A High-Lift Optimization Methodology for the Design of Leading and Trailing Edges on Morphing Wings

applied sciences

Article A High-Lift Optimization Methodology for the Design of Leading and Trailing Edges on Morphing Wings

Charalampos Themistokleous 1 , Nikolaos-Grigorios Markatos 2,3 , John Prospathopoulos 1,* , Vasilis Riziotis 1, Giorgos Sieros 4 and George Papadakis 5

1 Fluids Section, School of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechniou, GR 15780, Zografou, 157 80 Athens, Greece; [email protected] (C.T.); vasilis@fluid.mech.ntua.gr (V.R.) 2 School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 9 Heroon Polytechniou, GR 15780, Zografou, 157 80 Athens, Greece; [email protected] 3 School of Electronic and Electrical Engineering, Kingston Lane, Brunel University London, Uxbridge, London UB8 3PH, UK 4 iWind Renewables, Kleisthenous 166, GR 15344, Gerakas, 153 44 Attiki, Greece; [email protected] 5 School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou, GR 15780, Zografou, 157 80 Athens, Greece; papis@fluid.mech.ntua.gr * Correspondence: jprosp@fluid.mech.ntua.gr

Abstract: Morphing offers an attractive alternative compared to conventional hinged, multi-element high lift devices. In the present work, morphed shapes of a NACA 64A010 airfoil are optimized for maximum lift characteristics. Deformed shapes of the leading and trailing edge are represented through Bezier curves derived from locally defined control points. The optimization process employs the fast Foil2w in-house viscous-inviscid interaction solver for the calculation of aerodynamic Citation: Themistokleous, C.; characteristics. Transitional flow results indicate that combined leading and trailing edge morphing Markatos, N.-G.; Prospathopoulos, J.; may increase maximum lift in the order of 100%. A 60–80% increase is achieved when morphing Riziotis, V.; Sieros, G.; Papadakis, G. A High-Lift Optimization is applied to leading edge only—the so-called droop nose—while a 45% increase is obtained with Methodology for the Design of trailing edge morphing. Out of the stochastic optimization algorithms tested, the Genetic Algorithm, Leading and Trailing Edges on the Evolution Strategies, and the Particle Swarm Optimizer, the latter performs best. It produces the Morphing Wings. Appl. Sci. 2021, 11, designs of maximum lift increase with the lowest computational cost. For the optimum morphed 2822. https://doi.org/10.3390/ designs, verification simulations using the high fidelity MaPFlow CFD solver ensure that the high lift app11062822 requirements set by the optimization process are met. Although the deformed droop nose increases drag, the aerodynamic performance is improved ensuring the overall effectiveness of the airfoil Academic Editor: Jérôme Morio design during take-off and landing.

Received: 4 February 2021 Keywords: high-lift devices; morphing; design optimization; droop nose; trailing edge ﬂap Accepted: 16 March 2021 Published: 22 March 2021

Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in published maps and institutional affil- High lift devices such as leading edge (LE) slats and trailing edge (TE) flaps have been iations. effectively employed for many years to increase the lifting performance of aircraft wings during take-off and landing. Conventional slats and flaps used by the aviation industry today consist of movable, split, or hinged mechanical sections with or without slotted surfaces. A modern alternative to the abovementioned articulated multi-element high lift devices is based on shape morphing concept, like the droop nose LE and the morphed Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. TE. Morphed wings can attain similar aerodynamic performance characteristics to those This article is an open access article of conventional high lift devices by tailoring the shape (both mean-line curvature and distributed under the terms and local thickness distribution) of the wing airfoil sections through the deployment of smart conditions of the Creative Commons material elements at the LE and TE regions. Attribution (CC BY) license (https:// Smart materials have the property of altering their shape when variable external creativecommons.org/licenses/by/ conditions are imposed (e.g., Shape Memory Alloys—SMAs—deform through heating 4.0/). and cooling). In this way, varying LE and TE shapes can be achieved using lightweight

Appl. Sci. 2021, 11, 2822. https://doi.org/10.3390/app11062822 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 2822 2 of 23

distributed actuators, acting locally on the smart material elements. By following the above approach vulnerable complex and heavy mechanical parts can be avoided. Additionally, a seamless surface geometry is attained eliminating large gaps or slots of multi-element devices, which constitute the main source of high frequency noise emission during take-off and landing. A typical example of morphing control is the droop nose airfoil, which compared to the traditional slat, exhibits lower drag levels (better aerodynamic performance) and better stall characteristics. However, the sole use of the droop nose wing cannot ensure as high levels of the maximum lift coefficient (CL,max) and the stall angle of attack (AoA) as those obtained by two- or three-elements slats. Research on high-lift devices has been carried out for several decades. Gillis and McKee [1] studied the behavior of NACA23012 airfoil in a wind tunnel, using a Maxwell LE slat and a slotted and a split flap. Their target was to determine the optimum slot gap and investigate the aerodynamic characteristics of the airfoil for several deflection angles of both types of flap. The optimum slot gap of 0.0175% of the chord increased the CL,max and stall AoA considerably. The effectiveness of the slat, however, dropped off with increasing flap deflection. Axelson and Stevens [2] investigated the aerodynamic performance of NACA 64A010 with slat in a wide range of Mach and Reynolds numbers. Two families of slats were employed, one with the LE slat extended forward along the airfoil chord line, and the other with the slat extended forward and displaced below the chord line. At lower Mach number tests, the highest maximum lift values were measured with the slat nose displaced below the wing chord line. However, at Mach numbers higher than 0.7, adverse effects such as a large increase in drag and in the zero lift angle of attack resulted with the slat nose below the airfoil chord line. Until the beginning of 1990, substantial research effort focused on the determination of the optimum position for the different flap and slat types. Investigations included many different experiments, using either flap or slat or their combination. Moreover, the effect of high lift devices at different flight stages, speeds, and altitudes was analyzed. Rudolph [3] made an extensive review of the available high-lift devices, their functions, and design criteria. He appraised the high-lift systems of the commercial aircrafts of his time, presented personal study results, and developed a weight and cost model. He concluded that there was still margin for improvement in terms of weight, cost, and aerodynamic performance, pointing to the direction of reduced complexity, increased reliability, and lower weight, while maintaining or improving aerodynamic performance. Monner et al. [4] developed the concept of a “smart” slat without slot as an alternative to the conventional droop nose utilized on A380 aircraft. The main emphasis of the new device was on realizing a structure/system solution for a smooth leading surface, which can be deflected into a typical high lift application. The approach employed flexible panels on the upper and lower surface of the device in order to provide a continuous curvature distribution over the device chord. The simulation results proved the feasibility of the device and pointed out the need for materials with improved strength for morphing applications. Burnazzi and Radespiel [5] evaluated the potential of gapless droop nose devices designed for improving the aerodynamics of airfoils with active high lift. Flexible droop nose configurations were obtained by smoothly morphing the baseline LE shape. Increasing the stall angle of attack and reducing the power required by the active high-lift system were the main objectives. The most promising droop nose configurations were compared with a conventional slat device as well as with the clean LE. Another droop nose investigation was carried out by Wang et al. [6], who performed CFD simulations to study the aerodynamic characteristics of the two-element airfoil with droop nose, based on the British NHLP 2-D L1T2 three-element airfoil with slat. Numerical results showed that the three-element airfoil with slat produced a higher CL at the tail-scrape angle, as well as a much higher CL,max than the two-element airfoil with droop nose. On the other hand, the two-element airfoil with droop nose had a much higher critical aerodynamic factor at the second-segment 2 climb stage (lift to drag ratio when CL = CL,max/1.13 ). By deflecting the Fowler flap and the spoiler, the aerodynamic characteristics of the two-element airfoil with droop nose were Appl. Sci. 2021, 11, 2822 3 of 23

improved; CL at the tail-scrape angle surpassed that of the three-element airfoil with slat, maintaining at the same time a higher lift-to-drag ratio. However, the three-element airfoil still exhibited a higher CL,max due to larger stall AoA. Tian et al. [7] introduced the flexible variable camber TE flap to improve flight efficiency during take-off, cruise, and landing. A multi-objective optimization with a numerical simulation method indicated that, compared to the traditional flap, CL was considerably increased during take-off and landing (8% and 1.3% respectively), whilst under cruise state, the lift to drag ratio reached a maximum increment of about 30% over a wide range of CL. Finally, in their recent research, Magrini and Benini [8] performed an aerodynamic optimization of a morphing leading edge based on the class/shape transformation technique, incorporating a procedure to keep the arc length of the curve constant in order to limit the axial stress of the deformed shapes. The main benefit of the morphing leading edge was a substantial drag reduction, whereas the lift was less positively affected. A metamodel-assisted optimization loop through the use of an artificial neural network was proven suitable for reducing the computational effort. In previous investigations, morphing is applied and assessed either on the LE or on the TE region of an airfoil. In the present work, combined morphing of both LE and TE regions is applied and the potential of increasing the CL,max of a typical aircraft airfoil is investigated. The target of the work is twofold: First, the development of a fast methodology capable of delivering the optimum morphed airfoil designs in terms of the CL,max increase and second, the physical understanding of the effect of morphing on the CL,max and the stall AoA through a detailed flow field inspection. As baseline airfoil, the typical in aircraft applications, NACA 64A010, has been selected. Morphed LE and TE shapes are approximated using Bezier curves derived from a number of 11 and 9 control points, locally defined in the vicinity of the LE and TE regions, respectively. The vertical positions of the control points constitute the design variables of a design optimization loop, which is thereafter followed in order to obtain the LE and TE shapes, which provide maximum CL and stall AoA. The procedure of defining the search space of the control points is described in detail and may constitute a guideline for future works on design optimization. In the optimization process, candidate morphed solutions are requested to satisfy certain geometrical constraints regarding smoothness of the surface curvature, while their aerodynamic performance characteristics (lift and drag polars) are obtained using Foil2w, a fast viscous-inviscid interaction in-house solver [9]. The objective function of the optimization process is the maximum lift coefficient CL,max. In order to reduce the computational cost and render the entire process efficient, three different categories of optimization algorithms are tested. Apart from the traditional and most commonly used in design problems, Genetic Algorithm, the alternative of Evolutionary Strategies and the more modern Particle Swarm Optimizer are applied and evaluated. Optimum designs stemming from the design optimization loop are verified through simulations using the in-house Unsteady Reynolds Averaged Navier-Stokes (URANS) solver MaPFLow [10]. In addition to the verification purpose, CFD aerodynamic simulations (a) provide better understanding and physical insight of the local flow characteristics over the modified portions of the airfoil section, which are directly linked to the resulting distributed and overall aerodynamic loads and (b) estimate the change in the aerodynamic performance (CL/CD) using the morphed airfoils. Hence, CFD simulations serve the scope of interpreting the results obtained, but also ensure the overall aerodynamic effectiveness of the design. The following text is organized as follows: In Section2, the optimization procedure along with the solvers employed for the flow simulations are described. The application and the results of the methodology are presented in Section3. Section4 refers to the verification of the aerodynamic results obtained through the optimization chain for selected morphing designs using the CFD solver. Finally, in the last section, the main conclusions of the work are reported and discussed. Appl. Sci. 2021, 11, 2822 4 of 23

