Experimental and Numerical Investigations of the Hydrodynamic Characteristics, Twine Deformation, and Flow Field Around the Netting Structure Composed of Two Types of Twine Materials for Midwater Trawls

THIERRY Bruno Nyatchouba Nsangue, TANG Hao, 2), 3), 4), 5), *, ACHILE Njomoue Pandong,XU Liuxiong, 2), 3), 4), 5), ZHOU Cheng, 2), 3), 4), 5), and HU Fuxiang

Experimental and Numerical Investigations of the Hydrodynamic Characteristics, Twine Deformation, and Flow Field Around the Netting Structure Composed of Two Types of Twine Materials for Midwater Trawls

THIERRY Bruno Nyatchouba Nsangue1), TANG Hao1), 2), 3), 4), 5), *, ACHILE Njomoue Pandong6),XU Liuxiong1), 2), 3), 4), 5), ZHOU Cheng1), 2), 3), 4), 5), and HU Fuxiang7)

1),,201306,2),201306,3),,201306,4),,,201306,5),,201306,6),,,2701,7),,,108-8477,

Nettings are complex flexible structures used in various fisheries. Understanding the hydrodynamic characteristics, de- formation, and the flow field around nettings is important to design successful fishing gear. This study investigated the hydrodynamic characteristics and deformation of five nettings made of polyethylene and nylon materials in different attack angles through numeri- cal simulation and physical model experiment. The numerical model was based on the one-way coupling between computational fluid dynamics (CFD) and large deflection nonlinear structural models. Navier-Stokes equations were solved using the finite volume ap- proach, the flow was described using the-shear stress turbulent model, and the large deflection structural dynamic equation was derived using a finite element approach to understand the netting deformation and nodal displacement.The porous media model was chosen to model the nettings in the CFD solver. Numerical data were compared with the experimental results of the physical model to validate the numerical models. Results showed that the numerical data were compatible with the experimental data with an average relative error of 2.34%, 3.40%, 6.50%, and 5.80% in the normal drag coefficients, parallel drag coefficients, inclined drag coefficients, and inclined lift coefficients, respectively. The hydrodynamic forces of the polyethylene and nylon nettings decreased by approxi- mately 52.56% and 66.66%, respectively, with decreasing net solidity. The drag and lift coefficients of the nylon netting were appro- ximately 17.15% and 6.72% lower than those of the polyethylene netting. A spatial development of turbulent flow occurred around the netting because of the netting wake. However, the flow velocity reduction downstream from the netting in the wake region in- creased with increasing attack angle and net solidity. In addition, the deformation, stress, and strain on each netting increased with in- creasing solidity ratio.

netting;-shear stress turbulent (SST) model; porous media model; large deflection nonlinear structural model;hydrodynamic characteristics

1 Introduction

Understanding the hydrodynamic characteristics of net- tings and the flow behavior around them is important to improve the design, performance, and reliability of fish- ing gear, such as net cage, trawl net, and purse seine. The netting structure, which is a main component in the net pro-duction industry, should be improved to minimize the drag of fishing gear. The hydrodynamic properties of the net- ting and the flow characteristics around it influence fish- ing gear efficiency with reduction in hydrodynamic forces, which are two major reasons for tremendous improvementsin fishing gear R&D worldwide (Tang., 2017a; Thierry., 2020a, 2020b).

Sea trials and flume tank experiments are widely used in designing fishing gear, but they are expensive in terms of resources and time (Thierry., 2020a). Numerical simulation allows the analysis of different designs at an early stage and the preselection of the best design to be test-ed experimentally, hence saving resources. This approach is effective for interpreting and predicting fishing net per- formance (Lee., 2005, 2011; Priour, 2009; Li., 2015; Thierry., 2019; Zou., 2020). The porous media method is widely used with improved computer ability in numerical simulation. The interaction betweenthe fluid and fishing gear was investigated between po- rous media and lumped-mass models. By contrast, velo- city mitigation (Løland, 1991; Zhan., 2006) is too simple to calculate the flow field. Patursson. (2010) used the porous media method to analyze the flow field distribution around the mesh at different attack angles and to determine the drag coefficient in the porous media mo- del. Zhao. (2013a, 2013b), Bi. (2014a, 2014b, 2015), and Bi and Xu (2018) also proposed a numerical approach based on the joint use of the porous media mo- del and the lumped mass model to solve the fluid-struc- ture interaction problem between the flow and flexiblenettings based on the iteration concept, leading to a steady flow field around the flexible netting. Chen and Christen- sen (2016) developed a numerical model based on Open FOAM to study the steady current flow through planar net panels and circular fish cages. Yao. (2016) proposed a hybrid volume method based on CFD simulation to ana- lyze the interaction between the netting mesh and the sur- rounding flow field. This proposed method was used to cal- culate the interaction between the fluid and net cage un- der large deformation conditions. Devilliers. (2016) implemented a solver for the fluid-structure interaction analysis of current flow through netting structures and pre- dicted the netting deformation, approximating the net as a set of rigid bars. The static and dynamic behavior of the netting should be analyzed based on reliable estimates of its hydrodynamic coefficients. Chen and Christensen (2017) developed a numerical model for the fluid-structure in- teraction analysis of flow around a net panel and an aqua- culture net cage based on the coupling between the porous media model and the lumped mass structural model in OpenFOAM. Bi. (2017) developed a numerical mo- del to evaluate the effect of cylindrical cruciform patterns on fluid flow and examined the effect of mesh orientation on the netting drag and flow around meshes T0 and T45. Martin. (2018) implemented a numerical model for the determination of the deformed shape of nets fixed in a stiff frame based on discretization of the net in mass po- ints connected to straight bars and provided a detailed de- rivation of the tension element method for physical net structure. Martin. (2020) developed a numerical mo- del based on the Lagrangian approach for the coupled si- mulation of fixed net structures in a Eulerian fluid model by solving the Reynolds-averaged Navier-Stokes (RANS) equations in an Eulerian fluid domain. They compared the loads and velocity reductions behind the net with the avail- able measurements and demonstrated the superior perfor- mance of the proposed model over other existing approa- ches for various applications. Recently, Tu. (2020)have used a 3D multirelaxation time lattice Boltzmann mo- del to simulate the flow field around a planar net in a con- stant current. They demonstrated that the flow velocity at- tenuation is mainly related to the net solidity of the planar net with a high velocity-drop in response to high net so- lidity.

