3.1 Aircraft model

The baseline mesh and geometry used during the current study was generated using the Computer Aided Design (CAD) data of the N47-05 aircraft model utilised during the wind-tunnel experiments. Since the current study employs the ARA TWT wind-tunnel test results as the reference, a mesh was generated for the wind-tunnel scale, i.e. Re = 5.3 × 10⁶ (see Table 2 for mesh characteristics).

3.2 Solver settings

The RANS solver settings applied during the simulations are summarised in Table 3. Two different turbulence models, the original Spalart-Allmaras one-equation model (SAO)^{(Reference Spalart and Allmaras9)} and the SSG/LRR Reynolds-stress model (SSG)^{(Reference Cécora, Radespiel, Eisfeld and Probst10)}, were utilised during the simulations.

Computations were performed for a constant Mach number of M _∞ = 0.85 and the nominal Reynolds number used in the wind tunnel during the TWT test campaign. Transition was set during the simulations in order to mimic the transition setting adopted in the TWT, see Table 1.

3.3 Deformation estimation

In the analysis of the clean wing simulation results presented in Ref. Reference Hahn6, it has been shown that the wing deformation is an important factor affecting the flow characteristics. Since the deformation increases with the wing loading, larger deformations are expected at high angles of attack, which is the main region of interest of the study presented in this paper.

The wing bending and twisting experienced by the model during the test campaign in TWT is unknown; however, the deformation measurements are available for the ETW experiments. Since the same model has been used in both test campaigns, it was anticipated that its structural behaviour is similar under load and only marginal differences, due to temperature variations, are present.

Therefore, the wing stiffness can be derived, based on the known aerodynamic loading and the deformation measured at Re = 32.5 × 10⁶ during the ETW test campaign. It is then possible to combine the wing structural properties and the loading measured at TWT to estimate wing deformation in the ARA wind tunnel.

The approach discussed above is applied for different angles of attack. The overall deformation of the wing is shown in Fig. 3, while the span-wise twisting distribution is illustrated in Fig. 4. It can be observed that the twist is essentially the same for both the clean wing and the wing with attached VGs for incidences up to α = 2.0°. The situation is more complicated for higher angles of attack. Whereas, in the case with VGs, the deformation continues to increase withα– the clean wing is actually less deformed at α = 3.0°than at α = 3.5°. This is due to the boundary-layer separation effects, which reduce the wing loading at higher incidences. The overall difference between twist of the wing tip for the clean wing configuration and the wing with VG configuration at α = 3.5° is approximately Δα = 0.15°.

Figure 3. Estimated model deformations at TWT flow conditions.

Figure 4. Estimated model twist at different incidences for TWT flow conditions.

4.2 Un-deformed wing results

During the initial stage of the study, lift polars were computed for both the clean wing configuration and for the wing with attached VGs, using both SAO and SSG turbulence models. As initial results from the SAO model gave a poor performance, which was likely due to the inappropriateness of the eddy viscosity hypothesis for vortex modelling, it was decided to not examine a two-equation eddy viscosity model, such as k-ω SST.

The clean wing simulations converged to the specified convergence limits regardless of the chosen turbulence model. On the other hand, C_L oscillations are observed in the convergence history for simulations with VGs at high angles of attack α =3.5° (see Fig. 6 for a representative convergence history plot). The lift coefficient predicted by the SAO model is C_L = 0.784±0.004, and for the SSG model is C_L = 0.780±0.005. This corresponds to an uncertainty of around ± 0.5% for both turbulence models. Because of the small uncertainty levels, some information about general flow behaviour can still be extracted from the obtained solutions. Still, a level of variability is expected in the obtained results and has to be taken into account (see Ref. Reference Zastawny12 for details). Note that the analyses presented in this paper are based on the results from the final iteration of each TAU simulation.

Figure 6. Typical convergence history for the simulations at α = 3.5°.

The lift-and-drag polars obtained during the CFD simulations, along with the referenced wind-tunnel data, are illustrated in Fig. 7. Regardless of the chosen turbulence model and whether VGs are present, the predicted lift coefficient remains virtually identical up to the onset of separation estimated at α = 2.0°, which agrees with observations presented in the clean wing study^{(Reference Hahn6)}.

Figure 7. Force coefficient polars for the simulated model.

