Developments in Modeling Techniques for Reliability Design of Aquaculture Cages: A Review

Abstract

Offshore aquaculture is gaining traction due to space limitations in nearshore waters, more pristine water, cooler temperatures, and better waste dispersal. This move has spurred the development of new technologies for offshore aquaculture. Despite the numerous analysis methods for designing aquaculture infrastructure, limitations and challenges remain in modeling the influence of fish cages on flow fields and in addressing fluid–structure interaction. This paper presents a comprehensive review of analysis methods and modeling techniques applied in the design of offshore aquaculture systems, emphasizing the structural reliability analysis. This review includes statistical and predictive analysis of extreme sea conditions, evaluation of environmental loads and hydrodynamic analysis, structural reliability modeling and assessment, and seabed geotechnical responses to mooring anchors. For each design consideration, the relevant theories and applicability are elaborated upon and discussed. This review provides valuable insights for engineers involved in the development and design of offshore aquaculture infrastructure.

1. Introduction

The aquaculture industry is one of the major suppliers of animal proteins for mankind. According to the report from the Food and Agriculture Organization [1], harvested fish account for at least 20% of the animal protein intake for 3.3 billion people. Over recent years, the rapid development of fisheries has created considerable economic value due to population growth and increased food demand. The data from DNV [2] demonstrate that from 1990 to 2020, the outputs from marine aquaculture increased by 5.8 times to 29 million tons per year, and it is expected to grow to 74 million tons in 2050. In view of this growth trajectory, offshore fish farming is becoming an emerging innovation for providing a large and sustainable production of fish to meet the continuously growing population demands. When compared to traditional nearshore sites, offshore ones offer higher efficiency and reduced environmental impact because of the larger water column, more pristine water, cooler water temperature, and better return on investment from a larger scale of fish production [3]. However, numerous design difficulties exist within offshore fish farming that have not been clearly outlined or thoroughly investigated; consequently, fish farmers harbor reservations about transitioning to offshore fish farming due to these uncertainties [4]. As a result, this has driven technological innovation in the analysis and design of offshore aquaculture infrastructure [5].

Karsnitz, et al. [6] defined the design process as “a systematic problem-solving strategy, with criteria and constraints, used to develop many possible solutions to solve a problem or satisfy human needs and wants and to narrow down the possible solutions”. Therefore, a good offshore fish cage design should provide adequate and suitable living spaces for fish during the service life, the structures should have sufficient strength to resist extreme environmental loads, and they should be easy to maintain. As shown in Figure 1a, a typical aquaculture system consists of a floating collar or frame, net chambers with weights, mooring line systems, and anchor foundation, all of which require meticulous consideration for their reliability in complex metocean environments. These analyses involve meteorological and oceanographic predictions, environmental load evaluations, hydrodynamic/hydroelastic analysis, structural reliability analysis, and geotechnical response analysis (Figure 1b). It is clear that a wealth of theoretical and technical support is needed not limited to just aquaculture but also spanning other fields of marine and ocean engineering.

At present, there are limited literature reviews regarding the modeling techniques applied to aquaculture infrastructure designs. This paper will present a comprehensive review of this topic. The overall layout will include the determination of the design’s environmental conditions through metocean statistics (Section 2), environmental load evaluation and hydrodynamic analysis (Section 3), modeling and analysis for structural reliability (Section 4), and modeling and analysis for the geotechnical responses of mooring anchors (Section 5). These contents may provide valuable references for the practice of offshore fish farming.

2. Design Environmental Conditions

Site selection is a key factor in any marine aquaculture activity not only to ensure the project’s success and product quality but also to resolve conflicts regarding land or water resources [7]. The site selection of a fish farm requires suitable geography, seabed topography, and environmental factors that will maximize fish growth and welfare. Chu et al. [4] proposed the following definition of an offshore farming site based on spatial and environmental conditions:

• Located in an exclusive economic zone and at least 3 km away from the coastline;
• Water depth is greater than 50 m or three times the depth of the fish cage;
• Mean current velocity is within 0.1 m/s to 1 m/s;
• Significant wave height is greater than 3 m.

When compared to coastal areas within sheltered bays, offshore seas naturally have stronger waves, currents, winds, and more extreme sea conditions during storms. Intense wave actions at offshore sites can damage both cage installations and their anchor points, potentially causing harm to the fish population and fish escape as well; while a suitable current is necessary for fish farming in cages to ensure oxygen supply and the removal of organic waste, excessive flow rates can have negative effects on both the cage infrastructure and the well-being of the fish [4].

Table 1. Norwegian aquaculture site classification scheme for waves and currents [8].

Degree of Exposure	Hs (m)	Tp (s)	Uc (m/s)
Small	0.0–0.5	0.0–2.0	0.0–0.3
Moderate	0.5–1.0	1.6–3.2	0.3–0.5
Medium	1.0–2.0	2.5–5.1	0.5–1.0
High	2.0–3.0	4.0–6.7	1.0–1.5
Extreme	>3.0	5.3–18.0	>1.5

presents a classification scheme of waves and currents according to the Norwegian aquaculture standard [8]. Offshore sites are characterized by high to extreme degrees of exposure.

2.1. Metocean Statistics and Predictions

Similar to most offshore structures, fish cages must withstand extreme sea conditions throughout their design service life. Borgman [9] introduced the concept of an encounter probability, wherein the probability of extreme sea states exceeding the design conditions within the design service life is determined as follows:

$$P\left(X\ge X_R\right)=1-{\left(1-\frac{1}{T_R}\right)}^{T_L}$$

(1)

where X is the corresponding sea state stochastic variables, such as wave conditions and current velocities; X_R is the design’s sea states for the specified recurrence period T_R and the design’s service life T_L. The recurrence period T_R needs to be extrapolated from the long-term probability distribution of the metocean conditions by extending the observational duration of statistical samples to two to three times [10]. Additionally, attention must be given to more frequent sea conditions due to their relevance to structural fatigue response [11] or fish welfare. Therefore, statistical and probability distribution modeling of sea states is crucial.

2.1.1. Wave Analysis

Wave statistics include long-term statistics and short-term statistics. The former is utilized to describe the distribution of sea state variables such as significant wave height H_s, while the latter describes the distribution of individual wave parameters within a stationary stochastic duration. (Sample distribution characteristics do not change over time.) The short-term distribution of linear wave heights has been demonstrated to follow the Rayleigh distribution in intermediate water and deep water, but the applicability of statistical models employed to characterize deep ocean waves to waves in shallower waters still remains an open question [12]. Battjes and Groenendijk [13] proposed a composite Weibull distribution to describe the wave height distributions on shallow foreshores based on theoretical and experimental data. Also, Alkhalidi and Tayfun [14] believed that the generalized Boccotti distribution performs best for shallow-water data, especially for large wave heights, i.e., H > H_s, as fitted to the shallow-water data. Based on the Rayleigh distribution, the mean of top n% wave height H_n% in a recording sequence has the following relationship with H_s:

$$\frac{H_{n\%}}{H_s}=\sqrt{-\mathrm{l}\mathrm{n}\left(\frac{n}{100}\right)}/1.42$$

(2)

Hitherto, there is no theoretical basis for determining the probability distribution of significant wave heights [15]. The choice of distribution only needs to exhibit unimodal and nonsymmetrical characteristics [11]. Various probability distribution models have been used to fit the observational datasets of H_s, including the three-parameter Weibull distribution, the lognormal distribution, and the Gamma distribution [15]. Also, according to Nguyen, et al. [16], the significant wave height H_s and the peak wave period T_p can be determined by using a wind-induced wave with limited fetch models by the Coastal Engineering Research Center (U.S) [17]:

$$\begin{array}{c}\frac{g{H}_s}{{{\tau }_w}^2}=0.283\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[0.53{\left(\frac{gh}{{{\tau }_w}^2}\right)}^{\frac{3}{4}}\right]\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left\{0.00565{\left(\frac{g{F}_e}{{{\tau }_w}^2}\right)}^{\frac{1}{2}}/\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[0.53{\left(\frac{g{F}_e}{{{\tau }_w}^2}\right)}^{\frac{3}{4}}\right]\right\}\\ \frac{g{T}_p}{{{\tau }_w}^2}=7.54\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[0.833{\left(\frac{gh}{{{\tau }_w}^2}\right)}^{\frac{3}{8}}\right]\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left\{0.0379{\left(\frac{g{F}_e}{{{\tau }_w}^2}\right)}^{\frac{1}{3}}/\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[0.833{\left(\frac{g{F}_e}{{{\tau }_w}^2}\right)}^{\frac{3}{8}}\right]\right\}\end{array}$$

(3)

in which g is the gravity acceleration; h is the water depth; ${\tau }_w=0.71{U_{10}}^{1.23}$ is the wind stress factor, where U₁₀ is the local wind speed at 10 m height above the mean water level; F_e is the maximum fetch distance. A core challenge associated with such idealized parameter models is the complexity in estimating fetch in irregular domains [18]. Information regarding the duration growth in water with finite depth is often not available; consequently, there is a need for fetch–duration conversion relationships to address the issue of deriving duration-limited functions from fetch-limited ones [19]. Eyhavand-Koohzadi and Badiei [19] presented a summary of all conversion equations developed for deep-water waves. Furthermore, the use of these formulas is rather restricted to situations where there are no refraction and diffraction problems, and the fetch can be determined rather clearly.

Extreme wave heights can be directly extrapolated from the probability distribution by fitting the entire dataset, that is, the initial distribution method. However, this method usually leads to high bias due to the inability to reasonably describe the tail of the distribution [12]. There are two approaches commonly employed in extreme value analysis, i.e., the block maxima (BM) approach and the peak over threshold (POT) approach. The BM method considers the distribution of the maxima order statistic, such as the annual maximum value of H_s, fitted with the generalized extreme value (GEV) distribution, but it is limited by the waste of data since there is only one data point in each block [20]. The POT method needs to define a threshold and fit the generalized Pareto (GP) distribution to the samples that exceed the threshold. The POT method requires the determination of an appropriate threshold, in which a threshold that is too high cannot generate enough samples, resulting in a greater estimated variance, but a threshold that is too low cannot reasonably describe the characteristics of extreme values. Bommier [20] presented a detailed guideline on the choice of thresholds. Some scholars (e.g., [11,21]) proposed the use of a two-part model to fit the distribution of H_s; that is, the core part (interval of dense observations) is as described by a distribution fitted to the entire dataset, while the tail is estimated by the POT method.

