## 2 Computational Framework
### Governing Equations and Solution Details
The momentum-thickness Reynolds number transport equation was used to capture the nonlocal effect of freestream-turbulence intensity and pressure gradient at the boundary-layer edge, which indicates where transition onset occurs, and is defined as:  
\[\frac{\partial\left(\rho\widetilde{\rm R}e_{\rm\theta t}\right)}{\partial t}+ \frac{\partial\left(\rho\rm Uj\widetilde{\rm R}e_{\rm\theta t}\right)}{ \partial\chi_{\rm j}}=\rm P_{\rm\theta t}+\frac{\partial}{\partial\chi_{\rm j }}\left[\sigma_{\rm\theta t}(\mu+\mu_{\rm t})\frac{\partial\widetilde{\rm R}e _{\rm\theta t}}{\partial\chi_{\rm j}}\right] \tag{4}\]  
The transport equation for intermittency is used to trigger the transition process (\(\gamma>0\)) and is defined as:  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial\left(\rho\rm Uj\gamma \right)}{\partial\chi_{\rm j}}=\rm P_{\gamma}-\rm E_{\gamma}+\frac{\partial} {\partial\chi_{\rm j}}\left[\left(\mu+\frac{\mu_{\rm t}}{\sigma_{\rm f}} \right)\frac{\partial\gamma}{\partial\chi_{\rm j}}\right] \tag{5}\]  
However, when the boundary separates, the modification of intermittency to unity is defined as:  
\[\gamma_{\rm sep}=\rm min\bigg{\{}8\max\bigg{[}\left(\frac{\rm Re_{\rm v}}{2. 193\rm Re_{\rm\theta c}}\right)-1,\,0\bigg{]}e^{-\left(\frac{\rm Re_{\rm v}}{ 15}\right)^{4}},5\bigg{\}}\rm F_{\rm\theta t} \tag{6}\]  
In the above equations, (5), \(\rm F_{\rm length}\), and \(\rm Re_{\rm\theta c}\) are two key functions, in which the former controls the transition extent and the later determines the onset of transition. The source term \(\rm P_{\gamma}\) is activated when the local strain-rate Reynolds number exceeds the local transition-onset criterion. The destruction source term \(\rm E_{\gamma}\) enables the relaminarization prediction when the transition-onset criterion is no longer satisfied, and vanishes in the fully turbulent regime. A complete description of the model is available in the article by [PERSON] et al., 2006 [19]. This model has been used by numbers of researchers for low \(\rm Re\) transitional flows. For example, a detailed study on two specific parameters (\(\rm F_{\rm length}\) and \(\rm F_{\rm onset}\) are used in the intermittency equation for controlling the length of transition region and onset location of transition, respectively) was shown by [PERSON] [21]. [PERSON] [22] conducted a validation study for the model for low \(\rm Re\) number flows. According to the comparison between the numerical and experimental data, it was shown that the \(\gamma-\rm Re_{\rm 0}\) model accurately predicted the location and extent of the two-dimensional laminar separation bubble. [PERSON] [23] also studied the two-dimensional airfoils using the transition model, showing that the transition model was accurate in the preturbulent regions. A comparative study of four airfoils at low Reynolds numbers using the different transition modeling methods, namely, \(\gamma-\rm Re_{\rm 0}\) and \(\rm e^{N}\), was performed by [PERSON] [24]. A blended first- and second-order accurate scheme was implemented for the spatial discretization, which switched from the latter to the former in regions of steep spatial gradients, based on the boundedness principle of [PERSON] [25]. SIMPLEC algorithm [26] with a staggered grid and the second order-accurate central difference method was enforced, and such pressure-velocity coupling used coordinated under-relaxations for the momentum and pressure corrections to improve the convergence which was inherently slow in the original SIMPLE method. An implicit second-order quadratic backward approximation with an iterative procedure was realized for temporal discretization. The nonlinear coefficients were updated within each inner loop while the outer loop advanced the solution in time. The entire computational domain consisted of a rotational zone containing the propeller and a stationary zone containing the MAV wing and fuselage, as shown in Figure 1. The multiple zones were connected with each other through non-conformal mesh [27].