Differences in the calculated C_L value can be observed for higher incidences. While the SAO model predicts the same value of lift coefficient with and without VGs, a clear impact of the VG array presence is observed when the SSG model is employed. It is worth noting, however, that a significant difference between the turbulence models is observed in the clean wing simulations, as the C_L obtained with the SSG model is considerably lower than that from the SAO model. This indicates that the SAO model does not successfully capture the flow behaviour at large angles of attack. These observations are in line with the results of the HGR-01 aerofoil at stall simulations presented in Ref. Reference Cécora, Radespiel, Eisfeld and Probst10, in which neither of the turbulence models was able to match the experimental results, but the SSG solution was noticeably closer.

The drag polar shown in Fig. 7(b), indicates that the VGs are reducing the drag coefficient at a higher lift coefficient. This holds both for the SSG simulations and the wind-tunnel measurements. More details about the flow behaviour can be extracted upon inspection of the skin friction coefficient plots and oil flow photographs at α = 3.5°, shown in Fig. 8. The c_f distributions obtained with the SAO model strongly differ from the SSG results for both configurations. The figure clearly indicates a completely separated outboard wing flow behind the shock for the clean configuration simulated with the SAO model, and a much more complicated but less separated flow structure observed for the SSG simulations. Furthermore, it is quite obvious that the SAO model fails to capture the global effects of the VGs, as the flow behaviour is modified only in the outboard region of the wing. The SSG model, in contrast, produces flow patterns much closer to the ones observed in the oil flow pictures taken during the wind-tunnel test campaign. The vortices shed from the VGs result in a more energetic boundary layer, which is less prone to separation.

Figure 8. Wind-tunnel oil flow photographs and CFD skin friction distributions at α = 3.5°.

The pressure distribution profiles shown in Fig. 9, allow for a more quantitative analysis to be conducted. The clean wing pressures are significantly different for the two turbulence models. The shock location predicted by the SSG model is upstream of the one observed in the SAO simulation for the clean configuration. This is one possible explanation for the reduced lift computed with the SSG model, even though the SSG results predict smaller boundary-layer separation. Moreover, a very strong region of separation is observed near the trailing edge with the SAO model, resulting in additional lift generated by the wing.

Figure 9. Pressure distributions obtained with different turbulence models at α = 3.5°.

Inspection of the pressures with VGs shows a number of interesting features. In general, the aft shock movement predicted by the SSG model agrees with the wind-tunnel observations, while no clear pattern in the SAO solutions can be identified. Also, the impact of the VGs on the lower surface pressures is captured in the SSG results, but not in the SAO simulation. Both models, however, predict different trailing-edge pressure to that measured during the wind-tunnel test campaign. This is probably due to the fact that deformation is not considered in this case, and it is expected that the results will be closer to the experiment when the deformation is taken into account.

In addition to the aforementioned simulations, a test to assess whether the mesh refinement had a significant effect on the obtained flow solutions was performed by adding the VG sources to the clean wing geometry, i.e. local refinement of volume mesh without addition of VG surfaces. The results indicate that, while there was some impact on the SAO solutions, the SSG simulations were indifferent to the addition of VG sources. A polar was also simulated and is shown in Fig. 10, where the VG mesh contained four cells across the vane thickness rather than two cells as in the standard mesh,. The fact that small differences are observed between the results obtained with different meshes, when compared to the overall difference between the clean wing and VG configurations, justifies the observations made in this paper.

Figure 10. Lift coefficient polar for the SSG simulations with different VG grid refinement levels.

4.3 Deformed wing results

The solutions presented in the previous section used the un-deformed wing as the baseline geometry. However, at high incidences, the aero-elastic effects significantly contribute to the flow behaviour, changing its character, especially on the outboard portions of the wing. Therefore, simulations in which the wing deformation is included are necessary in order to obtain a more quantitative comparison with the wind-tunnel measurements.

The modified wing shape was estimated, based on the approach outlined in Section 3. The calculated bend-and-twist were used as inputs to a TAU process that generated a new, deformed computational grid.

As in the case of the previously presented analyses, two turbulence models (SAO and SSG) were employed to simulate flows past a clean wing and the wing with attached VGs. A polar was generated for each run. Since the wing shape was different at every angle of incidence, each simulation was initialised with free-stream conditions, which led to longer convergence times.

As before, the SSG simulation with VGs did not converge fully and high-frequency oscillations in the lift coefficient at high incidences were encountered. The magnitude of the fluctuations is similar to the un-deformed case and is equal to ±0.2% of the average lift coefficient value. Still, the SSG model is able to capture some of the VG impact on the flow, which is indicated by the differentC_L value.