The drag and lift coefficients can be determined either by examining the local flow velocity to the twines (Mori- son., 1950; Løland, 1991) and obtaining values corre- sponding to the local Reynolds number, or by using uni- form hydrodynamic coefficients applied to every part of nets on the basis of experimental data (Balash., 2009).In the latter case, Kawakami (1964) analyzed the resistance to currents and proposed a simple predicted formula for the drag coefficient. Aarsnes. (1990) conducted tested net- ting panels and cage systems and developed formulas for drag and lift forces owing to the constant current and mo- deled the effect of flow direction. Zhan. (2006) ex- perimentally examined the effect of Reynolds number, mesh pattern, and flow direction on the drag force of planar and circular nettings. Kristiansen and Faltinsen (2012) proposed a screen-type force model for the viscous hydrodynamic load on net, in which the net is divided into a slew of flat net panels, or screens and used the curve fitting method to obtain the drag and lift coefficients of the netting as a func- tion of solidity ratio, attack angle, and the drag coefficient of twines of the netting in steady current, depending on the Reynolds number. Tang. (2018) studied the differ- ences in the hydrodynamic force coefficients between knot- less and knotted polyamide-netting panels and found that the hydrodynamic coefficient of the knotted netting is 1.23– 1.35 times greater than that of the knotless netting. They used the curve fitting method to obtain the drag and lift coefficients of the polyamide netting. Tang. (2019) investigated the hydrodynamic characteristics of netting design with various twine materials at small attack angles and found that the drag coefficients of nylon knotless net- ting are approximately 8.4% and 7% lower than those of knotless polyethylene netting and knotless polyester net- ting, respectively. Kebede. (2020) experimentally in- vestigated the hydrodynamic properties of helix ropes com- pared with conventional polyethylene and nylon ropes, de- monstrating that helix ropes produce increased lifting force (L) for conventional ropes of the same diameter, verifying their description of self-spreading ropes.

The effect of flow field around the netting on the hydro-dynamic characteristics and the netting deformation is still unclear. Only the effect of attack angle on the flow field around the netting has been previously studied. In addi- tion, previous numerical studies did not consider netting structures, such as twine diameter, mesh size, twine mate-rials, and mesh opening, in the design and deformation du- ring flow field simulation. Therefore, the effect of net so- lidity and twine material on the netting deformation was not investigated previously. In addition, the remaining is- sue for direct application of the porous media model on the flow around the netting is finding the porous resistance coefficients in the Darcy-Forchheimer equation, and the fitting procedure requires available measured drag and lift forces for each individual netting for different incoming velocities and attack angles, limiting its application in prac- tical design. Therefore, the numerical model considering the fluid-structure interaction using the porous media mo- del and the nonlinear finite element model must be imple- mented in one-way coupling for the 3D simulation of the netting. An alternative approach must be provided to calcu- late the porous resistance coefficients, expressing them as a function of net solidity, attack angle, and Reynolds num- ber.

This study aimed to analyze the effects of net solidity, twine material, and attack angle on the hydrodynamic per-formance and deformation of the netting and the flow field around it. A numerical model was developed based on the combination of the-shear stress turbulent (SST) mo- del and porous media model with a large deflection non- linear finite element model for flexible netting. The results obtained from the numerical model were validated with the experimental data obtained from the flume tank. These findings are expected to improve the fishing gear hydro- dynamic performance and the numerical study of netting.

2 Materials and Methods

The main concept of the proposed numerical model was to combine the porous media model with the-SST mo- del and the large deflection nonlinear structural model to simulate the interaction between fluid and netting using one-way coupling techniques. The netting in the fluid sol- ver was simplified entirely as a porous media model. The six-degree-of-freedom beam element was applied to simu- late the netting, in which the beam element was connect- ed with rotable joints. A large deflection nonlinear model based on the finite element method was applied to analyze the deformation of the flexible netting in the current using ANSYS static structural-mechanical software. The-SST model was used to describe the flow field around the net- ting using ANSYS Fluent.

2.1 Numerical Fluid Flow Model

Fig.1 shows the netting structure with the diamond meshused in the numerical model. Different nettings used in the numerical simulation are a series of separate cylinders connected to each other for flow obstruction. In the fluid domain, the netting structure is represented by a sheet of porous media.

Fig.1 Diamond netting meshes applied in the present nu- merical model (a) Net-1, (b) Net-5, and (c) Net-4.

2.1.1 Governing equations and-turbulence model

In this study, a computational fluid dynamics (CFD)-based scheme was used to solve the interaction between the flow and netting in the numerical water tank. The govern- ing equations of the flow model were solved using the fi- nite volume method together with the SST-and porous media models. The motion of the incompressible flow was described by the system of RANS equations with Reynold stress approximated by the eddy viscosity model.

Since the fluid in this model is a single component and does not involve heat exchange, the law of mass conserva-tion and momentum conservation is needed to describe the fluid motion. These equations can be expressed in the ten- sor form and Cartesian coordinate as follows:

whereis the time average pressure andis the turbulent kinetic energy.

In our simulation, the SST-model (Menter, 1994) was used to model the Reynolds stress. This model is a com- bination of two transport turbulence equations combining the original-and Wilcox-models (Menter, 1994). The use of-formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sublayer. Hence, the SST-model can be used as a low-Reynolds turbulence model without any extra damping functions.This model formu- lation also switches to a-behavior in the free stream, thereby avoiding the oversensitivity of the-model to the inlet-stream turbulence properties.In addition, this mo-del produces slightly high turbulence levels in regions with large normal strain, such as regions with strong accelera- tion. The SST-model solves two transport equations, the turbulent kinetic energyand the turbulent dissipa- tion rate, as follows:

The function and constant are:

is the distance to the nearest wall. In the near wall region,1=1, whereas it goes to zero in the outer region.

The empirical constants of the SST-model are as fol- lows:

The numerical model of the governing equation was sol- ved using a 3D pressure-based Navier-Stokes solver. The semi-implicit method for pressure-linked equations-consis-tent scheme algorithms (Doormaal and Raithby, 1984) wasused to account for pressure-velocity coupling. The discre-tization scheme for pressure was conducted using PRESTO (PREssure STaggering Option), the discretization scheme for momentum and intermittency was conducted using asecond-order upwind scheme, and the transient formulation was conducted using second-order implicit. The simulation is defined as having reached a converged solution when all residuals reduce to less than 0.0001. For the time, the transient method with a time step of 0.001s was used du- ring calculation.

2.1.2 Description of porous media model

The porous media fluid model proposed by Patursson (2008) and Zhao. (2013a, 2013b) was introduced in this study to simulate the netting. To determine the hydro- dynamic force of the porous structure, the source term re- presenting the porous media resistance used in Eq. (2) canbe described by the Darcy-Forchheimer equation as follows:

whereDrepresents the porous viscous resistance coeffi- cient matrix andCis the porous inertial resistance coef- ficient matrix. Porous media resistance coefficients are de- fined as follows:

whereDandCrepresent the normal viscous and inertial resistance coefficients, respectively,DandCrepresent tan- gential viscous and inertial resistance coefficients, respec- tively.