2. Methodology Three representative stochastic algorithms are integrated into the optimization process, all included in the Inspyred library of Python [11]: Genetic Algorithm (GA), Evolution Strategies (ES) and Particle Swarm Optimizer (PSO). All of them are members of the Evolutionary Algorithms (EA) [12,13] and share a great number of common features. They are population-based performing with best-to-survive criteria. EA algorithms are considered suitable for design optimization of airfoils, therefore they have been selected in the present work instead of other methods, such as deterministic gradient-based algorithms, which may present convergence difficulties during the optimization process. GA [14] is the most traditional category, especially in the fields of Aerodynamics and Turbomachinery. It resembles the members of population with chromosomes and applies the typical operators of crossover and mutation to evolve to the next generation. ES algorithm [15] applies the mutation and recombination operators not only on the design variables, but also on endogenous strategic parameters such as the mutation strength. It usually employs encoding of design variables in the field of real and not binary numbers as GA does. PSO algorithm [16] has been in application for the last two decades. It represents each solution as a particle and the population of solutions is called a swarm of particles. The best positions of each particle and of the swarm are continuously updated and the velocities are adjusted based on the experiences of the particles. The procedure stops when the position of the swarm satisfies the criterion for the objective function. In the following the optimization problem is outlined. This requires definition of the objective function, the design variables of the problem and the necessary geometric design constraints, which ensure that physically acceptable solutions are obtained, while at the same time prevent unnecessary computations for geometrically inconsistent candidate solutions. In the present analysis, as objective function the maximum lift coefficient (CL,max) of the morphing airfoil shape is chosen, under specific conditions with respect to the freestream, i.e., specific values of Reynolds (Re) and Mach (Ma) numbers. The baseline airfoil (undeformed shape of wing cross section) is chosen as the starting design of the optimization process and its coordinates are normalized with respect to the chord. In order to maximize CL,max, the shape of the baseline airfoil is modified at the LE and TE regions, defined by the intervals (0, 0.25) and (0.8, 1) of the x-coordinate, respectively. The design variables are the y-coordinates of the control points of the Bezier curves used in approximating the airfoil morphing shapes. The control points are only allowed to move vertically. In the present work, the Bezier curves approximation method has been chosen due to its flexibility in representing smooth geometries, such as airfoil deformations, by altering the positions of the control points. Curvature and slope constraints are imposed to avoid irregularities and discontinuities of the derived geometries and to ensure a smooth connection between deformed and undeformed parts. These constraints are related to: (a) The slope variations of the airfoil geometry. Large changes in the geometry gradients should not be permitted and this is checked by the product of the gradients (first derivatives) of consecutive segments of the discretized geometry. In the case that negative values of this product less than −10−3, are obtained, the current airfoil geometry is rejected. (b) The maximum curvature of the LE geometry, which undergoes the highest deformations. An upper limit is set for the curvature of the upper and lower parts of the deformable LE geometry. The curvature is calculated with respect to the second and the first derivative.

The viscous-inviscid ﬂow solver used for the prediction of CL and CD polars of the designed shapes during the optimization procedure is the in-house Foil2w code [9]. It is a low-cost viscous-inviscid interaction model, formulated on the basis of a standard panel Appl. Sci. 2021, 11, 2822 5 of 23

method combined with a vortex blob approximation of the wake, in which the “equivalent → inviscid ﬂow (EIF)” velocity u e is expressed in the form:

→ →0 →∗ u e = u + u (1)

→0 →∗ where u is the velocity of the purely potential flow and u is the correction term that →0 needs to be added in order to account for viscous effects. The potential flow part u is simulated on the basis of a standard panel formulation by distributing singularities along the airfoil geometry (piecewise constant sources and constant vorticity), while the wake is represented by free vortex particles, which are convected with the local flow velocity. →∗ The modeling of the viscous correction part u is also based on an integral representation (integration over the airfoil SB and the wake SW): → → →∗ → 1 Z →∗ → →∗ → → r → 1 Z →∗ → r → u ( x ; t) = u ( s ; t) − u ( s ; t) k × ds( s ) + [[u ]] ( s ; t) ds( s ) (2) 2π n τ r2 2π n w r2 SB SW

→∗ →∗ where u n is the normal correction velocity on SB and [[u n]] the normal correction velocity →∗ jump on SW, while u τ is the tangential correction velocity on SB (note that tangential correction velocity jump in the wake is omitted). The viscous flow solution (the integral boundary layer quantities, displacement thickness, and momentum thickness) is obtained by solving the unsteady integral boundary layer equations as formulated in steady-state form by Drela and Giles [17] and re-casted in their unsteady form by Riziotis and Voutsinas [9]. The viscous-inviscid interaction coupling is achieved through the normal to the wall and the wake shear layer transpiration velocity distribution that represents the mass flow difference between the real viscous flow and the equivalent inviscid flow within the boundary layer, given by:

→∗ d ∗ →∗ d ∗ upper d ∗ lower u = (u τδ ) and [[u ]] = (u τδ ) + (u τδ ) (3) n ds e, n w ds e, ds e, As seen in Equation (3), in the wake, the transpiration velocity jump is calculated by summing up the mass deficit of the upper and lower side of the wake shear layer. The boundary layer solution is supplemented by a transition prediction model based on the eN spatial amplification theory [18]. It is noted that Foil2w has been thoroughly validated against measurements on an airfoil undergoing harmonic pitching motion [9]. The flow chart of the optimization process is illustrated in Figure1. Random combinations (ensembles) of control points are generated and create Bezier curves that approximate the LE and TE regions. They represent an initial population of solutions (geometries). Every single airfoil is evaluated using Foil2w: the polar CL-AoA is predicted and the maximum lift coefficient value CL,max is compared to the one of the previous airfoil (starting value of CL,max is that of the baseline airfoil). If CL,max increases, geometry and control points (design variables) are saved and a new population (generation) is generated, based on the best solutions of the current set studied. The final morphing shapes selected through the optimization process are verified through high fidelity CFD simulations. The URANS CFD solver employed for the validation of the Foil2w predictions is the in-house MaPFlow code [10]. It is a multi-block MPI enabled compressible solver equipped with the Eriksson’s preconditioner [19] to handle regions of low Mach flow. The discretization scheme for the convective fluxes is cell centered and makes use of the Roe approximate Riemann solver [20]. In space, the scheme is 2nd-order accurate defined for unstructured grids and applies the Venkatakr- ishnan’s limiter [21]. In time, the scheme is 2nd-order accurate and implicit utilizing dual time stepping to facilitate convergence. The solver is equipped with the Spalart-Allmaras (SA) [22] and the k-ω SST [23] eddy viscosity turbulence models. Regarding transition, Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 23 Appl. Sci. 2021, 11, 2822 6 of 23

garding transition, three models have been implemented, the correlation γ-Reθ model of γ threeMenter models [23], the have Granville/Schlichting been implemented, transi the correlationtion method-Re describedθ model ofin Menter[24], and [24 the], the eN N Granville/Schlichtingmodel described in [25]. transition MaPFlow method has been described thoroughly in [25 validated], and the against e model experimental described indata [26 in]. MaPFlowmany steady has beenand unsteady thoroughly airfoil validated simulations against including experimental simulation data in manyof flow steady con- and unsteady airfoil simulations including simulation of flow control devices such as vor- trol devices such as vortex generators [26] and TE flaps [27]. In [27] both MaPFlow and tex generators [27] and TE flaps [28]. In [28] both MaPFlow and Foil2w, predictions were Foil2w, predictions were validated against measurements in static (steady) and oscillat- validated against measurements in static (steady) and oscillating (unsteady) TE flap cases. ing (unsteady) TE flap cases.

Figure 1.1. Flowchart of the optimization process. CP stan standsds for for Control Control Point. Point. The The superscript superscript 0 0 on on CL,maxdenotes the previous ensemble of CPs and geometry. CL,max denotes the previous ensemble of CPs and geometry. 3. Optimization Results For thethe optimizationoptimization process, NACA 64A010 is used, as it isis aa commoncommon airfoilairfoil forfor aircraft applications.applications. TheThe ambient ambient flow flow conditions conditions employed employed in thein the numerical numerical analyses analyses are Mare = M 0.22 = 0.22 and and Re =Re 6 =× 610 × 610, corresponding6, corresponding to to typical typical take-off take-off and and landing landing conditions conditions of of a shorta short to to medium medium range range aircraft aircraft with with flight fligh speedt speed of of 75 75 m/s m/s andand aa chordchord lengthlength ofof 1.21.2 mm (close(close toto thethe tiptip ofof thethe aircraftaircraft wing).wing). Separate Bezier curves are producedproduced for thethe TETE andand thethe LELE regionsregions consideringconsidering aa separate group of of control control points points for for each each one one of ofthem. them. Both Both the thesuction suction and andpressure pressure side sidegeometry, geometry, either either of the of LE the or LE the or TE the region, TE region, are represented are represented by the by same the same curve. curve. The Thex-coordinates x-coordinates of the of control the control points points used usedin the in present the present analysis analysis are pre-selected are pre-selected as illus- as illustratedtrated in Figure in Figure 2 and2 and they they are are not not modified modified throughout throughout the the optimization optimization procedure. procedure. A Apreliminary preliminary study study is made is made to determine to determine the optimum the optimum number number of control of control points pointsfor the forLE the LE and TE regions. A total of nine control points is found sufficient for reproducing Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 23

