Latest revision as of 14:26, 7 May 2024

Abstract

Existing experimental methods or traditional numerical methods struggled to separate the resistance caused by a vessel's movement and the impact of waves. However, the impact of waves significantly affected the stability and safety of Unmanned Underwater Vehicles (UUVs). To summarize the influence of different attitudes and sea conditions on ocean wave impact, this work first proposed a method to separate the influence of ocean waves. This method used the unsteady finite volume method to analyze the forces on the vehicle in both wave and calm water environments. It also employed a separation formula to isolate the wave impact based on a separation equation. Then, this work selected a specific UUV as the research subject and applied this method to separate the wave impact under various attitudes and sea conditions. The results were then reconstructed. Finally, the effects of various factors on wave impact were summarized, including wave height, wave period, pitch angle, heading angle, velocity, and diving depth.

Keywords: UUV, wave influence, numerical method, characteristics

1. Introduction

Ocean waves are a type of periodic vibration that propagates in a certain direction to form waves. With an astonishing energy accumulation, it greatly affects the maneuverability and stability of ships. Therefore, the 28th ITTC conference in 2017 even established a special committee on maneuverability in waves [1], marking the impact of waves on ships as an important research direction in the future field of ship hydrodynamics [2]. Most current researches on the impact of wave environments on ships focuses on vessels with free surfaces, such as the work by Carrica et al., who achieved free turning and zigzag maneuvering movements of self-propelled ships under wave conditions [3]. Wang et al. extended the numerical simulation of self-propelled ship maneuvering movements to complex wave conditions using overlapping grid technology [4]. Submarines and other underwater vehicles can avoid the impact of free surface waves by diving below the ‘wave base’ [5-6]. Therefore, scholars both domestically and internationally have mainly focused their researches on the effects of internal waves on the attitude and load characteristics of submarines [7-9].

However, most UUVs have small displacement and small turning inertia, and due to their operational depth, they cannot completely avoid the impact of free surface waves below the ‘wave base’ in most cases. Especially in some harsh sea conditions, waves can even lead to mission failure for UUVs. Considering the force characteristics in the wave environment and responding to them in the control module of UUVs is the most important and economical method to improve their stability and safety [10]. However, existing experimental methods or traditional numerical methods find it difficult to separate the resistance generated by the motion of the vehicle from the influence of ocean waves. As a result, it is difficult to summarize the impact of different attitude conditions and sea conditions on the effect of waves

Therefore, this work takes a certain UUV as the research object, uses the unsteady finite volume method to analyze the forces on the vehicle in both wave and calm water environments. Subsequently, a separation formula for the impact of ocean waves was established, and based on this formula, the impact of the waves was isolated. At the same time, this work uses fitting methods to reconstruct the numerical results and studies the characteristics of UUVs being influenced by ocean waves, summarizing the influence of wave height, wave period, pitch angle, heading angle, velocity, and depth on the variation of ocean wave impact. Due to the low center of gravity of the unmanned underwater vehicle, the roll angle generated during the experiment is small and less than 8 degrees, which results in strong lateral stability. Therefore, the impact of the ocean wave impact caused by the roll angle is not significant.

2. Establishment of a separation model for wave influence

2.1 Mathematical model

In this work, a description of the gas-liquid two-phase flow, based on the VOF (Volume of Fluid) model, is implemented, where seawater is the primary phase and air is the secondary phase. The interface between the two phases is handled using the geometric reconstruction method. Since the purpose of this work does not involve the study of the temperature field, the finite volume method is used to solve mass-conservation Eq. (1) and momentum conservation Eq. (2), and the ${\displaystyle SST-k\omega }$ model is employed as the turbulence equation [11]:

Here, ${\displaystyle {\alpha }_{w}}$ represents the volume fraction of seawater (${\displaystyle {\alpha }_{w}=V_{w}/V_{cell}}$, ${\displaystyle V_{w}}$ represents the unit volume occupied by seawater, ${\displaystyle V_{cell}}$ represents the total volume of seawater), ${\displaystyle {\rho }_{w}}$ represents the density of seawater, ${\displaystyle v}$ represents velocity vector, ${\displaystyle t}$ represents time, ${\displaystyle p}$ represents pressure, and ${\displaystyle g}$ represents gravity vector. In Eq. (2), ${\displaystyle \rho }$ represents density and ${\displaystyle \mu }$ represents dynamic viscosity, both are calculated by Eqs. (3) and (4), respectively:

where ${\displaystyle {\alpha }_{\mbox{a}}}$ represents the volume fraction of air (${\displaystyle {\alpha }_{a}=1-{\alpha }_{w}}$), ${\displaystyle {\rho }_{a}}$ represents the density of air, ${\displaystyle {\mu }_{w}}$ represents the dynamic viscosity of seawater, and ${\displaystyle {\mu }_{a}}$ represents the dynamic viscosity of air.

