An ALE Numerical Method for the Waterline Evolution of a Floating Object with Non-linear Shallow-water Equations

In this work, a numerical method is introduced for the study of nonlinear interactions between free-surface shallow-water flows and a partly immersed floating object. At the continuous level, the fluid's evolution is modeled with the nonlinear hyperbolic shallow-water equations. The description of the flow beneath the object reduces to an algebraic and nonlinear equation for the free-surface, together with a nonlinear differential equation for the discharge. The object's motion may be either prescribed, or computed as a response to the hydrodynamic forcing. In the later case, with heaving, surging and pitching allowed in the horizontal one-dimensional case, these equations are supplemented with the Newton's second law for the object's motion, involving the force and torque applied by the surrounding fluid, and parts of this external forcing are regarded as an added-mass effect. At the discrete level, we introduce a discontinuous Galerkin approximation, stabilized by a recent a posteriori Local Subcell Correction method in the vicinity of the solution's singularities and loss of admissibility. The motion of the fluid-structure contact-points is described with an Arbitrary-Lagrangian-Eulerian strategy, resulting in a global algorithm that ensures the preservation of the water-height positivity at the sub-cell level, preserves the class of motionless steady-states even when the object is allowed to evolve freely, and the Discrete Geometric Conservation Law. Several numerical computations involving wave and floating object interactions are provided, showing the robust computation of the air-water-body contact-points dynamics.