Study on the Optimal Design for Cavitation Reduction in the Vortex Suction Cup for Underwater Climbing Robot

Abstract

In order to adhere to the wall stably in an underwater environment, a vortex suction cup that injects high-pressure water inside via two axisymmetrically side-distributed inlets to create a negative pressure area in the center is the necessary component for the underwater climbing robot (UCR). However, the suction force of this vortex suction cup is reduced and periodically unstable due to unstable cavitation. The aim of this paper is to propose a cavitation reduction optimization method for vortex suction cups and to verify the effectiveness of the optimization. Analyses of this vortex flow, including streamlines, pressure, and cavitation number fluctuations, were carried out by the introduced computational fluid dynamics (CFD) simulating methods based on the multiphase RNG $k-\epsilon$ model to study the periodic fluctuations of the suction force of the original suction cup and the optimized ones. Force measurement and vortex observation experiments were conducted to compare the suction force of the original vortex suction cup and the optimized suction cup, as well as the cavitation and pressure fluctuation phenomenon. Results of simulation and experiments prove the existence of the effect of vortex cavitation on the suction performance and verify the rationality of optimization as well.

1. Introduction

In recent years, underwater climbing robots have been extensively developed so as to replace humans to complete some dangerous and hazardous underwater operations, which include cleaning, maintenance, and inspection of bridges [1,2], dams [3,4,5], ship hulls [6,7,8] and so on. During the research of underwater climbing robots, choosing which kind of suction method is one of the main points that researchers should consider [9,10]. There are various types of adhesion methods that include magnetic adhesion [11], vacuum suction, vortex suction, Bernoulli suction [12], adhesion using gripping equipment [13], propulsion adhesion [14], biomimetic adhesion [15], and so on.

In order to suck on the non-ferromagnetic surface, non-ferromagnetic adhesion methods are applied, among which vacuum suction is the most widely used method. However, the vacuum suction cup is in direct contact with the wall to retain vacuum sealing. Therefore, it has some disadvantages. The existence of frictional force leads to the abrasion of sealing skirts. Moreover, the suction ability is greatly affected by the roughness of the adsorbed surface. Vortex suction and Bernoulli suction [16,17,18] are both non-contact suction methods and have the characteristics of flexible mobility and low sensitivity to the wall material. Vortex suction cup mainly includes fluid activated suction device and electrically activated suction device. It was found by Li et al. [19] that compared to vortex suction cup, the Bernoulli suction cup needs larger flow rates and causes more air power losses so as to generate the same suction force. Iio S. et al. [20] analyzed the relationship between the sucking pressure and the flow dynamics when gap distance from the pneumatic suction cup to a work piece changes, which is useful to guide the design of the suction cup. In 2013, Li et al. [21] proposed an electrically activated swirl gripper that generates swirling air flow to create an upward-lifting force. In 2020, Chen et al. [22] focused on the modeling and design of the height of the blade, which drives the air rotating at high speed within a chamber to create and maintain a negative pressure distribution.

In addition, the complex internal flow field inside the suction cup’s cavity and the cap clearance between the suction cup and wall were also described to reveal the mechanism of the suction device. In 2013, Wu et al. [23] measured the two-dimensional flow field for vortex gripper using the particle image velocimetry method, which provided the theory support of simulation study for pneumatic vortex gripper and the structure optimization. Li et al. [24] analyzed the velocity field and pressure field inside the pneumatic vortex gripper with vortex suction cup in detail by using the numerical fluid dynamics method and revealed the relationship between the velocity and pressure distribution. However, most of the analysis of the vortex field is aimed at the pneumatic vortex suction device. When the vortex suction cup is used underwater, serious cavitation would be formed in the cavity of the suction cup because of the negative pressure in the center of the cavity. Therefore, cavitation is one of the important factors to be considered when analyzing the fluid flow and the suction force at the interface. In the preliminary study, the cavitation model was not included in the numerical simulation of the suction cup. The pressure distribution and the values of suction force obtained by the numerical analysis are rather different from the experimental phenomena and measured values. Moreover, the simulation without cavitation could not explain the phenomenon of axial flow surge observed in the experiment, so it cannot describe the flow characteristics and analyze the suction performance of underwater vortex suction cups precisely. However, in the past research, the analysis of vortex flow cavitation is mainly about the cavitation in the inducer of turbopumps and hydroturbines [25,26,27]. Kang et al. [28] clarified the mechanism of cavitation instabilities in inducer based on analyses of the velocity field under cavitation instabilities.

This paper introduces a cavitation reduction optimization method of the vortex suction cup to improve the suction performance in detail. Through the previous research of the author, which was on the suction performance of the suction cup, it can be found that cavitation has a negative effect on the suction capacity of the vortex suction cup, which reduces the suction capacity and at the same time aggravates the instability of suction. First, an optimization method of the vortex suction cup is designed to solve the problem, that is, increasing the angle between the inlet of the suction cup and the horizontal plane of the original vortex suction cup to several different angles. Then the CFD simulations with the cavitation model were brought out to simulate the fluid flow condition more accurately. Force measurement and flow observation experiments were also carried out. A comparison of the experimental results and the simulation results between the original suction cup and the optimized suction cup was made to show the effectiveness of the optimization method of the vortex suction cup.

