1. Introduction
Vortex Generators (VGs) are passive flow control devices, whose objective is to delay or remove the flow separation, transferring the energy generated from the outer region to the boundary layer region. They are small vanes placed before the expected region of separation of the boundary layer. They are usually mounted in pairs, with an incident angle with the oncoming flow. Regarding their shape, VGs can be of various geometries, but they are mainly triangular or rectangular. Their height is typically similar to the boundary layer thickness where the VG is applied, in order to ensure a good interaction between the boundary layer and the vortex generated in the VG. However, since tall VGs lead to high drag forces, VGs with smaller heights than the local boundary thickness, i.e., Sub-Boundary-Layer Vortex Generators (SBVGs), are used in many applications, see Ashill et al. [
1,
2]. Aramendia et al. [
3,
4] comprehensively reviewed the available flow control devices, including VGs, and Lin [
5] conducted an in-depth review of the control of flow separation in the boundary layer using SBVGs.
Since their introduction by Taylor [
6] in the late 1940s, VGs have been used in a wide range of industries for numerous applications. Among these industries, aerodynamics and thermodynamics are the most remarkable. Øye [
7] and Miller [
8] implemented VGs on 1 MW and 2.5 MW wind turbines, respectively. Heyes and Smith [
9] added VGs of numerous shapes on the wing tip of an aircraft, and Tai [
10] studied the effect of Micro-Vortex Generators (MVGs) on V-22 aircraft. All of them showed a significant aerodynamic performance improvement when implementing VGs.
Another sector in which VGs are widely used is thermodynamics. Currently, major efforts are being made in the thermodynamics field to increase heat and mass transfer, see Agnew et al. [
11]. For this reason, numerous authors have implemented in different thermodynamic systems. For example, Joardar and Jacobi [
12] studied the heat transfer and pressure drop of a heat exchanger before and after the addition of VGs, showing an increase of the heat transfer coefficient between 16.5% and 44% with a single VG pair and between 30% and 68.8% with 3 VG pairs.
Although many studies use experimental techniques, the use of Computational Fluid Dynamics (CFD) tools for performing numerical studies is becoming a very popular choice for studying VGs. CFD studies of VGs are currently focused on two main goals. The first goal is to optimize the position and distribution of VGs, see the studies of Subbiah et al. [
13] and Yu et al. [
14]. The second goal is to analyze the swirling vortexes generated on the wake behind the VG, see the studies of Carapau and Janela [
15] and Sheng et al. [
16] about this phenomenon.
Many authors have studied SBVGs using CFD. Ibarra-Udaeta et al. [
17] and Martinez-Filgueira et al. [
18] analyzed the vortices generated by rectangular vane-type SBVGs on a flat plate under negligible pressure gradient flow conditions. They analyzed VGs with heights equal to 0.2, 0.4, 0.6, 0.8, 1, and 1.2 δ and incident angles equal to 10°, 15°, 18°, and 20°. Fernandez-Gamiz et al. [
19] studied three different SBVGs with heights of 0.21, 0.25, and 0.31 δ and an incident angle equal to 18°. Gutierrez-Amo et al. [
20] analyzed a rectangular, a triangular, and a symmetrical NACA0012 SBVG. Fully resolved mesh modelling technique was used in all the mentioned studies, and all of them showed good agreements with experimental data.
The main disadvantage of the fully resolved mesh model is the fine mesh that requires to accurately capture the physical phenomena, especially in the near-VG region and in the wake behind the VG. For that reason, numerous authors [
21,
22,
23] have implemented alternative models. The majority of these models are based on the BAY model developed by Bender et al. [
24], which models the force produced by a VG. Errasti et al. [
25] implemented the jBAY source-term model developed by Jirasek [
26] in vane-type SBVGs under adverse pressure flow conditions and showed accurate results in terms of vortex path, vortex decay, and vortex size.
