## 2 Simulation Setup
### Solving Methods for Flow Simulations  
The principle of fluid dynamics necessitates adherence to three fundamental conservation laws: the law of mass conservation, the law of momentum conservation (expressed  
Figure 3: Surface position division of NACA2412 airfoil.  
Figure 2: Distribution of \(\beta\) on the NACA2412 airfoil surface.  
through the Navier-Stokes equations), and the law of energy conservation. Their respective mathematical expressions are denoted by Equations (2)-(4):  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=S_{m} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}(\rho u_ {i}u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ij}}{ \partial x_{j}}+\rho g_{i}+F_{i} \tag{3}\]  
\[\frac{\partial(\rho T)}{\partial t}+div(\rho uT)=div\bigg{(}\frac{k}{C_{p}} gradT\bigg{)}+S_{T} \tag{4}\]  
In the provided equation, \(\rho\) represents fluid density, \(t\) denotes time, \(x_{i}\) represents the length in the i-direction, \(u_{i}\) corresponds to the velocity components in the i-direction, and \(S_{m}\) stands for a user-defined source term. The variable \(p\) signifies the static pressure of the fluid element, \(\tau_{ij}\) denotes the stress tensor on the element's surface, \(g_{i}\) and \(F_{i}\) respectively represent the gravitational force and external volumetric forces in the i-direction, \(C_{p}\) is the specific heat capacity, \(T\) represents the static temperature, \(k\) is the fluid heat transfer coefficient, and \(S_{T}\) accounts for internal heat generation within the fluid and viscous dissipation. These conservation equations, along with various other fluid dynamic equations, collectively form a set of nonlinear partial differential equations using numerical methods to approximate analytical solutions.  
When simulating the flow near the grooves, [PERSON] et al. [39] conducted a comparative analysis between the numerical methods of Reynolds Averaged Navier-Stokes (RANS) and Direct Numerical Simulation (DNS). The findings revealed that the RANS method not only was cost-effective but also met the required accuracy. Furthermore, the work by [PERSON] et al. [40] demonstrated that, in comparison with experimental results, the RANS method also could effectively simulate the realistic flow around airfoils. Consequently, in this study, the RANS approach was employed for flow simulation computations. Within the RANS, the velocity is decomposed into time-averaged velocity and instantaneous velocity, introducing Reynolds stress terms in the equations. It is necessary to employ turbulence models to close the system of equations. [PERSON] et al. [41] compared the experimental results and evaluated the accuracy of the k-omega SST model in predicting parameters such as airfoil pressure, lift, and drag coefficient at low Reynolds numbers, believing that it is suitable for MAVs. Combining their research and considering the sensitivity of the k-omega SST model to flow separation and the capability to accurately capture near-wall flow characteristics [42], it was selected for the numerical simulation.  
The transport equations of the model are as follows, where \(P_{k}\) and \(P_{\omega}\) are the generating terms, and k and \(\omega\) are turbulent kinetic energy and dissipation coefficient, respectively.  
\[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=\frac{1} {\rho}P_{k}-\beta\kappa\omega+\frac{1}{\rho}\frac{\partial}{\partial x_{j}} \Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{ \partial x_{j}}\Bigg{]} \tag{5}\]  
\[\frac{\partial\omega}{\partial t}+u_{j}\frac{\partial\omega}{\partial x_{j}}= \frac{1}{\rho}P_{\omega}-\beta\omega^{2}+\frac{1}{\rho}\frac{\partial}{ \partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\omega}}\bigg{)} \frac{\partial\omega}{\partial x_{j}}\Bigg{]} \tag{6}\]  
Subsequently, the accuracy of the numerical simulation method was validated in the following sections.