Computational study of the vortex path variation with the VG height

An extensive range of conventional, vane-type, passive vortex generators (VGs) are in use for successful applications of flow separation control. In most cases, the VG height is designed with the same thickness as the local boundary layer at the VG position. However, in some applications, these conventional VGs may produce excess residual drag. The so-called low-profile VGs can reduce the parasitic drag associated to this kind of passive control devices. As suggested by many authors, low-profile VGs can provide enough momentum transfer over a region several times their own height for effective flow-separation control with much lower drag. The main objective of this work is to study the variation of the path and the development of the primary vortex generated by a rectangular VG mounted on a flat plate with five different device heights h = δ, h1 = 0.8δ, h2 = 0.6δ, h3 = 0.4δ and h4 = 0.25m, where 5 is the local boundary layer thickness. For this purpose, computational simulations have been carried out at Reynolds number Re = 1350 based on the height of the conventional VG h = 0.25m with the angle of attack of the vane to the oncoming flow β = 18.5°. The results show that the VG scaling significantly affects the vortex trajectory and the peak vorticity generated by the primary vortex.


Introduction
Due to large energy losses associated with boundary-layer separation, flow separation control has become a very important issue for several industrial applications in the field of fluid mechanics. The most important reason of flow-separation is the lack of momentum in the boundary layer, thus usually the primary option in trying to control the flow separation is the installation of vortex generators because they have the advantage of being cost-effective and simple to set-up and manufacture. Flow control devices can be used to increase both free-shear and wall-bounded flows by extending the effective area through which transport occurs, by setting off resonant flow instabilities, by advancing laminar to turbulent transition and by enhancing the turbulence once the shear flow is already turbulent, Gad-el-Hak [1]. The effective generation of secondary flows and/or recirculation areas is an additional tool with which to enhance mixing in both laminar and turbulent flows. The use of vortex generators is a successful example of this strategy.
Many models for the generated vortices have been presented over the years. Theoretical models include, for example, the one by Smith [2] and a model presented by Velte et al. [3] that was developed and applied to show the helical symmetry of the vortices generated by a passive rectangular vane-type vortex generator. Regarding models incorporated into codes, most are based on the BAY-model by Bender et al. [4] such as the actuator VG model (AcVG) developed in [5], which introduces body forces using source terms in the momentum and/or energy equations to simulate the presence of a vane.
VGs are usually triangular or rectangular vanes inclined at an angle of attack to the oncoming flow and dimensioned in relation to the local boundary layer thickness to allow for the best interaction between the generated vortex wake and the boundary layer, and are usually placed in groups of two or more upstream of the flow separation area, Anderson [6]. For several flowcontrol applications, conventional VGs first introduced by Taylor [7,8,9] have been successfully applied onto aircraft wings for flow control that enhance mixing of the boundary layer and thus transferring high momentum fluid closer to the wall, thereby suppressing or delaying separation, Schubauer [10]. Furthermore, the experimental work carried out by Velte [11] and the corresponding simulations made by Fernández-Gámiz et al. [12] showed that the primary vortex produced by a rectangular VG mounted on a flat plate achieves self-similarity for both axial and azimuthal velocities. In most applications, VGs are designed with the device height h on the order of the local boundary layer thickness δ and are mounted normal to the surface with an angle of incidence to the oncoming flow to generate streamwise longitudinal vortices. Nevertheless, in some flow-control application, the residual drag associated to these relatively large δ-scale VGs could be excessive.
A practical and fairly simple method to improve the VG efficiency is by generating the streamwise vortices using vanes with reduced device height. Lin et al. [13] demonstrated that by reducing the height of the conventional VGs to a fraction of the local boundary layer, the devices still provide enough momentum transfer downstream of the VG, especially in turbulent boundary layers where the velocity profile is relatively full. These low-profile (sub-δ-scale) VGs were successfully implemented on multi-element high-lift airfoils to control flow separation on the flap. Yao et al. [14] made a comparison of the effects on the vortex trajectory between a low-profile VG with the device height approximately equal to 20 percent of the boundary layer thickness and a conventional VG with h = δ. Furthermore, the experimental and theoretical study carried out by Ashill et al. [15] revealed a successful delaying of the shock-induced separation on a transonic airfoil by the implementation of Sub Boundary Layer Vortex Generators (SBVG). It was found that wedge SBVGs give significant increases in lift and reductions in drag. According to Lin [16], the low-profile VGs are one of the best options when the flow-separation locations are relatively fixed and the devices can be mounted reasonably close upstream of the separation.