The cell-set model is another alternative model, which consists of leveraging the previously generated mesh to build the desired geometry. Besides the advantages that the cell-set model provides over the fully resolved mesh model, there are not many studies in which this model has been applied. Ballesteros-Coll et al. [
21,
27] used the cell-set model to generate Gurney flaps and microtabs on DU91W250 airfoils, and Ibarra-Udaeta et al. [
28] modelled conventional vane-type VGs with this model.
The goal of the present paper is to evaluate the accuracy of the cell-set model applied on SBVGs with heights of 0.2, 0.4, 0.6, 0.8, and 1 δ. For that purpose, CFD simulations of SBVGs on a flat plate in a negligible streamwise pressure gradient flow conditions are conducted using the fully resolved mesh model and the cell-set model, and the results obtained with the fully resolved mesh model are taken as benchmark. With the purpose of testing the cell-set model with RANS and LES, both models are used to conduct the simulations.
The remainder of the manuscript is structured as follows:
Section 2 provides a general description of the used numerical domain and meshing models.
Section 3 explains the results obtained in the present work. Finally,
Section 4 summarizes the main conclusions reached from the results and future directions.
3. Results and Discussion
In this study, the vortices generated in the wake behind the VGs have been analyzed. Moreover, an exhaustive analysis of the primary vortex has been performed, studying its path, size, and strength and the wall shear stress behind it.
For the interpretation of RANS results, the last obtained values have been considered, whereas for the results of LES simulations, the average values of 2 s of simulation have been considered, after the flow is completely developed.
Parallel computing with 56 Intel Xeon 5420 cores and 45 GB of RAM were used to carry out all the simulations. Simulations performed with fully resolved mesh modelling were run for about 47 h using the RANS turbulence model and for around 184 h using the LES model. In contrast, simulations in which cell-set modelling was applied were run for approximately 28 h for RANS and 111 h for LES.
3.1. Vortex Structure Regimes in the Wake
As measured by Velte [
37], two basic vortex mechanism appear in the wake behind the VG. The main vortex system is composed of a primary vortex (P), which is formed on the wing tip, and a horseshoe vortex, which is generated from the rollup vortex around the LE of the VG. This horseshoe vortex is divided in two sides, the pressure side (Hp) and the suction side (Hs). As the primary vortex is stronger than these sides, the primary vortex pulls the suction side. As the primary vortex and the pressure side have the same sign, the pressure side remains undisturbed.
The secondary vortex structure is created by the local separation of the boundary layer in the lateral direction between the primary vortex and the wall. Due to the dragging of the suction side by the primary vortex, the primary vortex becomes stronger, the boundary layer region grows, and finally detaches, forming a discrete vortex (D).
Figure 7 shows a representation of the primary and secondary vortexes.
Nevertheless, Velte [
37] showed that these structures can vary, depending on the incident angle and the height of the VG.
Figure 8 displays a comparison of the vortical structures predicted by the numerical simulations with both studied models at a distance of 5 δ from the VG Trailing Edge (TE), and the vortical structures measured by Velte [
37].
Regarding the main vortex structure, the results show that both RANS and LES are able to accurately predict the primary vortex with the fully resolved mesh model and the cell-set model. In LES simulations, the horseshoe vortex, which is expected to appear in VGs with heights above 0.4 δ, is predicted for all the heights, including 0.2 δ. In contrast, RANS is not able to capture the pressure side of this horseshoe vortex for heights below 0.8 δ.
The largest discrepancies between the simulations and the experimental results appear in the secondary vortex structure. This vortex structure is expected to appear in VGs whose heights are below 0.4 δ. In LES, this vortex is only visible for H = 0.2 δ with the fully resolved mesh model. This vortex is not predicted in RANS for neither height.
Despite showing different values, the fully resolved mesh model and the cell-set model predict very similar vortexes in terms of vortex shape and direction.
3.2. Vortical Structure of the Primary Vortex
The Q-criterion [
38] method has been used to compare qualitatively the primary vortex generated by each VG in terms of shape and size. This method visualizes structures of the flow, and its value is defined by
, where Ω is the spin tensor and S the strain-rate tensor. As the value of Q is set at Q = 2500 s
−2, the vortical structures are displayed.
