*3.1. Aerodynamic Hysteresis Loops*

Figure 6 shows the calculated aerodynamic hysteresis loops with and without VGs. During the upstroke process, the aerodynamic coefficients of the airfoil with single-row and double-row VGs were relatively close. VGs significantly delayed the onset of dynamic stall. The *Cl* of clean airfoil began to diverge from the linear regime at α = 16◦, implying the start of flow separation (Figure 6a). However, the *Cl* with VGs well followed the linear theory until α = 22◦. At the high AOAs, the strong

dynamic stall vortex motion caused large fluctuations in the aerodynamic coefficients of the clean airfoil. Figure 6 also indicates that the degree of these fluctuations can be decreased by VGs.

**Figure 6.** Aerodynamic hysteresis loops of the NREL S809 airfoil with and without VGs. Solid lines denote increasing angles of attack (AOA), and dashed lines indicate decreasing AOA. (**a**) *Cl*; (**b**) *Cd*; (**c**) *Cm*.

During the downstroke process, the aerodynamic coefficients showed a clear difference between single-row and double-row VGs. The downstroke process from αmax to αmin can be further divided into three parts:

• From αmax to α = 25◦, double-row VGs quickly restored the decreases in *Cl* and *Cd* in comparison with single-row VGs. This suggests that the second-row VGs impacted greatly on the massive flow separation when the airfoil began to pitch down.


Table 3 provides the dynamic-stall parameters extracted from the hysteresis loops in Figure 6. The definition of aerodynamic pitch damping ζCm is given by:

$$\zeta\_{\mathbb{C}m} = -\oint \mathbb{C}\_m d\alpha / \left( 4A^2 \right) \tag{2}$$

A high ζCm also implies a high torsional aeroelastic stability. If the ζCm decreases to a negative value, the amplitude of airfoil pitch will increase rapidly, and then the flutter occurs unless restrained [17].


**Table 3.** Dynamic-stall parameters of the airfoil with and without VGs.

Single-row and double-row VGs increased the *Cl*,max of NREL S809 airfoil by 40% and 49%, respectively. This is because double-row VGs could further delay the onset of dynamic stall. Both single-row and double-row VGs dramatically reduced the *Cm*,min by almost 70%. The reason is that VGs hindered the forward motion of the center of pressure with the trailing-edge flow separation effectively suppressed.

Table 3 also indicates a large decrease of 64% in the ζCm of the airfoil with single-row VGs. Therefore, single-row VGs can reduce the torsional aeroelastic stability, thereby likely causing the airfoil flutter. In this regard, double-row VGs are better to only reduce the ζCm from 0.153 to 0.124.

#### *3.2. Flow Field Developments*

Figure 7 illustrates the flow field developments around the airfoil with and without VGs. Three AOAs of 9.83◦, 18.75◦, and 27.67◦ were chosen to represent the three typical degrees of flow separation: fully attached flow, trailing-edge (TE) separated flow, and massively separated flow, respectively. The clean airfoil flow also showed wider separation zones during the downstroke process than during the upstroke process. This manifests that the aerodynamic hysteresis was attributed to the retarded flow reattachment when the airfoil pitched down.

During the upstroke process, both single-row and double-row VGs eliminated the TE separation vortex at α = 18.75◦. The *Cl* of airfoil with VGs was therefore dramatically increased (Figure 6a). At α = 27.67◦ (↑), there were three separation vortices on the upper surface of the clean airfoil. Two small separation vortices were located on the first half chord, and one large separation vortex was near the trailing edge. Furthermore, single-row and double-row VGs produced a fourth small TE separation vortex. This small vortex crowded out the large separation vortex, thereby leading to a high TE suction peak (Figure 8a). Surprisingly, the second-row VGs seemed to bring about an undesirable effect to reduce the TE suction peak.

During the downstroke process, the leading-edge (LE) and TE separation vortices shed into the wake alternately. At α = 27.67◦ (↓), the airfoil flow field with single-row VGs was highly distorted, because the LE separation vortex was hardly attached to the surface. Consequently, the suction value with single-row VGs was greatly decreased, even lower than that of the clean airfoil (Figure 8b). Double-row VGs, however, suppressed the LE flow separation effectively, and hence kept a high suction

on the first half chord. Additionally, both single-row and double-row VGs avoided the secondary separation vortex near the trailing edge.

