Structural Design and Horizontal Wave Force Estimation of a Wall-Climbing Robot for the Underwater Cleaning of Jackets

Abstract

Currently, divers face significant safety risks when cleaning marine organisms from the steel structures of offshore underwater platform jackets. Consequently, utilizing robots instead of divers to carry out underwater biofouling removal operations will be an important development direction for the underwater maintenance of offshore platforms in the future. In this study, a wall-climbing robot was designed to clean marine organisms from the underwater surface of a platform jacket leg. The overall structure of the underwater cleaning wall-climbing robot is introduced, including the cleaning actuator and the variable curvature-adapted connecting rod mechanism. The corresponding relationship between the variable curvature-adapted connecting rod mechanism and the jacket leg is analyzed in detail. The variable curvature-adapted connecting rod mechanism was optimized using a genetic algorithm to ensure that the underwater cleaning wall-climbing robot can adapt to a minimum diameter of 1 m for the jacket leg. By drawing on Airy wave theory and random wave theory, the Airy wave parameters for waves were analyzed under different sea conditions, considering practical application scenarios. By using Fluent software 2022, a 2D numerical wave tank was constructed to simulate waves under various sea conditions, and the wave surface shapes for different sea states were determined. By building on the Morison equation, a method for calculating the horizontal wave forces on the underwater cleaning wall-climbing robot using the equivalent area and equivalent volume is proposed. By using the two aforementioned methods, the horizontal wave forces on the underwater cleaning wall-climbing robot under specific sea states were determined. The horizontal wave forces of the underwater cleaning wall-climbing robot under different sea conditions were analyzed and simulated in a 3D numerical wave tank. By comparing the theoretical analysis results with the numerical simulation results, where the maximum difference at the extreme points is approximately 11%, the feasibility of the proposed horizontal wave force estimation method was verified.

1. Introduction

When the jacket of an offshore platform is immersed in seawater for a long time, numerous marine organisms, such as barnacles, mussels, and algae, adhere to its surface [1]. The attachment of marine organisms increases the stress area of the underwater steel structure of the jacket, aggravates the load caused by waves and ocean currents, and weakens its bearing capacity [2,3]. At the same time, because the metabolic secretions of marine organisms are generally acidic, the corrosion of the surface protective layer of underwater steel structures is aggravated, and their service life is shortened [4,5]. Therefore, regularly cleaning the jacket of the offshore platform is very important for its safe operation. At present, the cleaning of marine organisms from jackets is almost always done by divers. Operational ROVs carry a high-pressure water gun to clean marine organisms from jackets only when the water depth exceeds the diver’s operating limit [6].

Divers possess high flexibility in cleaning marine organisms, as they can carefully clean complex structures and thoroughly remove marine organisms attached to the surface of the underwater steel structure of the jacket. However, divers face the following severe problems when cleaning off marine organisms: (1) High risk: Divers need to carry various pipelines when cleaning off marine organisms underwater, which seriously affects their mobility. At the same time, whether using a high-pressure water gun or a cavitation water gun to clean marine organisms from a jacket underwater, many floating objects are produced during operation, which seriously affect divers’ sight and bring great hidden dangers to their safe operation. (2) Limited time: Divers’ underwater operation time is influenced by multiple factors such as sea conditions, temperature, and visibility, which seriously restricts their operation time. Several divers often need to work continuously for two or three months to clean underwater marine organisms from an offshore platform jacket. (3) Limited operation depth: At present, the operation depth of air diving is generally 50 m. If the water depth exceeds 50 m, saturated diving is needed. This diving method has a high cost and limited operation time [7,8].

With the continuous development of robot technology, researchers have gradually performed research work on using robots to clean underwater marine organisms from offshore platform jackets. At present, the underwater robots used to clean marine organisms from jackets have three main forms: (1) ROVs: An ROV carries a high-pressure water gun to clean the underwater jacket. The advantage of this operation method is that the ROV has high mobility, enabling it to perform underwater cleaning tasks in complex positions. It also offers long operation times and a wide range of operational depths. However, this method also has several disadvantages. The main issues are the poor performance of ROVs in rough seas and the difficulty in precisely controlling their movement and attitude to clean specified areas. Additionally, the high-pressure water’s recoil force makes it challenging for the ROV to maintain a fixed target distance, affecting the cleaning effectiveness. Moreover, the ROV cannot thoroughly clean the parts of the jacket legs located within the wave splash zone [9]. (2) Pipe-holding robots: For a jacket without obstacles (such as an anode block) on the surface, a pipe-holding robot composed of several concentric rings and moving mechanisms can be used to clamp the jacket surface, and several high-pressure water guns are fixed on the ring surface to clean off the marine organisms attached to the jacket surface. The advantages of this operation method are its simple structure, high operational efficiency, ease of operation, and low cost. The main disadvantage is that the robot requires a high-quality working surface without obstacles, such as anode blocks and inclined support rods, thus limiting the application range of pipe-holding robots [10]. (3) Wall-climbing robots: Wall-climbing robots adsorb onto the jacket surface by the principle of permanent magnet adsorption. With high-pressure water guns installed at the front end or both sides of the jacket, the robot can control the movement and clean off the marine organisms attached to the jacket surface. The main advantages of the wall-climbing robot for cleaning the jacket are its high reliability, minimal impact from external sea conditions during operation, long operation times, wide operational range, and ease of operation for personnel. The disadvantages of this method are the complex structure of the wall-climbing robot, poor obstacle-crossing ability, and relatively low technological maturity [11].

