**2. Methodology**

#### *2.1. Approach*

Figure 1 schematically illustrates the methodology used in this study. An Unsteady Reynolds Averaged Navier–Stokes (URANS)-based CFD model was developed to replicate the physical Wigley hull model test of Song et al. [27]. The different hull roughness conditions were modelled using the modified wall-function approach with the roughness function model (Δ*U*+) of Song et al. [25]. The CFD simulations were performed with different hull conditions and compared with the towing test results [27]. Finally, the local wall shear stress and the roughness Reynolds number on the hull surfaces were examined to be correlated with the findings.

**Figure 1.** Schematic illustration of the current methodology.

#### *2.2. Numerical Modelling*

2.2.1. Mathematical Formulations

The URANS method was used to solve the governing equations using STAR-CCM+. The averaged continuity and momentum equations for incompressible flows can be given as

$$\frac{\partial(\rho \overline{u}\_i)}{\partial x\_i} = 0 \tag{1}$$

$$\frac{\partial(\rho \overline{u}\_i)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho \overline{u}\_i \overline{u}\_j + \rho \overline{u\_i' u\_j'}\right) = -\frac{\partial \overline{p}}{\partial \mathbf{x}\_i} + \frac{\partial \overline{\tau}\_{ij}}{\partial \mathbf{x}\_j} \tag{2}$$

where, *ρ* is the fluid density, *ui* is the time-averaged velocity vector, *ρu i u <sup>j</sup>* is the Reynolds stress, *p* is the time-averaged pressure, *τij* is the mean viscous stress tensor components. This viscous stress for a Newtonian fluid can be expressed as

$$\overline{\pi}\_{ij} = \mu \left( \frac{\overline{\partial \overline{u}\_i}}{\overline{\partial x\_j}} + \frac{\overline{\partial \pi}\_j}{\overline{\partial x\_i}} \right) \tag{3}$$

In which, *μ* is the dynamic viscosity. The governing equations were discretised using the finite volume method with a second-order convection scheme and a first-order temporal discretisation. The *k*-*ω* SST turbulence model [30] was adopted to capture the turbulent flow, which combines the advantages of the *k*-*ω* and the *k*-ε turbulence models. For the free surface effects, the Volume of Fluid (VOF) method was used.

#### 2.2.2. Modified Wall-Function Approach

The roughness effect causes a downward shift of the velocity profile in the turbulent boundary layer. This downward shift is often called the "roughness function", Δ*U*+. With the roughness function, Δ*U*+, the log-law of the turbulent boundary layer can be written as

$$
\Delta U^{+} = \frac{1}{\kappa} \ln y^{+} + B - \Delta U^{+} \tag{4}
$$

In which, *U*<sup>+</sup> is non-dimensional velocity defined as the ratio between the mean velocity, and the frictional velocity (i.e., *U*<sup>+</sup> = *U*/*Uτ*). *U<sup>τ</sup>* is defined as *τw*/*ρ* where *τ<sup>w</sup>* is the wall shear stress and *ρ* is the fluid density. *y*<sup>+</sup> is the non-dimensional length defined as *yUτ*/*ν*, in which *y* is the normal distance from the wall and *ν* is the kinematic viscosity. *κ* is the von Karman constant (=0.42) and *B* is the log-low intercept.

The roughness function, Δ*U*<sup>+</sup> is a function of roughness Reynolds number, *k*+, defined as

$$k^{+} = \frac{kU\_{\pi}}{\nu} \tag{5}$$

In which, *k* is the roughness height of the surface. The modified wall-function Equation (1) can be employed in CFD simulations to predict the roughness effect in the flow over the rough wall.

Song et al. [24] evaluated the roughness function of the sand grit (60/80 grit aluminium oxide abrasive powder) from the results of flat plate towing tests. Later on, a mathematical model of the roughness function (i.e., roughness function model) was proposed by Song et al. [25] to be used in CFD simulations.

As the same sand grit was used for the Wigley hull towing test [27], the same modified wall-function approach [25] was employed in this study to model the heterogeneous hull roughness conditions in CFD simulations. As proposed by Song et al. [25], the roughness function model for the 60/80 grit sand grain surface can be written as

$$
\Delta \mathcal{U}^+ = \begin{cases} 0 & \rightarrow & k^+ < 3 \\ \frac{1}{\kappa} \ln \left( 0.49k^+ - 3 \left( \frac{k^+ - 3}{25 - 3} \right) \right)^{\sin \left( \frac{\pi}{2} \frac{\log(k^+ / 3)}{\log(25/3)} \right)} & \rightarrow & 3 \leq k^+ < 25 \\ & \frac{1}{\kappa} \ln (0.49k^+ - 3) & \rightarrow & 25 \leq k^+ \end{cases} \tag{6}$$

where *k*<sup>+</sup> is the roughness Reynolds number based on the peak roughness height over a 50 mm interval (i.e., *k* = *Rt*<sup>50</sup> = 353 μm). As shown in Figure 2, the roughness function model of Song et al. [25] agrees well with the experimental roughness function [24].

**Figure 2.** Experimental roughness function of Song et al. [24] and the roughness function model of Song et al. [25] Equation (6).

#### 2.2.3. Geometry and Boundary Conditions

The Wigley hull is a parabolic hull form represented as

$$y = \frac{B}{2} \left[ 1 - \left(\frac{2x}{L}\right)^2 \right] \left[ 1 + \left(\frac{z}{T}\right)^2 \right] \tag{7}$$

where, *L*, *B* and *T* are the length, waterline beam and the draught of the model. In the current CFD simulations, the Wigley hull was modelled using the principal particulars used for the physical towing tests of Song et al. [27] as shown in Table 1.



As shown in Figure 3, the different hull roughness conditions tested by Song et al. [27] were modelled in CFD simulations. It is of note that the four regions of the Wigley hull have the same longitudinal length (i.e., *L*/4).

**Figure 3.** Different hull conditions of the Wigley hull simulations.

Figure 4 depicts the computational domain and the boundary conditions of the Wigley hull simulations used in this study. The computational domain size was chosen to be similar to that used by Dogrul et al. [31]. The velocity inlet boundary conditions were used for the inlet, top and bottom boundaries while the pressure outlet boundary condition was used for the outlet boundary. The centre midplane was defined as a symmetry plane. The no-slip wall boundary condition was used for the hull surface whilst simultaneously using different wall-functions. In other words, the modified wall-function, i.e., Equation (1) was used for the rough surfaces and the smooth-type wall-function (i.e., Equation (1) with Δ*U*<sup>+</sup> = 0) is used for the smooth surfaces. The model was free to heave and sink in the simulations.

**Figure 4.** Computational domain and the boundary conditions of the Wigley hull simulation.
