Open AccessArticle

Numerical Study on the Influence of Drift Angle on Wave Properties in a Two-Layer Flow

Xiaoxing Zhao

^1,2,

Liuliu Shi

^1,2,*

and

Eryun Chen

^1,2

School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, Shanghai 200093, China

Author to whom correspondence should be addressed.

J. Mar. Sci. Eng. 2024, 12(12), 2139; https://doi.org/10.3390/jmse12122139

Submission received: 29 October 2024 / Revised: 18 November 2024 / Accepted: 21 November 2024 / Published: 23 November 2024

(This article belongs to the Section Ocean Engineering)

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Figure 1
Schematic of the DARPA SUBOFF bare hull model. "> Figure 2
Schematic of the computational domain. "> Figure 3
Grids in the vertical central plane. "> Figure 4
Grids in proximity to the submarine’s surface. "> Figure 5
Rankine ovoid model. "> Figure 6
Comparison of numerical and experimental results [<a href="#B24-jmse-12-02139" class="html-bibr">24</a>]. "> Figure 7
Free surface wave of the Rankine ovoid. "> Figure 8
Distribution of free surface waves at a submergence depth of h = 1.1 D. "> Figure 9
Distribution of free surface waves at a submergence depth of h = 2.0 D. "> Figure 10
Free surface wave profiles at different submergence depths. "> Figure 11
Internal surface wave profiles at different submergence depths. "> Figure 12
Lateral waveforms at different streamwise locations. "> Figure 13
Distribution of surface pressure along the length of the submarine within the horizontal center plane. "> Figure 14
Distribution of surface pressure along the length of the submarine within the vertical center plane. "> Figure 15
Distributions of the convergence and divergence of surface velocity at the free surface. ">

Versions Notes

Abstract

This study examines the influence of drift angle on the wave and flow field generated by a submarine navigating through a density-stratified fluid. Employing a numerical methodology, this research computed the viscous flow field around the SUBOFF bare hull under conditions of oblique shipping maneuvers. The analytical framework relies on the Reynolds-Averaged Navier–Stokes (RANS) equations, supplemented by the Re-Normalization Group (RNG) k-ε turbulence model and the Volume of Fluid (VOF) method. The initial phases of this study involved verifying grid convergence and the accuracy of the numerical methods used. Subsequently, numerical simulations were performed across a spectrum of drift angles while maintaining a fixed Froude number of F_n = 0.5, with submergence depths set at 1.1 D and 2.0 D. The analysis focused on the wave profiles at both the free surface and the internal surface. The results indicate that the presence of a drift angle produces significant alterations in the characteristics of the free surface and internal surface when compared with straight-ahead motion. Specifically, the asymmetry in the flow field is enhanced, and the variability in the roughness of the free surface is pronounced.

Keywords:

stratified fluid; drift angle; SUBOFF; oblique shipping maneuvers

1. Introduction

The investigation of the hydrodynamic characteristics of ships and underwater vehicles, along with their interactions with the free surface, holds significant importance in enhancing the performance of marine vessels. Sun et al. [1] conducted simulations evaluating the hydrodynamic performance of the KRISO Container Ship (KCS) at various drift angles. Their findings revealed that an increase in drift angle is associated with elevated resistance coefficients, lateral force coefficients, and yaw moment coefficients. Additionally, the uniformity of the propeller disk’s flow field deteriorates, and the wave patterns at different broadsides of the hull become asymmetrical. Various researchers, including Farkas et al. [2], Grlj et al. [3], and Dogrul [4], have assessed the effects of turbulence models and scale effects on ships’ hydrodynamic performance. Dai et al. [5] explored how head waves and oblique waves influence hydrodynamic performance. In contrast, Hou et al. [6] employed different optimization algorithms aimed at improving ships’ hydrodynamics.