To study the influence of sea condition parameters, this work utilizes the fifth-order Stokes wave theory for wave generation. This theory accounts for the nonlinear effects induced by the free liquid surface, making it more complex and practical for engineering applications than the linear wave theory [12]. The wave surface equation ${\displaystyle \eta (x,t)}$ and the potential function ${\displaystyle \phi (x,t)}$ are shown as follows:

where ${\displaystyle C}$ is the wave velocity, and the coefficient ${\displaystyle \lambda }$ and wave number ${\displaystyle k}$ are solved by combining the following equations:

where ${\displaystyle H}$ represents the height of the ocean waves, ${\displaystyle L}$ represents wavelength, ${\displaystyle d}$ represents total depth, ${\displaystyle L_{0}={\frac {gT^{2}}{2\pi }}}$, ${\displaystyle T}$ represents the wave period, and the coefficients ${\displaystyle A_{i}}$, , and ${\displaystyle C_{j}}$ can be referred to in references [13-14].

2.2 Calculation model

The subject of this work is a certain UUV, with an overall length of 10.6 m and a main body diameter of 1.18 m (Figure 1). To investigate the force characteristics of the main body under wave conditions, this work performs geometric cleanup on the rudder and propeller. Based on the geometric model, two sets of grid models are established using an identical grid division strategy (Figure 2). The grid model on the left is used to calculate the forces on the UUV under wave conditions, while the grid model on the right is for calculating the forces in a quiescent water state [15-16]. Both models use Cartesian grids for division, with a refined grid strategy applied to the surrounding area of the UUV, resulting in a total of 4 million grids.

Figure 1. UUV geometric model

(a)                                         (b)

Figure 2. Mesh model. (a) Wave environment (profile display mode). (b) Still water environment

Subsequently, the mesh models are initialized based on the VOF (Volume of Fluid) model (Figure 3), where the gas-liquid interface in the wave environment model is described by the wave surface equation shown in Eq. (5). For both environmental grid models, aside from the top boundary, the remaining boundary conditions are set as velocity inlet boundaries, with a horizontal velocity of ${\displaystyle {\frac {\partial \phi (x,z,t)}{\partial x}}}$ and a vertical velocity of ${\displaystyle {\frac {\partial \phi (x,z,t)}{\partial z}}}$. The density of seawater ${\displaystyle {\rho }_{w}}$ is 1024 kg/m³, and the density of air ${\displaystyle {\rho }_{a}}$ is 1.18 kg/m³.

Figure 3. Multiphase fluid volume contour (left: wave environment; right: still water environment)

The bottom employs two sets of grid models with the top using pressure outlet boundary conditions, where the atmospheric pressure is set at 1 atm. The pressure fields for both environmental grid models are initialized using the hydrostatic pressure theory (Figure 4).

Figure 4. Pressure contour (left: wave environment; right: still water environment)

Subsequently, the SIMPLE algorithm is used to calculate the forces acting on the UUV under two different environments. The calculation timestep is 0.01 seconds, with a total computation time of 15 seconds. After each timestep's iterations are completed, the wave influence force and torque are separated using Eq. (8):

where the superscript ${\displaystyle left}$ denotes the wave environment, and the superscript ${\displaystyle right}$ denotes the quiescent water environment, each corresponding to the models on the left and right of Figure 2, respectively. ${\displaystyle p_{e}}$ represents the flow field pressure of the UUV wall element ${\displaystyle e}$, ${\displaystyle n_{e}}$ represents the normal direction vector of the UUV wall element ${\displaystyle e}$ in the local coordinate system ${\displaystyle x_{sub}^{}}$, ${\displaystyle {\tau }_{e}}$ represents the shear stress experienced by the UUV wall element ${\displaystyle e}$, ${\displaystyle a_{e}}$ represents the area of the wall element ${\displaystyle e}$, and ${\displaystyle d_{e}}$ represents the moment arm distance vector from the centroid to the wall element ${\displaystyle e}$ in the local coordinate system ${\displaystyle x_{sub}^{}}$.