and TE regions. A total of nine control points is found sufficient for reproducing the sharp TE shape of the airfoil section with reasonable detail. Less points result in significant geometrical irregularities, demanding additional constraints that is not obvious to determine. More points increase the possible combinations and the computational cost of the optimization procedure. Two of the control points are placed at the connections between the undeformed and the morphing parts of the airfoil (one on the upper and another on the lower side respectively) and remain fixed throughout the optimization process. The remaining points are divided between the upper and lower side (three on the upper side, three on the lower side, and one at the TE point) and they are only allowed to move vertically. As shown in Figure 2b, expected deflections in the TE region will be relatively small and therefore it is sufficient to simply distribute control points at equal distances along the x-axis (Figure 2b). The y-coordinates of the seven movable control points constitute the design variables of the optimization loop for the TE morphing shape. A similar approach is followed for the LE region. Again, two control points remain fixed, connecting the undeformed with the morphing part, and seven control points are distributed on the two sides, three on the upper and four on the lower side, as shown in (Figure 2a). Again, the y-coordinates of the seven control points constitute the design Appl. Sci. 2021, 11, 2822 variables of the optimization loop. However, there are some differences compared 7to of the 23 approach followed at the TE region: First, more control points are needed close to the LE in order to maintain the rounded shape of the LE region. Therefore, the control point 4 of the lower side (CP4) must be located very close to the LE control point (CP5). Further- themore, sharp on TEthe shapeupper ofside the two airfoil extra section control with points reasonable ex1, ex2 detail. are added Less points(Figure result 2a). The in significantx,y-coordinates geometrical of the irregularities,ex1 additional demanding control point additional are derived constraints through that linear is not extrapola- obvious totion determine. to the x,y-coordinates More points increase of CP4 the and possible CP5. The combinations x,y-coordinates and the of computational the ex2 additional cost ofcontrol the optimization point are defined procedure. so that Two the ofcurvature the control of the points lower are side placed (points at theCP3,CP4,CP5) connections is betweensimilar to the that undeformed of the upper and side the (points morphing CP5,ex1,ex2), parts of themaintaining airfoil (one thus on the the roundness upper and of anotherthe LE. Since on the the lower y-coordinates side respectively) of the ex1, and ex2 remain points fixeddepend throughout on the y-coordinates the optimization of the process.control points The remaining 3, 4, 5, they points are arenot dividedconsidered between as independent the upper anddesign lower variables. side (three Second, on the an upperadditional side, control three on point, the lowerCP1, is side, needed and at one the at lower the TE side, point) close and to the they quarter are only of allowedchord, in toorder move to vertically. avoid abrupt As shown gradient in Figure changes2b, expectedbetween deflectionsthe morphing in theand TE the region undeformed will be relativelygeometries small that andmay therefore cause flow it isdisturbances. sufficient to An simply overall distribute number controlof 11 control points points at equal are distances along the x-axis (Figure2b). The y-coordinates of the seven movable control used for determining the Bezier curve of the LE region (Figure 2a). points constitute the design variables of the optimization loop for the TE morphing shape.

(a) (b)

FigureFigure 2. 2.Deﬁnition Definition of of the the x-coordinates xcoordinates ofof thethe controlcontrol pointspoints inin ((aa)) thethe leading leading edge edge (LE)region (LE)region and and ( b(b)) the the trailing trailing edge edge (TE)(TE) region. region. The The black black dashed dashed arrows arrows indicate indicate that that control control points points are are expected expected to beto shiftedbe shifted downwards downwards with with respect respect to the to the original geometry. Fixed control points are shown with green color. The additional control points ex1 and ex2, on the original geometry. Fixed control points are shown with green color. The additional control points ex1 and ex2, on the upper upper side of the LE, are depicted with squares. side of the LE, are depicted with squares.

A similar approach is followed for the LE region. Again, two control points remain fixed, connecting the undeformed with the morphing part, and seven control points are distributed on the two sides, three on the upper and four on the lower side, as shown in (Figure2a). Again, the y-coordinates of the seven control points constitute the design variables of the optimization loop. However, there are some differences compared to the approach followed at the TE region: First, more control points are needed close to the LE in order to maintain the rounded shape of the LE region. Therefore, the control point 4 of the lower side (CP4) must be located very close to the LE control point (CP5). Furthermore, on the upper side two extra control points ex1, ex2 are added (Figure2a). The x,y-coordinates of the ex1 additional control point are derived through linear extrapolation to the x,y-coordinates of CP4 and CP5. The x,y-coordinates of the ex2 additional control point are defined so that the curvature of the lower side (points CP3,CP4,CP5) is similar to that of the upper side (points CP5,ex1,ex2), maintaining thus the roundness of the LE. Since the y-coordinates of the ex1, ex2 points depend on the y-coordinates of the control points 3, 4, 5, they are not considered as independent design variables. Second, an additional control point, CP1, is needed at the lower side, close to the quarter of chord, in order to avoid abrupt gradient changes between the morphing and the undeformed geometries that may cause flow disturbances. An overall number of 11 control points are used for determining the Bezier curve of the LE region (Figure2a). Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 23

Appl. Sci. 2021, 11, 2822 8 of 23

The y-coordinates of the 14 variable control points (7 for the TE and 7 for the LE regions)The vary y-coordinates within specific of the ranges, 14 variable which control constitute points an (7input for theparameter TE and of 7 forthe thedesign LE regions)process. varyA proper within selection specific of ranges, the above which ranges constitute ensures an that input acceptable parameter LE and of the TE designshapes process.are generated A proper during selection the optimization of the above loop ranges and ensures that no that redundant acceptable computational LE and TE shapes cost is areadded generated to the duringcomputations. the optimization Proper selection loop and consists that no in redundant choosing computationalthe smallest possible cost is addedintervals to through the computations. which overlapping Proper selection of the allowable consists y-coordinate in choosing therange smallest of neighboring possible intervalscontrol points through is whichavoided. overlapping In this way of thethe allowablenumber of y-coordinate non-physical range solutions, of neighboring such as control“wavy” points geometries is avoided. is reduced. In thisway Even the though number non-physical of non-physical solutions solutions, may such be asfinally “wavy” ex- geometriescluded through is reduced. the geometrical Even though constraints, non-physical unnecessary solutions design may computations be finally are excluded added throughwhen large the geometricalintervals are constraints, considered. unnecessary An initial estimation design computations of the intervals are added level whenis ob- largetained intervals using a reference are considered. deformation An initial shape estimation of the original of the airfoil intervals geometry level for is obtainedgiven LE usingand TE a referencedeployment deformation angles: The shape mean of theairfoil original line is airfoil deformed geometry through for givena polynomial LE and TEre- deploymentlationship with angles: respect The to mean the distance airfoil line from is deformed the LE (or through TE) (see a polynomialAppendix A), relationship whilst the withinitial respect thickness to the at distance each x-position from the LEis (orretained. TE) (see In Appendix the presentA), whilstwork, thethe initial LE and thickness TE de- atployment each x-position angles are is retained.chosen as In 12° the and present 14°respectively. work, the LE and TE deployment angles are ◦ ◦ chosenIn asthe 12 TEand region, 14 respectively.the shape of the reference deformation is used to define the mean valueIn of the each TE region,control thepoint’s shape interval. of the reference For CPs deformation8,9,13, and 14 is used(numbering to define of thepoints mean is valueshown of in each Figure control 2b), point’sthe difference interval. between For CPs the 8, 9,y-coordinates 13, and 14 (numbering of the baseline of points (unde- is shownformed) in and Figure the2 b),deformed the difference airfoil between is used theas the y-coordinates interval width. of the For baseline the TE (undeformed) point 11, the andinterval the deformedwidth is kept airfoil small is used and ascorresponds the interval to width. a ±3% Fordeviation the TE with point respect 11, the to interval the TE widthdeployment is kept angle small andof 14°. corresponds For points to 10 a ±and3% 12 deviation, the interval with respect width tois theset TE1.5 deploymenttimes larger ◦ anglethan that of 14 of. the For TE points point 10 11. and The 12, selected the interval rang widthes for isthe set TE 1.5 control times largerpoints thanare depicted that of the in TEFigure point 3b. 11. In The the selectedLE region, ranges higher for overall the TE controldeformations points are depictedrequired in Figureorder 3tob. sub- In thestantially LE region, affect higher the overall overall loading deformations of the ai arerfoil. required On the in upper order toside substantially the shape of affect the theref- overallerence loadingdeformation of the is airfoil. used to On define the upper the lower side the bound shape of of each the referencecontrol point’s deformation interval, is usedwhilst to on define the thelower lower side bound the reference of each controldeformation point’s is interval, used to whilstdefine on the the upper lower bound side the of referenceeach control deformation point’s interval. is used toFor define CPs 1,2,3,6, the upper and bound 7 (numbering of each control of points point’s is shown interval. in For CPs 1, 2, 3, 6, and 7 (numbering of points is shown in Figure2a), the difference between Figure 2a), the difference between the y-coordinates of the baseline (undeformed) and the the y-coordinates of the baseline (undeformed) and the deformed airfoil is used as the deformed airfoil is used as the interval width. For LE points 4 and 5, the interval width interval width. For LE points 4 and 5, the interval width corresponds to a 25% deviation corresponds to a 25% deviation with respect to the LE deployment angle of 12°. The se- with respect to the LE deployment angle of 12◦. The selected ranges for the LE control lected ranges for the LE control points are depicted in Figure 3a. points are depicted in Figure3a.

(a) (b)

FigureFigure 3.3.Range Range ofof variationvariation ofof designdesign variablesvariables atat ((aa)) thethe LELE regionregion andand ((bb)) thethe TETE region region (indicated (indicated by by the the blue blue bar). bar). TheThe initialinitial NACANACA 64A010 64A010 is is depicted depicted in in red red and and the the airfoil airfoil geometry geometry after after the the reference reference deformation deformation is is depicted depicted in in green. green.