In fact, univariate analysis is insufficient for load and response calculations [22] due to significant correlations between wave parameters, for example, wave heights, periods, and directions. As the number of variables increases, establishing an appropriate joint distribution model becomes increasingly challenging, even in the bivariate case [12]. Typically, at least a joint distribution of significant wave height and wave period is required, but other environmental factors such as wind, currents, surges, and tides may also be relevant [22]. Currently, there are two main methods for joint modeling, namely the conditional method and the Copula-based method. For the bivariate analysis, the former describes the joint probability distribution function (PDF) as the product of a marginal PDF for the first variable and the conditional PDF of the other variable under the given first variable [23]. The latter uses Copula functions to describe the dependency structure between the two variables and only needs to specify their marginal distributions [22]. A comparison of these two approaches for describing the joint distribution of H_s and the mean zero up-crossing wave period T_z can be found in [22,24]. Copula-based models have been already comparable to conditional models, which are widely applied in engineering practice, but it is still a challenge to unequivocally determine the best candidate model [22].

It is also possible to extrapolate from multivariate distributions to extreme sea states under changing recurrence periods, i.e., environmental contours (Figure 2a). The environmental contour identifies a specific area in the environmental parameter space, referred to as the “design region”, where a structure that can endure all environmental conditions within that area has a probability of failure equal to or lower than a predetermined value [25] (as illustrated in Figure 2b). Different techniques for constructing environmental contours are based on distinct assumptions regarding the shape of the failure region (the portion of the environmental parameter space where the structure fails), and as a result, they define the exceedance of the environmental contour differently [25]. Approaches for defining environment contours include the inverse first-order reliability method (IFORM), the direct sampling method (DSM), the inverse second-order reliability method (ISORM), and the highest density region method (HDRM) [25]. Because the failure region is assumed to be convex, the IFORM and DSM may overestimate (if the failure region is convex) or underestimate (if the failure region is concave) the true failure probability; both the ISORM and HDRM are defined in terms of the total probability outside the contour, which results in the failure probability always being conservative and requiring no assumptions about the shape of the failure area [25].

2.1.2. Currents

The motion of the ocean surface is dominantly driven by the wind, resulting in the formation of ocean current components. Therefore, surface currents are strongly correlated with local wind speeds. Chu [26] found that the magnitude of surface current velocities follows the Weibull distribution by deriving the stochastic differential equations of the Ekman layer forced by winds. However, Campisi-Pinto, et al. [27] proposed a statistical test to assess the validity of the assumed wind speed and current velocity distributions across the globe; by analyzing the surface currents acquired using satellite altimetry, the generalized Gamma distribution was found to be more suitable for describing the surface wind speed and current velocity.

In the real marine environment, ocean currents are indeed not uniformly distributed vertically. From the ADCP observation in the northern North Sea, the current direction distribution changes minorly with water depth, but the current velocity decreases with deeper water depth, which indicates a spatial variation in ocean current conditions [28]. Therefore, it is necessary to determine their vertical profiles. In the North Sea, the currents are mainly caused by the tide and to a limited extent by wind forces. Jonathan, et al. [29] and Bore, et al. [30] proposed a procedure of joint modeling to simulate the characteristics of extreme current vertical profiles under arbitrary T_R:

• Principal component analysis (PCA) is applied to decompose the total current velocity U_c into two uncorrelated and orthogonal main-component U_M and minor-component U_m;
• Decoupling the total current into tidal and residual current components is carried out through harmonic analysis;
• For the marginal modeling of each residual component, the tail is fitted by the GP distribution, but the core part uses an empirically estimated distribution;
• The structural dependence of these residual components can be determined by the conditional extremes model by Heffernan and Tawn [31].

Bore et al. [30] also proposed a method to derive the design current velocity profile. For example, for a vertical cylinder, the N-year design drag force can first be estimated, and then, the corresponding current velocity profile can be derived, yielding the equivalent force.

Moreover, for a more accurate analysis of marine structures, joint probability distributions of different meteorological ocean parameters have received increasing attention over the past decades [32]. In order to perform a more accurate analysis of extreme environmental loads on marine structures, Bruserud et al. [32] established a bivariate distribution model for significant wave height and current velocity magnitude; it is recommended to establish a conditional model for the current velocity magnitude using a lognormal distribution based on the marginal distribution of H_s. This model is also applied to Monte Carlo simulation of joint significant wave heights and current speeds for the ultimate and accidental limit state load estimation according to the Norwegian design standard in [32].

2.2. Data Availability

For extreme value analysis in meteorology and oceanography, it is often necessary to have a data-recorded duration of more than 20 years [33], and observational data commonly suffer from missing data due to uncontrollable factors. Therefore, reanalysis or hindcast data generated through numerical simulations can serve as datasets for statistical analysis. Some open-source global or regional wave hindcast datasets include ERA-5 [34], NOAA WaveWatch III [35], IOWAGA [36,37], and CAWCR [38]. BRAN [39] also provides hindcast datasets for global ocean currents. However, the outputs of these global or regional hindcast datasets typically have relatively large spatial and temporal resolutions, which may not offer sufficient predictive accuracy for the selected fish farm sites. Karathanasi, et al. [40] quantitatively compared the statistical distribution differences and fitting performance of hindcast datasets from ERA5 and wave buoy measurement data in the Mediterranean region. The comparison by Karathanasi et al. [40] revealed that the hindcast dataset had better predictions for H_s when compared to T_p, and the prediction accuracy was notably influenced by spatial location.

On the other hand, data-driven models also play a crucial role in the statistical analysis of metocean data. For regression analysis, compared to traditional statistical models, data-driven models based on machine learning algorithms have advantages in handling nonlinearity and multivariate problems. Machine learning algorithms can establish relationships between the input and output features through supervised learning. Tang and Adcock [41] and Adnan, et al. [42] predicted sea surface elevation data for short-term wave analysis using random forest models. Some artificial neural network (ANN) models, such as backpropagation (BP) ANN [43,44,45] and long short-term memory (LSTM) ANN [46], are also applied to preprocess, calibrate, or forecast the data used for long-term wave analysis. Notably, the performance of data-driven models depends on the quality and sample size of the training data, the model algorithm, and the selection of the model parameters, indicating there are still challenges and exploration opportunities for engineering applications.

2.3. Far Field Analysis

The accurate prediction of wave conditions and ocean currents is crucial for the design of aquaculture infrastructure. Simulating the sea conditions in real ocean environments is filled with challenges. Along nearshore and offshore areas, complex wave transformations occur due to variations in bathymetry and irregular geometric shapes of coastlines. The complexity of wave transformations is mainly attributed to the nonlinear interactions among waves, leading to an uneven wavefield [47]. Additionally, the presence of ocean currents can alter wave characteristics such as wavelength, wave height, and wave energy spectrum; the velocity distribution in the flow field is also modified by wave motion and the radiation stress generated by waves [48]. This constitutes the interaction between waves and currents.

Phase-averaging models based on the spectral wave theory to approximately describe the wave statistical parameters, while being unable to provide a detailed temporal wave field, are often suitable for application in large, open seas due to their lower computational costs [49] (e.g., WAM [50], WW3 [51], SWAN [52], and MIKE 21SW [53]). These models are all third-generation wind wave models, but WAM and WW3 are more suitable for global or regional wave prediction, while SWAN and MIKE 21 SW include shallow-water physics packages. In addition, the spectral wave model can be coupled with the two/three-dimensional incompressible Reynolds-averaged N-S (RANS) equation that satisfies the assumptions of Boussinesq and hydrostatic pressure to simulate wave–current interactions (e.g., open-source framework COAWST [54] and commercial software MIKE 21/3 [55]). However, phase-averaging methods have limited ability to describe some nonlinear phenomena such as strong diffraction and reflection in the scattered archipelago outside fjords and deep channels [47,56].

The commonly used models for phase solving in wave propagation in coastal areas include the mid-slope equation, Boussinesq-type equations, and non-hydrostatic shallow-water equations. The open-source program SWASH [57] based on the non-hydrostatic shallow-water equations may be a good alternative for the Boussinesq models, especially for modeling fish cages. SWASH enhances its frequency dispersion by augmenting the number of layers instead of escalating the order of derivatives of the dependent variables, as is done in Boussinesq-type wave models. This is likely the primary factor contributing to the superior robustness and speed of SWASH when compared to any other Boussinesq-type wave model [57]. These numerical models are derived from the shallow-water equations and utilize various techniques to improve the representation of dispersion relationships in intermediate- to deep-water conditions [47].

Offshore fish farming is normally located in the intermediate- or deep-water region. The 3D non-hydrostatic RANS equations can be used to model complex wave patterns in arbitrary water depths but are limited by the tremendous computational cost. The fully nonlinear potential flow models may be a computationally efficient method for simulating wave propagation in deep water. BEM has been used to solve the wave propagation and breaking in a variable bathymetry, but the method is not optimal for large-scale domains due to the fully populated unsymmetrical matrix in the BEM (reviewed by Wang et al. [47]). Another technique for solving nonlinear potential flow models is the higher-order spectral method (HOS), in which only the discretization of the free water surface boundary is required, and the application of the fast Fourier transform (FFT) greatly improves the computational efficiency. Several robust open-source codes for the HOS model have been developed, such as HOS-NWT [58] and HOS-Ocean [59]. Also, the coupling between the CFD and HOS solvers can be achieved through a wrapper program, i.e., Grid2Grid [60], to transfer the far-field outputs from the HOS solver to the CFD solver. However, the HOS model requires determining the general solution to the Laplace equation, which limits its application to complex bathymetry. Wang et al. [47] developed a solver FPNP for the nonlinear potential flow model in the finite difference method-based open-source framework Reef3D. This model can efficiently describe wave breaking and was employed in a case study for the wavefield around a fish farm.

Nevertheless, the aforementioned models usually ignore the effect of fish cages when simulating the flow field in a large domain. To capture the impacts of fish farm structures on water currents, Michelsen, et al. [61] and Broch, et al. [62] applied an N-S equation-based ocean model SINOMD to simulate the current around a fish farm site, where the porous blockage effect of the cages was modeled by reducing the flow rate based on the volume occupied by the cages and by adding a drag term to the momentum equation.

Based on the review of hydrodynamic fish farming analyses, whether for near-field or far-field hydrodynamic analysis, considering the impact of the fish farm on the flow field is crucial for fish farming design. CFD models based on the Navier–Stokes equations can more accurately capture the wake and wave disturbances caused by net blockage effects. On the other hand, potential flow theory can provide an approximate description of large-scale fluid dynamics at a lower computational cost. Therefore, adopting a hybrid approach for hydrodynamic modeling is an inevitable trend for offshore aquaculture infrastructure design in the future.

