Analysis of Submarine Motion Characteristics in Mesoscale Vortex Environment Based on the Arbitrary Lagrange–Euler Method

Abstract

The special eddy field of mesoscale vortices plays an important role in the global shipping process. The statistical morphology of mesoscale vortices observed via global satellites and the numerical simulation of the ocean are applied to the simulation of computational fluid dynamics, which can more truly reflect the influence of mesoscale vortices on the motion characteristics of underwater vehicles. In this paper, the ALE (Arbitrary Lagrangian–Eulerian) finite element method is used to simulate the random vortex of a submarine in three dimensions (horizontal x, vertical z, height y) and establish quantitative submarine movement characteristics. Our results show that with an increase in mesoscale vortex strength, the effects on the submarine’s speed and displacement increase, but the overall effect is still limited. In the 300 m transmission simulation, the velocity effect is within ±2 m/s, and the displacement effect is within 4 m. The simulation results can be applied to the route optimization algorithm of underwater vehicle automatic navigation and provide a reference for energy consumption calculations and route safety evaluations.

1. Introduction

Mesoscale vorticity, as a significant oceanic phenomenon widely observed in the world’s oceans, has huge spatiotemporal scale and complex dynamic characteristics. In general, such vortices can persist in the marine environment for tens to hundreds of days, and their horizontal scale can span tens to hundreds of kilometers. The persistence and wide spatial distribution of mesoscale vortices make them important parts of the ocean circulation system that exert a profound and multi-dimensional influence on the structure and characteristics of the ocean flow field that they affect [1].

The significance of mesoscale vortices in the ocean flow field cannot be underestimated. They reshape the distribution of matter and energy in the marine environment in unique ways. By adjusting the local water mass, mesoscale vortices can significantly affect the spatial distribution of thermohaline characteristics, resulting in significant differences between the inner and outer vortices. Such differences are not only reflected in changes in water temperature and salinity values but also have extensive and far-reaching impacts on marine ecosystems, biological habitat distribution, and biological migration patterns [2]. At the macro level, mesoscale vortices play an important role in modulating the dynamic mechanism of ocean circulation systems. They can interact with the background flow field, affect the large-scale circulation structure of the ocean, and thus regulate the transport path and efficiency of matter and energy in the ocean. Substances such as nutrients and plankton will migrate and reorganize with the movement of vortices, which greatly changes the material basis and energy flow mode of marine ecosystems, and their widespread existence has played an important role in global marine transportation practice [3].

In the study of the fluid–structure coupling during submarine underwater motion, the boundary fit method [4], the non-boundary fit method [5], and the meshless method [6] are the three main processing methods. Boundary fit methods, such as the arbitrary Lagrange–Euler method [7] (ALE), are numerical methods that accurately capture the details of the interface flow field by rigorously matching the fluid and solid grids. However, the disadvantage of this approach is that it requires frequent grid updates, resulting in higher computational costs and challenges when dealing with large problems. Non-boundary fit methods, such as the immersion boundary method, avoid the need for grid updating by adopting a non-matched mesh strategy, making them particularly suitable for dealing with large deformation problems [8]. However, this method needs to interpolate the interface’s physical quantity, so the precision of the simulation will be to some extent limited. Meshless methods, such as smooth particle fluid dynamics [9] (SPH), circumvent mesh deformation and interpolation problems by means of node discretization, but the computational load is significantly increased, and the demand for computational resources is higher. Considering factors such as simulation accuracy and computing resources, the ALE method takes the finite volume method as its core, and through a series of fine numerical calculation steps, such as operator separation [10], governing equation discreteness [11], smooth substep [12], convection substep [13], and contact coupling [14], it not only performs well in dealing with convection problems; it also effectively guarantees the definition of physical quantities and interface precision, showing a unique advantage in the study of submarine fluid–structure coupling characteristics.

In summary, the influence of mesoscale vortex on a submarine is mainly reflected in the following aspects: It affects the concealment of the submarine, especially by changing the sound velocity profile and current disturbance. It may lead to the change of submarine speed and the influence of sailing stability, especially the strong disturbance of vortex. The noise propagation and detection capability of submarines is affected, especially the disturbance of ocean turbulence caused by vortices. Submarine channel selection and ocean monitoring are affected by mesoscale vorticity and need to be adjusted according to real-time data. Understanding and coping with the effects of mesoscale vortices is an important factor in submarine design and tactical deployment, especially when conducting covert missions or conducting operations in complex waters. In view of the importance and widespread existence of mesoscale vortices in the ocean flow field, it is increasingly important to study their effects on the motion of submerged bodies under different intensities or with different properties. Therefore, on the basis of previous research on ship dynamics computer simulation, ALE dynamic grid technology is used to analyze the dynamics of several mesoscale vortex parameters, as summarized via statistical analyses. The motion characteristics of a submerged body under the influence of several working conditions are given quantitatively, and the safety of the submerged body motion is qualitatively analyzed. The purpose of this study is to provide a scientific basis and reference for the research into and application of submerged body motion in the mesoscale vortex environment. This research not only helps to understand the motion characteristics of a submarine in the mesoscale vortex environment but also can be applied to route optimization in underwater vehicle automatic navigation applications and provide references for energy consumption calculation and route safety evaluation.

