Accuracy of the Cell-Set Model on a Single Vane-Type Vortex Generator in Negligible Streamwise Pressure Gradient Flow with RANS and LES

Passive flow control devices are included in the design of wind turbine blades in order to obtain better performance and reduce loads without consuming any external energy. Vortex Generators are one of the most popular flow control devices, whose main objective is to delay the flow separation and increase the maximum lift coefficient. Computational Fluid Dynamics (CFD) simulations of a Vortex Generator (VG) on a flat plate in negligible streamwise pressure gradient conditions with the fully-resolved mesh model and the cell-set model using Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS) were carried out, with the objective of evaluating the accuracy of the cell-set model taking the fully-resolved mesh model as benchmark. The implementation of the cell-set model entailed a considerable reduction of the number of cells, which entailed saving simulation time and resources. The coherent structures, vortex path, wall shear stress and size, strength and velocity profiles of the primary vortex have been analyzed. The results show good agreements between the fully-resolved mesh model and the cell-set mode with RANS in all the analyzed parameters. With LES, acceptable results were obtained in terms of coherent structures, vortex path and wall shear stress, but slight differences between models are visible in the size, strength and velocity profiles of the primary vortex. As this is considered the first application of the cell-set model on VGs, further research is proposed, since the implementation of the cell-set model can represent an advantage over the fully-resolved mesh model.


Introduction
Wind energy has become a key source of electricity generation for the change to a cleaner and sustainable energy model. In order to compete in energy production and cost against the traditional energies, improvement and optimization of wind turbines is required.
To improve the efficiency of wind turbine blades, both active (such as trailing edge flaps, air jet vortex generators and synthetic jets) and passive devices (such as vortex generators, microtabs, spoilers, fences, serrated trailing edge and Gurney flaps) can be added. Aramendia et al. [1] and Aramendia-Iradi et al. [2] thoroughly reviewed both active and passive flow control devices. Among these two types of devices, passive devices can be highlighted, since they improve performance of wind turbine blades without consuming any external energy.
Vortex Generators (VGs) are one of the most popular passive flow control devices due to their simplicity, effectiveness and low cost for both production and installation. They are small vanes which analyzed the flow structures of 25 different configurations of VG pairs by LES, and Saha et al. [17] investigated the coherent structures of a vane-type rectangular VG pair with a high incident angle.
Colleoni et al. [18] developed an innovative design of a solar plate receiver with VGs, and carried out CFD simulations using both RANS and LES turbulence models. The results show that LES allows for a better understanding of the behavior of the generated vortices and thermal exchanges.
This paper aims to analyze the capability of the cell-set model to reproduce the physics of the wake downstream of a VG on a flat plate in a negligible streamwise pressure gradient flow. As the above-mentioned studies demonstrate, the fully-resolved mesh model provides accurate results in VG problems. Therefore, CFD simulations are carried out using the fully-resolved mesh model and the cell-set model, and the obtained results are compared, in order to evaluate the accuracy of the cell-set model taking the fully-resolved mesh model as benchmark. RANS and LES turbulence models are selected in order to study the accuracy of the cell-set model with both turbulence models.

Numerical Setup
Numerical simulations were carried out in order to analyze the accuracy of cell-set model in a VG on a flat plate in a negligible streamwise pressure gradient flow. To conduct the simulations, Star CCM+v14.02.012 [19] CFD commercial code was used.

Computational Domain
The computational domain consists of a block with a rectangular VG located on its lower surface. The height of the VG (H) is equal to 0.25 m and the length (L) is twice its height. Two different incident angles (α) are considered, 18 • and 25 • . The lower surface of the block and the surfaces of the VG are considered walls with no-slip conditions, the upstream part of the domain is set as inlet and the downstream part of the domain is set as outlet. The rest of the surfaces are considered symmetry planes, in order not to affect the flow. The computational domain and the distance from the inlet to the VG position were designed in order to have the boundary layer thickness (δ) at the VG position equal to the height of the VG (δ = H). More information about the computational domain is shown in Figure 1.
This paper aims to analyze the capability of the cell-set model to reproduce the physics of the wake downstream of a VG on a flat plate in a negligible streamwise pressure gradient flow. As the above-mentioned studies demonstrate, the fully-resolved mesh model provides accurate results in VG problems. Therefore, CFD simulations are carried out using the fully-resolved mesh model and the cell-set model, and the obtained results are compared, in order to evaluate the accuracy of the cell-set model taking the fully-resolved mesh model as benchmark. RANS and LES turbulence models are selected in order to study the accuracy of the cell-set model with both turbulence models.