The rest of the paper is organized as follows. Section 2 is dedicated to describing the optimization method and basis of the vortex suction cup, the numerical simulation process of the fluid inside the suction cup’s cavity, and the gap clearance between the suction cup and wall. Force measurement and vortex observation experiment apparatus and methods are also proposed. In Section 3, the validation of the optimization method of the vortex suction cup is proposed by comparing the suction results of the original suction cup and the optimized suction cup. Also, the application on underwater robots is shown in this section. Finally, conclusions and future work are provided in Section 4.

2. Research Methods

2.1. Geometry, General Parameters, and Optimization Basis

The sketches of the underwater vortex suction cup are shown in Figure 1. Figure 1a,b show the structure of the original vortex suction cup. In order to connect with the water pipe conveniently, an L-shaped inlet channel is made, and the inlet connected with the water pipe is opened on the upper surface of the suction cup. The part of the inlet that connected with the cavity is horizontal. High-pressure water is injected into the cylindrical cavity through the nozzles. The water flows into the cavity through the horizontal part of the inlet and rotates inside the cavity. Figure 1c,d show the optimized vortex suction cup we designed, which is made up of an alloy cylinder with a cylindrical cavity and two tangential nozzles. The axisymmetrically distributed nozzles are at a certain angle to the horizontal. The optimization is changing the angle between the inlet and the horizontal plane from 0° to an angle that is larger than 0°. Negative pressure is formed in the central area of the cavity by the centrifugal force of the high-speed swirling flow and Bernoulli law, thus leading to an adhesion force to the surface, which is close to the suction cup.

The two axisymmetrically distributed tangential high-pressure water inlets of the original suction cup based on the vortex suction theory are horizontal, while the inlet fluid of another suction cup/gripper is to enter the cavity along the axial direction. Referring to the Bernoulli suction cup to optimize the original suction cup, the original suction cup is optimized by changing the direction of the axisymmetrically distributed tangential high-pressure water inlet to a certain angle $\theta$ with the horizontal direction. The changing law between the inlet included angle $\theta$, and the value of the suction force is analyzed quantitatively in this article. Therefore, the suction state of the vortex suction cup with an inclined angle between the inlet and the horizontal can be simplified and comprehended as the superposition of the above two types of suction cups. That is, the inlet flow $Q$ of the optimized suction cup is comprehended by dividing it into horizontal component $Q_h$ and vertical component $Q_v$. From the research of pressure distribution of vortex suction cup [29]. The pressure distribution is approximately $p=\frac{1}{2}\rho {\omega }^2{r}^2+p_0$, and the pressure distribution of tangential direction is shown in Figure 2a. Moreover, from the research of pressure distribution of Bernoulli gripper [16], the approximation of pressure distribution is in Equation (1):

$$\frac{\partial P}{\partial r}=\left\{\begin{array}{c}\frac{3G^2}{10{\pi }^2{r}^2{h}^2\rho }(\frac{1}{r}+\frac{1}{P}\frac{\partial P}{\partial r})-\frac{6\mu G}{\pi r{h}^3\rho }\\ \frac{16G^2}{63{\pi }^2{r}^2{h}^2\rho }(\frac{1}{r}+\frac{1}{P}\frac{\partial P}{\partial r})-\frac{0.079{\mu }^{1/4}{G}^{7/4}}{4{\pi }^{7/4}{r}^{7/4}{h}^3\rho }\end{array}\begin{array}{c}{R}_e<2000\\ \begin{array}{l}\\ R_e\ge 2000\end{array}\end{array}\right\}$$

(1)

where $G$ is the supply mass flow rate, $P$ is the pressure, $r$ is the radial position, $\rho$ is the fluid density, $h$ is the gap height, $R_e$ is the Reynolds number. The force that the Bernoulli gripper applies to the workpiece can be calculated by the following equation:

$$F=\frac{1}{4}\pi d^2{P}_{0t}+{\displaystyle {\int }_{d/2}^{D/2}2\pi rPdr}$$

(2)

where $P_{0t}$ is the total pressure acting on the center of the workpiece, d and D are the diameter of the inlet and the griper, respectively. The pressure distribution of the Bernoulli gripper is shown in Figure 2b. The fluid flow collides with the contact surface to create a pressure peak at the center. After entering the gap between the suction cup/ gripper and the contact surface, it decelerates in the radial direction abruptly, and the inertial effect increases the pressure predominantly. Then when the radius increases, the inertial effect weakens, and the viscous effect becomes the dominant part that leads to the pressure drop. The pressure reaches the atmospheric pressure when the radius equals the radius of the suction cup/gripper.

So when choosing different $\theta$, It can be simply understood as a combination of two types of suction cups in different proportions. The variation of $\theta$ (from small to large) can be understood that the proportion of the suction method of injecting high-pressure fluid along the axial direction varies from small to large. The pressure of the area between the cavity and contact surface ($r<\frac{d}{2}$) increases from a larger negative pressure, while in the area ($\frac{d}{2}<r<\frac{D}{2}$), the negative pressure decreases. So when $\theta$ is a certain value, the total suction force of the suction cup reaches a peak value. Therefore, in the article, the vortex suction cup with horizontal inlets is optimized by increasing the angle between the inlet of the suction cup and the horizontal plane to $10°,15°,20°,25°$ and $30°$. By comparing the suction force and the fluctuation amplitude of the above-mentioned series of optimized vortex suction cups and the original suction cup with horizontal inlets, the most adequate inclined angle of inlets of the suction cup with different usage conditions can be selected.

Some specific parameters are listed in

Table 1. Parameters of the vortex suction cup (unit: mm).