In Eq. (12), the first term proposed in Darcy’s law (1856) was used for low Reynolds numbers. In this case, the vis- cous forces are dominant, and a linear relationship was ob- tained between the flow velocity and pressure drop through the porous material. However, when the flow velocity in- creases, the relationship between the pressure drop and ve- locity becomes nonlinear. The second term by Forchhei- mer (1901) was added to model this regime.

When the drag and lift coefficients were obtained, the drag and lift forces acting on a netting can be calculated from the Morison equation:

whereCis the drag coefficient,Cis the lift coefficient,is the netting area, andis the angle between the flow direction and the netting .

According to Bi and Xu (2018), the porous media re- sistance forces can be obtained by the following equa- tions:

whereis the porous media thickness. During our study, we used five values of porous media thickness 20, 30, 40, 50, and 60mm, to confirm the research of Zhao. (2013b) that porous media thickness does not affect numerical si- mulation results.

In our study, porous media coefficients were calculated from the curve fit between the hydrodynamic forces of net- ting obtained experimentally and corresponding current ve- locities using the least squares method.

The porous media coefficients for different nettings are presented in Fig.2. The normal porous media resistance co-efficients (DandC) for each netting increase with increas- ing attack angle. On average, the normal porous resistance coefficients of netting 3 are 53.70% and 3.25% greater than those of nets 1 and 2, respectively. Notably, the porous me- dia resistance coefficients of the polyethylene nettings are approximately 80.1% greater than those of the nylon net- tings.

Fig.2 Porous media resistance coefficients corresponding to different nettings and different attack angles.

Dwas neglected to propose a new formula to predict the porosity coefficients because, according to the experimen-tal investigation of Patursson (2008) and Bouhoubeiny (2012), the flow developing around the fishing net is turbulent flow. Therefore, Eq. (12) becomes:

Control volume analysis based on Newton’s second law was introduced to calculate the force acting on the fishing net in the current (Jensen., 2014). Thus, on the basis of the conservation of linear momentum in control volume,the force acting on the fluid from the porous media is equalto the momentum loss in the control volume. According to the method proposed by Paterson (2008), the force can be defined as:

whereis the CFD volume of the porous media zone.

The forces acting on the porous media with different attack anglescan be obtained as follows:

These forces should be equal to the netting force (Eq. (14)). Therefore, substituting the expression of the netting force and the force acting on the porous media produces the following relationship forCand C:

whereCandCdepend on the attack angle and net so- lidity.

2.2 Large Deflection Nonlinear Finite Element Method

2.2.1 Governing equation

Under the action of the current, the netting structure is considered as a large deflection nonlinear structure (Tang., 2017b). Thus, the equilibrium equation of the flexi- ble netting structure in the geometric nonlinear theory was obtained as a function of the netting shape after deforma- tion in the current. As shown in Fig.3, the forces acting on the netting structure are hydrodynamic forces (Fand F), the gravity force of the structureF, and buoyancy forceF. Thus, the basic governing equation of the nonlinear fi- nite element model applied to solve the large deflection nonlinear structural model (refer to Bathe. (1975) andTang. (2017b)) is described as follows:

where() is the displacement vector of the net structure and() is the nonlinear structural stiffness matrix, which is related to the unknown displacement vector(). Accord- ing to the virtual work principle, the governing equation of the finite element model for the netting structure can be gi- ven as follows:

where

The linear stain displacement transformation is defined as:

where

The nonlinear strain displacement transformation is define as:

2.2.2 Load transfer

To transfer the pressure data from the flow model to the structural model of the netting structure, all nodes on the netting surface were projected to the face of fluid-net- ting interaction wall element in the flow model according to the mapping rule-projecting each nodes in the target sur- face (netting boundary in the structural model) normal to the nearest mesh face in the source surface (fluid-netting interaction wall in the flow model) as shown in Fig.3b.

The pressure data were transferred from the flow model to the structural model of the netting structure as follows. All nodes on the netting surface were projected to the faceof the fluid-netting interaction wall element in the flow mo- del according to the mapping rule, namely, projecting each node in the target surface (netting boundary in the struc- tural model) normal to the nearest mesh face in the source surface (fluid-netting interaction wall in the flow model), as shown in Fig.3b. The transferred variableL(line loads) was linearly interpolated on the netting face by integrat- ing the pressure on the fluid-structure interface (FSI) along the outer perimeter of the netting as follows:

where Pnis the surface pressure on the fluid-structure in- terface and Φis the outer perimeter of the netting. The line load acting on each netting element was considered theinput hydrodynamic load and used to calculate the net- ting deformation by the large deflection nonlinear structural model.

2.2.3 Equilibrium iteration algorithm

On the account of the nonlinearity of the netting in the current, the Newton-Raphson iteration method was app- lied to solve Eq. (21) as follows:

(=0, 1, 2, 3, 4L), (32)

where

andis the number of iteration.

2.3 Coupling Between the CFD and Mechanical Structural Solvers

The process flowchart for a strong one-way coupling al- gorithm is shown in Fig.4. Initially, the fluid flow simula- tion was performed around the netting structure without considering the netting deformation using the SST-model and the porous media model until convergence is reached. The effect of the wall roughness of the netting sur-face was considered by a refined boundary layer. In the CFD solver, the implicitly formulated equations were solved by iterative techniques. The calculated hydrodynamic forces (water forces) at the interface from the fluid calculation (netting boundaries) were interpolated to the netting struc- ture mesh (mechanical structural solver). Then the struc- tural dynamic (netting configurations) calculations were per-formed until the convergence criterion is met. This pro- cess was repeated until the end time is reached.

2.4 Numerical Simulation Setup

The hydrodynamic forces acting on the porous and fle- xible netting structures 9.0m in length, 2.2m in width, and1.6m in depth were calculated adopting the numerical wa- ter tank dimension to reduce the boundary effect of the numerical water tank and avoid the iterative non-conver- gence of numerical simulation in ANSYS Fluent solver (Fig.5). The netting characteristics are listed in Table 1. Five velocity magnitudes and nine attack angles varying from 0˚ to 90˚ were used as listed in Table 2.

Fig.4 Solution algorithm for the coupling between water and netting in one-way.

Fig.5 Schematic of numerical flume water tank.