3. Environmental Loads and Hydrodynamic Analysis for Aquaculture Cages

In ocean environments, drag forces driven by excessive current velocities can cause severe fish net cage deformations and critical mooring loads [63]. The reduction in cage volume due to cage deformation affects the swimming space and increases the stress on the fish, resulting in a decline in growth or even increased fatality [64]. Additionally, the intensity and period of turbulence inside the net cage can also affect the maneuvering of fish while swimming [65]. In practice, current velocities in the range of 0.1–0.6 m/s are most suitable for salmon farming, but breeding salmon in environments where current velocities exceed 1 m/s is also not recommended (reviewed by Chu et al. [4]). However, currents also have several positive effects: First, they can accelerate the transport of deposited wastes generated by the fish farm and dilute them by the surrounding water. Helsley and Kim [66] conducted a field test in the wake region of a submerged biconical fish cage. Ammonia (NH4⁺) was used as a tracer to measure the mixing in the wake stream. Significant dilution occurred in the first two diameters downstream of the cage, and the concentration of NH4⁺ was reduced 10-fold. This indicated that the water in the wake region was replaced by fresh water from the outer region. Additionally, the slack in the net was also reduced because of the current action [63].

Waves are a common form of energy transfer in the sea. Excessive hydrodynamic loads caused by waves may not only damage cages and mooring systems but also injure fish or lead to fish escape (especially when the cage is over-squeezed or the fish suffers an impact) [4]. Unlike long-term deformation caused by a steady flow, the wave excitation force will cause periodic motions of the cage, which are collectively known as the structural dynamic response. When the wave frequency is close to the natural frequency of the structure causing the resonance, the structure will have a significant dynamic response. Therefore, in the design of offshore floating structures, it is essential to avoid the occurrence of resonance.

Laboratory tests or field measurements are the most direct and effective methods of hydroelastic analysis. However, physical simulations are limited by the design dimensions, equipment conditions, and laboratory costs. Scale models have a problem in that the scale of the hydraulic process and the scale of the elasticity of the structure are usually not the same; this is a complicating factor of which researchers should be very aware. Moreover, it is relatively difficult to monitor and measure structural responses in field experiments, which are susceptible to complex external factors and disturbances. Therefore, numerical modeling is an effective and flexible tool, but challenges exist when establishing the numerical model of a fish cage. Chen [67] highlighted the following three main aspects in which such challenges manifest:

• A net is a complex structure composed of numerous twines, and the design scale of fish cages in practice is usually greater than 10 m in diameter; therefore, detailed modeling is not feasible for the cages with actual scales in practice;
• The hydrodynamic modeling of fish cages is a multiscale problem. Microscopically, the diameter of normal net twine is approximately 2 mm to 5 mm, and determining the effects of viscosity and turbulence in the boundary layer requires a relatively small scale for spatial discretization. However, macroscopic analysis of the fish cage system requires a relatively large computational domain with millions or even billions of computing grids, which is also extremely difficult and expensive;
• The net material is a relatively flexible structure. In real sea conditions, considering the interaction of the flow field influenced by the structural deformation is essential.

These aspects imply that more appropriate modeling techniques should be developed for this complex system. In reviewing the relevant literature, various rational methods have been proposed for the hydrodynamic analysis of net cages. The remainder of this section will comprehensively elaborate upon the principles and details of these methods.

3.1. Environmental Load Evaluations

From a design perspective, the environmental loads acting on the entire aquaculture facility may be linearly superposed as follows:

$${\mathbf{F}}_{\mathrm{e}\mathrm{n}\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}=\sum _{n=1}^{NC}\left({{\mathbf{F}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}}^n+{{\mathbf{F}}_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{c}}}^n\right)$$

(4)

where NC is the number of cages in a fish farm, F_environment is the environmental load, F_windⁿ is the wind load on the n^th cage, and F_hydrodynamicⁿ is the hydrodynamic load driven by currents and waves on the n^th cage.

3.1.1. Wind Loads

Similar to the analysis of wind loads on an oyster farm by Nguyen et al. [16], the wind load component F_wind can be calculated based on the exposed area at the top of each cage and the instantaneous draft in the fish farm, following the recommended practice of the American Bureau of Shipping (ABS) in their “Rules for Building and Classing Mobile Offshore Drilling Units” [68]:

$$F_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}=\frac{1}{2}{\rho }_{\mathrm{a}\mathrm{i}\mathrm{r}}{C}_h{C}_s{U_{10}}^2{A}_e$$

(5)

where ρ_air is the density of air; C_h is the height coefficient to account for the increase in wind speed with elevation, and C_h = 1.0 when the height of the structure above the water is less than 15.3 m; C_s is the shape coefficient, which depends on the shape and orientation of the structure exposed above the water, as illustrated in Figure 3a; A_e is the projected area of the exposed structure relative to the direction of the wind force, as shown in Figure 3b. The specific values for these parameters were summarized in [3].

3.1.2. Hydrodynamic Loads

Calculations of the current loads and the corresponding structural dynamic responses can often be considered quasi-static during a tidal cycle [16]. The drag force acting on the net is generally believed to be determined by the angle of flow attack, solidity ratio S_n, Reynolds number R_e, and material properties, among others. The solidity ratio S_n indicates the density of the net mesh instead of the porosity, which is defined as the net projected area divided by the outline area [69,70]:

$$S_n=\frac{2d_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}{l_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}-{\left(\frac{d_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}{l_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}\right)}^2\approx \frac{2d_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}{l_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}$$

(6)

where d_twine is the diameter of the net twine, and l_twine, which is sometimes referred to as bar length or half mesh size, is the distance between the net twines, as illustrated in Figure 4. The flow through fish cage nets is, in general, Re~$\mathcal{O}$(10²~10³) [71]. The experimental investigation by Tang, et al. [72] illustrated that, for an inclined net panel in the current, the drag force increases with increasing angle between the incoming flow velocity vector and the tangential vector of the screen, but the lift force reaches its peak at an angle of 50°. Tang, et al. [73] conducted extensive physical water flume tests for the fluid resistance effect of net panels. Their results clearly demonstrated that the drag force increases with a greater solidity ratio S_n. When S_n was low (0.190), the relationship between the fluid resistance coefficients of the net panel and the Reynolds numbers R_e was similar to that of a cylinder. Therefore, for fish cage nets with a high solidity ratio, the interactions between the net twines cannot be ignored. It was also observed from the measured data that the effect of the net knot was negligible under a flow condition with low Reynolds numbers. Additionally, the discrepancy in the fluid resistance of nylon and steel is explained by the material hydrophilicity and surface roughness, while it is independent of the Reynolds number [73].

Different from constant currents, since the wave-generated flows are unsteady, the inertial effect of the fluid cannot be ignored. The Keulegan–Carpenter (KC) number is a dimensionless number used to describe the relationship between the viscous and inertial forces on an object in an oscillating flow field. Dong, et al. [74] investigated the hydrodynamic forces on the net panel in a wave flume. Their experimental results revealed the horizontal component of the wave force significantly depended on the KC number, and the horizontal component was greater than the vertical component. The fitting of experimental data by Hamelin, et al. [75] indicated that the hydrodynamic drag coefficient of the slatted screen in an oscillatory flow was close to the results in a steady flow when KC > 100. This implies that the fluid inertial force is minor in this case. Furthermore, the calculations performed by Zhao, et al. [76] proved that the inertia effect can be ignored when the KC number of the net exceeds 150. Usually, the cage being immersed under the free water surface helps reduce the imposed wave loads. The experimental results of Liu, et al. [77] illustrated that the tension of the mooring cables and the movement of the floating collar were significantly weakened as the diving depth of the cage increased; however, when the fish cage reached a certain depth, the attenuation trend tended to stabilize. Based on the results of Liu et al. [77], one-third of the water depth was determined to be the optimal submergence depth for the fish cage. In addition, the theoretical solutions of Su, et al. [78] and Mandal and Sahoo [79] demonstrated that the peak of the wave loads appeared near the top of the net chamber in a disturbed wavefield.

In addition, if the disturbance of the fish cage to the flow field is ignored, the wave water particle velocity and current velocity can be linearly superposed [80,81], or the wave–current interaction is equivalent to a new wave [82,83,84]:

$$\begin{array}{c}\frac{L}{L_s}={\left(1-\frac{U_c\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }}{C}\right)}^{-2}\frac{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(kh\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_{s}h\right)}\\ \frac{H}{H_s}={\left(1-\frac{U_c\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }}{C}\right)}^{0.5}{\left(\frac{L}{L_s}\right)}^{0.5}{\left(\frac{N}{N_s}\right)}^{0.5}{\left(1-\frac{U_c\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }}{C}\frac{2-N}{N}\right)}^{0.5}\end{array}$$

(7)

where L and H represent the wavelength and wave height in current, respectively; L_t and H_t represent wavelength and wave height in still water; C = C_t + U_ccos α, and ω = ω_t + U_ccos α; C_t and ω_t represent wave velocity and wave angular frequency in still water; k = 2π/L, and k_t = 2π/L_t; N = 1 + [2kh/sinh(2kh)], and N_t = 1 + [2k_th/sinh(2k_th)].

Owing to the complexity of net mesh configuration, empirical formulas based on theoretical analyses and experimental investigations are usually simple and practical, so they are often applied for estimating hydrodynamic forces on fish cage nets. Herein, two commonly used empirical formulas are discussed, namely the Morison-type equation and the screen-type methods.

(A)

Morison-type equation

The net structure can be treated as consisting of mesh bars (as seen in Figure 5a). The Morison equation [85] is an empirical formula that is commonly used to evaluate the hydrodynamic force acting on slender bars, where the force is divided into two components, namely the drag forces F_D and the inertial forces F_I. According to Brebbia and Walker [86] and Faltinsen [87], for movable members, the Morison equation can be generalized as follows:

$$\mathbf{F}={\mathbf{F}}_D+{\mathbf{F}}_I=\frac{1}{2}\rho C_d{d}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}\left|\mathbf{u}-\dot{\mathbf{x}}\right|\left(\mathbf{u}-\dot{\mathbf{x}}\right)+\rho C_m{V}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}\frac{d\mathbf{u}}{dt}-\rho \left(C_m-1\right)V_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}\ddot{\mathbf{x}}$$

(8)

where u is the flow velocity at the midpoint of the net twine; x is the displacement at the midpoint of the net twine; ρ is the fluid density; V_twine is the volume of the twine; C_d is the drag coefficient; C_m is the added mass coefficient. In Equation (8), the first term on the right-hand side represents the drag force, while the second term is the inertial force component resulting from the Froude–Kriloff force (undisturbed wave pressure force) [87], and the third term is the inertial force component due to the added mass effect.

