Experimental Study on Nonlinear Motion Characteristics of Clean and Biofouling Aquaculture Cage Array in Waves

Abstract

This paper aimed to understand the nonlinear dynamic responses that arise from the interaction between waves and a biofouling aquaculture cage array. To that end, physical model tests of a biofouling aquaculture cage array in a 1 $\times$ 3 configuration in regular waves were conducted. Wave steepness values of 1/60, 1/30, and 1/15 were considered within the frequencies spanning from low- to high-frequency bands. Then, the nonlinear dynamic responses of the system were systematically decomposed into four successive orders of components. This approach allowed for a thorough assessment of the inherent nonlinearity within the biofouling system by analyzing each individual order. The results highlight that the first-order harmonic component was the predominant contributor influencing the nonlinear dynamic response, while the higher-order harmonic components remained crucial in their contribution to the overall nonlinear dynamic response of the system. In particular, the harmonics within the low-frequency regime exhibited significantly greater nonlinearity compared with those in the high-frequency regime. A notable decrease in the amplitude of the harmonic component could be identified in the low-frequency regime due to the damping from the biofouling. The comprehensive analysis of the nonlinear dynamics within the biofouling system provides insights into optimizing the performance of aquaculture systems.

1. Introduction

Aquatic foods offer a practical solution to satisfy the increasing global population and income-driven demand [1,2]. Aquaculture, as the major source of these foods, has experienced a significant surge in recent decades. Specifically, marine aquaculture often incorporates multiple cages, known as cage arrays, and has expanded into more exposed, open-ocean areas to address environmental and spatial challenges [3]. Nevertheless, these open-ocean fish farms are highly vulnerable to high-energy waves and biofouling conditions. As such, research on the wave–structure interactions of offshore aquaculture systems with biofouling cages has become the primary task for engineers aiming to safely design aquaculture farms.

Biofouling is one of the labor-intensive and cost-challenging problems related to sustainable marine aquaculture [4]. It can severely increase the weight of the cage net [5], cause physical damage to aquaculture systems and cultured organisms [4,6], and disrupt oxygen supplies and nutrient exchange [7,8]. Consequently, it is imperative to consider local hydrodynamic conditions, oxygen supplies, and nutrient exchange when designing aquaculture farms [9]. It has been reported that the common fouling organisms include colonial hydroids (Tubularia sp.) and blue mussels (Mytilius edulus), typically representing soft and hard marine biofoulings [10,11]. Over the past two decades, there have been limited studies on marine fouling’s effects on the hydrodynamic behaviors of offshore aquaculture net cages. However, researchers recognize the economic implications associated with fouling organisms, which not only affect the structural integrity of net cages but also influence the overall productivity of aquaculture operations. Recent studies have highlighted the correlation between the extent of biofouling and the reduction in water flow rates, leading to suboptimal conditions for fish growth and health. As the aquaculture industry continues to expand, understanding these dynamics becomes increasingly critical for sustainable management practices.

Previously, studies on biofouling’s effect on net panels were primarily conducted under current conditions, with significant emphasis on physical model testing. Swift et al. [10] experimentally studied the impacts of different biofoulings on the drag force of net panels. The results revealed that the drag coefficient of the biofouled netting increased by 6% to 240% compared with that of clean netting. Lader et al. [12] conducted towing tank experiments on the hydrodynamic drag on fouled nettings, using artificial hydroid fouling to represent the fouled colonial hydroid (Ectopleura larynx). The results highlighted that the presence of fouled colonial hydroid increased the hydrodynamic drag on the netting. Gansel et al. [13] examined the drag on nets fouled with blue mussel (Mytilus Edulis) and sugar kelp (Saccharina Latissima) through flume tank experiments. The results suggested that existing models can be directly used to estimate the effects of additional mussel and kelp fouling on drag. Bi et al. [14] studied the drag and velocity reduction of hydroid-fouled nets using laboratory experiments. Their results showed that biofouling can increase the hydrodynamic load on nets by more than 10 times. Chen et al. [15] conducted extensive physical model tests of nettings subjected to various types of biofoulings. Consequently, the authors proposed empirical formulae for predicting the drag coefficients of netting panels under severe fouling conditions.

