Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring
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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; benedikt.grueter@tum.de (B.G.); haichao.hong@tum.de (H.H.) * Correspondence: sachs@tum.de 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 Figure 1 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 Figure 1 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) Upper Lower curve curve inin the the region no windofregion, high wind withspeed, a flightwith a flight direction direction change fromchange from leeward to windward windward. 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 , . Figure1.1.High-speed Figure High-speeddynamic dynamicsoaring soaringtrajectory trajectoryand andshear shearwind windscenario scenarioatatridge. ridge. Formodel The dealing with the dynamic developed soaring for the shear windproblem scenariounder consideration, at ridges twodescribed comprises the flight me- chanics three windmodels areThe regions. developed. Oneinvolves shear layer model denoted by “3-DOF a rapid increase dynamics of the model” wind speed fromis based zero on point mass dynamics and the other model denoted by “energy to the value of the free stream wind speed which is denoted by Vw,re f . model” is based on the energy For characteristics dealing with thedeterminative in high-speed dynamic soaring problemdynamic soaring. Results under consideration, two applying the flight me- 3-DOF models chanics dynamicsaremodel are used developed. Onetomodel validate the energy denoted model.dynamics model” is based by “3-DOF on point mass dynamics and the other model denoted by “energy model” is based on the 2.1. 3-DOF energy Dynamics determinative characteristics Model in high-speed dynamic soaring. Results applying the 3-DOFThedynamics model are used to validate the energy motion of aerial vehicles in high-speed model. dynamic 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 axis is parallel to the wind speed, the is The motion horizontal and of is aerial vehicles pointing in high-speed downward. dynamic With regard to soaring can besystem, this reference mathematically the equa- described using a point mass dynamics tion of motion can be expressed as model. An inertial reference system which is presented in Figure 2 is applied where the xi axis is parallel to the wind speed, the yi is d /d =With horizontal and zi is pointing downward. − regard / −to this / reference system, the equation of motion can be expressed as d /d = − / − / dui /dt = − au1 D/m − au2 L/m d /d = − / − / + dvi /dt = − av1 D/m − av2 L/m dwi /dt = − aw1d /d − D/m = a w2 L/m + g (1) dxid =u /dt/d =i (1) dyi /dt = vi dℎ/d = − dh/dt = −wi The coefficients , , , and , are abbreviation factors used for describing rela- The coefficients tionships au1,2 , the regarding av1,2angles and aw1,2 , are and abbreviation factors used for describing relation- , yielding ships regarding the angles γa , µ a and χ a , yielding au1 = cos γa cos χ a au2 = cos µ a sin γa cos χ a + sin µ a sin χ a
= cos sin = cos sin sin − sin cos (2) Aerospace 2021, 8, 229 = −sin 4 of 26 = cos cos The angles and are determined by the following relations: av1 = cos γa sin χ a sin = − / av2 = cos µ a sin γa sin χ a − sin µ a cos χ a (2) tan aw1 = = − sin /( γa + ) (3) The angle is a control that is determined aw2 = cos µ aby costhe γa optimality conditions, described by Equation (A2) in the Appendix A. Figure 2. Speed vectors and inertial coordinate system. (The 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 and pointing in the opposite direction.). speed vector Vw and pointing in the opposite direction). Theangles The aerodynamic γa and χforces, and , are related to the airspeed vector , while the a are determined by the following relations: motion of the vehicle is described by the inertial speed vector sin =− γa = ( w,i /V , , a ) (4) tan χ a = which is related to the inertial reference i +, , vi /(u( system Vw )). The speed vectors and (3) are connected by the wind speed vector (Figure 2), yielding The angle µ a is a control that is determined by the optimality conditions, described by Equation (A2) in the Appendix A. = − (5) The aerodynamic forces, L and D, are related to the airspeed vector Va , while the The 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 (Figure 2), with no loss of generality. Thus, the wind speed vector can be written as ==(u(− Vinert i , vi ,,w, )T 0,0) (4) (6) With reference which is relatedto tothis the expression and to system inertial reference Equation (xi(3), , y, zthe following i ). The speedrelations vectors Vhold true a and for Vinert theconnected are airspeed by the wind speed vector Vw (Figure 2), yielding = V = ( + −, Vinert Vw, , ) (7a) (5) a The xi axis can be chosen to be in the opposite direction of the wind speed vector Vw (Figure 2), with no loss of generality. Thus, the wind speed vector can be written as Vw = (−Vw , 0, 0)T (6) With reference to this expression and to Equation (3), the following relations hold true for the airspeed Va = (ui + Vw , vi , w,)T (7a) and q Va = (ui + Vw )2 + v2i + wi2 (7b)