At present, the field of applying underwater cleaning wall-climbing robots focuses on the underwater cleaning of ships [12,13], and its research scope is limited to the structural design and magnetic circuit optimization of wall-climbing robots [14,15,16]. Research relating to underwater wall-climbing robots that clean marine organisms from offshore platform jackets is extremely rare. Therefore, this study designs a permanent magnet adsorption underwater cleaning wall-climbing robot that can adapt to different pipe diameters. With the high-pressure water gun installed at the front end of the robot, the marine organisms attached to the jacket surface can be cleaned. Regarding the anti-wave performance of wall-climbing robots that users are most concerned about, this study analyzes the wave surface shape under different sea conditions based on Airy wave theory and emphatically studies the horizontal wave force on a wall-climbing robot under different sea conditions, which provides the theoretical basis for the subsequent improvement of force analysis when designing an underwater cleaning wall-climbing robot. In addition, this study proposes a calculation method for estimating the wave load of an underwater cleaning wall-climbing robot and compares the outcomes with the numerical simulation results of Fluent software 2022, which verifies the feasibility of the proposed horizontal wave force estimation method.

The organizational structure of this paper is as follows: Section 2 introduces the overall structure of the underwater cleaning wall-climbing robot, which studies the corresponding relationship between the variable curvature-adapted connecting rod mechanism and the jacket leg. Section 3 discusses Airy wave theory and the PM wave spectrum to analyze the wave surface parameters of different sea state grades and simulate different sea state grades in a 2D numerical wave tank using Fluent software 2022, determining its wave surface shape. Section 4 studies the horizontal wave forces on an underwater cleaning wall-climbing robot under different sea conditions based on the Morison equation and compares these outcomes with those from a 3D numerical wave tank simulation. Section 5 summarizes the research and concludes by looking forward to follow-up research directions. This study innovatively applies wave theory from ocean engineering to the analysis and design of wall-climbing robots. We analyze the horizontal wave forces acting on an underwater cleaning wall-climbing robot by considering the actual sea state, providing a theoretical foundation for the subsequent analysis of the robot’s adhesion stability and motion flexibility. Additionally, the Airy wave parameters derived from the real sea state in this study can also be used to analyze the impact of waves on other types of robots.

2. Robot Structure Design

The jacket-cleaning underwater wall-climbing robot designed in this study adopts the principle of permanent magnet adsorption [17]. Its structure is mainly composed of a cleaning actuator, a steering and moving mechanism, a variable curvature-adapted connecting rod mechanism, a permanent magnet adsorption unit, a control box, and a buoyancy block. The underwater cleaning wall-climbing robot has variable curvature adaptability, and its overall structure is shown in Figure 1.

The main parameters of the underwater cleaning wall-climbing robot are as follows: (1) Robot weight: 160 kg; (2) Robot dimensions (L × W × H): 1220 mm × 1020 mm × 570 mm; (3) Minimum working surface diameter: 1 m; (4) Single operation width: 700 mm; (5) Maximum working depth: 50 m; (6) Magnetic adsorption force: greater than 5000 N; (7) Motor power: 400 W for the drive servo motor and 200 W for the steering servo motor; (8) Operation control method: Wired remote control.

2.1. Cleaning Actuator

The underwater cleaning wall-climbing robot uses a high-pressure water operation mode to clean off the marine organisms attached to the jacket surface. The high-pressure water gun is clamped by a fixed block, installed on the circular guide rail in the cleaning executive mechanism, and connected to the reciprocating screw nut through a lifting connecting rod. The core design of the cleaning actuator adopts a reciprocating screw structure, as shown in Figure 2. This design mode can realize the reciprocating movement of the high-pressure water gun along the circular arc guide rail without changing the rotating direction of the motor. Compared with the common ball screw transmission structure, which needs to frequently change the rotating direction of the motor, the reciprocating screw structure can improve cleaning efficiency.

Because the high-pressure water gun always moves along the circular arc guide rail in a reciprocating motion, the included angle between the high-pressure water gun and the jacket surface, as well as the target distance in the jacket’s axial plane, remains unchanged. This setup ensures the consistency of the cleaning operation’s effect along the circumferential direction. The reciprocating cleaning attitude of the high-pressure water gun is depicted in Figure 3.

2.2. Variable Curvature-Adapted Connecting Rod Mechanism

The jacket leg of the offshore platform has many specifications, and its diameter is concentrated in the range of 1–3 m [18], so the underwater cleaning wall-climbing robot must be adaptable to variable curvature. The underwater cleaning wall-climbing robot designed in this study can adapt to jacket legs with diameters starting from a minimum of 1 m. The posture of the robot when adapting to jacket legs of different diameters is illustrated in Figure 4.

When the underwater cleaning wall-climbing robot adapts to the variable curvature jacket, if the center line of the driving wheel does not coincide with the diameter line of the jacket, only the edge of one side of the driving wheel has contact with the jacket, as shown in Figure 5a; this leads to an uneven force on the driving wheel in its width direction, a large force on the side of the driving wheel in contact with the jacket, and serious wear and influence on the movement of the robot. In order to solve this problem, a variable curvature-adapted connecting rod mechanism is designed in this study to ensure the center line of the driving wheel is consistent with the radial line of the jacket. Its principle is shown in Figure 5b.

The motion diagram for the variable curvature-adapted connecting rod mechanism in the underwater cleaning wall-climbing robot is depicted in Figure 6. The contact state between the underwater cleaning wall-climbing robot and the jacket leg is shown in Figure 5b; the extension line of the driving wheel center line ($DJ$ line in Figure 6) coincides with the center of the jacket leg. The underwater cleaning wall-climbing robot can adapt to different jacket legs by adjusting the range of $h$.

