Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring

aerospace

Article Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring

Gottfried Sachs *, Benedikt Grüter and Haichao Hong

Department of Aerospace and Geodesy, Institute of Flight System Dynamics, Technical University of Munich, Boltzmannstraße 15, 85748 Munich, Germany; [email protected] (B.G.); [email protected] (H.H.) * Correspondence: [email protected]

Abstract: Dynamic soaring is a flight mode that uniquely enables high speeds without an engine. This is possible in a horizontal shear wind that comprises a thin layer and a large wind speed. It is shown that the speeds reachable by modern gliders approach the upper subsonic Mach number region where compressibility effects become significant, with the result that the compressibility-related drag rise yields a limitation for the achievable maximum speed. To overcome this limitation, wing sweep is considered an appropriate means. The effect of wing sweep on the relevant aerodynamic characteristics for glider type wings is addressed. A 3-degrees-of-freedom dynamics model and an energy-based model of the vehicle are developed in order to solve the maximum-speed problem with regard to the effect of the compressibility-related drag rise. Analytic solutions are derived so that generally valid results are achieved concerning the effects of wing sweep on the speed performance. Thus, it is shown that the maximum speed achievable with swept wing configurations can be increased. The improvement is small for sweep angles up to around 15 deg and shows a progressive increase thereafter. As a result, wing sweep has potential for enhancing the maximum- speed performance in high-speed dynamic soaring.

Citation: Sachs, G.; Grüter, B.; Hong, Keywords: maximum-speed dynamic soaring; swept-wing glider conﬁguration; high-speed ﬂight H. Performance Enhancement by without engine Wing Sweep for High-Speed Dynamic Soaring. Aerospace 2021, 8, 229. https://doi.org/10.3390/ aerospace8080229 1. Introduction

Academic Editor: Gokhan Inalhan Dynamic soaring is a non-powered flight technique by which the energy required for flying is gained from a horizontal shear wind [1,2]. This type of wind shows changes in the Received: 29 June 2021 wind speed with the altitude. For sustained dynamic soaring, a minimum in the strength Accepted: 5 August 2021 of the wind shear is necessary [3]. Published: 19 August 2021 There are different modes of dynamic soaring which are associated with features of the shear wind. These features concern the strength of the shear in terms of the magnitude Publisher’s Note: MDPI stays neutral of the wind gradient, the vertical extension of the shear layer or the wind speed level. Shear with regard to jurisdictional claims in wind scenarios enabling high-speed dynamic soaring show a thin shear layer and a large published maps and institutional affil- difference in the wind speeds above and below the layer. This type of shear wind exists iations. at the leeside of ridges which features a region of large wind speed above the layer and a region of zero or low wind speed below the layer [4]. Theoretical studies and flight experience have shown that it is possible to achieve extremely high speeds with dynamic soaring by exploiting the wind energy relating to the Copyright: © 2021 by the authors. addressed shear layer at ridges [4–7]. The efficiency of dynamic soaring for transferring Licensee MDPI, Basel, Switzerland. wind energy into the kinetic and potential energy of the soaring vehicle manifests in the This article is an open access article fact that the speed of the vehicle is many times larger than the wind speed, yielding values distributed under the terms and of the order of 10 for the ratio of these speeds [8,9]. conditions of the Creative Commons The high-speed performance enabled by dynamic soaring has stimulated successful Attribution (CC BY) license (https:// speed record efforts over the years to increase continually the maximum-speed level, with creativecommons.org/licenses/by/ the result that the current record is 548 mph (245 m/s) [10]. This highlights the unique 4.0/).

Aerospace 2021, 8, 229. https://doi.org/10.3390/aerospace8080229 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 229 2 of 26

capability of dynamic soaring to enable extremely high speeds by a purely engineless flight maneuver. The experience gained in high-speed dynamic soaring of gliders at ridges is an item of interest for other engineless aerial vehicle types. This can relate to initial testing of dynamic soaring where the strong shear, easy access and relatively obstruction-free environment makes ridge shears attractive. The possibility of gaining energy from the wind so that engineless flight is feasible has stimulated research interest in using the wind as an energy source for technical applications. Evidence for this perspective is found in biologically inspired research and development activities directed at utilizing the dynamic soaring mode of albatrosses for aerial vehicles [11–18]. The aforementioned speed record translates to a Mach number of greater than Ma = 0.7. It can be assumed that this speed is not the top speed as for safety reasons the speed recording is in a section of the dynamic soaring loop after the glider has reached the lowest altitude and is climbing upwind again. Accordingly, the greatest Mach number of the loop may be even higher than that associated with the record speed of 548 mph [9]. Dynamic soaring at such high Mach numbers poses specific problems unique for soaring vehicles because the compressibility effects existing in the Mach number region above Ma = 0.7 have an adverse, limiting effect on the achievable maximum-speed performance. The reason for the addressed limiting effect on the speed performance is due to the drag rise caused by compressibility. The compressibility-related drag rise which is associated with the development of shock waves shows a rapid increase in the drag with the result that the lift-to-drag ratio falls abruptly [19]. The problem concerning the penalty in the speed performance caused by the compressibility-related drag rise is dealt with in [20] and it is shown that it can be so powerful as to yield a limitation in the achievable maximum-speed performance. The question is whether wing sweep is a solution for the maximum-speed limitation problem. The reason for this consideration is that wing sweep is a means for alleviating compressibility-related drag rise problems in aerospace vehicles by shifting the drag rise to higher Mach numbers. The purpose of this paper is to deal with increasing the maximum speed enabled by dynamic soaring in the high subsonic Mach number region. To that end, the problems associated with compressibility are addressed and solutions for increasing the maximum speed are derived. Wing sweep is considered as a suitable means to enhance the maximum- speed performance in high-speed dynamic soaring. The focus of this paper is on the development and use of appropriate mathematical models for the flight mechanics of the soaring vehicle at high speeds and the shear wind characteristics, including a suitable optimization method for achieving solutions of the maximum-speed problem. With these developments, analytic solutions are derived, and it is shown that and to what extent the maximum speed can be increased using wing sweep. It is found that the improvement is small for sweep angles up to around 15 deg and shows a progressive increase thereafter.

2. Flight Mechanics Modellings of High-Speed Dynamic Soaring The maximum-speed dynamic soaring problem under consideration is graphically addressed in Figure1 where a shear wind scenario at a ridge and a dynamic soaring trajectory are shown. The wind blows at high speed over the ridge, to the effect that there are three regions leeward of the ridge: upper region of high wind speed, thin shear layer showing transition of the wind speed from the region above the layer to the region below the layer and a lower region with no wind or only small wind speeds. The dynamic soaring trajectory presented in Figure1 has a shape that enables high speeds by gaining energy from the wind. The trajectory consists of an inclined closed loop showing four ﬂight phases (indicated by nos. 1 to 4), yielding: (1) Windward climb where the wind shear layer is traversed upwards. Aerospace 2021, 8, x FOR PEER REVIEW 3 of 26

(1) Windward climb where the wind shear layer is traversed upwards. Aerospace 2021, 8, 229 3 of 26 (2) Upper curve in the region of high wind speed, with a flight direction change from windward to leeward. (3) Leeward descent where the wind shear layer is traversed downwards. (2)(4) UpperLower curvecurve inin thethe regionno wind of region, high wind with speed, a flight with direction a ﬂight change direction from change leeward from to windwardwindward. to leeward. (3) Leeward descent where the wind shear layer is traversed downwards. The model developed for the shear wind scenario at ridges comprises the described (4) Lower curve in the no wind region, with a ﬂight direction change from leeward three wind regions. The shear layer involves a rapid increase of the wind speed from zero to windward. to the value of the free stream wind speed which is denoted by 𝑉,.

FigureFigure 1. 1.High-speed High-speed dynamic dynamic soaring soaring trajectory trajectory and and shear shear wind wind scenario scenario at at ridge. ridge.

TheFor modeldealing developed with the dynamic for the shear soaring wind problem scenario under at ridges consideration, comprises thetwo described flight me- threechanics wind models regions. are Thedeveloped. shear layer One involves model denoted a rapid increaseby “3-DOF of thedynamics wind speed model” from is based zero toon the point value mass of thedynamics free stream and windthe other speed model which denoted is denoted by “energy by Vw,re model” f . is based on the energyFor characteristics dealing with the determinative dynamic soaring in high-speed problem dynamic under consideration, soaring. Results two applying ﬂight me- the chanics3-DOF modelsdynamics are model developed. are used One to model validate denoted the energy by “3-DOF model. dynamics model” is based on point mass dynamics and the other model denoted by “energy model” is based on the energy2.1. 3-DOF characteristics Dynamics determinativeModel in high-speed dynamic soaring. Results applying the 3-DOFThe dynamics motion modelof aerial are vehicles used to in validate high-speed the energy dynamic model. soaring can be mathematically described using a point mass dynamics model. An inertial reference system which is pre- 2.1. 3-DOF Dynamics Model sented in Figure 2 is applied where the 𝑥 axis is parallel to the wind speed, the 𝑦 is The motion of aerial vehicles in high-speed dynamic soaring can be mathematically horizontal and 𝑧 is pointing downward. With regard to this reference system, the equa- describedtion of motion using can a point be expressed mass dynamics as model. An inertial reference system which is presented in Figure2 is applied where the xi axis is parallel to the wind speed, the yi is horizontal and zi is pointing downward.d𝑢/d𝑡 With = −𝑎 regard𝐷/𝑚 to this −𝑎 𝐿/𝑚 reference system, the equation of motion can be expressed as d𝑣/d𝑡 = −𝑎 𝐷/𝑚 −𝑎 𝐿/𝑚 dui/dt = −au1D/m − au2L/m d𝑤/d𝑡 = −𝑎 𝐷/𝑚 −𝑎 𝐿/𝑚 +𝑔 dvi/dt = −av1D/m − av2L/m d𝑥 /d𝑡 = 𝑢 (1) dwi/dt = −aw1D/m − aw2L/m + g

dxi/dt = ui (1) d𝑦/d𝑡 = 𝑣

dyi/dt = vi dℎ/d𝑡 = −𝑤 dh/dt = −wi The coefficients 𝑎,, 𝑎, and 𝑎, are abbreviation factors used for describing rela- The coefﬁcients au1,2, av1,2 and aw1,2 are abbreviation factors used for describing relation- tionships regarding the angles 𝛾, 𝜇 and 𝜒, yielding ships regarding the angles γa, µa and χa, yielding

au1 = cos γa cos χa

au2 = cos µa sin γa cos χa + sin µa sin χa Aerospace 2021, 8, x FOR PEER REVIEW 4 of 26

𝑎 = cos 𝛾 cos 𝜒

𝑎 = cos 𝜇 sin 𝛾 cos 𝜒 +sin𝜇 sin 𝜒

𝑎 = cos 𝛾 sin 𝜒

𝑎 = cos 𝜇 sin 𝛾 sin 𝜒 −sin𝜇 cos 𝜒 (2)

𝑎 = −sin 𝛾 Aerospace 2021, 8, 229 4 of 26

𝑎 = cos 𝜇 cos 𝛾

The angles 𝛾 and 𝜒 are determined by the following relations: av1 = cos γa sin χa sin 𝛾 =−𝑤/𝑉 av2 = cos µa sin γa sin χa − sin µa cos χa (2)

tanaw1 𝜒= =𝑣− sin/(𝑢γa+𝑉) (3)

The angle 𝜇 is a control that is determinedaw2 = cos µ abycos theγa optimality conditions, described by Equation (A2) in the Appendix A.