InIn thethe following,following, thethe threethree optimizationoptimization algorithmsalgorithms describeddescribed inin SectionSection2 2 are are tested tested forfor the case case of of combined combined LE LE and and TE TE morphing. morphing. InTable In Table 1, the1, theCL,max C valuesL,max values obtained obtained by the bythree the algorithms three algorithms are provided are provided as a function as a function of the number of the number of generations of generations of the optimi- of the optimization process. The optimization setup is the same for all three algorithms, meaning Appl. Sci. 2021, 11, 2822 9 of 23

that the same number of initial airfoils are created and the same number of offspring are produced at each generation. In the present work, the default number of offspring is used, defined as 8 times the number of the design variables (8 × 14 = 112). Therefore, the number of total fitness evaluations equals the product of the total generations multiplied by the number of offspring at each generation, plus the initial population. As an example, for the maximum number of generations (17), the number of fitness evaluations is 17 × 112 + 112 = 2016. For the current setup, PSO algorithm outperforms the other two algorithms as it predicts the solution with the maximum CL,max, while it requires the minimum number of generations (4). ES algorithm requires a significantly larger number of generations (17) to reach about the same CL,max level (2% lower). On the other hand, the optimum design predicted by the GA algorithm after 17 generations, has a maximum CL,max value 7% lower than that of the PSO algorithm. Comparing ES against GA, it is observed that the ES algorithm achieves a higher CL,max value at the expense of computational time, as it requires 17 generations in order to reach that solution. GA reaches a solution with a slightly higher CL,max (2.5% difference) at a lower number of generations (after only eight generations).

Table 1. Maximum lift coefﬁcients of the modiﬁed airfoils created during the parametric study of the number of generations.

CL,max Optimization Algorithm 2 Generations 4 Generations 8 Generations 17 Generations GA 2.02 2.07 2.12 2.12 PSO 2.2 2.27 2.27 - ES 2.0 2.01 2.07 2.22

The resulting optimum morphing shapes produced by the three algorithms are then compared in terms of the required airfoil shape deformation in order to reach their maximum CL,max. Shape deformation is measured in terms of the percentage of increase of the airfoil skin length at the morphing regions with respect to the corresponding lengths of the baseline undeformed airfoil. Table2 summarizes these percentages separately at the LE and TE regions as well as for the overall skin length. In general, two shape types of geometries are produced by the three algorithms, the ﬁrst type with the highest CL,max (>2.2) presenting larger deformations at the LE region (≈5.7%), and the second type with CL,max ≈ 2.1 presenting smaller deformations at the LE region (≈4.5%). Similar TE deformations are predicted for both types of morphed designs.The solution of GA after eight generations and the solution of ES after eight generations belong to the ﬁrst type. The solution of ES after 17 generations and the solution of PSO after 4 generations belong to the 2nd type.

Table 2. Deformations of the LE and TE regions and of the total geometry (length increase with respect to the original geometry) for the derived airfoils of ﬁrst and second type.

Deformations Airfoil Leading Edge Trailing Edge Overall PSO (4 generations, 1st type) 5.68% 3.54% 4.74% GA (8 generations, 2nd type) 4.49% 3.31% 3.98% ES (8 generations, 2nd type) 4.28% 3.48% 3.93%

Figure4 presents one morphed geometry per airfoil type; the PSO ﬁrst type after four generations and the ES second type after eight generations. In the subsequent analysis, the ES solution is preferred instead of the corresponding GA solution, since it is considered as more representative of the second type (smaller deformation at the LE region). Differences between the two shapes are highlighted in the focus plot of Figure5. The second type solution could qualify as preferred design, e.g., in the case of an upper limit on allowable Appl.Appl. Sci. Sci. 2021 2021, 11, 11, x, xFOR FOR PEER PEER REVIEW REVIEW 1010 of of 23 23

Appl. Sci. 2021, 11, 2822 10 of 23

TheThe second second type type solution solution could could qualify qualify as as pref preferrederred design, design, e.g., e.g., in in the the case case of of an an upper upper limitLElimit deformation, onon allowableallowable imposed LELE deformation,deformation, by constraints imposedimposed on material byby constraints propertiesconstraints(maximum onon materialmaterial stress propertiesproperties level) or (maximumactuator(maximum limitations. stress stress level) level) or or actuator actuator limitations. limitations.

FigureFigureFigure 4. 4.4. Final FinalFinal solutions solutionssolutions Particle ParticleParticle Swarm SwarmSwarm Optimizer OptimizerOptimizer (P (PSO) (PSO)SO) (four (four generations, generations, ﬁrstfirst first type) type) andand and ESES ES (eight (eightgenerations,(eight generations, generations, second second second type) type) along type) along withalong thewith with original the the original original airfoil. airfoil. airfoil.

(a(a) ) (b(b) ) FigureFigureFigure 5. 5.5. Comparison ComparisonComparison of of the the morphing morphing achieved achieved usingusing using thethe the PSOPSO PSO (four (four (four generations) generations) generations) and and and ES ES ES (eight (eight (eight generations) generations) generations) algorithms algorithms algorithms (a ) (LEa()a )LE regionLE region region and and and (b )( bTE(b) )TE region.TE region. region.

ItItIt is isis noted notednoted that thatthat in inin the thethe current currentcurrent setup, setup,setup, a aa rath ratherratherer “aggressive” “aggressive”“aggressive” design designdesign has hashas been beenbeen chosen chosenchosen in inin termstermsterms of ofof the thethe permitted permitted permitted deformations. deformations. AtAt At thethe the LELE LE region,region, region, the the the droop droop droop nose nose nose part part part corresponds corresponds corresponds to totheto the the 25% 25% 25% of of of the the the chord chord chord length, length, length, both both both on on on the the the upperupper upper andand and lowerlower lower sides,sides, sides, whilst whilst at at at the the the TE TE TE region regionregion thethethe deformable deformabledeformable flap ﬂapflap corresponds correspondscorresponds to toto the thethe 20% 20%20% of ofof the thethe chord chordchord length. length.length. In InIn some somesome of ofof the thethe previous previousprevious investigationsinvestigationsinvestigations [3–5], [[3–5],4–6], more moremore conservative conservativeconservative approaches approachesapproaches have havehave been beenbeen adopted adoptedadopted to toto avoid avoidavoid possible possiblepossible discontinuities,discontinuities,discontinuities, especially especiallyespecially on on thethe the upperupper upper side side side of of theof the the droop droop droop nose nose nose part. part. part. The The presentThe present present algorithm algo- algo- rithmcanrithm be can can easily be be easily adaptedeasily adapted adapted to such to to approachessuch such approaches approaches by keeping by by keeping keeping the control the the control control points points points 1 and 1 1 7and and ﬁxed 7 7 fixedatfixed the at atLE the the part LE LE part (Figurepart (Figure (Figure2a), 2a), as 2a), well as as well well asthe as as the pointsthe points points 8 and 8 8 and and 14 14 at14 at theat the the TE TE TE part part part (Figure (Figure (Figure2 b).2b). 2b). In In In thisthisthis way, way,way, the thethe deformable deformabledeformable parts partsparts of ofof the thethe LE LELE and andand TE TETE regions regionsregions are areare restricted restrictedrestricted to toto 20% 20%20% and andand 15%, 15%,15%, respectively.respectively.respectively. Moreover, Moreover,Moreover, anan an eveneven even smallersmaller smaller deformation defo deformationrmation is is permittedis permitted permitted for for for the the the upper upper upper side side side of theof of thedroopthe droop droop nose nose nose part. part. part. TheTheThe results resultsresults of ofof this thisthis more moremore conservative conservativeconservative approach approachapproach for forfor the thethe three threethree optimization optimizationoptimization algorithms algorithmsalgorithms areareare summarized summarizedsummarized in in Table Table3 .3. 3. It It canIt can can be be be observed observ observed that,ed that, that, keeping keeping keeping the the samethe same same number number number of generations, of of genera- genera- the PSO algorithm again provides the airfoil with the maximum C . Compared to the tions,tions, the the PSO PSO algorithm algorithm again again provides provides the the airfoil airfoil with with the the maximum maximumL,max C CL,maxL,max. .Compared Compared more “aggressive” approach, there is a 2.25% reduction in the C plus the fact that the toto the the more more “aggressive” “aggressive” approach, approach, there there is is a a 2.25% 2.25% reduction reduction inL,max in the the C CL,maxL,max plus plus the the fact fact double number of generations are needed to reach the optimum design. On the other hand, Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 23 Appl. Sci. 2021, 11, 2822 11 of 23

that the double number of generations are needed to reach the optimum design. On the smaller deformations are required to achieve that C , as shown in Table4. Figure6 other hand, smaller deformations are required to achieveL,max that CL,max, as shown in Table 4. Figurecompares 6 compares the two the different two different designs, designs, whilst Figure whilst7 a,bFigure focuses 7a,b onfocuses the deformed on the deformed parts at partsthe LE at andthe TELE regions,and TE respectively.regions, respective These plotsly. These conﬁrm plots the confirm smallest the deformations smallest defor- when mationsthe more when conservative the more setup conservative is used. It setup should is be used. noted It though,should thatbe noted in both though, setups, that proper in bothconstraints setups, ensuringproper constraints the continuity ensuring of the the geometry continuity have of beenthe geometry imposed, have as described been im- in Section2. The more conservative design could prove to be more efﬁcient in cases where posed, as described in Section 2. The more conservative design could prove to be more limitations of the permitted deformations are imposed by structural analysis. In the present efficient in cases where limitations of the permitted deformations are imposed by struc- work, such an analysis is not included, therefore the more aggressive design (design 1) tural analysis. In the present work, such an analysis is not included, therefore the more with the higher CL,max is adopted in the sequel. aggressive design (design 1) with the higher CL,max is adopted in the sequel.

Table 3. Maximum lift coefﬁcients of the airfoils generated using the three optimization algorithms Table 3. Maximum lift coefficients of the airfoils generated using the three optimization algorithms forfor the the more more conservati conservativeve design design approach approach (up (up to to 17 17 generations). generations). C CL,maxL,max OptimizationOptimization AlgorithmAlgorithm 22 Generations 4 4Generations Generations 8 8Generations Generations 17 17Generations Generations GA 1.981.98 2.0 2.05 2.05 2.05 2.05 PSO 2.19 2.16 2.22 2.22 PSOES 2.062.19 1.962.16 2.22 1.98 2.22 2.02 ES 2.06 1.96 1.98 2.02 Table 4. Deformations of the LE and TE regions and of the total geometry (length increase with respect Tableto the 4. original Deformations geometry) of the for LE the and optimum TE regions design and of of the the aggressive total geometry (design (length 1) and increase the conservative with respect(design to 2) the approachusing original geometry) the PSO for algorithm. the optimum design of the aggressive (design 1) and the conservative (design 2) approachusing the PSO algorithm. Deformations Airfoil Deformations Airfoil Leading Edge Trailing Edge Overall Leading Edge Trailing Edge Overall PSO,PSO, design 1 1 (4 (4 generations) generations) 5.68% 5.68% 3.54% 3.54% 4.74% 4.74% PSO, design 2 (8 generations) 4.70% 2.73% 3.84% PSO, design 2 (8 generations) 4.70% 2.73% 3.84%

FigureFigure 6. 6. FinalFinal solutions solutions of of PSO PSO algorithm algorithm using using the the aggressive aggressive (design (design 1, 1, four four generations) generations) and and the the conservativeconservative (design (design 2, 2, eight eight generations) generations) setup. setup.

