CFD Study of Submarine Hydrodynamics near the Free Surface in Snorkel Conditions

Abstract

Submarines are primarily designed for optimal performance while operating submerged, as they spend the majority of their operational time below the free surface. However, they also navigate at various depths near the free surface, such as during snorkel conditions or other shallow-water operations. Under snorkel conditions, as the sail depth decreases and the distance between the free surface and the top of the hull is reduced, a suction effect occurs, inducing an upward force on the submarine. Consequently, a comprehensive assessment of hydrodynamic forces at different depths and speeds is essential during the design phase to ensure stability and performance optimization. In this study, computational fluid dynamic (CFD) simulations were performed to analyze the heave and surge forces acting on the generic Joubert BB2 (BB2) submarine. The computed surge and heave forces, as well as the pitch moment, were validated against experimental data, showing discrepancies within approximately 12%. The influence of the free surface on these forces and moments was investigated, demonstrating trends consistent with both experimental measurements and numerical predictions. These findings confirm that CFD simulations serve as a reliable tool for predicting free-surface effects on submarines, offering valuable insights for the design process.

1. Introduction

Submarine design has undergone significant advancements over the past few centuries, beginning with the French Navy’s introduction of a diesel-powered submarine in 1863. Early submarine designs featured hull shapes similar to those of surface vessels. However, to reduce hydrodynamic resistance and enhance speed for tactical advantages, hull forms evolved into more streamlined and rounded shapes [1]. Modern submarines are primarily optimized for submerged operations, as they operate predominantly underwater. Nevertheless, they must also function near the free surface in scenarios such as snorkel operations and surface piercing. Under these conditions, reduced speeds are often required to minimize the risk of detection, which consequently impacts the vessel’s maneuverability and control [2].

Under snorkel conditions, as the distance between the free surface and the top of the hull decreases, a suction effect occurs, inducing an upward force on the submarine. The accurate prediction of these free-surface effects is crucial, particularly for determining the required hull angle of attack to ensure stability and control at low speeds. Despite the significance of understanding the hydrodynamic forces acting on submarines near the free surface, research in this area remains limited, largely due to security restrictions.

Various approaches have been employed to investigate the effects of the free surface on submarines. Park et al. [3] utilized linear theory to model wave excitation, suction, and current forces, simulating the motion of a submerged body near the free surface. Jung et al. [4] developed a nonlinear maneuvering model and computed hydrodynamic parameters using the boundary element method. Sarraf et al. [5] conducted model experiments and identified a critical depth level that significantly influences submarine behavior. Kwon et al. [6] performed captive model tests on a generic Joubert BB2 (BB2) submarine at varying depths near the free surface, analyzing the heave force induced by the “tail suction effect.” Their findings indicated that the heave force increases as the submarine approaches the free surface. Kim et al. [7] conducted model experiments and proposed a practical six-degree-of-freedom (6-DoF) maneuvering simulation model to enhance submarine motion predictions.

Advancements in computational hardware have facilitated the extensive use of computational fluid dynamics (CFDs) to analyze variations in hydrodynamic forces acting on submarines operating near the free surface. Griffin [8] introduced a hybrid approach that integrates Reynolds-averaged Navier–Stokes (RANS) simulations with an inviscid panel code to estimate submarine maneuvering characteristics under these conditions. However, due to the limited availability of experimental data on forces and moments for submarine hulls affected by significant free-surface effects, validation remains predominantly qualitative.

Boger and Dreyer [9] employed CFD with overset grids to predict hydrodynamic forces and moments acting on underwater vehicles. Their findings demonstrated strong agreement between the computed surface pressure distributions and experimental data for the DARPA SUBOFF submarine. Similarly, Coway et al. [10] utilized RANS-based CFD simulations to evaluate suction effects on the DST Group/MARIN generic BB2 submarine at various depths and speeds, revealing that minor variations in these parameters could significantly influence lift forces, potentially destabilizing the vessel. Amiri et al. [11] investigated the role of bow and aft shoulder wave interactions in the hydrodynamic behavior of shallowly submerged submarines. Vasileva and Kyulevcheliev [12] conducted a numerical study on the DARPA SUBOFF submarine, demonstrating that resistance increased as the vessel approached the free surface, although this effect diminished at higher speeds. Courdier et al. [13] extended this analysis to surfaced underwater vehicles, examining their hydrodynamic behavior across a range of speeds using CFD.

Further research has examined submarine dynamics in more complex free-surface environments. Carrica et al. [14] investigated self-propulsion performance under both calm water and head wave conditions. Wang et al. [15] analyzed the interactions between a submarine’s propeller wake and the free surface, while Dong et al. [16] conducted simulations of submarine operations in long-crested wave environments.