Figure 2. Speed vectors and inertial coordinate system. (The 𝑥 axis is chosen parallel to the wind Figure 2. Speed vectors and inertial coordinate system. (The x axis is chosen parallel to the wind speed vector 𝑉 and pointing in the opposite direction.). speed vector Vw and pointing in the opposite direction).

The aerodynamic forces, 𝐋 and 𝐃, are related to the airspeed vector 𝐕, while the The angles γa and χa are determined by the following relations: motion of the vehicle is described by the inertial speed vector

sin γa = −wi/Va 𝐕 =(𝑢,𝑣,𝑤,) (4)

which is related to the inertial referencetan χa =systemvi/(u (i𝑥+,𝑦,𝑧Vw)). The speed vectors 𝐕 and 𝐕(3) are connected by the wind speed vector 𝐕 (Figure 2), yielding The angle µa is a control that is determined by the optimality conditions, described by

Equation (A2) in the AppendixA. 𝐕 =𝐕 −𝐕 (5) The aerodynamic forces, L and D, are related to the airspeed vector Va, while the The 𝑥 axis can be chosen to be in the opposite direction of the wind speed vector motion of the vehicle is described by the inertial speed vector 𝐕 (Figure 2), with no loss of generality. Thus, the wind speed vector can be written as T V = (u , v , w,) (4) inert𝐕 =(−𝑉i i ,0,0) (6)

whichWith reference is related to to this the expression inertial reference and to systemEquation (x i(3),, y, zthei). Thefollowing speed relations vectors V holda and trueVinert for arethe connected airspeed by the wind speed vector Vw (Figure2), yielding 𝐕 =(𝑢+𝑉,𝑣,𝑤,) (7a) Va = Vinert − Vw (5)

The xi axis can be chosen to be in the opposite direction of the wind speed vector Vw (Figure2), with no loss of generality. Thus, the wind speed vector can be written as

T Vw = (−Vw, 0, 0) (6)

With reference to this expression and to Equation (3), the following relations hold true for the airspeed T Va = (ui + Vw, vi, w,) (7a)

and q 2 2 2 Va = (ui + Vw) + vi + wi (7b) Aerospace 2021, 8, x FOR PEER REVIEW 5 of 26

and

Aerospace 2021 8 , , 229 5 of 26 𝑉 = (𝑢+𝑉) +𝑣 +𝑤 (7b)

2.2.2.2. Energy Energy Model Model ForFor developing developing an an energy energy model model valid valid for thefor ﬂightthe flight mechanics mechanics of high-speed of high-speed dynamic dy- soaring,namic soaring, assumptions assumptions on the on trajectory the trajectory and speeds and speeds are made. are made. To this To this aim, aim, reference reference is madeis made to Figureto Figure3 which 3 which provides provides an obliquean oblique view view on on a dynamic a dynamic soaring soaring loop. loop. It It is is as- as- sumedsumed that that the the dynamic dynamic soaring soaring loop loop is is circular circular and and the the inclination inclination is is small. small. The The wind wind speedspeed is is supposed supposed to to be be smaller smaller than than the the inertial inertial speed speed and and the the airspeed airspeed by by an an order order of of magnitude,magnitude, yielding yielding Vw Vinert, Vw Va (8) 𝑉 ≪𝑉, 𝑉 ≪𝑉 (8) Furthermore, the shear layer thickness is assumed to be inﬁnitesimally small. Furthermore, the shear layer thickness is assumed to be infinitesimally small.

Figure 3. Oblique view on high-speed dynamic soaring loop. (The inclination of the trajectory rela- Figure 3. Oblique view on high-speed dynamic soaring loop. (The inclination of the trajectory tive to the horizontal is shown exaggerated for representation purposes). relative to the horizontal is shown exaggerated for representation purposes). When traversing the shear layer, there is an approximate speed increase of When traversing the shear layer, there is an approximate speed increase of (1/2)Vw cos γtr for(1/2)𝑉 the inertialcos𝛾 speed for the in inertial the leeward speed descent in the leeward (transition descent point (transition 1) point 1) 1 𝑉 =𝑉1+ 𝑉cos𝛾 (9) V = V + V2 cos γ (9) inert inert 2 w tr and for the airspeed in the leeward descent (transition point 2) and for the airspeed in the leeward descent (transition point 2) 1 𝑉 =𝑉 + 𝑉 cos𝛾 (10) 1 2 Va = Vinert + Vw cos γtr (10) where 𝑉 is the average inertial speed of 2the loop and 𝛾 is the flight path angle at the transition points. where Vinert is the average inertial speed of the loop and γtr is the ﬂight path angle at the The energy gain from the wind which is necessary for propelling the vehicle is transition points. achievedThe energy in the upper gain fromsection the of windthe loop which where is necessarythe wind is for blowing. propelling This themanifests vehicle in isan achievedincrease of in the the kinetic upper energy section between of the loop the two where transition the wind points is blowing. 1 and 2, yielding This manifests in an increase of the kinetic energy between the two transition points 1 and 2, yielding 𝑚 1 1 (11) 𝐸 =" 𝑉 + 𝑉 cos𝛾 −𝑉 − 𝑉 cos𝛾 =𝑚𝑉# 𝑉 cos𝛾 m 2 1 2 2 21 2 E = V + V cos γ − V − V cos γ = mV V cos γ (11) g 2 inert 2 w tr inert 2 w tr inert w tr

The kinetic energy increase, Eg, is equal to the total energy increase since there is no change in the potential energy. This is because the two transition points 1 and 2 are at the

same altitude. Furthermore, Eg is the net energy increase as the drag work is included in the difference of the total energy between the two transition points in the upper loop section. Aerospace 2021, 8, 229 6 of 26

The energy gain Eg compensates the drag work in the lower loop section. This is the requirement for energy-neutral dynamic soaring. The drag work in the lower loop section is given by

Z t2 WD = − DVinert dt (12) t1

where t1 and t2 refer to the beginning and end of the lower loop section (associated with the transition points 1 and 2). The drag work expression can be expanded using the following relation for the drag

C C D = D L = D nmg (13) CL CL

and assuming that the CD/CL ratio and the load factor n are constant. Thus

CD WD = − nmgs (14) CL where s = πRcyc (15) is the length of the lower loop section. The load factor in high-speed dynamic soaring is of the order of n = 100 [5,6,9]. For this n level, the following approximate relation holds true

2 V n = inert (16) Rcycg

Thus, the drag work, Equation (14), can be expressed as

CD 2 WD = −π mVinert (17) CL Taking the balance of the drag work and the energy gain into account

Eg + WD = 0 (18)

and using Equation (11), the average inertial speed of the cycle can be determined to yield

1 CL Vinert = Vw cos γtr (19) π CD The speed relationships presented in Figure3 show that the highest speed in the dynamic soaring cycle is in the leeward descent at the transition point 1, as given by Equation (9). Taking account of this and the fact, that Vinert reaches its greatest value at the maximum lift- to-drag ratio (CL/CD)max according to Equation (19), the following result on the maximum speed is obtained (|γtr| 1) 1 1 CL Vinert,max = + Vw (20) 2 π CD max This is an analytic solution for the maximum speed so that generally valid results concerning the effects of wing sweep can be achieved. Analyzing the relation Equation (20), the key factors for the maximum speed Vinert,max can be identiﬁed, yielding

(1) wind speed, Vw (2) maximum lift-to-drag ratio, (CL/CD)max Aerospace 2021, 8, 229 7 of 26

This result shows that there are two key factors. Only one of them is relating to the vehicle. This concerns the aerodynamic characteristics of the vehicle in terms of the maximum lift-to-drag ratio, (CL/CD)max. It is notable that there is no other vehicle feature (size, mass, etc.) that has an effect on Vinert,max. Regarding the wing sweep problem under consideration, the relation Equation (20) shows that (CL/CD)max is determinative. This means for the aerodynamics inﬂuence on the maximum speed that there is only one aerodynamic quantity in terms of (CL/CD)max that has an effect. As a result, the effect of wing sweep on the speed performance enhancement in high-speed dynamic soaring is determined by the dependence of (CL/CD)max on wing sweep.

2.3. Straight Wing Reference Configuration (Aerodynamics, Size and Mass Properties) A straight wing configuration which is representative for modern high-speed gliding vehicles is regarded as a reference and, therefore, will be dealt with first. This is because of two reasons: One reason is to show that there is a limitation of the achievable maximum speed for modern high-speed gliding vehicles which have a straight wing configuration. This limitation is due to compressibility in the high subsonic Mach number region by causing a substantial increase in the drag. The other reason is that the energy model can be validated with results obtained applying the 3-DOF dynamics model. Thus, the energy model can be used to show the performance enhancement achievable with wing sweep for high-speed dynamic soaring. The aerodynamic forces L and drag D used in the 3-DOF dynamics model and in the energy model can be expressed as

2 D = CD(ρ/2)Va S

2 L = CL(ρ/2)Va S (21) The compressibility-related drag rise in the high subsonic Mach number region manifests in the drag coefficient CD. Usually, for gliders, the drag coefficient CD shows a dependence only on the lift coefficient CL, but not on the Mach number Ma, i.e., CD = CD(CL). This is because the speed of glider type vehicles is comparatively low and associated with the incompressible Mach number region. Concerning high-speed dynamic soaring, however, modern gliding vehicles fly at speeds that can reach the upper subsonic Mach number region. Here, compressibility exerts substantial effects on the aerodynamic characteristics of aerial vehicles, including gliders [9]. An important effect of compressibility in this Mach number region for the performance problem under consideration is a significant rise in the drag. This compressibility-related drag rise manifests in the drag coefficient such that CD shows a dependence on the Mach number Ma, in addition to that of the lift coefficient CL, yielding

CD = CD(CL, Ma) (22)

For the high-speed gliding vehicle dealt with in this paper, the drag modelling is graphically addressed in Figure4. The drag characteristics are shown in terms of drag polars which present the drag coefficient depending on the lift coefficient and the Mach number. This includes the modelling of the induced drag which is based on a quadratic 2 drag-lift relationship, CD = CL/(eΛ). The term e is the Oswald efficiency factor for which a value of e = 0.9 is regarded appropriate for the wing under consideration [21]. Furthermore, the contributions of the fuselage and the tail to the drag at zero lift are included in the drag modelling. This contribution amounts to 15 % of the zero-lift drag. Further modelling data concern the wing reference area, S = 0.51 m2, the aspect ratio, A = 22.5, and the vehicle mass, m = 8.5 kg. For the described modelling including the drag related compressibility effects, reference is made to existing vehicles and to experience in this field [6,8,22]. Aerospace 2021, 8, x FOR PEER REVIEW 8 of 26

Aerospace 2021, 8, 229 the vehicle mass, 𝑚=8.5 kg. For the described modelling including the drag related8 ofcom- 26 pressibility effects, reference is made to existing vehicles and to experience in this field [6,8,22].

Figure 4. Drag coefficient of straight wing configuration dependent on lift coefficient, 𝐶 and Mach Figure 4. Drag coefficient of straight wing configuration dependent on lift coefficient, CL and Mach number, 𝑀𝑎, [7]. number, Ma,[7].