Table 1 Structural parameters of the netting panels used in the numerical simulation

Table 2 Tested and numerical conditions

2.5 Boundary Condition, Initial Condition and Meshing

As shown in Fig.5, the left boundary of the numerical water tank was constituted by the velocity-inlet boundary condition, whereas the right boundary was described bythe outflow boundary condition. The other boundaries of the computational domain were modeled as zero shear stress wall. The fluid-netting interaction wall was modeled us-ing the no-slip wall boundary with consideration to the viscosity force between the water flow and the netting. In areas close to the netting, the wall function and boundary layer mesh were adopted to resolve the effect of the wall surfaces. The right-handed, 3D coordinate system was used as a coordinate system for the numerical model.The ori- gin of the coordinate system was set at the geometric cen- ter of the netting, and located in the middle of the nume-rical water tank. In the coordinate system,represents the flow direction (positive along of this direction),is per- pendicular to the flow direction on the horizontal plane (in depth), andis upward (perpendicular direction).

The inlet turbulence quantities (and) of the velocity- inlet boundary should be specified to initialize the-mo- del for the calculation of turbulence flow. In this study,andwere calculated as follows:

whereReis the Reynolds number based on the hydraulic diameter of the flume tank (D),is the turbulence inten- sity,is the turbulent length scale,Dwas assumed to be 1.6m, and=0.09 is the empirical constant in the SST-model. In this study,Revaries between 355211.7 and 1205981.7 andbetween 2.8% and 3.23% matching to those measured in the experimental flume tank without net- ting by Hu. (2004).

Unlike the other turbulent model, the-SST model was integrated through the viscous sublayer without the need for a two-layer approach. This function can be used for a+insensitive wall treatment by mixing the viscous sublayer formulation and the logarithmic layer formulation based on+. Thus, in consideration of the low Reynolds number, a ground rate of 1.2,+=20, and the maximum cell of boundary layer of 20mm were set around the netting. The maximum cell size in the computational domain was set to control the distribution.

To achieve the optimal mesh type and grid size, unstruc- tured full-tetrahedral grids were applied to generate the mesh of numerical simulation water tank and the netting. The curvature size function with a curvature normal angle of 12˚ was applied to refine the mesh close to the netting us- ing the grid generation tool ICEM-CFD 16.0. An example of the mesh size distribution of the numerical water tank with the netting at an attack angle of 90˚ is shown in Fig.6a, and the total number of grid elements is 10376963. For the structure model, bar elements were also divided by unstruc-tured full-tetrahedral grids with a total number of grid nodes(netting 2) of 24882895 determined by the independent grid analysis. Fixed support boundaries were defined at the top, sides, and bottom of the net (Fig.6b).

Fig.6 Tetrahedral mesh distribution for the numerical simulation of the netting in the current.

2.6 Physical Model Experimental Test

Physical model tests were performed to measure hydro- dynamic forces of the nettings and the flow field around the netting in the flume tank at Tokyo University of Ma-rine Sciences and Technology to verify the numerical simu-lation model. The flume tank working section was 9.0m long, 2.2m wide, and 1.6m deep, with a depth of water of approximately 1.35m. The impellers were 1.6m in diame- ter, delivering a flow speed varying from 0.1ms−1to 2.0ms−1; the acceleration flow varied from 1.0cm2s−1to 5.0cm2s−1; the reciprocating oscillation flow of velocity amp- litude was±0.5ms−1.

2.6.1 Experimental test for the netting hydrodynamic forces measurements

For controlled netting hydrodynamic force measure- ments and attack angles in the flume tank, the netting was fixed to the steel frame with a dimension of 500mm×500mm, as shown in Fig.7. This steel frame was fixed to one connection point containing a load cell of six components (Fig.7).The load cells used had a capacity of 5kg per cell and were fabricated by Denshi Kogyo Co., Japan. The load cells were calibrated and zeroed at the beginning and end of testing.A total of nine attack angles (netting inclination) and five incoming velocities were setup experimentally (Table 2).Data were sampled at 20Hz for 20s, and the ex- perimental parameters are as follows: fresh water temper- ature of 9˚, water density of 999.8kgm−3, and kinematicviscosity of 0.01341cm2s−1. Detail about the experimental process can be found in the study by Tang. (2018).

The physical test was conducted for five nettings each with a dimension of 490mm×490mm, a diamond mesh, various materials, and net solidity under the same experi- mental conditions as listed in Table 1.

The Reynolds number represents the ratio of inertial and viscous forces and can be described using Eq. (35):

whereis the flow velocity,is the kinematic viscosity andis the twine diameter.

The drag and lift coefficients were calculated using Eqs. (36) and (37):

The hydrodynamic forces (Hyd) on the netting were found by subtracting the averaged measurements for each flow velocity on the frame from the averaged measurements for each flow velocity on the frame and netting:

The relative error between the experimental and nume- rical results was calculated as shown below to evaluate the applicability of the numerical method:

whereF(Exp)irepresents the flume tank experiment drag force data andF(Num)irepresents the numerical simulation drag force result.

2.6.2Acoustic doppler velocimeter (ADV) mea- surement system for the flow around netting

The flow field around the netting was measured using an acoustic doppler velocimeter (ADV, Nortek, USA) in different positions behind the netting, mostly in the wake region, but also outside the wake (Fig.8). The flow velo-city at different measurement points in the water was mea- sured using an ADV with a specified accuracy of 1mms−1(Fig.8). At each measurement point, velocity data were sam- pled for 10s at a sampling rate of 4Hz. Thus, the three components of velocity data were assessed every 10s. Themeasurement was performed at an incoming velocity of 90cms−1, and an attack angle of 90˚. And the measurement was run three times to assess the repeatability of the mea- surement.

Fig.8 Schematic of the model setup for measurement of flow field.

3 Results

3.1 Effect of Net Solidity and Twine Material on the Netting Deformation

Figs.9b and 9d show the comparisons of the nylon net- ting deformation between the numerical and experimental results by Li and Chen (2016). The results indicate that the netting deformation from one-way coupling numerical simulation is compatible with that from the physical mo- del tests. The deformation of each netting was great at the center of the netting compared with the other parts of the netting because the four sides of the netting were fixed (Figs.9a and 9b). The deformation of the polyethylene net- ting (net 2) was 93.12% greater than that of the nylon net- ting (Fig.9c). Thus, the stiff nylon material has limited de- formation capacity compared with the polyethylene netting.The increase of net solidity increased the netting defor- mation, resulting in a drag force. As shown in Fig.10, the stress and strain on each netting increased with increasing solidity ratio. On average, the maximum engineering stressand strain of the polyethylene netting (net 1) were 56.70% and 87.53%, respectively, which are greater than those of the nylon netting (net 4). The engineering stress-strain curve was observed from the numerical study of tensile strength of the polyethylene and nylon nettings (Fig.10e). The en- gineering stress increased with increasing equivalent strain.In addition, the curves followed a linear elastic theory be-cause the Young’s modulus and Poisson’s ratio of each net-ting material used in the numerical simulation were con- stant in the whole analysis following Hooke’s law (shown in Fig.10e).