Submarines, which operate at varying depths, face hydrodynamic performance influences from factors like submergence depth, speed, and maneuverability. As a submarine approaches the free surface, the amplitude of free surface waves increases, which elevates resistance and power demand. Conversely, a deeper submergence depth decreases the impact of free surface waves on the submarine’s hydrodynamics [7,8,9]. Moreover, underwater oblique shipping maneuvers introduce significant effects from parameters like drift angle and pitch angle. Zhang et al. [10] conducted a computational study on slender axisymmetric bodies during steady turning maneuvers, comparing the distributions of longitudinal lateral force, total forces, and moments across various drift angles. Amiri et al. [11] emphasized the importance of leeward vortical flow structures on the hydrodynamic behavior of underwater vehicles at a moderate drift angle. Ling et al. [12] noted that pitch angle can cause asymmetric pressure distributions on different surfaces of the SUBOFF model, and the non-linear characteristics of forces and moments increase significantly at larger pitch angles due to crossflow separation. Their further comparisons [13] between submarines’ hydrodynamic properties near the free surface versus those near the seabed indicated that the seabed’s influence on hydrodynamic coefficients is less significant than that of the free surface.

Most research regarding submarines’ straight-ahead and oblique shipping maneuvers has been limited to homogeneous flows, neglecting density stratification effects. Seawater’s density varies due to temperature, salinity, and pressure, leading to vertical stratification that affects the surrounding flow field, particularly regarding internal wave generation. Internal waves are pivotal for the stability and operability of underwater vehicles, also impacting stealth capabilities. Chang et al. [14] examined internal waves in stratified flows, observing that higher Froude numbers correspond to longer internal wavelengths and reduced internal Kelvin angles. Studies by Ma et al. [15], Liu et al. [16], and Huang et al. [17] demonstrated that in two-layer flows, the influence range of free surface waves broadens compared with that of homogeneous flows, and density stratification enhances the amplitude of free surface waves, consequently increasing resistance on submarines. Li et al. [18] found that the maximum amplitude of free surface waves decreases with submergence depth, while the wavelength remains stable at maximum cruising speeds. Additionally, model geometry can affect the internal wave characteristics. Zou et al. [19] used numerical simulations to explore the internal wave behaviors of two tail configurations, indicating that tail linearization impacts the amplitude of internal waves at higher Froude numbers. Cao et al. [20] conducted an analysis of the propagation and evolution of submarine wakes in continuously stratified fluids. Their findings indicated that under conditions of strong density stratification, the overall wave pattern becomes more divergent, accompanied by pronounced anisotropy between crests and troughs. Additionally, they observed that as density stratification intensifies, both the propagation distance and duration of the wake can increase significantly. Conversely, Shi et al. [21] investigated the wake characteristics of the SUBOFF model under various drift angles in stratified fluids. While they acknowledged the influence of drift angle on the flow field during oblique shipping maneuvers, the impact of drift angle on free surface roughness was not addressed.

Despite numerous studies focusing on the hydrodynamic and free surface wave characteristics of underwater vehicles operating in stratified fluid, the existing research primarily emphasizes straight-ahead motion. Consequently, there is a notable gap in understanding how drift angles affect free surface waves and flow field during oblique shipping maneuvers. Such movements may generate more complex flow patterns and wave characteristics. To bridge this knowledge gap, the present study employed numerical simulation methodologies to investigate the influence of drift angle on the free surface waves and flow field characteristics of submarines during oblique shipping maneuvers. In this study, a numerical model encompassing submarines’ oblique shipping maneuvers in a two-layer flow setup was developed. The accuracy of the numerical model was verified by comparing the experimental results with those predicted by the Rankine ovoid model. Following the validation process, a numerical investigation was conducted on the flow field of a submarine executing oblique shipping maneuvers, utilizing the Reynolds-Averaged Navier–Stokes (RANS) equations in conjunction with the Volume of Fluid (VOF) model. Subsequently, a detailed analysis of the surface waveforms was carried out.

2. Numerical Methods

2.1. Model Geometry

This study utilized the SUBOFF scale model, developed by the US Defense Advanced Research Projects Agency (DARPA), which is a scaled version of the original design. The main hull was defined as an axisymmetric slender body, measuring a total length (L) of 4.356 m. The hull’s various sections included a fore-body measuring 1.016 m, a parallel middle body of 2.229 m, and an after-body with a length of 1.111 m. The maximum diameter (D) was recorded to be 0.508 m, as illustrated in Figure 1.

The navigation speed was set at 3.27 m/s, resulting in a Froude number (F_n) of 0.5. The drift angle, which is the angle between the incoming flow direction and the hull’s axis, is essential for ensuring operational efficiency. To improve the submarine’s stability, decrease the impact of significant current disturbances, and lower the risk of detection during complex maneuvers—such as navigating narrow channels or avoiding obstacles—this analysis emphasized three specific drift angles, namely 0°, −2°, and −4°, for the numerical simulations. Furthermore, the submergence depth was expressed as the vertical distance from the submarine’s axis to the free surface. For this investigation, two fixed submergence depths were selected: h = 1.1 D and 2.0 D.

