Aerodynamics of non-slender delta and reverse delta wings:Wing thickness,anhedral angle and cropping ratio

Go¨ktug?KOC? AK, Mehmet Metin YAVUZ

aTurkish Aerospace Industries, Inc.Aerodynamics Chief Engineer at TF-XProject, Ankara 06980, Turkey

bMechanical Engineering Department, Middle East Technical University, Ankara 06800, Turkey

KEYWORDSAerodynamic coefficients;Anhedral;Cropping;Leading-edge vortex;Longitudinal static stability;Non-slender delta wing;Non-slender reverse delta wing;Stall;Three-dimensional surface separation

AbstractThe effects of thickness-to-chord（t/c）ratio,anhedral angle（δ）,and cropping ratio from trailing-edge（Cr%）on the aerodynamics of non-slender reverse delta wings in comparison to nonslender delta wings with sweep angle of 45 ° were characterized in a low-speed wind tunnel using force and pressure measurements.The measurements were conducted for total of 8 different delta and reverse delta wings.Two different t/c ratios of 5.9% and 1.1%, and two different anhedral angles ofδ=15° and 30° for non-cropped and cropped at Cr=30% conditions were tested.The results indicate that the reverse delta wings generate higher lift-to-drag ratio and have better longitudinal static stability characteristics compared to the delta wings.The wing thickness has favorable effect on longitudinal static stability for the reverse delta wing whereas longitudinal static stability is not influenced by wing thickness for the delta wing.For reverse delta wings, the anhedraled wing without cropping has adverse effect on aerodynamic performance and decreases the lift-to-drag ratio.Cropping in anhedraled wing causes significant improvement in lift-to-drag ratio,shift in aerodynamic and pressure centers towards the trailing-edge,and enhancement in longitudinal static stability.

1.Introduction

The non-slender delta wing configurations, having sweep angles less than 55°, have recently drawn great attention since these planforms have been employed in a variety of air vehicles including Unmanned Air Vehicles (UAV), Micro Air Vehicles(MAV),and Unmanned Combat Air Vehicles(UCAV)due to the continuous need for performance improvements as well as achieving the economic and environmental aims of the aviation industry.Although the earlier studies focus on aerodynamic characteristics of slender delta wings,1researchers turned their attention to the non-slender planforms acknowledging that these planforms have larger surface areas suitable for flight systems equipment layout at expense of deteriorated maximum lift and stall angle.2The distinct and complex flow structures around non-slender delta wings have pronounced effects on both flight performance and stability, which lures researchers to address unresolved issues emanating from the flow instabilities and flight control problems.

The flow over a delta wing is characterized by two counterrotating vortices shed from the leading-edges.3–8The strong Leading-Edge Vortices (LEV) energize the flow,which in turn delay the stall angle of the wing.However, these vortices undergo a sudden expansion due to the rapid deceleration of the axial velocity at the vortex core known as vortex breakdown at a sufficiently high angle of attack,1,9,10which results in lift deterioration11and increased pitch-up moment.12High vortical flow at the leeward side of the wing with low pressure at the vortex core provides suction effect and enhances the lift at the expense of high drag1,13,14.

The velocity fluctuations are enhanced due to the high gradients in the cross-flow velocities and additional lift arises from the increased mean velocities and high suction peaks on the leeward side of the delta wing accompanied with the occurrence of the fully developed leading-edge vortex system as well as high maximum stall angle of attack.15At sufficiently high incidences,delta wing vortical structure loses its characteristics and the pre-stall regime of the non-slender delta wings is characterized by the large-scale and three-dimensional surface separation.16The suction peaks decreases near stall, the shear layer reattachment to the leeward surface becomes no longer possible,17the velocity components at the vortex core are greatly reduced.18The rapid loss of lift and stall occurs due to the collapse of the leading-edge vortices at high angle of attack conditions,where delta wing aircrafts generally operate during take-off, landing, combat maneuvers, and atmospheric re-entry.19Therefore,delaying the vortex breakdown,which is generally associated with the stall,18is quite crucial for enhanced delta wing performance20,21.

The flow patterns of non-slender delta wings are substantially different compared to slender ones and indicate strong reattachment of the vortex pattern.22,23,24The position of the vortex structure is closer to the wing surface causing an interaction between the vortex and the boundary layer,25which might lead to the appearance of the second primary vortex.26The Reynolds number dependence of the flow structure,which is relatively less critical for slender delta wings,27is significant when the secondary primary vortex appears on the non-slender planform28.

