# Nonlinear Free-surface Flows

Numerical Tools:

The use of computational tools to predict hydrodynamic performances of ships has gained a lot of popularity in recent years. Models based on potential flow theory have historically been among the most successful to assess the wave pattern around a ship hull in presence of a forward ship motion. In this framework, the assumptions of irrotational flow and inviscid fluid reduce the Navier Stokes incompressibility constraint and momentum balance equations to the Laplace's and Bernoulli's equations, defined on a moving and unknown domain. This boundary value problem is tackled with a Mixed Eulerian--Lagrangian approach, in which the Eulerian field equations are solved to obtain the fluid velocities, which are then used to displace in a Lagrangian way the free-surface, and update the corresponding potential field values. In this framework, the Eulerian problem is expressed in boundary integral form, and it is typically discretized using the Boundary Element Method (BEM). The velocity field and Bernoulli's equation provide a kinematic boundary condition for the Lagrangian evolution of the free-surface, and a dynamic boundary condition for the evolution of the potential field. In Mola et al. (2013) and (2014) we present a new stabilized semi-Lagrangian potential model for the simulation of three dimensional unsteady nonlinear water waves generated by a ship hull advancing in calm water. The resulting integro-differential boundary value problem is discretized to a system of nonlinear differential-algebraic equations, in which the unknowns are the positions of the nodes of the computational grid, along with the corresponding values of the potential and of its normal derivative. Time advancing of the nonlinear differential-algebraic system is performed using implicit Backward Differentiation Formulas (BDF) with variable step size and variable order, and the collocated and iso-parametric BEM discretization of the Laplace's equation has been implemented using the open source C++ library (www.dealii.org). These results are related to the OpenSHIP and OpenViewSHIP projects (www.openship.it).