Based on the relationship between the projections of the connecting rod length in the $x$ and $y$ directions shown in the mechanism motion diagram, the following relational expression can be derived [19]:

$$\left\{\begin{array}{c}(R+l_6)\sin (90-{\theta }_6-{\theta }_3)=l_7\cos ({\theta }_3)+l_4\cos ({\theta }_4)+l_5\\ l_1+l_2\cos ({\theta }_2)=l_3\cos ({\theta }_3)+l_4\cos ({\theta }_4)+l_5\\ h=l_2\sin ({\theta }_2)+l_3\sin ({\theta }_3)-l_4\sin ({\theta }_4)\end{array}\right.$$

(1)

The initial design parameters of the variable curvature-adapted connecting rod mechanism for the underwater cleaning wall-climbing robot are as follows: ${\theta }_2={30}^{°}$, $l_1=155 \mathrm{mm}$, $l_2=235 \mathrm{mm}$, $l_3=188.48 \mathrm{mm}$, $l_4=100 \mathrm{mm}$, $l_5=86.5 \mathrm{mm}$, $l_6=329 \mathrm{mm}$, $l_7=140.01 \mathrm{mm}$, and ${\theta }_6={68.2}^{°}$. As shown in Equation (1), when the robot adapts to a jacket leg with a diameter of 1 m, the slider displacement, $h$, is 62 mm. For a jacket leg with a diameter of 3 m, $h$ is 109 mm. When designing the variable curvature-adapted connecting rod mechanism for the underwater cleaning wall-climbing robot, a reasonable slider displacement is crucial. It is especially important to ensure that $h$ is not too small when the robot adapts to the minimum jacket leg. Additionally, when the robot adapts to jacket legs with different diameters, the adjustment range of $h$ should be as large as possible. Therefore, it is necessary to optimize the slider displacement. As shown in Figure 6, $h$ is influenced by multiple parameters. Considering that $l_2$, $l_4$, $l_5$, and ${\theta }_2$ are not affected by the frame and wheel dimensions and are easy to adjust during the design process, these parameters were chosen as optimization variables for the slider displacement. First, the influence of the jacket radius, $R$, and the parameters $l_2$, $l_4$, $l_5$, and ${\theta }_2$ for $h$ was analyzed, as shown in Figure 7:

As shown in Figure 7, the effects of different parameters on slider displacement in the variable curvature-adapted connecting rod mechanism vary significantly. Given that slider displacement is influenced by multiple variables, this issue can be transformed into a multi-parameter, multi-objective optimization problem. Constraints can be established based on the range of changes for $l_2$, $l_4$, $l_5$, and ${\theta }_2$, where $185\le l_2\le 260$, $80\le l_4\le 120$, $70\le l_5\le 100$, and ${20}^{°}\le {\theta }_2\le {60}^{°}$. The objective function can be defined such that slider displacement, $h$, should be 62 mm when the radius of the jacket leg is 500 mm and 125 mm when the radius is 1500 mm. Genetic algorithms can be used to solve this problem [20], and the results of the solution are presented in

Table 1. Parameters of the connecting rod mechanism.

Variable Name	Pre-Optimization	Post-Optimization
$l_2$	235 mm	214.1211 mm
$l_4$	100 mm	81.997 mm
$l_5$	86.5 mm	83.5987 mm
${\theta }_2$	30°	24.6536°

.

Based on the connecting rod parameters before and after optimization, the relationship between slider displacement and the radius of the jacket leg is illustrated in Figure 8. As shown in Figure 8, when the underwater cleaning wall-climbing robot adapts to a jacket leg with a diameter of 1 m, $h$ is 64.272 mm, and when the robot adapts to a jacket leg with a diameter of 3 m, $h$ is 124.51 mm. The adjustment range of the slider displacement increased by about 28.17% compared to that before optimization.

3. Wave and Sea State Analysis

At present, the research on underwater wall-climbing robots focuses on structural design and magnetic circuit optimization, and research on the hydrodynamic part of underwater wall-climbing robots is rare. In the literature, the research on the hydrodynamic part of the underwater wall-climbing robot is limited only to the analysis of the steady-state flow on the velocity of the flow field near the underwater wall-climbing robot and the distribution of the fluid pressure on the robot surface [21,22,23]. The findings of this research are still far from solving the core problem of the anti-wave performance of underwater wall-climbing robots, which is the aspect that users are most concerned about. The ability of the underwater wall-climbing robot to work normally under various sea conditions has also not been fully addressed. This study innovatively applies wave theory to the study of an underwater wall-climbing robot and analyzes the horizontal wave force of an underwater cleaning wall-climbing robot at a specific sea state level combined with the actual sea wave conditions.

3.1. Mathematical Description of Waves

The mathematical characteristics of waves are shown in Figure 9.

The main characteristics of waves are as follows: (1) wave height, $H$: the vertical distance between the crest and the adjacent trough; (2) wavelength, $\lambda$: the horizontal distance between the wave crest and adjacent wave trough; (3) wave amplitude, $A$: the vertical distance from the peak to the adjacent static water surface (wave amplitude $A$ is half of the wave height $H$); and (4) still water depth, $d$: the vertical distance between the still water surface and the bottom. In addition, the wave number $k$, wave circle frequency $\omega$, wave period $T$, and wave frequency $f$ can be defined, as shown in the following equation [24]:

$$k={\displaystyle \frac{2\pi }{\lambda }}\hspace{1em}\hspace{1em}\hspace{1em}\omega =2\pi f={\displaystyle \frac{2\pi }{T}}$$

(2)

According to the different mathematical characteristics of waves, the wave theories commonly used in engineering mainly include Airy wave theory, Stokes wave theory, cnoidal wave theory, and solitary wave theory [25,26]. For the convenience of engineering applications, Airy wave theory is used to study the influence of waves on the jacket of offshore platforms in Port Hydrographic Standards [27]. Wind, ocean currents, and the number of jacket legs in a marine environment can all influence wave propagation, which, in turn, affects the wave forces acting on underwater cleaning wall-climbing robots. Given that there are currently no published studies specifically addressing the wave forces on wall-climbing robots, we have opted to simplify the complex marine environment by disregarding external factors such as wind, ocean currents, and the number of jacket legs. We focus solely on the wave forces exerted on the underwater cleaning wall-climbing robots using Airy wave theory.