and Aerospace 2021, 8, 229 5 of 26 = ( + ) + + (7b) 2.2.Energy 2.2. EnergyModel Model Fordeveloping For developinganan energy energy model model valid valid for flight for the the flight mechanics mechanics of high-speed of high-speed dynamic dy- namic soaring, assumptions on the trajectory and speeds are made. To this soaring, assumptions on the trajectory and speeds are made. To this aim, reference is aim, reference is made made to Figure to Figure 3 which 3 which provides provides an an oblique oblique view view onon a dynamic a dynamic soaringloop. soaring loop.ItItisisas- as- sumedthat sumed thatthe thedynamic dynamicsoaring soaringloop loopisiscircular circularandandthe theinclination inclinationisissmall. small.The Thewindwind speedisissupposed speed supposedtotobe besmaller smallerthan thanthetheinertial inertialspeed speedand andthetheairspeed airspeedby byan anorder orderofof magnitude,yielding magnitude, yielding Vw Vinert , Vw Va (8) ≪ , ≪ (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). Whentraversing When traversing the shear the shear layer, layer, there there is an is an approximate approximate speed increase speed of (1/2)increase Vw cos γtrof (1/2) cos for the inertial speed in the leeward descent for the inertial speed in the leeward descent (transition point 1) (transition point 1) 1 = 1+ cos (9) Vinert = V inert + Vw2 cos γtr (9) 2 and for the airspeed in the leeward descent (transition point 2) and for the airspeed in the leeward descent (transition point 2) 1 = + cos (10) 1 2 Va = V inert + Vw cos γtr (10) where is the average inertial speed of 2the loop and is the flight path angle at the transition points. where V inert 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. achieved in the upper The energy section gain from theofwind the loop whichwhere the wind isfor is necessary blowing. This the propelling manifests vehicleinis an increase of the kinetic energy between the two transition points 1 and 2, achieved in the upper section of the loop where the wind is blowing. This manifests yielding in an increase of the kinetic energy between the two transition points 1 and 2, yielding = " 1 + cos 2 − 1 − cos 2= # cos (11) m 2 1 2 2 1 Eg = V inert + Vw cos γtr − V inert − Vw cos γtr = mV inert Vw cos γtr (11) 2 2 2 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
CD C
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 inert
n= (16)
Rcyc g
Thus, the drag work, Equation (14), can be expressed as
CD 2
WD = −π mV inert (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
V inert = Vw cos γtr (19)
π CD
The speed relationships presented in Figure 3 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 V inert 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 concern-
ing 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 enhance- ment 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 D = CD (ρ/2)Va2 S L = CL (ρ/2)Va2 S (21) The compressibility-related drag rise in the high subsonic Mach number region mani- fests 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, com- pressibility 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 Figure 4. 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 drag-lift relationship, CD = CL2 /(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, = 8.5 kg. For the described modelling including the drag related8 of 26 com- pressibility effects, reference is made to existing vehicles and to experience in this field [6,8,22]. Figure4.4.Drag Figure Dragcoefficient coefficientofofstraight straightwing wingconfiguration configurationdependent dependenton onlift coefficient,C and liftcoefficient, andMach Mach L number, , [7]. number, Ma, [7]. Theenergy The energymodel, model,Equation Equation(20), (20),shows showsthat thatthere thereisisonly onlyoneoneaerodynamics aerodynamicsquantityquantity that is determinative for the effect of wing sweep on the that is determinative for the effect of wing sweep on the maximum speed performance.maximum speed performance. Thisisisthe This themaximum maximumlift-to-drag lift-to-drag (C( ratio ratio / ) . Thus, the dependence of ( / ) L /CD )max . Thus, the dependence of (CL /CD )max ononthe theMach MachnumbernumberMa is determinative is determinative for for thethe wing wing sweepsweep problem problem under under considera- consideration. tion. The The dependence dependence of (CLof /C( ) D max / ) on Ma on can be can be determined determined by an by an examination examination of of the the drag drag polars depicted polars depicted in Figure 4. in Figure 4. Resultsare Results arepresented presented in in Figure Figure 5 where 5 where (CL /C( D/ ) dependent )max dependenton Ma on is is plotted. plotted. The (C L /C( The / )curvecurve D )max has a shape has a shape that can thatbecan be subdivided subdivided into twointoparts. two parts. In the Inleftthe left curve curve part, (Cpart, L /CD ( )max / is) constant is constant and shows and the shows the highest highest level. Constancy level. Constancy of (CL /C ofD )( / ) max with regard with toregard to that Ma implies implies there is that no there effect is of no effect of compressibility. compressibility. Thus, the leftThus, part ofthetheleft (CLpart /CDof the )max ( / can curve ) be curve relatedcanto be therelated to the incompressible incompressible Ma region. In the region. right In the right curve part, curve there part, is a there is a decrease continual continualofdecrease (CL /CDof )max / ) rapidly ( which which rapidly leads leadsvalues. to small to small values. This This de- decrease of (Ccrease Aerospace 2021, 8, x FOR PEER REVIEW L /CD ) ofmax( is/ ) an effect is of an effect of compressibility, compressibility, associated with associated the dragwith the drag rise rise described 9 ofde- above. 