# Fast Prediction of Flow Field around Airfoils Based on Deep Convolutional Neural Network
## 3 Results and Discussion
### Investigation of the Extensionality of the CNN-DCNN Model  
All the previous results and analyses are based on flow conditions with fixed Reynolds numbers (\(3\times 10^{6}\), \(5\times 10^{6}\) or \(7\times 10^{6}\)), as the far-field velocity of airfoil changes continuously in practice, we study the extensionality of the CNN-DCNN model with unseen Reynolds numbers not available in the training datasets. Specifically, the input parameters of the three cases analyzed in Section 3.3 are reused here with modified Reynolds numbers, namely \(Re=2.5\times 10^{6}\) (case 4), \(Re=6\times 10^{6}\) (case 5) and \(Re=8\times 10^{6}\) (case 6). Compared with the training dataset, case 4 is an interpolated case while the other two are extrapolated cases. Figure 16 shows the predicted flow fields of the three extended cases, in comparison with those by CFD calculation. It is appreciated that the model predictions are quite close to the results obtained by much more expensive CFD calculations. According to the relative errors displayed in the last column of Figure 16, the largest deviation appears in case 4, as the NACA9712 airfoil has the largest camber and its position along the chord is closer to the trailing edge compared with the other two cases. The prediction accuracies for the three cases are reported in Table 9, which confirm that airfoils with larger deformation turn to decreasing the prediction accuracy of the CNN-DCNN model, possibly due to more complex flow structures.  
Figure 15: POD/CNN-DCNN comparison of the reconstructed physical field (case 3: NACA6212, AOA = \(3^{\circ}\), \(Re=7\times 10^{6}\)): (**a**) pressure field; (**b**) velocity field.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Pressure Field (\%)** & **Velocity Field (\%)** \\ \hline CNN-DCNN & 99.1913 & 99.7574 \\ POD with 5 basic modes & 86.4541 & 93.2564 \\ POD with 10 basic modes & 92.8026 & 95.6101 \\ POD with 20 basic modes & 96.9733 & 97.6766 \\ POD with 30 basic modes & 98.6831 & 98.9191 \\ POD with 50 basic modes & 99.9648 & 99.9925 \\ POD with 70 basic modes & 99.9825 & 99.9969 \\ \hline \hline \end{tabular}
\end{table}
Table 8: CNN-DCNN/POD comparison of accuracy for case 3.  
Following the routine of Section 3.2, contour lines of pressure and velocity fields of the extended cases are explicitly given in Figure 17, so that discrepancies in spatial flow structures are more visible. For both pressure and velocity, the CNN-DCNN model managed to reproduce contour lines with good accuracy even in the separation area,  
Figure 16: The true (CFD) and predicted flow fields of three cases under the extended Reynolds numbers: (**a**) case 4: NACA9712, AOA = 0\({}^{\circ}\), \(Re\) = 2.5 \(\times\) 10\({}^{6}\); (**b**) case 5: NACA2412, AOA = 5\({}^{\circ}\), \(Re\) = 6 \(\times\) 10\({}^{6}\); (**c**) case 6: NACA6212, AOA = 3\({}^{\circ}\), \(Re\) = 8 \(\times\) 10\({}^{6}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Accuracy** & **Pressure Field (\%)** & **Velocity Field (\%)** \\ \hline Case 4 (\(Re\) = 2.5 \(\times\) 10\({}^{6}\)) & 98.6469 & 99.5619 \\ Case 5 (\(Re\) = 6 \(\times\) 10\({}^{6}\)) & 98.8817 & 99.7842 \\ Case 6 (\(Re\) = 8 \(\times\) 10\({}^{6}\)) & 99.1802 & 99.7544 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Accuracy for three cases under the extended Reynolds numbers.  
suggesting good extensionality of the proposed model when dealing with unseen flow situations, at least in the investigated range of Reynolds numbers.