# Numerical Investigation of Jet Angle Effect on Airfoil Stall Control
## 1 Introduction
It, however, comes at a price of non-negligible reduction in the aerodynamic lift. Furthermore, the effect of the jet angle on the flow control was not separated from other jet parameters including the jet location and the jet size. It seems that the effect of the jet angle on the flow control performance needs to be investigated systemically. Compared to the high jet angle of the chord-normal jet, a shallow angle would allow the jet to reside in the boundary layer longer and to energize the flow more, which helps to delay the stall and improve the aerodynamic performance of a lifting body. The flow control performance of a jet flow is primarily determined by three major parameters: the jet location \(x_{jet}\), the jet angle \(\theta_{jet}\) and the jet momentum, normally given by the jet momentum coefficient \(c_{\mu}=\frac{m_{jet}\tilde{v}_{jet}}{\rho_{\infty}L_{\infty}^{2}bc}\) where \(\tilde{m}_{jet}\) is the mass flow rate of the jet, \(v_{jet}\) is the jet velocity, and \(b\) is the span length of a wing. Based on previous studies with steady-blowing jet [16; 19; 20; 30] and oscillatory jets [31], an upstream jet location near the leading edge \(x/c\simeq 0.1\) outperforms downstream locations. Flow control studies with pulsed jets [21; 22; 23] also indicate that positions near the leading edge \(0.05\leq x_{jet}/c\leq 0.15\) are able to mitigate the flow separation further when compared to a slightly downstream location of \(x/c=0.2\). A significant jet momentum is required to achieve meaningful control authority over the separation boundary layer [15]. [PERSON] and [PERSON] [32] and [PERSON] [33] presented that the steady-blowing jet can be detrimental, if the momentum coefficient \(c_{\mu}\) is \"low\", i.e., \(c_{\mu}\lesssim 0.01\). A large jet momentum up to \(c_{\mu}\approx 0.05\) can improve the performance of a lifting body, increasing both the lift coefficient \(c_{l}\) and the rate \(dc_{l}/dc_{\mu}\). Thus many studies of jet blowing (steady or unsteady) for flow control have been conducted with the jet momentum in the order of the magnitude of \(c_{\mu}=\mathcal{O}(0.01)\)[16; 17; 18; 19; 21; 22; 23]. A concise review of the blowing parameter can be found in [34; 35]. Following the experiments [16; 17; 18], the jet momentum coefficient \(c_{\mu}=0.06\) is used in this study, satisfying the earlier founding of the minimum requirement for the jet momentum. In contrast to the jet location and the jet momentum, the impact of the jet angle on the separation control has not been systematically investigated in the literature. Recent studies on steady-blowing jet [16; 17; 18; 19] primarily focus on a large jet angle \(\theta_{jet}\) with respect to the wing surface. Although a small jet angle was also used in the computations of [19], other jet parameters including the jet location were also varied at the same time. It was not straightforward to distinguish the effect of the jet angle on the stall control from other jet parameters. Therefore, in the current study, of the primary interest is the effect of the jet angle on the dynamic stall. The hypothesis in this study is that a jet ejected at a shallow angle toward the airfoil surface can add the jet momentum more to the boundary layer, compared to a larger jet angle. Two distinct jet angles are selected to test this hypothesis. A small enough angle is determined for a tangential jet which ejects almost tangentially to the airfoil surface. A large jet angle normal to the chord line is used for a chord-normal jet. Those two angles are \(\theta_{jet}=22^{\circ}\) and \(80^{\circ}\) in this study. The shallow angle \(\theta_{jet}=22^{\circ}\) is suggested for a practical limit for the fabrication of a jet actuator in [PERSON] et al. [21; 22]. Computational methods are followed in Section 2. In Section 3, computational results of the nominal airfoil are compared to relevant literature data [19; 36; 37; 38]. Then, stall control with the variation of the jet angle is investigated for both the static and dynamic stalls. Conclusions are followed in Section 4.