26 Thus, scribedtheabove. right part the(right of the Thus, CL /CD part the ( )maxofcurve can/ be )related to the curve compressible can be related to Mathe region. com- pressible region. The lift coefficient associated with ( / ) , denoted by ∗ , is also presented in ∗ Figure 5. The relation between and ( / ) is given by ∗ ( ) = (23) ∗ ( ) The ∗ curve has a shape that can also be subdivided into two parts. The left curve part where ∗ is constant and highest, can be related to the incompressible region. In the right curve part, ∗ shows a decrease to approach considerably smaller values which is due to compressibility. Thus, the right curve part can be associated with the com- pressible region. Figure 5. Maximum lift-to-drag ratio, ( / ) and associated lift coefficient, ∗ , dependent on Figure 5. Maximum lift-to-drag ratio, (CL /CD )max and associated lift coefficient, CL∗ , dependent on Mach number, , (relating to drag polar in Figure 4. Mach number, Ma, (relating to drag polar in Figure 4. 3. Maximum Speed Achievable with Straight-Wing Configuration The objective of this Chapter is to validate the energy model developed in the previ- ous 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
Aerospace 2021, 8, 229 9 of 26 The lift coefficient associated with (CL /CD )max , denoted by CL∗ , is also presented in Figure 5. The relation between CL∗ and (CL /CD )max is given by CL∗ ( Ma) CL = (23) CD CL∗ ( Ma) CD max 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 pre- vious 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 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 Figure 6 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 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 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 Aerospace 2021, 8, x FOR PEER REVIEW 11 of 26 Figure 6. Dynamic soaring loop optimized for maximum speed, , = 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 tra- jectory projection on the - 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 contrib- utes to the validation of the energy model. Figure 7. Side and top views of dynamic soaring loop optimized for maximum speed, , = 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, and the Mach number, , are addressed by presenting their
Aerospace 2021, 8, 229 11 of 26 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. 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, Vinert and the Mach number, Ma, are addressed by presenting their12time Aerospace 2021, 8, x FOR PEER REVIEW of 26 histories in Figure 8. Figure 8. Time histories of speeds, and , and Mach number, , 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, , = 271.8 m/s. optimized for maximum speed, Vinert,max = 271.8 m/s. 3.2. The Maximum-Speed speed curve Performance andisRelated chosenKey Factors (the time scale such that the maximum speed V inert,max = 271.8 m/s The is maximum speed achievable at the beginning t = 0) shows in high-speed dynamic that Vinert features one soaring, the, downward oscillation , is central for of part thewhich subject is under longer consideration than the upward in this paper. part. There are, of The minimum according Vinert is at toaEquation point of the (20), two key where trajectory factorsthe forvehicle , has : the passed wind thespeed, lowest , and altitude the and maximum is climbing lift-to-drag upwind ratio, again. ( The / )role . ofBecause compressibility of their for keythe problem factor role, under the effects of and consideration ( / evident becomes ) on addressing, , the willMach number Ma be analyzed in detail.reached during the Emphasis cycleon is put andtheputting role ofthis( in/ relation ) with as this the keydragfactorpolar characteristicsfor is determinative of thethe vehicle. effect ofThewing timesweephistory of Ma on the presented maximum in Figure speed achieva-8 shows ble. This the Ma curve thatanalysis comprises is considered two oscillations a means for validating which theare rathermodel. energy similar. Furthermore, the MaResultslevel ison sothe high that it relates maximum speed to are the presented compressible region9ofwhere in Figure the dragthe coefficient dependence CDof throughout, the on entire is cycle. shown. This The means range with selected reference for to is Figure such as 4 to that cover thea drag wide rise due spectrum toofcompressibility wind speeds possible is fully at effective. ridges, Thus, with the theobjective maximum-speed to achieve performance results generally is negatively valid for influenced at all points high-speed dynamic 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 maximum achieve solutionsspeed achievable for determining in high-speed