In recent years,forward-swept-wing particularly the reverse delta wing,which is an inverted form of the regular delta wing,has also been a major field of interest.These planforms,which are commonly utilized as ground effect vehicles or for ground proximity applications, offer aerodynamic benefits compared to regular delta wings in terms of generating increased lift force at low speeds, hence reducing required power and noise levels during take-off and landing.12In addition, the flow irregularities such as crossflow and attachment line instabilities as well as the primary modes of transition on swept wings were reported minimal on reverse delta wing configuration.29,30Further, the favorable aerodynamic characteristics of the reverse delta wing for efficient supersonic flight were also endorsed.31A variable forward-sweep wing concept, which combines the beneficiary aspects of forward-swept wings and unswept wings,by positioning the wing between its unswept and full-forward positions, was also proposed.32It is stated that this versatile aircraft concept is capable of landing on short runways,having a large payload, and overcoming the increased drag at transonic and supersonic speeds while achieving desirable handling quality, control, and stability characteristics.

The studies into the low-speed aerodynamics of delta wings go back to the pioneering work conducted by Alexander Lippisch,14whereas early investigations into the aerodynamics of reverse delta wings conducted by NACA were dated back to 1947.31Elsayed et al.,33and Altaf et al.34characterized the flow of Λ=75odelta and reverse delta wings using PIV and force measurements, in which they reported that the reverse delta wing had a higher lift-to-drag ratio resulting from the lower lift and lower drag values.The vortex flow structure and aerodynamics of a reverse delta wing having a sweep angle of Λ=65owere investigated with PIV measurements, flow visualization, and force measurements.12The flow visualization results showed that leading-edge vortices seen in delta wings were replaced by a unique ‘‘a(chǎn)rm-and-fist”leading-edge tip vortex pattern as well as the multiple spanwise vortex filaments, which caused the stall of reverse delta wings, whereas the stall of delta wings was related to the breakdown of leading-edge vortex.The PIV measurements showed that the RDW vortex was positioned above and outboard of the wing,which moved inboard the spanwise direction as it progressed downstream in contrast to the outboard movement of leading-edge-vortex of the delta wing suggesting that the upper surface acted like a wake generator and RDW vortices were not the primary source of the lift.12The force measurements showed that reverse delta wing had higher lift as well as the higher lift-to-drag ratio at low angle of attacks(α ≤10o)compared to the delta wing.12The effect of sweep angle on the flow structure was examined by Ref.35and the comparison of the vortex flow and aerodynamic characteristics over slender（Λ =65o）and nonslender （Λ =50o）delta and reverse delta wings at low Reynolds number （Re=11000）was made.It was found that reverse delta wings, when compared to delta wings, exhibited similar aerodynamic and vortex flow characteristics but lower maximum lift and delayed stall angle regardless of the slenderness.

To reduce the deteriorated aerodynamics and stability characteristics of non-slender delta and reverse delta wings, active and passive flow control techniques along with the geometrical modifications are utilized.Enhancing the shear layer or postponing the vortex breakdown and eradication of disorganized three-dimensional flow characteristics on the leeward side of the wing are the primary focus for delta wings.Energizing the flow utilizing either steady or unsteady blowing or suction were utilized as active flow techniques by.36,37,38,39,40Passive control techniques and geometrical modifications including flexible wing structure,41bioinspired edge modifications,42leading-edge shape,43,27,44trailing-edge attachment,45,46,47thickness effect,48,49and passive bleeding50,51have been adopted and investigated thoroughly.Considering the aerodynamic performance enhancements for reverse delta wings,adaptations of different geometrical features have been frequently utilized.Lee52adopted Gurney flap-like strips on both leading and trailing edges with different strip heights on a reverse delta wing with sweep angle of 65 degree.The force measurement results showed that the addition of the trailingedge strips resulted in a leftward shift of the lift curve with respect to angle of attack accompanied with significant increases in lift and lift-to-drag ratio while leading-edge strips postponed the stall with the penalty of decreased lift-to-drag ratio.Lee et al.53investigated the effect of anhedral on a reverse delta wing with sweep angle of 65°.The anhedral angle on aerodynamic characteristics showed a monotonic behavior such that as the anhedral angle increases, lowered lift and liftto-drag ratio were obtained.Lee and He54reported that the contribution of the trailing apex region of the reverse delta wing was negligible, hence cropping of the wing from the trailing-edge side might be effectively used for weight reduction without a major loss in the lift.Therefore, sole cropping of the reverse delta wing, as well as the combination of cropping, anhedral, Gurney flap-like strips, and winglets, were employed on a reverse delta wing with sweep angle of 65°and effects of those geometrical modifications on the vortex flow and aerodynamic characteristics were extensively investigated.The results indicated that the lift of the cropped wing could be significantly improved with the presence of Gurney flap-like trailing-edge strips54.

Very few studies have addressed the flow characteristics of reverse delta wings along with the delta wing counterpart.In addition, most of the existent experimental studies on nonslender delta wings and reverse delta wings do not include the longitudinal static stability characteristics due to a lack of pitch moment data.Therefore, the current understanding of the non-slender reverse delta wings requires further investigations and comparisons with relatively well-established delta wing aerodynamics since these planforms can offer great potential for low-speed flights.They can be effectively used as ground effect vehicles as well as bioinspired micro air vehicles,which can be considered simplified planforms of the wings of the gliding animals such as fox bats, dragonflies, manta rays, and butterflies.