TheThe energy energy model, model, Equation Equation (20), (20), shows shows that that there there is is only only one one aerodynamics aerodynamics quantity quantity thatthat is is determinative determinative for for the the effect effect of of wing wing sweep sweep on on the the maximum maximum speed speed performance. performance. This is the maximum lift-to-drag ratio (𝐶 /𝐶 ) . Thus, the dependence of (𝐶 /𝐶 ) This is the maximum lift-to-drag ratio (CL/CD)max. Thus, the dependence of (CL/CD)max onon the the Mach Mach number numberMa 𝑀𝑎is determinative is determinative for for the the wing wing sweep sweep problem problem under under consideration. consideration. The dependence of (𝐶 /𝐶 ) on 𝑀𝑎 can be determined by an examination of the The dependence of (CL/CD)maxonMa can be determined by an examination of the drag polarsdrag polars depicted depicted in Figure in Figure4. 4. (𝐶 /𝐶 ) 𝑀𝑎 ResultsResults are are presented presented in in Figure Figure5 where 5 where(CL /CD)max dependent dependent on Ma on is plotted. is plotted. The (𝐶 /𝐶 ) (CTheL/ CD)max curve curve has a shapehas a shape that can that be can subdivided be subdivided into two into parts. two parts. In the In left the curve left curve part, (Cpart,L/C D(𝐶)max/𝐶is) constant is constant and shows and theshows highest the highest level. Constancy level. Constancy of (CL/ CofD )(𝐶max/𝐶with) regard with toregardMa implies to 𝑀𝑎 that implies there isthat no there effect is of no compressibility. effect of compressibility. Thus, the left Thus, part of the the left(C Lpart/CD of)max the curve(𝐶/𝐶 can) be curverelated can to be the related incompressible to the incompressibleMa region. In𝑀𝑎 the region. right In curve the right part, curve there part, is a continualthere is a decreasecontinual of decrease(CL/CD of)max (𝐶which/𝐶) rapidly which leads rapidly to small leads values. to small This values. decrease This de- of Aerospace 2021, 8, x FOR PEER REVIEW(CcreaseL /CD )ofmax (𝐶is/𝐶 an effect) is of an compressibility, effect of compressibility, associated with associated the drag with rise describedthe drag rise above.9 ofde- 26

Thus,scribed the above. right part Thus, of thethe( rightCL/C Dpart)max ofcurve the (𝐶 can/𝐶 be) related curve to the can compressible be related toMa theregion. compressible 𝑀𝑎 region. ∗ The lift coefficient associated with (𝐶/𝐶), denoted by 𝐶 , is also presented in ∗ Figure 5. The relation between 𝐶 and (𝐶/𝐶) is given by ∗ 𝐶 𝐶 (𝑀𝑎) = ∗ (23) 𝐶 𝐶𝐶 (𝑀𝑎) ∗ The 𝐶 curve has a shape that can also be subdivided into two parts. The left curve ∗ part where 𝐶 is constant and highest, can be related to the incompressible 𝑀𝑎 region. ∗ In the right curve part, 𝐶 shows a decrease to approach considerably smaller values which is due to compressibility. Thus, the right curve part can be associated with the compressible 𝑀𝑎 region.

∗ Figure 5. Maximum lift-to-drag ratio, (𝐶/𝐶) and associated lift coefficient, ∗𝐶 , dependent on Figure 5. Maximum lift-to-drag ratio, (CL/CD) and associated lift coefﬁcient, C , dependent on Mach number, 𝑀𝑎, (relating to drag polar in Figuremax 4. L Mach number, Ma, (relating to drag polar in Figure4.

3. Maximum Speed Achievable with Straight-Wing Configuration The objective of this Chapter is to validate the energy model developed in the previous Chapter. Thus, the energy model can be used to produce results on the maximum- speed performance. For validating the energy model, results achieved with the 3-DOF dynamics model are used. These results were produced applying the optimization method described in the Appendix A.

3.1. Maximum-Speed Performance of Straight Wing Reference Configuration As a reference for the wing sweep problem under consideration, the straight wing configuration described above is dealt with first. An objective is to show characteristic features of the trajectory and relevant motion variables in maximum-speed dynamic soaring. Another objective is to address the effect of compressibility, especially with regard to its limiting influence. For this purpose, a high wind speed scenario showing a wind speed of 𝑉, = 30 m/s is selected, with the result that compressibility is effective throughout the entire dynamic soaring cycle. Furthermore, the treatment of this section is also intended to give an illustrative insight into high-speed dynamic soaring in terms of a highly dynamic flight maneuver. Results are presented in Figure 6 which provides a perspective view on the optimized closed-loop trajectory. The maximum speed is obtained as 𝑉, = 271.8 m/s. The picture shows the spatial extension of the trajectory in the longitudinal, lateral and vertical directions and its relation relative to the wind field regarding the wind direction and the shear layer. The trajectory point where 𝑉, occurs is at about the end of the upper curve. This corresponds with the energy model.

Aerospace 2021, 8, 229 9 of 26

∗ The lift coefﬁcient associated with (CL/CD)max, denoted by CL, is also presented in ∗ Figure5. The relation between CL and (CL/CD)max is given by ∗ CL CL(Ma) = ∗ (23) CD max CD CL(Ma) ∗ The CL curve has a shape that can also be subdivided into two parts. The left curve ∗ part where CL is constant and highest, can be related to the incompressible Ma region. In ∗ the right curve part, CL shows a decrease to approach considerably smaller values which is due to compressibility. Thus, the right curve part can be associated with the compressible Ma region.

3. Maximum Speed Achievable with Straight-Wing Conﬁguration The objective of this Chapter is to validate the energy model developed in the previous Chapter. Thus, the energy model can be used to produce results on the maximum- speed performance. For validating the energy model, results achieved with the 3-DOF dynamics model are used. These results were produced applying the optimization method described in the AppendixA.

3.1. Maximum-Speed Performance of Straight Wing Reference Configuration As a reference for the wing sweep problem under consideration, the straight wing configuration described above is dealt with first. An objective is to show characteristic features of the trajectory and relevant motion variables in maximum-speed dynamic soaring. Another objective is to address the effect of compressibility, especially with regard to its limiting influence. For this purpose, a high wind speed scenario showing a wind speed of Vw,re f = 30 m/s is selected, with the result that compressibility is effective throughout the entire dynamic soaring cycle. Furthermore, the treatment of this section is also intended to give an illustrative insight into high-speed dynamic soaring in terms of a highly dynamic flight maneuver. Results are presented in Figure6 which provides a perspective view on the optimized closed-loop trajectory. The maximum speed is obtained as Vinert,max = 271.8 m/s. The picture shows the spatial extension of the trajectory in the longitudinal, lateral and vertical directions and its relation relative to the wind field regarding the wind direction and the shear layer. The trajectory point where Vinert,max occurs is at about the end of the upper curve. This corresponds with the energy model. Side and top views of the optimized trajectory are presented in Figure7. The side view shows that the trajectory projection comprises two lines which are nearly straight and close to each other. This means that the trajectory itself is virtually in a plane. Another feature is that the inclination of the trajectory is small. The top view reveals that the trajectory projection on the xi-yi plane shows a circular-like shape. This feature and the small inclination suggest that the circular characteristic also holds true for the trajectory itself. Furthermore, the side view shows that the extensions of the trajectory above and below the middle plane of the shear layer are practically equal. Aerospace 2021, 8, 229 10 of 26 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 26

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Figure 6. Dynamic soaring loop optimized for maximum speed, 𝑉, = 271.8 m/s. Figure 6. Dynamic soaring loop optimized for maximum speed, Vinert,max = 271.8 m/s. Side and top views of the optimized trajectory are presented in Figure 7. The side view shows that the trajectory projection comprises two lines which are nearly straight and close to each other. This means that the trajectory itself is virtually in a plane. Another feature is that the inclination of the trajectory is small. The top view reveals that the trajectory projection on the 𝑥 -𝑦 plane shows a circular-like shape. This feature and the small inclination suggest that the circular characteristic also holds true for the trajectory itself. Furthermore, the side view shows that the extensions of the trajectory above and below the middle plane of the shear layer are practically equal. The described trajectory characteristics are in accordance with the assumptions made for the trajectory of the energy model, as presented in Figure 3. This accordance contributes to the validation of the energy model.

Figure 7. Side and top views of dynamic soaring loop optimized for maximum speed, 𝑉 = Figure 7. Side and top views of dynamic soaring loop optimized for maximum, speed, 271.8 m/s. Vinert,max = 271.8 m/s. Compressibility exerts an essential influence on the maximum-speed performance for wind speeds of the current level. An insight on how this influence becomes effective can be gained considering the speed and Mach number characteristics. For this purpose, the inertial speed, 𝑉 and the Mach number, 𝑀𝑎, are addressed by presenting their time histories in Figure 8. The speed curve (the time scale is chosen such that the maximum speed 𝑉, = 271.8 m/s is at the beginning 𝑡=0) shows that 𝑉 features one oscillation the downward part of which is longer than the upward part. The minimum of 𝑉 is at a point of the trajectory where the vehicle has passed the lowest altitude and is climbing upwind again. The role of compressibility for the problem under consideration becomes evident addressing the Mach number 𝑀𝑎 reached during the cycle and putting this in relation with the drag polar characteristics of the vehicle. The time history of 𝑀𝑎 presented in Figure 8 shows that the 𝑀𝑎 curve comprises two oscillations which are rather similar. Further- more, the 𝑀𝑎 level is so high that it relates to the compressible region of the drag coefficient 𝐶 throughout the entire cycle. This means with reference to Figure 4 that the drag rise due to compressibility is fully effective. Thus, the maximum-speed performance is negatively influenced at all points of the trajectory.

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The described trajectory characteristics are in accordance with the assumptions made for the trajectory of the energy model, as presented in Figure3. This accordance contributes to the validation of the energy model. Compressibility exerts an essential inﬂuence on the maximum-speed performance for wind speeds of the current level. An insight on how this inﬂuence becomes effective can be gained considering the speed and Mach number characteristics. For this purpose, Aerospace 2021, 8, x FOR PEER REVIEWthe inertial speed, Vinert and the Mach number, Ma, are addressed by presenting their12 time of 26 histories in Figure8.