2. Data

Ocean observation data come from a wide range of sources and usually include ocean remote sensing data, crewless platform monitoring data, and in-situ crewed survey data. The core data requirement for the fluid–structure coupling calculation of submerged bodies in this paper is the ocean calculation flow field in the whole water layer, which requires a large quantity of grid data covering multi-layer water depth temperature and salinity; as such, we here adopt ocean reanalysis data. The data used in this paper are sea surface height data, thermohaline reanalysis data, and part of the in-situ observation data derived during the period from 1 January 2007 to 31 December 2020.

2.1. Ocean Reanalysis of Climate State Data

JCOPE2M (Japan Coastal Ocean Predictability Experiment 2 Modified) data are obtained by the Japan Marine Affairs Agency and comprise high-resolution reanalysis data. They cover the Northwest Pacific Ocean and comprise 46 layers assessed in full depth with a time resolution of 1 day. The grid resolution is 1/12°, and the assimilated sea surface temperature field, sea surface height anomaly data, and part of the Argo data are included, which have been applied by many scholars in measuring the mesoscale eddy temperature salt field and flow field with relatively high accuracy [15,16]

The WOA23 Climate State data represent a global marine environmental dataset that was released in 2023, covering temperature, salinity, dissolved oxygen, and other key climate variables since 1856, with a spatial resolution of 0.25 degrees latitude and longitude and featuring vertical full water depth climate state data measured from the sea surface to the seabed. The dataset not only updates 1.8 million ocean profiles but also provides reference values for multiple climate periods, which are widely used in ocean model building, numerical simulation validation, and satellite data comparisons. The WOA23 data reflect the trend of global climate change, such as rising greenhouse gas concentrations and increasing ocean acidification, providing important support for scientific research and policy-making [17,18].

2.2. Satellite Altimetry Data

Sea Level Anomaly (SLA) and geostrophic data were compiled at the Archiving Validation and Interpretation of Satellite Oceanographic Data Center, CNES, France. Lattice products were provided by AVISO [19,20]. The data were fused with those from several satellite altimetry measurements and interpolated onto a 1/4° × 1/4° grid via Mercator projection, with a time resolution of 7 d, before being interpolated to 1 d.

3. Methods

3.1. Identification of Mesoscale Vortices and Calculation of Geostrophic Currents

Since the opening of access to ocean satellite altime data in 1992, the identification of mesoscale vortices has roused great interest in scholars [1,21,22]. At present, the mainstream methods of mesoscale vortex identification mainly include the physical parameter method [23], the flow field geometry method [24], the machine vision method [25], etc. However, due to differences in data sources, the method of quality control used for identification, and the different evaluation criteria used for vortex identification results, we cannot confirm which recognition algorithm is most suitable for this paper without experiments. Therefore, this paper uses the above algorithms to determine the data to be used and adopts the artificial evaluation method used by oceanographic experts to evaluate them so as to determine the best scheme.

$$u^{\prime }=-{\displaystyle \frac{g}{f}}{\displaystyle \frac{\partial h^{\prime }}{\partial y}} , v^{\prime }=-{\displaystyle \frac{g}{f}}{\displaystyle \frac{\partial h^{\prime }}{\partial x}}$$

(1)

where $u^{\prime }$ and $v^{\prime }$ are the meridional and zonal components of the surface geostrophic flow field (m/s), respectively, and $h^{\prime }$ is the sea surface height (m).

In this paper, the outlier value of the sea surface height at the vortex center is taken as the main index to determine the strength of the mesoscale vortex. In order to better assess the force, speed, and displacement of the submarine under different strengths of the vortex, we here build a vortex model according to the three working conditions set out in

Table 1. Anomalous values of sea surface height at the center of the mesoscale vortex under three working conditions.

	Type 1	Type 2	Type 3
Amplitude (m)	0.3	0.6	1.0
Radius of Vortex (m)	10,000	13,000	20,000

. Based on the anomalies in the vortex center sea surface height shown in Table 1, the warm salt field data were matched with the eddies, which fluctuate by 10% in the JCOPE2M reanalysis data. After removing the annual average value, the warm salt field data were averaged and superimposed into the average warm salt field at the same position in the data period (2007–2020) so as to enable the warm salt field’s final construction. Three working conditions are considered in this paper, as shown in Table 1.

For the construction of the flow field of a typical vortex, we adopt the method of calculating the flow field from the water distribution layer to form a three-dimensional model. Specifically, we do not consider the vertical material exchange in the internal environment of the ocean, which means that it does not represent the flow rate in the vertical direction, only showing it in the horizontal direction.