AA closer closer look look at at the the connection connection points points between between the the morphing morphing and and the the undeformed undeformed partsparts of of the airfoilairfoil revealsreveals small small discontinuities discontinuities of of the the geometry, geometry, more more evident evident at the at lower the lowerside ofside the of LE the region LE region and theand upperthe upper side side of the of the TE region.TE region. These These discontinuities discontinuities can can be besmoothed smoothed out out by by imposing imposing additional additional fixedfixed controlcontrol pointspoints between those already already fixed fixed (green(green points points in in Figure Figure 2)2) and and the the points points with with numbers numbers 1, 1,7, 7,8, 8,and and 14 14(Figure (Figure 2).2 ).The The effecteffect of such of such a treatment a treatment is shown is shown in Figure in Figure 8. 8It. is It isobserved observed that that a smoother a smoother geometry geometry is achievedis achieved locally, locally, through through the theaddition addition of the of theintermediate intermediate control control points, points, which which keep keep the continuitythe continuity of the of first the derivative first derivative at the at connections. the connections. As expected, As expected, the effect the effectis more is morepro- pronounced at the lower side of the LE and the upper side of the TE, where the larger discontinuities were present. Appl.Appl. Sci. Sci.2021 2021, 11, x11 FOR, x FOR PEER PEER REVIEW REVIEW 12 of12 23of 23

nouncednounced at theat the lower lower side side of theof the LE LE and and the the upper upper side side of theof the TE, TE, where where the the larger larger dis- dis- continuitiescontinuities were were present. present. TheThe effect effect of ofthe the combined combined LE/TE LE/TE morphing morphing on onthe the CL -AoACL-AoA polars polars is depictedis depicted in in FigureFigure 9. The9. The airfoils airfoils designed designed from from the the PSO PSO and and ES ESalgorithms algorithms are are compared compared against against the the initialinitial NACA NACA 64A010 64A010 airfoil airfoil and and the the one one with with the the reference reference deformation, deformation, which which was was used used as aas starting a starting point point for for the the definition definition of theof the control control points’ points’ intervals intervals (Figure (Figure 3). 3).First, First, stall stall is is delayed,delayed, occurring occurring at aat higher a higher angle angle of attackof attack, in, comparisonin comparison to bothto both the the original original NACA NACA 64A01064A010 and and the the reference reference deformation deformation airfoils airfoils (17° (17° for for PSO, PSO, 14° 14° for for ES, ES, 12° 12° for for reference reference deformation,deformation, 10° 10° for for original original NACA NACA 64A010). 64A010). Second, Second, CL,max CL,max increases increases considerably, considerably, from from 1.11.1 (original (original NACA NACA 64A010) 64A010) to 2.27to 2.27 and and 2.07 2.07 for for the the PSO PSO and and the the ES ESairfoils, airfoils, respectively. respectively. Appl. Sci. 2021, 11, 2822 A significantA significant increase increase is alsois also observed observed with with respect respect to tothe the reference reference deformation deformation 12airfoil of airfoil 23 thatthat presents presents a C aL,max CL,max of 1.84.of 1.84.

(a) (a) (b)( b) FigureFigureFigure 7.7. Comparison Comparison7. Comparison ofof the theof the morphingmorphing morphing achievedachieved achieved withwith with thethe the aggressiveaggressi aggressive (designve(design (design 1,1, four four1, four generations)generati generations)ons) andand and thethe the conservativeconservative conservative (design(design(design 2,2, eighteight 2, eight generations)generations) generations) setupsetup setupusing using using the the PSOthe PSO PSO algorithm. algorithm. algorithm. ((aa)) LE (LEa) region LEregion region and and and ( b(b)) TE (TEb) region. TEregion. region.

(a) (a) (b)( b)

FigureFigureFigure 8.8. Improvement Improvement8. Improvement onon theonthe the continuitycontinuity continuity ofof thetheof the optimizedoptimized optimized PSOPSO PSO airfoilairfoil airfoil geometrygeometry geometry atat thetheat the connectionconnection connection pointspoints points betweenbetween between thethe the morphing and the undeformed parts. (a) Focus on the connection points of the LE region. (b) Focus on the connection morphingmorphing and and the the undeformed undeformed parts. parts. (a) Focus(a) Focus on theon the connection connection points points of theof the LE region.LE region. (b) ( Focusb) Focus on theon the connection connection pointspoints of theof the TE TEregion. region. points of the TE region.

The effect of the combined LE/TE morphing on the CL-AoA polars is depicted in Figure9. The airfoils designed from the PSO and ES algorithms are compared against the initial NACA 64A010 airfoil and the one with the reference deformation, which was used as a starting point for the deﬁnition of the control points’ intervals (Figure3). First, stall is delayed, occurring at a higher angle of attack, in comparison to both the original NACA 64A010 and the reference deformation airfoils (17◦ for PSO, 14◦ for ES, 12◦ for reference ◦ deformation, 10 for original NACA 64A010). Second, CL,max increases considerably, from 1.1 (original NACA 64A010) to 2.27 and 2.07 for the PSO and the ES airfoils, respectively. A signiﬁcant increase is also observed with respect to the reference deformation airfoil that presents a CL,max of 1.84. In order to distinguish the effect of LE and TE morphing on the overall lift change, additional optimizations are performed applying each one of the two morphings separately. Both PSO and ES algorithm are employed, while the maximum number of generations Appl. Sci. 2021, 11, 2822 13 of 23

is kept the same with the previous analyses (four generations for PSO and eight gener- Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 23 ations for ES), to derive results directly comparable with those of the combined LE/TE shape optimization.

FigureFigure 9. 9. EffectEffect of of the the combined combined LE LE droop droop nose/TE nose/TE flap ﬂap optimization optimization on onthe thelift liftcoefficient coefﬁcient polar. polar. ComparisonComparison between thethe geometriesgeometries derived derived from from the the PSO PSO and and Evolution Evolution Strategies Strategies (ES) (ES) optimization optimi- zationalgorithms, algorithms, the original the original geometry, geometry, and the and reference the reference deformation. deformation. Flow simulations Flow simulations were performed were performed using Foil2w. using Foil2w.

InFrom order the to comparison distinguish of the CL effectpolars of in LE Figures and TE 10 morphing and 11, it ison observed the overall that: lift change, additional optimizations are performed applying each one of the two morphings sepa- • When only LE droop nose is applied, stall is significantly delayed (20◦ for PSO-airfoil rately.and Both at 18PSO◦ for and ES-airfoil ES algorithm compared are empl to 10oyed,◦ for thewhile original the maximum airfoil), whilst number the of zero-lift gener- ationsangle is kept of attackthe same remains with the almost previous unchanged analyses (similar (four togenerations the effect offor a PSO multi-element and eight generationsslat). As for a result,ES), to a 60–80%derive results increase direct (dependingly comparable on the optimizationwith those of method the combined used) of LE/TE shape optimization. the CL,max is achieved. • FromWhen the only comparison TE flap is applied, of CL polars the effective in Figures10 camber and line 11, increase it is observed at the TEthat: region results • Whenin the shiftingonly LE of thedroop entire nose CL polaris applied, towards stall lower is AoA significantly (similar again delayed to the (20° effect for of PSO-airfoila multi element and at flap). 18° Infor this ES-airfoil case, stall compared occurs atto a10° lower for anglethe original of attack airfoil), (8◦ for whilst both theairfoils). zero-lift Furthermore, angle of attack the slowing remains down almost in theunchanged rate of the (similar lift increase to the starts effect earlier of a multi-elementin the morphed slat). airfoil As a as result, compared a 60–80% to the increase baseline (depending undeformed on shape.the optimization This is an methodindication used) of an of earlierthe CL,max separation is achieved. of the flow at the TE region, triggered by the local • Wheneffect ononly pressure TE flap by is the applied, high geometric the effect curvature.ive camber The line CL,max increaseincrease at the in thisTE region case is resultslimited in to the 45%. shifting of the entire CL polar towards lower AoA (similar again to the • effectWhen of both a multi morphing element mechanisms flap). In this are case, applied, stall occurs the two at a effects lower areangle superimposed of attack (8° for(though both airfoils). not linearly) Furthermore, and the overall the slowing increase do inwn C inL,max theis rate maximized of the lift (88–107%). increase starts • earlierMeasurements in the morphed of the lift airfoil coefficient as compared polar for to anthe extended baseline andundeformed drooped slatshape. configura- This is antion indication on the same of an NACA earlier 64A010 separation are also of the presented flow at inthe Figures TE region, 10 and triggered 11. The by Mach the ◦ localnumber effect and on the pressure flap deployment by the high of thegeometric experiment curvature. [1] are 0.25The andCL,max 14.5 increase, respectively, in this casevery is close limited to the to simulated45%. setup. The comparison shows a significant improvement • Whenof the CbothL,max morphingwhen the mechanisms suggested droop are appl noseied, morphing the two is effects applied, are even superimposed if the lower × 6 (thoughReynolds not number linearly) of theand experiment the overall isincrease considered in CL,max (3.5 is maximized10 ). (88–107%). • MeasurementsPercentages of CofL,max the liftincrease coefficient for the polar individual for an extended LE droop and nose drooped and TE flap,slat config- as well as foruration the combined on the morphingsame NACA are summarized64A010 are also in Table presented5. The C inL,max Figuresincrease 10 and with 11. respect The to theMach reference number deformation and the flap airfoil deployment (combined of morphing) the experiment is 23.5% [1] andare 0.25 12.3% and for 14.5°, the PSO re- and ESspectively, airfoils, respectively.very close to the simulated setup. The comparison shows a significant improvement of the CL,max when the suggested droop nose morphing is applied, even if the lower Reynolds number of the experiment is considered (3.5 × 106). Appl.Appl. Sci. Sci. 20212021,,, 1111,,, 2822xx FORFOR PEERPEER REVIEWREVIEW 14 of 2314 of 23

FigureFigure 10. 10. EffectsEffects of ofthe the individual individual LE LEdroop droop nose nose and and TE flap TE ﬂapmorphing morphing and their and theircombination combination on on thethe lift lift coefficient coefﬁcient polar. polar. PSO PSO algorithm algorithm with with four four generationsgenerations generations (560(560 evaluations)evaluations) (560 evaluations) waswas employedemployed was employed forfor for the individual as well as the combined morphing optimization. Flow simulations were performed thethe individual individual as as well well as asthe the combined combined morphing morphing optimization. optimization. Flow Flowsimulations simulations were performed were performed using Foil2w. Measurements for extended and drooped slat are reported in [28]. using Foil2w. Measurements for extended and drooped slat are reported in [2].