Fig.9 Comparison of the deformation of (a) polyethylene (net 1) and (b) nylon (net 4) nettings between the numerical simulation and (c) physical model test of Li and Chen (2016), and (d) netting deformation curves.

Fig.10 (a) and (b), Strain of polyethylene (net 1) and nylon (net 4) nettings; (c) and (d), engineering stress of nylon (net 4) and polyethylene (net 1) netting; (e), engineering stress-strain curves.

3.2 Comparison of Drag Coefficients of Netting at Normal and Parallel Attack Angles Between Nu- merical Simulation and Experimental Results

The calculated normal and parallel drag coefficients us- ing numerical simulation were significantly higher than theexperimental data (Figs.11 and 12). The comparisons of drag coefficients and drag forces at normal and parallel attack angles for each netting between the numerical simulation and experimental data presented a satisfactory result, indi- cating that the porous coefficients used were accurate and the numerical simulation agreed well with the experimen- tal data. The mean relative errors for the normal drag force were less than 0.7%, 0.5%, 1.6%, 2%, and 1% for nets 1, 2, 3, 4, and 5, respectively (Fig.12c), whereas those for the parallel drag force were 2.5%, 2.6%, 3%, 2%, and 3.4% for nets 1, 2, 3, 4, and 5, respectively (Fig.12d).

3.3 Effect of Net Solidity, Netting Material, and Reynolds Number on the Drag Coefficient at Normal and Parallel Attack Angles

The normal drag coefficient of each netting systemati- callyincreasedwith increasing net solidity, and the paral- lel drag coefficient increased with decreasing net solidity in the numerical simulation and experimental results, ex- cept that of net 1, which was lower than that of netting 4 (Fig.11).This exception can be attributed to the difference in the material and twine area. In specific, netting 4 had a lower twine diameter and greater mesh size than net 1, al- lowing it to have a lower twine material. The drag force of each netting increased with increasing flow velocity, and the parallel drag force increased with the solidity ratio un- der the same flow velocity (Fig.12). The normal and paral- lel drag coefficients decreased with increasing Reynolds number (Fig.11). In general, the use of nylon material with a lower twine diameter and greater mesh size (netting 4) and the polyethylene netting with greater mesh size allow- ed the netting to have a lower drag force, eventually im- proving the efficiency of fishing gear in terms of hydrody- namic force.

Fig.11 Comparison of the normal and parallel drag coefficient of netting as a function of Reynolds number between the present numerical simulation and flume tank experiment data.

Fig.12 Comparison of normal (a) and parallel (b) drag force of netting between the present numerical simulation and flume tank experiment data, and relative errors (c) and (d) as a function of flow velocity (Sn represents the solidity ratio).

On the basis of the present numerical and experimental data, the nonlinear, least-squared error regression model was applied to predict the normal and parallel drag coef- ficients of the polyethylene and nylon nettings as a func- tion of Reynolds number and net solidity.

whereis the net solidity.

3.4 Comparison of Drag and Lift Coefficients ofNetting at Inclined Attack Angles Between Nu- merical Simulation and Experimental Results

The numerical results of the drag and lift coefficients were compatible with those obtained experimentally in the flume tank (Figs.13 and 14). On average, the numerical re- sults were greater than those of the experimental results,with a difference of 2.5% for the inclined drag coefficientsand 3.6% for the lift coefficients. The maximum relative errors between the numerical and experimental inclined drag coefficients were 2.3%, 5.2%, 3.8%, 6.5%, and 3.5% for nets 1, 2, 3, 4, and 5, respectively, whereas those be- tween the numerical and experimental lift coefficients were 4.8%, 4.92%, 5.8%, 5.97%, and 5.7% for nets 1, 2, 3, 4, and 5, respectively.

Fig.13 Comparison of netting drag coefficients with different net solidity ratios as a function of the attack angle for each flow velocity between numerical simulation and flume tank experiment data.

Fig.14 Comparison of netting lift coefficients with different net solidity ratios as a function of the attack angle for each flow velocity between numerical simulation and flume tank experiment data.

3.5 Effect of Net Solidity, Attack Angle and Twine Material on Drag Coefficient and Lift Coefficient of the Netting

The drag coefficient increased with increasing attackangle and flow velocity in the numerical simulation and ex- perimental results (Fig.13). The drag coefficients of the po-lyethylene nettings (nets 1, 2, and 3) increased with in- creasing net solidity. When the Reynolds number was in- creased, the drag coefficients of the five nettings decreased, and the polyethylene netting with a higher twine diameter and mesh size (net 1) had lower drag coefficients and great-er Reynolds number than the other nettings, except for net-ting 5 (Fig.13). The drag coefficient of the nylon netting decreased with increasing net solidity at an attack angle less than 60˚, but the drag coefficients of the two nettings (nettings 4 and 5) were very close with a gap of 3.5% be- tween 5˚ and 60˚ (Fig.13).

The lift coefficient decreased with increasing net soli- dity and increased with increasing flow velocity (Fig.14). The polyethylene nettings had greater lift coefficients and less effective drag than the nylon nettings (Fig.14). The lift coefficient reached a culminating point at an attack angle of 45˚ and subsequently decreased with increasing attack angle (Fig.14).

Thus, the nonlinear least-squared error regression model was applied to predict the inclined drag and lift coeffi- cients for the polyethylene and nylon nettings as a func- tion of Reynolds number, solidity ratio, and attack angle with the present numerical data as follows:

3.6 Validation of the Flow Field Around Netting by Experimental Results

Fig.15 shows the flow velocity field downstream to the netting measured at differentanddirections. The flow velocity field in the centerline downstream the netting in- creased with the increase in coordinate values in theanddirections. For the numerical and experimental results, the flow velocity reduction along the centerline downstream to the netting varied from 5% to 15%. The flow velocity fields obtained in the numerical simulation were basically consistent with the experimental results. The relative error between the numerical simulation and experimental results ranged from 0.4% to 6.3%, indicating that the fluid-struc- ture interaction including the porous media model based on on-way coupling was effective for calculating the flow velocity variation around netting (Table 3). Results showed that the flow velocity was homogeneous for different wa- ter depths (Fig.15). In addition, the flow velocity increasedas the measurement point moved away from the netting (Table 3) because the fluctuation-dissipation due to the net- ting was more developed when the measuring point was close to the netting.

Fig.15 Flow velocity field curve along the centerline downstream of the netting at different y (perpendicular to the flow direc- tion on the horizontal plane) and z (upward). ‘Exp’ and ‘Num’ represent the experimental and simulation data, respectively.