3.2. The Governing Equation of Fluid–Structure Coupling Under ALE Description

3.2.1. Governing Equation

In this paper, the explicit dynamic flow coupling ALE (Arbitrary Lagrangian–Eulerian) method is used to simulate fold modeling, and the flow field equation is based on a mass equation, a momentum equation, and an energy equation [26], which are respectively shown as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}=-\rho {\displaystyle \frac{\partial v_i}{\partial x_i}}-w_i{\displaystyle \frac{\partial \rho }{\partial x_i}}$$

$${\displaystyle \frac{\partial v_i}{\partial t}}={\sigma }_{ij,j}+\rho b_i-\rho w_i{\displaystyle \frac{\partial v_i}{\partial x_j}}$$

(2)

$$\rho {\displaystyle \frac{\partial E}{\partial t}}={\sigma }_{ij}{v}_{i,j}+\rho b_i{v}_i-\rho w_j{\displaystyle \frac{\partial E}{\partial x_j}}$$

where $v_i$ represents material velocity; $w_i$ indicates the relative speed $w_i=v_i-u_i$; $u_i$ represents the speed of the grid; ${\sigma }_{ij}$ represents the stress tensor ${\sigma }_{ij}=-p{\delta }_{ij}+\mu \left(v_{i,j}+v_{j,i}\right)$; $b_i$ represents unit volume force; ${\delta }_{ij}$ represents the $Kronecker \delta$ function. The grid governing equation is

$${\displaystyle \frac{\partial f\left(X_i,t\right)}{\partial t}}={\displaystyle \frac{\partial \left(x_i,t\right)}{\partial t}}+w_i{\displaystyle \frac{\partial \left(x_i,t\right)}{\partial t}}$$

(3)

where $X_i$ represents the Lagrange coordinates; $x_i$ represents the Euler coordinates; and $w_i$ represents relative speed. The structural control equation is

$$M\ddot{w}+C\dot{w}+Kw=F$$

(4)

The velocity of the material is determined based on the structural dynamics of the shell. Here, $M,C, \mathrm{and} K$ represent the element mass, damping modulus, and elastic modulus, respectively. $F$ is the resultant force applied on the shell element. It is worth mentioning that the difference between the descriptions given by the ALE, pull, and European methods is that ALE can describe the material domain boundary as accurately as the Laplace grid, while the material domain information (such as the velocity of the material, stress, etc.) is kept in the reference grid (Euclidean description). When the material domain is seriously deformed, the mesh describing it can be updated to maintain good mesh quality, while the information of the material domain given in the reference grid is updated by means of convection. The three categories of difference are illustrated in Figure 1.

3.2.2. Update of the Substep

When the mesh deforms with the movement of the material ($\chi =X$), the internal mesh is seriously deformed too, but the material domain boundary and its node position can still be determined (only incompressible seawater and no other substances are found in the reference domain of this project). Therefore, the material domain boundary is here always the reference grid boundary, and the number of nodes and density of the reference grid boundary are pre-divided. Therefore, the reconstruction of the grid can be regarded as a boundary value problem [27]. The so-called boundary value problem seeks to specify the value of $x_i\left({\xi }_i\right)$ on the boundary of the regular calculation domain and then obtain the coordinates on the physical plane corresponding to each point in the calculation domain by solving the $Laplace$ differential equation. For three dimensions, the differential equation is

$$\left\{\begin{array}{l}{\alpha }_1{\partial }_{{\xi }_1{\xi }_1}x+{\alpha }_2{\partial }_{{\xi }_2{\xi }_2}x+{\alpha }_3{\partial }_{{\xi }_3{\xi }_3}x+2{\beta }_1{\partial }_{{\xi }_1{\xi }_2}x+2{\beta }_2{\partial }_{{\xi }_1{\xi }_3}x+2{\beta }_3{\partial }_{{\xi }_2{\xi }_3}x=0\\ {\alpha }_i={\partial }_{{\xi }_i}{x}_1^2+{\partial }_{{\xi }_i}{x}_2^2+{\partial }_{{\xi }_i}{x}_3^2\\ {\beta }_1=({\partial }_{{\xi }_1}x\cdot {\partial }_{{\xi }_3}x)({\partial }_{{\xi }_2}x\cdot {\partial }_{{\xi }_3}x)-({\partial }_{{\xi }_1}x\cdot {\partial }_{{\xi }_2}x){\partial }_{{\xi }_3}{x}^2\\ {\beta }_2=({\partial }_{{\xi }_2}x\cdot {\partial }_{{\xi }_1}x)({\partial }_{{\xi }_3}x\cdot {\partial }_{{\xi }_1}x)-({\partial }_{{\xi }_2}x\cdot {\partial }_{{\xi }_3}x){\partial }_{{\xi }_1}{x}^2\\ {\beta }_3=({\partial }_{{\xi }_3}x\cdot {\partial }_{{\xi }_2}x)({\partial }_{{\xi }_1}x\cdot {\partial }_{{\xi }_2}x)-({\partial }_{{\xi }_3}x\cdot {\partial }_{{\xi }_1}x){\partial }_{{\xi }_2}{x}^2\end{array}\right.$$

