The Dynamic Response of a 3D Mussel Dropper in Waves

Abstract

Offshore mussel cultivation is gaining popularity, with the longline technique emerging as a popular method for cultivation. The longline technique entails multiple backbones from which mussel lines that loop around it are suspended and which are kept afloat by buoys. Overdulve Offshore Services designed the Cees Leenaars, a semi-submersible mussel farm that employs the longline method for mussel cultivation. However, the configuration of the longlines and the offshore location of the farm introduce several challenges. The mussel will be exposed to high loads due to wind, current, and waves. Therefore, the question remains if the flow through the mussel farm will be sufficient to provide nutrients to the mussels. Hence, it is of importance to model the behaviour of the mussel droppers in waves.

The model will be constructed by first reviewing the motion of a single 3D pendulum, followed by the motion of a double 3D pendulum. For both situations the Lagrangian method is used to construct the equations of motion. Semi-Implicit Euler is used as integration scheme for the discretization of time. The results provide insights into the chaotic and complex characteristics of pendulums.

Thereafter, a model for a string will be constructed and this model is eventually expanded to a model for a mussel dropper by adding constraints to the last pendulum of the string. The method used for these models was different than for the model for a single and double pendulum due to instability issues. For the string Newton's second law was used to determine the equations of motion, since using this method made it easier to apply external forces and constraints to the system. Instead of a semi-implicit integration scheme an implicit one was used, resulting in an improved stability of the system. Due to the switch from semi to a fully implicit integration scheme, a nonlinear solver was build to solve the set of nonlinear equations at every time step. Later on, it was noticed that the implicit scheme accounted for too much numerical damping. In order to reduce the numerical damping, the Crank-Nicolson method combined with the theta-method was applied.

After constructing the model, an experiment was conducted for validating the model and for determining the damping coefficient of the system, which was still unknown. The experiment consisted of a physical model of the dropper and a camera which recorded the motion of the physical model. A code was written to determine the position of the pendulums in each frame and these positions were calculated relative to the angle each pendulum made with the horizontal axis. The experimental setup was reconstructed by the numerical model and compared in order to determine a value for the linear and quadratic damping coefficients. The model overdamped the amplitude of the motion right after the dropper was released from its equilibrium position. This could be due to numerical damping, due to errors in the experiment or due to a faulty translation of the experiment to the model. This provides the numerical model with not enough damping to let the motion die out, but it better represents the behaviour of the dropper for larger motions. These coefficients were found to be most suitable because from an engineering perspective it is safer to underestimate the damping in the system and assume higher amplitudes of the motion.

Lastly, the 3D model is used to simulate a mussel dropper in waves, as was the aim of this research. The dimensions of the mussel dropper are chosen to represent a real-life mussel dropper. The sea state to which the mussel dropper is subject will be one occurring every once in 25 year in the North Sea. The mussel dropper is simulated using seventeen pendulums. One pendulum is reviewed to determine the response of the mussel dropper. The time signal of this pendulum and the response spectrum of the pendulum show how the pendulum reacts to the incoming waves. A phase space plot is constructed to visualize the chaotic and complex behavior of the mussel dropper in waves.