Zhou et al 2024a - Revision history

Rimni at 13:26, 7 May 2024

2024-05-07T13:26:00Z

Rimni at 13:24, 7 May 2024

2024-05-07T13:24:11Z

Rimni at 13:23, 7 May 2024

2024-05-07T13:23:36Z

Rimni at 13:22, 7 May 2024

2024-05-07T13:22:45Z

Rimni: /* 3.3 The effect of pitch angle */

2024-05-07T13:21:09Z

‎3.3 The effect of pitch angle

Rimni: /* 3. Analysis of the changes in wave influence */

2024-05-07T13:15:27Z

‎3. Analysis of the changes in wave influence

Rimni at 13:13, 7 May 2024

2024-05-07T13:13:46Z

Rimni at 13:11, 7 May 2024

2024-05-07T13:11:46Z

Rimni: /* 1. Introduction */

2024-05-07T13:09:55Z

‎1. Introduction

Rimni at 12:22, 7 May 2024

2024-05-07T12:22:21Z

@@ Line 70: / Line 70: @@
 |}
-where,  <math>{\alpha }_\mbox{a}</math> represents the volume fraction of air (<math>{\alpha }_a=1-{\alpha }_w</math>),  <math>{\rho }_a</math> represents the density of air,  <math>{\mu }_w</math> represents the dynamic viscosity of seawater, and  <math>{\mu }_a</math> represents the dynamic viscosity of air.
+where  <math>{\alpha }_\mbox{a}</math> represents the volume fraction of air (<math>{\alpha }_a=1-{\alpha }_w</math>),  <math>{\rho }_a</math> represents the density of air,  <math>{\mu }_w</math> represents the dynamic viscosity of seawater, and  <math>{\mu }_a</math> 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  <math>\eta (x,t)</math> and the potential function  <math>\phi (x,t)</math> are shown as follows:
@@ Line 102: / Line 102: @@
 |}
-where,  <math>C</math> is the wave velocity, and the coefficient  <math>\lambda </math> and wave number  <math>k</math> are solved by combining the following equations:
+where  <math>C</math> is the wave velocity, and the coefficient  <math>\lambda </math> and wave number  <math>k</math> are solved by combining the following equations:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 117: / Line 117: @@
 |}
-where,  <math>H</math> represents the height of the ocean waves,  <math>L</math> represents wavelength,  <math>d</math> represents total depth,  <math>L_0=\frac{gT^2}{2\pi }</math>,  <math>T</math> represents the wave period, and the coefficients  <math>A_i</math>,  <math>B_j</math>, and  <math>C_j</math> can be referred to in references [13-14].
+where  <math>H</math> represents the height of the ocean waves,  <math>L</math> represents wavelength,  <math>d</math> represents total depth,  <math>L_0=\frac{gT^2}{2\pi }</math>,  <math>T</math> represents the wave period, and the coefficients  <math>A_i</math>,  <math>B_j</math>, and  <math>C_j</math> can be referred to in references [13-14].
 ===2.2 Calculation model===
@@ Line 206: / Line 206: @@
 |}
-where, the superscript  <math>left</math> denotes the wave environment, and the superscript  <math>right</math> denotes the quiescent water environment, each corresponding to the models on the left and right of [[#img-2|Figure 2]], respectively.  <math>p_e</math> represents the flow field pressure of the UUV wall element  <math>e</math>,  <math>n_e</math> represents the normal direction vector of the UUV wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>,  <math>{\tau }_e</math> represents the shear stress experienced by the UUV wall element  <math>e</math>,  <math>a_e</math> represents the area of the wall element  <math>e</math>, and  <math>d_e</math> represents the moment arm distance vector from the centroid to the wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>.
+where the superscript  <math>left</math> denotes the wave environment, and the superscript  <math>right</math> denotes the quiescent water environment, each corresponding to the models on the left and right of [[#img-2|Figure 2]], respectively.  <math>p_e</math> represents the flow field pressure of the UUV wall element  <math>e</math>,  <math>n_e</math> represents the normal direction vector of the UUV wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>,  <math>{\tau }_e</math> represents the shear stress experienced by the UUV wall element  <math>e</math>,  <math>a_e</math> represents the area of the wall element  <math>e</math>, and  <math>d_e</math> represents the moment arm distance vector from the centroid to the wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>.