(5)

After the mesh is updated, the flow field information of the new mesh needs to be updated via the convection method. Based on the core idea of the finite volume method [28,29] (FVM method), the flux $\varphi$ (including mass, momentum, and other flow field information) can be updated using the following governing equation:

$${\displaystyle \frac{\partial \varphi }{\partial t}}+{\displaystyle \frac{\partial c_i\varphi }{\partial x_i}}=0$$

(6)

For three-dimensional problems, the interpolated definition of the discrete format of the convection term can be written in the following flux form:

$${\varphi }_A^{n+1}{V}_A^{n+1}={\varphi }_A^n{V}_A^n+{\displaystyle \sum }_{j=1}^6{f}_j^{\varphi }$$

(7)

In the formula, subscript $A$ is the cell, $j$ is the cell adjacent to $A$, $V$ is the cell volume, and $f_j^{\varphi }$ is the flux through the adjacent cell.

$$\begin{array}{l}{f}_j^{\varphi }={\displaystyle \frac{c_j}{2}}({\varphi }_j^{-}+{\varphi }_j^{+})+{\displaystyle \frac{|c_j|}{2}}({\varphi }_j^{-}-{\varphi }_j^{+})\\ {\varphi }_j^{+}=S_{j+{\displaystyle \frac{1}{2}}}^n(x_j^n+{\displaystyle \frac{1}{2}}\Delta t\cdot c_j-x_{j+{\displaystyle \frac{1}{2}}}^n)+{\varphi }_{j+{\displaystyle \frac{1}{2}}}^n\\ {\varphi }_j^{-}=S_{j-{\displaystyle \frac{1}{2}}}^n(x_j^n+{\displaystyle \frac{1}{2}}\Delta t\cdot c_j-x_{j-{\displaystyle \frac{1}{2}}}^n)+{\varphi }_{j-{\displaystyle \frac{1}{2}}}^n\end{array}$$

(8)

In Equation (8), $S_{j+{\displaystyle \frac{1}{2}}}^n$ is the slope with second-order precision, which can be defined as follows ($S_{j-{\displaystyle \frac{1}{2}}}^n$ is the same):

$$\begin{array}{l}{S}_{j+{\displaystyle \frac{1}{2}}}^n={\displaystyle \frac{({\varphi }_{j+\frac{3}{2}}^n-{\varphi }_{j+\frac{1}{2}}^n)\Delta x_j^2+({\varphi }_{j+\frac{1}{2}}^n-{\varphi }_{j-\frac{1}{2}}^n)\Delta x_{j+1}^2}{\Delta x_j\Delta x_{j+1}\left(\Delta x_j+\Delta x_{j+1}\right)}}\\ \Delta x_j=x_{j+{\displaystyle \frac{1}{2}}}^n-x_{j-{\displaystyle \frac{1}{2}}}^n\hspace{1em}\Delta x_{j+1}=x_{j+{\displaystyle \frac{3}{2}}}^n-x_{j+{\displaystyle \frac{1}{2}}}^n\end{array}$$

(9)

The above discrete scheme is the MUSCL scheme with second-order accuracy, as proposed by Van Leer in 1977 [30,31], which is often referred to as the Van Leer MUSCL scheme.

3.2.3. Coupling Method Based on Penalty Function

Coupling between the structure and the flow is carried out via the penalty function method [32]. Its basic structure is as follows: at each time step, one must first check whether each slave node penetrates the main surface; if there is no penetration, no processing is done. If there is penetration, a large interface contact force is introduced between the slave node and the penetrated main surface, the magnitude of which is proportional to the penetration depth and the stiffness of the main surface. In short, a spring is applied between the node and the surface that may be penetrated to prevent overall penetration, and the spring force is denoted as part of the external force $f^{ext}$. This algorithm is easy to program and offers uncountable value noise and accurate momentum conservation, and as such, and is widely used in many explicit finite element codes.

In the contact problem, the master and slave nodes need to be defined first. Suppose that the element that may be contacted is called the face (or the main face) and is represented by $S_i$, while $c_i$ and $c_{i+1}$ are the two side vectors of the face $S_i$ at the main node $m_S$. $g$ is the vector from the primary node $m_S$ to the slave node $n_S$, m is the external normal unit vector of the surface $S_i$, and s is the projection vector of vector g on surface $S_i$. A node in the solid grid acts as the slave node $n_S$, and the primary node is the matter integration point in the flow field grid (Figure 2) rather than a node on the flow field grid.