the maximum dynamic speed.soaring, Vinert,line The solid , is central maxcurve shows for the subject under consideration in this paper. There the results using the energy model. Concerning this model, solutions were constructed are, according to Equation (20), two key factors using Equation (20). Both for V inert, max : the wind speed, V , and the maximum curves are close to each other and overlap in parts, w lift-to-drag ra-to , tio, ( C the effect L /C D ) that . Because of their key factor role, maxthe results of the 3-DOF dynamics model and the energy the effects of V w and ( C model /C ) L approaches D max on Vvirtually inert, max , will agree. beThis analyzed suggests in detail. that the Emphasis energy modelis putEquation on the role (20)ofis(confirmed. CL /CD )max as this key factorThe is determinative fact that both model for the effect of wing approaches sweepagree virtually on the maximum manifests speedaspects. in more achievable. This This analysis is considered a means for validating the energy will be shown in the following sections where a detailed treatment on these aspects is model. Results on the maximum speed are presented in Figure 9 where the dependence of presented. Vinert, max on Vw isthe Examining shown. The range curveselected for Vw is characteristics insuch Figure as to cover 9, the lefta part wideofspectrum the curve , of wind speeds possible at ridges, with the objective to achieve results generally valid for shows a comparatively large and constant gradient, whereas the gradient in the right part high-speed dynamic soaring. is substantially reduced. Relating this property of the , gradient to the Mach number 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 , curves can be related to the incompress- ible 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, , , dependent on wind speed, . Figure 9. Maximum speed, Vinert,max , dependent on wind speed, Vw . Thedotted The reasonline forcurve the fact that the shows there are two results using different the 3-DOF in the model. partsdynamics , curve is due Concerning to ( / ) . In the left part of the , this model, the optimization procedure that is described in the Appendix A wascurve, ( / ) is constant soapplied that, ac- cording to Equation (20), the gradient is also to achieve solutions for determining ,the maximum speed. The solid line curve shows the constant, whereas the , gradient results usingin the theright energy curve part Concerning model. is substantially thisreduced. The cause model, solutions for the were reductionusing constructed of the Equation , gradient (20). Both is duemax Vinert, ( / are to curves ) close . Thistoiseachbecauseother( and / overlap ) shows in parts,a decrease to the in thethat effect compressible the results of Mach numberdynamics the 3-DOF region. According model andtothe Equation (20), a approaches energy model reduction in ( / ) agree. virtually leads Thistosuggests a corresponding that the energyreduction model in Equation, . (20) Thisismeans that ( / ) confirmed. is the The cause fact for thattheboth limitation in the achievable model approaches virtually agree, values. manifests in more aspects. This will be Further shownevidence and understanding in the following sections where of the effect a detailed / ) of ( treatment on these on can be , aspects provided is presented. when analyzing the actual lift-to-drag ratio, / , during the dynamic soaring cycle and comparing Examining the Vinert,thismax with the characteristics curve maximum lift-to-drag in Figure ratio, 9, the ( left / part ) of , of thethe same curve cycle. aResults shows on that issue comparatively largeareand plotted constantin Figure gradient,10. whereas the gradient in the right part isAs each of reduced. substantially / and Relating( / ) this varies property during of thethe cycle, gradient Vinert,max the average to thevalues Machof number Ma indicated these quantities on the ordinate are considered shows that theThey as representative. change arein the gradient given by starts in a zone where compressibility begins to become effective 1 for the aerodynamic characteristics of the vehicle. Thus, the left part of the Vinert,max curves = ∫can be related d to the incompressible Ma (24) 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, 1 according to Equation (20), the Vinert, max = gradient ∫ is also constant, d whereas the Vinert, max (25) , gradient in the right curve part is substantially reduced. The cause for the reduction of the VBoth gradient quantities inert, max is due have been (CL /CD )maxusing to determined . Thisthe is because 3-DOF dynamics(CL /CD )model max shows andathe decrease optimi- inzation the compressible Mach number method describes in the Appendix A. region. According to Equation (20), a reduction in (CL /CThe D )max leadspresented results to a corresponding in Figure 10 reduction addressinthe Vinert,max . This means relationships between thatthe (CLlift-to-drag /CD )max isratios, the cause for ( / ) the limitation in the achievable V values. and ( / ) and the wind speed, . Basically, each of the inert,max , ( / Further ) evidence and ( and / understanding ) curves shows of thetwoeffect parts. (CLleft