Experimental studies have also played a significant role in advancing this field. Dogrul [17] investigated scale effects by comparing propulsion characteristics between model- and full-scale submarines. Cao et al. [18] simulated submarine operations in stratified flows, assessing hydrodynamic performance and wake characteristics. Li et al. [19] demonstrated that the relative driving depth has a substantial impact on internal wave behavior. Guo et al. [20] developed a simulation method to achieve a steady turning motion, analyzing the resulting free-surface waves and flow fields.

Despite numerous numerical and experimental investigations, the interaction between submarines and the free surface remains inadequately explored. In particular, substantial variations in vertical forces and moments occur when a submarine operates near the free surface, highlighting the need for further analysis across a range of Froude numbers and submerged depths.

The primary objective of this study is to investigate the evolution of hydrodynamic forces and flow characteristics at varying Froude numbers and submerged depths. Additionally, this study aims to identify key parameters that effectively represent submerged depth. By analyzing variations in the flow field and fluctuations in forces, this research seeks to improve the understanding of factors influencing submarine behavior under snorkel conditions. This study is conducted in a fixed state, without allowing any motion of the submarine.

In this study, Star-CCM+ was utilized to compute the forces and moments acting on the BB2 submarine. The obtained results were compared with existing numerical data and experimental findings [6,10] to assess their accuracy. The paper is organized as follows: First, a detailed description of the submarine model is provided, followed by an overview of the computational methods employed. The subsequent section presents the computational results and their analysis. Finally, the key findings are summarized, and the main conclusions of the study are discussed.

2. Problem Description

In this research, the BB2 submarine was chosen as the reference vessel, as shown in Figure 1. The selection was primarily influenced by the availability of previously published experimental data [6]. The bare hull model has a scale ratio of 1/15, with a total model length ($L$) of 4.68 m. The longitudinal center of gravity is located at 2.153 m, and the vertical center of gravity is at 0.033 m. A summary of the main dimensions is provided in

Table 1. Principal particulars of the target submarine.

Item	Full Scale	Model Scale
Scale ratio	1	1/15
$L$ [m]	70.2	4.68
$B$ [m]	9.6	0.64
$D$ [m]	10.6	0.707
Displacement [m3]	4365.2	1.293
$L_{CG}$ [m]	32.29	2.153
$V_{CG}$ [m]	0.5	0.033

.

Computational fluid dynamic (CFD) simulations were performed to evaluate the surge ($X$) and heave ($Z$) forces across different depths and vessel speeds. The obtained results were then compared with existing numerical studies [10]. The non-dimensional surge ($X^{\prime }$) and heave ($Z^{\prime }$) forces are presented as follows [10]:

$$X^{\prime }={\displaystyle \frac{X}{\frac{1}{2}\rho u^2{L}^2}}$$

(1)

$$Z^{\prime }={\displaystyle \frac{Z}{\frac{1}{2}\rho u^2{L}^2}}$$

(2)

Furthermore, the pitch moment ($M$) was analyzed in comparison with model test results. The non-dimensional pitch moment ($M^{\prime }$) is defined as follows [10]:

$$M^{\prime }={\displaystyle \frac{M}{\frac{1}{2}\rho V^2{L}^3}}$$

(3)

In this study, $u$ and $\rho$ represent the velocity and fluid density, respectively. $L$ is the submarine length. To incorporate the influence of depth variation, the non-dimensional depth ($H^{\prime }$) and non-dimensional sail depth ($D^{\prime }$) of the submarine were utilized. $H^{\prime }$ denotes the maximum depth, while $D^{\prime }$ represents the lowest depth of the vessel. The definitions of $H^{\prime }$ and $D^{\prime }$ are as follows [10]:

$$H^{\prime }={\displaystyle \frac{d}{D}}$$

(4)

$$D^{\prime }={\displaystyle \frac{D_o}{D}}$$

(5)

The variable $d$ represents the depth below an equivalent free surface, while $D_o$ denotes the sail depth relative to the same reference surface. The maximum diameter of the submarine’s hull is represented by $D$, as shown in Figure 1. A right-handed Cartesian coordinate system, also depicted in Figure 1, was used for the results analysis. In this system, $X$, $Y$, and $Z$ correspond to the surge, sway, and heave forces, respectively, while $K$, $M$, and $N$ represent the roll, pitch, and yaw moments. Additionally, in this coordinate framework, the negative values of $X$ and $Z$ indicate surge and heave forces, respectively.