To better analyze the characteristics of wave influence, the wave torque in Eq. (8) is taken with the origin of the local coordinate system shown in Figure 3 as the center of moment. This local coordinate origin is located at the center of the bow of the vessel. The coordinates of the actual center of gravity of the UUV in the local coordinate system ${\displaystyle x_{sub}^{}}$ are (5.4, -1, 0). To obtain the torque at the actual center of gravity or to convert the torque at the center of gravity when the UUV mission changes, Eq. (9) is used for the torque transformation:

where ${\displaystyle M_{x}}$, ${\displaystyle M_{y}}$, and ${\displaystyle M_{z}}$ represent the components of the actual center of gravity torque ${\displaystyle M_{wave}}$ in the hull coordinate system, while ${\displaystyle F_{x}}$, ${\displaystyle F_{y}}$, and ${\displaystyle F_{z}}$ are the components of ${\displaystyle F_{wave}}$ in the hull coordinate system. ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ represent the distances from the actual center of gravity to the origin ${\displaystyle x_{sub}^{}}$.

To ensure that the UUV remains centered in the mesh at all times, after each timestep calculation, the grid and local coordinates shown in Figure 2 are translated as a whole by displacement ${\displaystyle v_{sub}\cdot \Delta t}$. Here, ${\displaystyle v_{sub}}$ represents the horizontal projection vector of the UUV's velocity in the global coordinate system.

3. Analysis of the changes in wave influence

In order to obtain the influence of different wave heights, wave periods, UUV pitch angles, heading angles, velocities, and depths on the variation of wave forces, a uniform sampling method is adopted within the range of the aforementioned operational and sea condition parameters to establish a computational sample database in this work (Table 1). Through various combinations, the number of samples reaches 511,758.

After calculating the aforementioned samples, this work fits the results to obtain a fitting model, which enables the output of wave influence force and torque for any parameter combination. In actual engineering, variations in the UUV's internal mission modules can cause the actual center of gravity to differ from the initially designed one. The fitting results can also provide wave influence force and torque for any center of gravity, aiding in the balancing process [17].

The computational results are compared with the fitting results in Figure 5, revealing a high degree of overlap between the fitting and CFD results. The CFD results show oscillations in the first second due to the flow field not having formed a stable structure yet. These oscillations are not conducive to the subsequent control module input or the synthesis of complex wave spectra for wave influence. Therefore, it is crucial to fit and reconstruct the fitting model for all the CFD results of the samples mentioned in Table 1. To verify the accuracy of the fitting model, any parameters within the range described in Table 1 are randomly selected for CFD calculations and are then compared with the fitting results in this work. The comparison shows that the results from the random test samples also highly coincide with the fitting model (Figure 6).

Figure 5. Comparison of impact on marine environment (${\displaystyle {\theta }_{attack}=0{\mbox{°}}}$, ${\displaystyle {\theta }_{yaw}=0{\mbox{°}}}$, ${\displaystyle v_{sub}=0}$kn, ${\displaystyle d_{sub}=12.5}$m, ${\displaystyle a_{wave}=1.25}$m, ${\displaystyle t_{wave}=5.5}$s)

Figure 6. Comparison of impact on marine environment (${\displaystyle {\theta }_{attack}=2.023{\mbox{°}}}$, ${\displaystyle {\theta }_{yaw}=20.23{\mbox{°}}}$, ${\displaystyle v_{sub}=2.023}$kn, ${\displaystyle d_{sub}=25}$m, ${\displaystyle a_{wave}=2.023}$m, ${\displaystyle t_{wave}=6.069}$s)

3.1 Effect of wave height

A comparison of the changes in the marine environmental influence under different wave heights is shown in Figure 7, where ${\displaystyle M_{wave}}$'s center of moment is the center of gravity of the UUV during a specific mission, and ${\displaystyle M_{wave}}$ in subsequent analysis refers to the torque experienced by the center of gravity in the hull coordinate system. To facilitate comparison, the conditions shown in Figure 7 are as follows: ${\displaystyle t_{wave}}$ is 6s, ${\displaystyle d_{sub}}$ is 15m, ${\displaystyle {\theta }_{attack}}$ is 0°, ${\displaystyle {\theta }_{yaw}}$ is 45°, and ${\displaystyle v_{sub}}$ is 0kn. Thewave heights ${\displaystyle a_{wave}}$ are 0.25m, 0.75m, 1.25m, 1.75m, and 2.25m, respectively.