Numerical Setup
Numerical simulations were carried out in order to analyze the accuracy of cell-set model in a VG on a flat plate in a negligible streamwise pressure gradient flow. To conduct the simulations, Star CCM+v14.02.012 [19] CFD commercial code was used.

Computational Domain
The computational domain consists of a block with a rectangular VG located on its lower surface. The height of the VG (H) is equal to 0.25 m and the length (L) is twice its height. Two different incident angles (α) are considered, 18° and 25°. The lower surface of the block and the surfaces of the VG are considered walls with no-slip conditions, the upstream part of the domain is set as inlet and the downstream part of the domain is set as outlet. The rest of the surfaces are considered symmetry planes, in order not to affect the flow. The computational domain and the distance from the inlet to the VG position were designed in order to have the boundary layer thickness (δ) at the VG position equal to the height of the VG (δ = H). More information about the computational domain is shown in Figure 1.
For data extraction 12 points and 12 spanwise planes normal to the streamwise direction are considered. The planes are located from 3H to 25H of the LE (Leading Edge), separated 2H between each other. The points are located on the planes, downstream the TE (Trailing Edge) of the VG, at a height of 1H from the wall.

Physics Models
A three-dimensional incompressible flow with constant values of 1.225 kg/m 3 for density (ρ) and 1.85185 × 10 4 m 2 /s for kinematic viscosity ( ) is considered. The free stream velocity ( ) of the flow is set at 20 m/s, which means that the Reynolds number (Re) is around 27,000 according to Equation (1).
Two different turbulence models were selected for the simulations, Menter's k-ω SST (Shear Stress Transport) [20] RANS-based turbulence model for the simulations in which the incident angle is equal to 18° and LES Smagorinsky SGS (sub-grid-scale) [21] model for the simulations in which the incident angle is equal to 25°. Allan et al. [22] analyzed the flow field of a single VG on a flat plate with two different angles of attack, and showed that RANS-based simulations agree well with the experimental data, except for short distances downstream of the vane. In their study, a better performance of the SST turbulence model was found in comparison with other RANS turbulence models. RANS and LES turbulence models were selected for two main reasons. The first reason is that performing simulations with these two turbulence models allows one to analyze the accuracy of the cell-set model in both RANS and LES. The second reason is that, as shown by Urkiola et al. [5], when working with high incident angles, RANS is not able to accurately capture flow characteristics; therefore, LES is more appropriate when α = 25°.
For RANS-based simulations, a full second order linear-upwind scheme for the domain discretization has been used [23]. In LES, the large scales of the turbulence are directly resolved everywhere in the flow domain, and the small-scale motions are modelled. In contrast to the RANS equations, the equations that are solved for LES are obtained by a spatial filtering rather than an averaging process. Therefore, the variables of the flow are divided into a filtered value and a subgrid value. The filtered values are inserted in the Navier-Stokes equations. To give closure to these equations, the SGS stress tensor is modelled by an SGS model.

Fully-Resolved Mesh Model
A structured mesh of around 11.5 million hexahedral cells was generated. In order to enhance the accuracy of the results in the area near the VG, the cell density is much higher in this region. For data extraction 12 points and 12 spanwise planes normal to the streamwise direction are considered. The planes are located from 3H to 25H of the LE (Leading Edge), separated 2H between each other. The points are located on the planes, downstream the TE (Trailing Edge) of the VG, at a height of 1H from the wall.

Physics Models
A three-dimensional incompressible flow with constant values of 1.225 kg/m 3 for density (ρ) and 1.85185 × 10 −4 m 2 /s for kinematic viscosity (v) is considered. The free stream velocity (U ∞ ) of the flow is set at 20 m/s, which means that the Reynolds number (Re) is around 27,000 according to Equation (1).
Two different turbulence models were selected for the simulations, Menter's k-ω SST (Shear Stress Transport) [20] RANS-based turbulence model for the simulations in which the incident angle is equal to 18 • and LES Smagorinsky SGS (sub-grid-scale) [21] model for the simulations in which the incident angle is equal to 25 • . Allan et al. [22] analyzed the flow field of a single VG on a flat plate with two different angles of attack, and showed that RANS-based simulations agree well with the experimental data, except for short distances downstream of the vane. In their study, a better performance of the SST turbulence model was found in comparison with other RANS turbulence models. RANS and LES turbulence models were selected for two main reasons. The first reason is that performing simulations with these two turbulence models allows one to analyze the accuracy of the cell-set model in both RANS and LES. The second reason is that, as shown by Urkiola et al. [5], when working with high incident angles, RANS is not able to accurately capture flow characteristics; therefore, LES is more appropriate when α = 25 • .
For RANS-based simulations, a full second order linear-upwind scheme for the domain discretization has been used [23]. In LES, the large scales of the turbulence are directly resolved everywhere in the flow domain, and the small-scale motions are modelled. In contrast to the RANS equations, the equations that are solved for LES are obtained by a spatial filtering rather than an averaging process. Therefore, the variables of the flow are divided into a filtered value and a sub-grid value. The filtered values are inserted in the Navier-Stokes equations. To give closure to these equations, the SGS stress tensor is modelled by an SGS model.