H (mm)	H (mm)	D (mm)	D1 (mm)	D (mm)	d1 (mm)	Φ	L (mm)
32	18	100	50	1	4	160°	17.8

. The cylindrical suction cup is 32 mm thick (H), the outer diameter of which is 100 mm (D). The cylindrical cavity is 18 mm in height (h), and its diameter is 50 mm (D₁). The nozzles are two axisymmetric distributed holes with a diameter of 1 mm (d), which are 4 mm (d₁) away from the top of the cavity. Angle Φ is the inclination between the conical channel and horizontal plane. From the top view of the vortex suction cup, the distance between the inlet channel and the center of the suction cup is 17.8 mm (l). Other than the same parameters shown in Table 1, compared with the original suction cup, the inlet of the optimized vortex suction cup is not horizontal but at an angle of $\theta$ to the horizontal. As shown in Figure 1a,b, it could be found that the vortex flow inside the cavity and gap of the original suction cup has one rotation center. However, it can be seen from Figure 1c,d that the vortex flow inside the cavity and gap of the optimized suction cup has two rotation centers due to the presence of an inlet incidence angle $\theta$. Therefore, the area of negative pressure of the optimized suction cup is significantly larger, thus leading to a larger suction force. We measured the suction force of six test vortex suction cups with varying inclined angles (see

Table 2. Nomenclature of the test suction cup.

No.	Original Suction Cup	Suction Cup- $\mathit{\theta }$10	Suction Cup- $\mathit{\theta }$15	Suction Cup- $\mathit{\theta }$20	Suction Cup- $\mathit{\theta }$25	Suction Cup- $\mathit{\theta }$30
Inclined angle of the inlet	$0°$	$10°$	$15°$	$20°$	$25°$	$30°$

).

2.2. Numerical Simulation

2.2.1. Mathematical Model and Numerical Algorithm

The swirling flow in the vortex sucker can produce vortices and result in complex turbulent motion and severe cavitation. Therefore, the liquid inside the cavity cannot be regarded as an incompressible single phase. The liquid should be considered a homogeneous and compressible mixed medium of water liquid and vapor. Taking use of compressible Navier-Stokes equations, the continuity and momentum equations of the flow inside the vortex sucker are as follows:

$$\frac{\partial {\rho }_m}{\partial t}+\frac{\partial \left({\rho }_m{u}_i\right)}{\partial x_i}=0$$

(3)

$$\frac{\partial \left({\rho }_m{u}_i\right)}{\partial t}+\frac{\partial \left({\rho }_m{u}_i{u}_j\right)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_i}\left[\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3}\mu \frac{\partial u_j}{\partial x_j}{\delta }_{ij}\right]$$

(4)

The definition equations of ${\rho }_m$ and ${\mu }_m$ are ${\rho }_m={\rho }_w{\alpha }_w+{\rho }_v{\alpha }_v$ and $\mu ={\mu }_m+{\mu }_{tb}={\mu }_w{\alpha }_w+{\mu }_v{\alpha }_v+{\mu }_{tb}$; where $\rho$, $\mu$ and $\alpha$ are the density, dynamic viscosity, and volume fraction of each phase, respectively; subscripts $w$,$v$, and $m$ denote the water liquid phase, the vapor phase, and the mixture phase; $u$ and $p$ are the velocity and pressure; ${\mu }_{tb}$ is the turbulent viscosity, which is defined from several turbulent models.

Standard $k-\epsilon RNG$ model is a commonly used model to analyze turbulence. In this model, ${\mu }_{tb}=\rho C_{\mu }{k}^2/\epsilon$, where $k$ and $\epsilon$ are the turbulent kinetic energy ($m^2/s^2$) and turbulent kinetic energy dissipation rate ($m^2/s^3$), $C_{\mu }=0.0085$ [30]. However, this model is originally devoted to fully incompressible fluids, which only takes into account the fluid compressibility by mean density changes. Thus, some unstable cavitating phenomena are neglected. Considering the need to analyze the highly compressible two-phase mixed flow, some modifications of the standard $k-\epsilon RNG$ model is required, and the turbulent viscosity should be defined as:

$${\mu }_t=f({\rho }_m)C_{\mu }{k}^2/\epsilon$$

(5)

where

$$f({\rho }_m)={\rho }_v+{\left[\frac{\left({\rho }_m-{\rho }_v\right)}{\left({\rho }_w-{\rho }_v\right)}\right]}^n·({\rho }_w-{\rho }_v)$$

(6)

choose n = 10.

Moreover, the liquid-vapor mass transfers due to cavitation are solved along with the transport equation-based cavitation model:

$$\frac{\partial \left({\rho }_{\nu }{\alpha }_{\nu }\right)}{\partial t}+\nabla \cdot \left({\rho }_{\nu }{\alpha }_{\nu }u\right)={\dot{m}}_{vap}-{\dot{m}}_{con}$$

(7)

where the model of mass transfer rate for vaporization and condensation proposed by Zwart [31].

