## 3 Validation of Numerical Simulation Method
### Validation of Aerodynamic Force  
Referring to reference [25], the numerical simulation of the NACA0012 airfoil at ultra-low Reynolds number of 5,300 is carried out. The range of the angle of attack of the airfoil is \(\alpha=0^{\circ}\)-\(25^{\circ}\), chord length of airfoil is \(c=0.1\) m. The experiment is conducted in a water tunnel, and the water flow velocity is \(U=0.053\) m/s. Because the experimental velocity of the water tunnel is too small, the incompressible flow condition is transformed into low-speed incompressible air flow according to the criterion of Reynolds similarity. The size of the airfoil is scaled to 1/40 of the original size, namely \(c^{\prime}=0.0025\) m. The inflow velocity of air inlet boundary is \(V=30.96\) m/s. In order to verify the influence of the quality of computation mesh, five angles of attacks of the NACA0012 airfoil were calculated and compared with the experimental values. Figure 3 represents the assessment between the experiments and the results of the CFD computations with three different meshes: 52,500 (coarse), 73,000 (medium), and 95,000 (fine) cells. The results of the mesh dependency study is shown in Table 1. Exp represents the experiment results, R the ratio of errors between Fine and Exp. All lift coefficients converged monotonically for all angles of attack and the value of R is less than 0.15. As illustrated in Figure 3, the numerical results obtained with the fine mesh predicted the experimental lift well. Therefore, this higher resolution mesh was used for the current computations. It should be noted that because the due to value of lift at \(2^{\circ}\) angle of attack is small, the error of the numerical calculation result is relatively large. The computational mesh contains 95,000 cells, and the first layer grid element to wall distance is \(d_{1}=6.53\times 10^{-4}c\), \(y^{+}=0.25\). Figure 2: Comparison of numerical calculation and experiment results of pressure distribution. CFD: computational fluid dynamics; EXP: experiment. A definite curve of the decay law of experimental turbulence intensity at \(Tu\) = 6% is given in reference [25], as is shown in Figure 4a. The range between airfoil leading edge and trailing edge is X/\(M\) = 14-16. A decay law of turbulence intensity curve is fitted by changing turbulence scale and turbulent dissipation rate in numerical calculation and is shown in Figure 4b, where the range between airfoil leading edge and trailing edge is \(X\) = \(-\)0.0325--0.03. The comparison between the numerical and experimental results of different turbulence intensities is depicted in Figure 5. Figure 5a indicates that the lift coefficient fits well within the angle of attack of \(10^{\circ}\) at \(Tu\) = 6%. As the angle of attack increases, there is no definite stall phenomenon at \(\alpha\) = 12\({}^{\circ}\)-15\({}^{\circ}\), which is slightly different from the experimental results. As a result of the lack of corresponding decay law of experimental turbulence intensity at \(Tu\) = 0.6%, the calculation error is large. Even so, the trends of the lift and drag coefficients are consistent with the experimental results. \begin{table}
\begin{tabular}{c c c c c c} \hline
**AOA** & **Coarse** & **Medium** & **Fine** & **Exp** & **R** \\ \hline
2 & 0.1324 & 0.1256 & 0.1123 & 0.098 & 0.146 \\
5 & 0.3276 & 0.3207 & 0.2950 & 0.282 & 0.046 \\
8 & 0.6323 & 0.6218 & 0.6224 & 0.612 & 0.017 \\
10 & 0.7314 & 0.7421 & 0.7507 & 0.785 & 0.044 \\
12 & 0.7231 & 0.7748 & 0.7893 & 0.856 & 0.078 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of the mesh independency study. AOA: angle of attack; R: ratio of errors between Fine and Exp. Figure 4: Decay law of turbulence intensity along the flow direction. (**a**) Turbulence intensity change in experiment; (**b**) Turbulence intensity change in numerical simulation. Figure 3: Lift coefficient of numerical simulations versus experiment data of the NACA0012 airfoil. The airfoil characteristics are further analyzed in the following section.