3.2. Analysis of Robot Wave Condition

Because the free surface kinematic and dynamic conditions in the basic governing equations of the water wave problem are nonlinear and must be satisfied on the unknown free surface [28], as shown in the following formula, solving these equations is extremely difficult.

$$\begin{array}{l}{\nabla }^2\varphi =0\\ {\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }t}}+{\displaystyle \frac{{\left|\nabla \varphi \right|}^2}{2}}+g\eta =0\\ {\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }y}}={\displaystyle \frac{\mathsf{\partial }\eta }{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }x}}{\displaystyle \frac{\mathsf{\partial }\eta }{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }z}}{\displaystyle \frac{\mathsf{\partial }\eta }{\mathsf{\partial }z}}\end{array}$$

(3)

In order to solve the basic governing equations of the water wave problem, researchers assume the ratio of wave amplitude to wavelength is small, meaning $A/\lambda \ll 1$, and the waves conforming to this characteristic belong to Airy waves [29]. If the wave is an Airy wave, the change in the liquid surface position, $\eta$, is small, and the boundary condition on the unknown free surface can be simplified to satisfy a static water surface. According to the above simplification, the velocity potential of an Airy wave $\varphi$, the wave surface equation $\eta$, the horizontal velocity of fluid particles $u$, the vertical velocity of fluid particles $v$, the horizontal acceleration of fluid particles $a_x$, and the vertical acceleration of fluid particles $a_y$ can be obtained by applying potential flow theory, as shown in

Table 2. Relation of Airy wave parameters [30].

Name	Wave Parameter Relation
Velocity potential	$\varphi ={\displaystyle \frac{gA}{\omega }}\sin (kx-\omega t){\displaystyle \frac{\cosh k(y+d)}{\cosh kd}}$
Wavefront equation	$\eta =A\cos (kx-\omega t)$
Horizontal velocity	$u=A\omega {\displaystyle \frac{\cosh k(y+d)}{\sinh kd}}\cos (kx-\omega t)$
Vertical velocity	$v=A\omega {\displaystyle \frac{\sinh k(y+d)}{\sinh kd}}\sin (kx-\omega t)$
Horizontal acceleration	$a_x=Agk{\displaystyle \frac{\cosh k(y+d)}{\sinh (kd)}}\sin (kx-\omega t)$
Vertical acceleration	$a_y=-Agk{\displaystyle \frac{\sinh k(y+d)}{\sinh (kd)}}\cos (kx-\omega t)$

. According to Equation (2) and Table 2, the calculation of wave parameters according to Airy wave theory does not only require known wave wavelengths or wave periods.

According to the Beaufort scale [31,32], the effective wave heights and wave surface characteristics of sea conditions of classes 1–6 are shown in

Table 3. Beaufort scale for sea state [32].

Sea State	Effective Wave Height ${\mathit{H}}_{\mathit{s}}$ (m)	Name	Description of Sea State Characteristics
0	0	No waves	The sea surface is as smooth as a mirror.
1	$H_s<0.1$	Microwaves	Corrugation.
2	$0.1\le H_s<0.5$	Small waves	The waves are very small, and the peaks begin to break, but the waves are not white.
3	$0.5\le H_s<1.25$	Light waves	The waves are not big, but they are very eye-catching. The peaks of the waves break, and white waves form in some places.
4	$1.25\le H_s<2.5$	Medium waves	The waves have evident shapes, and white waves are formed everywhere.
5	$2.5\le H_s<4.0$	Big waves	Tall peaks appear, the waves occupy a large area above the peaks, and the wind begins to cut off the waves on the peaks.
6	$4.0\le H_s<6.0$	Huge waves	The stripped waves on the crest begin to stretch into bands along the slope of the wave.

. Because the effective wave height of sea conditions 0–2 is small and the water flow velocity is low, it has a minimal influence on the underwater cleaning and climbing robot. When the sea condition is above level 6, outdoor operation is prohibited according to relevant specifications, so this study mainly studies the influence of waves on underwater cleaning wall-climbing robots under sea conditions levels 3–6. Because the Beaufort scale only provides the effective wave height of waves at different sea state levels but not the corresponding wave wavelength or wave period, this study calculates the wave period by combining the Pierson–Moskowitz spectrum using random wave theory [33] and the PM spectrum density function, as follows:

$$S(\omega )={\displaystyle \frac{0.78}{{\omega }^5}}\exp (-{\displaystyle \frac{3.11}{{\omega }^4{H}_s^2}})$$

(4)

When studying the random characteristics of waves, the moments of each order of the wave spectrum are often used [34], as shown in the following formula:

$$m_r={\displaystyle {\int }_0^{\infty }{\omega }^r·S(\omega )d\omega }$$

(5)

According to random wave theory, the expression of the average wave period is shown in the following:

$$\overline{T}=2\pi \sqrt{{\displaystyle \frac{m_0}{m_2}}}$$

(6)

where $\overline{T}$ is the average period of the waves, and $m_2$ and $m_0$ are the second moment and the zero moment of the wave spectrum, respectively.

The relationship between the effective period and the average period of waves is shown in the following:

$$T_s\approx 1.2\overline{T}$$

(7)

where $T_s$ is the effective period of the waves, and $\overline{T}$ is the average period of the waves.