**Figure 7.** Streamlines and pressure fields around the airfoil with and without VGs.

**Figure 8.** *Cont.*

**Figure 8.** Pressure distributions of the airfoil with and without VGs. (**a**) α = 27.67◦ (↑); (**b**) α = 27.67◦ (↓); (**c**) α = 18.75◦ (↓).

At α = 18.75◦ (↓), a large LE separation vortex appeared in the clean airfoil flow. The decreased height of separation vortex and the downstream movement of the vortex core suggest that the clean airfoil flow began to reattach gradually. However, both single-row and double-row VGs caused a tertiary vortex. This separation vortex was even detached from the upper surface due to double-row VGs, so that the suction on the upper surface and the *Cl* of airfoil with double-row VGs were vastly reduced (Figure 8c). Interestingly, although double-row VGs were positioned on the upper side, they significantly affected the *Cp* distribution on the lower side. This could decrease the pressure difference between the upper and lower sides. At α = 9.83◦ (↓), Figure 7 also implies that the second-row VGs further accelerated the flow reattachment and hence resulted in a high LE suction peak.

#### *3.3. Boundary-Layer Velocity Profiles*

Figures 9 and 10 show the non-dimensionalized streamwise velocity profiles when the AOA reached 27.67◦ during the upstroke and downstroke processes, respectively. Note that α = 27.67◦ means the airfoil flow fell into the deep dynamic-stall process. Boundary-layer velocity profiles at *x*/*c* = 10% and *x*/*c* = 75% can represent the LE and TE separation vortices, respectively.

**Figure 9.** Streamwise velocity profiles at α = 27.67◦ (↑). *Sn* denotes the normal distance away from the wall surface, and *u* the streamwise velocity. (**a**) *x*/*c* = 10%; (**b**) *x*/*c* = 75%.

During the upstroke process, the streamwise velocity profiles with single-row and double-row VGs were quite close (Figure 9). This is attributed to the similar flow fields around the airfoil with single-row and double-row VGs at α = 27.67◦ (↑) (Figure 7). Figure 9 also indicates that VGs increased the height of LE separation vortex and the severity of TE reverse flow. Nevertheless, the external flow was effectively accelerated by VGs, thereby producing a high suction value (Figure 8a).

**Figure 10.** Streamwise velocity profiles at α = 27.67◦ (↓). (**a**) *x*/*c* = 10%; (**b**) *x*/*c* = 75%.

During the downstroke process, the streamwise velocity profiles also showed a clear difference between single-row and double-row VGs (Figure 10). At *x*/*c* = 10%, although the boundary-layer thicknesses with double-row VGs and without VGs were close, the external flow was effectively accelerated due to double-row VGs (Figure 10a). This suggests that double-row VGs made the LE flow withstand a higher adverse pressure gradient. However, the boundary-layer thickness with single-row VGs was decreased, because the LE separation vortex seemed to be detached from the wall surface (Figure 7). At *x*/*c* = 75%, the boundary-layer thicknesses from high to low was in the sequence of single-row VGs, clean, and double-row VGs. This sequence also determined the severity of TE flow separation. Interestingly, double-row VGs could effectively counteract the adverse pressure gradient and then suppress the TE flow separation, but single-row VGs could not (Figure 8b). This finding highlights the great impact of the second-row VGs during the downstroke process.

#### **4. Conclusions**

This paper gives a flow analysis of deep dynamic stall of the NREL S809 airfoil controlled by single-row and double-row VGs. VGs were fully resolved, and URANS simulations were conducted with the transitional SST *k*-ω eddy viscosity model. Based on this study, several conclusions were reached as follows.


This paper also provides a performance assessment of VGs in controlling highly unsteady aerodynamic loads of a wind turbine airfoil. This study may contribute to understanding the deep dynamic stall controlled by single-row and double-row VGs. Future work should concentrate on the effect of passive VGs on a rotating blade undergoing dynamic stall.

**Author Contributions:** C.Z. conceived of the research and wrote the manuscript. C.Z. and J.C. conducted the data collection. T.W. and W.Z. contributed technical guidance and revised the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is funded by the National Key Research and Development Project under grant No. 2019YFB1503701-02, CAS Key Laboratory of Wind Energy Utilization under grant No. KLWEU-2016-0102, and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

**Acknowledgments:** The authors would like to express their gratitude to the conference chairs of SEGT 2019 for recommendation of this publication in *Energies*.

**Conflicts of Interest:** The authors declare no conflict of interest.