Fully-Resolved Mesh Model
A structured mesh of around 11.5 million hexahedral cells was generated. In order to enhance the accuracy of the results in the area near the VG, the cell density is much higher in this region. Figure 2 shows the refined mesh near the VG. This mesh was used for both cases (α = 18 • and α = 25 • ), rotating it depending on the desired incident angle. The General Richardson Extrapolation method [24] applied to lift and drag forces of the VG was performed in order to verify sufficient mesh resolution for RANS. This method estimates the value of the analyzed parameter when the cell quantity tends to infinite. For more information about the General Richardson Extrapolation method, check Almohammadi et al. [25]. According to Stern et al. [26], a convergence study requires a minimum of three mesh solutions. Apart from the above-mentioned mesh (which is considered the fine mesh), a medium mesh and a coarse mesh were generated. Furthermore, as recommended, geometrically similar grids were used along with structured grid refinement. The results show that the convergence condition (R) is between 0 and 1, which means that the tendency is a monotonic convergence. In addition, the estimated values (RE) of lift and drag forces are close to the values obtained from the simulations with the fine mesh. Thus, the mesh resolution is adequate for RANS. Table 1 shows the results of the General Richardson Extrapolation. Verification of sufficient mesh resolution for LES was performed based on the criterion stablished by Kuczaj et al. [27]. According to their study, the mesh resolution (∆ = 3 √ V cell ) should at least be in order of the Taylor length-scale (λ) to completely solve the Taylor length-scale. Taylor length-scale is obtained from the Taylor expansion coefficient of the autocorrelation function about the origin. The autocorrelation function defines the relation between data samples in a time series, on average, to the preceding data points. This function is useful to analyze how turbulent a flow is. The more turbulent the flow, the lower the relationship between the data. Consequently, the auto-correlation function will approach zero faster. From the autocorrelation function, the Taylor time-scale is calculated, and finally, the Taylor length-scale is estimated by the Taylor hypothesis [28]. For more information about the Taylor length-scale calculation process, check [27,29]. This method has been applied in the previously-mentioned 12 points. The autocorrelation functions of different points are shown in Figure 3a and the comparison between the Taylor length-scale and mesh resolution is plotted in Figure 3b. The results show that autocorrelation functions tend progressively to 0. This is due to the negligible streamwise pressure gradient conditions that have been assigned to the flow. This has resulted in a significant difference between Taylor length-scale and mesh resolution, which fulfills the criteria proposed by Kuczaj et al. [27]. Hence, the mesh is suitable for LES simulations.

Cell-Set Model
In the present work, the accuracy of the cell-set model applied on VGs is studied. To implement this model, a mesh without VG is generated, then the required geometry (in this case the VG) is defined. To define the VG, the mesh is leveraged by using the cells where the VG should be located. Once the geometry is defined, the cells which are around the geometry are selected, a new cell-set region is created with those cells and a wall boundary is assigned to that region. For more information about the cell-set model implementation process, check Ballesteros-Coll et al. [11]. Figure 4 shows the cell-set construction of the VG with the cell-set model. In comparison with the fully-resolved mesh, a coarser mesh can be designed when using the cell-set model, saving computational time and resources. In this case, the meshes generated with this model consist of around 7.2 million cells.

Results and Discussions
In this paper, the vortexes generated by a vane-type VG on a flat plate are analyzed with the objective of comparing the results obtained with the fully-resolved mesh model and the cell-set model. With that aim, the primary vortex path, size, strength and velocity profiles have been analyzed. In addition, the coherent structures and wall shear stress have been studied.
For all the simulations, parallel computing with 56 Intel Xeon 5420 cores and 45 GB of RAM were used. Simulations in which the fully-resolved mesh model was used were run for 170,334 s using the RANS turbulence model and 663,595 s using the LES model. In contrast, simulations in which the cell-set model was used were run for 103,120 s for RANS and 401,070 s for LES.