First, the node $n_S$ and its nearest primary node $m_S$ are searched, and all the surfaces related to the primary node $m_S$ are found, after which the surfaces that may come into contact are judged. Under normal circumstances, the master and slave nodes will not coincide. According to the following formula, the surfaces that may be contacted are judged:

$$\left\{\begin{array}{c}\left(c_i\times s\right)\cdot \left(c_i\times c_{i+1}\right)>0\\ \left(c_i\times s\right)\cdot \left(s\times c_{i+1}\right)>0\end{array}\right.$$

(10)

where the normal vector and projection vector outside the surface is

$$\begin{gathered} m={\displaystyle \frac{c_i\times c_{i+1}}{\left|c_i\times c_{i+1}\right|}} \\ \hspace{1em}s=g-\left(g\cdot m\right)m \end{gathered}$$

(11)

The flow field is generally discrete when using eight-node hexahedral elements (Figure 3), whose shape function is as follows:

$$N_I={\displaystyle \frac{1}{8}}\left(1-{\xi }_I\xi \right)\left(1-{\eta }_I\eta \right)\left(1-{\zeta }_I\zeta \right) \begin{array}{c}I=1,2,\dots ,8\end{array}$$

(12)

The velocity and spatial coordinates of the primary node $m_S$ can be obtained by interpolation via Function (13), as follows:

$$\begin{gathered} v_{m_S}\left({\xi }_{m_S},t\right)=v_I\left(t\right)N_I\left({\xi }_{m_S}\right) \\ {x}_{m_S}\left({\xi }_{m_S},t\right)=x_I\left(t\right)N_I\left({\xi }_{m_S}\right) \end{gathered}$$

(13)

To achieve fluid–structure coupling, two vectors need to be defined—the mean external normal unit vector $m_{average}$ and the permeability displacement vector $d_p^n$. $m_{average}$ is the average of the external normal unit vector (calculated in Formula (14)) of the shared slave node $n_S$ on the face $S_i$. Regarding $d_p^n$, when $t=0$, $d_p^0=0$; when $t=n+1$, it is

$$d_p^{n+1}=d_p^n+v_{rel}^{n+{\displaystyle \frac{1}{2}}}\cdot \Delta t \begin{array}{c}{v}_{rel}^{n+{\displaystyle \frac{1}{2}}}=\end{array}{v}_{n_S}^{n+{\displaystyle \frac{1}{2}}}-v_{m_S}^{n+{\displaystyle \frac{1}{2}}}$$

(14)

Only when $m_{average}\cdot d_p^n<0$ is the coupling force $f_{couple}$ calculated. According to Newton’s third law, the contact point $C$ on the surface $S_i$ also enacts a reaction force of the same size in the opposite direction, which can act on all nodes on the surface equally;

$$f_{coupleI}=-N_I\left({\xi }_{m_S}\right)f_{couple}$$

(15)

According to the characteristics of the shape function, it can be proven that the sum of the contact forces of the nodes on the surface is equal to that of the contact forces acting on the slave nodes,

$${\displaystyle \sum }_{I=1}^8{f}_{coupleI}=-{\displaystyle \sum }_{I=1}^8{N}_I\left({\xi }_{m_S}\right)f_{couple}=-f_{couple}\hspace{1em}{\displaystyle \sum }_{I=1}^8{N}_I\left({\xi }_{m_S}\right)=1$$

(16)

The team uses the penalty function method to calculate the contact. The submarine surface is the fluid–structure coupling contact surface, and the penalty function coefficient tracks the relative displacement d between the Lagrange node (structure) and the Euler fluid material position. Here, we must check the penetration of each structural unit node into the surface of the fluid material; if the structural unit node does not show penetration, do not perform any operation; if penetration occurs, the interfacial force F will be distributed to the Euler fluid unit node. The size of the interface force is proportional to the number of penetrations that occur,

$$F=k_i\cdot d$$

(17)

3.3. Submarine and Pool Modeling

In the traditional numerical wave-generating pool, we need to encrypt the grid at the fluid interface to improve the accuracy of the wavefront analysis, but the global grid division is too dense, resulting in a waste of computing resources, and the fixed encryption area cannot adapt to the dynamic movement of the wave, resulting in analysis errors. This project adopts adaptive mesh partitioning technology to set the initial large-size global grid by use of the input parameters and to capture the dynamic encryption of the moving area based on the interface, such that the grid can be adjusted in real time with the generation and propagation of the interface and internal solitary wave, reducing the calculation burden and improving the accuracy of waveform analysis (Figure 4).

This project takes the submarine model SUBOFF of the Defense Advanced Research Projects Agency (DARPA) of the United States as the research object [33]. The main boat length in the original model is 4.356 m, and the front body length is 1.016 m. The parallel body length is 2.229 m, the rear body length is 1.111 m, and the submarine’s maximum rotating diameter is 0.508 m. The attached body includes a command platform, a vertical tail, and a horizontal tail. The length of the command platform is 0.368 m, and the upper part is a convex top cover. The horizontal height of the bottom of the top cover is 0.459560 m. The four tail sections are NACA0020 airfoil profiles, which are symmetrically arranged at the tail of the boat.