3. Computational Methods

3.1. Governing Equations and Numerical Method

The fundamental equations governing steady incompressible viscous fluid flows are derived from the principles of mass and momentum conservation, which can be expressed as follows [21,22]:

$${\displaystyle \frac{\mathit{\partial }{\overline{u}}_i}{\mathit{\partial }{x}_i}}=0$$

(6)

$$\rho \left({\displaystyle \frac{\mathit{\partial }{\overline{u}}_i}{\mathit{\partial }t}}+{\overline{u}}_j{\displaystyle \frac{\mathit{\partial }{\overline{u}}_i}{\mathit{\partial }{x}_j}}\right)=-{\displaystyle \frac{\mathit{\partial }\overline{p}}{\mathit{\partial }{x}_i}}+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{x}_j}}\left(\mu {\displaystyle \frac{\mathit{\partial }{\overline{u}}_i}{\mathit{\partial }{x}_j}}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)+S_i$$

(7)

In these equations, ${\overline{u}}_i$ and $\overline{p}$ represent the time-averaged velocity and pressure, respectively. The symbol $\mu$ denotes the dynamic viscosity of the fluid. The terms $\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ and $S_i$ correspond to the Reynolds stress and the source term, respectively.

In CFD simulations, accurately capturing the interface between water and air poses considerable challenges, particularly at high Froude numbers, where significant errors may occur. To mitigate this issue, it is crucial to utilize a method capable of precisely resolving the free surface. Consequently, the multiphase flow was modeled using the Volume of Fluid (VOF) approach, which efficiently tracks the free surface. In the VOF method, the phase is represented by the volume fraction ($\alpha$) corresponding to the air and water phases. The transport equation for the volume fraction is given as follows [23]:

$${\displaystyle \frac{\mathit{\partial }\alpha }{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }\alpha {\overline{u}}_i}{\mathit{\partial }{x}_i}}=0$$

(8)

The mixture’s density and viscosity depend on the proportion of water and air within a given cell. These properties are defined as follows [23]:

$$\rho =\left(1-\alpha \right){\rho }_{air}+\alpha {\rho }_{water}$$

(9)

$$\mu =\left(1-\alpha \right){\mu }_{air}+\alpha {\mu }_{water}$$

(10)

Additionally, turbulence effects near the wall are critical for ensuring the accuracy of the simulation, as turbulence within the boundary layer significantly influences the results. To capture these effects, turbulence was modeled using the Reynolds Stress Turbulence Model (RSTM), which employs the Reynolds stress transport equation to account for the complex interactions in turbulent flow. This model was selected to closely analyze the turbulent structure generated by the submarine’s sail and its interaction with the hull.

The CFD simulations were conducted using Star-CCM+. For spatial discretization, the central differencing scheme was applied to the diffusion terms, while a high-order upwind differencing scheme was used for the convection terms. To ensure a sharp representation of the interface between water and air, the Modified High-Resolution Interface Capturing (MHRIC) scheme was implemented. Additionally, pressure–velocity coupling was handled using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [24,25]. The convergence criteria for all solution variables were 0.0001.

3.2. Boundary Conditions and Domain Extent

The computational domain is illustrated in Figure 2, with its dimensions determined based on the guidelines provided in the ITTC 2017 procedure [26]. The domain extends 1.5 times the vessel’s overall length (LLL) in the forward direction, 2.5 $L$ in the aft direction, 3.5 $L$ horizontally, and 3.2 $L$ vertically. The inlet, top, bottom, and side boundaries are all designated as velocity inlets, whereas the outlet boundary is defined as a pressure outlet. The bottom boundary in the present study is not the sea bottom and is the midpoint of the water depth. For the top and boundaries, the influence of the selected boundary condition is negligible. At the velocity inlet, the specified vessel speed is imposed, while at the pressure outlet, the static pressure is applied, and the velocity follows a zero-gradient condition [27]. To minimize wave reflections at the boundaries, wave damping zones are implemented, each extending 1 $L$ in length.

3.3. Grid Convergence Test

The computations used CPUs, with a total of 512 cores (Manufacturer: AMD; Model; EPYC 7742; Santa Clara, CA, USA). One computation took about 3 h. The computational mesh was constructed using a trimmed mesh approach [27]. To accurately resolve the turbulent boundary layer flow along the hull surface, prism layers were included. A total of six prism layers were generated, extending outward from the hull surface to provide the adequate resolution of near-wall flow characteristics.

A grid convergence study was performed to identify the optimal grid resolution using Richardson extrapolation [28]. This assessment utilized three different grid configurations—coarse, medium, and fine—while focusing on the heave force and pitch moment as key evaluation parameters. The mesh refinement followed a refinement ratio of $r_G=\sqrt{2}$, ensuring uniform refinement across successive grid levels. Both the near--hull and background grids were also refined. The number of computational cells for each grid configuration is summarized in

Table 2. Grid size and number for grid convergence test.