ofThe )max featuring /CDparts on Vinert,max can be a constant , provided when analyzing the actual lift-to-drag ratio, C L /C level are associated with the incompressible Mach number region, whereas the right parts D , during the dynamic soaring cycle and comparing involving a continualthis with the decrease aremaximum associatedlift-to-drag ratio, (CL /Cregion. with the compressible D )max , of the same cycle. A Results on that issue are main result is that the ( / ) plotted in Figure 10. and ( / ) curves are close to each other , and overlap in parts. This means that the actual ( / ) values virtually agree with the maximum lift-to-drag ratio, ( / ) , . These characteristics of the lift-to-drag ratios ( / ) and ( / ) , are of importance for the energy model. This is because the fact that ( / ) virtually agrees
Aerospace 2021, 8, x FOR PEER REVIEW 14 of 26 Aerospace 2021, 8, 229 with ( / ) , supports the assumption made for the energy model. Accordingly, 13 of 26it is assumed for the energy model that the maximum lift-to-drag ratio ( / ) holds true for the maximum speed , , Equation (20). Figure10.10.Lift-to-drag Lift-to-dragratio ratioin inmaximum-speed maximum-speed dynamic soaring. ( dynamic soaring. (CL/ /CD) ) : actual lift-to-drag ra- Figure av : actual lift-to-drag tio, 3-DOF dynamics model; ( / ) , : 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. AsThe main each of role CL /C that and( / (CL)/CD )plays as the determinative aerodynamics quantity for D max varies during the cycle, the average values of the maximum speed is further these quantities are considered as representative. substantiated andTheyconfirmed are given by the by broader treatment pre- sented in the following. For this purpose, three different ( / ) configurations are dealt with and their effects on CL , Z tcyc are1 analyzed CL and compared. The range selected = dt (24) for ( / ) is larger than the CD( av/ )tcyc range 0 CD characteristic for high-speed gliding vehicles. and Results are presented in Figure 11 which Z tshows the relationship between , , CL 1 cyc C and ( / ) where the notation = ( / ) , L refersdtto the maximum lift-to-drag (25) CD max,av tcyc 0 CD max 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 model Both quantities have been determined using the 3-DOF dynamics , and the curve optimiza- which are re- tion latedmethod to the describes incompressiblein the Appendix and compressibleA. Mach number regions, respectively. In the left The curve results parts,presented there are in Figure large 10 address differences between the three between the relationships , the lift-to-drag curves (for each of the (solid ratios, CL /Cand D )max,av dotted and (CL /C lines) thatD )are av and due the to wind the speed, effect of ( V w/ . ) Basically, . Theseeach of the differences (Creflect L /CD )themax,av and strong ( C /C influence L D ) avof ( curves / ) shows two on parts., The, left parts according featuring to the a energyconstant model level are associated with the incompressible Mach number Equation (20). This manifests in a particular manner for the case of the 50% decrease in region, whereas the right parts involving ( / ) a continual decrease as the related are, associated with the compressible curve practically does not reach region.the Mach number A main result is region where compressibility isthat the ( C /C ) L effective. and ( C D max,av FurtherL to the /C ) curves are D avleft curve parts, closecomparing to each other the and solidoverlap in parts. true lines holding Thisformeans that themodel the energy actual and(CL /C ) av values theDdotted lines virtually holdingagree true with the3- for the maximum DOF dynamics lift-to-drag modelratio,shows (CLthat )max,av /CDthey . virtually agree. These characteristics of the lift-to-drag Concerning the right curve parts, there are two ratios (Ceffects L /CD )valid av and for(Cthe L /C D )max,av are curves of , importance of both ( / ) for the energy model. This is because the cases reaching the compressible Mach number av fact that ( C L /C D ) virtually agrees region. Firstly, the (C with , D )max,av L /Ccurves supports (for each pairtheofassumption solid and madelines) dotted for the energy show no model.aAccordingly, longer large difference it isbut assumed for the energy model that are close to each other. Secondly, the increase of the maximum lift-to-drag ratio ( C of both L( / /C ) ) D max holds cases , true for the maximum speed V , Equation (20). is substantially reduced and there is practically a limitation of the achievable inert,max , The These values. main role two that effects(CLare /CtheD )max plays result of as thethe determinative decrease of ( / aerodynamics ) quantity due to compressi- for the maximum speed is further substantiated and confirmed by the broader treatment bility (Figure 5). This means that ( / ) plays the primary role in limiting the achiev- presented in the following. For this purpose, three different (CL /CD )max configurations able , values. are dealt with and their effects on Vinert,max are analyzed and compared. The range se- lected for (CL /CD )max is larger than the (CL /CD )max range characteristic for high-speed gliding vehicles.