Lift and polars generated for the deformed wing are shown in Fig. 11. It can be observed that, even though the predicted lift-and-drag coefficient values are closer to the experimental measurements, there are differences throughout the incidence range between the wind tunnel data and simulated coefficients. Explanations for this fact are explored in Ref. Reference Hahn6, and the most plausible causes can be summarised as:

• • Wind-tunnel wall interference and its influence on the flow past the half-model,

• • Local junction separations at the wing root during the wind-tunnel test,

• • Limited accuracy of the deformation prediction method,

• • Limited ability of the CFD to capture accurately flow behaviour at large angles of attack.

Figure 11. Lift coefficient polars for the simulations with a deformed model.

Nevertheless, current results show that the wing deformation under aerodynamic loads is an important factor and cannot be omitted in this type of study.

Focusing on the simulations obtained, it can be concluded that both turbulence models perform nearly identically up to α = 2.0°, where the onset of separation is identified. No VG influence on the forces is visible for low incidences. Their impact, however, starts to emerge for higher angles, although it is only captured by the SSG model, which is in line with the observations made in the previous section.

The SSG turbulence model results indicate that the predicted lift coefficient increase, caused by application of VGs, is smaller than for the un-deformed case. This is indeed confirmed by Fig. 12, where the change of the lift coefficient induced by the VGs is shown. Although, when the deformation is considered, ΔC_L is expected to be closer to the experimental measurements, the opposite is true. The polar data seems to indicate, however, that the difference is more likely to be a result of inaccuracies in the clean wing behaviour prediction.

Figure 12. Change in the lift coefficient due to application of the VGs.

The skin friction distribution plots shown in Fig. 13 differ strongly from their equivalents (Fig. 8), obtained when the deformation was not considered. While the SAO model predicts a fully separated outboard flow behind the shock for both configurations, the SSG turbulence model leads to much more complicated flow behaviour. The clean wing flow is widely attached with only small regions of separation visible in the skin friction image. On the other hand, the impact of the application of VGs can be easily identified by the presence of clear vortex trails. The boundary layer behind the VG vanes is, however, less energetic than the un-deformed simulations presented earlier, which is indicated by lower values of the skin friction coefficient c_f .

Figure 13. Wind-tunnel oil flow pictures and CFD skin friction distributions obtained during deformed wing simulations at α = 3.5°.

The quantitative analysis offered by the inspection of pressure distributions shown in Fig. 14 allows for a better understanding of the flow behaviour. Firstly, it can be observed that the CFD predictions do not match the data acquired during the wind-tunnel test. Although the suction peak c_p values are generally in a good agreement with the experiments, the shock position remains different both in the SAO and SSG results. For the clean wing simulations, the shock is located aft of the experimental measurements; however, its downstream movement, when the VGs are present, is not captured to the full extent. Focusing on the SSG results, it can be observed that the trailing-edge pressures computed with this turbulence model are similar to the experimental values for all configurations at Section 5, but only for the clean wing at Section 6. In other cases, the CFD predicts trailing-edge separation when none is observed in the wind tunnel. Section 5, however, shows an encouraging result, as the pressure distribution from the VG simulation is in line with the experimental measurement.

Figure 14. Pressure distributions obtained with different turbulence models during deformed wing simulations at α = 3.5°.

The analysis of the pressure distributions seems to indicate that the main limitation of the current study is the fact that the clean wing simulations are not similar to the experimental data. As was stated before, CFD solutions are expected to encounter problems when large incidences are considered, even if the SSG turbulence model is adopted. Nevertheless, the behaviour of the VG simulations seems to indicate that it is possible to simulate the VGs with the SSG turbulence model and to obtain reasonable flow prediction.

5.1 Chimera method

An alternative to conformal meshing for the fully resolved mesh around the VGs is to adopt the Chimera method, which employs an unmodified base mesh superposed with an overset Chimera block, used to generate a refined mesh around the VG vanes. In this approach, the VGs are still meshed directly; however, their mesh can be largely decoupled from the baseline wing mesh.

Due to its unique ability to combine independent meshes, Chimera meshing appears to be a promising approach for modelling flows with VGs. Some advantages of this technique are:

• • Once the Chimera block is generated, no modifications are required to the base clean wing geometry.

• • The approach allows for efficient modifications of the VG geometry inside the Chimera block.

• • In theory, the mesh generation for the VG and the wing is easier and much faster than in the case of fully resolved VGs.