The wave parameters calculated from Table 2 and Table 3 and Formulas (3)–(7) for sea state levels 3–6 are shown in

Table 4. Sea state class parameters.

Sea State	Effective Wave Height ${\mathit{H}}_{\mathit{s}}$ (m)	Effective Period ${\mathit{T}}_{\mathit{s}}$ (s)	Wavelength $\mathit{\lambda }$ (m)	Circular Frequency $\mathit{\omega }$ (${\mathbf{s}}^{-1}$)	Wavenumber $\mathit{k}$ (${\mathbf{m}}^{-1}$)
3	1.25	4.7680	35.4982	1.3178	0.1770
4	2.5	6.7430	70.9964	0.9318	0.0885
5	4.0	8.5293	113.6170	0.7367	0.0553
6	6.0	10.4462	170.2760	0.6015	0.0369

.

3.3. Numerical Simulation of an Airy Wave

The most commonly used method to study wave problems is to conduct model experiments in physical tanks. Waves in physical tanks are produced by rocking plates or pushing plates [35]. However, due to the long time and large space required to build physical tanks, the construction cost is also extremely high, so seeking a research method that can replace physical tanks to analyze wave problems is necessary. With the development of computer technology, researchers have created a numerical wave tank for analyzing wave problems. Compared with a physical tank, the construction of a numerical wave tank takes less time, and the analysis is more convenient. The wave generation modes in numerical wave tanks are mainly active term wave generation and boundary wave generation [36]. In this section, by using the Open Channel wave-making method of CFD commercial software Fluent, a 2D Airy wave numerical wave tank under sea states 3–6 is discussed to verify the wave height generated by the numerical wave tank and determine the position of the underwater cleaning wall-climbing robot in the numerical wave tank during the subsequent calculation of horizontal wave force. The theory of water waves holds that when the wave is an Airy wave, according to the relationship between still depth and wavelength, it can be divided into the following: (1) deep water waves, when the still depth is greater than half of the wavelength; (2) shallow water waves, when the still water depth is less than one-twenty-fifth of the wavelength; and (3) medium depth waves, when the relationship between the still depth and wavelength is between the above two conditions [37]. The sea area where the offshore platform jacket is located is generally about 100 m deep. The sea area where the jacket is located belongs to the deep water wave condition in an Airy wave, according to the wavelengths in Table 4. Therefore, the still water depth should be greater than half of the corresponding wave wavelengths when creating a numerical wave tank for sea states 3–6. The dimensions of the 2D numerical wave tank corresponding to sea states 3–6 are shown in

Table 5. Dimensions of the 2D numerical wave tank.

Sea State	Length (m)	Height (m)	Still Water Depth (m)
3	213	30	25
4	426	45	40
5	681	75	65
6	1020	120	100

.

In order to avoid wave reflection at the end boundary of the numerical wave tank, the wave-absorbing area in the numerical wave tank is set to two wavelengths. At the same time, to ensure ideal wave generation in the numerical wave tank and save calculation resources, a reasonable grid layout is very important. In this study, ICEM software 2022 was used to mesh the fluid domain of the numerical wave tank. The mesh is refined in the wave generation area and gradually coarsens in the non-wave generation area and wave absorption area. This mesh-division method can save calculation time and help wave attenuation in the wave absorption area. Figure 10 presents the grid distribution of the three-level sea state 2D numerical wave tank.

In order to monitor the relationship between wave height and time at different positions in the free liquid surface in a numerical wave tank, several virtual wave height meters are set up in the numerical wave tank along the wave propagation direction, and the wave height time-history curve can be generated by reading the data of virtual wave height meters in Fluent software. The simulation results of the Airy wave surface shape in a class 3 sea state are shown in Figure 10, and the time-history curves of wave height at each virtual wave height meter are shown in Figure 11. Figure 10 and Figure 11 show that wave height has a decreasing trend along the wave propagation direction. Because this study mainly studies the influence of wave force on underwater cleaning wall-climbing robots, the attenuation of wave height along the wave propagation direction will not be studied in depth for the time being. Figure 11e,f demonstrate that the wave-absorbing area plays a satisfactory role in wave absorption and avoids a reflection wave at the end boundary of the numerical wave tank, which affects the wave height of numerical wave-making. The 2D numerical wave tank of Airy waves for class 4–6 sea conditions was studied. The wave height time-history curve is shown in Figure 12. Figure 11a and Figure 12 show that the error between the wave-making wave height and the theoretical wave height at the half wavelength position is within 5%. According to the wave height time-history curve of the virtual wave height meter, when analyzing the horizontal wave force on the underwater cleaning wall-climbing robot, the distance between the underwater cleaning wall-climbing robot and the entrance of the numerical wave tank was set to be half of the corresponding wavelength.

4. Wave Force Analysis of Wall-Climbing Robot

The effects of waves on objects in the ocean mainly manifest in the following four aspects [38]: (1) viscosity effect: caused by fluid viscosity; (2) additional mass effect due to fluid inertia; (3) scattering effect: caused by the scattering of waves by the object itself; and (4) free surface effect: caused by the large ratio of the height and size of objects to the still water depth. The scattering effect and free surface effect are collectively called the diffraction effect. According to the main actions of waves on objects in the ocean, three primary methods are used to calculate the wave force on objects in the ocean [39]: the Morison equation, Froude–Krylov theory, and diffraction theory. In ocean engineering, the method of calculating wave force is usually determined according to the scale of object size relative to wave size. When the object is smaller than the wavelength, such as isolated pile legs and underwater oil pipelines, their existence has no substantial influence on wave motion, and the wave mainly acts on them according to the viscous effect and additional mass effect. For such small-scale objects, the wave force is mainly calculated by the Morison equation, which was proposed by Morison et al. in 1950 and later called the Morison equation [40].