Aside from physical model tests, numerical models have also been used to examine biofouling’s impact on the hydrodynamics of aquaculture netting systems. Cornejo et al. [9] conducted a high-resolution Large Eddy Simulation (LES) to assess the hydrodynamic effects of different levels of biofouling on fish cage aquaculture netting. They found that heavily biologically contaminated net cages result in substantial alterations in both the direction and magnitude of local flow velocities. This alteration impacts not only the immediate vicinity of the nets but also modifies the circulation dynamics of the entire fjord. More recently, Yu et al. [16] studied the drag and flow field of aquaculture nets subjected to different levels of biofouling using Large Eddy Simulations (LESs). The results indicated that biological fouling significantly affected the drag coefficient and flow field of the netting; however, it was largely independent of the form of fouling. As the level of biological fouling increases, both the length of the net flow area and the extent of vortex formation noticeably increase, adversely impacting the living conditions within the aquaculture nets.

Furthermore, alongside the flow conditions, studies on biofouling’s effect on net cages were also conducted under wave conditions. Bi et al. [17] examined the influence of biofouling on net cages using the porous media fluid model. They proposed empirical formulae for the flow velocity reduction and hydrodynamic loads on a biofouling net cage. Then, the porous media fluid model was expanded to a square array of biofouling net cages to study the wave attenuation inside and around it [18]. Recently, the wave attenuation and hydrodynamic forces of marine-fouled floating aquaculture cages were examined, respectively, in regular and irregular wave conditions [11,19]. Nevertheless, research concerning a biofouling aquaculture cage array remains sparse, and the nonlinear dynamic response of a biofouling aquaculture cage array is not completely understood. To resolve these issues, this paper experimentally studied the nonlinear dynamic response of a clean and biofouling aquaculture cage array in waves.

The rest of this paper is organized as follows. Section 2 details the experimental setup of the biofouling aquaculture cage array. Section 3 provides the experimental results and discussion of this paper. Finally, the concluding remarks and significant findings are summarized in Section 4.

2. Materials and Methods

Experiments involving the biofouling aquaculture cage array in a 1 $\times$ 3 configuration under regular waves were conducted at the water flume in the Ningbo Institute of Dalian University of Technology, Ningbo, China. The dimensions of the flume are 30 m (length) $\times$ 2 m (width) $\times$ 1.2 m (height) with a constant water depth of 0.8 m. The waves are generated from a hinged flap-type wave generator.

2.1. Physical Model

The physical model of each biofouling net cage is designed with a scale $\mathrm{\Lambda }=1/50$ and it corresponds to a specific prototype with a standard diameter of 40 m. For wave tank/flume experiments of offshore structures, the scale of 1/50 is typically remarked in a range of scales from 1/10 to 1/100 [20]. The physical model consists of a floating collar, the netting chamber with the bottom weights, and the mooring systems. Previous studies on the similarity favor the Froude similarity except for the netting [21,22,23,24,25]. Thus, components of the physical model and the wave conditions in the present study are scaled based on the Froude similarity, ensuring that the model accurately represents the dynamics of the prototype.

Under the assumption of zero axial tension of the floating collar, the governing equation can be considered the dynamic equation of an Euler–Bernoulli beam. This partial differential equation (PDE) is related to the vertical displacement and the bending stiffness EI, where E is the Young’s modulus and I is the second area moment of the cross-section. Combining the natural frequency of the system, we finally yield the bending stiffness for the model scale, which is ${\mathrm{\Lambda }}^5$. The diameter and length of the floating cage are scaled by $\mathrm{\Lambda }$, and the mass per unit length is scaled by ${\mathrm{\Lambda }}^2$. Regarding the scale effect, the solidity ratio and hanging ratio of the model nets need to be consistent with the prototype nets (nylon, Sn = 0.36). Combined with the specifications of the existing nets of the fishing net manufacturer, we chose the nets with a twine diameter of 2 mm and a mesh size of 10 mm to be used as the model nets in the experiments. Detailed specifications and characteristics of the physical model are illustrated in

Table 1. Parameters of the physical model.

