Numerical Simulation Study on Ship–Ship Interference in Formation Navigation in Full-Scale Brash Ice Channels

Abstract

Formation navigation in brash ice channels is increasingly utilized by merchant vessels in the Arctic and Baltic Sea, offering benefits such as improved efficiency and reduced carbon emissions. However, ship–ship interference poses a significant challenge to this method, impacting resistance performance. This paper presents full-scale simulations using the CFD–DEM coupling method in brash ice channels, which is validated by comparing simulation results with ice tank measurements. By varying the distance between two ships from 0.05 to 5 ship lengths, ship–ship interference in full-scale brash ice channels is analyzed using the CFD–DEM coupling strategy. The study examines hydrodynamic and ship–ice interactions, ice resistance effects, and simulation results. It is found that ship-to-ship distance significantly influences the velocity field, dynamic pressure distribution on the hull, and hydrodynamic interaction forces. Distances less than one ship length result in increased water resistance for the forward ship and decreased resistance for the rear ship. The forward ship demonstrates favorable interference with the ice accumulation of the rear ship. When distances are less than two ship lengths, the ice resistance of the forward ship remains mostly unaffected, while the ice resistance of the rear ship decreases as the distance decreases. These insights enhance our understanding of ship–ship interference in formation navigation, aiding in the optimization of brash ice channel navigation strategies.

1. Introduction

With the continuous exploitation of Arctic resources and the increasing demand for shipping in the Arctic and Baltic Sea, the number of merchant vessels navigating in ice-covered waters is on the rise. The current winter navigation system in the Baltic Sea employs assisting icebreakers and state authorities controlling the regulations and traffic restrictions in addition to the merchant fleet [1]. Formation navigation of these vessels in brash ice channels has become a major mode of navigation (see Figure 1). Assisting icebreakers are piloted in front and ice-strengthened vessels follow behind. However, the issue of mutual interference between ships during formation navigation cannot be ignored, particularly when it comes to the effects on resistance and heading performance. These factors directly impact the efficiency and safety of formation navigation. Therefore, conducting in-depth research on the phenomenon of resistance interference between ships navigating in brash ice channels is necessary.

The brash ice fragments in these channels are relatively small and less likely to break, allowing us to assume that the possibility of the ice breaking is negligible [2]. The discrete element method (DEM) is highly suitable for simulating brash ice, due to its inherent discreteness. Additionally, the motion of brash ice is influenced by the viscosity of water, and the water resistance in brash ice channels cannot be overlooked. Thus, employing the CFD–DEM coupling method to simulate ship–ship interference in ice-breaking channels is highly appropriate.

Numerous papers have already been published on the numerical simulation of resistance performance under pack conditions, including brash ice channels, using the computational fluid dynamics–discrete element (CFD–DEM) coupling method [3,4,5,6]. However, these simulations are based on a model scale, and the correction method from model-scale results to full-scale results is an important issue. The traditionally employed scaling methods, such as maintaining the Froude and Cauchy similitudes, might not be the optimal solution in tests conducted in broken ice, as the interaction of ice fragments becomes a significant factor in ship resistance [1,7]. Thus, conducting full-scale simulations proves to be an effective method. Mucha [8] simulated resistance, propulsion, and local flow fields in full-scale brash ice channels, comparing one-way and two-way CFD–DEM simulations. The study reported that using two-way coupling did not yield noticeable differences in ship resistance predictions. Zhang et al. [9] simulated resistance in full-scale brash ice channels and compared the results with experimental results, the FSICR formula, and Dubrovin’s formula. The findings indicated that the CFD–DEM coupling method could provide reasonable resistance results in full-scale brash ice channels. Furthermore, Polojärvi et al. [10] simulated local ice loads in an ice-floe field at full scale using 3D DEM and compared them with measurements taken at full scale. Therefore, full-scale simulation is applicable in brash ice channels and ice-floe fields, with a primary focus on resistance and ice load. However, research progress on the mutual interference of resistance performance between ships in the Arctic fleet navigation formation remains relatively limited.