Figure 8. Time histories of speeds, 𝑉 and 𝑉, and Mach number, 𝑀𝑎, of dynamic soaring loop Figure 8. Time histories of speeds, Vinert and Va, and Mach number, Ma, of dynamic soaring loop optimized for maximum speed, 𝑉, = 271.8 m/s. optimized for maximum speed, Vinert,max = 271.8 m/s. 3.2. Maximum-Speed Performance and Related Key Factors The speed curve (the time scale is chosen such that the maximum speed Vinert,max = 271.8 m/sThe ismaximum at the beginning speed achievablet = 0) shows in high-speed that Vinert features dynamic one soaring, oscillation 𝑉, the downward, is central partfor ofthe which subject is under longer consideration than the upward in this part. pa Theper. minimumThere are, ofaccordingVinert is atto aEquation point of the(20), trajectorytwo key wherefactors the for vehicle 𝑉, has: passedthe wind the speed, lowest 𝑉 altitude, and the and maximum is climbing lift-to-drag upwind again. ratio, (𝐶The/𝐶) role . ofBecause compressibility of their forkey thefactor problem role, underthe effects consideration of 𝑉 and becomes (𝐶/𝐶 evident) on addressing𝑉,, thewill Mach be analyzed number Main detail.reached Emphasis during the is put cycle on and the putting role of this (𝐶 in/𝐶 relation) as with this thekey drag factor polar is determinative characteristics for of thethe vehicle.effect of Thewing time sweep history on the of Ma maximumpresented speed in Figure achieva-8 showsble. This that analysis the Ma curveis considered comprises a means two oscillations for validating which the are energy rather model. similar. Furthermore, the MaResultslevel is on so the high maximum that it relates speed to are the presented compressible in Figure region 9 of where the drag the coefﬁcientdependenceCD of throughout𝑉, on the 𝑉 entire is shown. cycle. The This range means selected with reference for 𝑉 is to such Figure as 4to that cover the a dragwide rise spectrum due toof compressibility wind speeds possible is fully at effective. ridges, Thus,with the the objective maximum-speed to achieve performance results generally is negatively valid for inﬂuencedhigh-speed at dynamic all points soaring. of the trajectory. The dotted line curve shows the results using the 3-DOF dynamics model. Concern- 3.2. Maximum-Speed Performance and Related Key Factors ing this model, the optimization procedure that is described in the Appendix A was ap- pliedThe to maximumachieve solutions speed achievablefor determining in high-speed the maximum dynamic speed. soaring, The solidVinert line, max curve, is central shows forthe the results subject using under the consideration energy model. in Concerning this paper. Therethis model, are, according solutions towere Equation constructed (20), two key factors for V : the wind speed, V , and the maximum lift-to-drag ra- using Equation (20). Bothinert, max𝑉, curves are closew to each other and overlap in parts, to tio,the( CeffectL/C Dthat)max the. Because results of of the their 3-DOF key factordynamics role, model the effects and the of Venergyw and model(CL/C approachesD)max on Vvirtuallyinert, max, willagree. be This analyzed suggests in detail.that the Emphasis energy model is put Equation on the role (20) of is( confirmed.CL/CD)max as this key factorThe isfact determinative that both model for the approaches effect of wing virtua sweeplly agree on the manifests maximum in more speed aspects. achievable. This Thiswill analysis be shown is considered in the following a means sections for validating where a the detailed energy treatment model. on these aspects is presented.Results on the maximum speed are presented in Figure9 where the dependence of Vinert, max on Vw is shown. The range selected for Vw is such as to cover a wide spectrum Examining the 𝑉, curve characteristics in Figure 9, the left part of the curve ofshows wind a speeds comparatively possible atlarge ridges, and withconstant the objectivegradient, towhereas achieve the results gradient generally in the validright forpart high-speed dynamic soaring. is substantially reduced. Relating this property of the 𝑉, gradient to the Mach number 𝑀𝑎 indicated on the ordinate shows that the change in the gradient starts in a zone where compressibility begins to become effective for the aerodynamic characteristics of the vehicle. Thus, the left part of the 𝑉, curves can be related to the incompressible 𝑀𝑎 region, whereas the right part can be related to the compressible one.

Aerospace 2021, 8, 229 12 of 26 Aerospace 2021, 8, x FOR PEER REVIEW 13 of 26

Figure 9. Maximum speed, 𝑉,, dependent on wind speed, 𝑉. Figure 9. Maximum speed, Vinert,max, dependent on wind speed, Vw.

TheThe dotted reason line for curvethe fact shows that thethere results are two using different the 3-DOF parts dynamics in the 𝑉, model. Concerning curve is due thisto (𝐶 model,/𝐶) the. optimization In the left part procedure of the 𝑉, that is described curve, (𝐶 in/𝐶 the) Appendix is constantA was so applied that, ac- tocording achieve to solutions Equation for (20), determining the 𝑉, the maximumgradient is speed. also constant, The solid whereas line curve the shows 𝑉, the resultsgradient using in the the right energy curve model. part Concerningis substantia thislly reduced. model, solutions The cause were for the constructed reduction using of the 𝑉 (𝐶 /𝐶 ) (𝐶 /𝐶 ) Equation, (20). gradient Both isVinert due, max to curves are close. This to is each because other and overlap shows in parts, a decrease to the effectin the that compressible the results ofMach the 3-DOFnumber dynamics region. A modelccording and to the Equation energy model(20), a approachesreduction in virtually(𝐶/𝐶) agree. leads This to suggests a corresponding that the energy reduction model in 𝑉 Equation,. (20)This is means conﬁrmed. that (𝐶/𝐶) is theThe cause fact for that the both limitation modelapproaches in the achievable virtually 𝑉, agreemanifests values. in more aspects. This will beFurther shown evidence in the followingand understanding sections whereof the effect a detailed of (𝐶 treatment/𝐶) on on 𝑉, these aspects can be isprovided presented. when analyzing the actual lift-to-drag ratio, 𝐶/𝐶, during the dynamic soaring cycleExamining and comparing the Vinert this, max withcurve the characteristicsmaximum lift-to-drag in Figure ratio,9, the (𝐶 left/𝐶 part) of, of the the curve same showscycle. aResults comparatively on that issue large are and plotted constant in Figure gradient, 10. whereas the gradient in the right part isAs substantially each of 𝐶 reduced./𝐶 and Relating(𝐶/𝐶) this varies property during of the theVinert cycle,,max gradientthe average to the values Mach of numberthese quantitiesMa indicated are considered on the ordinate as representative. shows that the They change are ingiven the gradient by starts in a zone where compressibility begins to become effective for the aerodynamic characteristics of the 𝐶 1 𝐶 vehicle. Thus, the left part of the V curves= ∫ can be related d𝑡 to the incompressible Ma(24) inert𝐶 ,max 𝑡 𝐶 region, whereas the right part can be related to the compressible one. andThe reason for the fact that there are two different parts in the Vinert, max curve is due to (CL/CD)max. In the left part of the Vinert, max curve, (CL/CD)max is constant so that, 𝐶 1 𝐶 according to Equation (20), the V =gradient∫ is also constant, d𝑡 whereas the V 𝐶 inert, max 𝑡 𝐶 inert, max(25) gradient in the right curve part is substantially, reduced. The cause for the reduction of the VBothinert, maxquantitiesgradient have is due been to determined(CL/CD)max using. This the is because3-DOF dynamics(CL/CD) modelmax shows and athe decrease optimi- inzation the compressible method describes Mach in number the Appendix region. A. According to Equation (20), a reduction in (CL/CTheD)max resultsleads presented to a corresponding in Figure 10 reduction address in theVinert relationships,max. This means between that the(C Llift-to-drag/CD)max is the cause for the limitation in the achievable V values. ratios, (𝐶/𝐶), and (𝐶/𝐶) and the inertwind,max speed, 𝑉 . Basically, each of the Further evidence and understanding of the effect of (CL/CD) on V can be (𝐶/𝐶), and (𝐶/𝐶) curves shows two parts. The left partsmax featuringinert,max a constant providedlevel are whenassociated analyzing with the incompressible actual lift-to-drag Mach ratio, numberCL/CD region,, during whereas the dynamic the right soaring parts ( ) cycleinvolving and comparing a continual this decrease with the are maximum associated lift-to-dragwith the compressible ratio, CL/C region.D max, of the same cycle. Results on that issue are plotted in Figure 10. A main result is that the (𝐶/𝐶), and (𝐶/𝐶) curves are close to each other and overlap in parts. This means that the actual (𝐶/𝐶) values virtually agree with the maximum lift-to-drag ratio, (𝐶/𝐶),. These characteristics of the lift-to-drag ratios (𝐶/𝐶) and (𝐶/𝐶), are of importance for the energy model. This is because the fact that (𝐶/𝐶) virtually agrees

Aerospace 2021, 8, x FOR PEER REVIEW 14 of 26

Aerospace 2021, 8, 229 with (𝐶/𝐶), supports the assumption made for the energy model. Accordingly,13 of 26 it is assumed for the energy model that the maximum lift-to-drag ratio (𝐶/𝐶) holds true for the maximum speed 𝑉,, Equation (20).

Figure 10. Lift-to-drag ratio in maximum-speed dynamic soaring. (𝐶/𝐶): actual lift-to-drag ra- Figure 10. Lift-to-drag ratio in maximum-speed dynamic soaring. (CL/CD)av: actual lift-to-drag tio, 3-DOF dynamics model; (𝐶/𝐶),: maximum lift-to-drag ratio, 3-DOF dynamics model. ratio, 3-DOF dynamics model; (CL/CD)max,av: maximum lift-to-drag ratio, 3-DOF dynamics model.

The main role that (𝐶/𝐶) plays as the determinative aerodynamics quantity for As each of CL/CD and (CL/CD)max varies during the cycle, the average values of thesethe maximum quantities speed are considered is further assubstantiated representative. and Theyconfirmed are given by the by broader treatment presented in the following. For this purpose, three different (𝐶/𝐶) configurations are dealt with and their effects on 𝑉C are1 analyzedZ tcyc C and compared. The range selected ,L = L dt (24) for (𝐶/𝐶) is larger than theCD (𝐶av/𝐶)tcyc range0 C Dcharacteristic for high-speed gliding vehicles. and Results are presented in Figure 11 whichZ tshows the relationship between 𝑉,, CL 1 cyc CL 𝑉 and (𝐶/𝐶) where the notation= (𝐶/𝐶), refersd tto the maximum lift-to-drag(25) C t 0 C ratio configuration applied soD farmax and,av usedcyc as reference.D Themax results show that it can again Bothbe distinguished quantities have between been determined the left and using right the parts 3-DOF of each dynamics 𝑉, model curve and thewhich optimiza- are re- tionlated method to the describesincompressible in the Appendixand compressibleA. Mach number regions, respectively. In the left Thecurve results parts, presented there are inlarge Figure differences 10 address between the relationships the three 𝑉 between, thecurves lift-to-drag (for each ratios,of the (solidCL/C andD)max dotted,av and lines)(CL /thatCD )areav anddue theto the wind effect speed, of (𝐶Vw/𝐶. Basically,). These each differences of the ( ) ( ) CreflectL/CD themax ,strongav and influenceCL/CD av ofcurves (𝐶/𝐶 shows) on two 𝑉 parts., The, according left parts featuringto the energy a constant model levelEquation are associated (20). This with manifests the incompressible in a particular Mach manner number for region,the case whereas of the 50% the rightdecrease parts in involving(𝐶/𝐶) a continual as the related decrease 𝑉, are associated curve practically with the compressible does not reach region. the Mach number regionA main where result compressibility is that the (C isL/ effective.CD)max,av Fuandrther(CL to/C theD)av leftcurves curve are parts, close comparing to each other the andsolid overlap lines holding in parts. true This for means the energy that the model actual and(CL /theCD dotted)av values lines virtually holding agree true withfor the the 3- maximumDOF dynamics lift-to-drag model ratio, shows(C Lthat/C Dthey)max virtually,av. agree. These characteristics of the lift-to-drag ratios (C /C ) and (C /C ) are of Concerning the right curve parts, there are two effectsL D validav for theL 𝑉D,max,av curves importance for the energy model. This is because the fact that (CL/CD) virtually agrees of both (𝐶/𝐶) cases reaching the compressible Mach number avregion. Firstly, the with (CL/CD) supports the assumption made for the energy model. Accordingly, it 𝑉, curvesmax,av (for each pair of solid and dotted lines) show no longer a large difference is assumed for the energy model that the maximum lift-to-drag ratio (CL/CD) holds but are close to each other. Secondly, the increase of 𝑉, of both (𝐶/𝐶max) cases true for the maximum speed Vinert,max, Equation (20). is substantially reduced and there is practically a limitation of the achievable 𝑉, The main role that (CL/CD)max plays as the determinative aerodynamics quantity values. These two effects are the result of the decrease of (𝐶/𝐶) due to compressi- for the maximum speed is further substantiated and conﬁrmed by the broader treatment bility (Figure 5). This means that (𝐶/𝐶) plays the primary role in limiting the achiev- presented in the following. For this purpose, three different (CL/CD)max conﬁgurations able 𝑉, values. are dealt with and their effects on Vinert,max are analyzed and compared. The range selected for (CL/CD)max is larger than the (CL/CD)max range characteristic for high-speed gliding vehicles.