The growth of a vapor bubble in water liquid is solved by the Rayleigh-Plesset equation:

$$R_b\frac{d^2{R}_b}{d{t}^2}+\frac{3}{2}{\left(\frac{d{R}_b}{dt}\right)}^2+\frac{2\sigma }{R_b}=\frac{P_v-P}{{\rho }_l}$$

(8)

In which $R_b$ represents the radius of bubble, $\sigma$ represents the surface tension coefficient, $P_v$ represents the vapor pressure. Neglecting the second-order terms and the surface tension term simplifies the above expression.

$$\frac{d{R}_b}{dt}=\sqrt{\frac{2}{3}\frac{P_v-P}{{\rho }_l}}$$

(9)

$$\frac{d{m}_b}{dt}=4\pi R_b^2{\rho }_v\sqrt{\frac{2}{3}\frac{P_v-P}{{\rho }_l}}$$

(10)

$${\dot{m}}_{vap/con}{=\mathrm{F}}_{vap/con}\frac{3{\alpha }_v{\rho }_{\nu }}{R_b}\sqrt{\frac{2}{3}\frac{\max \left(p_{\nu }-p,0\right)}{{\rho }_l}}$$

(11)

The nucleation site density would decrease when the vapor volume fraction increases. Therefore, the ${\alpha }_{\nu }$ is replaced by ${\alpha }_{nuc}(1-{\alpha }_{\nu })$ during vaporization, where ${\alpha }_{nuc}$ is the volume fraction of nucleation site.

$${\dot{m}}_{vap}{=\mathrm{F}}_{vap}\frac{3{\alpha }_{nuc}(1-{\alpha }_{\nu }){\rho }_{\nu }}{R_b}\sqrt{\frac{2}{3}\frac{\max \left(p_{\nu }-p,0\right)}{{\rho }_l}}$$

(12)

$${\dot{m}}_{con}{=\mathrm{F}}_{con}\frac{3{\alpha }_{\nu }{\rho }_{\nu }}{R_b}\sqrt{\frac{2}{3}\frac{\max \left(p-p_{\nu },0\right)}{{\rho }_l}}$$

(13)

where ${\mathrm{F}}_{vap}$ and ${\mathrm{F}}_{con}$ are the empirical calibration coefficient of vaporization and condensation, vapor pressure $P_v$ is influenced by local turbulent pressure fluctuation, which can be described by saturated vapor pressure at the current test temperature.

$$p_v=p_{sat}+\frac{1}{2}(0.39){\rho }_{m}k$$

(14)

The pressure distribution and flow situation of the fluid in the vortex sucker’s cavity can be obtained by combing the continuity and momentum equations of the flow, the modified $k-\epsilon RNG$ turbulent model, and the transport equation-based cavitation model.

2.2.2. Boundary Conditions and Mesh Generation

To simulate the pressure distribution and swirl flow situation of the vortex suction cup model, a calculation domain is first established for the area of fluid flow in the cavity of the vortex suction cup, which is shown in Figure 3a. The model of vortex suction cup designed in SolidWorks is imported to the workbench, and then the cavity of the suction cup is filled. The solid parts of the suction cup and wall area are removed, and the remaining part is the required fluid calculation domain. As shown in Figure 3b, the whole model is divided into blocks with different-size meshes to ensure computational accuracy and reduce the computational complexity as well.

The setting of boundary conditions is a key step in CFD simulation and calculation. The boundary conditions are shown as follows: The model inlet is two tangential inlet channels, and its boundary condition is set as pressure inlet, that is, a series of pressure values from 5 to 15 MPa; the model outlet is the outer side of the buffer fluid domain, and the exit boundary condition is set as pressure outlet, which is set to 0 (gauge pressure) since the pressure here is close to the atmospheric pressure. The Zwart-Gerber-Belamri cavitation model is chosen because it is more robust and converges quickly. The liquid temperature was chosen as 25 °C. The air content of the liquid (dissolved air in water) is 0.023 g/kg.

The density-based solver has a better ability to solve the compressible fluid, but it is not compatible with the multiphase flow model selected. The pressure-based solver is chosen, which is based on the pressure correction strategy. The coupled algorithm is generally robust and converges faster, particularly for cavitating flows in rotating machinery, so that it is selected for the pressure-velocity coupling. The spatial discretization schemes are the PRESTO! Scheme used for the pressure, the second-order upwind schemes for the momentum, the first order upwind schemes for the turbulence kinetic energy, turbulence dissipation rate. To achieve convergence and numerical efficiency, the following values of relaxation factors are chosen in

Table 3. Values of relaxation factors in CFD solution controls.

Courant Number	Relaxation Factors for Density	Relaxation Factors for Vaporization Mass	Relaxation Factors for Pressure
30	1	1	0.5

.

2.3. Experimental Setup

In order to compare the suction force of the vortex suction cup before and after optimization, as well as the axial and radial precession instability caused by cavitation, the suction force measurement test platform and the vortex phenomenon observation test platform were used to measure the suction force of the test suction cups under different gap clearances and inlet pressures and to observe the cavitation in the cavity of the suction cup.

2.3.1. Apparatus and Method for Suction Force Measurement

A sketch of the apparatus to measure the suction force of the suction cup is shown in Figure 4. It is designed to study the change in the suction performance of the suction cup when the inlet pressure of the suction cup and the gap clearance between the bottom surface of the suction cup and the wall is varied. The device consists of three parts: outer frame, elevator platform, and suction cup. The outer frame is designed to fix the force sensor, the suction cup, and the base plate of the elevator platform. The elevator platform (MTS-LJ100, moving range: 0~20 $\mathrm{mm}$, accuracy: 50 $\mathsf{\mu }\mathrm{m}$, Anying Optical Instrument Co., Ltd., Shandong, China) is composed of a fixed baseplate, the lead screw mechanism, and a mobile top-plate that is linked to an alloy sheet. The force sensor (ARIZON 3041, range: 0~500 N, accuracy: 0.2%, Arizon Technology Co., Ltd., Yangzhou, China) is stalled between the upper beam and the suction cup to measure the adhesion force generated by the vortex suction cup. At the same time, the data are transmitted to a PC and saved online. Below the vortex suction cup is an alloy sheet set to simulate the wall. The apparatus needs to be placed in a tank in which water should floor the suction cup so as to achieve the adhesion function. The inlets of the suction cup are connected with the high-pressure water pump through high-pressure water pipes, and the pressure value of the suction cup’s inlet is varied by adjusting the pressure value of the pump. The gap clearance between the suction cup and the wall is altered by turning the handle on the elevator platform and measured by reading the caliper. The data of measured adhesion force are recorded when choosing gap values of 1, 1.5, and 2 mm, and different inlets’ pressure values.