Our earlier research on ship resistance in brash ice channels focused on the model scale using the CFD–DEM coupling strategy. The simulation results and observed phenomena were found to be in good agreement with the test results and observed phenomena [6]. Subsequently, we further studied the self-propulsion performance in model-scale brash ice channels by integrating the discretized propeller model [2]. This approach allowed us to obtain information on the thrust, developed power, and propulsion efficiency, as well as the ice load on the hull and propeller. However, for formation navigation in brash ice channels, full-scale simulation is more appropriate than model-scale simulation.

Initially, we conducted a full-scale simulation of a polar ship using the CFD–DEM coupling method in brash ice channels and compared the results with tank test results. Building on this foundation, we assume that the formation navigation of two identical polar ships in brash ice channels. Several simulations were carried out to investigate the impact of ship-to-ship distance on ship performance. These simulations encompassed different distances between the ships, ranging from 0.05 to 5 ship lengths. Finally, we analyzed and discussed the simulation results and phenomena related to the formation navigation.

2. Numerical Method

In the simulations, the fluid was considered to be an incompressible Newtonian fluid that adheres to the continuity and momentum conservation equations. Heat exchange between the fluid and discrete ice particles was disregarded. As the standard Reynolds-averaged Navier–Stokes (RANS) equations account for turbulent fluid, the shear stress transport (SST) $k-\omega$ turbulence model was employed to close the equations. Further details regarding the specific numerical schemes applied can be found in Ferziger et al. [11]. The CFD–DEM coupling strategy employed in this study models the fluid phase using RANS equations, while the translational and rotational motions of discrete particles are described by Newton’s and Euler’s second laws of motion, respectively.

2.1. CFD–DEM Coupling

In the ship–ice–water interaction process, various mechanical properties and heat transfer/exchange processes between water and ice come into play. However, this paper primarily focuses on the mechanical properties while disregarding heat transfer and exchange. It assumes that heat is in balance between the discrete ice particles and the fluid. The model incorporates different force components acting on the particles, including gravitational force, contact forces between colliding particles, and fluid–particle interaction forces. The CFD–DEM formulation and the coupling of mass and momentum between phases are as follows [12]:

$$\frac{\partial ({\rho }_f{\epsilon }_f)}{\partial t}+\nabla \cdot ({\rho }_f{\epsilon }_f\overrightarrow{u})=0,$$

(1)

$$\frac{\partial ({\rho }_f{\epsilon }_f\overrightarrow{u})}{\partial t}+\nabla \cdot ({\rho }_f{\epsilon }_f\overrightarrow{u}\cdot \overrightarrow{u})=-{\epsilon }_f\nabla p-\nabla \cdot ({\epsilon }_f{\overleftrightarrow{\tau }}_f)+{\rho }_f{\epsilon }_f\overrightarrow{g}-\overrightarrow{F},$$

(2)

where ρ_f and ε_f are, respectively, the density and volume fraction of the fluid term in the control volume. The relationship between the variables is ${\epsilon }_f=1-{\epsilon }_p$ and ${\epsilon }_p={\displaystyle \sum _{i=1}^{n_p}{V}_{p_i}/\Delta V}$ as follows: ε_p represents the volume integral number in the control volume for the discrete ice particle term; $V_{p_i}$ and ∆V represent the volume of the discrete ice particles numbered i and the total volume of the regional control volume, respectively. The average velocity of the fluid is denoted by $\stackrel{\rightharpoonup }{u}$, p represents the mean value of pressure, and $\stackrel{\rightharpoonup }{F}$ represents the volume average of the resistance of the particles to the surrounding fluid in the discrete ice term of the control volume. This resistance includes factors such as resistance, pressure-gradient force, shear stress, and other interaction forces between the fluid term and the discrete ice term.

The stress tensor of the fluid ${\overleftrightarrow{\tau }}_f$ is expressed as

$${\overleftrightarrow{\tau }}_f=-u_f(\nabla \stackrel{\rightharpoonup }{u}+{(\nabla \stackrel{\rightharpoonup }{u})}^{\prime })+(\frac{2}{3}{\mu }_f-k)(\nabla \cdot \stackrel{\rightharpoonup }{u})\overleftrightarrow{\delta },$$

(3)

where $\overleftrightarrow{\delta }$ is the unit tensor, and $u_f$ and k are dynamic and dilatational viscosities of the fluid, respectively. It should be noted that for an ideal monoatomic gas (extended to ideal gases), k is zero, and, for an incompressible fluid, $\nabla \cdot \stackrel{\rightharpoonup }{u}=0$.