 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 [[#img-3|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  <math>x_{sub}^{}</math> 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:
@@ Line 224: / Line 224: @@
 |}
-where,  <math>M_x</math>,  <math>M_y</math>, and  <math>M_z</math> represent the components of the actual center of gravity torque  <math>M_{wave}</math> in the hull coordinate system, while  <math>F_x</math>,  <math>F_y</math>, and  <math>F_z</math> are the components of  <math>F_{wave}</math> in the hull coordinate system.  <math>x</math>,  <math>y</math>, and  <math>z</math> represent the distances from the actual center of gravity to the origin  <math>x_{sub}^{}</math>.
+where <math>M_x</math>,  <math>M_y</math>, and  <math>M_z</math> represent the components of the actual center of gravity torque  <math>M_{wave}</math> in the hull coordinate system, while  <math>F_x</math>,  <math>F_y</math>, and  <math>F_z</math> are the components of  <math>F_{wave}</math> in the hull coordinate system.  <math>x</math>,  <math>y</math>, and  <math>z</math> represent the distances from the actual center of gravity to the origin  <math>x_{sub}^{}</math>.
 To ensure that the UUV remains centered in the mesh at all times, after each timestep calculation, the grid and local coordinates shown in [[#img-2|Figure 2]] are translated as a whole by displacement  <math>v_{sub}\cdot \Delta t</math>. Here,  <math>v_{sub}</math> represents the horizontal projection vector of the UUV's velocity in the global coordinate system.
@@ Line 283: / Line 283: @@
 ===3.1 Effect of wave height===
-A comparison of the changes in the marine environmental influence under different wave heights is shown in [[#img-7|Figure 7]], where,  <math>M_{wave}</math>'s center of moment is the center of gravity of the UUV during a specific mission, and  <math>M_{wave}</math> 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 [[#img-7|Figure 7]] are as follows:  <math>t_{wave}</math> is 6s,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0kn. Thewave heights  <math>a_{wave}</math> are 0.25m, 0.75m, 1.25m, 1.75m, and 2.25m, respectively.
+A comparison of the changes in the marine environmental influence under different wave heights is shown in [[#img-7|Figure 7]], where  <math>M_{wave}</math>'s center of moment is the center of gravity of the UUV during a specific mission, and  <math>M_{wave}</math> 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 [[#img-7|Figure 7]] are as follows:  <math>t_{wave}</math> is 6s,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0kn. Thewave heights  <math>a_{wave}</math> are 0.25m, 0.75m, 1.25m, 1.75m, and 2.25m, respectively.
 <div id='img-7'></div>
@@ Line 327: / Line 327: @@
 |}
-where,  <math>t_{Force}^0</math> represents the initial phase difference, while <math>t_{Force}^x</math>,  <math>t_{Force}^y</math>, and  <math>t_{Force}^z</math> correspond to the phase differences in the periodic variations of  <math>F_x</math>,  <math>F_y</math>, and  <math>F_z</math>, respectively.
+where  <math>t_{Force}^0</math> represents the initial phase difference, while <math>t_{Force}^x</math>,  <math>t_{Force}^y</math>, and  <math>t_{Force}^z</math> correspond to the phase differences in the periodic variations of  <math>F_x</math>,  <math>F_y</math>, and  <math>F_z</math>, respectively.
 The three components of  <math>M_{wave}</math> are influenced not only by  <math>F_{wave}</math> but also by the difference between the force center (<math>F_{wave}</math>) and the actual center of gravity (Eq. 9). Therefore, this paper only conducts qualitative and quantitative analysis on the variation pattern of  <math>F_{wave}</math>.

