## Numerical Simulations
### Dispersion of Exhaust Smoke Gases
#### 3.2.2 Grid Uncertainty Assessment  
To determine the appropriate level of the grid system, a grid dependency test was performed by adopting the procedure proposed by [PERSON] et al. of the American Society of Mechanical Engineers [19]. The effects of the grid systems on the NO\({}_{2}\) concentration measured 1 m after the exhaust pipe were examined, as shown in Figure 6. Only NO\({}_{2}\) was considered because the NO contained in the exhaust smoke reacted with oxygen and easily transformed into NO\({}_{2}\).  
For the numerical simulations, the typical operating conditions were assumed as follows: the load of the main engine was 25% of the maximum continuous rating, that of the auxiliary engine was 100%, and the head wind was 15 knots. For the boundary conditions of the exhaust gas pipes, 300 ppm and 900 ppm of NO\({}_{2}\) concentrations from the main and auxiliary engines, respectively, were assumed to be mixed with air. The corresponding mass-flow rates for the two concentrations above were 8000 and 6000 kg/h, respectively.  
Three levels of unstructured grid systems were investigated, for which the total numbers of grids were 807,000, 1,711,000 and 3,559,000. The grids near the exhaust pipes are shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Boundaries**} & \(\
u\) & \(p\) & \(k\) & \(\epsilon\) & \(\gamma_{i}\) \\ \hline \multirow{2}{*}{Ship} & Hull & No slip & \multirow{2}{*}{Zero gradient} & \multicolumn{2}{c}{Zero gradient} \\  & Pipe inlet & Fixed value & & & Fixed value \\ \hline \multirow{4}{*}{Domain} & Inlet & Fixed value & Zero gradient & \multirow{2}{*}{Fixed value} & \multirow{2}{*}{Zero gradient} \\  & Outlet & Zero gradient & Fixed value & & Zero gradient \\ \cline{1-1}  & Sides & Fixed value & Zero gradient & & Fixed value \\ \cline{1-1}  & Top & & & Zero gradient & \\ \cline{1-1}  & Bottom & Zero gradient & Free slip & & Zero gradient \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary conditions.  
Figure 5: Computational domain and boundaries with zoomed area near the pipes in a pink rectangular.  
Figure 6: NO\({}_{2}\) concentration measurement point for grid dependency test.  
The representative cell (h) ratio between each grid system was maintained at approximately 1.28, whereas the expansion ratio of the spatial grid was maintained constant. The apparent order (\(p\)) of this method can be expressed as  
\[p=\frac{1}{\ln(r_{21})}|\ln|e_{32}/e_{21}|+q(p)|, \tag{4}\]  
\[q(p)=\ln\left(\frac{r_{21}^{p}-1\cdot sgn\left(\frac{e_{32}}{e_{21}}\right)}{r _{32}^{p}-1\cdot sgn\left(\frac{e_{32}}{e_{21}}\right)}\right), \tag{5}\]  
where \(r_{21}=h_{2}/h_{1}\) and \(r_{32}=h_{3}/h_{2}\). Meanwhile, \(e_{32}=\phi_{3}-\phi_{2}\)and\(e_{21}=\phi_{2}-\phi_{1}\), where \(\phi_{k}\) is the solution for the kth grid. The NO\({}_{2}\) concentration was measured at a position equal to the exhaust pipe height and 1.5\(L\) away from the stern.  
The order was calculated using a fixed-point iteration. The extrapolated values were calculated as follows:  
\[\phi_{ext}^{21}=\left(r_{21}^{p}\phi_{1}-\phi_{2}\right)/\left(r_{21}^{p}-1, \right) \tag{6}\]  
Figure 7: Grid systems of (a) coarse, (b) medium and (c) fine levels near exhaust pipes for grid dependency tests.  
The approximate relative error (\(e_{a}^{21}\)), extrapolated relative error (\(e_{ext}^{21}\)), and fine-grid convergence index were calculated, in addition with the apparent order (GCI\({}_{fine}^{21}\)).  
\[e_{a}^{21}=\left|\frac{\phi_{1}-\phi_{2}}{\phi_{1}}\right|, \tag{7}\]  
\[e_{ext}^{21}=\left|\frac{\phi_{ext}^{12}-\phi_{2}}{\phi_{ext}^{12}}\right|, \tag{8}\]  
\[\text{GCI}_{fine}^{21}=\frac{1.25e_{a}^{21}}{r_{21}^{p}-1}, \tag{9}\]  
The error due to the grids was estimated to be approximately 4%, as shown in Table 2, where the details of the grid convergence tests are listed.