The current study aims to characterize non-slender delta wings and reverse delta wings in a low-speed wind tunnel using force balance and surface pressure measurements and discuss the effects of geometrical features and modifications such as thickness-to-chord ratio, anhedral, and cropping on aerodynamic performance as well as longitudinal flight characteristics.For this purpose, eight wing models with sweep angle of Λ=45owere designed and manufactured.Two base delta wing and two base reverse delta wing models representing thick and thin configurations having t/c=1.1% and 5.9%,and additional four thin reverse delta wing models with t/c=1.1% as well as having different anhedral angles δ=00; 15o; and 30oand cropping ratios Cr=0%; and 30%are tested at Re=90000 for angles of attack varying between 0?≤α ≤35?.The evaluation of the results is made using pressure coefficient Cpand aerodynamic coefficients of drag CD;and lift CLforces, pitch moment coefficient Cmas well as derived parameters such as aerodynamic center in pitch Xaand pressure center Xp.

2.Experimental methods

Experiments were performed in a low-speed,suction type,and open-circuit wind tunnel,located in Fluids Mechanics Laboratory of Mechanical Engineering Department at Middle East Technical University.The wind tunnel has a transparent test section of 750 mm wide, 510 mm deep, and 2000 mm long,with a contraction ratio of 8:1.The Reynolds number of Re=9×104was used for the experiments, calculation of which was based on the baseline wing chord length of 135 mm regardless of the cropping condition.The maximum turbulence intensity was found to be less than 1% at the free stream velocity for the corresponding Re=9×104.The maximum blockage ratio was below 2.7% at the highest angle of attack over the entire test matrix.

Eight different wings with a sweep angle of 45° were used.The wing geometries are detailed in Fig.1 where wings are labelled with numbers 1 and 2 for delta wings and reverse delta wings, respectively.The first two columns of Fig.1 provide geometric parameters including the wing thickness, edge type,anhedral angle （δ ）and cropping ratio (Cr%), while the third column defines the parameters such as base wingspan (S),quarter-span (s), base wing chord line (c), cropped chord line(ccr), sweep angle, cropping percentage (Cr%), and indicates freestream directions for delta and reverse delta wings.Anhedral angle and wing cropping are explained on the bottom right schematic of Fig.1.Anhedral angle is the downward angle from the chord or cropped chord lines.The wings 2c and 2d configurations are the anhedraled reverse delta wings in absence of cropping whereas the 2e and 2f configurations are the cropped versions of the wings 2c and 2d.The cropped trailing apex region is shown with the dashed area.The uncropped wings had a chord length of 135 mm and span of 270 mm whereas cropped wings had a chord length of 94.5 mm.Base delta wings and base reverse delta wings were geometrically identical for thick (1a and 2a) and thin (1b and 2b) configurations, respectively.

Thicknesses of the wings were 8 mm and 1.5 mm with corresponding thickness-to-chord ratios of t/c=5.9% and 1.1%for thick and thin configurations.The base thick wings(1a and 2a) were manufactured using rapid prototyping of fine polyamide PA2200 with a thickness of 0.15 mm.All three edges of these two wings were sharp and beveled symmetrically with a 45°angle.Only base thick delta wing(1a)had 18 pressure taps distributed symmetrically at the chordwise distance of x/c=0.5.The symmetric bevel condition enabled that both sides of the wing could be used as suction and pressure sides.Therefore, 1a wing had pressure taps, which were present on only one side of the wing.The diameter of these taps on the surface of the wing was 1.5 mm.The thin wings (1b, 2b, 2c,2d,2e,2f)were made of an aluminum flat plate and had sharp and cut edges at all sides with no bevel on both leading and trailing edges.

The angle of attack was measured using a digital inclinometer, having an accuracy of ±0.1°.The surface pressure measurements on both suction and pressure sides were conducted using a 16-channel pressure scanner, which was equipped with a 16-piezo-resistive transducer with the range of 0–2.5 kPa.The scanner had an accuracy of 0.05% FS (full scale).The pressure data were collected at a sampling rate of 500 Hz for 10 seconds.Preliminary tests were carried out to ensure the symmetry along with the spanwise pressure distribution, hence the measurements for only one half of the wing were performed.Pressure coefficient Cpwas calculated using Eq.(1).The maximum uncertainty values for pressure coefficient CP, are ±0.0509 and ±0.0275 for suction and pressure sides.The combined uncertainty analysis of pressure measurements is also discussed in the following section.

Fig.1 Schematic representations of wings including delta wing, reverse delta wing, anhedraled and cropped.