Index	Layer Number	Layer Expansion Ratio	Mean y+	Maximum Skewness	Minimum Skewness	Normalized Grid Size (m)	Number of Grid (×106)
Coarse grid	6	1.3	100	120	0	0.092	1.8
Medium grid	6	3	90	100	0	0.074	3.1
Fine grid	6	3	80	80	0	0.059	6.3

, with the refinement patterns depicted in Figure 3. The total cell count for the coarse, medium, and fine grids is 1.8 × 10⁶, 3.1 × 10⁶ and 6.3 × 10⁶, respectively.

The variation in CFD results between the medium and fine grids is represented as ${\epsilon }_{21}=S_2-S_1$, while the difference between the coarse and medium grids is given by ${\epsilon }_{32}=S_3-S_2$. Here $S_1$, $S_{\mathrm{s}}$, and $S_3$ are the solution variables for the fine, medium, and coarse grids, respectively. Using these differences, the convergence ratio is computed according to Equation (11). The convergence behavior is evaluated through the convergence rate, which is determined using Richardson extrapolation, as defined in Equation (12) [28].

$$R_G={\displaystyle \frac{{\epsilon }_{21}}{{\epsilon }_{32}}}$$

(11)

$$\begin{array}{c}\left(a\right) R_G>1, \mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d} \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\\ \left(b\right) 0<R_G<1, \mathrm{M}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\\ \left(c\right) R_G<0, \mathrm{O}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\end{array}$$

(12)

As part of the grid convergence study, the heave force and pitch moment were analyzed at a Froude number ($Fr=u/\sqrt{gL}$ = 0.26), with the results summarized in

Table 3. Test results of grid convergence.

Variable	Heave Force (N)	Pitch Moment (N-m)
$S_1$ (Fine grid)	18.42	10.15
$S_2$ (Medium grid)	18.20	10.14
$S_3$ (Coarse grid)	17.40	9.21
${\epsilon }_{21}$	−0.224	−0.010
${\epsilon }_{32}$	−0.787	−0.928
$R_G$	0.285	0.011

. The computed grid convergence ratio ($R_G$) for all variables indicates that monotonic convergence was achieved, as the values of $R_G$ lie within the range of 0 to 1. Consequently, the medium grid was selected for use in this study.

4. Results and Discussion

The simulation conditions, including vessel speeds ($Fr$), sail depth ($D^{\prime }$), and overall depth ($H^{\prime }$), are presented in

Table 4. Simulation conditions for various depths.

Parameters	Values
Froude number ($Fr$)	0.16, 0.21, 0.26, and 0.31
Non-dimensional sail depth ($D^{\prime }$)	0.057, 0.198, and 0.378
Non-dimensional depth ($H^{\prime }$)	1.4, 1.8, 2.2, and 3.0

. For the depth variations, three different depths and three sail depths were considered in the analysis.

The impact of the submarine’s submerged sail depth on the surge force ($X^{\prime }$), heave force ($Z^{\prime }$), and pitch moment ($M^{\prime }$) acting on the hull was investigated. Figure 4 and Figure 5 present a comparison of the surge and heave responses. The obtained surge and heave forces exhibit trends that align closely with those observed in the model tests [10]. The analysis revealed that as the hull approaches the free surface, the pitch moment increases. This is due to the influence of the free surface, which allows for pitch variation. Under snorkel conditions, pitch stability plays a crucial role in submarine operations. As shown in Figure 6, the CFD results closely follow the trends observed in the model tests. Additionally, the heave force increases as the submarine moves closer to the free surface. Due to free-surface effects, the pitch moment exhibits nonlinear behavior, increasing as the submarine approaches the free surface. In summary, the comparison confirms that the present simulations provide a reliable method for evaluating the free-surface effects on submarines.

The influence of different depths and vessel speeds was examined by evaluating the non-dimensional surge force ($X^{\prime }$) and heave force ($Z^{\prime }$) under various conditions. The results, presented in Figure 7 and Figure 8, illustrate the effects of these parameters. The simulation conditions, including vessel speeds and depths, are summarized in Table 4. The findings indicate that the trends observed for surge and heave forces closely align with those reported in the reference study [6], as shown in Figure 7 and Figure 8.