The drag coefficient C_d is dependent on the Reynolds number R_e, where the plotted curves of C_d versus R_e can be found in [88]. For the knot part of the net, Fredheim [89] suggested a drag coefficient C_d in the range of 1.0 to 2.0 when it is modeled as a sphere. However, in real flow fields, the incoming flow is not always perpendicular to the net bar, so the drag force F_D needs to be decomposed into a normal component F_Dn and a tangential components F_Dt:

$$\begin{array}{c}{\mathbf{F}}_{Dn}=\frac{1}{2}\rho C_{dn}{d}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}\left|{\mathbf{u}}_n-{\dot{\mathbf{x}}}_n\right|\left({\mathbf{u}}_n-{\dot{\mathbf{x}}}_n\right)\\ {\mathbf{F}}_{Dt}=\frac{1}{2}\rho C_{dt}{d}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}\left|{\mathbf{u}}_t-{\dot{\mathbf{x}}}_t\right|\left({\mathbf{u}}_t-{\dot{\mathbf{x}}}_t\right)\end{array}$$

(9)

where ${\mathbf{u}}_n$, ${\mathbf{u}}_t$, ${\mathbf{x}}_n$, and ${\mathbf{x}}_t$ are the velocity components in the normal and tangential directions of the flow and structures; the empirical formulas for fitting the normal drag coefficient C_dn and the tangential drag coefficient C_dt are summarized by Cheng, et al. [90] (as presented in

Table 2. Hydrodynamic coefficients in Morison equations for net twines when 10² < Re < 10⁴ (reviewed by Cheng et al. [90]).

Cdn	Cdt	References
1.2	0.1	[91]
1.3	-	[92]
$\left\{\begin{array}{c}{10}^{0.7}{R_e}^{-0.3}(R_e<200)\\ 1.2(R_e>200)\end{array}\right.$	0.1	[93]
$1.1+4{R_e}^{-0.5}$	$\pi \mu \left(0.55\sqrt{R_e}+0.084{R_e}^{2/3}\right)$	[94,95,96,97]
$-3.2891\times {10}^{-5}\left(R_e{S_n}^2\right)2+0.00068(R_e{S_n}^2+1.4253)$	-	[98]

).

Furthermore, the added mass coefficient C_m is usually taken as 0.5 for spheres and 1 for cylinders, according to Chen, et al. [99]. Nevertheless, under wave actions, the effects of KC numbers are also significant. Therefore, Zhao et al. [76] and Sarpkaya and Isaacson [100] demonstrated the factors that determine C_d and C_m through a nondimensional analysis as follows:

$$\left(C_d,C_m\right)=f\left(K_C,\frac{K_C}{R_e},\frac{{\mu }_s}{d_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{e}}}\right)$$

(10)

where μ_s is the material roughness coefficient of the net twine. The experimental measurements of Dong et al. [74] indicated that the C_d decreases, but C_m increases as the KC number increases. Furthermore, the net solidity ratio S_n might be the fourth factor influencing these hydrodynamic coefficients [89].

The deformation of net panels or cages under the action of currents has been simulated using Morison-type equations, producing reasonable results [97,99,101,102,103,104]. For the wave action, the Morison equation is valid for twine elements with a diameter much smaller than the wavelength, and the diffraction effect on the free water surface is also ignored [97]. Zhao et al. [76] compared the calculated results of the Morison equation with experimental data for net panels in a wave flume, where there were no significant discrepancies. The authors highlighted that, when evaluating the wave force, the influence of the KC number and inertial force can be ignored and that it is feasible to use only the drag coefficient C_d related to the Reynolds number R_e. However, as the conclusions of Zhao et al. [76] are not consistent with the experimental results obtained by Dong et al. [74], further studies are still required to clarify the effects of the KC number and inertial force.

The Morison-type equation provides a simple method of calculating the hydrodynamic load on the net cage, and the aforementioned experimental and numerical results indicate that it is feasible in engineering practice. However, the Morison equation still has some drawbacks: For example, the interaction between the net twines is ignored (e.g., the twine-to-twine wake interaction effects). The flow around the net twine elements is usually assumed not to be disturbed by the interaction with other twines [97]. Notably, Berstad et al. [70] suggested that a corrected flow velocity surrounding the net mesh could be determined by the conservation of fluid momentum. The influences of R_e and KC still require further discussion. When a vortex street is formed in the wake behind the twine cylinder, the cylinder will also impose a periodic lift excitation force perpendicular to the incoming flow direction, which is related to R_e and KC. (KC is only for oscillating flow.)

(B)

Screen-type method

Another hydrodynamic force empirical model for fish cage nets is the screen-type method. Kristiansen and Faltinsen [105] proposed a general, efficient, and easily applicable technique to quantitatively express this method. The net structure can be regarded as the composition of several super elements (the shading area in Figure 5b) whose centers are consistent with the knot centers. Each super element can be divided into four planar screens. The drag force F_D and lift force F_L acting on each screen can be decomposed into a normal force F_N perpendicular to the plane and a tangential force F_T tangential to the plane. The force magnitude can be described as follows:

$$\begin{array}{c}{\mathbf{F}}_D=\frac{1}{2}\rho C_D\rho A_{\mathrm{s}}{\left|{\mathbf{u}}_{\infty }-\dot{\mathbf{x}}\right|}^2,{\mathbf{F}}_L=\frac{1}{2}\rho C_L\rho A_{\mathrm{s}}{\left|{\mathbf{u}}_{\infty }-\dot{\mathbf{x}}\right|}^2\\ {\mathbf{F}}_N=\frac{1}{2}\rho C_N\rho A_{\mathrm{s}}{\left|{\mathbf{u}}_{\infty }-\dot{\mathbf{x}}\right|}^2,{\mathbf{F}}_T=\frac{1}{2}\rho C_T\rho A_{\mathrm{s}}{\left|{\mathbf{u}}_{\infty }-\dot{\mathbf{x}}\right|}^2\end{array}$$

(11)

where ${\mathbf{u}}_{\infty }$ is the undisturbed incoming flow velocity, A_s is the area of the screen, and the coefficients C_D, C_L, C_N, and C_T satisfy the following relationships by linearizing:

$$\begin{array}{c}{C}_D=C_N\mathrm{c}\mathrm{o}\mathrm{s}\alpha +C_T\mathrm{s}\mathrm{i}\mathrm{n}\alpha \\ C_L=C_N\mathrm{s}\mathrm{i}\mathrm{n}\alpha -C_T\mathrm{c}\mathrm{o}\mathrm{s}\alpha \end{array}$$

(12)

C_D and C_L can be expanded in a Fourier series dependent on the attack angle of flow α:

$$\begin{array}{c}{C}_D\left(\alpha \right)=C_D\left(0\right){{\displaystyle \sum }}_{n=1}^{\infty }{a}_{2n-1}\mathrm{c}\mathrm{o}\mathrm{s}\left[\left(2n-1\right)\alpha \right]\\ C_L\left(\alpha \right)=C_L\left(\frac{\pi }{4}\right){{\displaystyle \sum }}_{n=1}^{\infty }{b}_{2n}\mathrm{s}\mathrm{i}\mathrm{n}\left(2n\alpha \right)\end{array}$$

(13)

C_N and C_T are the functions determined by the Reynolds number R_e and the solidity ratio S_n:

$$\begin{array}{c}{C}_N\left(\alpha \right)=\frac{C_d\left(R_e\right)S_n\left(2-S_n\right)}{2{\left(1-S_n\right)}^2}{\mathrm{c}\mathrm{o}\mathrm{s}}^2,0\le \alpha \le \frac{\pi }{4}\\ C_T\left(\alpha \right)=\frac{{4C}_N\left(\alpha \right)}{8+C_N\left(\alpha \right)}{\mathrm{c}\mathrm{o}\mathrm{s}}^2,0\le \alpha \le \frac{\pi }{4}\end{array}$$

(14)

In Equation (13), the unknown constants a_2n−1 and b_2n (n = 1, 2, …, ∞) can be calibrated using experimental data, and it is usually an acceptable approximation to retain two or three truncation terms. Some calibrated parameters or empirical formulas employed by previous scholars about the coefficients C_D and C_L are summarized in

Table 3. Hydrodynamic coefficients in screen-type method for net panels.

a2n−1 (n = 1, 2, and 3)	b2n (n = 1, 2, and 3)	Reference
(0.9, 0.1, 0)	(1, 0.1)	[105]
(0.9, 0.1, 0)	(1.1, 0.1, 0.12)	[106]
(0.9725, 0.0139, 0.0136)	(1.2291, 0.1116)	[107]

. The screen-type model greatly compensates for the drawback of the Morison-type equation that ignores the interaction between net twines. It has been proven to be a reasonable approximation when evaluating the hydrodynamic loads on nets [80,105,106,107]. In structural modeling, its principle is to replace the bar elements with equivalent super elements, so it can be more easily coupled with computational fluid dynamics (CFD) models [63], as discussed in the subsequent section.

However, few researchers currently consider the effect of inertial forces and the KC number under waves in the screen-type model; therefore, this method only performs more reasonably for constant current actions. For the net panel, Lader, et al. [108] compared the measured wave forces in a water tank with the results calculated using the Morison equation and screen-type model; by contrast, the screen-type model only provided a favorable evaluation for the net with high solidity ratios. Therefore, the Morison equation still has potential in the evaluation of wave forces. Actually, the accuracy of the Morison equation model and screen-type model in predicting the hydrodynamic loads should be both highly dependent on the accurate selection of the hydrodynamic coefficients.

A summary regarding the adopted hydrodynamic force models and structural models in different software packages or codes is presented in

Table 4. Numerical software or codes for dynamic analysis of fish cages [90].