Table 3Flow velocity field at the measurement point from numerical simulation and experimental test (ms−1)

3.7 Numerical Simulation Results of Flow Field Around Netting

A small flow-velocity-reduction region was observed up-stream of the netting, whereas a large flow-velocity-reduc- tion region was observed downstream from the netting at different attack angles called the wake region (Fig.16). This wake region is divided into three zones. The first zone near the netting is the lower-velocity zone having low veloci- ties varying from approximately 0.40to 0.60(Fig.16). The second zone corresponds to the medium-velocity zone with flow velocities varying from approximately 0.70to 0.910. Finally, the third zone corresponds to the high- velocity zone with flow velocities varying from approxi- mately 0.910to 0.990. The velocity reduction regions upstream and downstream from the netting become small- er with increasing inclination angle, and more than 20% velocity reduction was found in the wake (Fig.16). When the attack angle was greater than 45˚, the inclined netting structure caused the largest wake in length and width.

Analysis of the velocity fields clearly indicates the spa- tial development of turbulent flow corresponding to the tur- bulence due to the netting wake (Fig.16). A significant flowvelocity reduction close to the netting occurred because of the blockage of the netting structure, thus limiting the passage of the flow through the mesh of the netting. In consideration that theflow direction is unidirectional and has no obvious diversion around the netting structure be- cause of the large porosity of netting, the velocity field re- duction downstream from the netting is mainly because of the weak passage of the flow through the mesh of netting(Fig.16).

Fig.16 Flow velocity distribution in the x-z plan y=0 around the netting (α=0.170) for each attack angle calculated by numerical simulation for an incoming velocity of 1ms−1.

As shown in Fig.17, the velocity reduction was invari- ably upstream of the netting and decreased slightly in the wake further downstream as the inclination angle was in- creased from 0˚ to 60˚. A distinct decrease in the velocity reduction distinctly decreased as the inclination angle was increased to 90˚. The magnitude of the velocity variation fluctuated along the centerline at attack angles >45˚. These fluctuations are mainly because of the effect of wake from the mesh opening of the netting. Notably, the minimum ve- locity reduction downstream the netting along the center- line was about 3.5% at an attack angle of 0˚, and the ma- ximum velocity reduction was approximately 33% at an attack angle of 90˚ (Fig.17). Fig.17 also shows that at=0.35, 0.5, 0.6, 0.55, 0.32, 0.58, 0.16, and 0.2m, and>0.19m, the velocity reductions reached their maximum at the attack angles of 0˚, 5˚, 10˚, 15˚, 20˚, 30˚, 45˚, 60˚, and 90˚, respectively, indicating that the wake zone and turbu- lence are important at these positions.

Fig.17 Numerical results of flow velocity along the cen- terline of netting (α=0.170) at different attack angles. The vertical lines represent the position of the fishing net.

4 Discussion

The design parameters that effectively influence the hy- drodynamics of netting are net solidity, twine material, and flow velocity. Net solidity is defined as the ratio of the projected surface area of the netting and the outline area re- ferring to the total area enclosed by the netting, and flow velocity determines the extent of the netting exposed to flow.This study demonstrated that the numerical simulation and experimental flume test provide valuable knowledge con- cerning the porous media coefficients, hydrodynamic co- efficients, flow field distribution, and netting deformation.In specific, a homogenous compartment was found between the present numerical simulation and physical test in the flume tank in predicting the main performance parameters of netting, such as drag coefficients, lift coefficients, net- ting deformation, and the flow field around it.Taking the results in this study and the results reported previously by Patursson. (2010), Zhao. (2013a, 2013b), Bi. (2014a, 2014b), Chen and Christensen (2017), and Chen and Yao (2019), it can be indicated that the numerical si- mulation is an efficient method for predicting the netting hydrodynamic performance and the netting deformation and for studying flow field around the netting despite the fact that the theoretical results must be validated by experimen- tal data. Different from the investigation by Bi. (2014b),Chen and Christensen (2017), and Tang. (2017b), the present study shows that the changes in the twine mate- rials, net solidity, and attack angle can strongly affect the hydrodynamic coefficients of the netting, the deformation of the netting, and the flow around it not only using CFD but also combining it with the finite element method (largedeflection nonlinear structural model) in one-way coup- ling with the netting design in 3D. Therefore, in this study, the hydrodynamic forces were calculated to investigate the hydrodynamic behavior of the flow field around the net and the netting porous characteristics. In addition to the cal- culation of the deformation of netting, other factors consi- dered were as follows: netting material parameters and the pressure acting on the netting. However, the results of this study were better than those obtained by Bi. (2014b) and Chen and Christensen (2017) who combined the po- rousmedia and the lumpedmass models to simulate the flow around the netting and those obtained by Tang. (2017b) who used FSI in one-way coupling to study the netting deformation.

A comparison of the deformation results with the ex- perimental results of Li and Chen (2016) showed slight differences in the response of the structure. Probably it is caused by the simplification of the frame as a cylinder in the simulation without considering its mass in the struc- tural solver, or by the hydrodynamic and flow field ef- fects. Bi. (2014b) and Tang. (2017b) also found differences between the results of experiment and the ANSYS model. Thus, we can conclude that the netting deformation with bending stiffness is affected by many fac- tors, such as elasticity of the netting material, its construc- tion, density, the hydrodynamic force acting on netting, and its flexible motion. The deformation of the nylon net- tings was lower than that of the polyethylene nettings. There- fore, the netting deformation is affected by the hydrody- namic forces, which consequently affect the net solidity, flow velocity, and attack angle. The hydrodynamic forces increased with increasing net solidity. The net solidity and drag force and deformation of the polyethylene nettings were greater than those of the nylon nettings used. This trend was confirmed by Chen and Christensen (2017) and Myrli (2017) that the deformation of the flexible netting increases with increasing incoming velocity. We hypothe- sized that the deformation of the nylon netting is lower than that of the polyethylene netting because the deforma- tion for the rigid netting of nylon relative to polyethylene is less influenced by the hydrodynamic parameters at large deformation than by the structural properties. If the defor- mation of the netting in response to hydrodynamic forces can be accurately predicted by only using the structural pa- rameters, then it must be because the structural parame- ters alone can capture the behavior of the deformation at large deflections (Gansel., 2013).