Coherent Structures
To identify the coherent structures of the flow, the Q-criterion method [30], represented by isosurfaces, is applied. The value of Q has been set at 2500 s −2 . These isosurfaces can be shown in Figure 5.
Velte et al. [31] showed by PIV (Particle Image Velocimetry) measurements that two vortical structures appear in the wake behind the vane. The primary vortex appears at the top of the VG and follows the direction of the flow. The secondary vortical structure is a horseshoe vortex which appears at the lower part of the VG. This secondary vortex is divided into two sides, the pressure side and the suction side. As the suction side of the horseshoe vortex has the opposite sign to the primary vortex, at the early stage of its generation, the primary vortex sweeps the secondary vortex. Therefore, at the early stage of its generation, the secondary vortex does not follow the direction of the flow.
Both RANS simulations predict the primary vortex, but not the secondary one, and with both models similar predictions of the primary vortex are obtained.
In LES simulations, both the primary and the secondary vortex are predicted, but despite predicting the two vortical structures, the predictions of the cell-set model are not as good as those of the fully-resolved mesh model. The primary vortex predicted by the cell-set model is narrower and longer than the one predicted by the fully-resolved mesh model. In addition, the secondary vortex does not look as uniform as with the fully-resolved mesh model, so the pressure and suction side of the vortex are not accurately defined. Hence, the secondary vortex is not dragged by the primary vortex. In both RANS and LES simulations with the cell-set model, small turbulences appear on the surface of the VG. These turbulences are attributed to the fact that the surface of the VG is not straight with this model.

Vortex Path
To analyze the vortexes, the axial velocity of the flow in the previously-mentioned 12 spanwise planes is considered. In RANS simulations, the instantaneous values are considered, whereas in LES simulations, the mean values of 2 s after the flow are fully developed. Figure 6 displays the axial velocities. The results show that as the flow distances from the VG, the value of the axial velocity diminishes and the primary vortex grows. In accordance with Yao et al. [32], the shape of this vortex is not circular.
In RANS, similar results are obtained with the fully-resolved mesh model and the cell-set model, but in contrast with the coherent structures, the axial velocity shows the pressure side of the secondary vortex. In LES, as shown by the coherent structures, the primary vortex is narrower and the velocity in its center is bigger with the cell-set model, especially far from the VG.
Despite being similar in shape, the displacement of the primary vortex from the VG is not equal with both models. Hence, the position and trajectory of the primary vortex is studied. Figure 7 shows the vertical (Figure 7a) and lateral (Figure 7b) displacement of the center of the primary vortex (the point where the peak vorticity appears). Near the VG, the vertical displacements show that the primary vortex center is located under the height of the VG, but as the flow displaces from the VG, the primary vortex goes up. This is more notable when α = 25 • . With RANS, the vertical displacement near the VG is lower with the cell-set model, but it increases more as the flow distances from the VG. With LES, the trend is equal with both models, but slight differences in values are visible, being the vertical displacement of the cell-set model larger.
Regarding the lateral displacement, in all the cases the vortex center displaces flow the VG as the flow goes ahead. Near the VG, the displacements are very similar in both cases, but as the flow goes forward, the differences between models increase with LES. With RANS the results remain concordant along the whole path.
The results of both the vertical and lateral displacement are in accordance with the results obtained by Fernandez-Gamiz et al. [33].

Velocity Profiles
Velte et al. [34,35] showed that the primary vortex produced by a VG possess helical symmetry, and therefore the axial (u x ) and azimuthal (u θ ) velocity profiles are interrelated. To analyze the velocity profiles of the vortex, a horizontal line that crosses the center of the vortex is considered. Figure 8 shows the axial and azimuthal velocity profiles at 5H, 15H and 25H from the VG LE. These velocity profiles are normalized with the streamwise velocity.  The obtained velocity profiles describe the interrelation proposed by Velte et al. [34,35], but some disturbances are observed owing to the influence of the secondary vortex. The main disturbances are visible in the axial velocity profiles, which are not symmetric due to the influence of the pressure side of the secondary vortex.
In RANS, simulations almost equal to velocity profiles are obtained in the analyzed planes. In LES, though, despite obtaining similar results near the VG, bigger values of the velocity are obtained with the cell-set model, especially far away from the VG. However, the differences are relatively small, and the trend is almost the same.