For surge forces, a reduction in non-dimensional depth ($H^{\prime }$) leads to an increase in surge forces, particularly when $H^{\prime }$ is below 2.2. This behavior is primarily attributed to the rise in wave-making resistance at shallower depths. Conversely, when $H^{\prime }$ exceeds 2.2, the wave-making resistance becomes negligible, resulting in minimal variations in surge force. In regions where $H^{\prime }$ > 2.2, the general trend of increasing resistance with higher speeds is observed. However, as H’ decreases, the influence of speed on resistance diminishes. While the effect of the free surface on the surge force weakens as $H^{\prime }$ surpasses 2.2, its impact on the heave force persists at greater depths, as shown in Figure 8. Unlike the surge force, the heave force remains relatively constant regardless of speed variations when the water depth increases. In other words, at greater depths, the surge force stabilizes to a constant value proportional to hydrostatic pressure. The free surface tends to exert a lifting effect on the submarine. As the submarine approaches the free surface, the fluid velocity above the hull increases relative to the velocity below the hull due to the reduced fluid volume between the hull and the free surface. This velocity difference generates an upward lift force on the hull, with the heave force reflecting the suction effect induced by this lift. As shown in Figure 8, the heave force generally increases with higher Froude numbers and lower non-dimensional depths ($H^{\prime }$). However, a notable crossover occurs when $H^{\prime }$ < 1.8 at $Fr=0.31$. This behavior is likely due to nonlinear effects in free-surface dynamics induced by higher vessel speeds.

Figure 9 presents the pressure coefficient contours on the hull and the corresponding wave patterns generated at different depths for $Fr=0.26$. These results further confirm that wave-making resistance becomes more pronounced as the water depth decreases. When $H^{\prime }=2.2$ and $H^{\prime }=3.0$, only minimal wave formation is observed, which likely accounts for the negligible wave-making resistance under these conditions. Additionally, the pressure coefficient contours on the hull appear nearly identical across these cases. In the absence of significant wave-making resistance, the non-dimensional surge force remains approximately constant across different Froude numbers.

Figure 10 illustrates the variation in the free surface at the $y$ = 0 plane for different depths and Froude numbers. Similarly to the wave contours presented in Figure 9, the free-surface elevation increases as H’ decreases, with a particularly significant rise observed at $H^{\prime }$ = 0.31. At this depth, the free surface undergoes considerable fluctuations, leading to pronounced nonlinear effects. Furthermore, as the Froude number increases, the variations in free-surface elevation become more pronounced.

Figure 11 presents the pressure coefficient contours on both the hull and the free surface above it, as viewed from the side. The pressure coefficient distributions at the bow and stern exhibit similar trends, regardless of variations in the submerged depth. This suggests that wave-making resistance plays a significant role in influencing the surge force. In contrast, the pressure coefficient in the z-direction, which affects the heave force, shows notable differences in the midsection of the hull. As the submerged depth increases, the pressure difference between the upper and lower surfaces in this region decreases. Furthermore, the impact of depth variations becomes less pronounced at greater depths.

Figure 12 depicts the vortices surrounding the hull, visualized using the Q-criterion. The vortices primarily originate behind the sail plane. At shallow submerged depths, these vortices exhibit vertical oscillations, rising and falling in response to variations in the free surface.

5. Concluding Remarks

CFD simulations were performed to evaluate the influence of free-surface effects on the BB2 submarine at various submerged depths. A grid dependency analysis was conducted to ensure numerical convergence. The obtained results were then compared with existing numerical predictions and experimental data. The numerical methods were carefully selected based on the comparison with experimental data and the results of the grid dependency analysis.

The surge and heave forces predicted by the CFD simulations exhibited trends consistent with both numerical predictions and experimental data. Analysis of the surge forces revealed that as the non-dimensional depth ($H^{\prime }$) decreased, surge forces increased, particularly for $H^{\prime }<2.2$. This trend was attributed to the rise in wave-making resistance at shallower depths. Conversely, for $H^{\prime }>2.2$, wave-making resistance became negligible, leading to minimal variations in surge forces. The heave force generally increased with higher Froude numbers and lower non-dimensional depths ($H^{\prime }$). However, a significant crossover was observed at $H^{\prime }<1.8$ and $Fr=0.31$, likely due to nonlinear effects in free surface dynamics at higher vessel speeds. Vortex structures were observed behind the sail plane, moving in synchronization with the height of the generated waves. Under snorkel conditions, pitch stability played a crucial role in submarine operations. The analysis showed that pitch moments increased as the submarine approached the free surface, a trend that closely aligned with experimental observations.

These comparisons validate that CFD simulations, using the selected numerical methods, provide an effective and reliable approach for assessing free-surface effects on submarines. Future studies will focus on investigating the hydrodynamic forces and flow dynamics influencing a submarine as it ascends toward the free surface during snorkel operations.