4.1. Introduction to Morison Equation

Morison et al. believed that the horizontal wave force $f_H$ acting on any height of a cylinder consists of two parts: (1) the horizontal drag force $f_D$ produced by the horizontal velocity of wave water points and (2) the horizontal inertia force $f_I$ generated by the horizontal acceleration of wave water point motion. The horizontal wave force $f_H$ acting on the cylinder at any height per unit height is as follows [40]:

$$f_H=f_D+f_I={\displaystyle \frac{1}{2}}{C}_D\rho A{u}_x\left|u_x\right|+C_M\rho V_0{\displaystyle \frac{d{u}_x}{dt}}$$

(8)

where $C_D$ is the drag force coefficient, $\rho$ is the density of the fluid, $A$ is the projected area of the unit length cylinder perpendicular to the flow direction, $u_x$ is the velocity of the fluid particle in the horizontal direction, $C_M$ is the quality coefficient, $V_0$ is the column volume per unit length, and $d{u}_x/dt$ is the acceleration of the fluid particle in the horizontal direction.

The recommended values of the drag force coefficient and mass force coefficient are shown in

Table 6. Coefficient table for the Morison equation.

	Port Hydrographic Standards
Wave theory adopted	Airy wave theory
$C_D$	circular cross-section 1.2; square cross-section 2
$C_M$	circular cross-section 2; square cross-section 2.2

.

4.2. Horizontal Wave Force Analysis

According to Airy wave theory and the Morison equation, the wave force of an underwater cleaning wall-climbing robot in the horizontal direction of wave propagation was analyzed. In order to simplify the calculation, the underwater cleaning wall-climbing robot was made equivalent to a cube with length dimension $L$, width dimension $B$, and height dimension $J$, as shown in Figure 13.

According to the Morison equation, the drag force $f_{DX}$ of wave flow around the underwater cleaning wall-climbing robot in the horizontal direction is shown in Equation (9), and the wave inertia force $f_{IX}$ is shown in Equation (10).

$$f_{DX}={\displaystyle \frac{1}{2}}{C}_D\rho B{u}^2$$

(9)

$$f_{IX}=C_M\rho V{\displaystyle \frac{du}{dt}}=C_M\rho LB{\displaystyle \frac{du}{dt}}$$

(10)

Table 2 presents the water point velocity $u$ and acceleration $a_x$ of an Airy wave in the horizontal direction, as shown in Equations (11) and (12).

$$u=A\omega {\displaystyle \frac{\cosh k(y+d)}{\sinh kd}}\cos (kx-\omega t)$$

(11)

$$a_x={\displaystyle \frac{du}{dt}}=Agk{\displaystyle \frac{\cosh k(y+d)}{\sinh kd}}\sin (kx-\omega t)$$

(12)

When the underwater cleaning wall-climbing robot is below the free surface, the distances between its upper and lower end surfaces and the static water surface are $y_1$ and $y_2$, respectively. Because the height of the robot is $J$, $y_1+J=y_2$, and the wave force $F_{HX}$ on the underwater cleaning wall-climbing robot in the horizontal direction is shown in the following equation:

$$\begin{array}{l}{F}_{HX}=F_{DX}+F_{IX}={\displaystyle {\int }_{-y_2}^{-y_1}{f}_{DX}dy}+{\displaystyle {\int }_{-y_2}^{-y_1}{f}_{IX}dy}={\displaystyle {\int }_{-y_2}^{-y_1}\frac{1}{2}{C}_D\rho B{u}^{2}dy}+{\displaystyle {\int }_{-y_2}^{-y_1}{C}_M\rho BL\frac{du}{dt}dy}\\ F_{HX}={\displaystyle {\int }_{-y_2}^{-y_1}\frac{1}{2}{C}_D\rho B{[A\omega \frac{\cosh k(y+d)}{\sinh kd}\cos (kx-\omega t)]}^{2}dy}+\\ \hspace{1em}\hspace{1em}+{\displaystyle {\int }_{-y_2}^{-y_1}{C}_M\rho BLAgk\frac{\cosh k(y+d)}{\sinh kd}}\sin (kx-\omega t)dy\end{array}$$

(13)

Equation (13) shows that when the external dimensions of the underwater cleaning robot are determined, the main factors that determine the horizontal wave force on it are the velocity and acceleration of water points and the position of the robot in the wave depth. Because the velocity and acceleration of the water points in waves are not only related to time but also change with water depth, by taking the class 3 sea state as an example, Table 2 and Table 4 and Figure 14 show the velocity and acceleration of water points in the horizontal direction.

Figure 14 shows that the horizontal velocity and acceleration of water points in Airy waves change periodically with time and decay with an increase in water depth. According to Airy wave theory, the ratios of the velocity of any point in the wave flow field to the velocity of the static water surface and the ratios of the acceleration of any point in the wave flow field to the acceleration of the static water surface in the horizontal direction are shown in the following equation [24].

$${\displaystyle \frac{u}{U_0}}={\displaystyle \frac{\cosh k(y+d)}{\cosh kd}}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{a_x}{a_{x0}}}={\displaystyle \frac{\cosh k(y+d)}{\cosh kd}}$$

(14)

For deep water waves, the above formula can be simplified to Equation (15).

$${\displaystyle \frac{u}{U_0}}={\displaystyle \frac{\cosh k(y+d)}{\cosh kd}}\approx e^{ky}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{a_x}{a_{x0}}}={\displaystyle \frac{\cosh k(y+d)}{\cosh kd}}\approx e^{ky}$$

(15)

For a specific class of sea conditions, Table 4 shows that the wavenumber $k$ is a small quantity, and because the vertical dimension $J$ of the underwater cleaning wall-climbing robot is generally less than 1 m, it can be considered a small quantity and negligible. Therefore, the horizontal velocity and acceleration of water points close to the free surface in the wave flow field and within the height dimension range of the underwater cleaning wall-climbing robot are values that do not change with depth, e.g., $U_0$ and $a_{x0}$. From the above assumptions and Table 2 and Table 4, $U_0$ and $a_{x0}$ are shown in

Table 7. Parameters of wave horizontal velocity and horizontal acceleration under specific sea conditions.