For ship–ice–water coupling interactions, the average volume $\stackrel{\rightharpoonup }{F}$ of particles in the discrete ice term is expressed as [2]:

$$\stackrel{\rightharpoonup }{F}=\frac{1}{V_{cell}}{\displaystyle \sum _{i=1}^{n_p}\stackrel{\rightharpoonup }{F_i}}=\frac{1}{V_{cell}}{\displaystyle \sum _{i=1}^{n_p}(\stackrel{\rightharpoonup }{F_{di}}+\stackrel{\rightharpoonup }{F_{bi}}+\stackrel{\rightharpoonup }{F_{li}}+\stackrel{\rightharpoonup }{F_{ai}}+\stackrel{\rightharpoonup }{F_{pi}})},$$

(4)

where n_p is the number of discrete ice particles in each fluid control volume, u_f is the dynamic viscous coefficient of the fluid, $\overleftrightarrow{\delta }$ is the unit tensor, and $\stackrel{\rightharpoonup }{F_{di}}$, $\stackrel{\rightharpoonup }{F_{bi}}$, $\stackrel{\rightharpoonup }{F_{li}}$, $\stackrel{\rightharpoonup }{F_{ai}}$, and $\stackrel{\rightharpoonup }{F_{pi}}$ are the drag force, buoyancy force (included in the pressure-gradient force), lift force, additional mass force, and pressure-gradient force of the fluid term acting on the ice particles, i, in the discrete ice term, respectively.

2.2. Particle Contact Model in the DEM

The contact-force model utilized in the DEM typically adopts a variant of the springdashpot model. This model involves a spring that generates a repulsive force, pushing the particles apart, while the dashpot accounts for viscous damping and allows for the simulation of collision types beyond perfect elasticity. At the contact point, the forces are represented by a pair of spring-dashpot oscillators. The normal force is modeled by a parallel linear spring-dashpot, while the tangential force direction in relation to the contact-plane normal vector is represented by a parallel linear spring-dashpot combined with a slider. In both cases, the spring accounts for the elastic response, while the dashpot accounts for the energy dissipation during collisions [13].

In this study, we have chosen a computationally efficient and accurate linear spring contact model based on the work of Coudall and Strack [14].

The contact-force model is shown in Figure 2, where F_n is the normal force, and F_t is the tangential force.

The contact force between two particles is given by

$$F_{contact}=F_{nij}+F_{tij},$$

(5)

where F_nij is the normal force, and F_tij is the tangential force.

The normal force is

$$F_n=-K_n{d}_n-N_n{v}_n,$$

(6)

where K_n is the normal spring stiffness, d_n is the normal overlap of the contact point, N_n is the normal damping, and v_n is the normal component of the sphere surface velocity at the contact point.

The expression for the tangential force is

$$F_t=\{\begin{array}{l}-K_t{d}_t-N_t{v}_t, \left|K_t{d}_t\right|<\left|K_n{d}_n\right|C_{fs}\\ \frac{\left|K_n{d}_n\right|C_{fs}{d}_t}{\left|d_t\right|}, \left|K_t{d}_t\right|>\left|K_n{d}_n\right|C_{fs}\end{array},$$

(7)

where K_t is the tangential spring stiffness, d_t is the tangential overlap of the contact point, N_t is the tangential damping, v_t is the tangential component of the sphere surface velocity at the contact point, and C_fs is the friction coefficient between particles.

3. Ice Tank Test Description

The model test of a polar ship in a brash ice channel was conducted at the Hamburg Ship Model Basin (HSVA) [15]. An ice-strengthened Panamax bulk carrier model was utilized in the ice tank. The ship model was scaled down to a ratio of 1:30.682. The experiment involved a towed propulsion test, and

Table 1. Main particulars.

Principal Hull Data	Abbreviation	Full Scale	Model Scale
Scale	λ	1	30.682
Length between perpendiculars	Lpp (m)	217.00	7.073
Length at scantling draft	Lwl_Scantl. (m)	221.07	7.205
Length at ballast draft	Lwl_Ballast. (m)	214.35	6.986
Breadth molded at DWL	B (m)	32.25	1.051
Scantling draft	TFScantl. (m)	14.73	0.480
	TAScantl. (m)	14.73	0.480
Ballast draft	TFBallast (m)	7.15	0.233
	TABallast (m)	5.05	0.165

provides the principal dimensions of the vessel in both full and model scales.

