Elsevier

Ocean Engineering

Volume 35, Issue 13, September 2008, Pages 1372-1380
Ocean Engineering

Numerical model of a mussel longline system: Coupled dynamics

https://doi.org/10.1016/j.oceaneng.2008.05.008Get rights and content

Abstract

The longline is modelled using lumped masses and tension-only springs including structural damping. The mussel culture is modelled as cylinders attached to the main line and the equations are formulated for the coupled dynamics of the main line, buoys and mussel socks using Kane's formalism. Surface waves are described by Stokes’ second-order wave theory. The hydrodynamic loads are applied via a Morison's equation approach using the instantaneous relative velocities and accelerations between the fluid field, the longline and the attached buoys and mussel masses. The algorithm is presented and the equations are solved using the Runge–Kutta routine “ode45” in MATLAB. Outputs include position, orientation and velocity of all components and tension in all line segments. The numerical model may be used to predict the dynamics of longline systems using drag coefficients determined from field measurements. We expect that the results will be useful for checking and optimizing shellfish aquaculture designs prior to installation and for modifying existing designs to safeguard against failure.

Introduction

The authors have undertaken a significant extension of their previous work on the numerical modelling of aquaculture longline configurations which are based on arrangements found in operating shellfish farms located in the Gulf of St. Lawrence in Canada. In our previous work (Raman-Nair and Colbourne, 2003), the main line was modelled using a lumped parameter representation of cable dynamics with the effects of all attachments (buoys, mussel culture, concrete weights) modelled as applied loads. The present study involves the coupled dynamics of the main line and the attached buoys and mussel culture. The dynamics of the concrete blocks is not modelled and their effect on the main line is applied via a line stiffness. The simulations can be used to determine the configuration of a longline system in current and waves and to assess the loads appearing at the anchoring points, thus evaluating the adequacy of the anchors and identifying the potential for anchor slippage (or dragging). It is expedient to use Kane's equations (Kane and Levinson, 1985) for formulating the equations of motion for this multibody system. We present the results of some simulations predicting the loads in and shapes assumed by the longlines under various loading scenarios. We assume that the hydrodynamic loads are primarily due to added-mass effects and drag. In this regard, we allow for loading due to an arbitrary fluid velocity and acceleration field which is assumed to be undisturbed by the system. This allows for the inclusion of wave and current effects via the use of the Morison et al. approach (Chakrabarti, 1987). A series of experimental trials were conducted by Biorex Inc. and the Institute for Ocean Technology to directly measure the drag of key components of the shellfish longline structures.

Section snippets

System configuration

A diagram of the system to be analysed is given in Fig. 1. The origin of inertial coordinates is an arbitrary point O and the inertial frame is denoted by N with unit vectors N1,N2,N3. The longline is anchored at points A0 and An+1 and is composed of an assembly of the sub-systems Sk(k=1,,n) illustrated in Fig. 2. Sub-system Sk consists of node Ak on the main line with an attached buoy Bk (sphere or cylinder) and suspended cylinder Yk. In addition, the mass of the segment halves on either

Kinematics

We need to express the quantities q˙ikA,q˙ikB,q˙ikY in terms of the generalised speeds, where the dots indicate differentiation, with respect to time t. SincevAk=i=13q˙ikANi(k=1,,n)we haveq˙ikA=uikA(i=1,2,3;k=1,,n)We describe the kinematics for mussel sock Yk only, the procedure being similar for buoy Bk. For body Yk, the standard relations for space-three 1-2-3 rotation angles are given by Kane et al. (1983) asq˙1kY=u1kY+s2kYc2kY(u2kYs1kY+u3kYc1kY)q˙2kY=u2kYc1kY-u3kYs1kYq˙3kY=1c2kY(u2kYs1k

Forces on main line

The system is subjected to gravity and fluid forces. The fluid field is described by either a current velocity field or a wave velocity and acceleration field (in directions N1 and N2) using standard formulae for a Stokes second-order wave.

Forces on mussel socks and buoys

We will describe the computation for a mussel sock. The procedure for the buoys is similar.

Equations of motion

The generalised inertia and active forces for sub-system Sk (Fig. 2) are assembled asFrk*S=Fr*Ak+Fr*Bk+Fr*YkFrkS=FrAk+FrBk+FrYk(r=1,,15;k=1,,n)where superscripts A,B and Y refer to the main line, buoy and mussel sock, respectively, and superscript S refers to sub-system Sk. In order to solve for the accelerations, the generalised inertia forces are written in the formFrk*S=-MrkSu˙rkS+brkS(r=1,,15;k=1,,n)whereMrkS=VrkA+(VrkB+WrkB)+(VrkY+WrkY)brkS=(φrkB+ψrkB)+(φrkY+ψrkY)The quantities in Eq.

Test problem

We consider the simple system, illustrated in Fig. 3, consisting of two nodes on the mainline each carrying a buoy and mussel sock. Each of the segments OA,AB,BC has stiffness k and unstretched length L0. The submerged weight of each mussel sock is WY and of each mainline node is WA. The buoyancy of each buoy is FB acting upwards.

Typical results

The design of a typical longline used by mussel farmers in Cascapedia Bay, Quebec is described below based on the information they provided to Biorex Inc. and the data collected by Biorex on their culture gears. A diagram of the system is illustrated in Fig. 1. The longline has distance 201 m between the anchors. The line has diameter 0.019 m, weight per unit length (in air) without biofouling 0.19 kg/m, modulus of elasticity 1 GPa. The longline is modelled by n=113 nodes, labelled Ak(k=1,,n).

Conclusions

A numerical model of the three dimensional dynamics of a submerged mussel longline system has been presented. The model includes the coupled dynamics of the longline, buoys and mussel culture, which is an extension of the authors’ previous work in this area. The authors’ previous model will not capture effects due to the dynamics of the buoys and mussel culture, such as wave-induced motions. The previous model can be used for steady-state current provided the forces applied to the mainline due

Acknowledgements

This research project was partly funded by the Société de l’Industrie maricole du Québec Inc. (SODIM) and two provincial departments (Quebec, Canada): Agriculture, Pêcheries et Alimentation (MAPAQ) and Développement économique, Innovation et Exportation (MDEIE). The authors would like to thank  Mr. Craig Kirby (Institute for Ocean Technology) for assisting with the field measurements of the drag coefficients. The environmental consulting firm Biorex Inc. (Quebec, Canada) has used the numerical

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