FigureFigure 11. 11. EffectsEffects of ofthe the individual individual LE LEdroop droop nose nose and and TE flap TE ﬂapmorphing morphing and their and theircombination combination on on the lift coefficient polar. ES algorithm with eight generations (1008 evaluations) was employed for thethe lift lift coefficient coefﬁcient polar. polar. ESES algorithm algorithm with with eigh eightt generations generations (1008 (1008 evaluations) evaluations) was employed was employed for for the individual as well as the combined morphing optimization.optimization. FlowFlow simulationssimulations werewere performedperformed the individual as well as the combined morphing optimization. Flow simulations were performed using Foil2w. Measurements for extended and drooped slat are reported in [28]. using Foil2w. Measurements for extended and drooped slat are reported in [2].

Percentages of CL,max increase for the individual LE droop nose and TE flap, as well Table 5. Relative increase inas CL,max for the applyingcombined optimized morphing LE are droop summarized noose, TE inTable flap, and 5. combined The CL,max morphing increase with therespect PSO and ES algorithms. Simulationto the of reference the flow field deformation was performed airfoil by (combine Foil2w. d morphing) is 23.5% and 12.3% for the PSO and ES airfoils, respectively. Optimization CL,max % Difference with NACA 64A010

Algorithm LE Droop Nose TE Flap Combined LE Droop Nose TE Flap Combined ES 1.76 1.60 2.07 60.22 45.44 88.31 PSO 1.99 1.60 2.27 80.96 46.02 106.96 Appl. Sci. 2021, 11, 2822 15 of 23

A closer look at the pressure distribution (presented in Figure 12) around the morphed shapes and the undeformed baseline NACA 64A010 can provide information for the physical interpretation of the attained CL,max increase. Pressure distributions are plotted for AoAs corresponding to the maximum value of the lift coefficient. This is at 17.5◦ and 14◦ for the morphed shapes predicted by PSO and ES algorithms, respectively, and 10◦ for the baseline NACA 64A010 (see Figure9). Apparently, the larger extent of the high under-pressure pocket on the suction side in the vicinity of the LE region drives the increase in the maximum CL. The PSO airfoil exhibits a suction pressure peak of about 25% higher than that of the ES airfoil. This explains the higher CL,max value obtained by this airfoil. Although the suction peak of the baseline airfoil is larger than those of both morphed shapes, the pressure of the baseline airfoil quickly recovers within a very short distance (less than 5%) from the LE point. The under-pressure of the morphed shapes remains high up to 10% of the chord distance from the LE. It is noted that for both morphed shapes, at the CLmax AoA, the flow remains almost attached over the whole suction side. A limited separation is obtained for both morphed airfoils at about 80% of the chord (seen as a plateau of the pressure coefficient beyond this chordwise position). Another interesting observation is related to the pressure variation at the connection points between the deformed and undeformed airfoil parts, where it is desirable to avoid steep flow disturbances. The plots of the pressure coefficient in Figure 12 do not present significant irregularities, confirming that the imposed curvature and slope constraints, integrated into the optimization procedure, prevent the occurrence of significant geometric discontinuities. Moreover, in the PSO airfoil, the additional fixed control points between the morphed and the undeformed part improve the continuity of the geometry at the connections (see Figure8), and this effect is also reflected on the CP distribution (at least at the two connection points of the suction side and the TE connection point of the pressure side). It is noted that compared to the ES, the PSO airfoil presents a smoother pressure distribution in the vicinity of the LE connection Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 23 point on both sides (upper and lower), despite the larger deformation undergone by the PSO morphed airfoil at the LE region.

FigureFigure 12. 12. PressurePressure coefficient coefficient distribution for the airfoilsairfoils producedproduced byby thethe PSOPSO andand ESESoptimization optimiza- tionalgorithms algorithms at the at maximumthe maximum CL angles CL angles of attack. of attack. Flow Flow field field was was simulated simulated using using Foil2w. Foil2w. 4. Validation with a High Fidelity Solver 4. Validation with a High Fidelity Solver To verify the results obtained by Foil2w, the flow around the modified geometries is To verify the results obtained by Foil2w, the flow around the modified geometries is also simulated using MaPFlow, a CFD URANS solver. In both codes, transition modeling also simulated using MaPFlow, a CFD URANS solver. In both codes, transition modeling is realized using the same model eN. For the CFD simulations, the computational grid is is realized using the same model eN. For the CFD simulations, the computational grid is generated using the ANSYS ICEM software [29]. Special care is taken to generate a grid sufficiently dense within the boundary layer region and in the vicinity of the LE and TE regions. The distance of the first node from the surface is 10−6 chords (y+≈1).In Figure 13, the mesh for the PSO-airfoil is depicted, whilst a similar mesh was created for the ES-airfoil. It consists of 606 wrap around nodes and 180 nodes in the normal to the surface direction. Out of the 180 nodes in the normal direction, 120 are distributed from the surface up to a distance of 0.13 chords using a geometrical progress with ratio 1.08.

Figure 13. Computational grid (C type) for PSO airfoil, constructed using ICEM, ANSYS (2020). Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 23

Figure 12. Pressure coefficient distribution for the airfoils produced by the PSO and ES optimization algorithms at the maximum CL angles of attack. Flow field was simulated using Foil2w.

4. Validation with a High Fidelity Solver Appl. Sci. 2021, 11, 2822 16 of 23 To verify the results obtained by Foil2w, the flow around the modified geometries is also simulated using MaPFlow, a CFD URANS solver. In both codes, transition modeling is realized using the same model eN. For the CFD simulations, the computational grid is generatedgenerated using using the the ANSYS ANSYS ICEM ICEM software software [29]. [29]. Special Special care care is is taken taken to to generate generate a a grid grid sufficientlysufﬁciently dense dense within within the the boundary boundary layer layer region region and and in in the the vicinity vicinity of of the the LE LE and and TE TE regions.regions. The The distance distance of thethe ﬁrstfirst nodenode from from the the surface surface is is 10 10−6−6chords chords (y (y+ +≈≈1).In1). In Figure Figure 13, 13 , thethe mesh forfor the the PSO-airfoil PSO-airfoil is depicted,is depicted, whilst whilst a similar a similar mesh wasmesh created was forcreated the ES-airfoil. for the ES-airfoil.It consists It of consists 606 wrap of around606 wrap nodes around and nodes 180 nodes and in180 the nodes normal in tothe the normal surface to direction.the sur- faceOut direction. of the 180 Out nodes of the in the 180 normal nodes direction,in the normal 120 aredirection, distributed 120 are from distributed the surface from up the to a surfacedistance up of to 0.13 a distance chords usingof 0.13 a chords geometrical using progressa geometrical with ratioprogress 1.08. with ratio 1.08.

FigureFigure 13. 13. ComputationalComputational grid grid (C (C type) type) for for PSO PSO airf airfoil,oil, constructed constructed using using ICEM, ICEM, ANSYS ANSYS (2020). (2020).

Figure 14 compares the MaPFlow and Foil2w predictions of the CL-AoA polars for the optimized morphed airfoils. The MaPFlow and Foil2w polars for the original airfoil are also presented in the plot. Higher CL,max values are systematically predicted by the CFD solver for the morphed airfoils, although the opposite applies for the baseline NACA 64A010. In general, compared to the original airfoil, the differences between the two codes increase, as the airfoil geometry is deformed. This is due to the fact that, as the geometry becomes more complex, and taking into account that transitional effects are included, the results become more sensitive to the simulation of viscous flow phenomena, such as the prediction of the transition locations and the appearance of separation. The maximum lift coefficient increases from 1.1 to 2.5 and 2.3 for the PSO and ES airfoils, respectively. This is due to the fact that in the CFD simulations stall takes place at higher angles of attack (19◦ for both PSO and ES). It is noticeable that in MaPFlow predictions the ES airfoil exhibits ◦ higher CL values than the PSO airfoil at AoAs in the range 5–15 , below the CL,max AoA. In aeronautical applications, this region is outside the operation points of interest and it is probable that morphing is applied differently at the lower AoAs. In the case of wind energy applications, though, the operational region is broader indicating that a larger part of the polar should be taken into account. In Table6, percentages of the maximum lift coefficient increase for the designed airfoils, as predicted using Foil2w and MaPFlow, are summarized and compared. It is underlined that CFD predictions indicate that PSO-airfoil reaches a 140% increase in CL,max, while ES-airfoil reaches a 125% increase. Therefore, MaPFlow exhibits an even higher increase in CL,max (more than 100%), suggesting that the results of the optimization using the lower fidelity Foil2w solver are conservative at least for the particular airfoil studied in the present work. It is worth noting that contrary to the morphed airfoils, the CLmax predicted by the CFD code for the baseline airfoil is slightly lower than the one provided by Foil2w. Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 23

Figure 14 compares the MaPFlow and Foil2w predictions of the CL-AoA polars for the optimized morphed airfoils. The MaPFlow and Foil2w polars for the original airfoil are also presented in the plot. Higher CL,max values are systematically predicted by the CFD solver for the morphed airfoils, although the opposite applies for the baseline NACA 64A010. In general, compared to the original airfoil, the differences between the two codes increase, as the airfoil geometry is deformed. This is due to the fact that, as the geometry becomes more complex, and taking into account that transitional effects are included, the results become more sensitive to the simulation of viscous flow phenomena, such as the prediction of the transition locations and the appearance of separation. The maximum lift coefficient increases from 1.1 to 2.5 and 2.3 for the PSO and ES airfoils, respectively. This is due to the fact that in the CFD simulations stall takes place at higher angles of attack (19° for both PSO and ES). It is noticeable that in MaPFlow predictions the ES airfoil exhibits higher CL values than the PSO airfoil at AoAs in the range 5–15°, below the CL,maxAoA. In aeronautical applications, this region is outside the operation points of interest and it is probable that morphing is applied differently at the lower AoAs. In the case of wind energy applications, though, the operational region is broader indicating that a larger part of the polar should be taken into account. In Table 6, percentages of the maximum lift coefficient increase for the designed airfoils, as predicted using Foil2w and MaPFlow, are summarized and compared. It is underlined that CFD predictions indicate that PSO-airfoil reaches a 140% increase in CL,max, while ES-airfoil reaches a 125% increase. Therefore, MaPFlow exhibits an even higher increase in CL,max (more than 100%), suggesting that the results of the optimization using the lower fidelity Foil2w solver are conservative at least for the particular airfoil studied in the present work. It is worth noting that contrary to the morphed airfoils, the Appl. Sci. 2021, 11, 2822 17 of 23 CLmax predicted by the CFD code for the baseline airfoil is slightly lower than the one provided by Foil2w.

