The characteristic parameters of fish models applied in the simulation.
Abstract
In net cage hydrodynamic analysis, drag force of net is dependent on the physical dimensions of the net cage, the Solidity ratio, the Reynolds number and the projected area of the net, which is illustrated in numerous previous researches. However, rare studies attempt to investigate the effect of fish behavior. Thus a netfluid interaction model and a simplified fish model were proposed for analyzing the effects of fish behavior on the net cage. A series of physical model tests were conducted to validate the numerical model, which indicates models can simulate the stocked net cage in the current accurately. The simulation results indicate that circular movement of fish leads to a low pressure zone at the center of net cage, which causes a strong vertical flow along the center line of the net cage. The drag force on the net cage is significantly decreased with the increasing fish stocking density.
Keywords
 SST komega model
 finite element method
 net cage
 fish behavior
1. Introduction
A slew of researches show that the drag force (
In this chapter, two types of fish distribution patterns were analyzed: night distribution, fish is mainly gathered in the water surface and fish density is decreased with increasing water depth; and day distribution, most fishes perched near to the bottom of net cage and fish density is increased with increasing water depth. And two kinds of fish group structures are considered: circular (
Figure 1
) = polarized swimming in a circular movement; oncurrent (
Figure 1
) = swimming toward the current without forward movement. The latest progress of net cage numerical simulation method by our research group will be introduced in detail. The contents of the progresses include the following parts:
2. Numerical modeling approach
The numerical modeling for analyzing the flow field through the stocked net cage and the deformation of net cage is to combine the
2.1. Flow around net cage
The
2.1.1. The kω SST model
The governing equations describing the
Continuity equation:
Momentum equation:
where
The
2.1.2. Boundary conditions
The boundaries of the numerical flume are constituted by a free surface, three wall surfaces, an inlet surface and an outlet surface. The fsi_wall is modeled as a nonslip wall with considering the roughness of the net twine. To consider the roughness of the surface of the actual net twine, the physical roughness height
The governing equations for describing the flow field are solved by a threedimensional pressurebased NavierStokes solver. The SIMPLEC algorithms in [4] are employed to treat the pressurevelocity coupling. The discretization scheme for pressure, momentum, turbulent kinetic energy is carried out using a second order upwind scheme.
2.2. Structural model
The threedimensional net cage is divided into discrete elements, which the net mesh is modeled as the bar element connected with spherical hinge, as shown in Figure 2 . The connection force between the net bar and the spherical hinges including two parts: the tangential force and the normal force.
2.2.1. Large deformation nonlinear structure model
The flexible fish net will experience geometricnonlinear deformation under the action of the hydrodynamic loads, thus the LDNS model (in [5]) is applied here to describe the deformation of flexible fish net. The connection constraints are given as follows:
where a local Cartesian coordinate,
where
2.2.2. Load transfer and grid generation
To transfer pressure data from the flow model to the structural model, all nodes on the net twines surface are projected to the fsi_wall element in the flow model according to the mapping rule—projecting each nodes in the target surface normal to the nearest mesh face in the source surface as shown in Figure 3 . The transferred variable is linearly interpolated on the source face by linear twodimensional shape function.
2.3. Fish model
As shown in Figure 4 , the fish model includes the fish body and the fish tail. The fish body suffers the drag force and the fish tail creates the propulsion force. The resulting force of the model includes the propulsion force and the drag force, given as follows:
where the
where
2.4. Grid generation
An example of computational grids for a plane net is shown in Figure 5(a) . Tetrahedral grids exist in the majority computational area, and pentahedral prism grids are adopted to refine the meshes near the fluidsolid boundary. Ten boundary mesh layers are adopted to generate boundary grids around fish model as shown in Figure 5(b) . Bar elements are divided by hexahedral grids with uniform grid and spherical hinges are divided using tetrahedral grids as shown in Figure 5(c) .
3. Experimental validations of model
For fish model validation, a rigid fish model with different behaviors is simulated and the results are compared with the formula proposed in [6]. To validate the net cage model, a series of net cage experiment are conducted a flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. The flume is 22 m long, 0.45 m wide and 0.4 m water depth in the tests.
3.1. Validation of the fish model
A single rigid fish model is modeled in the flow as shown in Figure 6 . The size and weight of fish are same as small fish model (in Table 1 ). The specified wall shear stress in Eqs. (7) and (8), is used to simulate different fish behaviors. There are three operating variables: fish yaw angle, flow velocity and fish swimming velocity, used to investigate the effects of the fish behavior on the flow field and the net deformation. Thus three experiment groups are produced to validate fish model (referring to Table 2 ). Figure 7 shows that simulation is in good agreement with Eq. (9).
Items  Small fish  Large fish  

Real fish  Fish model  Real fish  Fish model  
Fish length (cm)  40  6.7  50  8.5 
Fish surface area (cm^{2})  –  1.235  –  2.112 
No. 