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Results are presented in Figure 11 which shows the relationship between Vinert, max, Vw and (CL/CD)max where the notation (CL/CD)max,re f refers to the maximum lift-to-drag ratio configuration applied so far and used as reference. The results show that it can again be distinguished between the left and right parts of each Vinert,max curve which are related to the incompressible and compressible Mach number regions, respectively. In the left curve parts, there are large differences between the three Vinert,max curves (for each of the solid and dotted lines) that are due to the effect of (CL/CD) . These differences Aerospace 2021, 8, x FORmax PEER REVIEW 15 of 26 Aerospace 2021, 8, x FOR PEER REVIEWreflect the strong influence of (CL/CD)max on Vinert , max, according to the energy model15 of 26 Aerospace 2021, 8, x FOR PEER REVIEW 15 of 26 Equation (20). This manifests in a particular manner for the case of the 50% decrease in (CL/CD)max as the related Vinert,max curve practically does not reach the Mach number region where compressibility is effective. Further to the left curve parts, comparingHere, the again, the findings of the energy model agree with those of the 3-DOF dynam- Here, again, the findings of the energy model agree with those of the 3-DOF dynam- solidHere, lines again, holding the findings true for of the the energy energy model model and agree the with dotted thos linese of the holding 3-DOF trueics dynam- model. for the This is particular important for the compressible Mach number region because ics model. This is particular important for the compressible Mach number region because ics3-DOF model. dynamics This is particular model shows important that they for the virtually compressible agree. Mach number regionthe because limitation in 𝑉, is related to that Mach number region. the limitation in 𝑉, is related to that Mach number region. the limitation in 𝑉, is related to that Mach number region.

Figure 11. Maximum speed, 𝑉,, dependent on wind speed, 𝑉 and on maximum lift-to-drag Figure 11. Maximum speed, 𝑉,, dependent on wind speed, 𝑉 and on maximum lift-to-drag FigureFigure 11. 11. MaximumMaximum speed, speed, 𝑉 Vinert,max, dependent, dependent on on wind wind speed, speed, 𝑉 V w andand on on maximum maximum lift-to-dragratio, lift-to-drag (𝐶 /𝐶 ) . Results obtained using the energy model, Equation (20). ratio, (𝐶 /𝐶 ) . , Results obtained using the energy model, Equation (20). ratio,ratio, (𝐶(C/𝐶/C) ) . . Results Results obtained obtained using using theenergy the energy model, Equationmodel, Equation (20). (20). Results obtained using the 3-DOF dynamics model. L D Resultsmax obtained using the 3-DOF dynamics model. Results obtained Results using obtained the 3-DOF using the dynamics 3-DOF model.dynamics model. In summary, the analysis in this Chapter shows that the results of the energy model In summary, the analysis in this Chapter shows that the results of the energy model InConcerning summary, the the analysis right curve in this parts, Chapter there sh areows two that effects the results valid for of the Venergyinert,virtuallymax modelcurves agree with the results of the 3-DOF dynamics model. The results on the maxi- virtually agree with the results of the 3-DOF dynamics model. The results on the maxi- virtuallyof both agree(CL/ CwithD)max thecases results reaching of the the3-DO compressibleF dynamics Machmodel. number The results region. on the Firstly,mum maxi- speed the and related quantities are close to each other and overlap in parts. This sug- mum speed and related quantities are close to each other and overlap in parts. This sug- mumVinert speed,max curves and related (for each quantities pair of solidare close and to dotted each lines)other showand overlap no longer in parts. a large Thisgests difference sug- that the energy model is validated so that it can be used in lieu of the 3-DOF dynam- gests that the energy model is validated so that it can be used in lieu of the 3-DOF dynam- gestsbut that are closethe energy to each model other. is Secondly, validated the so increasethat it can of beVinert used,max inof lieu both of the(CL 3-DOF/CD)maxics dynam- model.cases is ics model. icssubstantially model. reduced and there is practically a limitation of the achievable Vinert,max values. These two effects are the result of the decrease of (CL/CD)max due to compressibility4. Increase of Maximum-Speed Performance by Wing Sweep (Figure4. Increase5). This of meansMaximum-Speed that (C /C Performance) plays the by primaryWing Sweep role in limiting the achievable 4. Increase of Maximum-Speed LPerformanceD max by Wing Sweep 4.1. Aerodynamic Characteristics of Swept Wing Glider Configurations V4.1. Aerodynamicvalues. Characteristics of Swept Wing Glider Configurations 4.1. inertAerodynamic,max Characteristics of Swept Wing Glider Configurations Here, again, the ﬁndings of the energy model agree with those of the 3-DOF dynamicsThe results of the previous treatment on the limitation of 𝑉, for straight wing The results of the previous treatment on the limitation of 𝑉, for straight wing model.The results This is of particular the previous important treatment for the on compressible the limitation Mach of 𝑉, number for region straight becauseconfigurations wing the show that the decrease of (𝐶/𝐶) caused by compressibility is the rea- configurations show that the decrease of (𝐶/𝐶) caused by compressibility is the rea- configurations show that the decrease of (𝐶/𝐶) caused by compressibility isson the for rea- this limitation. The limitation begins to become effective at the Mach number at limitationson for this in Vlimitation.inert,max is relatedThe limitation to that Machbegins number to become region. effective at the Mach number at son for this limitation. The limitation begins to become effective at the Mach numberwhich theat compressibility-related drag rise begins. That drag rise is associated with two whichIn summary,the compressibility-related the analysis in this drag Chapter rise begins. shows That that thedrag results rise is of associated the energy with model two which the compressibility-related drag rise begins. That drag rise is associated withMach two numbers [21]. One is the critical Mach number, 𝑀𝑎 , which is the Mach number at virtuallyMach numbers agree with [21]. the One results is the of critical the 3-DOF Mach dynamics number, model. 𝑀𝑎 , Thewhich results is the on Mach the maximum number at Mach numbers [21]. One is the critical Mach number, 𝑀𝑎 , which is the Mach numberwhich theat speed of sound is reached at some point of the vehicle. The other one is the drag speedwhich and the related speed of quantities sound is are reached close toat eachsome other point and of the overlap vehicle. in parts.The other This one suggests is the thatdrag whichthe energy the speed model of sound is validated is reached so that at some it can point be used of the in lieuvehicle. of the The 3-DOF other dynamicsone isdivergence the model.drag Mach number, 𝑀𝑎, which is the Mach number at which the drag coefficient divergence Mach number, 𝑀𝑎, which is the Mach number at which the drag coefficient divergence Mach number, 𝑀𝑎, which is the Mach number at which the drag coefficientstarts to rise rapidly. 𝑀𝑎 is slightly greater than 𝑀𝑎 . starts to rise rapidly. 𝑀𝑎 is slightly greater than 𝑀𝑎 . starts to rise rapidly. 𝑀𝑎 is slightly greater than 𝑀𝑎 . A possible solution of the drag rise problem is wing sweep. This is illustrated in Fig- A possible solution of the drag rise problem is wing sweep. This is illustrated in Fig- A possible solution of the drag rise problem is wing sweep. This is illustratedure in 12 Fig- where the lift-to-drag ratio, 𝐶 /𝐶 , of a swept wing configuration and of a corre- ure 12 where the lift-to-drag ratio, 𝐶 /𝐶 , of a swept wing configuration and of a corre- ure 12 where the lift-to-drag ratio, 𝐶 /𝐶 , of a swept wing configuration and ofsponding a corre- straight wing configuration representative for glider type vehicles are shown. sponding straight wing configuration representative for glider type vehicles are shown. sponding straight wing configuration representative for glider type vehicles areThe shown. planform of the swept wing configuration is presented in Figure 13. The planform of the swept wing configuration is presented in Figure 13. The planform of the swept wing configuration is presented in Figure 13. Figure 12 shows the Mach number region where compressibility causes the 𝐶 /𝐶 Figure 12 shows the Mach number region where compressibility causes the 𝐶 /𝐶 Figure 12 shows the Mach number region where compressibility causes thedecrease 𝐶 /𝐶 of the two wing configurations. Concerning the straight wing configuration, the decrease of the two wing configurations. Concerning the straight wing configuration, the decrease of the two wing configurations. Concerning the straight wing configuration,𝐶 /𝐶 the decrease begins at about 𝑀𝑎 ≈ 0.7 ÷ 0.75 and the 𝐶 /𝐶 ratio shows thereafter a 𝐶 /𝐶 decrease begins at about 𝑀𝑎 ≈ 0.7 ÷ 0.75 and the 𝐶 /𝐶 ratio shows thereafter a 𝐶 /𝐶 decrease begins at about 𝑀𝑎 ≈ 0.7 ÷ 0.75 and the 𝐶 /𝐶 ratio shows thereaftersteep gradient a to reach rapidly smaller values. Concerning the swept wing configuration, steep gradient to reach rapidly smaller values. Concerning the swept wing configuration, steep gradient to reach rapidly smaller values. Concerning the swept wing configuration,the 𝐶 /𝐶 decrease begins at about 𝑀𝑎 ≈ 0.81 ÷ 0.83. Thereafter, the drag decrease the 𝐶 /𝐶 decrease begins at about 𝑀𝑎 ≈ 0.81 ÷ 0.83. Thereafter, the drag decrease the 𝐶 /𝐶 decrease begins at about 𝑀𝑎 ≈ 0.81 ÷ 0.83. Thereafter, the drag showsdecrease a behavior corresponding with that of the straight wing configuration in terms of shows a behavior corresponding with that of the straight wing configuration in terms of shows a behavior corresponding with that of the straight wing configuration ina terms similarly of steep gradient. Comparing the two cases provides an indication of the possible a similarly steep gradient. Comparing the two cases provides an indication of the possible a similarly steep gradient. Comparing the two cases provides an indication of theshift possible in the drag rise to higher Mach numbers for wings of glider type vehicles. shift in the drag rise to higher Mach numbers for wings of glider type vehicles. shift in the drag rise to higher Mach numbers for wings of glider type vehicles.

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4. Increase of Maximum-Speed Performance by Wing Sweep 4.1. Aerodynamic Characteristics of Swept Wing Glider Conﬁgurations

The results of the previous treatment on the limitation of Vinert,max for straight wing configurations show that the decrease of (CL/CD)max caused by compressibility is the reason for this limitation. The limitation begins to become effective at the Mach number at which the compressibility-related drag rise begins. That drag rise is associated with two Mach numbers [21]. One is the critical Mach number, Macr, which is the Mach number at which the speed of sound is reached at some point of the vehicle. The other one is the drag divergence Mach number, Madd, which is the Mach number at which the drag coefficient starts to rise rapidly. Madd is slightly greater than Macr. A possible solution of the drag rise problem is wing sweep. This is illustrated in Figure 12 where the lift-to-drag ratio, CL/CD, of a swept wing configuration and of a Aerospace 2021, 8, x FOR PEER REVIEWcorresponding straight wing configuration representative for glider type vehicles16 areof 26 Aerospace 2021, 8, x FOR PEER REVIEWshown. The planform of the swept wing configuration is presented in Figure 13. 16 of 26

FigureFigure 12. 12.Effect Effect of of Mach Mach number, number,Ma 𝑀𝑎, on, on lift-to-drag lift-to-drag ratio ratio of of glider glider model model wings wings (from (from [23 [23]).]). Figure 12. Effect of Mach number, 𝑀𝑎, on lift-to-drag ratio of glider model wings (from [23]).

