As already mentioned in the previous section, two different-fidelity fluid models are used for the optimization and the performance assessment post-optimization. The lower-fidelity fluid model is the commercial IBL code MSES of , and the higher-fidelity model is the open-source CFD code . In this subsection, the aim is to present a brief validation of both models on the benchmark case from . Additionally, the CFD model will be validated against experimental results from the NHLP-90-L1T2 multi-element airfoil section .

The first setup consists of a DU97-W-300 base element and a 25 % chord length NACA-22 slat element . Illustrations of the profile and the CFD mesh are shown in Fig. . The base-element-based Reynolds number is Re=1.3×106. The turbulence intensity in the tunnel was around TI≈0.1%.

MSES solves the Euler equations on a discrete 2D grid, coupled with an integral boundary layer formulation. Transition is predicted by a semi-empirical exp⁡N envelope method. Some modifications have been made to this particular version of MSES to improve the stability of the code as detailed in . The mesh far-field distance was set to six chord lengths around the airfoil, and a vorticity correction is used in the far field. Otherwise, standard grid generation settings were used. To match the turbulence intensity in the wind tunnel, the amplification factor was calculated with the correlation Ncrit=-8.43-2.4ln⁡TI100≈8.2 from . For rough conditions, the boundary layer is tripped for both the slat and the main element at x/C=0.1% and x/C=10% on the suction and pressure sides, respectively.

For the turbulence modelling in OpenFOAM a steady-state Reynolds-averaged Navier–Stokes (RANS) model is employed. For the cases with rough conditions, the k-ω shear stress transport (SST) RANS turbulence model from is used. Since this model is fully turbulent, transition does not occur and the boundary layer around the profile is fully turbulent. By contrast, for the cases with natural transition the correlation-based γ-Reθ transition model from is used as an extension to the original k-ω SST model. Second-order schemes were used for the discretization of the momentum and continuity equations. For convergence reasons, bounded first-order schemes were used for the turbulence model transport equations.

For the mesh generation, first cell height fulfils y+<1 and the wall expansion ratio is chosen to be ϵwall-normal<1.2. As a result of a mesh convergence study, the domain extent of the O mesh was set to 400 chord lengths, and around 300 points were used along the airfoil surface. The mesh is shown in Fig. . To match the inflow conditions in the wind tunnel experiment, the inflow turbulence was set to TI=0.1% and the eddy viscosity ratio was νt/ν=10.

The results from the two models and the wind tunnel measurements are shown in Fig. . The profile in the wind tunnel was clean, but for the sake of comparison, the tripped configurations are shown in the figures as well. Close agreement between the lift predictions of MSES and OpenFOAM are observed for both the clean and the rough cases. MSES tends to underpredict drag compared to CFD, but that is expected for IBL formulations. However, both models only yield satisfactory predictions for the lift coefficients and significantly underpredict drag coefficients compared to the wind tunnel measurements. Because of the close agreement between the two models and because the two codes yielded very accurate results for the base element only, the author speculates that the discrepancy comes from either the experiment description or the experiment itself.

Yet, to investigate this discrepancy, the NHLP-90-L1T2 multi-element section from was considered as a second benchmark case. The profile is pictured in Fig. . The free-stream Reynolds number, Mach number, and turbulence intensity corresponding with the experiment are Re=3.52×106, Ma=0.197, and TI=0.01%, where the Reynolds number is based on the length of the element when slat and flap are retracted. Although the free-stream Mach number is low, compressibility of the airflow may affect the slat flow due to the high acceleration of the flow. Nevertheless, as will be seen later on, good agreement between experimental and numerical results is obtained by an incompressible solver.

The experiment is performed on a clean profile. Nevertheless, both the transitional and the fully turbulent simulations were carried out. The results are shown in Fig. . The results show an overprediction of the stall angle of about 6∘ and a slight underprediction of the lift and drag below the stall angle. But, overall, the results are satisfactory and show that the employed CFD model is capable of predicting both lift and drag with an acceptable error margin as long as only predominantly attached flow is considered.

Unfortunately, MSES would not converge on this benchmark case due to the sharp corners of the profile. Nevertheless, correspondence between the two models was already established in the previous benchmark case. And since both MSES and OpenFOAM capture the same trends in results, it is concluded that for this particular case MSES can be used as a surrogate for the higher-fidelity CFD model during the optimization procedure.