Components	Parameters	Model Scale	Full Scale
Floating collar diameter	D = 2R	0.8 m	40 m
Net cage diameter	D = 2R	0.8 m	40 m
Net length	L	0.5 m	25 m
Cross-section diameter of the floating collar	2a	20 mm	1.0 m
Mass per meter of the floating pipe	mf	0.127 kg/m	79.4 kg/m
Bending stiffness of the floating pipe	EI	27.59 Nm2	1.33 × ${10}^8$ Nm2
Diameter of net twines	dw	2 mm
Bar length of net twines	lw	10 mm
Net solidity ratio	Sn	0.36	0.36
Mass of net cage	Mnet	0.57 kg	5578 kg
Mass of bottom weight in the air	Mbw	16 × 30 g	16 $\times$ 1172 kg

.

2.2. Experimental Setup

The schematic representation of the experimental setup is illustrated in Figure 1. The origin of the Cartesian coordinate system is established at the free surface, where $z=0$. The net cages are constructed within the framework of the mooring system and the bridles, ensuring stability and proper positioning. Five wave probes are strategically installed to accurately measure the mean amplitude of wave height, with portions submerged underwater. Additionally, three charge-coupled device (CCD) cameras are employed to monitor the overall motion of the floating cages. The other two cameras, with a frame rate of 60 fps and a resolution of 1280 × 720 pixels, are used to record the general behaviors of the cage array under waves.

2.3. Test Conditions

Regular waves in period $T=0.8-1.2\mathrm{s}$ with an increment of 0.1 s are prescribed in this paper, and the ratio $\left(\lambda /D\right)$ between the wavelength and the diameter of the net cage is from 1 to 2.4. Thus, the period ranges cover the high-frequency and low-frequency band of the incident waves. Incident waves with wave steepness $H/\lambda =1/60,1/30$ and $1/15$ propagating along the $x$-axis direction are applied in all cases. It should be noted that $H/\lambda$ is an appropriate wave steepness for deep water waves. For the entire range of relative water depths, using the generalized nonlinear parameter $gH/c^2$ would improve the accuracy of nonlinearity classification [26].

These test conditions are summarized in

Table 2. A comparison of prescribed and measured wave heights (m).

T (s)			H/λ
	1/60		1/30		1/15
	$H^{\left(p\right)}$	$H^{\left(m\right)}$	$H^{\left(p\right)}$	$H^{\left(m\right)}$	$H^{\left(p\right)}$	$H^{\left(m\right)}$
0.8	0.017	0.0171	0.033	0.0327	0.067	0.0665
0.9	0.021	0.0206	0.042	0.0428	0.084	0.0846
1.0	0.026	0.0258	0.052	0.0510	0.104	0.1010
1.1	0.031	0.0311	0.063	0.0650	0.126	0.1258
1.2	0.037	0.0376	0.075	0.0755	0.150	0.1449

, where $H$ is the wave height and $\lambda$ is the wavelength. Before testing the physical model, it is suggested to conduct the wave calibration to avoid the interference of signal noise [27].

In the present work, the prescribed waves are approximated by linear wave theory [28]. The velocity potential $\varphi$, surface elevation $\eta$, and dispersion relation are defined as follows:

$$\varphi =-{\displaystyle \frac{g{\zeta }_a}{\omega }}{\displaystyle \frac{\cos \mathrm{h}k\left(h+z\right)}{\cos \mathrm{h}\nu d}}\sin \left(kx-\omega t\right)$$

(1)

$$\eta ={\zeta }_a\cos \left(kx-\omega t\right)$$

(2)

$${\mathsf{\omega }}^2=gk \tan \mathrm{h}\left(kh\right)$$

(3)

where ${\zeta }_a$ is the wave amplitude (equal to ½ of the wave height), $g$ is the gravitational acceleration, $\mathsf{\omega }$ is the circular frequency, $k$ is the wave number equal to $2\mathsf{\pi }/\lambda$, $\lambda$ is the wavelength, $h$ is the water depth, $z$ is the vertical position in the water column and $x$ is the horizontal position.

