# A Power Based Analysis for a Transonic Transport Aircraft Configuration through 3D RANS Simulations
## 3 Results and Discussion
### The Analysis of Power Imbalance Error
#### 3.1.1 Mesh Size Study and Subdomain Error Study  
This section presents the power imbalance error of the entire computational domain. In the first place, all the power input and output terms listed in Table 1 were evaluated. For the wing body configuration, the only power input term is the drag power, DV\({}_{\infty}\), while the output terms are the wake power at the outlet plane, \(\hat{\text{E}}_{w,\text{out}}\), and the viscous dissipation, \(\Phi\). Once the 3D flow field was obtained through numerical simulations, these power terms were evaluated according to their expressions, as given in Section 2.1. The power imbalance error of the entire computational domain, \(e_{domain}\), was assessed according to Equation (13). The input power, DV\({}_{\infty}\), is used as the reference power to calculate the relative error of \(e_{domain}\).  
A mesh size study is presented in Table 5. It shows that the relative differences in drag coefficient (with respect to the experiment result) were less than 3%. The aircraft drag was independent of the mesh size and its value agreed well with the experimental results. On the other hand, the value of \(e_{domain}\) ranged from 36.4% to 66.3%. Even though \(e_{domain}\) tended to decrease as mesh size increased, the errors obtained in the presented 3D simulation were much higher than the 4% value obtained in the previous incompressible 2D simulations [23]. In general, the error in 3D flow simulation was far from a satisfactory level. It is critical to identify the main courses of the high value of \(e_{domain}\).  
To find the major sources of power imbalance error, \(e_{domain}\), this section introduces a so-called subdomain error study. As introduced in Section 2.4, it is possible to inspect the power imbalance for an individual subdomain. In this study, all the power terms listed in Table 1 were assessed for the individual subdomains. With these inputs, the errors of the power imbalance for the individual subdomains, \(e_{sub}\), were computed according to Equation (14). For demonstration purposes, the results of the unstructured medium mesh are presented. The errors were plotted at the corresponding X location of the survey planes, as shown in Figure 10. The suberror denotes the error of an individual subdomain, while the bulk-error is defined as the accumulation of the suberrors from the first subdomain to the subdomain at the X location. This bulk error depicts how the error increases along the X axis. It is noted that the bulk error at the outlet plane simply equals \(e_{domain}\).  
This subdomain error study clearly showed that the error in the subdomain of the wing (denoted by \(e_{wing}\)) made the major contribution to \(e_{domain}\). The \(e_{wing}\) value was 26.8%, while \(e_{domain}\) was 43.0%. The contributions of the other subdomains were significantly low: The suberror for the frontal body was less than 5%. The suberror in the downstream region was less than 0.5%. The error associated with the wing, \(e_{wing}\), was mainly responsible for the high errors of \(e_{domain}\). This study suggests that efforts should be made to reduce the suberror associated with the wing, \(e_{wing}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh** & **Cell Number** & **C\({}_{\text{D}}\) (Relative Difference)** & \(e_{domain}\) \\ \hline Experiment [24] & - & 0.0295 & - \\ Unstructured coarse & 3.3 million & 0.0299 (1.5\%) & 66.3\% \\ Unstructured medium & 4.1 million & 0.0299 (1.5\%) & 43.0\% \\ Unstructured fine & 7.8 million & 0.0289 (\(-\)1.9\%) & 38.5\% \\ Structured medium & 5.1 million & 0.0303 (2.7\%) & 37.5\% \\ Structured fine & 8.4 million & 0.0298 (1.0) & 36.4\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mesh size study presented by the drag coefficient and the power imbalance error of the computational domain.