(a)                                                                                               (b) (c)                                                                                              (d) (e)                                                                                              (f)

Figure 7. Wave impact patterns at different wave heights. (a) Force along the ${\textstyle x}$-axis. (b) Torque around the ${\textstyle x}$-axis. (c) Force along the ${\textstyle y}$-axis. (d) Torque around the ${\textstyle y}$-axis. (e) Force along the ${\textstyle z}$-axis. (f) Torque around the ${\textstyle z}$-axis

From the comparison shown in Figure 7, it can be observed that when the UUV's velocity is 0 knots, the variation period of ${\displaystyle F_{wave}}$ is not affected by the wave height. And its period is exactly the same as the wave period. However, the amplitude of the period variation of ${\displaystyle F_{wave}}$ is linearly positively correlated with the change in wave height. Additionally, there is a phase difference in the periodic variation among the three components of ${\displaystyle F_{wave}}$. The phase difference in the period variation of ${\displaystyle F_{x}}$ is 1.5 seconds more than that of ${\displaystyle F_{y}}$ (i.e., ${\displaystyle t_{wave}{\mbox{/4}}}$), and the phase difference in the period variation of ${\displaystyle F_{x}}$ is another 1.5 seconds more than that of ${\displaystyle F_{y}}$ (i.e., ${\displaystyle t_{wave}{\mbox{/4}}}$).

After summarizing, the following equation can be obtained:

where ${\displaystyle t_{Force}^{0}}$ represents the initial phase difference, while ${\displaystyle t_{Force}^{x}}$, ${\displaystyle t_{Force}^{y}}$, and ${\displaystyle t_{Force}^{z}}$ correspond to the phase differences in the periodic variations of ${\displaystyle F_{x}}$, ${\displaystyle F_{y}}$, and ${\displaystyle F_{z}}$, respectively.

The three components of ${\displaystyle M_{wave}}$ are influenced not only by ${\displaystyle F_{wave}}$ but also by the difference between the force center (${\displaystyle F_{wave}}$) and the actual center of gravity (Eq. 9). Therefore, this paper only conducts qualitative and quantitative analysis on the variation pattern of ${\displaystyle F_{wave}}$.

3.2 Effect of wave period

Figure 8 presents a comparison of the changes in the marine environmental impact under different wave periods, with the specific conditions as follows: ${\displaystyle a_{wave}}$ is 1.25m, ${\displaystyle d_{sub}}$ is 15m, ${\displaystyle {\theta }_{attack}}$ is 0°, ${\displaystyle {\theta }_{yaw}}$ is 45°, and ${\displaystyle v_{sub}}$ is 0 kn. The wave periods ${\displaystyle t_{wave}}$ are 5s, 6s, 7s, 8s, and 9s, respectively.

(a)                                                                                               (b) (c)                                                                                              (d) (e)                                                                                              (f)

Figure 8. Wave impact patterns at different wave periods. (a) Force on the ${\textstyle x}$-axis. (b) Moment around the ${\textstyle x}$-axis. (c) Force around the ${\textstyle y}$-axis. (d) Moment around the ${\textstyle y}$-axis. (e) Force around the ${\textstyle z}$-axis. (f) Moment around the ${\textstyle z}$-axis

From the comparison shown in Figure 8, it can be observed that when the UUV's velocity is 0 kn, the variation period of ${\displaystyle F_{wave}}$ matches the wave period for each condition. The amplitude change of ${\displaystyle F_{wave}}$ exhibits a strong nonlinear growth relationship with the wave period, with an inflection point occurring at a certain wave period value. The phase difference variations of the individual components of ${\displaystyle F_{wave}}$ under different conditions shown in Figure 8 still comply with Eq. (10).

3.3 Effect of pitch angle

It presents a comparison of the changes in the marine environmental impact under different UUV pitch angles in Figure 9, with the specific conditions as follows: ${\displaystyle t_{wave}}$ is 6s, ${\displaystyle a_{wave}}$ is 1.25m, ${\displaystyle d_{sub}}$ is 15m, ${\displaystyle {\theta }_{yaw}}$ is 45°, and ${\displaystyle v_{sub}}$ is 0 kn. The pitch angles ${\displaystyle {\theta }_{attack}}$ are -8°, -4°, 0°, 4°, and 8°, respectively.