2.2. Governing Equations

The oblique shipping maneuvers of a submarine in a two-layer flow are governed by the three-dimensional incompressible continuity and momentum conservation equations, which are defined as follows

\frac{\partial u_{i}}{\partial x_{i}} = 0

(1)

\frac{\partial (ρ u_{i})}{\partial t} + \frac{\partial (ρ u_{i} u_{j})}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ \frac{\partial u_{i}}{\partial x_{j}} - ρ \bar{u_{i}^{'} u_{j}^{'}}) + S_{i}

(2)

where

x_{i}

and

x_{j}

are the displacement components;

u_{i}

and

u_{j}

are the time-averaged velocity components of the fluid;

μ

is the dynamic viscosity;

- ρ \bar{u_{i}^{'} u_{j}^{'}}

is the Reynolds stress, where

u_{i}^{'}

and

u_{j}^{'}

are the fluctuating components of the velocity; and

S_{i}

is the generalized source term.

2.3. Turbulence Model

To achieve closure of the Reynolds time-averaged equations, the introduction of a turbulence model is imperative, as the choice of this model directly impacts the accuracy of the resultant simulations [22]. In the present study, the RNG (Re-Normalization Group) k-ε model was employed as the turbulence model for numerical analysis. This model is noted for its strong accuracy in capturing phenomena associated with small drift angles. However, it is acknowledged that the model demonstrates significant discrepancies when compared with experimental results in scenarios involving larger drift angles [23].

The equation of turbulent kinetic energy

k

is:

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ k u_{i})}{\partial x_{i}} = \frac{\partial}{\partial x_{j}} (α_{k} μ_{e f f} \frac{\partial k}{\partial x_{j}}) + G_{k} + G_{b} - ρ ε - Y_{M} + S_{k}

(3)

The equation of turbulent dissipation rate

ε

is:

\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ ε u_{i})}{\partial x_{i}} = \frac{\partial}{\partial x_{j}} (α_{k} μ_{e f f} \frac{\partial ε}{\partial x_{j}}) + C_{1 ε} \frac{ε}{k} (G_{k} + C_{3 ε} G_{b}) - C_{2 ε} ρ \frac{ε^{2}}{k} - R_{ε} + S_{ε}

(4)

2.4. Computational Domain and Boundary Condition

This study presents a cubic computational domain established within a Cartesian coordinate system, as illustrated in Figure 2. The origin of the coordinates is positioned at the bow of the submarine, with the x-axis representing the streamwise direction, the y-axis denoting the lateral direction, and the z-axis indicating the vertical direction. To minimize the potential for backflow during the simulation, the length from the stern of the submarine to the outlet of the computational domain has been extended. Specifically, the distance from the bow to the inlet measured 3 L, while the distance from the stern to the outlet was established at 12 L, resulting in a total length of 16 L for the computational domain. Furthermore, to reduce wave reflection at the lateral boundaries, the width of the computational domain was defined as 10 L. The freshwater layer was characterized by a height of 0.46 L and a density of 998.2 kg/m³, whereas the saltwater layer had a height of 0.69 L and a density of 1024 kg/m³. Consequently, the total height of the computational domain was established at 1.38L. The submarine was positioned within the freshwater layers. The surface of the submarine was set as a no-slip wall, the domain outlet was configured as a pressure outlet with the pressure distribution defined by a user defined function (UDF), and the inlet was designated as a velocity inlet. The finite volume method was used to discretize the governing equations, and the second-order upwind discretization format was used for both convection and diffusion terms. The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm was used for pressure–velocity coupling, and the Volume of Fluid (VOF) multiphase flow model was used to capture the free surface.

2.5. Computational Grid

ANSYS ICEM was utilized for the generation of a structured mesh that encompasses the entire computational domain (Figure 3), with refinement implemented around the hull’s surface (Figure 4). The thickness of the first-layer grid near the wall was optimized to ensure that the mesh size on the submarine’s surface adhered to the requirement of 30 ≤ y+ ≤ 300. To facilitate wave generation, the interface between the two-layer flow was established. Additionally, to enhance the accuracy of wave capture, the mesh near the free surface and internal surface was refined with a growth rate of 1.1, as depicted in Figure 3. A grid independence study was performed using four distinct grid configurations, ranging from coarse to very fine. These grid sets, outlined in Table 1, were employed for numerical simulations conducted under specific conditions characterized by a drift angle of 0°, a submergence depth of 0.4572 m, and an incoming velocity of 2.225 m/s. The analysis revealed that the differences in total resistance between the grids with 6.63 million and 8.15 million elements were negligible. In consideration of the computational resource constraints, the fine grid configuration was selected for subsequent simulations.