The definite solution conditions of radiation potential ${\phi }_j\left(j =1,2,\cdots ,6\right)$ and diffraction potential ${\phi }_7$ are given as follows [29]:

Continuity equation:

$${\nabla }^2{\phi }_j=0\left(j =1,2,\cdots ,7\right)$$

(4)

Linearized free-surface condition:

$${\displaystyle \frac{\partial {\phi }_j}{\partial z}}-{\displaystyle \frac{{\omega }^2}{\mathrm{g}}}{\phi }_j=0\left(z=0,j =1,2,\cdots ,7\right)$$

(5)

Object surface condition:

$${\displaystyle \frac{\partial {\phi }_j}{\partial z}}=n_j\left(j=1,2,\cdots ,6\right),{\displaystyle \frac{\partial {\phi }_7}{\partial n}}=-{\displaystyle \frac{\partial {\phi }_0}{\partial n}}$$

(6)

Bottom condition:

$${\left.{\displaystyle \frac{\partial {\phi }_j}{\partial z}}\right|}_{z=-h}=0\left(j=1,2,\cdots ,7\right)$$

(7)

Infinity condition:

$$\underset{R\to \infty }{\lim }\sqrt{R}\left({\displaystyle \frac{\partial \phi }{\partial R}}-ik\phi \right)=0$$

(8)

where $n$ is the unit normal vector of the object surface and ${\phi }_0$ is the incident potential of the wave without disturbance of floating structures.

According to Haskind Relations, the wave force on the floating structure can be expressed as follows [30]:

$$F_{0_j}=-i\rho e^{-i\omega t}\underset{S}{{\displaystyle \iint }}\left({\phi }_0+{\phi }_7\right){\displaystyle \frac{\partial {\phi }_j}{\partial n}}\mathrm{d}S\left(j=1,2,\cdots ,6\right)$$

(9)

At zero speed, the relation between radiation potential and diffraction potential can be obtained via the second Green theorem:

$$\begin{array}{c}\underset{S}{{\displaystyle \iint }}{\phi }_7{\displaystyle \frac{\partial {\phi }_j}{\partial n}}\mathrm{d}S=\underset{S}{{\displaystyle \iint }}{\phi }_j{\displaystyle \frac{\partial {\phi }_7}{\partial n}}\mathrm{d}S\\ \underset{S}{{\displaystyle \iint }}{\phi }_7{\displaystyle \frac{\partial {\phi }_j}{\partial n}}\mathrm{d}S=-\underset{S}{{\displaystyle \iint }}{\phi }_j{\displaystyle \frac{\partial {\phi }_0}{\partial n}}\mathrm{d}S\end{array}\left(j=1,2,\cdots ,6\right)$$

(10)

Finally, the wave force can be written as follows:

$$F_{0_j}=-i\rho e^{-i\omega t}\underset{S}{{\displaystyle \iint }}\left({\phi }_0{\displaystyle \frac{\partial {\phi }_j}{\partial n}}+{\phi }_j{\displaystyle \frac{\partial {\phi }_0}{\partial n}}\right)\mathrm{d}S\left(j=1,2,\cdots ,6\right)$$

(11)

For the cage model, the dynamic equilibrium equation of structural dynamic response is as follows [31]:

$$\left[m\right]\left[\ddot{\eta }\right]+\left[c\right]\left[\dot{\eta }\right]+\left[k\right]\left[\eta \right]=\left[G\right]+\left[f_b\right]+\left[f_w\right]$$

(12)

where $\left[\eta \right]$, $\left[\dot{\eta }\right]$ and $\left[\ddot{\eta }\right]$ are the displacement, velocity and acceleration of the structure, respectively. $\left[m\right]$, $\left[c\right]$, $\left[k\right]$ are the generalized mass matrix, damping matrix, and stiffness matrix of the structure, respectively. The forces acting on the cage model are gravity $\left[G\right]$, buoyancy $\left[f_b\right]$ and wave excitation force $\left[f_w\right]$, respectively.

It is suggested not to place any models in the water flume to obtain the prescribed wave conditions for wave calibration. Measured wave data were filtered and compared to the corresponding prescribed wave for wave calibration. The comparisons between prescribed and measured wave heights without the presence of the models are exhibited in Table 2. $H^{\left(p\right)}$ and $H^{\left(m\right)}$ are defined as prescribed and measured wave heights in meters. Errors within 0.3% compared to the prescribed values are achieved after testing.