Figure 13. Planform of swept wing configuration used for Figure 12 (from [23]). Figure 13. Planform of swept wing configuration used for Figure 12 (from [23]). Figure 13. Planform of swept wing configuration used for Figure 12 (from [23]). The following assumptions are made for determining the effects of wing sweep on The following assumptions are made for determining the effects of wing sweep on the Figuremaximum-speed 12 shows theperformance Mach number of high-speed region where dynamic compressibility soaring: causes the CL/CD the maximum-speed performance of high-speed dynamic soaring: decrease(1) Energy of the based two wingmodel configurations. Concerning the straight wing configuration, the C(1)L/ CDEnergydecrease based begins model at about Ma ≈ 0.7 ÷ 0.75 and the CL/CD ratio shows thereafter a It is assumed that energy model for high-speed dynamic soaring, Equation (20), can steep gradient to reach rapidly smaller values. Concerning the swept wing configuration, be usedIt is for assumed determining that energy the effects model of wingfor high-s sweeppeed on dynamicthe maximum soaring, speed, Equation 𝑉, (20),. This can the CL/CD decrease begins at about Ma ≈ 0.81 ÷ 0.83. Thereafter, the drag decrease shows be used for determining the effects of wing sweep on the maximum speed, 𝑉,. This ais behavior based on corresponding the results concerning with thatof the the validation straight wing of the configuration energy model. in terms of a similarly is basedThe on energy the results model concerning Equation (20) the showsvalidation that ofthe the (𝐶 energy/𝐶) model. is the only aerodynamic (𝐶 /𝐶 ) quantityThe energythat has model an effect Equation on 𝑉, (20) shows. Thus, that the the effect of wing sweep is the only on the aerodynamic speed per- quantityformance that enhancement has an effect in onhigh-speed 𝑉, dynamic. Thus, the soaring effect ofcan wing be determined sweep on the by speedtaking per- the formancedependence enhancement of (𝐶/𝐶) in high-speed on wing sweepdynamic into soaring account can and be determinedby incorporating by taking this thede- dependencependence in Equationof (𝐶/𝐶 (20).) For on thatwing purpose, sweep theinto critical account Mach and number by incorporating and its relationship this de- pendencewith the wing in Equation sweep is(20). used For to that address purpose, the dependence the critical Mach of (𝐶 number/𝐶) and on its wing relationship sweep. with the wing sweep is used to address the dependence of (𝐶/𝐶) on wing sweep. (2) Critical Mach number, 𝑀𝑎 (2) Critical Mach number, 𝑀𝑎 For the critical Mach number, the following relation is used [19,24,25] For the critical Mach number, the following relation is used [19,24,25] 𝑀𝑎 𝑀𝑎 = , (26) , 𝑀𝑎, 𝑀𝑎 = cosΛ (26) , cosΛ where 𝑀𝑎, refers to the swept wing featuring a sweep angle of 𝛬 and 𝑀𝑎, refers to wherethe straight 𝑀𝑎, wing. refers to the swept wing featuring a sweep angle of 𝛬 and 𝑀𝑎, refers to the straightThe relation wing. described by Equation (26) is considered as suitable for wings of a large aspectThe ratio. relation This describedis because by wings Equa oftion a large(26) is aspect considered ratio show as suitable a characteristic for wings similarof a large to aspect ratio. This is because wings of a large aspect ratio show a characteristic similar to

Aerospace 2021, 8, 229 16 of 26

steep gradient. Comparing the two cases provides an indication of the possible shift in the drag rise to higher Mach numbers for wings of glider type vehicles. The following assumptions are made for determining the effects of wing sweep on the maximum-speed performance of high-speed dynamic soaring: (1) Energy based model It is assumed that energy model for high-speed dynamic soaring, Equation (20), can be used for determining the effects of wing sweep on the maximum speed, Vinert,max. This is based on the results concerning the validation of the energy model. The energy model Equation (20) shows that the (CL/CD)max is the only aerodynamic quantity that has an effect on Vinert,max. Thus, the effect of wing sweep on the speed performance enhancement in high-speed dynamic soaring can be determined by taking the dependence of (CL/CD)max on wing sweep into account and by incorporating this dependence in Equation (20). For that purpose, the critical Mach number and its relationship with the wing sweep is used to address the dependence of (CL/CD)max on wing sweep.

(2) Critical Mach number, Macr For the critical Mach number, the following relation is used [19,24,25]

Ma Ma = cr,0 (26) cr,Λ cos Λ

where Macr,Λ refers to the swept wing featuring a sweep angle of Λ and Macr,0 refers to the straight wing. The relation described by Equation (26) is considered as suitable for wings of a large aspect ratio. This is because wings of a large aspect ratio show a characteristic similar to airfoils the characteristic of which is determined by 2-D effects. Accordingly, the 3-D effects near the tip and the root of large aspect ratio wings tend to be small when compared with low aspect ratio wings. Large aspect ratios are generally used in aerodynamically efficient gliders [26]. This holds true also for gliders in high-speed dynamic soaring [10]. For the type of wings under consideration, the swept wing configuration shown in Figure 13 can be seen as representative and the swept wing configuration 1 (described below), shows an aspect ratio of Λ = 22.5.

(3) Maximum lift-to-drag ratio, (CL/CD)max

The relationship between (CL/CD)max and Ma shown in Figure5 holds true for the straight wing configuration. The shape of this curve is also used for the swept wing configuration 1 (Figure 14). For that purpose, it is assumed that the (CL/CD)max curve is horizontally shifted to higher Mach numbers according to the increase of Macr,Λ compared to Macr,0, using the relationship of Equation (26). This procedure is based on the following considerations: It is assumed that the swept wing configuration shows the same (CL/CD)max value as the straight wing configuration in the incompressible Mach number region. This assumption implies that the zero-lift drag coefficients of both wing configurations agree and that an analogue relationship between the lift-dependent drag coefficients holds true. For the equality of the lift-dependent drag coefficients, it is assumed that the spanwise lift distributions of the two wing configurations agree. This is considered to be achievable by an appropriate twist of the wing. A further aspect for the described procedure is that a main effect of wing sweep on Vinert,max concerns the beginning of the (CL/CD)max decrease where the compressibility- related drag rise starts to grow rapidly. This corresponds with the Vinert,max behavior at the transition of the left and right curve parts in the Vinert,max diagrams presented in Chapter 3. Here, the increase of Vinert,max comes nearly to a stop and there is a limitation for the achievable maximum speed. Aerospace 2021, 8, x FOR PEER REVIEW 17 of 26

airfoils the characteristic of which is determined by 2-D effects. Accordingly, the 3-D effects near the tip and the root of large aspect ratio wings tend to be small when compared with low aspect ratio wings. Large aspect ratios are generally used in aerodynamically efficient gliders [26]. This holds true also for gliders in high-speed dynamic soaring [10]. For the type of wings under consideration, the swept wing configuration shown in Figure 13 can be seen as representative and the swept wing configuration 1 (described below), shows an aspect ratio of 𝛬= 22.5.

(3) Maximum lift-to-drag ratio, (𝐶/𝐶)

The relationship between (𝐶/𝐶) and 𝑀𝑎 shown in Figure 5 holds true for the straight wing configuration. The shape of this curve is also used for the swept wing con- Aerospace 2021, 8, 229 figuration 1 (Figure 14). For that purpose, it is assumed that the (𝐶/𝐶) curve17 is of hor- 26 izontally shifted to higher Mach numbers according to the increase of 𝑀𝑎, compared to 𝑀𝑎,, using the relationship of Equation (26).

Figure 14. Swept wing configurations 1 and 2. Figure 14. Swept wing conﬁgurations 1 and 2.

4.2. SweptThis Wingprocedure Configurations is based on the following considerations: It is assumed that the swept wingFor configuration showing that shows and howthe same with (𝐶 wing/𝐶 sweep) value can enhanceas the straight the speed wing performance configuration inin high-speed the incompressible dynamic Mach soaring, number two region. swept This wing assumption configurations implies 1 and that 2 the schematically zero-lift drag presentedcoefficients in of Figure both 14wing are configurations dealt with. Each agree of theand two that configurations an analogue relationship features the between same sweepthe lift-dependent angles in terms drag of coefficientsΛ = 15◦ andholdsΛ true= 30. For◦.A the40 equality◦ sweep of angle the lift-dependent configuration drag is additionallycoefficients, addressed it is assumed in the that results the spanwise presented lift below. distributions The straight of the wing two dealt wing with configura- so far istions used agree. as reference This is forconsidered both swept to be wing achievable configurations. by an appropriate twist of the wing. SweptA further wing aspect configuration for the described 1 is supposed procedure to have theis that same a main span aseffect the referenceof wing sweep straight on wing.𝑉, Furthermore, concerns the it is beginning assumed of that the the (𝐶 wing/𝐶) areas decrease agree in where order the to keepcompressibility- the wing loadingrelated the drag same. rise Thestarts reason to grow is that rapidly. the wing This loading corresponds has significant with the effects𝑉, in high-speed behavior at dynamicthe transition soaring of sothe that left a and change right in curve this quantity parts in would the 𝑉, distort thediagrams comparison presented of the in sweptChapter and 3. straight Here, the wing increase configurations of 𝑉, [20 comes]. Equality nearly in to wing a stop span, andb thereandwing is a limitation area, S, impliesfor the thatachievable the aspect maximum ratio, Aspeed.= b2 /S, of the swept wing configuration 1 agrees with that of the reference straight wing configuration. With reference to this accordance and an appropriate4.2. Swept Wing spanwise Configurations lift distribution, it can be assumed that the two wing configurations have theFor same showing induced that drag.and how with wing sweep can enhance the speed performance in high-speedSwept wing dynamic configuration soaring, 2two is supposed swept wing to be configurations generated by rotating1 and 2 schematically the two halves pre- of thesented reference in Figure straight 14 are wing dealt until with. the Each intended of the angle two ofconfigurations sweep is reached. features Rotating the same the wingsweep halvesangles in in the terms described of 𝛬 manner = 15° and leads 𝛬 to = a30°. decreaseA 40° sweep in the angle wing configuration span according is additionally to

b2Λ = b0 cos Λ (27)

where b0 relates to the reference case. Regarding the wing area, it is assumed that there is no change so that swept wing area is the same as that of the reference wing. Thus, the wing loading agrees with that of the reference straight wing conﬁguration. The reason for keeping the wing loading unchanged is the same as said above. Decreasing the wing span and holding the wing area constant causes a reduction of the aspect ratio, yielding

2 A2Λ = A0 cos Λ (28)

where A0 relates to the reference case. The aspect ratio reduction leads to an increase of the induced drag when comparing the swept wing conﬁguration 2 with the reference straight wing conﬁguration. Aerospace 2021, 8, 229 18 of 26

4.3. Potential of Wing Sweep for Enhancing the Maximum-Speed Performance Based on the results on the validation of the energy model in the previous Chapters, it is assumed that this model is suitable for showing the potential of wing sweep for enhancing the maximum-speed performance. Thus, the maximum speed, Vinert,max, can be determined using Equation (20). For this purpose, the maximum lift-to-drag ratio Aerospace 2021, 8, x FOR PEER REVIEW(CL /CD)max of the wing conﬁguration under consideration is required as the decisive19 of 26

Aerospace 2021, 8, x FOR PEER REVIEWaerodynamic characteristic. On this basis, results and ﬁndings on the maximum speed19 of 26 are derived. Results are presented in Figures 15 and 16 which show the maximum speed, V , dependent on the wind speed, V , for the swept wing conﬁgurations 1 and 2 and inert,max30° and the 𝑉 curve for 𝛬w = 40° can serve as an indication for the perfor- for the straight wing, reference case. mance30° and potential the 𝑉, of larger curve sweep for angles. 𝛬 = 40° can serve as an indication for the performance potential of larger sweep angles.