Hydrodynamic Force Model	Software or Code	Structural Model	References
Morison-type	ANSYS	Truss	[110,111]
	ABAQUS	Truss, beam	[102,103,104,112]
	Aqua-FE	Truss	[95,97,113]
	AquaSim	Membrane	[70,114,115]
	DUT-FlexSim	Lumped-mass	[76,82,83,84,96,116,117,118]
	FhSim	Triangles/lumped-mass	[119,120]
	Orcaflex	Lumped-mass	[98,121,122]
	ProteusDS	Lumped-mass	[123]
Screen-type	FhSim	Triangles/lumped-mass	[119,120]
	SIMA	Truss	[124,125,126]
	MPSL	Lumped-mass	[127]
	Reef3D	Truss	[128,129,130]
	UniS-Aqua	Truss	[131]

. Moreover, Wang, et al. [109] introduced an improved screen force model that utilizes a blend of high-precision CFD and Kriging metamodeling. This innovative model offers greater flexibility compared to traditional theoretical or experimental variations of screen force models. The suggested approach demonstrates superior performance when forecasting drag and lift coefficients across a broad spectrum of nets and inflow velocities. This superiority arises from its incorporation of various roughness levels and additional geometrical characteristics of the net in its formulation. Most researchers directly adopt the water particle velocity of incoming currents or incident waves when applying empirical formulas to assess hydrodynamic loads on the net. However, in actual conditions, the flow velocity would be significantly reduced in the wake behind the net (i.e., the flow field is disturbed by the net), and the diffraction effect of waves cannot be ignored because a fish cage is a large-scale structure relative to the wavelength macroscopically. In addition, the influence of rigid body motions or elastic deformations of the structure on the flow field is an issue in need of attention. Therefore, establishing a hydrodynamic model to describe the nearfield flow around the cage is essential.

For traditional flexible floating fish pens, the considered floating collar is a single torus or multiple toruses. In this case, the collar can be treated as a slender rod structure, and hydrodynamic loads are estimated through the Morison equation [132,133,134,135]. When the scale or motion of the floating collar is large, the potential flow theory can capture perturbations in the flow field [136,137,138]. Nevertheless, engineering tools for net cages typically employ the strip theory combined with linear potential flow and drag force corrections from the Morison equation to estimate the floater loads induced by waves; this approach neglects crucial three-dimensional flow, frequency dependency, and nonlinear effects [139]. Therefore, it is necessary to introduce a near-field hydrodynamic analysis model for aquaculture cages.

3.2. Near Field Analysis

3.2.1. Net Blockage Effects and Wakes

The topic of wakes caused by the blocking effect of fish cage nets has received much attention in engineering. Neglecting the flow velocity reduction at the rear of the cage will result in an overestimation of the mooring force by up to 22% [125]. Moreover, in an array of net cages, those cages further downstream might experience less water exchange; thus, there would be lower concentrations of dissolved oxygen and increased waste pressure compared with the upstream cages, and the interaction of the wakes due to each cage would result in a stronger combined water blockage [140].

Cheng et al. [90] concluded that the wake effects around a cage resulted in twine-to-twine, net-to-net, and cage-to-cage interactions. Also, Gansel et al. [141] defined the following three regimes using the solidity ratio S_n for describing the flow around the cylindrical net cage (reviewed by Klebert et al. [140]):

• 0 < S_n ≤ 0.25: At this time, a large amount of water can pass through the net and will evoke a clear vortex street in the wake region;
• 0.25 < S_n ≤ 0.75: The interaction of the wakes causes an additional blocking effect as well as more water to be compressed around the cage;
• S_n > 0.75: The porous cage is currently similar to a solid cylindrical shell, and more water blockage and vortex shedding occur around it.

The mechanism of wakes caused by the net blockage effect is very complex, which may depend on Reynolds number, solidity ratio, flow attack angle, net mesh shape, etc., so the velocity reduction through a net panel is commonly described empirically. According to Moe-Føre et al. [104], the flow velocity is reduced as a geometric series when passing several net panels:

$$\mathbf{u}={\mathbf{u}}_{\infty }\prod _{i=1}^{n_p}{r}_i$$

(15)

where n is the number of passed net panels, and r_i is the velocity reduction factor of panel i. Løland [142] suggested an engineering approach for estimating the reduction factor r_i:

$$r_i=1.0-0.46C_D\left(\alpha =0\right)$$

(16)

However, this method cannot comprehensively describe the flow field affected by the fish cage, so it is necessary to conduct modeling of the flow field taking into account the net blocking effect.

(A)

Computational fluids dynamic models

In the computational fluids dynamic (CFD) models based on the Navier–Stokes (N-S) equations, it is a widely employed practice to treat the action of nets to the flow field as a porous medium source term, as in [67,71,131], wherein the porous drag resistance has a polynomial relationship with the flow velocity [131]:

$${\mathbf{S}}_i=\sum _{i=1}^n{C}_i{\mathbf{u}}^i$$

(17)

where the coefficient C_i can be calibrated using the Morison equation or screen-type method by the momentum conservation. Also, n = 2 is sufficient for the engineering practice (reviewed by Cheng et al. [131]), which is known as the Darcy–Fochheimer equation. The displacement of the flexible net can be achieved by refreshing the position of the source term on the static girds, but the displacement of the discrete porous zone around the net may cause overlapping areas, as in [143]. To avoid this issue, Cheng et al. [131] introduced an improved topological method. Nevertheless, Higuera [144] indicated that the resistance source term S_i cannot fully represent the flows through regular porous materials with low porosities (0.35–0.65) since the fluid is constrained and able to pass only through the voids left by the solid matrix of the material. Therefore, the N-S equations may require some modifications (volume averaging process), as in [143], to model the flow passing through a net.

However, Martin et al. [107] suggested that defining the cells affected by the net as the entire porous medium will lead to a region where the pressure loss is constant rather than a thin slice where the pressure drops immediately, like in reality. Therefore, a Lagrangian–Eulerian coupling algorithm was proposed by Martin et al. [107] in which the forces on the net are distributed over Lagrangian points to represent the geometry of the net in the fluids. In addition, Yao et al. [106] proposed a hybrid volume model, that is, adding a weight of the reacting force of the net on the fluid in each computational grid to the source term. This approach enables the adoption of larger computational grids to avoid excessive computational costs.

Notably, to calculate the hydrodynamic forces acting on the net by the screen-type method, the unperturbed incoming flow velocity ${\mathbf{u}}_{\infty }$ must be known. For the net located in the wake generated by the upstream net, ${\mathbf{u}}_{\infty }$ is unknown, so Martin et al. [107] proposed an approximation of ${\mathbf{u}}_{\infty }$:

$${\mathbf{u}}_{\infty }=\frac{C_D}{2\left(\sqrt{1+C_D}-1\right)}\mathbf{u}$$

(18)

Cheng et al. [131] also derived the following one:

$${\mathbf{u}}_{\infty }=\sqrt{\frac{2}{2-\left(C_D+C_L\right)}}\mathbf{u}$$

(19)

Furthermore, a fine computational grid is usually required to capture the pressure drop and the wakes around the net, which may require a tremendous computation cost, so Devilliers, et al. [145] proposed an algorithm for adaptive grid refinement near the net interface when interacting with the flow, which efficiently optimizes the computational costs of the CPU and memory.

(B)

Potential flow models

When describing the interaction between porous net and waves, potential flow theory is a commonly used model for determining the scattered or interfering wave fields. In such models, the net is typically equivalent to the interface between the external flow field and the internal flow field around the cage, where dynamic boundary conditions are described by the porous media theory. This mainly includes linear models and quadratic models.

The linear model implies that the flow through the perforated structure satisfies Darcy’s law, where the pressure gradient linearly depends on the normal velocity of the permeation flow. Also, for a thin-walled structure, assuming the pressure across its thickness is approximated linearly, the velocity potential on both sides of the net interface is as follows [78,79,146]:

$$\nabla {\mathsf{\Phi }}^{+}·\mathbf{n}=\mathrm{i}{k}_0\tau \left({\mathsf{\Phi }}^{-}-{\mathsf{\Phi }}^{+}\right),\mathrm{a}\mathrm{t}S\left(x,y,z\right)=0$$

(20)

where Φ⁻ and Φ⁺ represent the velocity potential on the leeward and windward sides of the porous structure, respectively; n is the unit normal vector on the structural body surface; $S\left(x,y,z\right)=0$ is the cross-section geometry equation of the cage; the porous effect parameter τ is given by Yu and Chwang [146]:

$$\tau =\frac{{\tau }_0\left(f_r+\mathrm{i}{f}_i\right)}{k_0{t}_s\left({f_r}^2+{f_i}^2\right)}={\tau }_r+\mathrm{i}{\tau }_i$$

(21)

where τ₀ = 1 − S_n is the porosity of the net; t_s is the net thickness; k₀ is the incident wavenumber; f_r and f_i are the linearized coefficients of the porous resistance effect and fluid inertial effect, respectively. In Equation (20), the real part τ_r represents the porous resistance effect of the medium, while the imaginary part τ_i means the fluid inertia effect.

Ito et al. [147] offered a set of empirical formulas for the τ_r and τ_i of nets as follows:

$$\left\{\begin{array}{c}{\tau }_r=\frac{1}{2\pi }\frac{\left[27.73/\left(k_{0}H/2\right)\right]{{\tau }_0}^2}{1+\left[0.551-0.01998/\left(k_{0}H/2\right)\right]{\tau }_0}\\ {\tau }_i=\frac{1}{2\pi }\frac{0.002579\left[27.73/\left(k_{0}H/2\right)\right]\left(k_0{l}_{\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{e}}/2\right){\left(k_{0}H/2\right)}^{-1.547}{{\tau }_0}^2}{1+\left[0.551-0.01998/\left(k_{0}H/2\right)\right]{\tau }_0}\end{array}\right.$$

(22)

where the formula of τ_i is only for the cube cage with a side length l_cube, and the added mass effects are considered due to the acceleration of the net.

As the linear model makes it relatively simple to derive an analytical solution, it has been adopted in the studies on the interaction between waves and flexible net cages, as in [78,79,148]. Nevertheless, Darcy’s law might be not applicable to problems where the openings are large and sharp enough to separate the flow, and the frictional force is negligible compared with the pressure difference force [149].