2.4. Data Processing

For a prescribed sinusoidal wave, the actual wave formed in the flume may not have a sinusoidal shape due to the gravity, flume bottom, and the wave generator effects. In this situation, Fourier series analysis is implemented to process the measured wave data. Additionally, the interval between two testing cases should be sufficiently long to prevent interference between tests. We use 8–10 min based on the flume dimensions of 30 m (length) $\times$ 2 m (width) $\times$ 1.2 m (height). Each test is repeated at least 3 times to ensure reproducibility.

The steady-state dynamic response is filtered with a band-pass filter to remove noise out of the prescribed frequency range. The lower and upper limits in frequency are set to 0.95/T and 1.05/T, 1.95/T and 2.05/T, 2.95/T and 3.05/T, 3.95/T and 4.05/T. In this manner, the first-, second-, third- and fourth-order harmonics of the nonlinear dynamic response can be obtained after filtering. The data processing method is described as follows:

(1)

The raw data are obtained from three charge-coupled device (CCD) cameras. The raw data, namely, time histories of surge and heave motions, are obtained by transferring the camera signals to motion signals using our custom-developed codes.

(2)

Filtering is conducted to obtain harmonic components of the nonlinear surge and heave motions (e.g., ${\eta }_{11}^{\left(\omega \right)},$ ${\eta }_{11}^{\left(2\omega \right)},{\eta }_{11}^{\left(3\omega \right)}$ and ${\eta }_{11}^{\left(4\omega \right)}$). The processing methodology is based on the Fourier series analysis conducted using our custom-developed codes.

(3)

The obtained harmonic components of the nonlinear dynamic responses are normalized by ${\eta }^{\left(n\omega \right)}/{\zeta }_a$, respectively, ($n=2,3,4$ and ${\eta }^{\left(n\omega \right)}$ is the harmonic component of the nonlinear dynamic response after filtering).

Three examples of the first harmonic components of heave response of the biofouling aquaculture cage array after filtering are depicted in Figure 2.

The filtered first-order harmonics of heave response under $T =1.0\mathrm{s}$, $H/\lambda =1/60$ are depicted in Figure 2. The range of the time series for filtering is from 0.30 (low limit) to 0.80 (high limit), as shown by the red horizontal lines. The curves of the time series appear smooth after the removal of the noise. The mean amplitude of the first-order harmonics of the heave response is in a reasonable range (which almost reaches the prescribed wave amplitude) after filtering. When we measure the data, we stop the data acquisition device before the wave generator stops, so the time series of the acceleration is suddenly cut. There are no particular differences between these three filtered heave responses, highlighting the reproducibility of the present method. On the other hand, the wave radiation and diffraction have little effect on the first-order harmonics of the heave response when the wave steepness is relatively small.

3. Results and Discussion

The nonlinear harmonics of the dynamic response (i.e., surge and heave) for the biofouling aquaculture cage array are experimentally examined under regular waves. The experiments involving an aquaculture cage array without biofouling are also conducted for comparison. Figure 3 demonstrates the dependence of non-dimensional harmonics of surge (${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{11}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{11}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{11}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 1 with and without biofouling. To facilitate a meaningful comparison of the magnitudes of the various harmonic terms, we apply normalization of these harmonic terms using the frequency $\omega$ and amplitude ${\zeta }_a$ of the incident regular waves.

As can be seen in Figure 3, the overall trends of the non-dimensional harmonics of surge (${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{11}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{11}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{11}^{\left(4\omega \right)}/{\zeta }_a$) show distinct patterns as the non-dimensional wave number $ka$ increases. As the order of higher-order components increases, their values exhibit a consistent trend of diminishing magnitude. This becomes especially evident when examining the fourth-order components within the aquaculture cage array that are affected by biofouling. The reduction in values highlights the significant impact of biofouling on these components, suggesting the need for further investigation into the implications of such interactions in aquaculture systems. Moreover, the first-order harmonic component, ${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a$, as the primary contributor to the nonlinear surge motion, increases alongside the non-dimensional wave number $ka$ regarding aquaculture cage with and without biofouling. In contrast, the other higher-order components do not exhibit a consistent trend; however, they still contribute significantly to the nonlinear surge motion. These higher-order components enhance the nonlinearity of the surge motion, thereby influencing the overall dynamics of the system. By identifying the conditions under which harmonic components are maximized, engineers could enhance the resilience of aquaculture cage systems against wave-induced excitations.

