**1. Introduction**

Although passive vortex generators (VGs) are very simple, they have been proven to suppress the flow separation effectively and then boost the aerodynamic performance of horizontal axis wind turbines (HAWTs) [1]. Conventional VGs are composed of some pairs of vanes sticking out from the surface, angled to the incoming flow [2]. The height of VGs is close to the boundary-layer thickness. The fundamental principle of VGs is to produce streamwise vortices. These vortices can reenergize the boundary layer to resist the adverse pressure gradient.

The effectiveness of VG designs is primarily determined by the evolution of streamwise vortices. This vortex evolution is further impacted by various VG parameters. Godard and Stanislas [3] measured the boundary layer flow of a two-dimensional bump with VGs, using stereo particle image velocimetry (PIV) and hot-film probes. They found the triangular VGs better than rectangular VGs in decreasing the

drag penalty at low angles of attack (AOAs). The counter-rotating configuration was also found better than the co-rotating one in generating upwash and downwash wake regions. Mueller-Vahl et al. [4] carried out wind-tunnel measurements of the NACA 63(3)-618 airfoil equipped with triangular VGs. Their research implied that decreasing the spanwise spacing of VGs could not only delay the onset of static stall, but also cause a high drag penalty. Baldacchino et al. [5] systematically studied the effect of VG parameters on the aerodynamic performance of DU97-W-300 airfoil using wind-tunnel experiments. They found that the vane height and chordwise location of VGs are the main factors in the airfoil performance. The chordwise location plays a significant role in the post-stall behavior of airfoil. Positioning VGs too downstream can result in an early abrupt stall, because VGs become very prone to be submerged in the separation zones [4,5]. Wang et al. [6] studied the effect of rectangular VGs on the performance of NREL S809 airfoil by URANS simulations. Compared to single-row VGs, they found double-row VGs to further suppress the flow separation and then further delay the static stall.

RANS-based simulations of airfoil flow with VGs are by far the most common, although some researches were performed by highly expensive DNS/LES-type simulations [7,8]. Nevertheless, RANS methods also need a large quantity of computational cost, because the boundary layer with VGs requires adequately fine resolution. To reduce the cost of fully resolved RANS method, the VG modelling is often simplified [9]. The idea is to add a flow-dependent forcing term to the momentum equations based on the thin airfoil theory. This method can successfully predict the aerodynamic performances of both the airfoil and blade with VGs [10,11]. However, this simplified modelling has to calibrate the key coefficient first.

VGs have succeeded in aerospace engineering and have been practically applied in wind turbine engineering (Figure 1). However, the HAWT blade flow controlled by VGs remains unclear. Most studies have focused on the two-dimensional steady airfoil flow controlled by VGs. In contrast, the blade flow is three-dimensional, rotational, and often becomes unsteady. The unsteady operating conditions are attributed to complicated environmental effects such as wind gust, turbulent inflow, and yaw misalignment [12,13]. The blade sections therefore undergo a time-varying AOA. If the AOA variation is dramatic enough, dynamic stall of the rotating blade will occur [14].

**Figure 1.** Applications of passive vortex generators (VGs) on the aircraft wings and wind turbine blades [15,16].

Dynamic stall is characterized by the shedding and passage of a strong vortical disturbance over the suction surface, thereby causing a highly nonlinear fluctuating pressure field [17]. Dynamic stall often means the unsteady blade loads and there is noticeable aerodynamic hysteresis. These unsteady aerodynamic forces are directly linked to the structure failures, reduced turbine life, and increased operating maintenance.

Therefore, some methods were proposed to control dynamic stall, including aerodynamic blowing [18], trailing-edge flap [19], co-flow jet [20], and plasma actuator [21]. These existing ways can be classified as active control techniques, which will introduce auxiliary power equipment. This leads to a more complicated design of blades. Consequently, active control techniques are often limited to wind turbine blades [22]. In contrast, passive control techniques can also improve the wind turbine performance without external energy expenditure, among which VGs are very cost-effective.

Nevertheless, the effect of VGs on dynamic stall has been rarely investigated and hence is still poorly understood. Our previous works [23,24] demonstrated that VGs could effectively suppress the flow separation of oscillating wind turbine airfoil, thereby attenuating the aerodynamic hysteresis. In this regard, double-row VGs are more effective than single-row VGs. Our previous works only focused on the light dynamic stall controlled by VGs. However, the deep dynamic stall is often accompanied by stronger vortex motions and severer flow separation.

This work aims to investigate the effect of single-row and double-row VGs on the deep dynamic stall. The URANS method is used to identify the unsteady airfoil flow characteristics with and without VGs. The aerodynamic hysteresis loops, flow structures, and boundary-layer velocity profiles are analyzed in detail to reveal the effect of VGs on deep dynamic stall.
