## Computational Analysis of Multiphase Ship Resistance including 6-DoF Motion using OpenFOAM

With the increasing capability of computational resources in the past two decades, computational fluid dynamics (CFD) became a crucial tool for analysis in naval hydrodynamics. As a consequence, CFD can be used to tackle a complex application like multiphase ship resistance as well even though the cost of simulation is still considerable.

In the scope of this project a transient multiphase solver is used to compute the ship resistance in turbulent incompressible flow including 6-DoF motion. The software of which this study is based is OpenFOAM, an Open Source object-oriented library for numerical simulations in continuum mechanics written in the C++ programming language.

This report presents the algorithm details used by OpenFOAM in terms of its computational complexity and provides a comparison of the results for various discretization schemes, boundary conditions and turbulence models.

The most convenient way to describe the fluid flow is to use Navier-Stokes equations. These equations are widely used in the fields that deal with flow problems. In a very general case Navier-Stokes equations consist of 7 partial differential equations (PDEs) and 7 unknowns. This set can be reduced by making simplifying assumptions. For an isothermal, incompressible, steady and 1D case we can express fluid flow using 2 equations:

$\frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \nu \nabla^{2} u_{i}$
$\frac{\partial u_{i}}{\partial x_{i}} = 0$

First equation can be derived by Newton's 2nd law and is known as the momentum equation. The second equation is an expression for mass conservation and is used to resolve the pressure term from the first equation.

The momentum equation needs additional terms for an accurate description of multiphase flow. According to Shan and Delaney (H. Shan and K. Delaney, Guide to navyFOAM - 2011) in a two-phase flow the momentum equation becomes:

$\frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{1}{Re} \nabla^{2} u_{i} + \rho g + \sigma \kappa \nabla \gamma_{i}$

where "\gamma" is the volume fraction, "g" is gravitational acceleration, "\sigma" is the surface tension coefficent and "\kappa" is the interface coefficient.

The convective transport equation for volume fraction is:

$\frac{\partial \gamma_{i}}{\partial t} + u_{j} \frac{\partial \gamma_{i}}{\partial x_{j}} = 0$

This equation is solved simultaneously along with momentum and continuity equations. The indicator for free surface, "\gamma" represents the phase fraction of each phase and is defined in the interval [0, 1] where 0 denotes air phase, 1 corresponds to the water phase and transitional cells between these two phases have values between 0 and 1. There is no mass or energy transport between two phases.

Due to the possible convergence problems that might arise because of the unboundedness of the convective transport equation, OpenFOAM uses a modified form of the volume fraction given above (see H. Rusche, Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions - 2002):

$\frac{\partial \gamma_{a}}{\partial t} + \nabla \cdot \left( \mathbf{U} \gamma_{a} \right) + \nabla \cdot \left[ \mathbf{U_{r}} \gamma_{a} \left( 1 - \gamma_{a} \right) \right]$