Another approach considers the pressure drop acting on the perforated structure to have a quadratic relationship with the normal velocity of the penetration flow:

$${\mathsf{\Phi }}^{-}-{\mathsf{\Phi }}^{+}=\frac{1}{2}K\rho \left|\nabla {\mathsf{\Phi }}^{-}·\mathbf{n}\right|\left(\nabla {\mathsf{\Phi }}^{-}·\mathbf{n}\right),\mathrm{a}\mathrm{t}S\left(x,y,z\right)=0$$

(23)

where K is the discharge coefficient. The nonlinear term in Equation (22) can be solved through iterations. Molin and Remy [150] suggested introducing a relaxation to accelerate the convergence:

$${\mathsf{\Phi }}^{-\left(j\right)}-{\mathsf{\Phi }}^{+\left(j\right)}=\frac{1}{2}K\rho \left|\nabla {\mathsf{\Phi }}^{-(j-\frac{3}{2})}·\mathbf{n}\right|\left(\nabla {\mathsf{\Phi }}^{-(j-\frac{3}{2})}·\mathbf{n}\right),\mathrm{a}\mathrm{t}S\left(x,y,z\right)=0$$

(24)

where (j − 3/2) means the averaged value between the previous two iterations (j − 1) and (j − 2). Molin [149] believed that the quadratic model should satisfy the following assumptions:

• The structural thickness is small and can be negligible relative to the flow scale;
• Flow separation occurs through the openings, resulting in a quadratic discharge;
• The openings are infinitely small and numerous.

The discharge coefficient K can be calibrated by using the experimental data and was reviewed and summarized by Molin [149] for different types of screens. Gjøsund and Enerhaug [151] presented the following empirical formula of K for the net panel in a steady flow with an attack angle α:

$$K\left(\alpha \right)=\left[\frac{7.0}{R_e}+\frac{0.9}{\mathrm{l}\mathrm{o}\mathrm{g}\left(R_e+1.25\right)}+0.005\mathrm{l}\mathrm{o}\mathrm{g}\left(R_e\right)\right]\frac{1-{{\tau }_0}^2}{{{\tau }_0}^2}\mathrm{s}\mathrm{i}\mathrm{n}\left(2\alpha \right)$$

(25)

The formulas based on the experiments in steady flow conditions may be applicable to the scenario in an unsteady flow as well when the KC number is large enough because C_D/C_D^steady ≈ 1 at this time [75].

The quadratic model may be more realistic than the linear model because it accounts for the flow separation due to large openings. However, according to the experimental investigation by Ito et al. [147], the linear model can still predict the flow through the net reasonably in specific circumstances. Furthermore, in practical situations, the viscous fluid effect should be considered due to the increased porosity, roughness, and thickness of the net due to bio-retention, which may be described better by Darcy’s law.

3.2.2. Wave Scattering and Interferences

The motion and blockage effect of the cage may also influence the wavefield. It can be observed that there are some “wake areas” on the leeward sides of cages along the direction of wave incidence, as in the studies of Selvan et al. [148] and Ma et al. [152,153]. The wave amplitude is attenuated, especially for the rear cages in an array. In the inner regions of these cages, the wave amplitude also has different extents of reductions.

Water waves can be described using the potential flow theory, thus providing a fast and relatively simple approximate solution to the flow field, but it cannot handle some phenomena well, such as the viscous effects, vortices, turbulence, etc. If the fluid is assumed to be irrotational and inviscid, the fluid velocity vector u (x, y, z, t) can be expressed as the gradient of the velocity potential Φ (x, y, z, t); that is, u = ∇Φ. In this scenario, the flow field can be governed by the Laplace equation due to mass conservation:

$$\frac{{\partial }^2\mathsf{\Phi }}{\partial x^2}+\frac{{\partial }^2\mathsf{\Phi }}{\partial y^2}+\frac{{\partial }^2\mathsf{\Phi }}{\partial z^2}=0$$

(26)

Because the wave motion is assumed to be a harmonic oscillation, the difficulty of deriving its solution depends on the processing of the boundary conditions. According to [154], at the free water surface elevation z = ξ, the kinematic boundary condition is as follows:

$$\frac{\partial \xi }{\partial t}=\left(1+{\left|\nabla \xi \right|}^2\right)W-\nabla \tilde{\mathsf{\Phi }}·\nabla \xi$$

(27)

The dynamic boundary condition is as follows:

$$\frac{\partial \tilde{\mathsf{\Phi }}}{\partial t}=-g\xi -\frac{1}{2}{\left|\nabla \tilde{\mathsf{\Phi }}\right|}^2+\frac{1}{2}\left(1+{\left|\nabla \xi \right|}^2\right)W^2$$

(28)

where $\tilde{\mathsf{\Phi }}=\mathsf{\Phi }\left(x,y,z=\xi ,t\right),$ and $W=\frac{\partial \mathsf{\Phi }}{\partial Z}\left(x,y,z=\xi ,t\right)$. For the combined dynamic and kinematic condition on the cage interface in Equations (19) or (22), a linear approximation on the structural body surface is assumed, as the boundary condition cannot be satisfied at the instantaneous position of the wet surface. This requires the deformation amplitude of the structure to be relatively small compared with its cross-sectional dimension by omitting the first- and higher-order terms in the Taylor expansion of the boundary condition [87]. For the internal field, the velocity potential can be expressed as given:

$$\mathsf{\Phi }={\displaystyle \sum }_{i=1}^{NC}{\displaystyle \sum }_m^{\infty }{A_m}^i\left(t\right){{\phi }_m}^i\left(x,y,z\right)$$

(29)

where the spatial component in different orders ${{\phi }_m}^i$ can be represented as a sum of the incident wave velocity potential and the scattered wave velocity potential due to diffraction and radiation. Different from the radiation potential of six degrees of freedom caused by the rigid body motion, the radiation potential caused by the hydroelastic deformation of the cage is the superposition of infinite modes.

The closed-form solution to the first-order velocity potential disturbed by the deformed cage or cage array has been derived through the eigenfunction expansion method by many scholars [78,79,148,152,153], but it is constrained by cages with smooth geometric shapes. The boundary element method (BEM) can handle cages with more complex geometric shapes but requires discretization of the domain boundary (e.g., [147]). Nevertheless, these studies are limited to the linear-wave cases. Notably, Ma et al. [152] derived the second-order mean drift wave force acting on the cage based on the solved first-order velocity potential; the results indicated that in the direction of the incident wave, the front-row cage and the rear-row cage have opposite mean drift wave forces. This information serves as a valuable point for the design of the mooring system. The harmonic polynomial cell (HPC) method is a field method developed by Shao and Faltinsen [155,156] to solve the Laplace equation with boundary conditions, along with initial conditions in the time domain and a periodicity condition in the frequency domain. A key feature of the HPC method is its utilization of higher-order local expressions that satisfy the Laplace equation, which implies a better accuracy compared to other low-order field methods [157,158].

3.2.3. Water Sloshing

For rigid closed-containment fish tanks deployed in highly energetic environments or exposed sites, the motion of the tank under wave excitation can cause a significant water-sloshing phenomenon inside it. Unlike porous net cages, in an impermeable fish tank, different events, such as an apparent increase in wave amplitude, occur inside them [152]. Moreover, there is an occurrence of different wave patterns for sloshing in a circular cylindrical tank involving planar waves, swirling waves, and irregular chaotic waves under various harmonic force amplitudes and forced frequencies [159]. The highest natural period of sloshing as a function of radius and liquid depths for an upright rigid cylindrical tank can be found in [125]. Recently, Wiegerink, et al. [160] proposed installing annular slosh-suppression blocks at the top end of rigid closed-containment fish tanks in order to reduce sloshing. This is a promising solution for closed-containment fish tanks that are to be deployed in exposed offshore sites.

Flexible semi-closed cages have been trialed, as they boast a low fish mortality rate, low escape rate, and the ability to monitor fish growth in a day–night cycle [161]. However, the hydroelastic and structural analysis and design of these flexible membrane cages still face great difficulties. The key challenge is to ensure that the membrane cage maintains its tensile capacity under varying hydrodynamics loads and internal/external water salinity conditions. Strand and Faltinsen [157,158] derived the linear wave response of a two-dimensional closed flexible fish cage through the coupled HPC solution of the potential flow model and the two-dimensional beam theory; the numerical results indicate that the resonance response in the cage is predicted to be infinite, and the elasticity of the tank has a minor influence on the water-sloshing frequencies. However, because of the relatively large excitation amplitudes involved in the cage, the resonant sloshing may involve critical nonlinear free water surface effects [125]. Severe sloshing originates from the lowest sloshing mode with nonlinear energy transfer [159]. Furthermore, the sloshing response is predicted to be infinite at the eigenfrequencies of the cage system through the linear potential flow theory instead of the finite amplitude in reality; thus, viscous damping and nonlinear free water surface effects should be included [162]. Especially, the viscous boundary layer damping is nonnegligible when wave breaking occurs [125].

Faltinsen and Timokha [159] developed a nonlinear multimodal method for describing global sloshing loads in a time-efficient manner without considering wave breaking in deep water. The multimodal approach depicts the free water surface elevation using a Fourier series and expresses the velocity potential as a combination of the product of generalized coordinates and the linear eigenmodes of sloshing; the Bateman–Luke formulation is employed to transform the boundary value problem into a set of nonlinear ordinary differential equations governing the generalized coordinates of the free-surface elevation [125].

3.2.4. Bio-Effects

It is believed that the fish in the cage will weaken and redirect the water flow, and the motions of the fish may generate extra water flow (reviewed by Klebert et al. [140]). Inoue [163] pointed out that compared with external measurements, the water velocity measured in cages containing fish was reduced by 19 to 69%. In most hydrodynamic modeling of fish farming, the influence on the flow field due to the fish school has not been considered. The CFD simulation conducted by Winthereig-Rasmussen, et al. [164] indicated an over-prediction of 50% when compared to the field measurement data of flow, which may be due to ignoring the effect of fish inside the cages. Nonetheless, the school patterns, swimming speeds, depth distributions, and fish densities vary with environmental circumstances and internal motivational factors (reviewed by Oppedal, et al. [165]). Also, high biomass affects water flows in cages in several ways, and it is still unclear how organisms themselves affect fluid velocity, flow patterns, and turbulence [140]. Therefore, the effect of fish schooling on water flow is usually quantitatively expressed through experimental investigations.

On the other hand, the net cages immersed in seawater will cause biofouling layers to attach to the nets over time. The physical experiments on bio-fouled net panels conducted by Swift, et al. [166] showed that the biofouling layer will increase the drag resistance of the net to the flow, varying from 6 to 240% of the clean net values. The metrics currently employed to quantify net fouling are percent net aperture (PNA) and percent net occlusion (PNO) [140]. Quantitatively determining the correlation between these metrics and the drag resistance coefficient may be a potential approach to modeling the hydrodynamic effects of biofouling.

