The present version of ** FLOW-3D** allows multiple moving objects to be connected via elastic ropes to other moving objects or fixed locations. It is assumed that the ropes are weightless and always straight when stretched. The rope tension is uniform along the ropes and is simply related to the rope extension and the spring constant. Rope dynamics and its effects on motion of the tethered objects are ignored. These assumptions have imposed limitations to the elastic rope model. For example, when simulating motion of a moored floating vessel, the model works for taut mooring lines but fails for catenary mooring lines. This is because a taut mooring line has light weight compared to the line tension and is nearly straight in shape, whereas a catenary line has heavy weight that significantly affects the line’s shape and dynamics.

## Model description

To overcome these limitations, a compliant mooring line model has been developed and implemented in ** FLOW-3D** version 11.1. The model considers gravity, buoyancy, fluid drag and tension force on the mooring lines. Full 3D dynamics of mooring lines is calculated as well as their dynamic interactions with tethered objects. The lines can be taut or slack, and their instantaneous shape is computed.

The model uses a finite segment approach to numerically solve the 3D mooring line dynamics. Each line is divided uniformly into a certain number of discrete segments. Using a sub-time step algorithm, location and velocity of the mass center for each segment are calculated by solving the dynamic equation of motion. The fluid drag forces in the normal and tangential directions of the segment are calculated following the quadratic drag law. The tension force at the joint of two neighboring segments is calculated by Hooke’s law using the segment’s spring constant and the line extension between the two mass centers. The shape of a mooring line is determined by the spatial distribution of the mass centers of all its segments. Dynamic coupling of the mooring lines with the moving objects is implemented in the way that the line dynamics provides tension force for the objects while the objects supply the tether end locations for the lines.

## Simulation results

The animation shows a simulation result of four slack mooring lines that are anchored to the seabed and tethered to a rectangular floating object drifting in a progressive wave. The object’s size is 64 m x 72 m x 30 m and its density is 500 kg/m^{3}. The incoming nonlinear wave is 10 m in height with a period of 8 s. The water depth is 70 m. Each mooring line has the undisturbed length of 85 m. The length density of the synthetic mooring lines is 5 kg/m and the spring constant is 2.4 x 10^{5} N/m. The normal and tangential fluid drag coefficients for the lines are 1.0 and 0.3, respectively. Initially, the mooring lines are in a static equilibrium with the catenary shape. Driven by waves, the object moves and draws the mooring lines in the direction of wave propagation. With time, the two upstream lines change from slack to taut and experience significant length extension, whereas the two downstream lines remain slack and fluctuate with the wave motion. At the end of the simulation, the average position of the object is maintained by the two taut mooring lines.

**Animation 1.** **A floating object driven by wave motion is restrained by mooring lines**

## Model applications

The mooring lines model has many engineering applications. One of the most important applications is the mooring systems for Floating Production Storage and Offloading vessels (FPSOs) in the offshore oil and gas industry. It can also be used to simulate mooring lines for floating Wave Energy Converters (WECs), floating wave breakers, anchored ships and many other moored structures, equipment and devices. With incorporation of the mooring line model, ** FLOW-3D** will help users to simulate real world problems better.