During the preparation process for the brash ice channel, a parental-level ice sheet of adequate thickness is prepared according to HSVA’s standard model ice preparation procedure. After that, the ice strip between the two cuts is manually broken up into relatively small ice pieces using special ice chisels, as shown in Figure 3a. Those channel sections, where the ice pieces remain in a regular pattern, are cautiously stirred in order to achieve the most realistic appearance of the brash ice channel. For the second test run, the ice pieces are rearranged in the channel. A load cell was installed at the bow to measure the pull force (FP) during the towed propulsion test, as illustrated in Figure 3b.

Model tests were conducted for both scantling and ballast drafts, representing loaded and ballast conditions, respectively. These tests focused on the upper ice water line and the lower ice water line. A single ice sheet was prepared to simulate the brash ice used in the towed propulsion tests. The ice thickness in the ice tank was selected to align with polar ships with ice-class FSICR IA. Specifically, the target brash ice thickness was set at 1.42 m in full scale and 46.3 mm in model scale. The model ship speed was fixed at 0.464 m/s, which corresponds to 5 kn in full-scale measurements. During the towed propulsion tests, the ship model was pulled by a carriage moving at a constant speed while being simultaneously propelled by its own propeller system. The test matrix detailing the experimental configurations is presented in

Table 2. Target brash ice thickness.

Ice Class	Loading Condition	Ship Speed (Full Scale) (kn)	Ship Speed (Model Scale)(m/s)	Target Ice Thickness (Full Scale) (m)	Target Ice Thickness (Model Scale) (mm)
FSICR IA	Loaded draft	5.0	0.464	1.42	46.3
FSICR IA	Ballast draft	5.0	0.464	1.42	46.3

. In a towed propulsion test, the model is pulled by the carriage running at a constant speed (as in a resistance test) and simultaneously driven by its propeller. During a test run, the rate of revolution is gradually changed, usually four times. The propeller’s lowest revolution rate is running close to the zero-thrust condition. The highest rate of revolution is chosen, so that the propeller thrust is clearly above the self-propulsion point (i.e., the model is pushing the carriage). Then, the full-scale resistance is extrapolated from model test results [15].

4. Numerical Simulation

4.1. Ship Model

The ship model used in this study is an ice-strengthened Panamax bulk carrier, which underwent testing in the HSVA ice tank, as illustrated in Figure 4. The primary focus of this paper is ship–ship interference during navigation in full-scale brash ice channels; thus, only the loaded draft condition was considered. A full-scale numerical simulation was employed and subsequently compared with the extrapolated results obtained from the model test.

4.2. Brash Ice Model

According to FSICR IAS, there is a 0.1 m thick consolidated layer of level ice. However, for FSICR IA, IB, and IC, there is no consolidated layer. The ice class in this paper is FSICR IA, so the brash ice channels do not consider the consolidated layer. The shape of the brash ice model is a significant factor influencing ice resistance (Xie et al., 2022). Another important distinction between level ice and brash ice channel resistance lies in the characteristics of the ice fragments moving along the vessel surface. The ice fragments in a brash ice channel exhibit a distinct form and possess different friction properties compared to freshly broken ice fragments from an intact ice sheet. In nature, the ice fragments within a brash ice channel undergo multiple cycles of breaking and freezing. Consequently, they become more solid, rounded (as depicted in Figure 5a), smoother, and are likely to induce less friction during interaction with the hull than freshly broken level ice fragments [6]. Therefore, the brash ice model is designed to have a polyhedral shape that closely resembles a circular form in full-scale brash ice channels (as shown in Figure 5b).

To ensure consistency between the numerical simulation and the ice tank test, the characteristic parameters of the brash ice model were set to approximate those of the brash ice used in the ice tank. Thus, the length of the brash ice models was chosen to be approximately 0.71 m. The characteristic parameters of the brash ice model are presented in

Table 3. Characteristic parameters of brash ice.