Aerospace 2021, 8, 229 14 of 26 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 )max . These differences Aerospace 2021, 8, x FOR PEER REVIEW reflect the strong influence of (CL /CD )max on Vinert, max , according to the energy model Aerospace 2021, 8, x FOR PEER REVIEW 15 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, comparing Here, the again, th Here, again, the findings of the energy model agree with those of the 3-DOF dynam- Here, solid again, lines the findings holding true for of thethe energy energy modeland model agree thewith those dotted of the lines 3-DOF holding dynam- ics true model. for theThis is p ics model. This is particular important for the compressible Mach number region because ics3-DOF model.dynamics This is particular important model shows for the that they compressible virtually agree. Mach number regionthe limitation in because the limitation in , is related to that Mach number region. the limitation in , is related to that Mach number region. Figure 11. Maximum Figure11. 11.Maximum speed, Maximumspeed, , dependent , , dependent onwind speed,V and windspeed, and on maximumlift-to-drag lift-to-drag Figure Figure 11. Maximum speed, Vinert,max , , dependent onon wind speed, and w on on maximum maximum ratio, ( / ) lift-to-drag . ratio, ( / ) . Results obtained using the energy model, Equation (20). ratio,( ratio, (C/ L /C)D )max .. Results Resultsobtained using obtained the energy using model, Equation the energy (20). model, Equation (20). Results o Results obtained using the 3-DOF dynamics model. Resultsusing Results obtained obtained using the the 3-DOF 3-DOFmodel. dynamics dynamics model. In summary, t In summary, thethe analysis in this Chapter areshows that the results of the energy model InConcerning summary, the right analysis curve parts, in this there shows Chapter twothateffects valid the results of the V for energy curves model virtually inert,max agree with virtually agree (CL /Cwith with the results of the 3-DOF dynamics model. The results on the maxi- of bothagree virtually D )maxthecases resultsreaching of the the3-DOF compressible dynamics Mach model.number The results region. Firstly, on the mum maxi- the and re speed mumVmum inert,max speed and speedcurves and related related (for each quantities pair quantities of solid areand are close close to each other dotted to each lines) other show andnooverlap and overlap longer ainlarge in parts. parts. This sug- difference This gests sug- that the energ gests but are that close the to energy each model other. is validated Secondly, the so that increase it ofcanV be used ofin lieu both of ( C the /C 3-DOF ) dynam- gests that the energy model is validated so that it can beinert,max used in lieu of the 3-DOFmax L D model.is cases dynam- ics ics model. reduced and there is practically a limitation of the achievable V icssubstantially model. inert,max values. These two effects are the result of the decrease of (CL /CD )max due to compressibility 4. Increase of Maxi 4. Increase of Maximum-Speed /CPerformance means that (CLPerformance byprimary Wing Sweep 4. (Figure Increase5).ofThis Maximum-Speed D )max plays bytheWing Sweep role in limiting the achievable 4.1. Aerodynamic Ch V4.1. Aerodynamic values. Characteristics of Swept Wing Glider Configurations 4.1.inert,max Aerodynamic Characteristics of Swept Wing Glider Configurations The results of t Here, again, the The results findings of the previousof the energy model treatment on theagree with those limitation of of ,the 3-DOF dynamics for straight wing The model. results This is of the previous particular treatment important for on the the limitation compressible of , for straight wing configurations show configurations show that the decrease of ( / ) caused by compressibility is thethe Mach number region because rea- configurations limitation show that in Vlimitation. the decrease of ( / ) caused by compressibility is the son rea- for this limitati son for this inert,max is related to that Mach The limitation begins number to becomeregion.effective at the Mach number at son forInthis limitation. The limitation begins to shows becomethat effective at the Mach number which at compres the which the compressibility-related drag rise begins. Thatthe summary, the analysis in this Chapter drag results rise isofassociated the energy model with two which the virtually compressibility-related agree with the results drag ofcritical