2.3.2. Apparatus and Method for Vortex Observation

The force measurement apparatus mentioned above is used to study the values and change regularities of adhesion force, but it is not able to observe the cavitation and unstable adhesive phenomenon; hence, another experimental apparatus needs to be designed. Because the suction cup needs to be injected with high-pressure water, it needs to be made of aluminum alloy instead of transparent material, which is not strong enough. Moreover, due to the existence of cavitation, there are many bubbles in the suction cup rotating cavity, which cannot be observed from the side and the top part, so the vortex phenomenon in the suction cup should be observed from the bottom. The arrangement of the vortex observation experimental setup is shown in Figure 5a, which consists of four parts: support frame, transparent tank, vortex observation apparatus, and high-speed camera placed just below the apparatus. The support frame is used to place the tank and vortex observation apparatus and fix the high-speed camera. The vortex observation apparatus is similar to the force measurement apparatus, but the elevator platform is placed upside down with the bottom plate fastened to the upper frame. The suction cup is fixed on the upper plate of the elevator platform, and the gap between the suction cup and the bottom surface of the transparent tank is regulated by turning the handle on the elevator platform. The bottom surface of the transparent tank is chosen to observe the vortex and cavitation phenomenon.

3. Results and Discussion

3.1. Verification of Numerical Simulation

The aforementioned CFD model is used to simulate the suction device in gap clearance of 1, 1.5, 2 mm when the pressure of the suction cup’s inlet is set to different values from 5 to 15 MPa. In order to verify the rationality of the theoretical model with cavitation and calculation results of the suction cup, the numerical calculation results under cavitating and non-cavitating conditions are compared with the experimental test results measured by the experimental setup mentioned in Section 2. To avoid repetition, we choose suction cup-$\theta$20 as the representative of the optimized suction cup. It is shown in Figure 6 and Figure 7 that the simulation results of suction force with the non-cavitation model are quite different from the experimental test results, which proves that the cavitation model applied in vortex suction cup analysis is necessary. As shown in Figure 8 and Figure 9, the simulation results of the suction force of the cavitation model are in reasonable agreement with the experimental test results measured by the apparatus mentioned in Section 2.3. By comparing the simulation results of the cavitation model and non-cavitation model, it can be found that the variation range of suction force with pressure will be smaller in the real situation, that is, when there exists cavitation. As shown in Figure 6 and Figure 8, the suction force of the original suction cup ranges from 110 to 250 N when the pressure of the suction cup’s inlet ranges from 5 to 15 MPa and the gap between the suction cup and the wall is set to 1, 1.5, 2 mm. With the increase in the inlet pressure, the suction force increases non-linearly in an approximate parabolic curve form progressively. When the gap is 1 mm, the suction force corresponding to each inlet pressure is less than that when the gap is 1.5 mm or 2 mm. When the inlet pressure is less than 11 MPa, the suction force of 1.5 mm gap is greater than that of 2 mm. In the other case, the suction force is greater when the gap is 2 mm. While in Figure 7 and Figure 9, the suction force of the suction cup-$\theta$20 ranges from 160 to 320 N when the pressure of the suction cup’s inlet ranges from 5 to 15 MPa and the gap between the suction cup and wall is set to 1, 1.5, 2 mm. At all inlet pressures, suction cup-$\theta$20 has the maximum suction force among the three gap clearances when the gap is 1.5 mm. Moreover, in the case of the same inlet pressure and gap clearance, the suction forces of suction cup-$\theta$20 are all larger than that of the original suction cup.

The relative error $\delta$ of simulation results is obtained to validate the reliability of simulation results, which is calculated by Equation (15).

$$\delta =\raisebox{1ex}{$\Delta $}\!\left/ \!\raisebox{-1ex}{$I$}\right.\times 100\%=\frac{\left|F_{\mathrm{experiment}}-F_{\mathrm{simulation}}\right|}{F_{\mathrm{experiment}}}\times 100\%$$

(15)

where $\Delta$ and $I$ are, respectively, the absolute error and the set true value, F_experiment and F_simulation denote the measured data in experiments and simulation values, respectively. The relative errors of different inlet pressures and gap clearances are shown in

Table 4. Relative errors (%) for the original suction cup.

	Gap Clearance (mm)	1	1.5	2
Inlet Pressure (MPa)
5		11.86	8.40	2.41
6		0.95	3.36	0.871
7		3.59	1.53	3.59
8		3.90	0.76	4.77
9		3.42	0.12	3.19
10		1.97	1.59	2.44
11		3.95	1.26	0.92
12		2.81	1.32	5.28
13		3.49	0.46	1.72
14		3.05	0.30	4.05
15		4.18	5.69	0.57

and

Table 5. Relative errors (%) for suction cup-$\theta$ 20.