The work presented here shows the effects of a preliminary analysis and a feasibility study to assess whether this method can be applied to the simulations of wings with VGs. Since only an initial study has been performed, it was decided to apply the Chimera technique to model only one of the simulated VGs. The second outboard vane was chosen (VG2), as it is expected to provide the most valuable insight into the properties and performance of the technique. All remaining VGs were meshed using the same mesh as in Section 4.

The process of Chimera mesh generation involves a number of steps. Initially, the model geometry is divided into the wing and the Chimera block geometries. Then, additional faces are generated, corresponding to a hole in the base mesh and the Chimera block boundaries. Afterwards, grids for the wing and the Chimera block are generated separately and combined. Finally, the merged grid is assessed to ensure that the overlap region size and the grid refinement are adequate and allow for successful interpolation of the flow variables between the meshes. The process has iterative character, as tuning the Chimera block size and grid refinement is necessary.

As mentioned above, the first step of the process is generation of an appropriate geometry. Figure 15 illustrates the Chimera block and the hole cut in the original geometry. The choice of dimensions was driven by the fact that the hole in the baseline geometry should isolate the VG from the base mesh and the Chimera block should be large enough to ensure that there are at least three cells in the overlap region. The height of the hole was set to eliminate the mesh pull-back in the baseline grid.

Figure 15. The chimera block and the hole in the baseline grid.

A polar, using the Chimera mesh, was generated for the un-deformed model; while, a single point at α = 3.5° was computed for the deformed wing. The lift coefficient polar is presented in Fig. 16. As before, high-frequency oscillations were observed at high incidences; however, they remain small relatively to the mean lift coefficient. The computed C_L at the highest incidence for the un-deformed case differs from the conformal mesh results obtained earlier. On the other hand, the results are identical when wing deformation is considered.

Figure 16. Lift coefficient polar for the simulations involving the chimera meshing.

Inspection of the skin friction distribution plots, presented in Fig. 17, indicates possible reasons for the observed differences. A clear impact of the application of the Chimera method can be identified for the un-deformed case. The vortex trail behind the second VG has disappeared and a region of separation has been observed instead. This leads to the loss of lift which can be seen in the polar plot. The results for the deformed case are identical to those observed in Fig. 13. This is potentially encouraging, as it indicates that the Chimera method can be successfully applied to model the VGs and offers the same quality of results as the simulations with fully resolved mesh around the VGs. However, it should be borne in mind that the impact of the deformation is to reduce the wing loading; hence, the Chimera method might still encounter problems at higher incidences.

Figure 17. Skin friction distributions at α = 3.5° obtained using the Chimera mesh.

Inspection of pressure distributions shown in Fig. 18 allows for more quantitative analysis to be performed. As expected the pressure distributions at Section 3 are nearly identical for all the cases. On the other hand, thec_p values, obtained using the Chimera meshing in the un-deformed simulation, differ significantly from their equivalents at the outboard sections. A significant movement of the shock for both of the outboard sections can be observed. Interestingly, the shock motion differs from section to section, and is downstream of the resolved case in Section 5 and upstream for Section 6, with the Chimera method offering a better prediction for Section 5 and the resolved mesh for Section 6. For the deformed case, it can be clearly seen that the pressures at all sections are in line with the earlier results.

Figure 18. Pressure coefficent distributions at α = 3.5° obtained using the Chimera mesh.

In summary, the results presented here illustrate that the Chimera method can be successfully applied as a promising alternative to a fully resolved VG technique. A deeper study is, however, necessary in order to assess the Chimera method performance when the wing loading is high.

5.2 TAU reduced-order VG model

Adaptation of both the VG modelling approaches presented above requires considerable effort in preparation of appropriate geometry and during the mesh generation process. This fact has been widely accepted and has driven effort to develop a reduced-order VG model, which would allow the influence of the VGs on the flow to be captured, without the full representation of the geometric details of the VG.

A number of reduced-order models have been presented in the literature. An overview of some of the possible techniques, with focus on large-scale applications, is presented in Ref. Reference Booker, Zhang and Chernyshenko3. In general, the available approaches can be divided into two groups: modelling the VG and modelling the vortex alone. In the first approach, additional forcing terms are inserted into flow equations in order to align the flow with the VG, as for example in Ref. Reference Bender13. As modelling the VG directly assumes a continuous fluid domain, the resulting pressure field will not be correct. Also, some grid refinement is required in order to enhance the method accuracy.