The above geometric model establishes a grid model (Figure 5), and the suboff model and peripheral grid are 440,000. At the same time, the hull is scaled to equal proportions, such that the total displacement of the hull is about 3000 t. The initial grid model following the overall assembly is shown in Figure 6. It is worth mentioning that the “fluid-structure coupling” here is actually a generalized representation, and the solid domain can use a six-degree-of-freedom rigid body model (considering attitude changes) or a finite element model (considering attitude changes and deformation) and the seawater has little impact on the body, and the calculation amount caused by the above two models has little impact. In order to describe the geometry of the submarine, the finite element mesh is used to describe the submarine shape, but the attitude change is calculated by using a six-degree-of-freedom model in the solid domain; that is, the solid domain element is assigned as a rigid body.

3.4. Numerical Modeling

In different life cycles, the mesoscale vortex usually has a relatively stable maximum velocity zone, and the velocity decreases with the increase in the radius outside the maximum velocity zone. In the inner part of the maximum velocity zone, the velocity decreases as the radius decreases. It is assumed that the mesoscale vortex is a two-dimensional flow field that changes only in the horizontal direction and is evenly distributed in the vertical direction; that is, in the vertical direction, the flow field distribution of each layer is the same as that of the ocean surface. The submarine moves horizontally through the eddy current (Figure 7), and its acceleration, speed, attitude, and shape will change under the influence of the eddy current.

Tetrahedral mesh is used to discretize the submarine model. Each finite element parameter model is defined according to material characteristics and submarine operating characteristics. The submarine is an elastic tetrahedral 10-node solid element, which is a quadratic element. The displacement interpolation function within the element is a quadratic complete polynomial, and the stress and strain represent a first-order complete polynomial inside the element. The displacement convergence speed is fast, and the element uses fewer nodes, with strong geometric adaptability and high precision.

For the fluid domain, we discretize the fluid domain with a hexahedral grid, which is gradually more encrypted as we move closer to the submarine (center point). The density and viscosity coefficient of seawater in a given fluid domain is given. The partial stress is not considered in the state equation, and the initial internal energy, initial relative volume, and parameters C4 and C5 in the fitting equation are defined. The section parameter of the fluid domain is the single point multi-matter ALE unit, which is the most accurate and comprehensive when dealing with the ALE problem. For hexahedral solid units, the hourglass model uses the standard LS-DYNA viscosity type with an hourglass coefficient of 1 × 10⁻⁹.

After this, the boundary conditions and initialization conditions of the system’s motion are defined. The non-reflective boundary conditions are set on the six boundary surfaces of the fluid domain. Considering the relative motion relationship between the submarine and the current, the initial motion of the submarine has a forward velocity of 5 m/s (in the direction of the submarine axis x), with an initial lateral velocity shown in the fluid domain (the initial velocity caused by the eddy current, in the direction of the fluid domain z).

The motion state of the submarine, when it moves horizontally through the mesoscale vortex, is obtained via iterative calculation until convergence. It is worth mentioning that the feature size of the mesoscale vortex space domain is not the same order of magnitude as the submarine size, and it can be considered that the flow is constant in the length of the submarine.

4. Result

4.1. The Effect of Eddy Currents on Submarine Speed and Displacement

Because the size of the flow field involved in mesoscale eddies in the marine environment is many times greater than the scale of the submarine’s geometric characteristics, it is unrealistic to calculate the complete passage of the submarine through a mesoscale eddy with a diameter of 100 km. Therefore, in this paper, the calculation domain is reduced to calculate only the flow field of the submarine crossing the mesoscale vortex center with a diameter of 300 m. In this paper, the three mesoscale vorticity parameters listed in Table 1 are taken as examples for calculation.

Firstly, the eddy current effect is analyzed for the case of type 1. In this case, the submarine essentially does not float up or dive down in the vertical y direction (Figure 8a,b) but only advances at a speed of 5 m/s (x direction) and is affected by type eddy currents (z-direction). Therefore, its forward speed fluctuates slightly around about 5 m/s (Figure 8c), while its displacement increases steadily (Figure 8d). After 60 s, it advances by 300 m. The speed in the forward direction of the submarine is about 5 ± 1 m/s. At the same time, the oscillating change in the directional velocity causes the integral results to cancel each other out; that is, the displacement is not affected by the eddy current and remains the same as it would be in the absence of an eddy current.

After entering the type 1 vortex zone, the velocity of the submarine in the z direction increases from 0.012 m/s to 0.19 m/s at first and then decreases. With the passage of the vortex, the velocity decreases steadily. When the submarine reaches the center of the vortex, the velocity almost drops to 0, which takes just 30 s. In 60 s, it reaches −0.12 m/s, as shown in Figure 9a.