The purpose of this subsection is to establish sensible boundary conditions for the slat dimensions in the two following subsections. Further, to reduce the computational cost of this step, a preliminary single-objective optimization of the slat position and dimension was carried out. The slat shape was fixed to correspond to the NACA-22 airfoil. As already mentioned, the slat position and dimension are parameterized using 4 degrees of freedom: the slat chord length cslat, the slat angle βslat, the distance between the slat and the main element hslat, and the streamwise position of the slat element sslat. However, to obtain sensible optimization results, either the slat chord length sslat or the gap width hslat needs to be constrained. Hence, two different fixed wall distances hslat/C=4% and hslat/C=8% were employed. The slat chord length was allowed to vary freely between 15%≤cslat/C≤40%. The slat angle and chordwise position were not constrained.

The local chord-based Reynolds number for the NREL 5 MW reference turbine at design conditions was calculated up to 40 % span and was found to vary between roughly 9 and 10 million . Hence, the design Reynolds number for the preliminary and also the final cases is chosen to be 10 million. Similarly, the free-stream Mach number was also calculated at design conditions and it was below 0.1 up to about 40 % span. Thus, an incompressible CFD solver, as is custom for wind energy applications, was used. However, in MSES compressible effects were considered, and hence the free-stream Mach number was set to 0.1 when using MSES.

The results are shown in Fig. . The outcome shows that, independent of the prescribed slat height hslat, the chord length of the optimal design converges to the upper bound, namely a slat chord length of 40 %. The performance coefficients of the two designs indicate that a larger gap between the slat and the main element also leads to higher lift and lift-over-drag ratios. However, this comes at the cost of a high positive pitching moment at the quarter-chord point of the main element. Since the slat would have to be attached to the base element, the quarter-chord point of the main element is a logical representative position for the calculation of the combined pitching moment of the profile.

Due to structural considerations, in the remainder, the slat height will be prescribed to be smaller than hslat/C<4%. Further, two different chord lengths will be investigated, namely cslat/C=30% and cslat/C=40%.

Now fixing the slat chord length and gap width, the results of the shape optimization of four different auxiliary slat elements are presented in the following. For the optimization, a two-objective formulation is used where the effects of the trade-off between maximum lift and maximum aerodynamic efficiency are highlighted. Further, three different design angles of attack are chosen, namely α=8∘, α=12∘, and α=20∘, where the second one is weighted the most. Also, the performance of the soiled profile is weighted much higher than the performance of the profile in clean conditions, since the first condition is more prevalent in reality. The general optimization settings as well as the boundary conditions for the different auxiliary slat designs are specified in Table . The choice of boundary conditions documented there will be explained in the following.

An optimized reference configuration referred to as configuration A with a slat chord length of cslat=40%, a gap width of h/C=4%, and a base profile thickness of tmax/C=40% is chosen. The base profile has the shape of the DU00-W2-401 profile. At this point, no structural constraints are considered. Hence, the only constraint that is imposed on the slat geometry is that, at each chordwise position, the thickness is larger than the trailing edge. The trailing-edge thickness is fixed to hTE,slat/C=0.2%. As previously mentioned, in theory, either the slat chord length or the gap width needs to be fixed to have a fully constrained optimization problem. However, the optimization results for both the preliminary and the actual designs tended to converge towards the upper bound for the gap width even if the bound was relatively large such as h/C=8%. Hence, the gap width was fixed in subsequent optimization runs to save computational cost.

To assess the influence of the gap size between the slat and the main element, the slat relative chord length, and the base profile thickness on the optimal slat design, the optimization procedure is also carried out for three modified design. For the second configuration named configuration B, the gap size is halved to h/C=2% compared to the reference design. For the third configuration referred to as configuration C, the slat chord length is reduced from cslat/C=40% to cslat/C=30%. Lastly, for the fourth design named configuration D, a thicker base profile with a thickness of tmax,base/C=48% is chosen. The base profile is a scaled version of the FFA-W3-360 profile which was used in the .