@@ Line 479: / Line 479: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |<math>v_{wave\_r}=\frac{2\cdot \pi }{k\cdot t_{wave}}+\cos \left(\frac{{\theta }_{yaw}\cdot \pi }{180}\right)\cdot v_{sub}\cdot \mbox{0}\mbox{.5144}</math>
+| style="text-align: center;" |<math>v_{wave\_r}=\frac{2\cdot \pi }{k\cdot t_{wave}}+\cos \left(\frac{{\theta }_{yaw}\cdot \pi }{180}\right)\cdot v_{sub}\cdot 0.5144</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)

@@ Line 479: / Line 479: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |<math>v_{wave\_r}=\frac{2\cdot \pi }{k\cdot t_{wave}}+\cos (\frac{{\theta }_{yaw}\cdot \pi }{180})\cdot v_{sub}\cdot \mbox{0}\mbox{.5144}</math>
+| style="text-align: center;" |<math>v_{wave\_r}=\frac{2\cdot \pi }{k\cdot t_{wave}}+\cos \left(\frac{{\theta }_{yaw}\cdot \pi }{180}\right)\cdot v_{sub}\cdot \mbox{0}\mbox{.5144}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)

@@ Line 281: / Line 281: @@
-===3.1 The effect of wave height===
+===3.1 Effect of wave height===
 A comparison of the changes in the marine environmental influence under different wave heights is shown in [[#img-7|Figure 7]], where,  <math>M_{wave}</math>'s center of moment is the center of gravity of the UUV during a specific mission, and  <math>M_{wave}</math> 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 [[#img-7|Figure 7]] are as follows:  <math>t_{wave}</math> is 6s,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0kn. Thewave heights  <math>a_{wave}</math> are 0.25m, 0.75m, 1.25m, 1.75m, and 2.25m, respectively.
@@ Line 331: / Line 331: @@
 The three components of  <math>M_{wave}</math> are influenced not only by  <math>F_{wave}</math> but also by the difference between the force center (<math>F_{wave}</math>) and the actual center of gravity (Eq. 9). Therefore, this paper only conducts qualitative and quantitative analysis on the variation pattern of  <math>F_{wave}</math>.
-===3.2 The effect of wave period===
+===3.2 Effect of wave period===
 [[#img-8|Figure 8]] presents a comparison of the changes in the marine environmental impact under different wave periods, with the specific conditions as follows:  <math>a_{wave}</math> is 1.25m,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0 kn. The wave periods  <math>t_{wave}</math> are 5s, 6s, 7s, 8s, and 9s, respectively.
@@ Line 361: / Line 361: @@
 From the comparison shown in [[#img-8|Figure 8]], it can be observed that when the UUV's velocity is 0 kn, the variation period of  <math>F_{wave}</math> matches the wave period for each condition. The amplitude change of  <math>F_{wave}</math> 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  <math>F_{wave}</math> under different conditions shown in [[#img-8|Figure 8]] still comply with Eq. (10).
-===3.3 The effect of pitch angle===
+===3.3 Effect of pitch angle===
 It presents a comparison of the changes in the marine environmental impact under different UUV pitch angles in [[#img-9|Figure 9]], with the specific conditions as follows:  <math>t_{wave}</math> is 6s,  <math>a_{wave}</math> is 1.25m,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0 kn. The pitch angles  <math>{\theta }_{attack}</math> are -8°, -4°, 0°, 4°, and 8°, respectively.
@@ Line 404: / Line 404: @@
 In short, the marine environmental impact varies minimally under different pitch angles.
-===3.4 The effect of heading angle===