The success of implementing low-profile VGs to achieve performance improvements in several aerodynamic applications led to an increased focus on providing detailed information in the wake downstream of these devices for a better understanding of the physics of the flow.  The main objective of this work is to study the variation of the path of the primary vortex generated by a rectangular VG mounted on a flat plate with five different device heights h = δ, h 1 = 0.8δ, h 2 = 0.6δ, h 3 = 0.4δ and h 4 = 0.2δ, where δ is the local boundary layer thickness and h i represents the height of the low-profile VGs for each case, as sketched in figure 1. For this purpose, computational simulations have been carried out using RANS method and at Reynolds number Re = 1350 based on the boundary layer thickness at the VG position of δ = 0.25m. The case consists of a single vortex generator on a flat plate with the angle of attack of the vane to the oncoming flow β = 18.5 • . The flow over the flat plate without the VG has been previously simulated to calculate the boundary layer thickness at the VG position.
This study of the vortex development of low-profile VGs can contribute to obtain a better understanding of the behaviour of the wake downstream of these types of vortex generating devices and, therefore, to take advantage of the benefits of these vanes, such as their intrinsic simplicity, cheap implementation and low drag.

Computational configuration
In this study, steady state simulations were carried out and performed with a structured finite-volume flow solver using, in this work, Reynolds-Averaged Navier-Stokes equations. The convective terms are discretized using the third order Quadratic Upstream Interpolation for Convective Kinematics (QUICK scheme), [17]. For these computations the k − ω SST (Shear Stress Transport) turbulence model by Menter [18] was used. A successful validation of steady RANS simulations against measurements is the one carried out by Allan [19], where the flow field about a single low-profile VG on a flat plate with two different vane angle-of-attack of 23 • and 10 • was computed using OVERFLOW, a flow solver code developed at NASA. The results of these simulations were compared to experimental data and showed that the CFD and experiments agreed well except for short distances downstream of the vane.  the position where local boundary layer thickness was equal to 0.25m, approximately forty two times the local boundary layer thickness (42δ) downstream of the inlet. The VG angle of attack to the oncoming flow is β=18.5 o and the Reynolds number based on the VG height h= 0.25m is Re= 1350, using an inflow velocity U ∞ = 1ms −1 and a density of 1 kgm −3 . The vane angle is very close to the optimum found by Godard et al. [20] in a parametric study optimizing separation control. The computational setup of the CFD simulations consists of a block structured mesh of 25 million cells with the first cell height (Δz/h)of 1.5×10 −5 normalized by the height of the conventional VG. In the immediate vicinity of the vane, the mesh has 6.5×10 6 cells, while the mesh downstream of the VG for capturing the wake has approximately 11.5× 10 6 cells, see figure 3(a,b). In order to resolve the boundary layer, cell clustering has been used close to the wall and the dimensionless distance from the wall is less than 1 (y + < 1). Verification of enough mesh resolution was performed by a mesh independency study, where the results obtained from the finest mesh (Grid 1 with 25 million cells) are compared with the results from the medium (Grid 2 with 12 million cells) and the coarsest mesh (Grid 3 with 6.5 million cells).

Results
Numerical simulations of the vortex generating vane on a flat plate were performed. Four dimensions of a single low-profile vane h 1 = 0.8δ, h 2 = 0.6δ, h 3 = 0.4δ and h 4 = 0.2δ at β=18.5 o were computed and compared with the results of the conventional VG h = δ. The extraction of the data from the computations was conducted in a similar way to the procedure described in [12], in planes normal to the wall downstream of the VG to capture the development of the vortex. Table 1 shows the height of the conventional VG h and the heights of the low-profile (subδ-scaled) VGs with the corresponding Drag coefficients C D . The heights of the low-profile VGs have been designed with a reduction factor of 20% with respect to the height of the conventional VG. However, this decreasing rate in the VG height is not followed by the Drag coefficients, as presented in the fifth column of the table 1. The Drag coefficient of the smallest VG h 4 , with a height of 20% of the conventional VG height, has the 44% of the Drag of the largest VG h. Note here that the increase of the Drag on an aerodynamic body due to the implementation of VGs is not only related to the Drag of the vanes themselves, but also to the change in the body boundary layer features downstream of the vanes position. In order to analyze the vortex evolution, up to three parameters were previously identified regarding vortex development from past studies on streamwise vortex embedded in turbulent boundary layers, Bray [21] and Yao et al. [14]. Therefore, the vortex generating devices are characterized by the peak streamwise vorticity | ω x | max , vortex circulation Γ and location of the vortex core. The peak vorticity is used to locate the center of the streamwise vortex in each plane position downstream of the VG and, subsequently, to determine the vortex path, which is the main objective of this work. According to Pearcy [22] and Ashill et al. [15], the circulation is a parameter associated to the vortex induced rate of mixing of the outer flow with the boundary layer. The trajectory of the primary vortex generated by a vane-type VG plays also a determining role in the mixing performance.