4. Structural Reliability and Modeling of the Fish Cage System

In the design of a fish cage, the most critical procedure is structural reliability to guarantee the safety of life and property. Under the hydrodynamic action, excessive deformation or fatigue caused by long-term vibrations will lead to the failure of structural strength. A reliable quality of materials and facilities can avoid potential safety hazards to the aquaculture system. Moreover, a key part of sustainable development involves balancing safety and engineering costs.

For any rigid bodies or micro-elements of continuum structures, their time domain dynamic responses in fluids can be described by Newton’s second law:

$$\left(\mathbf{M}+\mathbf{m}\right)\ddot{\mathbf{x}}+\mathbf{B}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}={\mathbf{F}}_{\mathrm{e}\mathrm{x}\mathrm{c}}\left(t\right)$$

(30)

where M is the mass matrix, m is the added mass matrix, B is the damping matrix, K is the stiffness matrix, and F_exc is the external excitation load matrix. For any components of the fish cage, several analytical or numerical solutions have been developed for the above governing equations.

4.1. Floating Collar

The floating collar or frame has the function of supporting the cage and generating buoyancy. It is usually made of high-density polyethene (HDPE), steel, or light concrete according to different applications, such as flexible floating circular rings, hull-shaped rigid floaters, and polygon rigid frames. Regarding the floating collar made of elastic high-density polyethylene (HDPE), its motion can be decomposed into rigid body motions with six degrees of freedom (DOF) and mode-superposed elastic deformations.

Rigid body motions mainly dominate the dynamic response of a collar made of steel or light concrete. The rigid body motion of a floating structure includes three translational motions, namely surge, sway, and heave, and three rotational motions, namely roll, pitch, and yaw. Under linear wave excitation, Equation (29) can be transferred into the complex domain, and the time-dependence term can be eliminated. Then, the six DOF motions of the collar can be directly solved through the eigenfunction expansion approach, as in [138]. The shape of the floating collar may affect the wave excitation, but the added mass, damping, and response amplitude operator are not significantly different, according to the analytical solution in [138]. Some BEM software or codes (e.g., ANSYS-AQWA [167] or WAMIT [168]) also provide numerical solutions to the second-order wave responses of rigid bodies. If the cage is under the action of irregular waves, the motion of the floating collar satisfies the Gaussian distribution through time–frequency transformation [169].

For the flexible collar made of HDPE, besides the rigid body motion of floating collars in waves, their elastic deformation cannot be ignored. The collar is usually modeled as a slender ring, and the Morison equation has been applied for the hydrodynamic load evaluation. Dong, Hao, Zhao, Zong and Gui [133] regarded the circular floating collar as a slender curved beam and used Euler’s law of motions and elastic beam theory to solve its rigid displacement and elastic deformation, respectively. The elastic deformation solutions could be expanded in a series of eigenmodes. For such initial value problems, the Runge–Kutta method is a popular time-discretized method. Dong et al. [133] found that a small elastic deformation occurs when the mooring cable is assembled symmetrically along the incident wave direction. They suggested that the out-of-plane stiffness should be enhanced to diminish the deformation. The mechanism of damping was omitted for the initial vibration state of structures in the study of Dong et al. [133], so Zhao, Bai, Dong, Bi and Gui [135] introduced a Rayleigh damping model, assuming the structure to have a low velocity. They found that the undamped case was not consistent with the realistic system and that the structural deformation tended to be the same for arbitrarily different damping rates when the time was sufficiently long. However, the modal superposition method cannot solve the nonlinear dynamic response of the structure. Huang, Guo, Tao, Hu, Liu, Wang and Hao [134] established a discretized lumped mass model to obtain the nonlinear dynamic response of the floating collar. Also, four-node shell elements were used to model the floating collar to simulate a more realistic stress distribution in the FEM (finite element method) model of Zhao, et al. [170]. Huang et al. [134] demonstrated that the maximum deformation of the collar occurred when the wave propagated in the direction of the mooring line; furthermore, increasing the cross-section of the collar suppressed the maximum strain, but the effect of its circumference was relatively small.

Notably, the wave-scattering effect may not be neglected when the ratio of the collar diameter to the wavelength is over 0.2, as in [87]. Fu and Moan [136] established a coupled model connecting the FEM model of collars and the potential flow BEM model of waves in the frequency domain. The results suggested that the rigid motion mode would dominate the dynamic response of floating in head seas but that the flexibility of the collar would contribute more to the structural response in oblique seas [136]. Li, Faltinsen and Lugni [139] and Gharechae, Ketabdari, Kitazawa and Li [137] also coupled the curved beam model with the wavefield solutions solved by the boundary integral equation and only considered the vertical motion of the collar, but the latter directly provided the analytical solutions. The experimental and analytical investigation of Gharechae et al. [137] indicated that the upstream floaters in an array had more significant responses than a single floater in head sea waves. Nevertheless, a comparison between the experimental and numerical results indicated that the high-order wave effect cannot be ignored for the motion of the floating ring, especially for phenomena above the third-order that cannot be explained by the perturbation theory; thus, the N-S equation might be required [139].

Also, fatigue failure is important to the limiting state of the floating fish cage when subjected to long-term wave exposure. Static analysis, vibration analysis, and fatigue analysis using FEM serve as dependable approaches for assessing the strength and integrity of the floating system structure [171]. For instance, in the FEM case presented by Liu et al. [171], after 10⁸ cycles of 100 kN loads, the reliability of the floating collar was only 53.1%. The stress range of fatigue can also be fitted to its short-term and long-term distribution through statistical analysis. The results of Bai, Zhao, Dong and Bi [132] indicated that the fatigue estimations based on the short-term stress-generalized extreme value distribution and gamma distributions were more accurate than the Rayleigh and Weibull distributions. Furthermore, long-term distributions are more conservative in estimating the fatigue life of floater systems.

4.2. Net Chamber

The net structure can be made of nylon wires with or without nodes and steel wires without nodes, and the mesh patterns include diamond mesh and square mesh. In analysis, bending stiffness is usually considered negligible when deformed.

A crucial design indicator is to avoid the excessive volume reduction of the net chamber caused by deformation. The volume-reduction factor C_v was given by Chen et al. [99] to quantitatively express the volume variation of the net chamber:

$$C_v=\frac{V_{\mathrm{n}\mathrm{e}\mathrm{t}}}{V_{\mathrm{d}\mathrm{e}\mathrm{f}}}$$

(31)

where V_net is the undeformed net chamber volume, and V_def is the deformed net chamber volume. V_def can be calculated by the volume-division method from Huang, et al. [172] and Chen et al. [99], respectively. Through the experimental investigation of the geometry of fish cages in a constant and uniform current, Lader and Enerhaug [173] observed that the maximum transverse displacement of the net chamber occurs at the bottom. Different from the quasistatic deformation under a constant flow, the net will generate a frequency-dependent dynamic response under wave actions. In the analytical solution of Su et al. [78] and Mandal and Sahoo [79], the maximum deflection amplitude of the net chamber occurred near the top restraint because of more significant surface wave effects.

Adding bottom sinker weights can effectively suppress the deformation but will increase the net tension and fluid drag. Moe et al. [102] also identified the net seam at the bottom of the cage as a potential hazard area because the internal forces could reach the design capacity by a FEM analysis. Chen et al. [99] suggested that increasing the stiffness of the net twine rather than the diameter and adopting an elliptical cross-section cage and a uniform sinker weight distribution are both beneficial for suppressing the volume reduction of the cage. Regular cleaning of biofouling on the net is also necessary. Lader, et al. [174] considered it essential to develop an “early warning” system for detecting any significant deformation of fish cage nets.

The discretized models of fish cage nets mainly include the lumped-massed model (e.g., [76,99,101,106,117,128]) and the truss-element model (e.g., [80,97,102,103,104,105]). The former equates the knots and twines of the net as several massed nodes connected by massless springs (as shown in Figure 6a). The latter models the net as several truss elements by the FEM (as presented in Figure 6b). In the commercial software AquaSim, a four-node membrane element is adopted to model the net [175]. These discretized models are usually based on time domain analysis. Nevertheless, for these discretized approaches, a simplification of the calculation model relative to the original net pattern is usually necessary to avoid excessive computations, i.e., the mesh-group method. A detailed derivation of the mesh-grouping method can be found in [90]. In addition, this type of structural modeling is often coupled with CFD models to achieve FSI; thus, the amount of computation will also increase.

Some researchers (e.g., [78,79,148]) have also adopted continuum vibration equations to describe the net motion in analytical solutions, such as the lateral deflection equations of strings or elastic beams or the membrane vibration equation. However, it is not reasonable to model the net chamber as a whole beam or string. First, the cross-section of the cage is assumed to retain a constant circular shape, and second, the stress variation due to structural deformation is assumed to be ignored. Ma et al. [152,153] proposed to apply the shell-membrane theory to describe the displacement of the net chamber. The net is modeled as a perforated thin shell without bending stiffness (as illustrated in Figure 6c), and the governing equations of the structural displacement can be derived by analyzing the motion and elastic constitutive relationship of the micro-element segment. This type of structural modeling is commonly coupled with potential flow models based on eigenfunction expansion methods to achieve FSI. Because of the application of frequency domain analysis and analytical solutions, the response of the system can be predicted simply and quickly. The disadvantage is that the derivation of the analytical solutions usually requires the assumption that the structure satisfies small deformations and elastic constitutive relations.

In addition to the numerical model, machine learning is also an effective and accurate approach for predicting the dynamic behavior of aquaculture cage systems, such as the application of ANN by Zhao et al. [116]. For this ANN model in Zhao et al. [116], the training data were generated by a lumped-mass method based the numerical model named DUT-FlexSim developed by [76,82,83,84,96,117,118], and hazard factors such as wave conditions were fed into the input layer for predicting the response of the fish cage through the BP algorithm. However, the quality of the training data and the selection of model hyperparameters directly determine the performance of the ANN.

Also, it is essential to consider the fatigue failure of the net structure when exposed to long-term wave excitation. Thomassen and Leira [176] indicated that the Rayleigh distribution and the two-parameter Weibull distribution can achieve a reasonable fit for the stress range of the fish cage net.

4.3. Mooring System

The function of a mooring system is to restrict the movement of the cage. Davidson and Ringwood [177] presented categories of some commonly employed mooring systems:

• Spread moorings: catenary mooring, multi-catenary mooring, and taut-spread mooring;
• Single-point moorings: catenary anchor leg mooring (CALM) and single-anchor leg mooring (SALM).