+===3.4 Effect of heading angle===
 It presents a comparison of the changes in the marine environmental impact under different UUV heading angles in [[#img-11|Figure 11]], with the specific conditions as follows:  <math>t_{wave}</math> is 6s,  <math>a_{wave}</math> is 1.25m,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°, and  <math>v_{sub}</math> is 0 kn. The heading angles  <math>{\theta }_{yaw}</math> are 0°, 45°, 90°, 135°, and 180°, respectively.
@@ Line 434: / Line 434: @@
 From the comparison shown in [[#img-11|Figure 11]], it can be observed that changes in the heading angle have no effect on the variation period of  <math>F_{wave}</math>. Its variation period is the same as the wave period. The heading angle significantly affects the two horizontal components of  <math>F_{wave}</math>,  <math>F_x</math> ([[#img-11|Figure 11]]a) and  <math>F_z</math> ([[#img-11|Figure 11]]e). When the heading angle  <math>{\theta }_{yaw}</math> is 0° or 180°, meaning the UUV is head-on or tail-on to the waves, the longitudinal force  <math>F_x</math> reaches its maximum amplitude variation, while the transverse force  <math>F_z</math>'s amplitude variation is zero. As the UUV begins to turn towards 90°, the amplitude variations of  <math>F_x</math> and  <math>F_z</math> start to inversely affect each other. When the UUV fully turns to 90°, the amplitude variation of  <math>F_x</math> becomes zero, while the amplitude variation of  <math>F_z</math> reaches its maximum. Moreover, the maximum amplitude variation of  <math>F_z</math>'s period is greater than that of  <math>F_x</math>'s period, as the UUV's frontal area is the largest at a heading angle  <math>{\theta }_{yaw}</math> of 90°, with the entire side exposed to wave forces. Regardless of the heading angle, the vertical component <math>F_y</math>'s amplitude variation remains unchanged. Additionally, the heading angle  <math>{\theta }_{yaw}</math> also affects the initial phase difference  <math>t_{Force}^0</math> of the period variation of  <math>F_{wave}</math>. When  <math>{\theta }_{yaw}</math> is at 0° and 180°, the difference in the initial phase difference is 0.8s. Similarly, the different initial phase differences  <math>t_{Force}^0</math> are also due to the paper's approach of using the bow center as the reference point for the UUV's heading angle changes.
-===3.5 The effect of velocity===
+===3.5 Effect of velocity===
 It presents a comparison of the changes in the marine environmental impact under different UUV velocities in [[#img-12|Figure 12]], with the specific conditions as follows:  <math>t_{wave}</math> is 6s,  <math>a_{wave}</math> is 1.25m,  <math>d_{sub}</math> is 15m,  <math>{\theta }_{attack}</math> is 0°, and  <math>{\theta }_{yaw}</math> is 45°. The velocities  <math>v_{sub}</math> are 0 kn, 1.25 kn, 2.5 kn, 3.75 kn, and 5 kn, respectively.
@@ Line 526: / Line 526: @@
 |}
-===3.6 The effect of diving depth===
+===3.6 Effect of diving depth===
 It presents a comparison of the changes in the marine environmental impact under different UUV depths in [[#img-14|Figure 14]], with the specific conditions as follows:  <math>t_{wave}</math> is 6s,  <math>a_{wave}</math> is 1.25 m,  <math>{\theta }_{attack}</math> is 0°,  <math>{\theta }_{yaw}</math> is 45°, and  <math>v_{sub}</math> is 0 knots. The depths  <math>d_{sub}</math> are 5 m, 10 m, 15 m, 20 m, and 25 m, respectively.