Sea State	Horizontal Velocity ${\mathit{U}}_0$	Horizontal Acceleration ${\mathit{a}}_{\mathit{x}0}$
3	$0.8239\cos (1.3178t)$	$-1.0855\sin (1.3178t)$
4	$1.1667\cos (0.9318t)$	$-1.0871\sin (0.9318t)$
5	$1.4756\cos (0.7367t)$	$-1.0866\sin (0.7367t)$
6	$1.8068\cos (0.6015t)$	$-1.0873\sin (0.6015t)$

when the sea state is 3–6.

When the upper end of the underwater cleaning wall-climbing robot is on the static water surface, the horizontal wave drag force of the underwater cleaning wall-climbing robot under sea state 3 is shown in the following equation:

$$F_{DX}={\displaystyle {\int }_{-J}^0\frac{1}{2}{C}_D\rho B{[0.8239\cos (1.3178t)]}^2}dy={\displaystyle \frac{1}{2}}{C}_D\rho {[0.8239\cos (1.3178t)]}^{2}BJ$$

(16)

The above formula is composed of three parts: (1) $C_D\rho /2$, which is a fixed value, and $C_D=2$, as can be seen from Table 6. For convenience of calculation, the seawater density is 1000 kg/m³. (2) ${[0.8239\cos (1.3178t)]}^2$ is the square of the horizontal velocity of free surface water quality points under specific sea conditions. (3) $BJ$ is the vertical projection of the underwater cleaning wall-climbing robot, where $B=0.8 \mathrm{m}$ and $J=0.8 \mathrm{m}$ are taken.

Because the actual structure of an underwater cleaning wall-climbing robot is complex, its projection in the vertical direction cannot reach $BJ$, so adding an equivalent area coefficient, ${\epsilon }_s$, which is related to the actual structure of the robot, is necessary, and ${\epsilon }_s=0.7$ is tentatively determined here. By substituting the horizontal velocity $U_0$ of water quality points in the specific sea state waves in Table 7 into Equation (16), the horizontal drag force of waves on the underwater cleaning wall-climbing robot can be obtained, as shown in

Table 8. Wave horizontal drag force for specific sea state levels.

Sea State	Horizontal Drag Force ${\mathit{F}}_{\mathit{D}\mathit{X}}$
3	${\epsilon }_s\times 434.4392{\cos }^2(1.3178t)$
4	${\epsilon }_s\times 871.1609{\cos }^2(0.9318t)$
5	${\epsilon }_s\times 1393.5330{\cos }^2(0.7367t)$
6	${\epsilon }_s\times 2089.2968{\cos }^2(0.6015t)$

.

When the upper end of the underwater cleaning wall-climbing robot is on the static water surface, the horizontal inertia force of waves on the underwater cleaning wall-climbing robot is shown in the following equation:

$$F_{IX}={\displaystyle {\int }_{-J}^0{C}_M\rho BL{a}_{x0}dy}=C_M\rho a_{x0}BLJ$$

(17)

The above formula is composed of three parts: (1) $C_M\rho$, which is a fixed value, and $C_M=2.2$. For convenience of calculation, the seawater density is 1000 kg/m³. (2) $a_{x0}$ is the acceleration of water quality points corresponding to specific sea conditions in the horizontal direction. (3) $BLJ$ is the volume of the underwater wall-climbing robot. If $B=0.8 \mathrm{m}$, $L=0.6 \mathrm{m}$, and $J=0.8 \mathrm{m}$, then $BLJ=0.384 {\mathrm{m}}^3$.

Because the actual structure of an underwater cleaning wall-climbing robot is complex, its volume cannot reach $BLJ$, so adding an equivalenting volume coefficient, ${\epsilon }_v$, which is related to the actual structure of the robot, is necessary, and ${\epsilon }_v=0.4$ is tentatively determined here. The horizontal acceleration $a_{x0}$ of water quality points in waves with the specific sea conditions in Table 7 is brought into Equation (17), and the horizontal inertia force of waves on the underwater cleaning wall-climbing robot can be obtained, as shown in

Table 9. Wave horizontal inertia force for specific sea state levels.

Sea State	Horizontal Drag Force ${\mathit{F}}_{\mathit{D}\mathit{X}}$
3	$-{\epsilon }_v\times 917.0304\sin (1.3178t)$
4	$-{\epsilon }_v\times 918.3821\sin (0.9318t)$
5	$-{\epsilon }_v\times 917.9597\sin (0.7367t)$
6	$-{\epsilon }_v\times 918.5510\sin (0.6015t)$

.

According to the Morison equation and Table 8 and Table 9, the wave force $F_{HX}$ of the underwater cleaning wall-climbing robot in the horizontal direction is shown in

Table 10. Horizontal wave forces for specific sea state levels.