Figure 15. Effect of wing sweep on maximum speed, 𝑉, , for swept wing configuration 1. Figure 15. Effect of wing sweep on maximum speed, Vinert,max, for swept wing conﬁguration 1. Figure 15. Effect of wing sweep on maximum speed, 𝑉,, for swept wing configuration 1.

Figure 16. Effect of wing sweep on maximum speed, 𝑉, , for swept wing configuration 2.

Figure 16. Effect of wing sweep on maximum speed, 𝑉,, for swept wing configuration 2. FigureFor 16. Effectswept of wing wing sweepconfiguration on maximum 2, there speed, is Valsoinert ,maxa possibility, for swept wingof enhancing conﬁguration the 2.maximum-speed performance. This is shown in Figure 16 which presents results on 𝑉 ResultsFor swept for the wing swept configuration wing conﬁguration 2, there 1is arealso provided a possibility by Figure of enhancing 15 which presentsthe, maxi- for swept wing configuration 2. Here, again,◦ it can◦ be◦ distinguished between left and𝑉 right Vmum-speedinert,max for three performance. sweep angle This cases is shown (Λ = 15in Figure, 30 , 40 16) which and for presents the straight results wing on reference , 𝑉, curve parts which relate to the incompressible and compressible Mach number cases.for swept Generally, wing configuration the four cases 2. show Here, as again, a characteristic it can be distinguished feature of the betweenVinert,max leftcurves and thatright regions. there𝑉, is a left curve curve parts part which with a relate large gradientto the incomp (the sameressible for alland cases) compressible that is associated Mach number with theregions. incompressibleA comparison Mach with number swept wing region configuration and a right curve 1 shows part that with the a reduced increase gradient in 𝑉, that is basically smaller. This holds true for the entire 𝑉 region. Furthermore, swept wing A comparison with swept wing configuration 1 shows that the increase in 𝑉, configuration 2 begins to become effective for increasing 𝑉 only at higher wind is basically smaller. This holds true for the entire 𝑉 region., Furthermore, swept wing speeds. In addition, swept wing configuration 2 shows smaller 𝑉 values than the configuration 2 begins to become effective for increasing 𝑉,, only at higher wind straight wing configuration in the incompressible Mach number region. This is due to the speeds. In addition, swept wing configuration 2 shows smaller 𝑉, values than the smallerstraight (𝐶wing/𝐶 configuration) values inin thatthe incompressibleregion which are Mach caused number by the region. higher This lift-dependent is due to the drag resulting from the lower aspect ratio. On the whole, swept wing configuration 2 is a smaller (𝐶/𝐶) values in that region which are caused by the higher lift-dependent lessdrag effective resulting means from forthe enhancinlower aspectg the ratio. maximum-speed On the whole, performance. swept wing configuration 2 is a less effective means for enhancing the maximum-speed performance. 5. Further Effects of Wing Sweep Important for High-Speed Dynamic soaring 5. Further Effects of Wing Sweep Important for High-Speed Dynamic soaring

Aerospace 2021, 8, 229 19 of 26

is associated with the compressible Mach number region. The right parts of the Vinert,max curves are of primary interest because the maximum-speed level is highest in these parts for each curve. The relationship between the four Vinert,max curves shows the effect of wing sweep on the achievable maximum speed. The main result is that wing sweep yields an enhancement of the maximum-speed performance by increasing, Vinert,max, when compared with the straight wing reference case. The effect of wing sweep can be rated as follows: − Sweep angles up to around 15◦ yield only a very small improvement. ◦ ◦ − The area between the Vinert,max curve for Λ = 15 and the Vinert,max curve for Λ = 30 shows that there are efficient possibilities for enhancing the maximum-speed perfor- ◦ mance by wing sweep. The area between the Vinert,max curve for Λ = 30 and the ◦ Vinert,max curve for Λ = 40 can serve as an indication for the performance potential of larger sweep angles. For swept wing configuration 2, there is also a possibility of enhancing the maximum- speed performance. This is shown in Figure 16 which presents results on Vinert,max for swept wing configuration 2. Here, again, it can be distinguished between left and right Vinert,max curve parts which relate to the incompressible and compressible Mach number regions. A comparison with swept wing configuration 1 shows that the increase in Vinert,max is basically smaller. This holds true for the entire Vw region. Furthermore, swept wing configuration 2 begins to become effective for increasing Vinert,max only at higher wind speeds. In addition, swept wing configuration 2 shows smaller Vinert,max values than the straight wing configuration in the incompressible Mach number region. This is due to the smaller (CL/CD)max values in that region which are caused by the higher lift-dependent drag resulting from the lower aspect ratio. On the whole, swept wing configuration 2 is a less effective means for enhancing the maximum-speed performance.

5. Further Effects of Wing Sweep Important for High-Speed Dynamic Soaring The problems dealt with in the previous Chapter are concerned with the speed performance enhancement achievable with wing sweep. There are other, non-performance topics that are important for high-speed dynamic soaring. One topic is the controllability of the vehicle or flyability at the extreme speed level which poses unique challenges, associated with short time periods [5,6,8]. Another topic relates to the loads acting on the vehicle which can be extraordinarily large because of rapid turns at extreme speeds possible in high-speed dynamic soaring, manifesting in large load factors [5,6]. A further topic is the air space required for dynamic soaring in terms of the shape and extension of the trajectory. The question is whether there are relationships between these topics and the wing configuration regarding straight or swept wing shapes. The problems associated with these non-performance topics can be dealt with addressing the following quantities: − cycle time − load factor − trajectory radius The treatment of the addressed topics in the following Sections is concerned with swept wing configuration 1 as this is the more effective means for enhancing the maximum- speed performance. The energy model has been validated so that it can be used in lieu of the 3-DOF dynamics model. Thus, the energy model is suitable for analyzing the addressed non- performance topics.

5.1. Optimal Cycle Time The following expression generally holds true for the time completing a trajectory loop cycle 2πRcyc tcyc = (29) Vinert Aerospace 2021, 8, 229 20 of 26

Applying the load factor relation in curved ﬂight

2 L C (ρ/2)V S n = = L inert (30) mg mg

and accounting for Equation (16), the cycle time of a loop in high-speed dynamic soaring can be expressed as m tcyc = 2π (31) CL(ρ/2)SVinert

The optimal loop for achieving the maximum speed Vinert,max is, according to Equation (20), performed at the maximum lift-to-drag ratio, (CL/CD)max, and thus at the associated lift ∗ coefﬁcient, CL. Using Equation (19) with |γtr| 1, the optimal cycle time is obtained as

2 m 1 tcyc = 4π ∗ (32) ρS CL(CL/CD)maxVw

This relation describes the effect of wing sweep on tcyc. That effect is due to (CL/CD)max Aerospace 2021, 8, x FOR PEER REVIEWand C∗ since each of them is dependent on wing sweep. 21 of 26 L A quantitative insight into the wing sweep effect can be provided examining Equation (32) for the vehicle under consideration. Results are presented in Figure 17 for the swept wing configuration 1 and for the straight wing configuration used as reference. The straight wing The wing sweep influence on 𝑡, Equation (32), manifests in such a way as to shift reference configuration features∗ a decrease of tcyc in the left curve part which is associated the beginning of the 𝐶 and (𝐶/𝐶) decrease to higher Mach numbers.∗ Thus, the with the incompressible Mach number region (constancy of (CL/CD) and C ) and an wing sweep effect on 𝑡 begins to becomes effective for swept wingmax configurationL 1 at increase in the right curve part which is associated with the compressible region (decrease a correspondingly higher∗ Mach number which in turn is associated with a higher wind of (CL/CD)max and CL). speed 𝑉.

Figure 17. Effect of wing sweep on the cycle time for swept wing configuration 1 in maximum-speed Figure 17. Effect of wing sweep on the cycle time for swept wing configuration 1 in maximum-speed dynamic soaring. dynamic soaring. 5.2. Load Factor The wing sweep influence on tcyc, Equation (32), manifests in such a way as to shift ∗ the beginningHigh-speed of thedynamicCL and soaring(CL/C Dinvolves)max decrease a rapid to flight higher maneuver Mach numbers. that shows Thus, a large the wingturning sweep rateeffect and a onlargetcyc inertialbegins speed. to becomes Thus, effectivethere are for high swept loads wing acting configuration on the vehicle, 1 at to athe correspondingly effect that normal higher accelerations Mach number of an which order inof turnmagnitude is associated of 100 withg are areached higher wind[8]. As speeda result,Vw .large loads are typical for high-speed dynamic soaring. A relation between the load factor, 𝑛, and the wind speed, 𝑉, in maximum-speed dynamic soaring can be derived using Equations (19) and (30) to yield 1 𝜌𝑆 ∗ 𝐶 𝑛= 𝐶 𝑉 (33) 2𝜋 𝑚𝑔 𝐶 ∗ This solution shows that the effect of wing sweep on 𝑛 is relating to (𝐶/𝐶) and 𝐶 , according to their dependence on wing sweep. The quadratic form of the (𝐶/𝐶) term in Equation (33) is indicative for an increased influence. Quantitative results on the effect of wing sweep are provided by Figure 18 which presents 𝑛 dependent on 𝑉 for the swept wing configuration 1 and for the straight wing reference configuration. The straight wing reference configuration shows an increase of 𝑛 in the left curve part and a decrease in the right curve par, corresponding to the incompressible and compressible Mach number regions and the related behavior of (𝐶/𝐶) ∗ and 𝐶 . The effect of wing sweep manifests in such a way that there is a difference in terms of larger 𝑛 values in the compressible Mach number region. This is due to the shift of the ∗ beginning of the 𝐶 and (𝐶/𝐶) decrease to higher Mach numbers resulting from wing sweep.