Figure 9 shows the representation of the primary vortex at 5 δ from the VG TE by means of the Q-criterion.
The results show that for α = 18°, the taller the VG, the larger the vortex. However, for α = 25°, although this also occurs, the differences between heights are smaller, with the size of the vortices being more similar than when α = 18°.
With RANS, even if they have the same circular shape, the vortexes predicted by the cell-set model are larger than the ones predicted by the fully resolved mesh model. With LES, generally, the vortexes predicted by the cell-set model are smaller than those predicted by the fully resolved mesh model. In this case, slight disparities between models are visible in terms of vortex shape, which are attributed to the unsteadiness of the flow.
3.3. Vortex Path
In order to analyze the vortex path of the primary vortex, the location of the center of this vortex is studied. According to Yao et al. [
39], the vortex center is the point in where the peak vorticity appears.
Figure 10 shows the vertical and lateral path of the primary vortex normalized with the height of each VG. The lateral path obtained in the present study for α = 18° is compared to the one obtained experimentally by Bray [
40].
The lateral path shows that in all the cases, the vortex tends to follow the flow direction, showing a linear trend. With both turbulence models, the lower the height of the VG, the greater the horizontal displacement. Good agreements are obtained with the experimental data reported by Bray [
40]. Near the VG, the lateral displacements are nearly equal, except with H = 0.2 δ and α = 18°, but as the flow distances from the VG, the differences between cases increase, most notably with α = 25°.
Corresponding the vertical path, the results show that the vortexes tend to collapse near y/H = 1, except for the case H = 0.2 δ, in which the vortex continues its upward climb as it distances from the VG, this is more noticeable with α = 18° than with α = 25°.
The comparison between the fully resolved mesh model and the cell-set model shows that in both cases, the same trend is followed with both models. With RANS, larger lateral displacements are obtained with the cell-set model, while with LES, the larger lateral displacements are obtained with the fully resolved mesh model. The cell-set model predicts larger vertical displacements with RANS and LES. For the highest VGs, the results are very similar with both models, but as the VG height decreases, the differences between models increase, especially with LES.
3.4. Vortex Size
The vortex size is analyzed by the Half-Life Radius (R
05) parameter developed by Bray [
40]. This parameter determines the distance between the vortex center and the point where the local vorticity is equal to
. As shown in
Figure 9, the vortex shape is not always circular, therefore, R
05 has been estimated by averaging the values in vertical and lateral directions.
Figure 11 shows the R
05 values normalized with δ on the wake behind the VG for the tested cases.
The results show that the greatest vortexes appear in the taller VGs. In RANS, the R05 increases linearly from the near-VG region, but in LES, R05 remains almost constant near the VG, and it starts increasing at 10 δ from the VG LE.
Despite showing the same trend, in all the cases, the cell-set model predicts smaller vortexes than the fully resolved mesh model, these differences are more notable with LES. The largest discrepancies between models are visible in the lower VGs, most notably with LES.
3.5. Vortex Strength
To quantify the vortex strength, vortex circulation (Γ) is considered. This parameter determines the capacity of the vortex to mix the outer flow with the boundary layer [
2]. According to Yao et al. [
39], the vortex circulation can be estimated by expression (2).
Figure 12 shows the vortex circulation normalized with the VG height and the flow streamwise velocity.
As expected, the vortex losses its strength as it distances from the VG. Close to the VG, the stronger vortices appear in the lower VGs. With RANS, as the distance between the VG and the flow increases, the results tend to collide with both models. In contrast, with LES, this trend is only visible with fully resolved mesh modelling, since with the cell-set model, despite following a falling tendency, the collision of the results is not achieved. With this turbulence model, the normalized circulations are considerably higher with the cell-set model than with the fully resolved mesh model for the lower VGs, but for the higher ones, the differences decrease.