FigureFigure 14. 14. CCLLpolarspolars produced produced by by Foil2w Foil2w (dotted (dotted lines) lines) and and MaPFlow MaPFlow (continuous (continuous lines) lines) for for the the PSO PSO and ES geometries, along with the CLpolars of the original airfoil. The marked regions denote the and ES geometries, along with the CL polars of the original airfoil. The marked regions denote the appearance of CL,maxand the operation points of interest. appearance of CL,max and the operation points of interest.

Table 6. Relative increase in CL,max for the PSO and ES airfoils, using Foil2w and MaPFlow.

C % Difference with NACA 64A010 Solver L,max NACA 64A010 PSO ES PSO ES Foil2w 1.098 2.271 2.067 106.96 88.31 MaPFlow 1.046 2.502 2.360 139.28 125.72

Velocity contour plots around the PSO airfoil are depicted in Figure 15, at the AoA where the maximum value of lift coefficient occurs (19◦). The flow patterns of Figure 15 confirm the discussion made in the previous section. The concurrent drooping and thickening of the airfoil nose over its forward 10% leads to a smoother acceleration of the boundary layer (lower pressure gradient), which is retained over a larger portion of the suction side forming a high velocity pocket (flow velocities 2.5 times higher than the free stream velocity) up to 10–20% of the chord. Furthermore, the overall flattening of the whole suction surface, driven by the thickening of the forward 7–20% of the LE region leads to retarded separation of the flow, which, as also predicted by Foil2w code, at this high AoA (19◦) is limited to the aft 20% of the airfoil section (low velocity pocket in the vicinity of the morphed TE part). Flow velocities depicted in the contour plot of Figure 15 exhibit a smooth variation at the connection parts, confirming again that the geometric constraints imposed in the optimization loop prevent geometric irregularities. In Figure 16a, the pressure coefficient distributions predicted by Foil2w and MaPFlow ◦ ◦ code at the CLmax AoA of the PSO airfoil are compared (19 for MaPFlow and 17.5 for Foil2w). Comparing the two distributions, two differences are distinct: (a) MaPFlow predicts slightly higher suction peak resulting in a higher CL,max and (b) Foil2w predicts a slightly more extended flow separation at the TE region (pressure plateau at the rear part of the airfoil) even at the lower angle of attack of 17.5◦. In Figure 16b, the pressure distribution predicted by MaPFlow is presented for the two optimized airfoils at the CL,max ◦ AoA (angle of CL,max 19 for both the PSO and the ES airfoils). Results of Figure 16b (predicted by MaPFlow code) are qualitatively very similar to those of Figure 12 (predicted by Foil2w). The main difference is that Foil2w predicts much lower suction peak for the ES airfoil, which justifies the lower CL,max predicted by the same code for the particular airfoil. Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 23

Table 6. Relative increase in CL,max for the PSO and ES airfoils, using Foil2w and MaPFlow.

CL,max % Difference with NACA 64A010 Solver NACA 64A010 PSO ES PSO ES Foil2w 1.098 2.271 2.067 106.96 88.31 MaPFlow 1.046 2.502 2.360 139.28 125.72

Velocity contour plots around the PSO airfoil are depicted in Figure 15, at the AoA where the maximum value of lift coefficient occurs (19°). The flow patterns of Figure 15 confirm the discussion made in the previous section. The concurrent drooping and thickening of the airfoil nose over its forward 10% leads to a smoother acceleration of the boundary layer (lower pressure gradient), which is retained over a larger portion of the suction side forming a high velocity pocket (flow velocities 2.5 times higher than the free stream velocity) up to 10–20% of the chord. Furthermore, the overall flattening of the whole suction surface, driven by the thickening of the forward 7–20% of the LE region Appl. Sci. 2021, 11, 2822 leads to retarded separation of the flow, which, as also predicted by Foil2w code, at18 ofthis 23 high AoA (19°) is limited to the aft 20% of the airfoil section (low velocity pocket in the vicinity of the morphed TE part). Flow velocities depicted in the contour plot of Figure 15 exhibit a smooth variation at the connection parts, confirming again that the geometric constraintsMaPFlow predictions imposed in conﬁrm the optimization that disturbance loop prevent of the geometric pressure at irregularities. the connection points is lower for the PSO airfoil.

(a) (b)

Appl. Sci. 2021, 11, x FOR PEER REVIEW ◦ 19 of 23 Figure 15. ((aa)) Velocity Velocity ﬂowflow ﬁeldfield ofof thethe PSO-airfoilPSO-airfoil atat AoAAoA == 1919° which corresponds toto thethe CCL,maxL,max. .Velocities Velocities are are normalized normalized with the free stream velocity of 75 m/s. ( (bb)) Focus Focus on on the the LE LE region. region.

In Figure 16a, the pressure coefficient distributions predicted by Foil2w and MaP- Flow code at the CLmax AoA of the PSO airfoil are compared (19° for MaPFlow and 17.5° for Foil2w). Comparing the two distributions, two differences are distinct: (a) MaPFlow predicts slightly higher suction peak resulting in a higher CL,max and (b) Foil2wpredicts a slightly more extended flow separation at the TE region (pressure plateau at the rear part of the airfoil) even at the lower angle of attack of 17.5°. In Figure 16b, the pressure distribution predicted by MaPFlow is presented for the two optimized airfoils at the CL,max AoA (angle of CL,max 19° for both the PSO and the ES airfoils). Results of Figure 16b (predicted by MaPFlow code) are qualitatively very similar to those of Figure 12 (predicted by Foil2w). The main difference is that Foil2w predicts much lower suction peak for the ES airfoil, which justifies the lower CL,max predicted by the same code for the particular airfoil. MaPFlow predictions confirm that disturbance of the pressure at the connection points is lower for the PSO airfoil.

(a) (b)

FigureFigure 16. (16.a) Comparison(a) Comparison between between Cp distributions Cp distributions along along PSO-airfoil, PSO-airfoil, simulated simulated by Foi2w by Foi2w and MaPFlow, and MaPFlow, at the maximum at the maxi- CL mumangles C ofL angles attack;( ofb )Cattack;(p distributionb) Cp distribution along the along PSO andthe PSO ES airfoils, and ES simulatedairfoils, simulated by MaPFlow, by MaPFlow, at the maximum at the maximum CL angle CL of attack.angle of attack.

TheThe whole whole design design process process has has been been based based on theon the CL,max CL,maxoptimization, optimization, which which iscritical is critical duringduring the the take-off take-off and and landing landing segments segments of the of flight.the flight. However, However, another another important important factor, fac- especiallytor, especially during theduring take-off the procedure,take-off proc isedure, the aerodynamic is the aerodynamic performance performance (CL/CD), which (CL/CD), is relatedwhich directlyis related to the directly fuel consumption to the fuel of consumption the aircraft. Most of the commercial aircraft. transport Most commercial aircraft aretransport optimized aircraft to fly withare optimized a small incidence to fly with at an a small altitude incidence close to at 11,000 an altitude m. While close climbing to 11,000 to orm. descendingWhile climbing from to this or descending optimal altitude from thethis aircraftoptimal will altitude travel the with aircraft a large will incidence, travel with therefore,a large willincidence, operate therefore, inefficiently will andoperate as a resultinefficiently will consume and as a a result lot of fuel.will consume It is expected a lot of thatfuel. the designedIt is expected airfoils that will the produce designed increased airfoils drag will becauseproduce of increased the highly drag deformed because shape of the at thehighly droop deformed nose LE. shape This is at confirmed the droop by nose the MaPFlowLE. This is predictions confirmed(Table by the7) MaPFlowwhich suggest predic- largertions C (TableD values 7) which for the suggest morphed larger airfoils CD values at the for CL,max the morphedAoA. However, airfoils theat the substantial CL,max AoA. However, the substantial lift increase mentioned in the above analysis compensates for the drag increase, resulting in improved aerodynamic performance as reported in Table 7. This confirms the overall aerodynamic effectiveness of the designed airfoils.

Table 7. Comparison of aerodynamic characteristics at the CL,max angle of attack. Lift, drag, angle of attack, and aerodynamic performance are presented for the initial NACA 64A010 and the morphed PSO and ES airfoils.

Airfoil CL,max AoA CD CL/CD NACA 64A010 1.05 11° 0.030 34.33 ES 2.36 19° 0.055 42.81 PSO 2.50 19° 0.043 58.72

In all aforementioned simulations, transitional conditions are considered. Potential increase of the surface roughness would force transition of the flow from laminar to turbulent to occur earlier, which generally leads to reduced maximum lift. In order to assess performance deterioration of the morphed sections due to earlier (forced) transition, fully turbulent flow simulations are performed with MaPFlow code. As seen in the marked regions of Figure 17 changing flow conditions from transitional to fully turbulent does not significantly affect the CL,max value of the baseline NACA 64A010 airfoil, but causes an appreciable reduction in the CL,max of both PSO and ES optimized morphed airfoils. Moreover, the CL,max AoA decreases by 1° for both airfoils. Overall, as opposed to the transitional case, in turbulent flow conditions the ES airfoil outperforms the PSO airfoil over the whole range of the simulated AoAs. As noted above, this effect could be Appl. Sci. 2021, 11, 2822 19 of 23

lift increase mentioned in the above analysis compensates for the drag increase, resulting in improved aerodynamic performance as reported in Table7. This conﬁrms the overall aerodynamic effectiveness of the designed airfoils.