Sea State	Horizontal Wave Force ${\mathit{F}}_{\mathit{H}\mathit{X}}$
3	${\epsilon }_s\times 434.4392{\cos }^2(1.3178t)-{\epsilon }_v\times 917.0304\sin (1.3178t)$
4	${\epsilon }_s\times 871.1609{\cos }^2(0.9318t)-{\epsilon }_v\times 918.3821\sin (0.9318t)$
5	${\epsilon }_s\times 1393.5330{\cos }^2(0.7367t)-{\epsilon }_v\times 917.9597\sin (0.7367t)$
6	${\epsilon }_s\times 2089.2968{\cos }^2(0.6015t)-{\epsilon }_v\times 918.5510\sin (0.6015t)$

.

4.3. Numerical Simulation of Horizontal Wave Force

Based on 2D Airy wave numerical simulation, building a 3D numerical wave tank is necessary to numerically simulate the wave force on the underwater cleaning wall-climbing robot. The dimensions of the 3D numerical wave tank corresponding to different sea conditions are shown in

Table 11. Dimensions of the 3D numerical wave tank.

Sea State	Length (m)	Width (m)	Height (m)	Water Depth (m)
3	213	8	30	25
4	426	12	45	40
5	681	16	75	65
6	1020	20	120	100

.

Compared with the 2D numerical wave tank, the main difference in constructing the 3D numerical wave tank lies in grid division. Here, the ICEM + Fluent Meshing assembly grid division technology was used to divide the numerical wave tank, in which the underwater cleaning wall-climbing robot and its small flow field were meshed by Fluent Meshing, and the other fluid domains of the numerical wave tank were meshed by ICEM. The meshed grid is shown in Figure 15. The advantage of assembly mesh generation technology is that it cannot only use Fluent Meshing to perform efficient mesh encryption processing for the underwater cleaning wall-climbing robot, such as boundary layer mesh generation, but also use ICEM to generate a high-quality poly-hex core mesh for the numerical wave tank fluid domain.

Because of the complex structure of the underwater cleaning wall-climbing robot and the large effective wave height, accurately measuring the horizontal wave force on the robot in the actual environment is difficult. In order to verify the feasibility of the above-mentioned method for analyzing horizontal wave force, the horizontal wave force on the underwater cleaning wall-climbing robot was simulated in a 3D numerical wave tank. The underwater cleaning wall-climbing robot and wave shape corresponding to a certain time for sea states 3–6 are shown in Figure 16. The virtual force sensor in Fluent can be read to determine the wave forces on the underwater cleaning wall-climbing robot in different directions and compare the horizontal wave forces calculated by the simulation, providing the theoretical analysis results, as shown in Figure 17.

Figure 17 illustrates a significant difference in the trend variations between the theoretical analysis results and the numerical simulation results, primarily because the Morison equation is a semi-theoretical, semi-empirical formula. When the wave is stable, the extremities of horizontal wave force generally align with the numerical simulation results, and the discrepancy between them increases with the wave level. For instance, when the sea state level is 6, the difference stabilizes at about 11%. Figure 17 also demonstrates that the method used to calculate the horizontal wave force of the underwater cleaning wall-climbing robot is highly feasible for engineering applications. Due to the insufficient theoretical foundation for analyzing the wave forces on underwater cleaning wall-climbing robots with jackets, this study numerically simulates the horizontal wave forces, studying the effect of the jacket on these forces. Figure 18 shows the horizontal wave forces on underwater cleaning wall-climbing robots adhered to jackets of different sizes under sea state levels ranging from 3 to 6. At the same sea state level, the horizontal wave force on the robot decreases as the jacket diameter increases.

5. Conclusions

In this study, we designed a permanent magnet adsorption wall-climbing robot for the underwater cleaning of jackets, which features variable curvature adaptability. We introduced the overall structure of the underwater cleaning wall-climbing robot and optimized the parameters of the variable curvature-adapted connecting rod mechanism using a genetic algorithm. In order to address the hydrodynamic challenges faced by underwater wall-climbing robots, particularly in wave hydrodynamics, we thoroughly analyzed the wave surface shape of Airy waves under sea states ranging from 3 to 6 and determined the corresponding wave surface parameters. By using these parameters, a 2D numerical wave tank was constructed to simulate Airy waves under these conditions. Based on the Morison equation, we proposed a theoretical method to calculate the horizontal wave force on the underwater cleaning wall-climbing robot using the equivalent area and volume, obtaining theoretical values under sea conditions 3–6. In order to validate this theoretical method, we performed numerical simulations of the horizontal wave force in a 3D numerical wave tank. The comparison between the results derived from the formula and simulation results shows that they are closely aligned at the extreme points, with a maximum difference of about 11% when the waves are stable, thus confirming the method’s feasibility.

Given the insufficient theoretical basis for analyzing wave forces on underwater cleaning wall-climbing robots equipped with jackets, we simulated these forces in a 3D numerical wave tank. The simulation results indicate that the horizontal wave force on a jacketed robot decreases as the jacket’s diameter increases under the same sea conditions.

When studying the wave resistance of underwater cleaning wall-climbing robots, specifically analyzing the impact of wave forces on the robots’ adhesion stability and movement flexibility, both the horizontal and vertical wave forces need to be considered simultaneously. In order to build on this study, further analysis is needed on the vertical wave force acting on the underwater cleaning wall-climbing robot. Additionally, an in-depth study of how the jacket attenuates the wave force on the robot is also necessary. Due to the significant impact of wave forces on the adhesion stability and mobility of the underwater cleaning wall-climbing robot, it is essential to fully consider these forces when designing the next generation of permanent magnetic adhesion devices and driving mechanisms for underwater cleaning wall-climbing robots. Ultimately, the wave resistance performance of the underwater cleaning wall-climbing robot will be determined by the magnetic adhesion force generated by the permanent magnetic adhesion device and the torque of the drive motor.