@@ Line 390: / Line 390: @@
 From the comparison shown in [[#img-9|Figure 9]], it can be observed that the pitch angle has a minimal effect on  <math>F_{wave}</math> . The variation period for different  <math>F_{wave}</math> is the same as the wave period, which is 6s. As the pitch angle changes from -8° to 8°, the amplitude variation of  <math>F_{wave}</math> shows a monotonic decrease. This phenomenon is primarily due to the gradual decrease in the depth at which the UUV's actual center of gravity is located (with a depth change exceeding 8%) as the UUV's pitch angle varies from -8° to 8° around its bow center point, leading to a gradual decrease in the amplitude variation of  <math>F_{wave}</math> ([[#img-10|Figure 10]]). Additionally, apart from the change in depth, the horizontal position of the UUV's center of gravity also shifts forward with the change in pitch angle ([[#img-10|Figure 10]]). This affects the initial phase difference  <math>t_{Force}^0</math> in Eq. (10), meaning  <math>t_{Force}^0</math> decreases as the absolute value of the pitch angle increases. Since the horizontal forward shift caused by the UUV's pitch angle change is not significant, the initial phase difference  <math>t_{Force}^0</math> does not change markedly. When the pitch angle is -8° or 8°, the initial phase difference  <math>t_{Force}^0</math> is 0.28579s, and when the pitch angle is 0°, the initial phase difference  <math>t_{Force}^0</math> is 0.28301s, with a difference of about 1%. The periodic phase differences of the three components of  <math>F_{wave}</math> still satisfy Eq. (10).
 <div id='img-10'></div>
@@ Line 401: / Line 399: @@
 |}
 In short, the marine environmental impact varies minimally under different pitch angles.

