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; 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.