**2. Mathematical formulation**

The slightly compressible Navier-Stokes equations written in 3-D Cartesian coordinates are given below in the conservative form with the volume fraction equation included.

$$\left(Q\_{\ell} + (E\_{\mathcal{C}} - E\_{\mathcal{D}})\_{\mathcal{X}} + (F\_{\mathcal{C}} - F\_{\mathcal{D}})\_{\mathcal{Y}} + (G\_{\mathcal{C}} - G\_{\mathcal{D}})\_{\mathcal{Z}} = H \tag{1}$$

$$
\rho = \rho\_w c + \rho\_{air} (1 - c) \tag{2}
$$

where

$$Q = \begin{pmatrix} p \\ \rho u \\ \rho v \\ \rho w \\ c \end{pmatrix}, \qquad H = \begin{bmatrix} 0 \\ \rho \left(\frac{M\_{\infty}}{F\_{n}}\right)^{2} h\_{x} \\ \rho \left(\frac{M\_{\infty}}{F\_{n}}\right)^{2} h\_{y} \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag{3}$$

$$E\_{\mathcal{C}} = \begin{pmatrix} pu \\ \rho u^2 + p \\ \rho vu \\ \rho vu \\ cu \end{pmatrix} \qquad G\_{\mathcal{C}} = \begin{pmatrix} pv \\ \rho uw \\ \rho vw \\ \rho vw \\ cw \end{pmatrix} \qquad F\_{\mathcal{C}} = \begin{pmatrix} pv \\ \rho uv \\ \rho uv \\ \rho wv \\ \rho wv \\ cv \end{pmatrix} \tag{4}$$

$$E\_{\mathcal{D}} = \frac{M\_{\upsilon}}{R\_L} \begin{bmatrix} 0\\ u\_{\mathcal{X}} \\ v\_{\mathcal{X}} \\ w\_{\mathcal{X}} \\ 0 \end{bmatrix} \quad G\_{\mathcal{U}} = \frac{M\_{\upsilon}}{R\_L} \begin{bmatrix} 0\\ u\_{\mathcal{Z}} \\ v\_{\mathcal{Z}} \\ w\_{\mathcal{Z}} \\ w\_{\mathcal{Z}} \\ 0 \end{bmatrix} \quad F\_{\mathcal{U}} = \frac{M\_{\upsilon}}{R\_L} \begin{bmatrix} 0 \\ u\_{\mathcal{Y}} \\ v\_{\mathcal{Y}} \\ w\_{\mathcal{Y}} \\ 0 \end{bmatrix} \tag{5}$$

Equation (1) is solved numerically together with the initial conditions (6), boundary conditions on the body surface (7), and free-stream boundary conditions (8), where *M*=0.2 for incompressible flows, and =1 (water) and =0 (air).

Initial conditions:

396 Computational Simulations and Applications

sphere without free surface and around a series-60 ship hull in order to verify the implemented code. The agreement between the numerical results and the experimental and numerical data from the literature indicates that the implemented code is able to reproduce correctly the drag coefficient, pressure field, velocity field, and the free-surface elevation

The slightly compressible Navier-Stokes equations written in 3-D Cartesian coordinates are

(1 ) *w air*

*<sup>p</sup> <sup>M</sup> <sup>h</sup> u F*

 

*w h*

2

*w p cw*

0

*uz*

0

*Ge vw*

*M*

*<sup>G</sup> <sup>v</sup> <sup>v</sup> <sup>z</sup> RL wz* 

Equation (1) is solved numerically together with the initial conditions (6), boundary conditions on the body surface (7), and free-stream boundary conditions (8), where *M*

> =0 (air).

*pw uw*

 

*Q EE FF GG H ev ev e v <sup>t</sup> x y <sup>z</sup>* (1)

2

*x*

*y*

<sup>2</sup>

*Fe v p*

*M*

 

*F v <sup>v</sup> <sup>y</sup> RL*

*pv uv*

*wv cv*

0

*uy*

0

*wy*

(4)

(5)

=0.2

0

*n*

0 0

*n*

*F*

2

*c c* (2)

(3)

given below in the conservative form with the volume fraction equation included.

 

,

*c*

2 *pu u p*

 

*wu cu*

0

*ux*

0

=1 (water) and

*Ee vu*

*M*

*<sup>E</sup> <sup>v</sup> <sup>v</sup> <sup>x</sup> RL wx* 

for incompressible flows, and

*Q H <sup>v</sup> <sup>M</sup>*

around a ship hull.

where

**2. Mathematical formulation** 

$$\begin{cases} p = 1\\ \mu = M\_{\infty} \\ \upsilon = 0 \\ w = 0 \\ c = \beta \end{cases} \tag{6}$$

Boundary conditions on the body surface:

$$\begin{cases} \left< \partial p \right> \left< \partial n = 0 \\ \mu = 0 \\ \upsilon = 0 \\ w = 0 \\ \left< \partial c \right> \left< \partial n = 0 \right. \end{cases} \tag{7}$$

Free stream boundary conditions:

$$\begin{cases} p = 1\\ \mu = M^{\circ} \\ \upsilon = 0 \\ w = 0 \\ c = \beta \end{cases} \tag{8}$$