Parameter	Value
Elastic Modulus E (GPa)	2.7
Poisson’s ratio γ	0.35
Ice–ship friction coefficient f	0.1
Density ρi (kg/m3)	920
Brash ice diameter (m)	approx. 0.71
Ice thickness (m)	1.42

.

4.3. Computational Domain

The computational domain used for comparing the test results in the brash ice channel is illustrated in Figure 6. The dimensions of the domain are as follows: −2.5 L_pp ≤ x ≤ 3 L_pp; −2.0 L_pp ≤ y ≤ 2.0 L_pp; −2.0 L_pp ≤ z ≤ 1.0 L_pp. The inlet, sides, top, and bottom boundaries were designated as velocity inlets, while the outlet was set as a pressure outlet, as depicted in Figure 6. In order to facilitate comparison with the experimental results, the hull and rudder were defined as no-slip walls. However, it is worth noting that the guidelines established by the Finnish Maritime Administration (FMA) recommend a brash ice channel width twice that of the ship’s beam, which is based on model tests and does not align with reality. In nature, the width of full-scale brash ice channels is not necessarily twice the width of the ship’s beam. In this paper, the width of the full-scale brash ice channel is determined by adding the ship’s width to twice the thickness of the ice [16].

The simulation employed the method of relative motion, where the ship remained stationary while the water and brash ice moved at the ship’s speed. The brash ice particles were introduced into the water through injectors, which determine the location, direction, and frequency at which the particles enter. Various types of injectors can be used, such as part injectors, point injectors, line injectors, table injectors, and surface injectors. For this study, a line injector was utilized, as illustrated in Figure 6.

In order to examine ship–ship interference during navigation in full-scale brash ice channels, an additional ice-strengthened Panamax bulk carrier was introduced. This resulted in two identical ships navigating in the brash ice channels, with one positioned at the front (referred to as Ship A) and the other at the rear (referred to as Ship B). The computational domain used for ship–ship interference is illustrated in Figure 7. The boundary conditions of the computational domain are the same as those in Figure 6. Simulations were conducted with varying distances between the two ships, labeled as L_distance, ranging from 0.05 to 5 ship lengths.

4.4. Grid Generation

The grid and wall Y+ utilized for comparison with the test results are depicted in Figure 8. A hexahedral trimmer and surface remesher were employed to generate the global volume grid while allowing for local refinements. The boundary layer grid was constructed using a prism layer grid. The complete grid of the computational domain is presented in Figure 8a. In order to ensure an accurate simulation of brash ice motion in water and the ship–ice contact load, further refinement was implemented for the hull surface mesh, the free surface mesh of the ship–ice contact area, as well as the stem and stern, as shown in Figure 8b,c. To resolve the turbulent boundary layer, wall prism layers were employed. For the full-scale simulation, the wall Y+ of the submerged section of the hull ranged between 30 and 260 before uniformly expanding to the outer volume mesh. The number of prism layers was set at 12, with a prism layer stretching factor of 1.5 and a total thickness of 1.86 m. The wall Y+ of the hull is illustrated in Figure 8d.

5. Comparison between Simulation Results and Test Results

For ice-class FSICR IA, the brash ice thickness was 1.42 m at full scale, and the ship’s speed was fixed at 5 kn according to the ice tank test. The full-scale resistance is extrapolated from the model test results [15]. Before delving into the study of ship–ship interference, it is crucial to explore the numerical simulation methods. This section focuses on conducting full-scale simulations in brash ice channels and comparing the results with the ice tank test to validate their accuracy.

Figure 9 illustrates the simulated resistance–time curves of the ship in a brash ice channel. The initial stage of the simulations takes place in ice-free waters, spanning from 0 to 250 s. During this stage, a time step of 0.1 s and first-order temporal discretization are employed to expedite convergence. The second stage commences when the water resistance stabilizes from the first stage. The arrangement of brash ice begins through the use of a part injector, which persists for approximately 250 to 350 s. To facilitate the arrangement process, the time step and temporal discretization remain the same as in the first stage. Subsequently, the ship model enters the brash ice channel. Throughout this stage, a time step of 0.05 s and second-order temporal discretization are utilized to achieve sufficient accuracy. The ice resistance reaches a steady state after 150 s. The total simulation time varies within the range of 500 to 900 s, following which the average values of total resistance, water resistance, and ice resistance are obtained. As depicted in Figure 9, the contact between the ship and the brash ice begins at 350 s, and the ice resistance gradually increases. Due to the stochastic nature of ship–ice collisions, the ice resistance exhibits fluctuations, thereby causing variations in the total resistance.