2.6. Numerical Validation

This subsection details the validation of the numerical model utilizing the Rankine ovoid in the Taylor pool [24]. The model had a total length of 1.3716 m and a maximum diameter of 0.196 m. A three-dimensional representation of the Rankine ovoid is illustrated in Figure 5. The numerical simulation of the flow field was carried out with a fixed depth of 0.4572 m and an established velocity of 2.225 m/s. The resulting free surface wave profile from the numerical simulation was compared with the experimental data, as shown in Figure 6. Notably, the wave height associated with the second crest reached its peak value, while the maximum trough was located near the stern of the underwater vehicle. The findings suggest that the numerical results generally align with the experimental data, despite a minor variance in wave amplitude. Overall, the observed trends indicate that the numerical methodologies employed in this study effectively capture the characteristics of free surface waves.

Figure 7 illustrates the free surface waveform obtained from this study. The crest located at the bow of the submarine indicates the presence of the Bernoulli hump. Following this, the waveform exhibits a Kelvin wave pattern, reminiscent of that generated by surface ships. This pattern consists of a combination of transverse and divergent waves, emphasizing the intricate fluid dynamics associated with the submarine’s movement through the water.

3. Results and Discussion

3.1. Influence of Drift Angle on Waves

The navigation of a submarine within a density-stratified fluid results in the generation of waves at two distinct interfaces: the air–freshwater interface and the freshwater–saltwater interface. This study provides a comparative analysis of the distribution of free surface waves at three drift angles: β = 0°, −2°, and −4°. As illustrated in Figure 8 and Figure 9, the waveforms produced by the submarine at submergence depths of 1.1 D and 2.0 D, respectively, offer critical insights into this phenomenon. When the submarine moves in a straight trajectory (as depicted in Figure 8a and Figure 9a), the surface waveforms exhibit a symmetrical distribution along the submarine’s axis. Throughout the submarine’s movement, these waveforms propagate continuously toward the rear. However, an increase in the drift angle leads to a more pronounced surface wave on one side of the submarine, resulting in higher wave heights on the starboard side compared with the port side, thereby enhancing wave asymmetry. The three-dimensional surface waves’ contours reveal that the free surface demonstrates distinct transverse and divergent wave characteristics, with the divergent waves’ height slightly exceeding that of the transverse waves. Notably, as the submarine approaches the free surface, indicative of a decrease in the submergence depth, the intensity of the waves increases, leading to greater wave amplitudes. Despite this increase in intensity, both the wavelength and the overall waveform remain consistent.

Figure 10 presents the wave profiles at the free surface for submergence depths of h = 1.1 D and h = 2.0 D. It is observed that the wavelength, along with the positions of the crests and troughs, exhibits minimal variation with changes in the drift angle, and the number of crests and troughs remains consistent. The maximum wave height for h = 1.1 D approximates 0.15 m, while for h = 2.0 D, it measures around 0.75 m, which is comparatively smaller. Additionally, slight variations in wave height are noted with varying drift angles, particularly in the troughs and crests near the stern for h = 1.1 D. As the drift angle increases, a corresponding increase in wave height is observed, which results in greater roughness of the free surface. This roughness may potentially affect the stealth performance of the submarine.

To further establish the accuracy of the free surface wave, the transverse wavelength was calculated utilizing the following formula:

λ = \frac{2 π}{g} \times U^{2}

(5)

At a model speed of 3.27 m/s, the theoretical wavelength is derived to be 6.845 m. Analysis of Figure 10a reveals that the first crest appears at x = 0.557 m, while the fourth crest is positioned at x = 19.730 m. By measuring the distance between the crests of the first three waves and calculating the average, the average wavelength from the numerical simulation is determined to be 6.391 m. This results in an error margin of approximately 6.6%. The error arises because Equation (5) is an empirical formula derived from deep-water linear wave theory. Determining the wavelength fundamentally involves solving the dispersion relation, which relies on several assumptions [25]; thus, it serves as a rough approximation of the physical problem and can introduce errors in calculation. Overall, the simulation results are consistent with the theoretical values.