In Fig.2, schematic representations of the top and side views for the force measurement system are given.Aerodynamic forces and moments were measured using an external force balance system.ATI Gamma Series 6-Axis Force and Torque sensor, which was calibrated according to SI-32–2.5 scheme, was installed out of the wind tunnel near the sidewall and attached to the wing with a strut to obtain drag, lift, and pitch moment.The aerodynamic and gravitational forces and moments existing over the bare strut were also measured for each angle of attack.These loads were subtracted from the measurements for the wings to exclude the effect of the strut on force and moment measurements.The pitch axis was the trailing-edge or trailing apex of the wing.The force and moment data were collected for the angles of attack 0?≤α ≤35?.The force and moment coefficients were calculated using Eq.(2) and Eq.(3).For all eight wings, reference area A and reference length L were the surface area and the chord length of the base delta wing 1b and equivalent to 0.0182 m2and 0.135 m, respectively.National Instrument NI-PCIe-6321 16-bit data acquisition (DAQ) card was equipped and coupled to LabVIEW software for digitization of the raw voltage data, which were collected at 10 kHz for 15 seconds for each data point.The maximum uncertainty values for aerodynamic force coefficients CDand CLas well as the lift-to-drag ratio CL/CDand pitch moment coefficient Cmare±0.0398, ±0.0629, ±0.3055, and ±0.0629, respectively.The combined uncertainty analysis of force measurements is also discussed in the following section.

Fig.2 Schematic representations of top and side views for the force measurement set-up.

The aerodynamic coefficients are plotted with respect to α for CL, CDand CL/CDtogether with drag polar CLvs CD.The longitudinal stability characteristics are evaluated by utilizing pitch moment data, where the positive moment acts to pitch the wing in the nose-up direction.For that purpose,four different charts are constructed.In the first chart, Cmdata is expressed at the trailing-edge and plotted with respect to angle of attack.In the second chart, Cmexpressed at trailing-edge is plotted versus CL.The slope of this curve is equal to the aerodynamic center in pitch Xa, which is the point where pitch moment is independent of angle of attack and expressed as Eq.(4).It provides the non-dimensional distance of the aerodynamic center of the associated wing from its trailing-edge,which is positive if the aerodynamic center lies between the leading-edge and trailing-edge of the wing and negative if the point lies downstream of the trailing-edge.In the third chart,the pitch moment at the associated wing center of gravity is plotted with respect to angle of attack.The slope of this curve measures the static margin of the wing and must be negative to possess positive longitudinal static stability.For the fourth chart, the nondimensional center of pressure location XP,which is the point where Cmis equal to zero and expressed in Eq.(5),is plotted with respect to angle of attack.It provides the non-dimensional location, normalized by the chord length c or cropped chord length ccrdepending on whether the wing is cropped or not,which in turn indicates 0 and 1 limiting values representing the leading-edge and trailing-edge of the corresponding wing.

3.Results and discussion

In this chapter,the pressure and force measurement results are provided.In the first part, pressure measurements on suction and pressure sides of the base thick delta wing (1a) are presented,and then the wing thickness effect is examined by comparing base thick and base thin delta(1a,1b)and reverse delta(2a, 2b) wings, using force measurements.In the second part,aerodynamic characteristics of geometrically modified reverse delta wings(2b,2c,2d,2e,and 2f)including variation of anhedral angles and cropping ratios are compared using force measurements.

3.1.Effect of wing thickness

3.1.1.Results of surface pressure measurements

Fig.3 Cpdistributions at suction and pressure sides of base delta wing (1a) at chordwise location of x/c=0.5 for angles of attack of α=17?and 23° at Re=9×104.

In Fig.3,Cpdistributions at suction(in blue)and pressure (in red) sides of the base thick delta wing (1a) at chordwise location of x/c= 0.5 and at angles of attack of α=17?, 23° are given,representing the pre-stall and stall regions.The horizontal axis of the charts indicates the spanwise distance at the chordwise location of x/c= 0.5, which is normalized with the corresponding local half span length.The details of the pressure measurements were not given in this study for the sake of interpretation of data,although,spanwise Cpdistributions indirectly indicate both strength and reattachment position of the leading-edge vortex.Therefore, the results of the surface pressure measurements were only examined as a separate validation study for the force measurements considering the stall angle as well as normal force trends both of which have to be consistent for pressure and force measurements.Considering the suction side curves, the apparent hump-like behavior at α=17oindicates the occurrence of the leadingedge vortex whereas Cpdistribution turns into nearly a flat distribution at α=23o, which is indicating an appearance of three-dimensional surface separation on the planform and quite in line with the stall condition of the wing that will be discussed in the following section.In addition, considering both the suction and pressure side curves, the area enclosed by the curves is the footprint of the normal force,which has the contribution from both the lift and drag forces.

3.1.2.Results of aerodynamic force measurements

In Fig.4,distributions of drag coefficient CD,lift-to-drag ratio CL/CD, lift coefficient CLas a function of angle of attack and drag polar for base thick and thin delta wings(1a,1b)and base thick and thin reverse delta wings (2a, 2b) are given,respectively.