The optimal slat designs are pictured in Fig. . Each time, three points on the Pareto front are shown. The three points correspond to the two extreme points and one point roughly in the middle of the Pareto front. For clarification see the example shown in Fig. . Figure  compares the different optimal slat designs in terms of slat angle, thickness, camber, and streamwise position. The figure also already contains the results from the integral design procedure that will be presented later on, so for now part of the results can be ignored.

Given the limited sensitivity analysis shown in Figs.  and , the following observations can be made.

General trends. The optimal slat design for all configurations and objective weightings are highly cambered, with cambers of the shown designs ranging between roughly 8% and 20%. This is a consequence of the previously mentioned dumping effect. Due to this, attached flow around the slat can be maintained even for a highly cambered profile. Moreover, the chordwise positioning of the slat is largely insensitive to the imposed design choices. The optimal streamwise position of the slat relative to the base element is mostly determined by the streamwise location of the suction peak on the base element. Since all the base profiles are classical thick wind turbine airfoils, the location of the suction peak is very similar. Namely, the profiles have a relatively blunt leading edge that is designed to have the transition point close to the airfoil nose. Hence, the optimal streamwise slat position is around xTE,slat/C≈10% for all the investigated constellations. Compared to more traditional high-lift configurations for aerospace applications, the streamwise location of the slat element is further aft. This is because usually for aerospace applications thinner profiles which have the suction peak further forward are used.

Moving on to a more detailed performance assessment of the optimal designs, the performance coefficients of the maximum lift and the maximum glide ratio designs as estimated by CFD are shown in Figs.  and , respectively.

Figure  shows a comparison between the lift predictions from MSES and OpenFOAM for the baseline configuration alongside the pressure distribution at α=16∘. As was previously alluded to, the optimizer seems to exploit weaknesses in the fluid model for the high-lift designs. Due to the setup of the optimizer, the flow model has to converge up to α=20∘. However, MSES tends to overpredict the stall angle by at least Δα≈5∘ compared to OpenFOAM. Further, as was already visible in the benchmark cases, the employed turbulence models for CFD do already overpredict the stall angle compared to measurements. For the rough profile at α=16∘, MSES predicts fully attached flow on both elements whereas OpenFOAM seems to indicate large separation zones on both elements. Hence, the profiles optimized for maximum lift at the design angles actually perform worse in terms of lift compared to the ones optimized for maximum glide ratio at the design angles. Thus, going forward only the designs optimized for maximum glide ratio will be considered. Due to the shortcomings of MSES in modelling the stall onset, it does not seem to be a suitable tool to be used in maximum lift design optimization. For the sake of completeness, the lift and drag polars for the high-lift designs are plotted in Appendix  in Fig. .

The performance of the profiles optimized for maximum glide ratio in Fig.  shows some common characteristics despite the different optimization boundary conditions. For the rough profiles, the stall angle lies close to α≈20∘, and maximum lift coefficients lie above cL>3. Beyond stall the lift coefficients steeply drop to cL≈1. The roughness sensitivity is strongly reduced compared to the base profile only, but this could also be a consequence of the low weighting of the clean performance in the objective formulation. Beyond α≈8∘, the glide ratio is larger than for the base profile only, and the glide ratio shows a much less pronounced peak in the glide ratio distribution. The combined pitching moment calculated at the quarter chord point of the main element is positive due to the forward location of the slat element. The force and momentum coefficients for the main element only, shown in Fig. , are obtained from CFD as well, even though measurements are available.

Compared to the baseline design, both a reduction in the gap between the main and the slat element and a curtailment of the slat chord length lead to lower lift and lift-over-drag values. As expected, an increase in the base element thickness leads to higher lift and drag coefficients. The largest spread in the performance coefficients between the different designs is seen in the pitching moment coefficient. The design with the curtailed chord length shows pitching moment coefficients of the same magnitude as the main element only. The pitching moment of the other designs is between 2 and 4 times higher than that of the base element only.

In this subsection, the results of an integral slat design procedure where the shape of both the main and the slat element is optimized simultaneously will be presented. Again, four different cases with the same boundary conditions as for the design of an auxiliary slat design will be used. The general optimization settings and the optimization boundary conditions are summarized in Table .