@@ Line 266: / Line 266: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:65%;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_SUN_859092175-image79.png|600px]]
+|style="text-align: center;padding:10px;"| [[Image:Draft_SUN_859092175-image79.png|700px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 5'''. Comparison of impact on marine environment (<math>{\theta }_{attack} =0\mbox{°}</math>, <math>{\theta }_{yaw}  =0\mbox{°}</math>,  <math>v_{sub} =0</math>kn,  <math>d_{sub}=12.5</math>m,  <math>a_{wave}=1.25</math>m,  <math>t_{wave} =5.5</math>s)

@@ Line 48: / Line 48: @@
 |}
-Here,  <math>{\alpha }_w</math> represents the volume fraction of seawater (<math>{\alpha }_w=V_w/V_{cell}</math>,  <math>V_w</math> represents the unit volume occupied by seawater,  <math>V_{cell}</math> represents the total volume of seawater),  <math>{\rho }_w</math> represents the density of seawater,  <math>v</math> represents velocity vector,  <math>t</math> represents time,  <math>p</math> represents pressure, and  <math>g</math> represents gravity vector. In Eq. (2),  <math>\rho </math> represents density and  <math>\mu </math> represents dynamic viscosity, both are calculated by Eqs. (3) and (4) respectively.
+Here,  <math>{\alpha }_w</math> represents the volume fraction of seawater (<math>{\alpha }_w=V_w/V_{cell}</math>,  <math>V_w</math> represents the unit volume occupied by seawater,  <math>V_{cell}</math> represents the total volume of seawater),  <math>{\rho }_w</math> represents the density of seawater,  <math>v</math> represents velocity vector,  <math>t</math> represents time,  <math>p</math> represents pressure, and  <math>g</math> represents gravity vector. In Eq. (2),  <math>\rho </math> represents density and  <math>\mu </math> represents dynamic viscosity, both are calculated by Eqs. (3) and (4), respectively:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 102: / Line 102: @@
 |}
-Here,  <math>C</math> is the wave velocity, and the coefficient  <math>\lambda </math> and wave number  <math>k</math> are solved by combining the following equations:
+where,  <math>C</math> is the wave velocity, and the coefficient  <math>\lambda </math> and wave number  <math>k</math> are solved by combining the following equations:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 117: / Line 117: @@
 |}
-Here,  <math>H</math> represents the height of the ocean waves,  <math>L</math> represents wavelength,  <math>d</math> represents total depth,  <math>L_0=\frac{gT^2}{2\pi }</math>,  <math>T</math> represents the wave period, and the coefficients  <math>A_i</math>,  <math>B_j</math>, and  <math>C_j</math> can be referred to in references [13-14].
+where,  <math>H</math> represents the height of the ocean waves,  <math>L</math> represents wavelength,  <math>d</math> represents total depth,  <math>L_0=\frac{gT^2}{2\pi }</math>,  <math>T</math> represents the wave period, and the coefficients  <math>A_i</math>,  <math>B_j</math>, and  <math>C_j</math> can be referred to in references [13-14].
 ===2.2 Calculation model===
@@ Line 142: / Line 142: @@
-Subsequently, the mesh models are initialized based on the VOF (Volume of Fluid) model ([[#img-3|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  <math>\frac{\partial \phi (x,z,t)}{\partial x}</math> and a vertical velocity of  <math>\frac{\partial \phi (x,z,t)}{\partial z}</math> . The density of seawater  <math>{\rho }_w</math> is 1024 kg/m³, and the density of air  <math>{\rho }_a</math> is 1.18 kg/m³.
+Subsequently, the mesh models are initialized based on the VOF (Volume of Fluid) model ([[#img-3|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  <math>\frac{\partial \phi (x,z,t)}{\partial x}</math> and a vertical velocity of  <math>\frac{\partial \phi (x,z,t)}{\partial z}</math>. The density of seawater  <math>{\rho }_w</math> is 1024 kg/m³, and the density of air  <math>{\rho }_a</math> is 1.18 kg/m³.
 <div id='img-3'></div>
@@ Line 164: / Line 164: @@
-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)
+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):
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 206: / Line 206: @@
 |}
-Here, the superscript  <math>left</math> denotes the wave environment, and the superscript  <math>right</math> denotes the quiescent water environment, each corresponding to the models on the left and right of [[#img-2|Figure 2]], respectively.  <math>p_e</math> represents the flow field pressure of the UUV wall element  <math>e</math>,  <math>n_e</math> represents the normal direction vector of the UUV wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>,  <math>{\tau }_e</math> represents the shear stress experienced by the UUV wall element  <math>e</math>,  <math>a_e</math> represents the area of the wall element  <math>e</math>, and  <math>d_e</math> represents the moment arm distance vector from the centroid to the wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math> .
+where, the superscript  <math>left</math> denotes the wave environment, and the superscript  <math>right</math> denotes the quiescent water environment, each corresponding to the models on the left and right of [[#img-2|Figure 2]], respectively.  <math>p_e</math> represents the flow field pressure of the UUV wall element  <math>e</math>,  <math>n_e</math> represents the normal direction vector of the UUV wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>,  <math>{\tau }_e</math> represents the shear stress experienced by the UUV wall element  <math>e</math>,  <math>a_e</math> represents the area of the wall element  <math>e</math>, and  <math>d_e</math> represents the moment arm distance vector from the centroid to the wall element  <math>e</math> in the local coordinate system  <math>x_{sub}^{}</math>.
-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 [[#img-3|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  <math>x_{sub}^{}</math> 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
+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 [[#img-3|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  <math>x_{sub}^{}</math> 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:
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 16: / Line 16: @@
 ==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].
+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

