elevation is observed due to constructive interference between ship waves. Notably, at a separation distance corresponding to the maximum wave resistance (w/B = 2), transverse waves show a noticeable increase near the stern compared to the wave field of a single ship.

3.3 Effect of the Number of Ships on the Left and Right Sides

In this section, wave resistance characteristics according to the number of ships in parallel arrangement were comprehensively examined. Fig. 13 presents the wave resistance for the middle ship as a function of the number of ships under three separation distance conditions (w/B = 1.5, 3.0, 4.5), based on the thin ship theory calculation results. Overall, the wave resistance tendency with respect to the Froude number was found to be similar despite an increase in the number of ships, and it converged to the wave resistance value for the parallel arrangement of infinite ships when the number of ships reached nine or more. At a low separation distance (w/B = 1.5), however, the peak and trough values of wave resistance varied depending on the number of ships. In particular, as the number of ships increased, the increment in wave resistance due to constructive interference exhibited a tendency to increase. For example, at a separation distance of 1.5, the peaks in wave resistance increased by 88.7%, 118.5%, and 113.6% compared to a single ship at Froude numbers of 0.227, 0.278, and 0.480, respectively, when the number of ships was three. However, these values increased by 102.9%, 174.3%, and 195.3% at Froude numbers of 0.222, 0.270, and 0.452, respectively, when the number of ships was nine. This result appears to be because waves generated by ships in parallel arrangement, beyond the two adjacent ships, influenced the middle ship at low separation distances. In the parallel arrangement of more than nine ships, wave interference effects decreased, and the peak values in wave resistance gradually converged as the separation distance of the last two ships increased. The trough values in wave resistance due to destructive interference showed slight variations depending on the number of ships, but no clear tendency was observed. Meanwhile, the influence of the number of ships diminished as the separation distance increased. Specifically, at a separation distance of 3.0, wave resistance increased by 64.3% at a Froude number of 0.408 when the number of ships was three. The overall trend remained identical when the number of ships was nine or more, except that wave resistance increased by 79.8% at a Froude number of 0.420. At a separation distance of 4.5, the peaks in wave resistance were 27.5% and 30.6% when the number of ships was three and nine, respectively, indicating the reduced impact of the number of ships. This phenomenon can be attributed to the gradually diminishing influence of waves generated by the ships on either side of the middle ship as the separation distance increased.
Fig. 14 shows the results of nondimensionalizing the wave resistance of the middle ship (R) by that of the single ship (R0) in contour plots, with the separation distance (w/B) represented on the x-axis and the Froude number (Fn) on the y-axis. The number of ships increases from three to nine across the figures from left to right, and the nondimensionalized wave resistance (R/R0) is indicated using contour plots within a range from zero to 2.5. High wave resistance is expressed in yellow, and low wave resistance is expressed in blue. Overall, the increase or decrease in wave resistance due to constructive and destructive interferences was relatively large and periodically observed with variations in ship speed at low separation distances. However, wave resistance change characteristics gradually decreased and converged to the wave resistance value of the single ship at high separation distances. Regarding the tendency of wave resistance with an increase in the number of ships, wave resistance change characteristics were similar at high separation distances, but changes in wave resistance caused by an increase in the number of ships were partially observed at low separation distances. In particular, wave resistance changes were distinctly observed at high Froude numbers. Operating conditions with a nondimensionalized wave resistance value of 2.5 or higher were observed at a Froude number between 0.45 and 0.52 and a separation distance of 1.7 or less when the number of ships was three. However, strong constructive interference occurred across a wider range, with a Froude number between 0.4 and 0.55 and a separation distance of 2 or less, when the number of ships was nine.
To additionally examine the impact of the number of ships in parallel arrangement, the ship wave geometry according to the number of ships was analyzed using CFD simulation. Fig. 15 shows the wave field formed around the hull as the total number of ships increased from three to five and seven, under a separation distance of w/B = 3 and Froude number conditions (Fn = 0.3, 0.4), based on the CFD simulation results. At a Froude number of 0.3, the wave field geometry formed at the stern of the middle ship was generally similar across different numbers of ships. Specifically, the change in the wave resistance of the middle ship due to the number of ships was very small (up to 1.3%), confirming that the variation in wave resistance with respect to the number of ships is insignificant at this Froude number. Fig. 16 compares the wave elevation around the middle ship as a function of the number of ships at a Froude number of 0.4. While the wave elevation at the bow was relatively similar regardless of the number of ships, the wave elevation at the stern was lower in the parallel arrangement of five and seven ships compared to the arrangement of three ships. Consequently, the wave resistance of the middle ship in the parallel arrangement of five and seven ships was 6.2% higher than that in the arrangement of three ships, which was consistent with the result (7.3%) from the thin ship theory.