Although, as mentioned before, the R05 values predicted with the fully resolved mesh model are higher, vortex circulation values are very similar with the RANS model. This is attributed to the consideration of the vorticity in the expression of the circulation. With LES, despite showing less differences in Γ than in R05, differences are significant for low VG cases.
3.6. Wall Shear Stress
Wall shear stress is a major parameter to quantify the capacity of the VG to delay the flow separation.
Figure 13 shows the values of the wall shear stress on the wake behind the VG for all the tested cases.
In the tested cases, the wall shear stress goes from a low value to a maximum value, which appears between x/δ = 3 and x/δ = 6, depending on the case. The lower the VG, the closer to the VG the maximum appears. Then, as expected, wall shear stress slightly decreases as it distances from the VG.
According to Godard and Stanislas [
41], the optimum angle for the maximum wall shear stress is around 18°, and the wall shear stress is not very sensitive to the aspect ratio. The obtained results are in accordance to these statements, since the values obtained with α = 18° are greater than the ones obtained with α = 25°, and in the majority of the cases, very similar values are obtained, specially far away from the VG.
With RANS, nearly equal locations of the maximums are obtained with both models. Considering the values, very similar values are obtained with both models, but the biggest deviations between models appear when H = 0.2 δ and H = 0.4 δ. With LES, larger discrepancies between models are visible regarding the locations of the maximums, since the cell-set predicts the maximum closer to the VG. The values of the maximums show that the cell-set underpredicts these values for the taller VGs (H = 0.8 δ and H = 1 δ) and overpredicts the values for the lower VGs (H = 0.2 δ and H = 0.4 δ). Far away from the VG, the cell-set model underpredicts the wall shear stress value for all the cases, except for H = 0.2 δ.
The differences visible on the lower VGs for the LES case are attributed to the fact that these VGs are located on the buffer layer region, where the viscous effects are dominant and the flow is strongly turbulent. In this region, the cell-set seems not to be able to provide accurate predictions of both the viscous and turbulent shear stresses.
4. Conclusions
Numerical simulations of 10 different SBVGs on a flat plate in a negligible streamwise pressure gradient flow conditions were conducted using the fully resolved mesh model and the cell-set model with RANS and LES turbulence models, with the objective of analyzing the accuracy of the cell-set model.
The meshes generated with the fully mesh model are made of around 11.5 million cells, while the meshes generated with the cell-set model are composed of around 7.2 million cells. This fact has resulted in savings of around the 40% in terms of computational time.
This study is mainly focused on analyzing the vortexes generated on the wake behind the VG. Therefore, the vortex structure regimes on the wake behind the VG; the path, size, and strength of the primary vortex; and the wall shear stress behind it have been studied. The results demonstrate that the cell-set model is able to predict the vortexes generated on the wake behind the VG. Regarding the primary vortex, nearly equal values of its path and fairly accurate predictions of its size have been obtained. The vortex size and strength show that the cell-set models overpredict the vorticity of the core of the primary vortex, but underpredicts its size, especially with LES. This is reflected in the large differences that appear in the R05, but close values obtained in Γ and wall shear stress.
The major agreements between models appear in the higher VGs, and the biggest disparities appear in the lower ones. This is attributed to the location of the VGs on the boundary layer, since the lower VGs (H = 0.2 δ) are located on the buffer layer and the higher ones (H = 0.8 δ and H = 1 δ) on the outer region. These discrepancies are more notable in LES.
In conclusion, it has been demonstrated that the cell-set model is suitable for RANS turbulence modelling with all the tested SBVGs. With LES, it is adequate for VGs whose height is around the boundary layer, but for lower VGs, the differences with the fully resolved mesh model are significant. Hence, the cell-set model presented in the current work seems to be not very accurate for vane heights within the buffer layer.
Since the cell-set model represents a great advantage in terms of computational and meshing time savings, additional research is proposed, applying the studied meshing model on VGs with different conditions and geometries, or using it for generating other devices. Furthermore, more investigations should be done in order to improve the accuracy of the cell-set modelled geometries with heights within the buffer layer.