NUMERICAL SIMULATIONS OF THE FREE ROLL DECAY MOTION FOR THE ITALIAN NAVY SHIP "NAVE BETTICA"

In this report the work that has been done for the 6DoF-RANS II/MOU project, tasl L.2 {it ''Previsione dell'andamento temporale del decadimento del moto di rollio e del flusso in corrispondenza delle alette antirollio alle stesse condizioni di prova realizzate al vero di cui Sublotto I, mediante il solutore RANSE non stazionario--Prediction of the roll decay motion and of the flow field around the antiroll fins at the same sea trials tests conditions as in subtask I, by using an unsteady RANS solver''} is presented. Numerical simulations have been carried out for three different Froude numbers; the steady flow around the vessel with a fixed heel angle, and the unsteady free roll decay of the vessel from an initial of heel angle of 10 degrees are computed. As well known, simulations carried out by means of the numerical solution of the unsteady Reynolds Averaged Navier--Stokes (URANSE) equations can be used for the analysis of manoeuvring, sea keeping related problems and, in general, for the prediction of the motion of a surface vessel as in the case under consideration here of the free roll decay of a ship. Some very recent results (see, for example, the works by Broglia et al., Carrica et al., Cura-Hackbaum, Hyman et al., Jacquin et al., Simonsen and Stern presented at least ONR Symposium held in Rome ~cite{26ONRrome}) have demonstrated its effectiveness. Similarly, a second order accurate URANSE solver was used for the simulation presented in the present paper. The solver is based on a finite volume discretization of the unsteady incompressible Navier--Stokes equations, with advection terms evaluated by means of a third order upwind scheme, and diffusive fluxes discretized by a second order centered scheme. Physical time-derivatives are approximated by a second order accurate, three--point backward finite difference formula. In order to both satisfy the divergence-free constrain on velocity field at every physical--time step, and to remove constrains on the physical time step, the time discretization is fully implicit. The resulting system of coupled non linear algebraic equation is solved by a dual- or pseudo- time integration. To this purpose, a pseudo-time derivative is introduced in the discrete system of equations and the asymptotic solution with respect to pseudo time is computed at each physical time step, with the physical time derivative acting as a source term. Convergence acceleration for the inner iteration is achieved by means of both local pseudo-time step and an efficient multigrid technique. Turbulent viscosity is computed by means of a one-equation turbulence model cite{spall}. A single--phase level set approach is applied for the simulation of the free surface evolution cite{LS_CaF}. In order to deal with complex geometries and multiple bodies in relative motion, dynamic overlapping grid capabilities are implemented in the algorithm. Vessel trajectory can be predicted by the integration of the 6DoF rigid body motion equations. The report is organized as follows: in the next section, the mathematical models with the proper boundary and initial conditions will be briefly recalled; then, the numerical schemes will be described, together with the single phase level set approach for free surface simulation and the space discretization technique applied to simulate the flow. Numerical results will follow.