Additionally, it has been observed that the harmonic components exhibit a decreasing trend as the wave steepness increases. This relationship highlights the intricate dynamics between waves and their harmonic components, suggesting that as the wave becomes less steep, the presence and intensity of higher harmonics tend to increase correspondingly. For the aquaculture cage array without biofouling, the peak of the non-dimensional harmonics is recorded at a wave steepness of 1/60, spanning from the first to the fourth orders. Although a definitive trend has not been established for the aquaculture cage with biofouling, the predominant trend remains consistent with that of the aquaculture cage array without biofouling. Specifically, the harmonic components tend to decrease as the wave steepness increases. Nevertheless, it is imperative to consider the species of different marine organisms that may influence these dynamics. These variables could affect the interactions between waves and the harmonic responses observed in biofouling and non-biofouling conditions.

Furthermore, the observed trends in the harmonics may also reflect the nonlinear effects that arise in different wave conditions. For the aquaculture cage array with biofouling at wave steepness of 1/60, when the non-dimensional wave number is less than 0.08, the second and third harmonic components are much higher than for any other wave steepness. Notably, the low-frequency effect could lead to this nonlinearity in the system. As the wave steepness changes, the energy transfer between different harmonic components causes enhanced resonances. This interaction between waves and harmonic behavior is crucial for understanding the complex dynamics of wave interactions in different biofouling conditions.

Figure 4 depicts the dependence of non-dimensional harmonics of heave (${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{33}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{33}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{33}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 1 with and without biofouling. Similarly to Figure 3, as the sequence of higher-order components increases, the values associated with these components exhibit a notable decrease in magnitude. This trend indicates that the contribution of these components becomes less pronounced with each successive order. Specifically, the fourth-order components within the biofouling aquaculture cage array could be overlooked as their values almost reach O(${10}^{-3}$). Regarding the first-order harmonic component, ${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a$, it remains the predominant contributor influencing the nonlinear heave motion, but it does not show an upward trend as the non-dimensional wave number $ka$ increases. On the contrary, this first-order component reaches its maximum value as the non-dimensional wave number approaches 0.08. Specifically, the first-order harmonic component, ${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a$, increases in the high-frequency regime and decreases in the low-frequency regime where the non-dimensional wave number is greater than 0.08. This can be attributed to the intricate interaction between the wave dynamics and the structural response of the system. The results indicate that the damping mechanisms become more pronounced at lower frequencies, reducing the harmonic component’s amplitude. The damping characteristics are highly dependent on the material properties (e.g., biofouling induced) and the inherent viscoelastic behavior, which can significantly influence the energy dissipation of waves.

In addition, for the aquaculture cage array without biofouling, the harmonic components exhibit a decreasing trend as the wave steepness increases. For the aquaculture cage with biofouling, the predominant trend remains aligned with that of the aquaculture cage array without biofouling. More specifically, it is evident that the harmonic components also tend to decrease in response to the increasing wave steepness. It has also been observed that the biofouling may not significantly influence the nonlinearity of the heave motion, since the higher-order harmonic components of the heave can be negligible, and this is largely attributed to the additional damping effects.

Figure 5 presents the dependence of non-dimensional harmonics of surge (${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{11}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{11}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{11}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 2 with and without biofouling. Similarly to cage 1, as the higher-order components increase, the corresponding values exhibit a decreased trend in magnitude. A significant inflection point is identified at the non-dimensional wave number $ka\approx 0.0725$. Beyond this threshold, the first- and second-order harmonic components of the non-dimensional surge motion exhibit a nearly linear increase. Conversely, these components begin to decline when the values fall below this inflection point. In comparison, the third- and fourth-order harmonic components of the non-dimensional surge motion consistently show a decreasing trend across all non-dimensional wave numbers. Moreover, it is important to note that the first-order harmonic component, ${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a$, continues to be the predominant contributor to the nonlinear surge motion, highlighting its critical role in the overall behavior of the system. The other higher-order components remain important in the overall contribution to the nonlinear surge motion.