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 ==References==
 [1] ITTC. Tasks and structure of the 29th ITTC technical committees and groups. Proceedings of the 28th International Towing Tank Conference. Wuxi, China, 1:393-408, 2017.
-[2] Wang J.H. and Wang D.C.. CFD simulations of ship maneuvering motion. Journal of Harbin Engineering University, 39(5):813-824, 2018.
+[2] Wang J.H.,  Wang D.C. CFD simulations of ship maneuvering motion. Journal of Harbin Engineering University, 39(5):813-824, 2018.
-[3] P.M. Carrica, F. Ismail, M. Hyman et al. Turn and zigzag maneuvers of a surface combatant using a URANS approach with dynamic overset grids. Journal of marine science and technology, 18(2):166-181, 2013.
+[3]  Carrica P.M.,  Ismail F.,  Hyman M. et al. Turn and zigzag maneuvers of a surface combatant using a URANS approach with dynamic overset grids. Journal of Marine Science and Technology, 18(2):166-181, 2013.
-[4] Wang J.H. and Wang D.C.. CFD simulations of free running ship under course keeping control. Ocean engineering,141:450-464, 2017.
+[4] Wang J.H.,  Wang D.C. CFD simulations of free running ship under course keeping control. Ocean Engineering, 141:450-464, 2017.
-[5] H.V. Thurman and A.P. Trujillo. "Essentials of Oceanography", Edition 5, Prentice Hall, pp.240-243, 2001.
+[5]  Thurman H.V.,   Trujillo A.P. Essentials of oceanography. Edition 5, Prentice Hall, pp.240-243, 2001.
-[6] F. Ghasemzadeh, A. Ursolov, M. Moonesun et al. Effective depth of regular wave on submerged submarine. Indian Journal of Geo Marine Sciences, 48(9):1476-1484, 2019.
+[6]  Ghasemzadeh F.,  Ursolov A.,  Moonesun M. et al. Effective depth of regular wave on submerged submarine. Indian Journal of Geo Marine Sciences, 48(9):1476-1484, 2019.
-[7] Li J., Zhang Q. and Chen T.. Numerical Investigation of Internal Solitary Wave Forces on Submarines in Continuously Stratified Fluids. Journal of marine science and engineering, 9(12):1374, 2021.
+[7] Li J., Zhang Q., Chen T. Numerical investigation of internal solitary wave forces on submarines in continuously stratified fluids. Journal of Marine Science and Engineering, 9(12):1374, 2021.
 [8] Cui J., Dong S., Wang Z. et al. Kinematic response of submerged structures under the action of internal solitary waves. Ocean Engineering, 196:106814, 2020.
-[9] Sun P., Li H.F. and Kang. X.B.. Numerical simulation on free motion response of a submarine induced by internal solitary wave. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 39(4):1-12, 2023.
+[9] Sun P., Li H.F., Kang. X.B. Numerical simulation on free motion response of a submarine induced by internal solitary wave. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 39(4):1-12, 34, 2023.
-[10] Dezheng Z, Lijing Y, Xuanwei C, et al.Robust Identification Algorithm for Unmanned Underwater Vehicles Dynamics Model Parameters[J]. Journal of Physics: Conference Series, 2338(1), 2022.
+[10] Dezheng Z., Lijing Y., Xuanwei C., et al. Robust identification algorithm for unmanned underwater vehicles dynamics model parameters. Journal of Physics: Conference Series, 2338(1), 2022.
-[11] M. Ajmi, N. Hnaien, S. Marzouk et al. Numerical investigation of heat transfer enhancement of an inclined heated offset jet. International Communications in Heat and Mass Transfer, 116, 2020.
+[11]  Ajmi M.,  Hnaien N.,  Marzouk S. et al. Numerical investigation of heat transfer enhancement of an inclined heated offset jet. International Communications in Heat and Mass Transfer, 116, 104682, 2020.
-[12] On Stokes wave solutions[J]. Proceedings of the Royal Society A, 478(2258): 20210732-20210732, 2022.
+[12] Zhao K.,  Liu P.L.-F. On Stokes wave solutions. Proceedings of the Royal Society A, 478(2258):20210732, 2022.
-[13] Zhu Y.R.. Analysis of the applicability of several wave theories. Coastal Engineering, 2(2):11-27, 1983.
+[13] Zhu Y.R. Analysis of the applicability of several wave theories. Coastal Engineering, 2(2):11-27, 1983.
-[14] J.D. Fenton. A fifth-order Stokes theory for steady waves. Journal of Waterway, Port, Coastal, and Ocean Engineering, 111:216-234, 1985.
+[14]  Fenton J.D. A fifth-order Stokes theory for steady waves. Journal of Waterway, Port, Coastal and Ocean Engineering, 111:216-234, 1985.
-[15] Zekai Z, Weishi M, Jun D, et al.Design and Implementation of a Modular UUV Simulation Platform[J]. Sensors, 22(20): 8043-8043, 2022.
+[15] Zekai Z., Weishi M., Jun D., et al. Design and implementation of a modular UUV simulation platform. Sensors, 22(20):8043-8043, 2022.
-[16] Zhiyuan H, Zheng W and Yang Y. Research on Three Dimensional Global Path Planning of Unmanned Underwater Vehicle[J]. Journal of Physics: Conference Series, 1894(1), 2021.
+[16] Zhiyuan H., Zheng W.,  Yang Y. Research on three dimensional global path planning of unmanned underwater vehicle. Journal of Physics: Conference Series, 1894(1), 2021.
-[17] Yan Z, Wang L, Zhang W, et al. Polar Grid Navigation Algorithm for Unmanned Underwater Vehicles[J]. Sensors, 17(7): 1599, 2017.<span id="_Hlk160563923"></span><span id="_nebF9F37563_C5E0_47BA_A3E6_AB43CB4808D0"></span><span id="_neb0F8BB81C_4ACB_4A46_A5D8_18B1A98389B0"></span>
+[17] Yan Z., Wang L., Zhang W., et al. Polar grid navigation algorithm for unmanned underwater vehicles. Sensors, 17(7):1599, 2017.