The fixed trailing-edge thickness of the slat element remains unchanged. The trailing-edge thickness of the main element is fixed to hTE/C=1%. Additionally, to make the design more realistic, additional thickness constraints are imposed on the main element. This is to ensure that there is enough space for the wing box. Analogous to , a minimum local thickness of t(x/C)/tmax>85% was enforced at two chordwise stations, namely at x1/C=15% and at x2/C=40%. This leaves space for a box length and height of at least lbox/C=25% and hbox/tmax=85%, respectively. For the design of the slat element structural constraints are ignored at this point since they are considered less critical because both the bending and the torsional loads on the slat element are lower than the loads on the main element.

The resulting optimal designs are shown in Fig. . Again three elements of the Pareto front are shown, the two most extreme ones and one roughly in the middle. At first sight, the designs look very similar to the ones from the previous subsection where only the slat element is optimized. The optimal slat designs show the same sensitivity to the boundary conditions as for the design in the previous subsection. However, their streamwise placement is much further forward than for the auxiliary slat design cases owing to the blunter leading edge of the main elements. The shape of the optimal main elements is much less sensitive to the aerodynamic efficiency of the design than the optimal shape and position of the slat element. However, this is at least partly a consequence of the structural constraints on the main element, because without the structural constraint the pressure side of the main element in particular looked very different. Nevertheless, the main element profiles look very similar to the results obtained by . A more detailed comparison between the designs obtained with the auxiliary and the integral design procedure will be shown in the next subsection.

The performance assessment of the optimal designs was again carried out using CFD. The result for the designs with the maximum glide ratio are shown in Fig. . Since the designs optimized for maximum lift suffer the same shortcomings as for the previous optimization round, they are not discussed and only pictured in Fig.  in Appendix .

Again the integral two-element designs optimized for maximum aerodynamic efficiency also show some common characteristics. The stall angle lies around α≈25∘, and a steep drop in the lift coefficient to about cL=1.5 is predicted post-stall by CFD. The glide ratios lie above the ones for the main element only for α>8∘. For some of the designs, this holds true even below that angle of attack. The glide ratio remains close to the maximum glide ratio for the angle of attack range of roughly 8∘<α<20∘. Further, a much lower roughness sensitivity than for the base element only is observed. Lastly, again the pitching moment coefficients show the largest dependence on the optimization boundary conditions.

When comparing with the performance of the designs with the fixed main element from Fig. , roughly the same sensitivity to the boundary conditions is observed, and some performance gains are obtained through a more integral design approach independent of the boundary conditions. Namely, the stall angle increases by about Δαstall≈5∘. Further, at α=20∘, an increase in lift coefficient and aerodynamic efficiency of at least a maximum of ΔcLmax≈0.5 and ΔGmax≈5 is observed, respectively. On the flip side, due to the higher lift the pitching moment also increases; at α=20∘, the pitching moment increases by about a factor of 2 for all the designs.

In this subsection, a more detailed comparison between the optimal designs from the auxiliary slat and the integral two-element procedure is presented.

Figure  shows the designs optimized for aerodynamic efficiency for the auxiliary and the integral runs. Three general trends are observed when comparing the integral with the auxiliary designs independent of the boundary conditions. First, the shape of the slat element excluding angle and position is not very dependent on the shape of the main element but highly dependent on the optimization boundary conditions. Second, the optimal main elements show a much blunter leading edge and a lower profile thickness beyond x/C≈35% compared to the standard wind energy airfoils used for the auxiliary slat design. The blunt leading edge leads to a forward shift of the suction peak, and hence a forward shift of the optimal slat position compared to the fixed main airfoils. The thinner shape of the main elements for the backward part of the profile may be a result of different wing box constraints; namely for the design of the wind turbine base airfoils, a larger box length was possibly assumed. A more cambered main element leads to higher lift from the main element only, which allows for less aggressive slat designs for the same total lift between the main and slat elements and higher stall angles.

In Fig. , the slat angle, thickness, camber, and streamwise trailing-edge location are shown for both the auxiliary and the integral design procedure. The two extreme elements and one element roughly from the middle of the Pareto front are shown. The range of the optimal slat thickness and camber is not strongly influenced by the choice of the design procedure. The slat angle and streamwise trailing-edge location are more dependent on the choice of the design procedure. As already mentioned, the blunter leading edge of the main elements resulting from the integral design procedure leads to a more forward optimal slat trailing-edge position. Additionally, the range in the optimal slat angle roughly halves when compared to the auxiliary design procedure, likely because a more forward position of the slat element reduces the influence of the shape of the base element on the optimal slat design.

