This master thesis presents the computation of free-surface flow phenomena with emphasis on ship hydromechanics. We perform a simulation of the heave and the pitch motion of the DTMB 5415M ship moving in head sea. To this purpose we employ an existing Arbitrary-Lagrangian-Eulerian residual-based variational multiscale method (ALE-RBVMS) to discretize the incompressible Navier-Stokes equations and use the level-set method to capture the interface. We validate the numerical results obtained from the ALE-RBVMS computation with experimental data. We show that these results significantly improve upon those acquired from other CFD codes. This demonstrates the viability of the framework for simulations in ship hydromechanics.