Figure 11 presents the wave profiles at the internal surface corresponding to varying drift angles at two distinct submergence depths. The overall trend of wave profiles in relation to the drift angle remains generally consistent. Notably, with the exception of the maximum trough located anterior to the stern, the other crests and troughs display more pronounced variations as the drift angle changes. As the drift angle increases, the maximum trough before the stern exhibits a gradual rise, whereas the maximum crest after the stern shows a gradual decline. Furthermore, as the waveform propagates towards the rear of the submarine, the crest in cases without a drift angle diminishes faster than in those with a drift angle. This phenomenon can be attributed to the introduction of asymmetry caused by the presence of the drift angle.

The profiles shown in Figure 12 depict the lateral waveforms captured at both the free surface and the internal surface for a submergence depth of h = 1.1 D. These observations were taken at various streamwise locations, specifically at x/L = 0.5, 1.5, 2.5, 3.5, 4.5, and 5.5. A notable trend identified from the analysis is that the amplitude of the lateral waveforms on the port side of the submarine is consistently lower than that on the starboard side. Furthermore, as the drift angle increases, a noticeable lateral shift occurs in the positions of the wave crests and troughs at the different streamwise locations. On the starboard side, the crests and troughs show an increasing pattern, while those on the port side demonstrate a decreasing pattern. Additionally, it is important to note that the waves at the free surface have slightly higher elevation values compared with those at the internal surface.

3.2. Pressure Distribution

Figure 13 illustrates the distribution of surface pressure along the length of the submarine within the horizontal center plane. The figures reveal an asymmetrical pressure distribution on the starboard and port sides of the submarine, which varies in accordance with different drift angles. At the submarine’s bow, the incoming flow contributes to a relatively elevated pressure. In contrast, as one progresses toward the after-body, there is a noticeable decline in pressure.

When a drift angle is introduced, the starboard side is impacted by the incoming flow, resulting in an increase in pressure as the drift angle becomes more pronounced. At the same time, the pressure of the port side decreases with the increase in the drift angle, which is lower than the pressure without drift angle. A comparison of Figure 13a,b indicates that, despite both depths exhibiting similar pressure distributions, the pressure variations in the parallel middle body at a depth of h = 2.0 D are less pronounced. This reduced variability is attributed to the submarine’s distance from the free surface, which results in minimal interaction with it. Overall, as the submarine descends deeper, the surface pressure tends to rise due to an increase in static pressure.

Figure 14 presents the surface pressure distribution along the length of the submarine within the vertical center plane. The data show that regardless of the drift angle, the pressure distribution retains a consistent shape, exhibiting negligible differences in the pressure differences between the upper and lower surfaces.

3.3. Convergence and Divergence of Surface Velocity

The convergence and divergence of surface velocity serves as an indicator for assessing the roughness of the free surface [26]. The divergence of surface velocity can be defined as follows:

d i v (V) = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}

(6)

As depicted in Figure 15, the free surface exhibits alternating patterns of bright and dark streaks, which indicates that the shipping maneuvers of submarines have a significant impact on the roughness of the free surface. A comparison of Figure 15a,b reveals an increase in the minimum and maximum values to −4.23 and 4.63, respectively. This observation suggests that the introduction of a drift angle intensifies the variation in convergence and the intensity of divergence, resulting in an escalation of surface roughness, as evidenced by the increased wave amplitudes presented in Figure 8. Moreover, it is noted that as the submergence depth increases, the intensity of both convergence and divergence diminishes, as shown in Figure 15c. This reduction implies a corresponding decrease in the roughness of the free surface with greater depths.

4. Conclusions

This study employed the SUBOFF bare hull model to investigate the flow field and wave characteristics in a two-layer flow, maintaining a fixed Froude number (F_n = 0.5) at submergence depths of h = 1.1 D and h = 2.0 D. This research focused on the effects of varying drift angles on the free surface and the internal surface, with the principal findings summarized as follows.

(1). In stratified fluid, increasing the drift angle results in enhanced wave propagation on the starboard side of the submarine, along with an increase in wave amplitude. Although the drift angle influences the wave structure, leading to more pronounced crests and troughs, the wave propagation observed on the free surface manifests as transverse and divergent waves. Notably, significant changes in the wave profile occur at the trough and crest near the stern, while the overall wavelength remains relatively constant despite variations in the drift angle.