With regard to normal and parallel drag coefficients, the results obtained by the present numerical model and those obtained experimentally showed good similarity with the relative errors lower than 3.5%. These relative errors are 6.2%, 25%, 6.5%, and 4.2% lower than those obtained by Patursson. (2010), Chen and Christensen (2016), Ci- fuentes and Kim (2017), and Tang. (2017b), respec- tively. The numerical simulation results obtained by Ci- fuentes and Kim (2017), Tang. (2017b), and this study revealed that the drag force of netting increases with in- creasing flow velocity. The experimental and numerical re- sults of this study showed that the use of nylon nettings, such as net 4 with a small twine diameter approximately1.62 and 2.58 times lower than those of nets 1 and 5, re- spectively, and the large mesh size approximately 1.97, 3.42, and 1.17 times greater than those of nets 2, 3, and 5, de- creased the normal drag forces by approximately 17%, 81.42%, 69.04%, and 66.66% than those of nets 1, 2, 3, and 4, respectively. This trend was confirmed by the expe- rimental study conductedby Ward. (2005), Kim. (2007), and Tang. (2017a, 2018), exhibiting that the decrease in the twine diameter and the increase in the mesh size reduce the drag of the fishing gear, thus decreasing the fuel consumption.According to this study and the pre- vious studies by Madsen. (2011) and Tang. (2017a, 2018), the normal drag coefficient of the netting increases with increasing solidity ratio and decreasing Reynolds num- ber, whereas the parallel drag coefficients increase with in- creasing solidity ratio.

The normal drag coefficient of the polyethylene nettings was greater than those of nylon nettings in the previous studies and this study, and the parallel drag coefficients de- creased with increasing net solidity (Fig.18). The predict- ed Eqs. (40) and (41) based on the numerical data and pre- dicted equations provided by Tang. (2017a) and Tang. (2018) are in agreement for predicting the drag co- efficient, particularly for the normal drag coefficients of the polyethylene nettings and the parallel drag coefficients of the nylon nettings in this study (Fig.18). The normal drag coefficient provided by Balash. (2009) was great-er than that obtained this study, but the value predicted from the formula developed by Zhan. (2006), which is a function of net solidity and attack angle, was lower than the present results. The values provided by the pre- dicted formulas of Kumazawa. (2012) were less than those of this study, whereas those provided by the pre- dicted formulas of Hosseini. (2001) were close with those obtained in this study, when the Reynolds number was less than 1000 and greater than 4000. These differ- ences may be due to the factors not considered during this numerical simulation in the previous experimental studies or may be due to the difference in material, net solidity, and experimental conditions. In addition, the predicted nor- mal and parallel drag coefficient obtained with the formu- las of Fridman (1973), Zhan. (2006), and Kristiansen and Faltinsen (2012), was close with a gap less than 20%. However, comparison of the results obtained with these predicted formulas and those obtained in this study re- vealed that the normal and parallel drag coefficients ob- tained in this study were approximately 68% greater than those previously obtained with the formulas of Fridman (1973), Zhan. (2006), and Kristiansen and Faltinsen (2012). These differences can be ascribed to the fact that the models obtained by Fridman (1973), and Kristiansen and Faltinsen (2012) were based on a square mesh, the mo- del obtained by Zhan. (2006) was based on mesh orientations T0 and T90, and the model in this study was based on a diamond mesh in knotless netting. It can also be due to the conclusion of Arkley (2008) that using T90 meshes (square mesh) compared with standard diamond meshes reduces the netting drag and the twine surface area of fishing gear and, hence, improves fishing gear perfor- mance.

Fig.18 Comparison of the normal and parallel drag coefficients of nettings as a function of Reynolds number in the present numerical simulation and previous study (Sn represents the solidity ratio).

The numerical results on the inclined hydrodynamic co- efficient were in unison with the experimental results. Dif- ferent from the investigation of Patursson. (2010), Zhao. (2013a, 2013b), and Bi. (2017), this study showed numerically that changing the net solidity, twine material, and attack angle can strongly affect the netting hydrodynamic coefficients using the porous media method with the 3D netting design. Thus, the results obtained ex- perimentally and numerically in this study showed that using the netting with a low solidity ratio and modifiedtwine diameter increases the inclined drag coefficient (Tang., 2019). This trend was confirmed by the experimen- tal study conducted by Patursson (2008) and Tang. (2018), who showed that the nylon netting with a low so- lidity ratio can have greater hydrodynamic coefficients. The results obtained in this study are significant compared withthose of Patursson. (2010) and those of Chen and Christensen (2016) with relative errors of 12.5% and 20%, respectively.

The results provided by Tang. (2017a, 2018) agree with the result of this study (Fig.19); however, the results of Aarsnes. (1990) were less than that of this study. Unlike the equation of Aarsnes. (1990) that only con- sidered the net solidity and attack angle, the present equa- tion considers the Reynolds number, net solidity, and at- tack angle. The results provided by Kumazawa. (2012) agrees with those provided in this study at attack angles less than 30˚. Meanwhile, at attack angles ranging from 30˚ to 90˚, those provided by the predicted formulas of Ku- mazawa. (2012) were greater than those provided by this study (Fig.19). The difference between these predict- ed formulas and those previously proposed can be ascribed to the fact that the present study used nettings with two materials (polyethylene and nylon), whereas previous stu- dies used only one material, such as the Dyneema netting by Kumazawa. (2012), the polyethylene netting by Tang. (2017a), and the nylon netting by Tang. (2018). The difference can be due to the simulation and experimental conditions. In addition, the inclined drag and lift coefficients obtained with the formulas of Aarsnes.(1990), Løland (1991), and Kristiansen and Faltinsen (2012)were very close with a gap lower than 10%. However, the results of this study when the net solidity was 0.206 agree with those obtained using the predicted formulas of Aars- nes. (1990), Løland (1991), and Kristiansen and Fal- tinsen (2012) when the attack angles were less than 10˚. At attack angles greater than 10˚, the results provided in this study were greater than those provided using these pre- dicted formulas. The difference can be ascribed to the fact that the models of Aarsnes. (1990), Løland (1991), and Kristiansen and Faltinsen (2012) were based on screen models, whereas the formulas obtained in the present stu- dy were based on the Morrison model. In the study by Cheng. (2020), the screen models provided a lower drag force on the netting compared with the Morrison mo- del. These differences were also due to the fact that the netting design with a square mesh has inherently lower drag coefficient than the netting design with a diamond mesh. This trend was confirmed by Bi. (2017), who showed that the drag coefficient of the square netting is 33.8% lower than that of the diamond netting.

Fig.19 Comparison of the drag and lift coefficients of nettings with different net solidity ratios as a function of the attack angle between the present numerical simulation and previous study (Sn represents the solidity ratio).