Affected by eddy currents, the path class of the submarine in the z-direction depicts an arc similar to the flow direction of the vortex and finally returns to the initial position in the z-direction, as shown in Figure 9b. In the horizontal z direction, the maximum change in the velocity of the submarine is 0.31 m/s, while the displacement eventually returns to the initial position after a maximum change of 1.82 m. The displacement and velocity changes in the xyz direction under the other three numerical simulation conditions are shown in

Table 2. Displacement and velocity changes in the xyz direction under three numerical simulation conditions.

	x		y		z
	Displacement Change (m)	Velocity Change (m/s)	Displacement Change (m)	Velocity Change (m/s)	Displacement Change (m)	Velocity Change (m/s)
Type 1	0~298.00	−3.01~1.05	−0.0098~0.0109	−0.0006~0.0500	0~1.82	−0.1200~0.1930
Type 2	0~298.00	−3.02~1.04	−0.0103~0.0103	−0.0004~0.0554	0~2.71	−0.1000~0.2570
Type 3	0~298.00	−3.02~1.03	−0.0107~0.0094	−0.00007~0.0569	0~3.61	−0.2400~0.3220

Note: Take the extreme value of the initial stability moment and the end stability moment.

.

In summary, when the submarine traverses a mesoscale vortex horizontally, the velocity fluctuation in the three directions increases with the increase in vortex velocity, but the vertical and forward directions of the submarine converge to the same value and are thus almost unaffected by the vortex velocity. In the horizontal direction, although the offsetting of velocity and displacement increases with the eddy’s velocity, the amplitude of change is small, and the displacement eventually returns to the original position. Therefore, the effect of vortex velocity on the speed and displacement of the submarine is small and can be ignored.

4.2. Velocity Vector Distribution of Flow Field

When the submarine moves in a complex flow field environment, a complex interaction occurs between the flow field and the submarine. This interaction is not one-way but bidirectional; that is, the motion of the submarine will affect the characteristics of the flow field, and the characteristics of the flow field will, in turn, affect the motion state of the submarine. This interaction is particularly evident when the submarine passes through a special flow field structure called an eddy current. According to the previous detailed analysis, the speed of the submarine in the horizontal direction shows an almost linear change when passing through the eddy current. Based on this characteristic, in order to further study the changing characteristics of the flow field around the submarine, we choose the flow field vectors of 20 s and 50 s, which are three representative moments, as the comparative research objects. We still use the three operating conditions shown in Table 1 for comparative simulation. These three operating conditions cover different parameter settings, which can reflect the changes in the flow field around the submarine when it passes through the vortex under different conditions.

First, we should consider Figure 10, which shows the vector distribution of the flow field around the submarine under the type 1 condition (20 s and 50 s, respectively). It can be clearly observed from the figure that under the type 1 condition, although the velocity of the submarine and the distribution form of the flow field around it fluctuate with the passage of time, they show high similarity between different moments. This similarity indicates that although there is an interaction between the vortex and the submarine, the distribution of the surrounding flow field is not greatly changed by this interaction. In other words, under this condition, the existence of eddy currents does not fundamentally change the overall layout and characteristics of the flow field around the submarine. Similarly, the velocity vector distribution of the submarine flow field under type 2 and type 3 conditions (Figure 11) is analyzed (20 s and 50 s).

We further analyze the velocity vector distribution of the submarine flow field under type 2 and type 3 conditions (20 s and 50 s are also selected). With the increase in vortex velocity, the vortex phenomenon near the middle of the submarine becomes more obvious under these two conditions. This change in the vortex is caused by the increased force imparted on the fluid surrounding the submarine as the vortex speed increases. At the same time, it can be found that the submarine moves back and enters a state of oscillation under the influence of the mesoscale vortex. The recoil and swing state reflect the influence of mesoscale vorticity on the trajectory and attitude of the submarine. In addition, the strength of the submarine’s wake also increases with the increase in vortex velocity. However, it is worth noting that despite these changes, from an overall speed perspective, it remains pretty much the same. Our results show that under these working conditions, although the mesoscale vortex has some effects on the flow field around the submarine, such as via the enhancement of the vortex, the change in the position and attitude of the submarine, and the change in the wake intensity, these effects do not significantly change the overall velocity of the flow field around the submarine, indicating that the mesoscale vortex has little impact on the overall velocity.

By studying the vector distribution of the flow field around the submarine at different time points under these three working conditions, we can more deeply understand the complex interaction between the flow field and the submarine when the latter passes through an eddy current and thus provide a strong basis for further research into the motion characteristics of a submarine in this complex flow field environment.

4.3. Force Analysis of Submarine Caused by Mesoscale Vortex

According to the analysis of the above results, we can see that the lateral force of the submarine is affected by the mesoscale vortex, and the influences of the other directional force and moment can be almost ignored. Based on the above conclusions, the lateral force is fitted in MATLAB R2024a.

