# A Parametric Study of Wind Pressure Distribution on Facades Using Computational Fluid Dynamics
## 3 Model Convergence
### Convergence of Peak Pressures  
Of primary interest are the maximum pressures that occur in the event of one-in-50-year wind conditions. To be consistent with Limit State Design practices used in the Eurocode, peak wind pressures were calculated using Equation (1) and are presented in Figure 4:  
\[95\%\;\mathrm{P}_{\mathrm{wind}}\leq[\mathrm{P}_{\mathrm{av}}+1.64\;\mathrm{P }_{\mathrm{rms}}] \tag{1}\]  
where \(\mathrm{P}_{\mathrm{wind}}\) represent the wind pressures, \(\mathrm{P}_{\mathrm{av}}\) is the time averaged pressure and \(\mathrm{P}_{\mathrm{rms}}\) is the root mean square error of pressures. Figure 4 illustrates that peak pressures acting on the critical fin only vary slightly due to changes in mesh size. The root mean square error statistic acts as the sample standard deviation of the pressure in each of the simulations and was calculated as:  
\[\textit{Sample}\;SD=RMSE=\sqrt{\sum_{n=n_{t_{0}}}^{n_{T}}\frac{(\overline{p}- p_{t})^{2}}{n_{t}}}kN/m^{2}. \tag{2}\]  
This was done with respect to several samples at a set frequency which is termed '\(n_{t}\)'. The duration of data sampled begins at initial time '\(t_{0}\)' and a final time of 'T'. The quantitative differences between each mesh were then examined to prove convergence with respect to the critical fin. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Y\({}^{+}\)** & **Y\({}^{+}\)** & **Y\({}^{+}\)** & **Co** & **Co** & **Co** \\  & **(Max)** & **(Mean)** & **(Variance)** & **(Max)** & **(Mean)** & **(Variance)** \\ \hline Mesh 1 (2 mm) & 290.1 & 212.3 & 830.6 & 38.1 & 27.7 & 12.4 \\ Mesh 2 (1 mm) & 157.2 & 112.0 & 276.6 & 38.4 & 19.2 & 12.8 \\ Mesh 3 (0.5 mm) & 78.8 & 58.3 & 63.1 & 33.4 & 24.2 & 8.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values for Y\({}^{+}\) and Courant Number. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Cells** & **Ratio** & \multicolumn{2}{c}{**Refinement (r)**} \\ \hline Mesh 1 (2 mm) & 78,962 & Mesh\({}_{1-2}\) & 1.414 & Mesh\({}_{1-2}\) & 1.189 \\ Mesh 2 (1 mm) & 111,688 & Mesh\({}_{2-3}\) & 1.376 & Mesh\({}_{2-3}\) & 1.173 \\ Mesh 3 (0.5 mm) & 153,652 & – & – & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of cells, Ratio and Refinement factors for three mesh types. Figure 4: _Cont._  
A key measure for determining convergence is the relative error indicator:  
\[e_{12}=\left|\frac{y_{1}-y_{2}}{y_{1}}\right| \tag{3}\]  
where \(y_{1}\) and \(y_{2}\) are the quantity of interest for Mesh 1 and Mesh 2, respectively. In general, for a solution to converge, a value between meshes must have the relation:  
\[e_{12}\geq e_{23}=\left|\frac{y_{2}-y_{3}}{y_{2}}\right| \tag{4}\]  
while also showing a persistent increase, or decrease, in measuring the quantity based on the mesh resolution. In Figure 5a, only three of the 17 sample points show a consistent decrease in peak pressure with greater mesh refinement. The average relative error between Mesh 1 and Mesh 2 (\(e_{12}\)) is 3.3%, and between Mesh 2 and Mesh 3 (\(e_{23}\)) is 12.3%. This increase in error with mesh density establishes a lack of convergence. However, the problem is not limited to this windward case, and the complexity associated with maintaining the continuity of airflow means that leeward suction influences these values. There was close agreement found between Mesh 2 and Mesh 3, relative to the pressure distribution associated with the coarsest mesh (Figure 5b). Sixteen of the 17 sample points demonstrated a reduction in the magnitude of suction pressure, with decreasing cell sizes. The exception occurred at a corner location that induces more instability, which corresponds to higher Courant Number readings than elsewhere. The mean relative error \(e_{12}\) was 29.2% and \(e_{23}\) was 13.8%. Therefore, peak suction pressures on the leeward face of the critical fin tended to converge, independent of mesh size. Figure 4: Peak wind pressures (**a**) Mesh 1 (2 mm), (**b**) Mesh 2 (1 mm), (**c**) Mesh 3 (0.5 mm).