By observing the location of the vortex center as a function of the downstream axis x, one can determine the trajectory of the vortex in both lateral (y) and vertical (z) directions.   The vortex paths in the vertical direction are represented in figure 5(b) As in the case of the lateral path, the vertical positions are also normalized by the corresponding VG height. Figure 5(b) shows that the vertical trajectories roughly collapse in the streamwise plane positions studied. This means that, in absolute terms, the vertical trajectory of the VGs decreases proportionally as the VG height decreases and stays roughly within the order of the corresponding device height, except for the case of the lowest profile h 4 , which behaves quite differently. In this case the slope of the vertical path is much higher than all other cases, thus at positions far away from the VG the elevation of the vortex center is very large in comparison with the other VGs. Note that the vortex location for every plane position remains within the boundary layer and stays approximately parallel to the wall, except for the case of the smallest vane h 4 .
The normalized vortex paths for the low-profile VGs follow the same trends for both lateral and vertical direction of the conventional VG except for the lowest VG h 4 . In this case, the slope of the vertical path is much higher than the other cases, above all far away the VG. In 1933 Prandtl argued that for the inner law, the boundary layer profile would depend upon wall stress τ w , fluid properties (density ρ and dynamic viscosity μ) and vertical distance from the wall z, but not upon freestream parameters:ū = f cn(τ w , ρ, μ, z). Conversely, Kármán reasoned in 1930 that, for the outer layer, the wall acts merely as a source of retardation, reducing local velocityū(z) below the freestream velocity in a manner independent of dynamic viscosity but dependent upon wall shear stress, boundary layer thickness and pressure gradient. Therefore, due to the low height of the vane, the vortex generated by the VG h 4 is very close to the inner part of the boundary layer where the viscous (molecular) shear is prevailing, thus there is strong interaction with the wall. According to White [23], what happens very near the wall is that the turbulence is damped out and the inner boundary layer is dominated by the viscous shear. This could be one of the reasons for the different behavior.
In addition to the above in the lateral and the vertical paths of the vortex, the maximum value of the streamwise vorticity has been studied. To illustrate the vortex decay, the streamwise distribution of the normalized peak vorticity (ω x,max h i )/U ∞ is plotted as a function of the nondimensional downstream distance x/δ for all cases studied. Figure 6(a) shows the decay of the vortex in the streamwise direction and confirms that the peak vorticity is rapidly attenuated downstream of the VG for all cases except for the conventional VG. In this case, the maximum value of the streamwise peak vorticity is reached immediately downstream of the device at the position x/δ = 3. The peak vorticity decays exponentially and inversely proportional to x/δ. As expected, the magnitude of the peak vorticity decreases as the height of the VG is reduced.   The strength of the vortex is evaluated in this work by the positive circulation (Γ + ) defined by the equation (1), which is calculated by finding the vortex center position from the peak vorticity over the area surrounding the vortex core in cross-flow plane positions normal to the wall, as sketched in figure 2, and it has been normalized by the corresponding VG height h i and by the free-stream velocity U ∞ . The resulting streamwise distribution of the normalized positive circulation Γ + /(h i U ∞ ) is represented in figure 6(b).
The non-dimensional positive circulation barely varies at the streamwise stations studied for the VG cases h, h 1 , h 2 and h 3 . Thus, this means that the non-dimensional positive circulation is independent of the vane height for cases where device heights are located in the outer part of the boundary layer, which is in agreement with Ashill et al. [15]. However, once again, the behaviour of the VG h 4 is significantly dissonant with respect to the other VGs. The non-dimensional circulation development of the VG h 4 in the streamwise direction is considerably different from the other low-profile VGs and its maximum value is reached at the streamwise station of x/δ = 4 and then decreases due to viscous dissipation and reaches values close to zero at positions far away from the VG. In general, the decay of the circulation is more linear in comparison with the decay of the peak vorticity. The trends of the circulation decay follow approximately similar trajectories, except for the lowest VG h 4 , as has been previously mentioned. Apparently, this occurs because the vortex generated by the lowest VG is very close to the wall and consequently close to the inner part of the boundary layer where the viscous shear is dominant. Viscosity does not control the whole shear layer but only a thin part close to the wall (the so-called near-wall layer) while viscous effects can be safely neglected in the outer layer.