Figure 12 shows the lateral force fitting of the submarine under different working conditions, from which it can be inferred that the influence of the mesoscale vortex on the force of the submarine is not very obvious (in Figure 12, the force oscillation of 0 s is slightly larger because the explicit method used is fundamentally different from the implicit algorithm adopted above; if the implicit algorithm is adopted, the forces at the beginning and end times will be predicted to be the same). The fitting expression is $F_z=p_1\cdot t^3+p_2\cdot t^2+p_3\cdot t+p_4$, and the parameters are shown in

Table 3. Fitting expression parameters under different working conditions.

	Type 1	Type 2	Type 3
$p_1$	−0.02011	−0.03023	−0.04035
$p_2$	2.071	3.116	4.16
$p_3$	−62.18	−93.64	−125.1
$p_4$	449.4	678.7	907.7

.

By comparing the parameters under different working conditions, it can be found that the coefficient in the fitting expression changes with the increase in the eddy speed. Specifically, the fitting parameters under the type 1 condition are relatively small, while those under the type 3 condition are larger. This indicates that the lateral force of the submarine increases more significantly when the vortex velocity is higher, and the overall parameters increase with the increase in the strength of the mesoscale vortex, which indicates that a change in the strength of the mesoscale vortex can affect the edge and internal flow field, as well as affecting the stress of the submarine to a certain extent.

In addition, with the increase in the strength of the mesoscale vortex, its force pendulum momentum in the opposite direction will also increase, which means that within the same movement time, in order to maintain the original motion state, the submarine will actively exert greater torque on itself to offset the lateral force generated by the mesoscale vortex, which will not only increase fuel consumption but also increase the momentum of the vortex. This will also present considerable challenges to maneuverability.

Another, it should be explained that in the early stage of the movement, the data fluctuated more severely, which was caused by the insufficient development of the flow field structure in the fluid-structure coupling process. These outliers belong to normal phenomena in numerical calculation. Under different working conditions, there are different vortex shedding, which cannot be preset in advance during the flow field initialization process and is completely dependent on dynamic changes.

5. Conclusions

The purpose of this study is to analyze submarine motion characteristics in a mesoscale vortex environment using the fluid–structure coupling method. The motion behavior of a submarine in a mesoscale eddy current field and the influence thereof are discussed using advanced numerical simulation technology combined with the theory of fluid mechanics and structural dynamics. Through the grid division, material property analysis, and fluid domain discretization of the submarine model, this study successfully simulates the motion state of the submarine under different vortex conditions and reveals the influence of the mesoscale vortex on the speed, displacement, and force of the submarine. In terms of methods, we have noticed that the Gertler and Hagen models [33,34] are also used in the studies of some research institutions. Since the computational power of this paper is limited, and these two models are difficult to use in the transformation of discrete grids, we will not use these models as a comparison.

In this study, we use adaptive meshing technology to reduce the waste of computing resources and improve the accuracy of waveform analysis. By comparing the motion characteristics of a submarine under different eddy velocities, it is found that the velocity fluctuation of the submarine in the horizontal direction increases with the increase in eddy velocities but converges to the same value in the vertical and forward directions and, as such, is almost unaffected by the eddy velocities. Our results show that although the vortex velocity has a certain effect on the velocity and displacement of the submarine, the change amplitude is small, and the displacement finally returns to its original place, so the effect of the vortex velocity on the overall motion of the submarine is limited.

In terms of flow field vector distribution, it is found that the overall layout and characteristics of the flow field around the submarine do not fundamentally change in the presence of mesoscale vortices. Although the increase in vortex speed causes the vortex phenomenon that forms near the midsection of the submarine to become more obvious, and the position and attitude of the submarine are also affected, the overall speed remains unchanged. This indicates that the mesoscale vortex has no significant effect on the overall velocity.

In addition, through a fitting analysis of the lateral force of the submarine, we find that the mesoscale vortex has a certain effect on the lateral force of the submarine, but with the increase in the vortex speed, the effect is not very significant. The fitting results show that with the increase in eddy current intensity, the lateral force on the submarine increases, but the overall impact is still limited. This indicates that the mesoscale vortex has a certain influence on the force of the submarine, but it is not the dominant factor.

In summary, the motion characteristics of a submarine in a mesoscale vortex environment are systematically analyzed using the fluid–structure coupling method in this study. Our results show that although the mesoscale vortex has some effects on the velocity, displacement, and force of the submarine, these effects are limited on the whole. The research results not only provide a theoretical basis for submarine motion prediction in a mesoscale vortex environment but also provide an important reference for further optimizing submarine design and improving maneuverability. Future research can further explore the motion characteristics of submarines in more complex mesoscale vortex environments so as to improve the survivability and combat effectiveness of submarines in complex marine environments.