	Gap Clearance (mm)	1	1.5	2
Inlet Pressure (MPa)
5		1.58	7.53	4.67
6		2.15	1.84	4.23
7		0.65	0.63	1.61
8		1.12	2.13	1.50
9		0.63	1.06	0.39
10		3.71	2.28	4.60
11		2.76	0.06	2.89
12		3.26	0.84	1.88
13		4.62	0.58	5.04
14		3.29	4.58	6.59
15		7.53	5.34	7.36

and Figure 10 and Figure 11.

In the current flow situation, the maximum error is found to be within 12%, and the simulation appears to be acceptable to analyze the dynamics of the swirling flow in the cylindrical suction cup.

3.2. Suction Force Analysis

As one of the characteristics of analyzing and testing the effect of cavitation reduction optimization of the vortex suction cup, the values of the suction force of the original suction cup and optimized suction cups need to be measured and compared. In order to choose the optimal value of $\theta$, which is the key parameter of the vortex suction cup, the test results of the series of optimized suction cups are compared with that of the original suction cups. Figure 12 presents the measured suction forces for the six test vortex suction cups ($\theta =0°,10°,15°,20°,25°,30°$) at given gap clearances (1, 1.5, 2 mm) and inlet pressures (5, 10, 15 MPa), and it can be observed that in most cases, the suction force increases significantly with increasing inlet pressures. Moreover, it is also shown in Figure 12 that the suction forces generated by optimized suction cups of which inclined angle is larger than 0° are larger than that original suction cup. As the inclined angle $\theta$ becomes larger, the suction force first increases and then decreases. When the gap clearance is 1 mm, suction forces of the vortex suction cups with inclined inlets are all larger than the original suction cup; the suction force of the suction cup of which inclined angle $\theta$ is 15° is largest under each inlet pressure. Next to it are suction cups of which the inclined angle $\theta$ is 20° and 25°. When the gap clearance is 1.5 mm, suction forces of the vortex suction cups with inclined inlets are larger than the original suction cup except when $\theta$ is 30°; suction force of the suction cup of which inclined angle $\theta$ is 20° is largest under each inlet pressure. Next to it are suction cups of which inclined angles $\theta$ are 25° and 15°. The suction force of the suction cup of which inclined angle $\theta$ is 30° remains unchanged under different inlet pressures. When the gap clearance is 2 mm, suction forces of the vortex suction cups with inclined inlets are larger than the original suction cup except when $\theta$ is 25°or 30°; suction force of the suction cup of which inclined angle $\theta$ is 15° and 20° is largest under each inlet pressure. In the case of small inlet pressure, suction forces generated by suction cups with different inclined angle $\theta$ are almost unchanged. Taken into comprehensive consideration of the suction force values of six test suction cups under three different gap clearances and varied inlet pressures, it shows that suction force generated by the suction cup of which inclined angle $\theta$ is 20°is largest in most conditions. By comparing all 6 kinds of suction cups under varying clearances and inlet pressures, the suction force of suction cup-$\theta$20 is maximum when clearance is 1.5 mm and inlet pressure is 15 MPa.

3.3. Suction Periodic Instability Analysis

Periodic oscillating wall pressure is induced by vortex precession and cavitation surge. The vortex center, that is, the lowest pressure area, does not coincide with the geometric central axis of the suction cup and rotates regularly around the central axis with the change of time because of vortex precession. Liang et al. [32] has referred to the details of vortex precession phenomena. Besides the suction force generated by vortex flow, an extra oscillating radial force is observed to be generated by the cavitation surge instability. From the analysis in Section 3.2, it can be seen that from the perspective of improving the suction force, suction cup-$\theta$20 generates the maximum suction force when the inlet pressure is 15 MPa, and the gap clearance is 1.5 mm. Therefore, this paper mainly takes suction cup-$\theta$20 as an example to analyze the periodic pressure instability of the optimized suction cups.

3.3.1. Simulation Analysis

According to the simulation results, it could be found that cavitation surge is severer with the increase in inlet pressure. Therefore, this paper takes the original suction cup and suction cup-$\theta$20 under the case that the gap clearance is 1.5 mm and the inlet pressure is 15 MPa as an example to illustrate the changing process of streamline, pressure, and cavitation number when the cavitation surge occurs in vortex suction cup. Moreover, the effectiveness of cavitation reduction optimization is verified by comparing the simulation results of the above-mentioned two vortex suction cups.