Vortex Size
The Half-Life Surface (S 05 ) parameter introduced by Gutierrez-Amo et al. [36] is used in the present study to analyze the vortex size of the primary vortex. This parameter is an alternative to the Half-Life Radius (R 05 ) parameter developed by Bray [37] for cases where vorticity distribution is not symmetrical. This method consists of determining the elliptical-shaped area in which the vorticity value is greater than half of the peak value (ω > ω peak 2 ). For that purpose, the vorticity distribution from points with the same spanwise coordinate as the vortex center is extracted to obtain the vorticity distribution normal to the flow direction. Then, four points are determined with a vorticity equal to ω peak 2 . Finally, these four points are fitted to an elliptical shape. Figure 9 shows the S 05 values of the primary vortex. The results of S 05 are compared with the experimental results obtained by Bray [37]. Since in that study R 05 is analyzed, S 05 has been calculated from the R 05 . As expected, and in accordance with the results of the vortex path, the half-life surface increases as the flow distances from the VG. In accordance with the previously-obtained results, the half-life surface remains nearly constant with the cell-set model with LES. In RANS, the half-life surfaces are very similar with both models, and good agreements with experimental data are obtained, especially near the VG.

Vortex Strength
The capacity of spinning that the vortex acquires is evaluated with the positive circulation (Γ + ) following this expression: In this study, a parameter called mean positive circulation (Γ + 05 ) introduced by Gutierrez-Amo et al. [36] is considered. This parameter is an alternative to the positive circulation, with the fact that the previously-calculated S 05 parameter is required for its calculation, as defined in expression (3). Figure 10 displays the mean positive circulation of the primary vortex. As shown in the coherent structures, with LES, the primary vortex with the cell-set model is much longer than with the fully-resolved mesh model. This means that the vorticity is bigger and has resulted in larger values of the mean positive circulation far away from the VG. In addition, the values remain nearly constant, and therefore, barely diminish. In RANS, very small differences between models are found.

Wall Shear Stress
As mentioned above, the main target of VGs is to delay the flow separation. The wall shear stress is a significant parameter to understand this phenomenon. Figure 11 shows the wall shear stress values obtained behind the VG TE. In RANS simulations, the instantaneous values are considered, whereas in LES simulations, the mean values of 2 s after the flow is fully developed. In all the cases, the wall shear stress goes from a low value to a maximum value, then decreases. This maximum value is obtained when x/H is around 7.
With both RANS and LES, accurate predictions of the location of the wall shear stress maximum value are obtained using the cell-set model, but the values are different. With RANS, the maximum values are nearly equal, but as the flow distances from the VG, the differences increase. With LES, in spite of following the same trend, the values are slightly lower in all the locations. These differences in values can be attributed to the displacement of the primary vortex, since the greater the displacement, the smaller the influence of the vortex behind the VG; and this leads to lower values of the wall shear stress. With both models, larger wall shear stress values are obtained when α = 18 • . Since Godard and Stanislas [38] showed that the maximum skin friction appears when the angle of attack is near to 18 • , these results were expected.

Conclusions
CFD simulations of a single VG on a flat plate in negligible streamwise pressure gradient conditions were performed with the fully-resolved mesh model and the cell-set model using RANS and LES turbulence models. The coherent structures, wall shear stress and path, size, strength and velocity profiles of the primary vortex were compared in order to evaluate the accuracy of the cell-set model. The implementation of the cell-set model has resulted in a decrease of cells from 11.5 million to 7.2 million, with savings of around the 40% in terms of computational time with both RANS and LES.
The results show a very good performance of the cell-set model using RANS-based turbulence models, since good agreements between the fully-resolved mesh model and the cell-set model in all the studied parameters were obtained.
With LES turbulence models, good predictions of the coherent structures, vortex path and wall shear stress were obtained using the cell-set model, but some discrepancies were obtained when analyzing the size, strength and velocity profiles of the primary vortex. All these differences appear because the cell-set model overpredicts the vorticity of the primary vortex when using LES, especially far away from the VG. Hence, vortex size predictions are lower, vortex strength predictions are higher and velocity profile predictions are higher in comparison with the fully-resolved mesh model. Nevertheless, except in the vortex size, where the disparities are considerable, the results are considered acceptable.
In conclusion, due to the good results obtained with RANS, the cell-set model is clearly suitable for RANS. With LES, the results are acceptable, but for the studied case, the fully-resolved mesh model outperforms the cell-set model. In any case, this is considered the first application of the cell-set model on VGs. Therefore, further research should be done to analyze the cases where the cell-set model provides more accurate results with LES, since this model can represent a great advantage over the fully-resolved mesh model due to the simulation time savings that its implementation entails. Funding: The authors are thankful to the government of the Basque Country and the University of the Basque Country UPV/EHU for the ELKARTEK20/78 and EHU12/26 research programs, respectively.