Yaw (rad) 


1  0  2.0, 3.0, 4.0, 6.0, 8.0, 10.0  2.0, 3.0, 4.0, 6.0, 8.0, 10.0  0  0 
2  3.0, 6.0, 6.2  0  3.0, 6.0, 6.2  0  0 
3  3.0  6.0  –  0 
– 
3.2. Validation of the net cage
To validate the deformation of net cage in the steady flow, a net cage model is tested in the flume. The diameter of net cage is 0.254 m and the net mesh is square with 20.0 mm mesh size and 1.2 mm twine diameter. The top of the net is mounted on a top steel ring shown in Figure 8 .
Figure 9 indicates that the cage deformation simulation is close to the experiment. Figure 10 indicates the drag force on the net cage from simulation is in good agreement with experiment and the relative error is less than 6.82%.
4. The numerical simulation of the stocked net cage in steady flow
To investigate the effect of fish behavior on the net cage, a 2.5 m long, 2.5 m wide and 0.7 m depth numerical water flume is adopted here, and the stocked net cage model with 0.35 m outer diameter and 0.35 m height is located at the center of the flume. Sinkers with 1 N weight are applied to maintain the shape of net cage. The net is mounted as square meshes, in which the twine diameter is 2.6 mm and the mesh bar length is 20 mm. The detailed experiment setup is shown in Table 3 .
No. 

Fish model 

Distribution  GS 



1  8  Small  0  Night/day  Circle  0.45 & 0.90 0.45 
0 0.45 
2  0 8 
– Small 
0.45  – Night/day 
– Circle 
– 0.90 
– 0 
3  0 8 
– Small 
0.93  – Square/rhombus 
– Oncurrent 
– 0.93 
– 0 
4  8 16 
Large  0.45  Night/day  Circle  0.90  0 
5  24, 32, 40  Large  0.45  Uniform  Circle  0.90  0 
4.1. Circular movement of fish in the still water
The effect of fish swimming on the flow pattern around the net cage in the still water is analyzed, in EG (experiment group) 1 in Table 3 . Figure 11 shows the flow pattern in the net cage for different fish distributions with 0.45 BL/s swimming speed. According to Newton’s second law, the centripetal force acting on fish need be balanced by the hydrodynamic force which pushes water away from fish. Thus the flow velocity near the inner boundary of fish is greater than that of the outer boundary in the horizontal plane, leading to a low pressure area in the center of the rotational movement of the fish. For the day distribution, water is pulled from above and below the fish swimming depth; for the night distribution, water is only pulled from below the fish swimming depth.
4.2. Effect of fish distribution on flow field and drag force
The effects of fish distribution on the flow field around the net cage and the drag force of the net cage are analyzed in EG 2 and 3. Figure 12 shows the flow field around the net cage for the low current velocity case (EG 3) and the fish motion in high current case (EG 4) has little influence on the downstream. Figure 13 shows the circular movement of fish has larger influence on the drag force acting on the net cage and the influence of oncurrent movement is little.
5. Conclusions
A netfluid interaction model and a simplified fish model are proposed for analyzing the effects of fish behavior on the flow field around the net cage and the deformation of the net cage. And the following conclusions can be drawn from the case study:
The fish circular movement around the net cage can produce a low pressure zone at the center of the net cage, causing a vertical water exchange along the center line of the net cage.
The circular movement of fish has significant influence on the downstream wake, and especially the lowvelocity zone. While the oncurrent movements of fishes affect little.
The drag force on the net cage is significantly decreased with the increasing fish stocking density. Little differences in the fish distribution are observed.
Acknowledgments
This work was financially supported by the National Natural Science Foundation (NSFC) Projects No. 51239002, 51409037, 51579037, and 51221961, China Postdoctoral Science Foundation (No. 2014M560211 and No. 2015T80254), the Fundamental Research Funds for the Central Universities No. DUT16RC(4)25 and Cultivation plan for young agriculture science and technology innovation talents of Liaoning province (No. 2014008).
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