Airfoil CL,max AoA CD CL/CD NACA 64A010 1.05 11◦ 0.030 34.33 ES 2.36 19◦ 0.055 42.81 PSO 2.50 19◦ 0.043 58.72

In all aforementioned simulations, transitional conditions are considered. Potential increase of the surface roughness would force transition of the flow from laminar to turbulent to occur earlier, which generally leads to reduced maximum lift. In order to assess performance deterioration of the morphed sections due to earlier (forced) transition, fully turbulent flow simulations are performed with MaPFlow code. As seen in the marked regions of Figure 17 changing flow conditions from transitional to fully turbulent does not significantly affect the CL,max value of the baseline NACA 64A010 airfoil, but causes an appreciable reduction in the CL,max of both PSO and ES optimized morphed airfoils. Appl. Sci. 2021, 11, x FOR PEER REVIEW ◦ 20 of 23 Moreover, the CL,max AoA decreases by 1 for both airfoils. Overall, as opposed to the transitional case, in turbulent flow conditions the ES airfoil outperforms the PSO airfoil over the whole range of the simulated AoAs. As noted above, this effect could be important importantto wind energy to wind applications, energy applications, in which the in blade which airfoil the encountersblade airfoil a broadencounters range ofa broad AoAs. rangeTable 8of summarizes AoAs. Table the 8 CsummarizesL,max predictions the CL,max for predictions transitional for and transitional turbulent conditionsand turbulent and conditionsthe relative and increase the relative with respect increase to the with baseline respect airfoil. to the Changing baseline fromairfoil. transitional Changing to from fully transitionalturbulent conditions to fully mitigatesturbulent theconditions relative increasemitigates of the 139% relative and 126%, increase for the of PSO 139% and and ES 126%,airfoils, for respectively, the PSO and to ES the airfoils, same level respectively, of 118% forto the both same airfoils. level of 118% for both airfoils.

FigureFigure 17. 17. ComparisonComparison between between fully fully turbul turbulentent and and transitional flow ﬂow C L polars,polars, for for the the PSO PSO and and ES ES airfoils,airfoils, as as well well as as for for the the NACA NACA 64A010. 64A010. The The marked marked regions regions denote denote the appearance of C L,max andand thethe operation operation points points of of interest. interest. Flow Flow simulations simulations performed performed using using MaPFlow. MaPFlow.

Table 8. Relative increase of CL,max of the PSO and ES airfoils, for fully turbulent and transitional flow, simulations performed by MaPFlow.

CL,max % Difference with NACA Flow Condition NACA PSO ES PSO ES Fully turbulent 1.044 2.276 2.277 118.01 118.06 Transitional 1.046 2.502 2.360 139.28 125.72

5. Discussion In the present study, a methodology for designing high-lift mechanisms for aircraft wings is developed, applying a combination of LE droop nose and TE flap morphing. Airfoils of increased maximum lift are produced through an optimization process, where smooth polynomial curves approximate the morphing parts in the LE and TE regions of the airfoil, while a viscous-inviscid fast interaction code is used for the prediction of the aerodynamic characteristics. Results indicate that significant increase in maximum lift coefficient and stall angles of attack are achievable. The contribution of each morphing mechanism (LE droop nose, TE flap) to the total CL,max increase is investigated. When only LE droop nose is applied stall is delayed and, as a result, lift coefficient increases allowing the airfoil to operate at higher angles of attack. On the other hand, when only TE flap is applied the effective camber line increase at the TE region results in the shifting of the entire CLpolar towards lower AoA. The entire level of the CL curve increases, but stall occurs at lower AoA. The combined effect is a delayed stall along with a shift of the CL curve to a higher level. The above conclusions are verified using a high fidelity CFD solver. CFD predictions exhibit an even higher CL,max increase due to delayed stall and higher pressure peaks at the suction side of the LE, suggesting that the optimization results derived with the fast Appl. Sci. 2021, 11, 2822 20 of 23

Table 8. Relative increase of CL,max of the PSO and ES airfoils, for fully turbulent and transitional ﬂow, simulations performed by MaPFlow.

CL,max % Difference with NACA Flow Condition NACA PSO ES PSO ES Fully turbulent 1.044 2.276 2.277 118.01 118.06 Transitional 1.046 2.502 2.360 139.28 125.72

5. Discussion In the present study, a methodology for designing high-lift mechanisms for aircraft wings is developed, applying a combination of LE droop nose and TE flap morphing. Airfoils of increased maximum lift are produced through an optimization process, where smooth polynomial curves approximate the morphing parts in the LE and TE regions of the airfoil, while a viscous-inviscid fast interaction code is used for the prediction of the aerodynamic characteristics. Results indicate that significant increase in maximum lift coefficient and stall angles of attack are achievable. The contribution of each morphing mechanism (LE droop nose, TE flap) to the total CL,max increase is investigated. When only LE droop nose is applied stall is delayed and, as a result, lift coefficient increases allowing the airfoil to operate at higher angles of attack. On the other hand, when only TE flap is applied the effective camber line increase at the TE region results in the shifting of the entire CL polar towards lower AoA. The entire level of the CL curve increases, but stall occurs at lower AoA. The combined effect is a delayed stall along with a shift of the CL curve to a higher level. The above conclusions are verified using a high fidelity CFD solver. CFD predictions exhibit an even higher CL,max increase due to delayed stall and higher pressure peaks at the suction side of the LE, suggesting that the optimization results derived with the fast viscous- inviscid interaction code are conservative at least for the current airfoil study. The imposed curvature and inclination constraints prevent the appearance of geometric irregularities and flow disturbances at the connections between the undeformed and the morphing parts. This is reflected on the predicted smooth pressure distributions and the velocity contours. The present work focuses on the achievable increase of the maximum lift, which is desirable during take-off and landing flight procedures; aerodynamic performance is not considered in the optimization process. However, the predictions of the CFD solver demonstrate that although morphing airfoils produce an increased drag, they are still capable of operating with much higher aerodynamic performance (CL/CD). This interesting outcome ensures the overall aerodynamic effectiveness of the designed airfoils and indicates the potential of the developed optimization methodology to be applied to other contiguous technologies, such as wind energy, where maximization of CL/CD is likely to be the most relevant design objective for maximum energy yield. Finally, fully turbulent flow simulations are performed to investigate the effect of a potential increase of the surface roughness. As expected, fully turbulent calculations predict lower values of the maximum lift. It is observed that in fully turbulent conditions, PSO and ES optimized airfoils produce a similar CL,max, whilst in transitional conditions, the PSO airfoil produces a considerably higher CL,max. Therefore, changing from transitional to turbulent conditions mitigates maximum lift increase to the same level for both airfoils.

Author Contributions: Conceptualization, V.R. and J.P.; methodology, C.T., N.-G.M., J.P., V.R., G.S. and G.P.; software, G.S. and G.P.; validation, C.T. and N.-G.M.; formal analysis, C.T. and N.-G.M.; writing—original draft preparation, C.T. and N.-G.M.; writing—review and editing, J.P. and V.R.; visualization, C.T., N.-G.M. and J.P.; supervision, J.P. and V.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Appl. Sci. 2021, 11, 2822 21 of 23

Informed Consent Statement: Not applicable. Data Availability Statement: The experimental data presented in Figures 10 and 11 are available in Reference [2] “Axelson, J.A.; Stevens, G.L. Investigation of a Slat in Several Different Positions on a NACA 64A010 Airfoil for a Wide Range of Subsonic Mach Numbers. Technical Note 3129; Ames Aeronautical Laboratory: Moffett Field, CA, USA, March 1954.” Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A The baseline deformation mentioned in Section3 is deﬁned considering that the deformed mean line can be represented by a polynomial curve. Deformation is considered only in the vertical direction (x-coordinates of the airfoil and thickness distribution do not change). The polynomial curve of m degree has the general form:

η(ξ) = αξm(ξ + β) (A1)

where ξ denotes the non-dimensional distance in the x-direction, η denotes the deflection in the y-direction and α, β are constants to be determined. Polynomial curve should be continuous at ξ = 0 (beginning of deformation) and smooth at ξ = 1 (end of deformation), therefore the following conditions must be fulfilled: (a) At ξ = 0, the value and the first derivative should be zero (applies). (b) At ξ = 0, the value and the first derivative should be zero (applies).

The above conditions suggest that β = −3 and that α = ηTE or α = ηLE, where ηTE and ηLE are the (maximum) deflections at TE and LE respectively. Assuming polynomial of second degree, m = 2, the relationships providing the dimensionless ξ coordinate along with the maximum deflection and the deflection at each x-position are summarized below:

- In the TE region, x > x0, x0 = 0.8:

ξ = (x − x0)/(xTE − x0) ηTE = (xTE − x0) tan δ (A2) 2 η = ηTE × ξ (3 − ξ)/2

- In the LE region, x < x0, x0 = 0.25:

ξ = (x0 − x)/(x0 − xLE) ηLE = x0 tan δ (A3) 2 η = ηLE × ξ (3 − ξ)/2

where δ is the deployment angle which defines the maximum deflection. The deformed mean line and the relative notations are depicted in Figure A1. Finally, the new y-coordinates, ydef of the deformed geometry are obtained by adding the deflection, η, to the y-coordinates of the original geometry, yorg:

ydef = yorg + η (A4) Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 23

along with the maximum deflection and the deflection at each x-position are summarized below:

- In the TE region, x > x0, x0 = 0.8:

ξ = x - x0 / x TE - x 0 

ηTE = x TE - x 0  tanδ (A2) 2 η = ηTE ×ξ 3 -ξ / 2

- In the LE region, x < 푥0, x0 = 0.25:

ξ = x0 - x / x 0 - x LE 

ηLE = x 0 tanδ (A3) 2 η = ηLE ×ξ 3 -ξ / 2

where δ is the deployment angle which defines the maximum deflection. The deformed mean line and the relative notations are depicted in Figure A1. Finally, the new y-coordinates, ydef of the deformed geometry are obtained by adding the deflection, η, Appl. Sci. 2021, 11, 2822 to the y-coordinates of the original geometry, yorg: 22 of 23

ydef = yorg + η (A4)

(a) (b)

Figure A1. DefinitionDeﬁnition of the baseline deformation at the ( a) TE region and ( b) LE region. The thick red line represents the deformed mean line line..

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