The strength of the mooring cable must be guaranteed to avoid cable breakage; therefore, it is critical to perform a reliability analysis of the uncertainty quantification system for the allowable strength [178]. The reliability assessment for the fish cage with one broken mooring line by Hou, et al. [179] indicated that the failure of one of the mooring lines has a substantial impact on the likelihood of failure for the other mooring lines, and when the gradual deterioration of the mooring system is not taken into account, conventional reliability methods tend to overestimate the failure probability of the mooring system. The numerical studies and fatigue reliability analysis by Hou, et al. [180] also indicated that zero initial preload (when the mooring cable is in a relaxed state initially) can effectively improve the fatigue reliability of the mooring cables.

In addition to the excitation caused by the drag and inertial effect of the flow, the mooring cables can also generate vortex-induced responses. When the vortex-release frequency is close to the natural frequency of the structure, the lock-in phenomenon will occur, where the structure responds violently. The condition for vortex-included resonance to occur is usually when the Strouhal number (the reciprocal form of the KC number) is approximately 0.16 [181]. Moreover, the negligence of the effect of embedded chains in the seabed may also lead to unreasonable predictions of the static and dynamic response of the mooring system [182]. Nonetheless, from the experimental investigations by He, et al. [183], the measured mooring forces when live fish are present in the currents exceed those in fish-absent conditions by a range of 10% to 28%, mainly as a result of the interplay between the fish and the net enclosure.

Chu et al. [3] categorized mooring analysis methods into the quasi-static method and the dynamic method. In the former, the tension in mooring lines is statically calculated, with the tension in each mooring line computed based on the maximum offset for each design condition [3]. The dynamic method directly simulates the frequency-domain or time-domain response of the mooring system. In most analyses of floating structures, the mooring system is equivalent to linear springs, which able to be simply included in the rigid body motion equation of the floating structure as constant stiffnesses. Liu et al. [110] and Selvan et al. [148] tried to calculate the mooring-restoring forces on fish cages as provided by springs of constant stiffness along the transverse direction in their analytical solutions as well. Another method is to directly model the mooring cables in the FEM, where the drag forces induced by waves and currents and the motions of the mooring lines are considered, and the tension information of the mooring line is directly given [184]. By comparing the two methods, Kim et al. [184] demonstrated that the FEM is more realistic because the complex nonlinear behavior can be solved, but this is time-consuming.

The reliability of fish cage mooring lines under the ultimate limit state needs to be determined using the generalized probability density evolution method (GPDEM) to assess the evolution of the probability information of mooring tension [178]. For fish cages with broken mooring lines, the redistribution of tension response in the mooring system after line failure can be analyzed through numerical simulations, and the reliability of other mooring lines under the ultimate limit state can be evaluated [179]. Moreover, a transient method based on the S-N curve approach was employed to perform a fatigue reliability analysis of the shackle chains of fish cages by Hou et al. (2018).

4.4. Coupling of Structural Components

The abovementioned review only focuses on the studies of single structural components in aquaculture systems. However, the fish cage system is a comprehensive system that combines the floating collar, net chamber, and mooring system, and it is usually designed as a multi-cage system. The interaction among the structural components is a noteworthy issue.

The numerical simulations of Li et al. [112] indicated that displacements of the floating collar coupled with a net chamber are obviously different from the results obtained with only the floater. In particular, the interaction between the floating collar and the net has a large impact on the horizontal motions of the system. For frame-type fish net cages, the existence of frame cylinders leads to a 9.2% increase in the fluid drag coefficient when the solidity ratio S_n > 0.347 [185]. The maximum local stress occurs where the net connects to the floater; thus, this is a critical element of the structural design of fish cages [174]. The gravity cage system exhibits small responses to high frequencies, which is characteristic of highly damped systems [118]. Furthermore, the stiffness of the heave, roll, and pitch provided by the mooring system is small compared with the hydrostatic stiffness, but the coupling stiffness of surge–pitch or sway–roll is considerable when the weight of the mooring cable or the wave drift force is not negligible [184]. Currently, Martin et al. [129] proposed a complete numerical framework in Reef3D for simulating the dynamics of open-ocean aquaculture structures in viscous fluids, and a case study applied this model to analyze a submersible steel-frame fish farm in [130].

In multi-cage systems, each cage is not in an isolated state but rather has a mutual interaction. For an array of net cages under wave actions, the maximum response does not necessarily occur in the upstream direction, and their frequent rolling and swaying will cause the mooring tension to be redistributed [186]. The numerical simulations of Xu et al. [83,84] revealed that the dynamic response of each cage has a significant phase difference and that the mooring tension and cage volume reduction of multiple-cage systems is much greater than those in single-cage systems and is related to the number of cages. In most cases, two adjacent ropes do not pull the cage together because the mooring force in one is usually much greater than that in the other; thus, two adjacent ropes are not always effective at reducing stress concentrations [111].

5. Geotechnical Responses of Mooring Anchors

Floating platforms are secured in position through a network of mooring lines, typically consisting of chains and anchors. These anchors come in primarily three variations: traditional drag embedment anchors, suction caisson anchors, and gravity anchors, all with dimensions of the order of approximately 10 m [187].

For offshore fish farms, suction anchors have the advantages of a simple design, strong vertical load-bearing capacity, and easy installation for the mooring system [188]. However, it is necessary to consider the interaction between embedded chains and the soil to analyze the structural response of the fish farm system systematically [182]. According to Hou et al. [182], the normal and tangential soil resistance Q and F per unit length of the embedded chain can be expressed as follows:

$$\begin{array}{c}Q=E_{n}d{N}_c\left(s_{u0}-\nu z\right)\\ F=E_{\tau }d\left(s_{u0}-\nu z\right)\end{array}$$

(32)

where E_n and E_τ are the normal and tangential multipliers for the effective width; d is the diameter of the embedded chain; N_c is the bearing capacity factor; s_u0 is the soil shear strength at the seabed; ν is the soil shear strength gradient; z is the embedded depth level from the seabed downwards. The numerical analysis using the lumped-mass model by Hou et al. [182] indicated that for large-scaled cage structures, the influence of embedded chains accounts for a small proportion; nonetheless, in a multi-cage system, the relationship between the soil and the anchors that are embedded in it may become more pronounced, potentially leading to trenching. This can consequently lead to a reduction in both the shear strength of the soil and the load-bearing capacity of the embedded anchors [182].

Geotechnical responses on the seabed will also affect the stability of the mooring anchor chain, which mainly includes soil liquefaction, seabed trenches, and scouring.

The primary focus in the planning of offshore foundations in Northern European regions has traditionally revolved around addressing environmental loads. Conversely, in regions with a high level of seismic activity, such as East Asia and Southern Europe, earthquake-related forces and the potential for liquefaction in loosely packed silt and sand could assume a critical role in influencing the stability and dependability of these structures [189]. Through the utilization of the SANISAND constitutive model for saturated sand in the FLAC3D software, Esfeh and Kaynia [189] determined that the anchor piles could undergo enduring lateral displacement and tilting as a consequence of soil liquefaction brought about by the combined influence of static mooring loads and seismic activity in nonlinear dynamic analysis.

The reciprocating motion of mooring lines may induce seabed trenches near the anchor, thereby reducing the anchoring capability [190]. Wang et al. [190] proposed a method that takes into account the dynamic behavior of mooring lines to predict the trench profile; this predictive model consists of two simulation modules. The first one is the mooring line dynamics module, which considers the dynamic interaction between the mooring lines and the seabed. The other module is the seabed excavation module, including assessments of tunnel stability and soil excavation resulting from it; the model structure shows that the main effect of the formed trenches is that the load direction at the pad eye tends to be horizontal.

Furthermore, it is essential to consider the erosion caused by waves and currents around anchor piles. Evaluating scour can be carried out using two potential methods: CFD calculations and hydraulic scale model tests, as discussed by Sumer and Kirca [187]. Typically, local scour occurs either in clear-water conditions or live-bed regimes, as outlined by Raudkivi and Ettema [191]. For wave-induced scour, the critical factor is vortex shedding at the rear of the pile, while steady flow-induced scour is primarily influenced by the horseshoe vortex formation in front of the pile and the compression of streamlines at the pile’s sides, as indicated by Yang et al. [192]. The depth of scour depends on several factors, including water particle orbit velocity, KC number, Shields number, pile roughness, and Reynolds number, but not the relative directions of waves and currents, as reviewed by Yang et al. [192]. Sumer and Kirca [187] provided various formulations for estimating scour around structures, including scour depth and time scale, under the assumption that the structure extends to significant depths within the sedimentary bed.

6. Conclusions

This article reviews the analysis methods and modeling techniques relevant to the design of offshore aquaculture infrastructures. Some theories and techniques borrowed from other marine or maritime engineering fields are also discussed. The focus is on the reliability analysis of the offshore structures, as it is crucial to ensure the serviceability of aquaculture facilities during the design and operational phases to avoid property and personnel safety issues. From this perspective, the article reviews and discusses the statistical and predictive analysis of extreme sea conditions during the facility’s design and operational phases; methods for assessing environmental loads and relevant hydrodynamic analysis for both near-field and far-field conditions; structural reliability and dynamic response analysis; and the geotechnical response of the seabed for anchoring mooring systems.

The abovementioned review reveals that there is a wealth of theoretical and technical support for the reliability analysis of offshore aquaculture systems, which can be drawn from aquatic engineering or other marine engineering fields. The biggest challenge still lies in how to devise a solution that reasonably and efficiently describes the impact of the fish cage on the flow field, especially considering the pore-blocking effect of the net. The hydrodynamic modeling based on hybrid techniques (combining the N-S equation and potential flow theory) may be a reasonable solution. This will accelerate the industrial application of hydrodynamic modeling for aquaculture equipment. Furthermore, the fluid–structure interaction simulation of the fish cage remains challenging due to the complexity of modeling fish cage nets in fluids.

By discussing existing analysis methods and their applicability, this paper aims to assist engineers in the development and design of offshore fish farming infrastructure, thereby contributing to the design standards for offshore aquaculture. Currently, the offshore aquaculture industry is at the forefront and still holds significant growth potential. This has also spurred the development of related emerging concepts and technologies such as integrated offshore platforms and digital twin systems. These advancements aim to further enhance production efficiency, sustainability, and environmental friendliness. Furthermore, the development of data-driven technology has the potential to have a significant impact on offshore aquaculture, including monitoring, optimization, decision making, and more. This will be a powerful tool applied to address challenges in the design and application of marine aquaculture systems. These future scopes are expected to bring more opportunities to the aquaculture industry and will continue to be explored and researched.