Considering the drag coefficient CDshown in the upper left chart,for both delta and reverse delta configurations the thick wings demonstrate higher CDvalues with lower rate of change at low angles of attack up to α=10ocompared to the thin wings.For delta wing configurations,similar drag coefficients are observed between α=10oand 20o.However, the thick delta wing 1a exhibits sudden drag increase, which is absent for the thin delta wing 1b, at the stall angle, which can be deduced from either the pressure results in Fig.3 or CLchart in Fig.4, and roughly to be between αs=22o-23o.Comparing the drag coefficients of the base thick 2a and base thin 2b reverse delta wings, the drag coefficients linearly increase with angle of attack and the thick configuration always exhibits higher drag coefficients at all angles of attack.

Considering the lift coefficient CLshown in upper right chart,the delta wings exhibit typical stall behavior with a sudden loss in lift, whereas the lift coefficients of the reverse delta wings reveal that the lift curve starts to indicate flat distribution after a certain angle of attack.The base thin delta wing 1b has a stall angle of αs=20owhereas stall onset is postponed to αs=22ofor base thick delta wing 1a with a relatively less maximum CL.For the thick reverse delta wing 2a, lift curve flattens around the maximum CLvalue.The thin reverse delta configuration 2b possesses consistent increase in lift coefficient,however, the rate of increase in CLreduces dramatically after α=10o.Considering the lift curve slopes of all four wings,thin configurations 1b and 2b have higher rate of increase in lift compared to thick ones 1a and 2a.

The CL/CDdistribution is shown in the lower left chart and primarily presents the aerodynamic performance of the wings.Considering the thickness effect, the thin wings 1b and 2b demonstrate superior aerodynamic performances compared to thick wings 1a and 2a at all angles of attack,where the thin reverse delta indicates the best performance among all configurations.Considering the slope of CL/CDdistributions for all four wings, the rate of increase of CL/CDfor reverse delta wings 2a and 2b are higher compared to ones for the delta wings 1a and 1b up to the angle of attack where the maximum CL/CDappears.In addition, the peak values of CL/CDfor delta wings 1a and 1b are postponed to higher angles of attack compared to reverse delta wings 2a and 2b.

The drag polar shown at the lower right chart of Fig.4,presents the ability of the wing to generate additional lift without increasing the drag, hence having a higher slope is desired.Comparing the delta wings 1a and 1b in the pre-stall region and the reverse delta wings 2a and 2b for all angles of attack,the drag polar demonstrates that thin configurations 1b and 2b generate relatively less drag for the same lift coefficient compared to thick configurations 1a and 2a.The behavior of the reverse delta wing 2b significantly deteriorates after CL=0.6 and lift generation ability comes with a high drag generation penalty while 2a cannot generate additional lift after CL=0.71.The slopes of the drag polar curves of the delta wings 1a and 1b are quite similar between CL=0.6 and 0.86 suggesting that amount of the drag drawback is similar for the same amount of additional lift.

Fig.4 Distributions of drag coefficient CD,lift-to-drag ratio CL/CD,lift coefficient CLas a function of angle of attack and drag polar for base delta wings (1a), (1b) and base reverse delta wings (2a), (2b).

In Fig.5, variation of moment coefficient Cmat Trailing-Edge (TE) and Center of Gravity (CG) as functions of angle of attack and lift coefficient CL, and non-dimensional center of pressure coordinate XPas a function of angle of attack for delta wings 1a and 1b and reverse delta wings 2a and 2b are given.

Considering the distribution of Cmat trailing-edge shown in the upper left chart,the thick configurations 1a and 2a induces lower slopes up to α=14ocompared to the corresponding thin wings 1b and 2b.For the delta wings 1a and 1b, the slope of Cmdistribution decreases significantly and becomes negative at the stall angle.However, this behavior is not evident for the reverse delta wings 2a and 2b since the rate of change of Cmis always positive.In addition, reverse delta wings 2a and 2b, possess relatively higher Cmcompared to the delta wings.The center of pressure XPlocation, which will be further discussed along with the lower right chart of Fig.5, might be effective on this behavior since XPlocated close to leadingedge results in higher trailing-edge moments for the same amount of lift.Therefore, the location of XPis tremendously important over the longitudinal stability characteristics of the wings.

The distribution of Cmat the trailing-edge as a function of CLis demonstrated in the bottom left chart of Fig.5.The location of the aerodynamic center in pitch Xacan be found with respect to the trailing-edge from the slope of the curve and the corresponding values of Xaare given in Table 1.The highest slope is seen for the thin reverse delta wing 2b demonstrating the most forward Xaposition and located in the vicinity of the leading-edge （Xa=0.033）.The reverse delta wings 2a and 2b have Xapositions closer to the leading-edge compared to delta wings 1a and 1b.The thickness has negligible effect on Xaposition for delta wings 1a and 1b,both of which have similar values around Xa=0.3, whereas Xaappears closer to the leading-edge for the thin reverse delta wing 2b compared to thick reverse delta wing 2a.

Fig.5 Distributions of moment coefficient Cmat TE and CG as functions of angle of attack and lift coefficient CL,and non-dimensional center of pressure coordinate XPas a function of angle of attack for base delta wings (1a), (1b) and base reverse delta wings (2a), (2b).