The CFD analysis of the flowing condition in the suction cup’s cavity and the gap between the suction cup and the wall is shown in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31. The sequence of the simulation results can be visualized in a three-dimensional view by taking two vertical axial cross-sections that are through the axis. In addition, simulation results can be visualized on the interface between the vortex suction cup and wall. Figure 13, Figure 15, Figure 17, and Figure 19 show the velocity vector, pressure, cavitation number, and velocity distributions of the original suction cup in vertical axial cross planes, respectively. Figure 14, Figure 16, Figure 18, and Figure 20 show the velocity vector, pressure, cavitation number, and velocity distributions of the original suction cup in the adhesion interface plane, respectively. Figure 21 shows the velocity vector plot of the original suction cup in sequential planes views (from top view and bottom view). Figure 22, Figure 24, Figure 26 and Figure 28 show the velocity vector, pressure, cavitation number, and velocity distributions of suction cup-$\theta$20 in vertical axial cross planes, respectively. Figure 23, Figure 25, Figure 27 and Figure 29 show the velocity vector, pressure, cavitation number, and velocity distributions of suction cup-$\theta$20 in the adhesion interface plane, respectively. Figure 30 shows the velocity vector plot of suction cup-$\theta$20 in sequential planes views (from top view and bottom view). First, the flowing condition of the original suction cup is analyzed. In most cases, the flow inside the vortex suction cup has a radial outward flow component, the velocity distribution of which is not uniform in the circumferential direction. Because of the cavitation instability, the flow would have periodic axial fluctuations; that is, the fluid in the suction cup’s cavity flows from the gap to the outside in the most time of one cycle, and afterward, a back-flow is generated, which means that the outside fluid flows into the cavity. Thus, periodic pressure fluctuation is generated on the contact surface. When a large amount of fluid flows out of the suction cup’s cavity at high speed, a large low-pressure region would appear at the position of high-speed swirl flow due to the large centrifugal force. When part of the fluid flows from the external into the suction cup’s cavity, of which the flow rate is slow, the absolute value of the pressure in the low-pressure region of the center becomes smaller. By observing the cavitation number contour, we can see the cavitation phenomenon of fluid in the suction cup’s cavity and gap. The small cavitation number in the suction cup’s cavity indicates that there is severe cavitation inside the suction cup, especially near the central axis of the suction cup. When a large amount of fluid flows out of the cavity at high speed, the cavitation at the center of the cavity bottom would be severer. In contrast, the cavitation in the bottom center would be relieved to a certain extent when part of the fluid flows from external to cavity.

Next, the optimized suction cup is analyzed. Compared with the original suction cup, the streamline and velocity distribution of suction cup-$\theta$20 are mostly similar. Unlike the original suction cup, which has one rotation center, suction cup-$\theta$20 has two small rotation centers, so the area of negative pressure is significantly larger, thus leading to a larger suction force. Moreover, the negative pressure area is around the center of rotation, and it exhibits axisymmetric distribution most of the time, which leads to better suction stability. The area of low cavitation number (smaller than 1) of fluid inside the cavity and the gap clearance between the suction cup and the wall is significantly reduced, which illustrates that increasing the inclined angle of the inlets to 20° can reduce the level of cavitation, thereby reducing the effect of cavitation on the suction performance of the vortex suction cup. The radial velocity distributions of each section from h = 3 mm to h = 18 mm are shown in Figure 31. It could be found that in the range of small radius, the velocity distribution is similar to the corresponding characteristic of the Lamb vortex to a certain extent.

3.3.2. Experimental Results

The cavitation surge of vortex flow can be observed by the time-varying pictures of the swirl flow streamline between the suction cup and the wall recorded by high-speed camera in the experiments. Taking original suction cup and suction cup-$\theta$20 under the case that the gap clearance is 1.5 mm and the inlet pressure is 15 MPa as an example, the photos of vortex flow that were taken by high-speed camera from the bottom are shown in Figure 32 and Figure 33. For the original suction cup, it can be seen that in one cycle, there exist two stages: the first stage is the larger amount of fluid flows from outside to the inside of the vortex suction cup. After that, the larger amount of fluid flows from the inside of the vortex suction cup to the outside and then back to the initial state (the arrows in Figure 32 indicate the flow direction of fluid). For suction cup-$\theta$20, the two changing stages are similar to those of the original suction cup. However, different from the original suction cup, the cavitation surge of suction cup-$\theta$20 is not obvious, and most of the time, the larger amount of fluid flows from the inside of the vortex suction cup to the outside (the arrows in Figure 33 indicate the flow direction of fluid). Suction cup-$\theta$20 has two small rotation centers, and the area of negative pressure is significantly larger. Moreover, the photos of test results of suction cup-$\theta$20 show that the negative pressure area is around the center of rotation, and it exhibits axisymmetric distribution most of the time, and so is the cavitation area, thus leading to better suction stability than that of the original suction cup.

The frequency of cavitation surge of vortex flow can be calculated by observing Figure 32 and Figure 33. The frequency of vortex flow surge in the simulation analysis is obtained by observing the images of the velocity vector of vortex flow on the suction wall.

Table 6. Frequencies of cavitation surge for original suction cup (unit: Hz).

	Gap Clearance (mm)	1		1.5		2
Inlet Pressure (MPa)		Test	Simulation	Test	Simulation	Test	Simulation
5		19.23	19.23	23.53	21.74	20.62	20.00
10		27.03	27.78	28.57	26.32	27.78	27.78
15		40.00	35.71	46.51	45.45	42.55	33.33

and

Table 7. Frequencies of cavitation surge for suction cup-$\theta$ 20 (unit: Hz).

	Gap Clearance (mm)	1		1.5		2
Inlet Pressure (MPa)		Test	Simulation	Test	Simulation	Test	Simulation
5		19.23	19.23	21.01	20.00	19.05	20.00
10		30.30	27.40	27.78	27.03	34.48	28.57
15		37.04	40.82	43.38	45.45	45.45	35.09

show the cavitation surge frequencies of high-speed camera observation experiments and simulations. The frequency of cavitation surge for original suction cup and suction cup-$\theta$20 is similar to each other. The comparison between the frequency from simulation and tests can also confirm the validity of the simulation for vortex flow cavitation surge phenomenon in vortex suction cup. The frequency of cavitation surge is larger under larger inlet pressure.