Table 4. Simulation results.

$\mathbf{Ice}\mathbf{Thickness}{\mathit{h}}_{\mathit{i}}$/m			Resistance Value/kN	Percentage (%)	Error (%)
1.42	Experimental results (N)	Total resistance	1075.2	----	----
	Simulation results (N)	Water resistance	144.4	14.0	----
		Brash ice resistance	887.6	86.0	----
		Total resistance	1032.1	100	−4.01

presents the simulated resistance components and a comparison with the test results. The simulated resistance components are the average values, taken from the resistance–time curve within the range of 500 to 900s. The analysis indicates that the disparity between the simulation and test results is about 4%, signifying the high accuracy of the simulation outcomes. The percentage of ice resistance in the total resistance amounts to 86.0%, underscoring that brash ice is the primary source of resistance in brash ice channels.

6. Analysis of Formation Navigation Simulation Results

The simulation results of ship–ship interference in full-scale brash ice channels are analyzed, considering the effects of longitudinal distances between ships, hydrodynamic interaction, and ship–ice interaction. To facilitate clearer presentation, analysis, and discussion, the longitudinal distances between ships are nondimensionalized as $l=\frac{L_{distance}}{L_{pp}}$. The primary focus is on the influence of longitudinal distances on resistance in brash ice channels. Therefore, the nondimensional longitudinal distance was investigated within the range of 0.05 to 5, specifically at values of 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 3.0, and 5.0. The numerical simulation cases are summarized in

Table 5. Cases of numerical simulations.

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
l	0.05	0.1	0.2	0.5	1.0	2.0	3.0	5.0

.

6.1. Analysis of Simulation Results

The total resistance in brash ice channels comprises ice resistance and water resistance, with water resistance further divided into pressure resistance and friction resistance. Figure 10 illustrates the relationship between resistance and the nondimensional longitudinal distance. The force pointing towards the stern is resistance (positive value), and the force pointing towards the bow is thrust (negative value).

Figure 10a depicts the variation in pressure resistance. It is evident that the pressure resistance of Ship A remains stable, slightly exceeding that of Ship B when $l\ge 1$. This is because the wake of Ship A is nearly dissipated, but still has a slight impact on the inflow of Ship B. When $0.2<l<1$, the pressure of Ship A gradually increases, while that of Ship B decreases. When $l\le 0.2$, the pressure of Ship A becomes negative (providing thrust), and the pressure of Ship B rapidly increases as $l$ decreases. This indicates that ship interference becomes increasingly significant.

Figure 10b presents the variation in friction resistance. When $l\ge 1$, the friction resistance of Ship A is stable, while that of Ship B slightly decreases as $l$ increases. This is due to the minimal impact of Ship A’s wake on the inflow of Ship B. When $l<1$, the friction resistance of Ship A gradually decreases, whereas that of Ship B increases as $l$ decreases. This occurs because Ship B obstructs the wake behind Ship A, resulting in increased inflow compared to the ship’s speed. Consequently, the friction of Ship A increases, while that of Ship B exhibits the opposite trend.

Figure 10c shows the variation in water resistance, which follows a pattern similar to pressure resistance. When $l\le 0.1$, the water resistance of Ship A becomes negative, indicating that the blocking effect of Ship B already provides thrust on Ship A.

Figure 10d depicts the variation in ice resistance. The rule governing ice resistance is less obvious due to the randomness of ship–ice interactions. Generally, the ice resistance of Ship A remains stable, ranging between 830 kN and 940 kN. This is because the ice resistance of Ship A primarily arises from the brash ice ahead and is unaffected by the brash ice behind. When $l\ge 2$, the ice resistance of Ship B is unaffected by the distance but is approximately 30% lower than that of Ship A. This occurs because a portion of the brash ice is squeezed out of the channel by Ship A. When $l<2$, the ice resistance of Ship B decreases as $l$ increases, as it encounters less brash ice. Particularly, when $l<0.5$, the ice resistance of Ship B decreases rapidly.

