# A Parametric Study of Wind Pressure Distribution on Facades Using Computational Fluid Dynamics
## 3 Model Convergence
### Convergence of Peak Pressures
In each figure, the left-hand portion of the image illustrates results for the entire building and the right-hand portion of the image is zoomed in on the critical fin. Figure 5: Peak Pressure Distributions for 3 mesh resolutions with respect to distance along the fin from its top edge (**a**) windward side (**b**) leeward side (**c**) net pressures. Peak pressures have a 95% confidence interval of not being exceeded within the simulation. In Figure 5c, 16 of the 17 sample points measure a consistent reduction in peak net pressures, with decreasing cell size. The exception occurs at the top part of the fin closest to the building where peak net pressures increase as mesh resolution increases. The mean relative error \(e_{12}\) is 18.9% while \(e_{23}\) is 13.1%. Therefore, resultant peak net pressures on the critical fin tend to converge, independent of the mesh size. Net pressures provide a reliable quantification of how airflow translates to loading on an obstacle. While there is variance in the windward pressure case, the resultant loading can account for the complexity of the flow. Therefore, proof of convergence is based on the calculation of net force versus mesh resolution. Net force is calculated from the trapezoidal integral of the net pressures. It is positive acting rightwards as per net pressures. The units for net force are defined as kilonewtons-per-meter of height (of the fin) in the elevation of the facade. The relative error for peak net force \(e_{12}\) was 19.9% while \(e_{23}\) equaled 13.1%. This methodology was extended to investigate mean and min forces as well. An L-shaped bend is apparent in Figure 6a and therefore it was clear that peak net force converges independent of mesh resolution. Mean force was also seen to converge. Minimum values represent outlier readings where in only 5% of instances, a smaller value was reported. The moment generated from the net force onto the connection between the bottom of the bracket and the top of the fin (in plan-view) was then calculated. For this purpose, using the net pressure distribution, the lever arm, \(\overline{x}\) was calculated as:  
\[\overline{x}=\frac{\sum_{n=1}^{n}f_{n}x_{n}}{\sum_{n=1}^{n}f_{n}} \tag{5}\]  
where \(f_{n}\) represents the area of each trapezoidal section and \(x_{n}\) is the distance from the top of the fin to the center of each trapezoidal section. The results, reflecting the position of the net force at a distance from the bracket's connection to the fin, are given in Table 3.  
To derive net moments, the following equation was utilized:  
\[Net\;Moment=F_{Net}\overline{x} \tag{6}\]  
where \(F_{Net}\) and \(\overline{x}\), are the net pressure derived resultant force and lever arm, respectively. As with peak net forces, net moments converge independent of changes in mesh resolution. For an \(e_{12}\) equal to 21.8%, there is an improvement in reducing to a 0.5 mm mesh at a value of 13.9% for \(e_{23}\) (Table 4). By increasing mesh resolution, the net moment decreases between each change (0.233 kNm/m in Mesh 1, 0.182 kNm/m in Mesh 2 and 0.157 kNm/m in Mesh 3). The resultant L-shaped curve is apparent in Figure 6b. As is also the case for net forces, average and minimum net moment values do not converge in the manner highlighted for peak loading conditions. Minimum values represent outlier data necessary for statistical analysis but are not useful for design--so proof of convergence is not required. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Peak (m)** & **Mean (m)** & **Min (m)** \\ \hline Mesh 1 (2 mm) & 0.250 & 0.239 & 0.347 \\ Mesh 2 (1 mm) & 0.244 & 0.237 & 0.286 \\ Mesh 3 (0.5 mm) & 0.242 & 0.239 & 0.225 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of level arm lengths based on mesh resolution and loading condition. However, as average net forces and moments are considered important in choosing wind loading for facade design, they should ideally converge.