4. Conclusions

In this study, a numerical analysis was conducted to examine the wave resistance characteristics for ships arranged in parallel. The wave resistance for the ONRT hull form in parallel arrangement was theoretically calculated using thin ship theory. Specifically, a series of calculations were performed to evaluate wave resistance changes for the middle ship by varying the number of ships, ship speed, and separation distance, all of which significantly influence wave interference effects. Additionally, direct numerical analysis was conducted through CFD simulations for the same scenario to investigate wave interference effects in detail, allowing for mutual validation and the identification of ship wave field. First, wave resistance characteristics were analyzed with respect to ship speed. As the ship speed increased, the wavelength of the ship waves also increased, leading to periodic occurrences of destructive and constructive interferences between ship waves. These interferences resulted in noticeable increases or decreases in wave resistance. However, in the CFD simulations, wave interference effects were relatively weaker compared to those predicted by thin ship theory because the simulations accounted for the precise ship geometry and viscosity effects. Next, wave resistance characteristics were examined as a function of separation distance under the same ship speed conditions. Wave resistance changes due to wave interference effects were prominently observed when the separation distance was reduced. Conversely, as the separation distance increased, the wave resistance values converged to those of a single ship, owing to the diminished wave interference effects caused by adjacent ships. Notably, under low ship speed conditions, multiple points with peak and trough values in wave resistance were observed, depending on the separation distance, due to the shorter wavelength of ship waves. In contrast, under high ship speed conditions, the number of wave interference effects decreased as the wavelength increased. Finally, wave resistance characteristics were comprehensively analyzed based on the number of ships in parallel arrangement. Overall, the wave resistance trend remained consistent regardless of the number of ships. However, changes in wave resistance as a function of the number of ships were partially observed at low separation distances and high ship speeds. In conclusion, further research is necessary to explore wave interference effects in various ship arrangements. More systematic validation studies will also be conducted through model experiments in future work.

Conflict of Interest

Bo Woo Nam serves as a journal publication committee member for the Journal of Ocean Engineering and Technology, but he did not have a role in the decision to publish this article. There are no potential conflicts of interest relevant to this article.

Fig. 1.
Waterplane view of the wedge-shaped body
ksoe-2024-082f1.jpg
Fig. 2.
Ship-generated waves of a single-ship (a) and a parallel arrangement of three ships (b) based on thin ship theory
ksoe-2024-082f2.jpg
Fig. 3.
Validation of wave resistance for a parallel arrangement of 15 ships
ksoe-2024-082f3.jpg
Fig. 4.
Hull shape of ONRT model
ksoe-2024-082f4.jpg
Fig. 5.
Mesh setup for CFD simulation
ksoe-2024-082f5.jpg
Fig. 6.
Variation in ship resistance with ship speed (w/B=3)
ksoe-2024-082f6.jpg
Fig. 7.
Wave fields at various ship speeds from CFD simulations (w/B = 3)
ksoe-2024-082f7.jpg
Fig. 8.
Comparison of wave fields between a single ship and parallel arrangement under constructive (top) and destructive (bottom) interference conditions
ksoe-2024-082f8.jpg
Fig. 9.
Comparison of wave elevations around the ship hull between a single ship and parallel arrangement
ksoe-2024-082f9.jpg
Fig. 10.
Variation in wave resistance with separation distance
ksoe-2024-082f10.jpg
Fig. 11.
Variation in ship resistance with separation distance (Fn = 0.3)
ksoe-2024-082f11.jpg
Fig. 12.
Wave fields with various separation distances (Fn = 0.3)
ksoe-2024-082f12.jpg
Fig. 13.
Wave resistance with various numbers of ships (Fn = 0.3)
ksoe-2024-082f13.jpg
Fig. 14.
Contour plots showing the variations in wave resistance for different numbers of ships
ksoe-2024-082f14.jpg
Fig. 15.
Wave fields with various numbers of ships (Fn = 0.3, w/B = 3)
ksoe-2024-082f15.jpg
Fig 16.
Comparison of wave elevations around the ship hull for different numbers of ships (Fn = 0.4, w/B = 3)
ksoe-2024-082f16.jpg
Table 1.
Main specifications of the ONRT model
Item Full scale Model (1/48.9)
Length – LWL (m) 154 3.147
Beam – BWL (m) 18.78 0.384
Depth – D (m) 14.5 0.266
Draft – T (m) 5.495 0.112
Displacement - △ (m3) 8,490,000 72.6
Block Coefficient - CB 0.535 0.535

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