Additionally, two plots have been created to visualize the effect of the vane height scaling on the vortex development at three plane positions downstream of the VG x/δ = 5, x/δ = 10 and   For all VG cases, the non-dimensional peak vorticity increases as the device height increases and the maximum values are reached at the streamwise station of x/δ = 5. The conventional VG h has much lager values of normalized peak vorticity in comparison with the lower profile VGs. The smallest values of peak vorticity correspond to the lowest vane h 4 . Probably, due to its reduced size it is not able to generate as much vorticity as the other vanes. As shown in figure  7(a), the low-profile VGs h 1 , h 2 , h 3 and h 4 follow the trend of the conventional VG h and it seems that the evolution of the non-dimensional peak vorticity (normalized by the corresponding VG height) increases exponentially with the device height.
Regarding the vortex strength, figure 7(b) presents the values of the non-dimensional circulation against the device heights at three stations, in the same manner as was previously done with the peak vorticity. It is clear in this plot that normalized circulation remains roughly constant at the three plane stations for all vanes, with the exception of the lowest VG h 4 . This shows that the non-dimensional positive circulation reached at these plane positions does not depend on the vane height and the variation of the normalized circulation with the device height can reasonably be ignored for devices that are located in the outer part of the boundary layer. This outcome is strongly in accordance with the results presented in Betterton et al. [24]. Once more, it is worth noting the different behavior of the lowest VG h 4 , which reaches the largest value of the normalized circulation at the first station 5δ and then decays rapidly until almost zero in the third station 15δ.
Of the five types of VG studied, h, h 1 , h 2 , h 3 and h 4 , the last one was the most effective in terms of producing non-dimensional positive circulation just downstream of the device at the streamwise station of 5δ. The vane h 4 produces similar or even larger normalized circulation up to the station 9δ but, thereafter, decreases rapidly until values close to zero. The non-dimensional positive circulation for the other cases stays almost constant at the three streamwise stations evaluated. This means that the normalized circulation does not depend on the device height in the streamwise distance studied from 5δ to 15δ for the cases h, h 1 , h 2 and h 3 . However, with a notable difference to the rest, the VG case h 3 has a drag coefficient 41% lower than the conventional one. In terms of VG efficiency (measured as non-dimensional circulation production) to additional drag, it seems that the device height h 3 could be a more efficient alternative to the conventional VG due to its much lower drag and because it produces a similar level of non-dimensional circulation.

Conclusions
Vortices generated by rectangular vane-type vortex generators of the same and lower height than the local boundary layer thickness on a flat plate have been investigated. The ultimate goal of this study is to contribute to obtain a better understanding of the behaviour of the primary vortex downstream of the so-called low-profile vortex generators and, consequently, to make the most of the main advantages of these vanes, such as their intrinsic simplicity, cheap implementation and low drag. Hence, computational simulations of these passive devices within a turbulent boundary layer flow over a flat plate at Reynolds number Re = 1350 based on the conventional VG height h = 0.25 m and free stream velocity U ∞ = 1 ms −1 have been carried out using the RANS method. The VG scaling significantly affects the trajectory and non-dimensional steamwise peak vorticity of the vortex generated by vane-type VGs. The normalized vortex trajectories of the low-profile VGs h 1 , h 2 and h 3 follow the general trends of the conventional VG h. However, the path of the vortex h 4 is considerably different for both lateral and vertical direction in comparison with the conventional VG and the other low-profile devices. One reason for this different behaviour could be attributed to the fact that the vortex generated by the vane h 4 is very close to the wall, probably interfering with the inner part of the boundary layer, where the viscous shear stress is dominant and, thus, there is considerable interaction with the wall.
In terms of the ratio effectiveness to additional VG drag, the case h 3 seems to be the most promising VG configuration. In comparison with the case of the conventional VG, h 3 produces the same or even larger amount of normalized circulation downstream of the vane with much lower device drag. The case of h 4 produces the highest non-dimensional circulation at stations close to the trailing edge of the VG, nevertheless, its rapid decay in the streamwise direction makes this configuration less efficient than the case of h 3 .
As a future work, it would also be interesting to study in detail how the modified boundary layer downstream the VG affects the total drag of an aerodynamic object (in addition to the parasitic drag of the VG itself) and to carry out more CFD simulations at higher Reynolds numbers. Of course, wind tunnel experiments to validate the present computations are highly desirable.