Aerospace 2021, 8, 229 21 of 26

5.2. Load Factor High-speed dynamic soaring involves a rapid ﬂight maneuver that shows a large turning rate and a large inertial speed. Thus, there are high loads acting on the vehicle, to the effect that normal accelerations of an order of magnitude of 100 g are reached [8]. As a result, large loads are typical for high-speed dynamic soaring. A relation between the load factor, n, and the wind speed, Vw, in maximum-speed dynamic soaring can be derived using Equations (19) and (30) to yield

1 ρS C 2 = ∗ L 2 n 2 CL Vw (33) 2π mg CD max ∗ This solution shows that the effect of wing sweep on n is relating to (CL/CD)max and CL, according to their dependence on wing sweep. The quadratic form of the (CL/CD)max term in Equation (33) is indicative for an increased influence. Quantitative results on the effect of wing sweep are provided by Figure 18 which presents n dependent on Vw for the swept wing configuration 1 and for the straight wing reference configuration. The straight wing reference configuration shows an increase of n in Aerospace 2021, 8, x FOR PEER REVIEWthe left curve part and a decrease in the right curve par, corresponding to the incompressible22 of 26 ∗ and compressible Mach number regions and the related behavior of (CL/CD)max and CL.

FigureFigure 18. 18.Effect Effect of of wing wing sweep sweep on on the the load load factor factor for sweptfor swept wing wing conﬁguration configuration 1 in maximum-speed 1 in maximum- speed dynamic soaring. dynamic soaring.

5.3. TrajectoryThe effect Radius of wing of sweepMaximum-Speed manifests Cycle in such a way that there is a difference in terms of largerThen airvalues space in required the compressible in high-speed Mach dynamic number region.soaring This is determined is due to the by shift the ofshape the ∗ beginningand extension of the of theCL andtrajectory.(CL/C InD) Figuremax decrease 6, top and to higher side views Mach of numbersan optimized resulting maximum- from wingspeed sweep. trajectory loop are presented, thus, providing information of the extensions in the longitudinal, lateral and vertical directions. A question is whether the wing configuration 5.3.in terms Trajectory of straight Radius or of Maximum-Speedswept wings plays Cycle a role regarding the required airspace through its relationshipThe air space with required the maximum in high-speed speed, dynamic𝑉, soaring, and the is wind determined speed, 𝑉 by. theThe shapetrajec- andtory extension radius is ofconsidered the trajectory. a quantity In Figure that6 ,can top be and used side to views address of anthe optimized air space issue. maximum- speedWith trajectory reference loop to are Equations presented, (16) thus, and providing (30), the radius information of the oftrajectory the extensions loop in in high- the longitudinal,speed dynamic lateral soaring and can vertical be expressed directions. as A question is whether the wing conﬁguration in terms of straight or swept wings plays a role regarding the required airspace through its 2𝑚 1 relationship with the maximum speed, V𝑅inert ,=max, and the wind speed, Vw. The trajectory(34) radius is considered a quantity that can be used𝜌𝑆 to address𝐶 the air space issue. ∗ Accounting for 𝐶 =𝐶 holding true for the maximum-speed loop, the relation of the optimal trajectory radius is obtained as 2𝑚 1 𝑅 = ∗ (35) 𝜌𝑆 𝐶 ∗ This relation shows that the effect of wing sweep on 𝑅 is due to 𝐶 , according to ∗ the influence of wing sweep on 𝐶 . A further outcome is that 𝑅 is not explicitly dependent on 𝑉, or 𝑉. A quantitative insight into the wing sweep effect is provided by Figure 19 which presents 𝑅 dependent on 𝑉 for the swept wing configuration 1 and for the straight wing reference configuration. In the left part of the diagram, 𝑅 is constant and the same for all cases. In the right part of the diagram, 𝑅 shows an increase. This increase ∗ ∗ is due to 𝐶 because 𝐶 depends on 𝑀𝑎 (Figure 5). The effect of wing sweep yields a difference to the straight wing only in the compressible Mach number region in as much as the increase of 𝑅 is reduced. This due to ∗ the shift in the beginning of the 𝐶 decrease to higher Mach numbers.

Aerospace 2021, 8, 229 22 of 26

With reference to Equations (16) and (30), the radius of the trajectory loop in high- speed dynamic soaring can be expressed as

2m 1 Rcyc = (34) ρS CL ∗ Accounting for CL = CL holding true for the maximum-speed loop, the relation of the optimal trajectory radius is obtained as

2m 1 Rcyc = ∗ (35) ρS CL ∗ This relation shows that the effect of wing sweep on Rcyc is due to CL, according to the ∗ influence of wing sweep on CL. A further outcome is that Rcyc is not explicitly dependent on Vinert,max or Vw. A quantitative insight into the wing sweep effect is provided by Figure 19 which presents Rcyc dependent on Vw for the swept wing configuration 1 and for the straight wing reference configuration. In the left part of the diagram, Rcyc is constant and the same Aerospace 2021, 8, x FOR PEER REVIEWfor all cases. In the right part of the diagram, Rcyc shows an increase. This increase is23 dueof 26 ∗ ∗ to CL because CL depends on Ma (Figure5).

FigureFigure 19. 19.Effect Effect of of wing wing sweep sweep on on the the trajectory trajectory radius radius forswept for swept wing wing conﬁguration configuration 1 in maximum- 1 in maxi- speedmum-speed dynamic dynamic soaring. soaring.

6. ConclusionsThe effect of wing sweep yields a difference to the straight wing only in the compressible MachDynamic number soaring region shows in as much a unique as the increaseflight performance of Rcyc is reduced. capability This by due enabling to the shift ex- ∗ intremely the beginning high speeds of the inC Lsheardecrease winds to involving higher Mach a thin numbers. layer that separates regions of large and small wind speeds, as they exist at ridges. It is found that the speed level now reached 6.in Conclusions dynamic soaring approaches the high subsonic Mach number region, with the result thatDynamic the compressibility-related soaring shows a unique drag ﬂightrise caus performancees a limiting capability effect on by the enabling achievable extremely speed highperformance. speeds in Wing shear sweep winds is involvingconsidered a as thin a means layer thatfor shifting separates the regionsbeginning of of large the andcom- smallpressibility-related wind speeds, asdrag they rise exist to higher at ridges. Mach It numbers. is found thatFor dealing the speed with level the nowrelated reached issues, inappropriate dynamic soaring flight mechanics approaches models the high are subsonic developed Mach and number a suitable region, optimization with the method result thatis used, the compressibility-related with the goal to determine drag the rise speed causes performance a limiting effect enhancement on the achievable by wing speedsweep. performance.Analytic solutions Wing are sweep derived is considered for describing as a the means effects for of shifting wing sweep. the beginning It is shown of thethat compressibility-relatedsmall sweep angles have drag only rise a tominor higher effect Mach and numbers. that larger For sweep dealing angles with are the necessary related issues,for achieving appropriate a significant ﬂight mechanics improvement. models In conclusion, are developed it is andpossible a suitable to enhance optimization the max- methodimum-speed is used, performance with the goal in tohigh-sp determineeed dynamic the speed soaring performance by wing enhancement sweep. by wing sweep. Analytic solutions are derived for describing the effects of wing sweep. It is Author Contributions: Conceptualization, G.S.; methodology, G.S. and B.G.; software, B.G.; validation, G.S., B.G. and H.H.; formal analysis, G.S., B.G. and H.H.; writing—original draft preparation, G.S.; writing—review and editing, G.S., B.G. and H.H. 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. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

aij coefficients b wing span CD drag coefficient CL lift coefficient D drag E energy g acceleration due to gravity

Aerospace 2021, 8, 229 23 of 26

shown that small sweep angles have only a minor effect and that larger sweep angles are necessary for achieving a signiﬁcant improvement. In conclusion, it is possible to enhance the maximum-speed performance in high-speed dynamic soaring by wing sweep.

Author Contributions: Conceptualization, G.S.; methodology, G.S. and B.G.; software, B.G.; validation, G.S., B.G. and H.H.; formal analysis, G.S., B.G. and H.H.; writing—original draft preparation, G.S.; writing—review and editing, G.S., B.G. and H.H. 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. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

aij coefficients b wing span CD drag coefficient CL lift coefficient D drag E energy g acceleration due to gravity h altitude J performance criterion L lift Ma Mach number m mass n load factor Rcyc loop radius S wing reference area s length t time u, v, wi speed components u control vector Va airspeed Vinert inertial speed Vw wind speed x longitudinal coordinate x state vector W work y lateral coordinate z vertical coordinate A aspect ratio χ azimuth angle γ flight path angle Λ sweep angle µ bank angle ρ air density

Appendix A. Formulation of Optimal Control Problem The problem of determining the maximum speed achievable with swept-wing gliders was solved using an optimization method capable of dealing with highly dynamic and unsteady ﬂight maneuvers [20]. Aerospace 2021, 8, 229 24 of 26

The optimization problem is to determine the maximum speed achievable with dynamic soaring. This means to ﬁnd out the closed-loop dynamic soaring trajectory that shows the maximum speed in the course of the loop. For this purpose, the following performance criterion is speciﬁed

J[x(t)] = Vinert(t) (A1)

For maximizing J[x(t)], the optimal control problem can be formulated as to determine the optimal controls T ˆ ∗ ∗ ∗ u (t) = [CL(t), µa (t)] (A2) the optimal states

∗ ∗ ∗ ∗ ∗ ∗ ∗ T x (t) = [ui (t), vi (t), wi (t), x (t), y (t), h (t)] (A3)

and the associated, optimal cycle time tcyc, subject to the dynamic system according to Equation (1) . x(t) = f(x(t), u(t)) (A4) to control and state constraints

CL,min ≤ CL ≤ CL,max (A5)

h ≥ hmin and to periodicity boundary conditions Ψ(x) = x tcyc − x(0) = 0 (A6)

The optimal control problem is solved using the direct optimal control tool FAL- CON.m [27]. To this end, a full discretization of the optimal control problem on the time grid t is performed, resulting in the discretized states x and controls u. The dynamic constraints given by the equations of motion are replaced by a set of defect equations (equality constraints)

tk+1 − tk c (x, u) = x + − x − [f(x + , u + ) + f(x , u )] = 0 (A7) eq,k k 1 k 2 k 1 k 1 k k at every point on the discretized time grid. These constraints represent a trapezoidal quadra- ture of the dynamic equations. Furthermore, all path constraints given in Equation (A5), i.e., box constraints on selected control and state variables, are evaluated on this grid, yielding   CL,min − CL,k cineq,k(x, u) =  CL,k − CL,max  ≤ 0 (A8) hmin − hk

for every point tk on the discretized time grid. Using the above-mentioned discretized approximations of the objective function, differential equations and constraints contained in the original optimal control problem, the Lagrangian function can be constructed for a nonlinear programming (NLP) problem:

T T T L = λ0 · Vinert t f + λk · ceq,k(x, u) + λΨ · Ψ(x) + µk · cineq,k(x, u) (A9) Aerospace 2021, 8, 229 25 of 26

The optimization software IPOPT [28] is utilized to ﬁnd the solution of the constructed NLP problem, which employs an interior-point method solving for the (Karush-Kuhn-Tucker) ﬁrst-order optimality conditions

(optimality) → ∇zL = 0 (A10)

(feasibility) → ∇λL = 0 (A11) T (complementarity) → µk · cineq,k(x, u) + τ = 0 (A12) where z represents the primal variables x, u, t f , λ and µ represent dual variables, i.e., the vectors of Lagrange multipliers to equality and inequality constraints, respectively and τ is the barrier parameter, which is driven to zero by the interior-point optimization software. The corresponding function values, gradient and Hessian matrices are supplied to IPOPT by FALCON.m during every iteration. After calculation of an efﬁcient step in the primal and dual variables, IPOPT returns an updated of these variables until Equations (A10)–(A12) are satisﬁed. The solution of the original optimal control problem can be reconstructed from the solution of the NLP.

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