Table 1 Arodynamic center in pitch Xa.

Considering the Cmat wing CG as a function of α shown at upper right corner of Fig.5,all four wings 1a,1b,2a,2b have positive slopes and positive Cm,which in turn result in unstable longitudinal stability characteristics.As the slope of this curve approaches to zero, the wing becomes insensitive to possible disturbances due to incremental changes in angle of attack.The wing thickness does not affect the stability characteristics of the delta wings 1a and 1b, accompanied by similar amount of moment around the CG as well as demonstrating similar slopes.The slope of the curve for reverse delta wing 2b is similar to the delta wings 1a and 1b up to α=5o, whereas for higher angle of attack, reverse delta wings 2a and 2b demonstrate similar slopes with Cmcurve of the thin wing 2b shifted upwards.Therefore,reverse delta wings 2a and 2b have higher static margin compared to delta wings 1a and 1b.The stability in longitudinal channel can be assured utilizing a negative Cmsource such as a horizontal stabilizer generating nose-down moment around CG.Considering the overall maneuverability and controllability requirements, desired behavior can be adjusted with the position of CG as well as the control power generated by the horizontal stabilizer.The moment generated by the horizontal stabilizer is directly proportional to the area of the wing and the thick reverse delta wing 2a has the lowest Cm.This can be interpreted as an advantage of the thick reverse delta wing 2a compared to other three wings 1a, 1b and 2b in terms of weight reduction since it needs the smallest negative moment around CG generated by the horizontal stabilizer, which can be achieved by a smaller horizontal stabilizer.In addition, a smaller horizontal stabilizer at the same longitudinal position results in smaller shift of the CG towards trailing-edge.

Considering the XPas function of angle of attack demonstrated in lower right corner of Fig.5, the wing thickness is effective on reverse delta wings such that the thick reverse delta wing 2b has XPposition closer to the leading-edge compared to the wing 2a for all angles of attack while a negligible shift toward leading-edge is seen for the thick delta wing 1a compared to thin delta wing 1b.As the angle of attack increases,XPgets closer to the leading-edge for the thick reverse delta wing 2a whereas no remarkable movement of the XPis seen for other three wings 1a, 1b and 2b.Considering the delta wings 1a and 1b, center of pressure XPis around x/c=0.35 and does not change with angle of attack.The aerodynamic centers of these two wings are Xa=0.317 and 0.333 and quite close to XP.Considering the ‘‘thin airfoil theory”and neglecting the small changes in XPwith angle of attack,the combined behavior of Xaand XPdemonstrate 2D symmetric airfoil characteristics such that center of pressure is coincident with aerodynamic center in pitch and it does not change with angle of attack, although Xa≌XP=0.35 for delta wings 1a and 1b but the ‘‘thin airfoil theory”dictates that Xa=XP=0.25.

In Fig.6, uncertainty values of the present work and comparisons of the present work with the data in literature are discussed.Representative absolute uncertainty analyses for Cpand CLof the base thick delta wing 1a are given as a function of angle of attack in the first row.The chart for the pressure measurements is constructed at the spanwise location of y/s=0.77 at x/c=0.5.In the second row, the CLvalues of the current study for the thick delta wing 1a and the thin delta wing 1b are compared with the results of Refs.2,43,48,49,51,which utilize delta wings having 45osweep angle with different t/c ratios, Reynolds number and leading-edge shapes.

The combined uncertainty levels of the Cpgiven in the upper left chart indicate that the uncertainty levels on the pressure side does not vary much with angle of attack and are around 0.025 for all angles of attack and whereas the uncertainty levels on the suction side increase up to α=15oto the 0.047 and monotonically decreases with further increase in angle of attack.Considering the CLshown in the upper right chart, the combined uncertainty levels show similar trend observed in the pressure measurements for the suction side such that the uncertainty levels increase with angle of attack up to αs=22oto 0.06 and monotonically decreases with further increase in angle of attack.

The comparison of the CLresults of the thick delta wing 1a with respect to related studies in literature is presented in the lower left chart of Fig.6.Earnshaw and Lawford43studied a delta wing having a slight stream-wise camber,whereas Ghazijahani and Yavuz49and Kestel et al.51studied a sharp-edged delta wing with 45° bevel angle on the windward side of the wing.Considering the CLdistributions, the lift curve slopes of all four results are similar whereas the maximum attainable CLas well as maximum CLat α=0oare achieved in the study of Ref.43,which can be attributed to leading-edge shape of the wing.The nonzero CLvalues at α=0ofor Refs.49,51are also expected to be due to asymmetric bevel conditions.At the lower right chart of Fig.6,the CLdistribution of the base thin delta wing 1b is compared with the results of Refs.2,48,49.Refs.2,49adopted a sharp-edged 45° bevel angle on the windward side whereas Kawazoe et al.48utilized rounded and semicircular leading-edge shape.Kawazoe et al.48states that rounded leading-edge shape is responsible of delaying stall and attributed it to the primary attachment line,which reaches to the wing centerline at higher angle of attack.The stall characteristics of the present study is similar to the results of Ref.48and both wings stall at the same angle of attack α=20o,which might be associated with the delaying stall mechanism due to the symmetric bevel.Considering the charts on the second row together for the consistency assessment of the results of the current study, the CLdistributions of the present study are quite in line with the representative studies in literature in terms of CLslopes, maximum CLvalues, and stall angles even though the studies include variation in leading-edge shapes and Reynolds numbers.

