*2.2. URANS Settings*

Dynamic stall of the airfoil is obtained by sinusoidal pitch oscillation about the quarter-chord axis. The sliding mesh method [26] is used to simulate the dynamic motion of airfoil. The instantaneous AOA follows the sinusoidal variation:

$$
\alpha = \alpha\_{\rm m} + A \sin(2\pi \theta) \tag{1}
$$

The reduced frequency is defined as *k* = π*fc*/*U*0. In this work, the deep dynamic stall condition of α*<sup>m</sup>* = 18.75◦, *A* = 10.3◦, and *k* = 0.078 is simulated following the wind-tunnel experiments [27].

In the mesh motion simulated by sliding mesh method, nodes rigidly move in a given dynamic zone, but the cells defined by these nodes will not deform. A sliding interface is also introduced to connect multiple cell zones. The sliding interface is updated and synchronized with the mesh motion to reflect the new positions. Therefore, the computational domain is divided into two subdomains. The inner is a rotating region, and the outer a stationary region. The interaction between these two regions is made through a cylindrical sliding interface at radius of 4.4c in this work.

The commercial software ANSYS/FLUENT 16.0 [26] is used to numerically solve the URANS equations. Table 2 provides the main URANS settings. Ekaterinaris and Platzer [28] found that the proper consideration of transitional flow effect can improve the predictive accuracy of aerodynamic hysteresis. Therefore, the turbulence is simulated by the SST *k*-ω eddy viscosity model [29] incorporated with the γ-Re<sup>θ</sup> transition model [30]. This turbulence modelling has been proven to be reliable in simulating the dynamic stall of wind turbine airfoils [31]. The time step is set to assure 540 steps over each cycle and 20 inner iterations per time step, based on our previous works [13]. Iterative convergence criterion is met by assuring the cycle-to-cycle force variations negligible.



#### *2.3. Validation of Numerical Modelling*

Due to the lack of experimental data of dynamic stall with VGs (unsteady-controlled), present numerical modelling has been validated against two sets of available experimental data: steady-controlled and unsteady-uncontrolled.

For steady-controlled data, the numerical modelling can reliably predict the pressure distributions of the DU97-W-300 airfoil with and without triangular VGs (Figure 4). Figure 4 also demonstrates that VGs are effective in suppressing the trailing-edge separated flow at α = 15◦, thereby leading to a high leading-edge suction and greatly increasing the lift coefficient *Cl*. VGs, however, have a marginal effect on the pressure distribution when the flow is fully attached at α = 10◦.

**Figure 4.** Calculated and measured pressure distributions of the DU97-W-300 airfoil with and without VGs. (**a**) Without VGs; (**b**) with VGs (*x*VG/*c* = 20%).

For unsteady-uncontrolled data, the obtained results also show a good agreement with the experimental data of the NREL S809 airfoil in light dynamic stall [23]. Moreover, Figure 5 suggests that the calculated aerodynamic hysteresis loops generally agree with the experimental data [27] and Johansen's CFD results [31] in deep dynamic stall. The hysteresis loops also show noticeable fluctuations at the high AOAs. This is due to the severe vortex shedding and passage over the suction surface. The unsteady aerodynamic forces are accurately predicted during the flow separation and flow reattachment processes. Consequently, present numerical modelling of the deep dynamic stall of the NREL S809 airfoil with VGs should be adequately correct.

**Figure 5.** *Cl*-α and *Cd*-α hysteresis loops of the NREL S809 airfoil undergoing deep dynamic stall (α*<sup>m</sup>* = 18.75◦, *A* = 10.3◦, and *k* = 0.078). (**a**) *Cl*; (**b**) *Cd*.

#### **3. Results and Discussion**
