*3.4. Estimation of Numerical Uncertainties*

Verification and validation of the numerical method for calculations of the bare hull resistance have been carried out. ITTC V & V (Verification and Validation) 2008 [34] A validation procedure was used to estimate the uncertainties of the numerical method.

In general, the numerical uncertainty *USN* includes the following aspects: uncertainty of iteration steps *UI*, the uncertainty of the grid space *UG*, the uncertainty of the time step *UT* and the other parameters uncertainty *UP*. For URANS solver HUST-Ship, *UI*, and *UP* could be ignored utilizing lots of iterations. For the specific computation in this study, *UG* and *UT* are of the most concern. Therefore, the numerical uncertainty *USN* can be expressed as follows:

$$\mathcal{U}\_{\rm SN}{}^2 = \mathcal{U}\_{\rm G}{}^2 + \mathcal{U}\_{\rm T}{}^2 \tag{18}$$

Systematic grid-spacing and time-step studies were carried out using the generalized Richardson extrapolation method according to the literature [35].

At first, the uniform parameter refinement ratio *rk* between solutions is assumed as:

$$
\sigma\_k = \frac{\Delta x\_2}{\Delta x\_1} = \frac{\Delta x\_3}{\Delta x\_2} \tag{19}
$$

in which Δ*x*1, Δ*x*2, and Δ*x*<sup>3</sup> are the space of the coarse grid, medium grid, and fine grid or time step, respectively. *S*1, *S*2, and *S*<sup>3</sup> are the calculation results obtained by fine, medium and coarse grid spacing or time step, respectively. ε<sup>21</sup> = *S*<sup>2</sup> − *S*<sup>1</sup> and ε<sup>32</sup> = *S*<sup>3</sup> − *S*<sup>2</sup> are the differences between solutions of medium-fine and coarse-medium grid spacings, respectively. The convergence ratio R is defined as:

$$R = \frac{\varepsilon\_{21}}{\varepsilon\_{32}} \tag{20}$$

There are three possible conditions:


When monotonic convergence is achieved, the Richardson extrapolation method can be used. The estimated numerical error δ*RE* and order of accuracy *PRE* can be calculated as:

$$
\delta\_{RE} = \frac{\varepsilon\_{21}}{r\_k^{P\_{RE}} - 1} \tag{21}
$$

$$P\_{RE} = \frac{\ln(r\_{32}/r\_{21})}{\ln r\_k} \tag{22}$$

*J. Mar. Sci. Eng.* **2020**, *8*, 745

The correction factor *CG* is defined as:

$$\mathbb{C}\_{\mathbb{G}} = \frac{r\_k P\_{RE} - 1}{r\_k^{P\_{TH}} - 1} \tag{23}$$

where *Pth* is an estimated value for the limiting order of accuracy as the spacing size goes to zero; generally, *Pth* = 2. The numerical error δ*SN*, benchmark result *SC* and uncertainty *UG*,*<sup>T</sup>* can be estimated from:

$$
\delta\_{\rm SN} = \mathcal{C}\_{\rm G} \cdot \delta\_{\rm RE} \tag{24}
$$

$$S\_{\mathbb{C}} = S - \delta\_{\text{SN}} \tag{25}$$

$$\|L\_{G,T} = \begin{cases} \left(2.4(1 - \mathcal{C}\_G)^2 + 0.1\right) |\delta\_{RE}|\_\nu & |1 - \mathcal{C}\_G| < 0.25\\ |1 - \mathcal{C}\_G| |\delta\_{RE}|\_\nu & |1 - \mathcal{C}\_G| \ge 0.25 \end{cases} \tag{26}$$

When *CG* is significantly less than or greater than 1, which means the solutions are far away from the asymptotic range, the numerical uncertainty *UG*,*<sup>T</sup>* can be calculated from:

$$\mathcal{U}\_{\rm G,T} = \begin{cases} \left( 9.6 \left( 1 - \mathcal{C}\_{\rm G} \right)^2 + 1.1 \right) |\delta\_{RE}| \downarrow |1 - \mathcal{C}\_{\rm G}| < 0.125\\ \left( 2 \left| 1 - \mathcal{C}\_{\rm G} \right| + 1 \right) |\delta\_{RE}| \downarrow |1 - \mathcal{C}\_{\rm G}| \ge 0.125 \end{cases} \tag{27}$$