Figure 10e presents the variation in total resistance. When $l\ge 2$, the total resistance of Ship A remains stable, ranging between 967 kN and 1072 kN. However, when $l<2$, the total resistance of Ship A decreases as $l$ decreases, primarily due to a rapid decrease in water resistance, which can even provide thrust. When $l\ge 2$, the total resistance of Ship B is unaffected by the distance but is less than 30% lower than that of Ship A. When $0.2<l<2$, the total resistance of Ship B gradually decreases as $l$ decreases, whereas when $l\le 0.2$, it increases gradually as $l$ decreases. This is because the water resistance of Ship B increases while the ice resistance decreases as $l$ decreases, and the proportion of water resistance gradually exceeds that of ice resistance with $l$ decreasing.

In summary, the distance between two ships during formation navigation significantly impacts the resistance performance of both ships. When $l\le 1$, the total resistance of Ship B can be reduced by more than 25%, while the total resistance of Ship B is less affected except for $l\le 0.2$. However, it is essential to maintain a sufficient distance to ensure navigation safety.

6.2. Hydrodynamic Interaction between Ships

Water resistance is a significant factor that affects ship performance, especially in brash ice channels. When ships navigate in close proximity, the interference between their bow and stern waves creates a larger wake, which can have a negative impact on ship performance. The distance between ships plays a crucial role in determining the level of interference in water resistance.

Figure 11 illustrates the velocity field and dynamic pressure distribution in the side view (Ship A on the right and Ship B on the left), while the dynamic pressure distribution in the front and back views is shown in Figure 12. In Figure 11a–d, it is evident that the inflow velocity field of Ship B near the free surface slightly differs from that of Ship A. As a result, the high-pressure zone at the bow of Ship B is slightly lower than that of Ship A, as depicted in Figure 12a–d. Consequently, the pressure resistance and friction resistance of Ship B are slightly lower than those of Ship A, as shown in Figure 10a,b. Moreover, the velocity field and dynamic pressure distribution are not affected by distance when $l\ge 1$. This is because the stern track energy of Ship A is almost completely dissipated by the time the stern track reaches Ship B.

Figure 11e demonstrates a visible change in the velocity field between Ship A and Ship B when $l=0.5$. The high-pressure zone at the bow of Ship B is lower than that of Ship A, as shown in Figure 12e. In Figure 11f–h, the change in the velocity field between Ship A and Ship B becomes increasingly apparent as the distance decreases when $l<0.5$. This indicates that Ship B increasingly affects the hydrodynamic performance of Ship A. Consequently, the bow high pressure of Ship B gradually decreases, and the stern low pressure of Ship A gradually increases as the distance decreases, as depicted in Figure 12f–h. Therefore, the pressure resistance and friction resistance of Ship A decrease, while those of Ship B increase as the distance decreases, as shown in Figure 10a,b.

6.3. Ship–Ice Interaction and Ship–Ship Interference Phenomenon

Figure 13 provides an illustration of the ship–ice interaction phenomenon when the distance between the ships is one ship length. From Figure 13a, it is evident that the closed ice wake of Ship B is longer compared to that of Ship A. Behind the ship, accumulations of brash ice break up, resulting in ice pieces floating freely and filling the channel in its wake. The closed length of the ice wake for Ship A, denoted as L_close, is approximately 20% of L_pp. Observing Figure 13b,c, it becomes apparent that ice accumulation at the bow of Ship A is notably more severe than that of Ship B. This disparity arises from a substantial amount of brash ice being forced down to the bottom of the level ice, consequently reducing the quantity of brash ice in front of Ship B. Figure 13d demonstrates the accumulation of brash ice around Ship A, forming a buildup in front of the bow, along the channel sides, and beneath the level ice.