Moreover, it has been observed that the harmonic components exhibit a decreasing trend as the steepness of the waves increases. For both aquaculture cage arrays with and without biofouling, the peak of the non-dimensional harmonics is recorded at a wave steepness of 1/60, which includes harmonics ranging from the first to the fourth order. Notably, the non-dimensional harmonic components associated with the surge motion exhibit pronounced nonlinearity. This can be attributed to the disturbance of the wave field as it propagates through the net cage, specifically cage 1. The observed nonlinearity in the surge motion harmonics may indicate potential vulnerabilities in cage design, particularly under small-amplitude wave conditions.

Figure 6 depicts the dependence of non-dimensional harmonics of heave (${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{33}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{33}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{33}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 2 with and without biofouling. Similarly to the findings shown in Figure 4, as the order increases, the corresponding values associated with these components decrease in magnitude. This trend indicates that the contribution of these components becomes less pronounced with each successive order. More specifically, the impact of the third- and fourth-order components within the biofouling aquaculture cage array is notably understated, as their values are considered negligible compared to the first-order components. This observation highlights the diminishing relevance of these higher-order components in the overall analysis of the system.

The first-order harmonic component, ${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a$, reaches its maximum value when the non-dimensional wave number is approximately 0.08. Similar to Figure 4, the first-order harmonic component exhibits an increase within the high-frequency regime while it declines in the low-frequency regime. The findings suggest that the damping induced by the biofouling becomes increasingly significant at lower frequencies, resulting in a notable decrease in the amplitude of the harmonic component. This highlights the critical influence of biofouling on the dynamic behavior of the system, particularly in the frequency-dependent responses. Moreover, the analysis suggests an intricate relationship where the properties of the biofouling itself may alter the wave field. As the frequency decreases, the viscoelastic properties of the biofouling become increasingly relevant, potentially leading to energy dissipation mechanisms that are not present at higher frequencies. This, in turn, enhances the nonlinearity of the heave motion within the system.

Figure 7 depicts the dependence of non-dimensional harmonics of surge (${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{11}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{11}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{11}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 3 with and without biofouling. It can be clearly observed that the values associated with the higher-order components exhibit a consistent increase in correlation with their respective orders. There is a slight difference from the previous trends in that no inflection point can be identified. The distribution of the harmonic components of the non-dimensional surge demonstrates an irregular and unpredictable pattern. Due to the biofouling effect, the first-order harmonic components of the non-dimensional surge motion show more pronounced nonlinearity. Moreover, it is essential to emphasize that the first-order harmonic component, ${\eta }_{11}^{\left(\omega \right)}/{\zeta }_a$, remains the primary factor influencing the nonlinear surge motion. This highlights its significant impact on the overall dynamics of the system. Additionally, it is worth mentioning that the higher-order components still play a vital role in contributing to the overall characteristics of the nonlinear surge motion, thereby enhancing the nonlinearity of the system.

Still, the harmonic components exhibit a decreasing trend as the steepness of the waves increases. The first-order non-dimensional harmonics at a wave steepness of 1/60 have the largest values for both aquaculture cage arrays with and without biofouling. Interestingly, the second-highest values are recorded at a wave steepness of 1/15, which presents a divergence from previously established findings in this paper. Thus, the biofouling significantly influences the hydrodynamic environment, leading to alterations in the wave field and ultimately impacting the overall performance of the system.

Figure 8 depicts the dependence of non-dimensional harmonics of heave (${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a,$ ${\eta }_{33}^{\left(2\omega \right)}/{\zeta }_a,{\eta }_{33}^{\left(3\omega \right)}/{\zeta }_a$ and ${\eta }_{33}^{\left(4\omega \right)}/{\zeta }_a$) on non-dimensional wave number $ka$ at wave steepness of 1/60, 1/30 and 1/15 for cage 3 with and without biofouling. Similarly to the findings shown in Figure 6, it is observed that as the order of the components increases, there is a corresponding decrease in the magnitude of the values associated with these components. This trend indicates that the contributions made by these components become less significant with each successive order. In addition, the influence of the third- and fourth-order components within the biofouling aquaculture cage array becomes subtle, as the values attributed to these components are insignificant compared to those attributed to the first-order components.

