# Numerical Study of Flow Separation Control over a Hump Airfoil Using a Novel Sweeping Jet Actuator
## 2 Set Up and Numerical Simulation Methods  
In the course of Sweeping jet actuator research, the prevailing design aspects encompass traditional dual-feedback actuators [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35], single-feedback actuators, and feedback-free actuators [38]. The shapes of these three models are depicted as shown in Figure 2. However, during the design process of these models, it is not feasible to decouple the oscillation frequency from dimensions or velocity ratio. In the author's previous research on the SWJ actuator [22], a combination of numerical simulations and experiments was used to study the internal flow of the SWJ actuator. This research verified that the frequency of the SWJ actuator is proportional to the momentum coefficient or velocity ratio. Figure 2: The shapes of three actuators: (**a**) double-feedback actuator [21], (**b**) single-feedback actuator, and (**c**) feedback-free actuator [38]. Therefore, in the current study, the design of the SWJ actuator model is innovative in that it replaces the feedback channel with synthetic jets while keeping other geometric dimensions and shapes unchanged. The positions of the two synthetic jets correspond to the locations of the feedback channel inlets in a traditional SWJ actuator, allowing for control of the jet at the inlet to achieve periodic sweeping. The dimensions of the novel SWJ actuator are shown in Figure 3.  
In this study, ICEM CFD was used for mesh generation. As shown in the Figure 4, the grid diagram of the Sweeping jet actuator is shown. The setting of boundary conditions is the same as previous research [38], the position of the synthesized jet is set as the velocity inlet, and the other settings are set as non-slip boundary conditions. In this study, the SST k-\(\omega\) turbulence model was chosen, and its equations are shown as follows:  
\[\frac{\partial(pk)}{\partial t}+\frac{\partial(\rho u_{i}k)}{\partial x_{j}}= \tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}-\beta^{*}\rho\omega k+\frac{ \partial}{\partial x_{j}}\bigg{[}(\mu+\sigma_{k}\mu)\frac{\partial k}{\partial x _{j}}\bigg{]} \tag{1}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u_{j}\omega)}{ \partial x_{j}}=\frac{\gamma}{\mu_{t}}\tau_{ij}\frac{\partial u_{i}}{\partial x _{j}}-\beta\rho\omega^{2}+\frac{\partial}{\partial x_{j}}\bigg{[}(\mu+\sigma _{\omega}\mu_{t})\frac{\partial k}{\partial x_{j}}\bigg{]}+2(1-F_{1})\frac{ \rho\sigma_{\omega}\omega^{2}}{\omega}\frac{\partial k}{\partial x_{j}}\frac {\partial\omega}{\partial x_{j}} \tag{2}\]  
\[\gamma=\frac{\beta_{i}}{\beta^{*}}-\frac{\sigma_{\omega}\kappa^{2}}{\sqrt{ \beta^{*}}} \tag{3}\]  
\[F_{1}=\tanh\Big{(}\phi_{1}^{4}\Big{)} \tag{4}\]  
Figure 4: Grid diagram of the Sweeping jet actuator. Figure 3: New Sweeping jet actuator model and dimensions. \[\phi_{1}=\min\left[\max\left(\frac{\sqrt{k}}{0.09\omega y^{*}},\frac{500\mu}{y^{2} \omega}\right),\frac{4\rho k}{\sigma_{\omega 2}D_{\omega}^{+}y^{2}}\right] \tag{5}\]  
Figure 2:0  
The expression for the model turbulent viscosity is shown in Equations (7)-(9) as follows:  
Figure 2:1  
Figure 2:2  
Figure 2:3  
In Equations (1)-(9), \(\tau_{ij}\) represents the Reynolds stress tensor, \(\Omega\) is the vorticity tensor, \(\mu\) is the laminar viscosity, \(\mu_{t}\) is the turbulent viscosity, and y is the distance to the wall. The model constants are as follows: \(a_{1}=0.31\), \(\beta^{*}=0.09\), \(\kappa=0.41\), \(\sigma_{k1}=0.85\), \(\sigma_{\omega 1}=0.5\), \(\sigma_{k2}=1.0\), \(\sigma_{\omega 2}=0.856\). The two-dimensional Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations are solved using the commercial computational fluid dynamics (CFD) software, Ansys Fluent v22.0. Air is used as the working fluid. A pressure-based, time-dependent solver with a constant time step of t = 10\({}^{-5}\) s is employed. In selecting the velocity for the synthetic jet, insights from the study [15] revealed that the velocity field within the feedback channel can be represented using sinusoidal or cosine functions.