(2). The drift angle creates an asymmetrical pressure distribution, resulting in higher bow pressure on the starboard side compared with the port side.

(3). Additionally, the introduction of a drift angle leads to an increase in free surface roughness, which exhibits a decreasing trend with increased submergence depth.

This analysis offers valuable insights into how drift angles influence the waveforms generated by submarines performing oblique shipping maneuvers in stratified fluid, which can inform the design and operation of submarines. However, it is important to note that this study utilizes a simplified two-layer flow model for stratification, whereas real-world density stratification is often more complex and can be characterized as mixed stratification. Furthermore, the scaled nature of the model requires consideration of scale effects in future research endeavors. The use of the SUBOFF model, being a bare hull representation, also limits the applicability of findings to real submarines that incorporate more complex appendages and propellers. Continued exploration of real-world models would significantly enhance the practical relevance of this work.

Author Contributions

Conceptualization, L.S. and E.C.; methodology, L.S. and X.Z.; software, X.Z.; validation, X.Z.; formal analysis, X.Z. and L.S.; investigation, X.Z. and L.S.; resources, L.S.; data curation, X.Z.; writing—original draft preparation, X.Z.; writing—review and editing, L.S; visualization, E.C.; supervision, E.C.; project administration, L.S.; funding acquisition, L.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research is supported by National Natural Science Foundation of China (Grant No. 12172227).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Some or all data and models used during this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no competing interests.

Nomenclature

D	Maximum diameter
F_n	Froude number
h	Submergence depth
k	Turbulent kinetic energy
L	Overall length
U	Velocity
x	Horizontal coordinate
y	Axial coordinate
z	Vertical coordinate
$β$	Drift angle
$ε$	Turbulent dissipation rate

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Figure 1. Schematic of the DARPA SUBOFF bare hull model.

Figure 2. Schematic of the computational domain.

Figure 3. Grids in the vertical central plane.

Figure 4. Grids in proximity to the submarine’s surface.

Figure 5. Rankine ovoid model.

Figure 6. Comparison of numerical and experimental results [24].

Figure 7. Free surface wave of the Rankine ovoid.

Figure 8. Distribution of free surface waves at a submergence depth of h = 1.1 D.

Figure 9. Distribution of free surface waves at a submergence depth of h = 2.0 D.

Figure 10. Free surface wave profiles at different submergence depths.

Figure 11. Internal surface wave profiles at different submergence depths.

Figure 12. Lateral waveforms at different streamwise locations.

Figure 13. Distribution of surface pressure along the length of the submarine within the horizontal center plane.

Figure 14. Distribution of surface pressure along the length of the submarine within the vertical center plane.

Figure 15. Distributions of the convergence and divergence of surface velocity at the free surface.

Table 1. Comparison of total resistance for grid independence.

Grid	Nodes (×10⁶)	Total Resistance (N)	Error (%)
Coarse	3.63	12.74
Medium	4.49	12.31	3.38
Fine	6.63	12.20	0.89
Very fine	8.15	12.19	0.08

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MDPI and ACS Style

Zhao, X.; Shi, L.; Chen, E. Numerical Study on the Influence of Drift Angle on Wave Properties in a Two-Layer Flow. J. Mar. Sci. Eng. 2024, 12, 2139. https://doi.org/10.3390/jmse12122139

AMA Style

Zhao X, Shi L, Chen E. Numerical Study on the Influence of Drift Angle on Wave Properties in a Two-Layer Flow. Journal of Marine Science and Engineering. 2024; 12(12):2139. https://doi.org/10.3390/jmse12122139

Chicago/Turabian Style

Zhao, Xiaoxing, Liuliu Shi, and Eryun Chen. 2024. "Numerical Study on the Influence of Drift Angle on Wave Properties in a Two-Layer Flow" Journal of Marine Science and Engineering 12, no. 12: 2139. https://doi.org/10.3390/jmse12122139

APA Style

Zhao, X., Shi, L., & Chen, E. (2024). Numerical Study on the Influence of Drift Angle on Wave Properties in a Two-Layer Flow. Journal of Marine Science and Engineering, 12(12), 2139. https://doi.org/10.3390/jmse12122139

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