rise the 3-DOF begins. dynamics That drag rise is associated with theMach two numbers [21] Mach numbers [21]. One is the Mach number,model. , The whichresults is theonMach maximum number at Mach speed numbers and [21].quantities One is theare critical closeMach number,and overlap , which in is the Mach number which the at speed of which therelated speed of sound is reached to at each someother point of the vehicle. parts. This one The other suggests is thethat drag which the the energy speed model of sound is is reached validated so at that some it can point be of used the in vehicle. lieu of The the other 3-DOF one is dynamics the drag divergence model. Mach n divergence Mach number, , which is the Mach number at which the drag coefficient divergence Mach number, , which is the Mach number at which the drag coefficient starts to rise rapidly starts to rise rapidly. is slightly greater than . starts to rise rapidly. is slightly greater than . A possible solu 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 urein12Fig-where the li ure 12 where the lift-to-drag ratio, / , of a swept wing configuration and of a corre- ure 12 where the lift-to-drag ratio, / , of a swept wing configuration and ofsponding a corre- straight w 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 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 show Figure 12 shows the Mach number region where compressibility causes the / Figure 12 shows the Mach number region where compressibility causes the decrease/ of the two decrease of the two wing configurations. Concerning the straight wing configuration, the
Aerospace 2021, 8, 229 15 of 26 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 corresponding straight wing configuration representative for glider type vehicles Aerospace 2021, 8, x FOR PEER REVIEW 16 are of 26 shown. The planform of the swept wing configuration is presented in Figure 13. Aerospace 2021, 8, x FOR PEER REVIEW 16 of 26 Figure12. Figure 12.Effect EffectofofMach Machnumber, ,onon number,Ma, lift-to-drag lift-to-drag ratio ratio ofof glider glider model model wings wings (from (from [23]). [23]). Figure 12. Effect of Mach number, , 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 the Figure 12 showsassumptions maximum-speed performance the are Mach number of made for where high-speed region determining dynamic the effectscauses soaring: compressibility of wingthesweep CL /CD on the maximum-speed decrease of the two performance wing of high-speed configurations. Concerning dynamic the soaring: straight wing configuration, the (1) Energy based model C(1) L /CD decrease Energy basedbegins modelat 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 Itgradient to reach rapidly smaller values. Concerning the swept wing configuration, be usedisfor assumed that energy determining model the effects for of wing high-speed sweep on dynamic the maximum soaring, Equation speed, (20),. This , shows can the be C L used/C D fordecrease begins determining at the about effects Ma of ≈ wing 0.81 ÷ sweep 0.83. on Thereafter, the maximum the drag speed, decrease . This is based on the results concerning the validation of the energy model. , a behavior corresponding with that of the straight wing configuration in terms of a similarly is based Theon the results energy modelconcerning Equation (20)the shows validation that ofthethe ( energy / ) model. is the only aerodynamic The energy model Equation quantity that has an effect on (20), shows . that Thus, the the ( effect / of ) wing is the only sweep aerodynamic on the speed per- quantity formancethat has an effect enhancement in on high-speed , . Thus, the dynamic effect of soaring canwing sweep on the be determined byspeed takingper- the formance dependence enhancement of ( / )in high-speed on wing sweepdynamic soaring into account canandbe determined by taking by incorporating this the de- dependence of ( / (20). pendence in Equation ) onthat For wing sweep the purpose, intocritical accountMachandnumber by incorporating this de- and its relationship
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 per- formance 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 depen- dence 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] Macr,0 Macr,Λ = (26) 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 Figure 5 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.
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