# Fast Prediction of Flow Field around Airfoils Based on Deep Convolutional Neural Network
## 3 Results and Discussion
### Flow Field Prediction Analysis  
Following the methodology given in Section 2.2, the training datasets of the CFD results are fed to the CNN-DCNN model, with training parameters detailed in Section 3.1. Prediction of flow field of the trained model is then compared with the CFD results of the test datasets. Table 4 shows the mean accuracy of prediction for both pressure and velocity fields. Although both prediction accuracies are above 98%, the accuracy for velocity prediction is slightly better than that of the pressure. This is due to the non-negligible numerical differences between the pressure and velocity fields, despite being normalized before being fed into the neural network. Given the accuracy level of the model prediction, it is worth mentioning that prediction for a single case takes only 25 ms on a graphics processing unit (GPU), while a corresponding CFD calculation takes 3 min. Admittedly, the training process would spend 25 min before the proposed model become functional, as shown in Table 5, the CNN-DCNN framework might still be able to greatly reduce the computational cost of a pure CFD-based method, especially for aerodynamic design and optimization where a large number of parameters have to be studied and tested. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Hyperparameter** & **Learning Rate** & **Mini-Batch Size** & **Dropout** &
\begin{tabular}{c} **Regularization** \\ **Coefficient** \\ \end{tabular} \\ \hline Notation & \(\alpha\) & \(k\) & \(d\) & \(\lambda\) \\ Value & 0.0005 & 32 & 0.2 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Hyperparameters setting. Figure 9: Convergence history (sampled every 200 epochs) under different hyperparameters of the (**a**) learning rate, (**b**) batch-size, (**c**) random deactivation rate (dropout). Figure 10 compares the predicted and calculated flow fields of 3 airfoils with different geometries and AOAs, at the same Reynolds number of 5 \(\times\) 10\({}^{6}\). Visual inspections indicate that the predicted fields are almost identical to calculated ones for both pressure and velocity, with small relative errors for the majority part of the flow region. The predicted stagnation point of the leading edge is clearly visible, where the positive pressure reaches the maximum value and the airflow velocity drops to zero. For both predicted and calculated flow fields, we could find a local high-velocity region on the upper surface of the airfoil, as well as a long wake region behind the trailing edge. Typically, the maximum relative errors are found around the leading and trailing edges of the airfoil, due to the flow separation induced by the circulation and incoming flow. Specifically, the error band of velocity on the trailing edge of the airfoil almost coincide with the wake caused by flow separation. Therefore, the reasons for these errors may be the sharp numerical changes, but it should be noted that the errors appear much smaller. Comparison of the predicted and calculated field indicates that the proposed CNN-DCNN model is accurately able to predict the physical fields for the various airfoils, thus potentially substituting part of CFD workloads during aerodynamic design and optimization. Figure 11 compares of the gradient of the horizontal component of velocity post-processed from the calculated and predicted velocity field, which is closely coupled with viscous force, especially in the near wall region. As we can see from the relative error, the velocity gradient of the predicted field matches well with CFD calculation. The drag coefficient is one of the most widely used performance indicators for airfoil design, which measures the integral effect of pressure and velocity field around the airfoil. Table 6 gives the calculated and drag coefficient of the two cases in Figure 11, where the relative errors of the model prediction are below 1.3% compared with corresponding CFD calculations.