3.2.Aerodynamic characteristics of reverse delta wings

In Fig.7,distributions of drag coefficient CD,lift-to-drag ratio CL/CD, lift coefficient CLas a function of angle of attack and drag polar for reverse delta wings (2b, 2c, 2d, 2e, and 2e) are given, respectively.

Fig.6 Absolute uncertainty levels of Cpand CLof the base thick delta wing(1a)as a function of angle of attack and comparison of lift coefficient of base thick (1a) and base thin (1b) delta wings with the results of Refs.2,43,48,49,51.

Fig.7 Distributions of drag coefficient CD,lift-to-drag ratio CL/CD,lift coefficient CLas a function of angle of attack and drag polar for different reverse delta wing configurations (2b), (2c), (2d), (2e), (2f).

Considering the CDshown in the upper left chart, the sole effect of the anhedral angle on the reverse delta wing does not have a monotonic trend on drag since the anhedraled wing(δ=15o)2c has higher CDdistributions nearly for all angles of attack, but further increase in anhedral angle to δ=30odecreases CDand the anhedraled wing(δ=30o)2d has CDvalues in between the base wing 2b and the anhedraled wing 2c.When considering the wings 2c and 2e as well as the wings 2d and 2f separately, the sole effect of the cropping results in decreased CDdistributions for both anhedral angles δ=15?and 30?and downward shift of CDcurves, where the lowest CDdistributions are achieved with 2f wing configuration of δ=300and Cr=30%.

Considering the CLdistributions shown in upper right chart, the characteristic lift coefficient trend for the base reverse delta wing 2b is witnessed in all configurations such that lift curve slope decreases as the angle of attack increases without clear indications of stall formation and reduction in lift coefficient.The highest CLvalues and lift curve slopes are achieved with base reverse delta wing 2b.As the anhedral angle increases, both the lift coefficient and its rate of change decrease.Considering the effect of cropping at δ=30ofor wing configurations 2d and 2f, two different trends are observed for relatively low and high angles of attack.The cropped wing 2f indicates higher CLslopes compared to non-cropped wing 2d for angles of attack up to α=12o, on the contrary, the trend is complete opposite for higher angles of attack.

Considering the CL/CDshown in the lower left chart,aerodynamic performance of anhedraled and cropped wing 2e is superior compared to the other configurations.The maximum efficiency angle of attack of all five wings are close to each other and around α=5o.The sole effect of anhedral results in deterioration of the wing performance since CL/CDdistributions of the wings 2c and 2d are quite close to each other and the corresponding maximum value is significantly less than the one achieved with the base reverse delta wing 2b.Cropping tremendously improves the performance of the anhedraled wings 2c and 2d and results in higher CL/CDfor both wings 2e and 2f while the improvement is more prominent for δ=15owith respect to δ=30o.

Considering the drag polar chart demonstrated at the lower left corner, overall distributions of the wings 2b, 2c, and 2d indicate that drag penalty increases with increasing δ for the same amount of lift.Cropping causes significant improvement up to CL=0.6 and reverses the adverse effects of anhedral such that similar drag polar distributions are obtained among the wings 2b,2e,and 2f.However,the ability of generating lift without drag penalty deteriorates and distribution becomes nearly horizontal after CL=0.6.

Fig.8 Distributions of moment coefficient Cmat TE and GG as a function of angle of attack and lift coefficient CL, and nondimensional center of pressure coordinate XPas a function of angle of attack for different reverse delta wing configurations(2b),(2c),(2d),(2e), (2f).

In Fig.8,variation of moment coefficient Cmat TE and CG as functions of angle of attack and lift coefficient CL,and nondimensional center of pressure coordinate XPas a function of angle of attack for reverse delta wings (2b, 2c, 2d, 2e, and 2e)are given, respectively.

Considering the Cmat trailing-edge shown in the upper left chart,both the Cmand its slope decrease due to the increase in nose-down moment with the inclusion of anhedral for the wings 2c and 2d compared to the base wing 2b.Similarly,the cropping for the wings 2e and 2f causes significant drop in Cmand its slope compared to the base wing 2b and anhedraled wings 2c and 2d.