In the latter approach, the vortex is modelled independently of the underlying VG geometry, by superposition of the base flow with the velocity profile corresponding to the modelled vortex. This is useful when the details of the flow local to the VG are less important than the downstream impact of the vortex; the local near-vortex flow-field might not be represented accurately, because only tangential velocity components are inserted. A variant of this method was developed by Törnblom and Johansson^{(Reference Törnblom and Johansson14)}, who modify not only the momentum equations but also the Reynolds stress transport equations. This approach forms the basis of the reduced-order VG model currently available in TAU.

The method does not model the VGs directly. Instead, it approximates the vortices produced and adds their effect to the flow equations. The vortices are approximated by a Lamb-Oseen model, while their strengths are computed from lifting line theory. Since the VG model modifies the Reynolds stress transport equations, a simulation with the Reynolds Stress Model (RSM) is required in order to employ this technique.

Modelling of the vortices is achieved by generating a set of VG planes, defined by a number of wall points specified on wall surfaces, to which the modelled VGs are attached. Each wall point defines an infinite array of counter-rotating VG pairs (Fig. 19). The definition of the VG array contains information about the distance between the VGs in a pair (δ), vane height (h) and the distance between consecutive VG pairs (D). The incidence angle of the right hand side and left hand side vanes, their chords and some numerical parameters are also specified.

Figure 19. Geometry and flow induced by the VG array^{(Reference Törnblom and Johansson14)}.

The model's parameter specification is not straight-forward, as the model definition is somewhat ambiguous and not immediately applicable to the investigated set-up. Nevertheless, a number of test cases were prepared to evaluate its performance.

Although, in principle, the model should not require any additional mesh refinement, three meshes were investigated in order to assess its grid sensitivity. The clean wing mesh without any additional sources was used as a baseline case. Two additional grids, with refinement in the VG area, were also generated. Only un-deformed wing simulations at high angles of attack were performed using the VG model.

The resulting lift coefficient at α = 3.5° for different meshes is shown in Fig. 20. All the simulations converged to the set convergence criteria.

Figure 20. Lift polar obtained with TAU reduced-order VG model computed with different meshes.

Analysis of the lift coefficient values indicates that the base mesh solution did not capture the impact of the VGs on the flow behaviour. On the other hand, mesh refinement allows for the vortices to be resolved and leads to a visible increase in the C_L value. However, the lift increase obtained is considerably smaller than in the resolved VG case, and further increase of the refinement level does not improve the results.

The skin friction coefficient plot in Fig. 21 shows the character of the surface flow on the upper wing surface for the fine mesh simulation at α = 3.5°. Clear vortex traces can be identified downstream of the locations of the modelled VGs. Comparison of these results with the earlier observation from the resolved vanes indicates that the TAU VG model produces a much less energetic boundary layer behind the VGs, even though the separated flow region is, in fact, smaller.

Figure 21. Skin friction distribution obtained with the TAU reduced-order VG model computed with fine mesh at α = 3.5°.

The computed pressure distributions are shown in Fig. 22. The c_p values for the base mesh VG model are the same as for the clean wing simulations, which confirms the fact that there is no difference in the calculated force coefficients. The base mesh is too coarse for the vortex to be captured effectively; therefore, some level of refinement is clearly required.

Figure 22. Pressure distributions obtained with the TAU reduced-order VG model computed with different meshes at α = 3.5.

The impact of the modelled VGs is clearly visible in the pressure distributions of the outboard sections when the refined grids are applied. Aft shock motion can be identified; however, no clear pattern related to the mesh refinement level can be captured. Even though the VG model has clearly some influence on the flow behaviour, the obtained results are still closer to the clean wing results than to the resolved VG case. It is also worth noting that the lower surface pressures are virtually unaffected by application of the model, which is neither the case in the wind-tunnel experiments nor in the resolved VG simulations with the SSG model.

Despite the fact that the TAU VG model allowed some of the flow features induced by the VGs to be captured, and was the only method that resulted in a fully converged solution at high incidences, the C_L increase prediction offered by the model is not large enough. The specification of model parameters is somewhat ambiguous, and the obtained vortices are significantly weaker than those from other investigated approaches (Fig. 23). A number of additional tests, in which the model parameters (vane spacing, vortex core radius, etc.) were modified, did not lead to a significant improvement in results.

Figure 23. Vorticity contours at various distances behind the VG trailing edge. (a) 1xVG height behind the VG trailing edge. (b) 5xVG height behind the VG trailing edge.

Considering the analysis presented above and the complicated set-up procedure of the TAU VG model, it can be concluded that this approach is not yet mature enough for industrial application. With some improvements and additional tests, it is possible that the model could deliver a more promising performance.