The filtered data of suction force of the suction cup-$\theta$20, which is the most appropriate one among the five optimized suction cups, can be obtained in real time by a force measurement experiment. As shown before, suction cup-$\theta$20 generates the maximum suction force when the inlet pressure is 15 MPa, and the gap clearance is 1.5 mm, which is in line with the actual usage condition of the climbing robot. Therefore, this paper chooses the case that the inlet pressure is 15 MPa and the gap clearance is 1.5 mm to analyze the fluctuation of the suction force. The comparison of experimental results of suction force between the original suction cup and suction cup-$\theta$20 is shown in Figure 34 when the gap clearance is 1.5 mm, and the inlet pressure is 15 MPa. It can be seen in Figure 34 that the average suction force of suction cup-$\theta$20 is much larger than that of the original suction cup, while its amplitude of fluctuation is significantly smaller than that of the original suction cup. The average suction force of the original suction cup is about 223 N, while that of suction cup-$\theta$20 is about 304, which is 36% larger than the former value. The standard deviation of amplitude of fluctuation of the original suction cup is 1.0501, while that of suction cup-$\theta$20 is 0.8046, which is smaller than the former value. Moreover, the frequency of force fluctuation of the above two suction cups is almost the same, as shown in Figure 34. The above results are in accordance with the results of observation experiments. Therefore, it can be proved that the optimization of the vortex suction cup is also effective in reducing the instability of suction force.

3.4. Application on Underwater Climbing Robot

In order to validate the effectiveness of the optimized vortex suction cup, we designed the underwater climbing robot mounted with the vortex suction cup. The suction capability of the robot was tested. The climbing robot was immersed in the laboratory pool and placed near the wall. Then the high-pressure water pump was turned on to supply high-pressure water to the four vortex suction cups. It can be found that the underwater robot can stably suck on the wall of the laboratory pool and maintain continuous suction force, which is about 1200 N. During the suction process, the experiment member tried to drag the robot away from the wall several times, but the suction failure did not occur, which could also confirm the suction stability of the wall-climbing robot. As shown in Figure 35 and Supplementary Material Videos S4 and S5, four vortex suction cups were installed at the bottom of the robot and connected with the robot body through the connecting frames. The robot could automatically suck to the wall of the laboratory pool and the glass stably and firmly and could still maintain a suction state in case of large interference. In order to clearly show the distribution and operation of the suction mechanism when the robot sucked, glass was selected as the suction wall, and the picture of the suction mechanism of the robot was taken from the back of the glass when the robot was not sucked and sucked. By comparing Figure 35b,c, it can be observed that during the suction process, the flow condition of the four vortex suction cups in the suction cup’s cavity and the gap between the suction cup and the contact surface was consistent with that of the single vortex suction cup shown in the previous section.

4. Conclusions and Future Work

In this paper, the underwater vortex suction cup is designed to be mounted on the underwater climbing robot to create a non-contact suction force so as to improve the validity and surface adaptability of the suction cup and mobility of robots. At the same time, in order to solve the problem that severe cavitation occurs in the suction process of the vortex suction cup and has a negative effect on the suction performance, an optimization method was proposed, which is to increase the angle between the inlet of the suction cup and the horizontal plane. The negative effect of cavitation on suction performance is mainly reflected in reducing suction force and increasing fluctuation of the suction force, that is, the suction instability. Therefore, this paper expounds from the above two aspects and verifies the effectiveness of the optimization by comparing the numerical simulation and experimental results of the original suction cup and the optimized suction cup. Firstly, based on the analysis of the suction force, the force measurement experiment was carried out on the original suction cup and the optimized vortex suction cups. By comparing the suction force of a series of optimized suction cups, the optimal inclination angle of the suction cup’s inlet, i.e., 20 degrees, was selected. The comparison of the suction force between the optimized suction cup and the original suction cup can verify the effectiveness of the optimization in improving the suction force. Next, the suction periodic instability is analyzed. The CFD model with cavitation was built to obtain the streamline, pressure distribution, and cavitation in the suction cup’s cavity and the gap between the suction cup and wall. Then the simulation results of the original suction cup and the optimized suction cup are compared to observe the differences in the streamline, pressure distribution, and cavitation between the two suction cups shown above and to measure the frequency of suction fluctuation. After that, the force measurement experiment and vortex observation experiment were carried out on the original suction cup and the optimized suction cup. The vortex observation experiment shows that there are two rotation centers of vortex flow in the optimized suction cup, and the cavitation surge is significantly weakened. By recording the real-time suction force data of the force measurement experiment, it can also be found that the fluctuation of the suction force of the optimized suction cup is significantly reduced compared with that of the original suction cup. The consistency between experimental results and simulations shows that the reliability and effectiveness of the cavitation reduction optimization method mentioned in this paper. The simulation and experimental results reveal that the suction force of the optimized suction cup is 36% greater than that of the original suction cup, and the amplitude of fluctuation is significantly smaller than that of the original suction cup.

It is very important and necessary to conduct a further study on the adhesion performance of the suction mechanism containing an assembly of several suction cups of an underwater cleaning robot, which has not been performed in this paper. Due to the severe cavitation inside the suction cup’s cavity, the pressure distribution on the contact surface between the suction cup and wall is not uniform in the circumferential direction, and the suction force is periodically unstable in the axial direction, which is not conducive to the stable suction of robot mounted with the vortex suction cup. Therefore, in order to improve the suction force and suction stability of the cleaning robot, it is necessary to develop an improved design of the robot’s suction mechanism (e.g., choosing the appropriate number and arrangement of suction cups). Moreover, the usage conditions of the robot would also cause instability or even failure of the suction, such as the installation of a high-pressure cleaning device and the adhesion on the wall surface with curvature and raised obstacles/grooves.