In Figs. , , , and the pressure distributions for three angles of attack are shown for the designs optimized for maximum aerodynamic efficiency resulting from both the auxiliary and the integral design procedure. The pictured angles of attack are the two design angles α=12∘ and α=20∘, as well as an angle α=28∘ that is close to either stall or post-stall, depending on the design.

For the two baseline designs shown in Fig. , the profile resulting from the auxiliary optimization procedure shows higher suction peaks on the slat element and lower suction peaks on the main elements for all the pictured angles of attack compared to the profile resulting from the integral design procedure. This shows that the integral design procedure leads to profiles which better balance the inverse pressure gradients on the two elements. Hence, the flow around the resulting designs is expected to remain attached up to higher angles of attack.

Both profiles also show the same stalling behaviour as visualized in Fig.  for the clean reference configuration obtained from the integral design procedure. Initially, simultaneously the flow on both the slat and the main element begins to stall from the trailing edge. As the angle of attack is further increased and the separation line moves towards the leading edge, the wake of the slat becomes wider as well. At some point, the slat wake extent grows so much that the low-pressure area in the wake leads to a relaxation of the pressure gradient along the main element, and hence the flow on the main element reattaches. This is in accordance with the off-the-surface pressure recovery effect identified by and described in the introduction of the paper. Finally, at some point, both elements are fully separated. In fact, this is observed for all the designs optimized for maximum aerodynamic efficiency in this article. See Figs.  and in Appendix  for a visualization of this process for the configuration with the reduced chord length.

For the profiles with the smaller gap width shown in Fig. , the pressure distributions look very similar to the baseline case. The main difference is that for these two configurations, the slat is turned into the flow a bit more. As a consequence, the lift is a bit lower and the stall angle a bit higher compared to the baseline design.

The pressure distributions for the profiles with the curtailed slat chord length are shown in Fig. . The slat elements of these two profiles have the highest camber, the lowest thickness, and the highest slat angle when compared to designs obtained with the same optimization procedure. The interaction between all these partially counteracting effects leads to the highest suction peak on the main element and, as a consequence, the lowest pressure near the trailing edge on the slat element when compared to all the other profiles with the same main element thickness. This holds true despite the slightly lower lift coefficients when sized up with all the other designs. Consequently, compared to the other designs, the main element is producing a larger share of the overall lift loading. This explains why the pitching moment coefficients are significantly lower for the designs with the reduced chord length. Further, they also show the highest stall angles within the two categories. Given that, as shown in Figs.  and , the stalling mechanism is driven by the separation on the slat element, the reason for the higher stall angle could be the higher comparative circulation on the main element. This is because a higher circulation on the main element leads to a higher outflow velocity at the slat trailing edge and hence lower negative pressure gradients on the slat element.

Finally, Fig.  shows the pressure distributions for the two profiles with the thicker main element. As visible in Fig. , due to the higher base profile thickness, these two profiles have the highest lift coefficients and the lowest aerodynamic efficiency pretty much at all investigated angles of attack. However, beyond α>8∘, the pitching moment coefficients are roughly in the midfield between the highest values from the thinner designs with the regular slat chord length and the lower values from the thinner designs with the curtailed slat chord length. The origin of this is twofold. First, the pressure distributions show a slightly lower ratio of the slat to the main element lift loading for the designs with the thicker main element compared to the other ones. Second, for the integral design case, the optimal slat placement is a bit further aft compared to the other designs.

Some of the designs show very high suction peak values for the slat, which is an indication that locally compressibility effects may not be negligible despite the low free-stream Mach number of Ma∞≈0.1. Calculating the local Mach number from the incompressible flow field for all the CFD cases shows that for the designs and angle of attack configuration where the lift coefficient is higher than 4 the Mach number locally approaches 0.45. This is indeed very high, and it is recommended that in future publications, compressibility effects should be considered if a design optimization for high lift is carried out. Nevertheless, given that the turbulence model of the CFD solver is expected to overpredict the stall angle, it is not certain that such high Mach numbers will actually be reached in real life.