The Cmat trailing-edge as a function of CLis demonstrated in the bottom left chart.The locations of the aerodynamic center in pitch Xaof the reverse delta wings are given in Table 2.The location of the aerodynamic center in pitch Xais at the maximum forward position with respect to trailing-edge for the base reverse delta wing 2b.As the anhedral angle increases to δ=15ofrom the δ=0ofor the wing 2b,Xamoves towards the trailing-edge Xa= 0.112 for the wing 2c while further increase in anhedral angle results in movement of the Xatowards the leading-edge Xa=0.089 for the wing 2d.However, cropping results in monotonic shift of Xatowards tothe trailing-edge of the wings 2e and 2f compared to ones 2c and 2d.Among all five wings, wing 2e with δ=15oand Cr=30% induces the most aft positioned aerodynamic center Xa, which is Xa=0.195.

Table 2 Aerodynamic center in pitch Xaof reverse delta wings.

Considering the Cmat wing CG as a function of α shown at upper right corner of Fig.8,all five wings 2b,2c,2d,2e,and 2f have positive slopes and positive Cm, hence possess unstable longitudinal static stability characteristics.Introducing anhedral decreases the Cmat CG compared to the base wing 2b.However, further increase in anhedral angle from δ=15oto δ=30o, comparing the configurations 2c and 2d, do not induce further reduction in moment coefficient for angle of attack higher than α=10o.Cropping has a complex influence on anhedraled wings 2c and 2d such that it shifts the curve of the anhedraled wing 2c downward without changing the slope and results in a distribution seen for the wing 2e while it has no effect for the wing 2f compared to the wing 2d up to α=5obut decreases Cmbetween α=5oand 30owith significant decrease in the slope.Therefore,cropping has stabilizing effect for angle of attack higher than α=5ofor anhedraled wing δ=30obut has no effect for the wing with δ=15o.Considering the concepts of longitudinal static stability and aforementioned aerodynamic efficiency,the wing 2e offers the highest aerodynamic efficiency without any deterioration in longitudinal stability characteristics compared to the other four reverse delta wings 2b, 2c, 2d, and 2f.

The center of pressure XPas function of angle of attack is presented in the lower right corner of Fig.8.XPdoes not change with α for the base reverse delta wing 2b,while it moves toward to the leading-edge for the anhedraled wings 2c and 2b as well as the anhedraled and cropped wings 2e and 2f as the angle of attack increases.The slope of the XPis steeper for the cropped wings 2e and 2f compared to the non-cropped wings 2c and 2d.In addition, cropped wings 2e and 2f have the XPpositions closest to trailing-edge among all wing configurations.

4.Conclusions

In the present study, the effect of thickness-to-chord ratio on delta wings and reverse delta wings of sweep angle Λ=45°as well as aerodynamic characteristics of reverse delta wings with sweep angle of 45°subjected to geometrical modifications were studied using pressure and force measurements.The experiments were carried out in a low-speed wind tunnel using the base delta wings and base reverse delta wings planforms with t/c ratios 1.1% and 5.9% and reverse delta wings with anhedral angles δ=0; 15o, and 30oand cropping ratios Cr=0 and 30% for Reynolds number Re=9×104and angles of attack varying from 0 to 35 degrees.Considering the combined assessments of the pressure and force measurements, the principal findings are as follows:

(1) Considering the effect of wing thickness on aerodynamic performance of delta and reverse delta wings, aerodynamic characteristics of delta wings and reverse delta wings and their corresponding dependence on wing thickness are substantially different.The base thick delta wing 1a with symmetric bevel at the leading-edge could maintain a strong vortex structure up to α=20oand α=22oand achieve maximum CLvalue close to the base thin delta wing 1b.The base thin reverse delta wing 2b generates higher lift at low angles of attack with higher maximum efficiency compared to the base thin delta wing 1b.However, this is not witnessed when the thickness of the wing is increased.In addition, the reverse delta wings 2a, 2b have aerodynamic and pressure centers closer to the leading-edge and have better longitudinal static stability characteristics with higher static margin when compared to the delta wings 1a, 1b.The static margin of reverse delta wing increases significantly with increasing wing thickness whereas the wing thickness has negligible effect on it for delta wings.

(2) Considering the effect of geometrical modifications on aerodynamic performance of reverse delta wings, the sole effect of wing anhedral has deteriorating impact on the performance of reverse delta wings since it increases drag penalty, decreases the lift and lift curve slope and efficiency, although it promotes longitudinal static stability and shifts the aerodynamic and pressure centers towards the trailing-edge.Cropping of the anhedraled wings tremendously improves the deteriorated performance characteristics resulting in improvement in lift and efficiency, additional shift of aerodynamic and pressure centers towards the trailing-edge and further enhancement in longitudinal static stability and static margin.This might be due to elimination of the wing portion with cropping which neither contributes to lift generation capability nor creates nose-down pitch moment.Anhedraled and cropped wing 2e with δ=15oand Cr=30%possesses the highest CL/CDwith slight loss in lift at high angles of attack and the best longitudinal static stability characteristics.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank Og?uzhan Y?lmaz for his help with the experiments.The author(s)disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Turkish Aerospace Industries, Inc.and Middle East Technical University (No.BAP TEZ-D-302-2021-10725).