The first-order harmonic component, ${\eta }_{33}^{\left(\omega \right)}/{\zeta }_a$, attains its peak value when the non-dimensional wave number is approximately 0.08. The first-order harmonic component demonstrates an upward trend within the high-frequency range, while it experiences a decline in the low-frequency range. These observations indicate that the damping effects caused by biofouling become increasingly pronounced at lower frequencies, leading to a significant reduction in the amplitude of the harmonic component. This highlights the pivotal role that biofouling plays in influencing the dynamic behavior of the system, especially concerning its frequency-dependent responses. Moreover, the analysis reveals that as the wave number approaches this critical value, the interaction between the fluid dynamics and the biofouling becomes more complex. The biofouling can profoundly influence the surrounding wave field of the cage array. Subsequently, it influences the steepness of waves and the formation of wave harmonics, as shown in Figure 9.

In summary, the first-order harmonic component is the leading contributor to the nonlinear dynamic response of the aquaculture cage array, while the other higher-order components enhance the nonlinearity of the system. As the biofouling increases the weight of the net cage, the harmonic components associated with the bio-fouled aquaculture cage array are slightly reduced compared to those without biofouling. However, it is important to highlight that the presence of biofouling contributes to a significant enhancement in the nonlinearity in the dynamic response of the system, thereby affecting its overall performance and behavior. Experimental data suggest that the nonlinear responses are not merely a function of the harmonic components themselves, but also of the complex interaction between environmental factors and the structural integrity of the cage system.

4. Conclusions

This paper examined the nonlinear dynamic response of a biofouling aquaculture cage array configured in a 1 $\times$ 3 arrangement subjected to regular waves based on physical model tests. Three waves with steepness levels of 1/60, 1/30 and 1/15 were analyzed, with the frequencies spanning from low- to high-frequency bands. The nonlinear dynamic response of the system was systematically analyzed, focusing on both horizontal (surge) and vertical (heave) motions, which were subsequently decomposed into four separate orders of components. Through this analytical approach, the inherent nonlinearity of the biofouling system can be thoroughly assessed by examining each individual order.

The results highlight that the first-order harmonic component is the predominant factor influencing the nonlinear dynamic response, with an increase in the non-dimensional wave number regarding aquaculture cages with and without biofouling. Conversely, the higher-order harmonic components do not exhibit a uniform trend; however, they remain crucial in contributing to the overall nonlinear dynamic response. By identifying the conditions under which harmonic components are maximized, engineers could enhance the resilience of aquaculture cage systems, thereby improving their ability to withstand wave-induced excitations. Additionally, it has been observed that the harmonic components exhibit a decreasing trend as the wave steepness increases. This relationship reveals the intricate dynamics between waves and their harmonic components, suggesting that as the wave becomes less steep, the presence and intensity of higher harmonics tend to increase correspondingly. For the aquaculture cage array without biofouling, the peak of the non-dimensional harmonics is recorded at a wave steepness of 1/60, spanning from the first to the fourth orders. Notably, the harmonics within the low-frequency regime exhibit significantly greater nonlinearity than those found in the high-frequency regime. The results also indicate that the damping mechanisms exhibit a more significant effect at lower frequencies, resulting in a notable decrease in the amplitude of the harmonic component. The interaction between waves and biofouling cages can profoundly affect the energy dissipation of waves, highlighting the importance of understanding these interactions in wave dynamics and the behavior of biological organisms.

Understanding the nonlinear characteristics of wave interactions with biofouling structures provides insights into designing more resilient cage systems, ultimately enhancing the sustainability of aquaculture operations. Further research should focus on the technique incorporating motion measurement, active wave compensation control, and multi-system joint simulations, aiming to achieve a faster response time and higher compensation accuracy for offshore structures.