With regard to the flow field around the netting, the dif- ferent planes showed a velocity deficit related to the de- velopment of the turbulent flow through the netting, con- sisting of turbulence due to the netting wake. Thus, theflow-velocity-reduction zone is in most downstream of the netting and evolves as a function of attack angle. The flow velocity reductions around the netting obtained in this stu-dy were similar to those obtained by Parturssons. (2010)and Zhao. (2013a, 2013b). This study and previousstudies by Bouhoubeiny. (2014) and Druault. (2015) demonstrated that the turbulence flow developing around the netting can significantly affect the hydrodyna- mic forces of the netting. The difference between the re- sults obtained by this study and those by Bi. (2014a, 2014b) and Chen and Christensen (2016) can be ascribed to the different solver algorithms used. The flow velocity field in the wake zone was 0.65–0.96ms−1smaller than the incoming flow velocity. This flow velocity deficit in thewake zone results from a loss of momentum because of the netting drag and the energy exchange (Løland, 1993). The high velocity-reduction and energy exchange are caused by the blockage of the mesh opening on the passage of the flow around the netting, presenting a high incidence due to the inclination angle (Bouhoubeiny, 2012).In addition, the small mesh openings of the netting engender distur- bances caused by strong velocity reductions near the net- ting. These disturbances often relate to the presence of ob- stacles, such as the structure of the netting and the turbu- lent flow in the vicinity of the nodes, which are consider- ably near the netting (Bouhoubeiny., 2014). There- fore, the turbulence flow around the netting is less impor- tant and the netting drag decreases with increasing mesh size and attack angle (Tsukrov., 2011; Bi., 2017). Experimental studies by Bi. (2013) and the current numerical results showed that the main factors affecting the flow velocity distribution around the netting are the net solidity and the attack angle. The experimental study by Bi. (2013) revealed that the downstream of the net- ting, the velocity reduction, and the shielding effect in- crease with net solidity, decreasing the hydrodynamic co- efficients. In addition, the physical nettings with high twine diameter and low mesh size (higher solidity ratio) gene- rate large turbulence regions, and thus, great drag force (Chen and Christensen, 2016).

The proposed analytical method for finding optimal po- rous resistance coefficients yielded coefficients for which the model results are consistent with the experimental data. The model proposed in this study was based on two hypo- theses. The porous media resistance coefficient increased with increasing net solidity. This trend was confirmed by the numerical conducted studies of Zhao. (2013a, 2013b), Bi. (2014a, 2014b), and Bi and Xu (2018). The porous media viscous resistance coefficients (DandD) and tangential inertial resistance coefficients (C) of poly- ethylene nettings were greater than those of nylon net- tings because polyethylene nettings have lower density and Young’s modulus while greater drag than nylon nettings (Lowe, 1996; Tang., 2018). However, the porous nor-mal inertial resistance coefficients of the nylon nettings were greater than those of the polyethylene nettings be- cause the roughness of the nylon nettings was considera- bly greater than that of the polyethylene nettings. This trend was confirmed by studies by Casanova and Dwikar- tika (2013), Bi. (2014b), and Tang. (2017a), who exhibited that nylon nettings can have lower drag and greater stiffness than polyethylene nettings. The attack angle greatly affected the porous media resistance coefficients. In specific, the normal resistance coefficient increased with increasing attack angle, and the tangential resistance co- efficient was also affected by the attack angle. Thus, the attack angle should be considered in the empirical formu- las of porous media resistance coefficients and in the nu- merical simulation of the nettings, unlike in the previous studies by Zhao. (2014), Bi. (2015), and Chen and Christensen (2016), who only accounted for net soli- dity. On the basis of the transformation of Morison-type load model and the model proposed by Chen and Chris- tensen (2016), the following formulas were derived for the determination of porous media inertia resistance co- efficients by combining Eqs. (19), (40), (41), (42), and (43):

When the attack angle was lower than 10˚ and greater than 60˚, the formulas provided by Chen and Christensen (2016) agree well with the results of the present study (Fig.20b). When attack angles are between 10˚ and 60˚, the formulas proposed in this study to predict the tangential porous resistance coefficient is greater than those proposed by Chen and Christensen (2016). Notably, the formulas pro-posed in the present study are more particular compared with those proposed by Chen and Christensen (2016). In specific, the tangential coefficient in the proposed formu- la is determined from the lift coefficient, whereas Chen and Christensen (2016) only accounted the drag coefficient. Fig.20a shows that the results of the porous normal iner- tial resistance coefficients obtained by the formula estab- lished in this study were greater than those obtained by the formula of Chen and Christensen (2016).

Fig.20 Comparison of the normal and tangential porous resistance coefficients of netting with different net solidity ratios as a function of the attack angle between the present and previous studies.

5 Conclusions

The hydrodynamic variation, the netting deformation, and the flow field around the nettingat low and high at- tack angles were experimentally and numerically investi- gated. In the numerical model, the-SST turbulence mo- del and porous media model were combined with the large deflection nonlinear structural model in one-way coupling to simulate the flow field around the netting and the net- ting deformation.The hydrodynamic forces of polyethyl- ene and nylon nettings were numerically and experimen- tally evaluated. This study successfully analyzed the ef- fectsof meshopening and attack angle on the flow field and the effects of twine material and net solidity on the de- formation and hydrodynamic force of the netting. The po- rous media resistance coefficients were experimentally de- termined. The conclusions of this study are as follows:

1) A comparison with the experimental data shows that the one-way coupling model combining the porous media model with the large deflection nonlinear structural model is a reliable tool for simulating the interactions between the fluid and flexible structure of nettings. It can not only predict the hydrodynamic coefficients and netting defor- mation accurately but also characterize the flow around the netting and its effect on the netting performance.

2) The relative error between the present numerical mo- del and the experimental data is generally less than 5.8%, indicating that the numerical model can reproduce the ex- periments adequately. Thus, nonlinear equations are pro- posed to predict the drag and lift coefficients of the poly- ethylene and nylon nettings.

3) The polyethylene netting (net 2) has greater deforma- tion, stress, and strain than the nylon netting (net 4), indi- cating that the latter is stronger than the former. Therefore, the use of nylon nettings with low twine diameters and great mesh sizes provides a low hydrodynamic force.

4) The minimum velocity reduction downstream the net-ting along the centerline is approximately 3.5% at an at- tack angle of 0˚, and the maximum velocity reduction is ap-proximately 33% and varies from 35% when the attack angle is 90˚.The flow in the wake zone is completely tur- bulent, composed of the boundary layer flow and the tur- bulence due to the netting wake. In addition, the flow ve- locity field reduces with increasing attack angle.

5) Based on the transformation of the Morison-type load model and the model proposed by Chen and Christensen (2016) and the predicted formulas of hydrodynamic coef- ficients, nonlinear equations can be employed to predict porous resistance according to the variation in the attack angle and net solidity.

Acknowledgements

This study was financially sponsored by the National Natural Science Foundation of China (Nos. 31902426, 41 806110), the Shanghai Sailing Program (No. 19YF1419800), the National Key R&D Program of China (No. 2019YFD 0901502), and the Special Project for the Exploitation and Utilization of Antarctic Biological Resources of Ministry of Agriculture and Rural Affairs (No. D-8002-18-0097).

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. Tel: 0086-21-61900309 E-mail: htang@shou.edu.cn

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