To further investigate the effect of distance on ice resistance, an analysis was conducted on ship–ice interaction and ship–ship interference phenomena. Figure 14 illustrates these phenomena (Ship A on the right and Ship B on the left). In Figure 14a–c, it is evident that the ice accumulation at the bow of Ship B remains nearly unchanged when $l\ge 2.0$. However, the closed length of the ice wake of Ship B is longer compared to that of Ship A. This occurs because the amount of brash ice passing by Ships A and B is significantly reduced, resulting in a slower closure of brash ice at the stern of Ship B. Figure 14d–h demonstrate a gradual decrease in the amount of brash ice encountered by Ship B, which subsequently leads to a gradual decrease in ice resistance for Ship B, as shown in Figure 10d. Notably, when $l\le 0.1$, there is minimal accumulation of brash ice at the bow of Ship B. Therefore, it can be concluded that Ship A has a favorable interference with the ice accumulation of Ship B when $l<2.0$.

7. Discussion

In this paper, we propose a numerical simulation method to study ship–ship interference during navigation in full-scale brash ice channels and evaluate the impact of ship-to-ship distance on hydrodynamic and ice resistance. For the numerical simulations, a DELL workstation using 48 Xeon CPU cores at 2.69 GHz with 256 GB of RAM was used. The computational cost of the simulation is listed in

Table 6. Comparison of computational cost.

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
l	0.05	0.1	0.2	0.5	1.0	2.0	3.0	5.0
Grid cell count (millions)	235.6	235.6	236.6	238.6	241.7	246.2	251.0	257.6
Simulation time (hours)	48	48	49	54	64	89	121	206

. It is obvious that the computational domain increases and the grid cell count also increases as l increases. Since grid cell count is mainly due to the increase of volume grid count, and the prism layer grid count remains unchanged, the grid cell count does not increase sharply. At the same time, the computational cost increases with l increasing. The increasing trend is not a linear increase, but a non-linear increase, as shown in Figure 15, because the number of DEM ice fragments increases as l increases, leading to a further increase in computational cost.

The interaction between ships during formation navigation can significantly affect their performance. The distance between ships is a crucial factor that influences the resistance performance during formation navigation. Smaller distances between ships result in greater resistance interference due to the strong flow-field changes caused by the forward ship and the bow compression effect of the rear ship. When $l<1$, although the total resistance of both the forward and rear ships decreases as the distance decreases, there is a limit to how close the ships can be to ensure collision-free navigation. In harsh sea conditions, a larger distance between ships is necessary to ensure safe navigation. Therefore, further research is needed to determine the reasonable safety distance for emergency ship stopping, collision avoidance, and maneuvering.

8. Conclusions

This paper presents the study of ship–ship interference during formation navigation in brash ice channels. To validate the simulation results, full-scale simulations of a polar ship in brash ice channels were conducted and compared with ice tank tests. Subsequently, a formation navigation model of two identical ships was established in brash ice channels. Using the CFD–DEM coupling strategy, simulations were performed at eight distances between the ships, ranging from 0.05 to 5 ship lengths. Based on the analysis of the simulation results, the following conclusions can be drawn:

(1)

The difference in total resistance between the full-scale simulation and the ice tank test is about 4%. The simulation results of the total resistance at full scale align well with the test results.

(2)

The distance between ships during formation navigation has a significant impact on their resistance performance. When $l\le 1$, as the distance decreases, the water resistance of Ship A increases, while that of Ship B decreases. When $l\le 2$, the ice resistance of Ship A remains largely unaffected, whereas the ice resistance of Ship B decreases as the distance decreases. When $l\le 1$, the total resistance of Ship B can be reduced by more than 25%.

(3)

The influence of distance on the velocity field between the two ships and the dynamic pressure distribution on the hull differs. When $l\ge 1$, the velocity field and dynamic pressure distribution on the hull are not significantly affected by distance. When $l\le 0.5$, the effects become more pronounced.

(4)

Ship–ice interaction and ship–ship interference are greatly influenced by the distance between the ships. The length of the ice wake behind Ship B is longer than that behind Ship A, approximately equal to about 20% of L_pp. Notably, when $l<2.0$, Ship A exhibits favorable interference with the ice accumulation of Ship B.

In summary, this study demonstrates the importance of considering ship–ship interference during formation navigation in brash ice channels. The results highlight the significant impact of distance on the resistance performance, velocity field, dynamic pressure distribution, ship–ice interaction, and ship–ship interference. Future research should continue to examine the impact of ship-to-ship distance on ship performance, gain a deeper understanding of ship–ship interference mechanisms, and employ effective technologies and measures to mitigate interference effects. This will improve economic benefits and ensure safe navigation during formation navigation.