Analysis of take-off performance of a Prandtlplane 300 seat civil transport aircraft

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DIPARTIMENTO DI INGEGNERIA CIVILE E

INDUSTRIALE

Corso di Laurea Magistrale in Ingegneria Aerospaziale

Analysis of Take-Off Performance

of a PrandtlPlane 300 seat

civil transport aircraft

Supervisors:

Prof. Vittorio Cipolla

Prof. Aldo Frediani

Karim Abu Salem

Candidate:

Mario Bianchi

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List of Figures

1.1 3D rendering of the Prandtlplane configuration proposed within

PARSIFAL . . . 14

1.2 Elevators in a Prandtlplane configuration . . . 15

1.3 Three-view drawing of the Prandtlplane configuration pro-posed within PARSIFAL . . . 16

2.1 Take-off phases and related distances . . . 18

2.2 Forces acting on the aircraft during ground roll . . . 18

2.3 Forces acting on the aircraft during rotation . . . 20

2.4 Center of gravity position with respect to the point of contact with the runway . . . 21

2.5 Inertial forces due to the rotational motion of the aircraft cen-ter of gravity . . . 23

2.6 Forces acting on the aircraft during transition-to-climb . . . . 24

3.1 Numerical procedure for the simulation of ground roll . . . 29

3.2 Numerical procedure for the simulation of rotation . . . 30

3.3 Numerical procedure for the simulation of transition-to-climb . 31 3.4 Flight path and rate of climb for an unchecked maneuver . . . 32

3.5 Two-stage maneuver . . . 33

3.6 Flight path and rate of climb for a two-stage maneuver . . . . 35

3.7 Assumed pitch-attitude time history . . . 36

3.8 Angular speeds time histories resulting from different assumed pitch-attitude time histories . . . 37

3.9 Numerical procedure for the simulation of transition-to-climb by assuming a predetermined pitch-attitude time history . . . 38

3.10 Flight path and rate of climb resulting from the selected pitch-attitude time history . . . 39

3.11 Elevator deflection time histories resulting from different sim-ulation methods . . . 39

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4.1 Mirroring procedure for ground effect evaluation . . . 42

4.2 Lifting surface as modeled in a VLM . . . 44

4.3 Induce drag evaluated in AVL compared with that predicted

by Wieselsberger . . . 45

4.4 Airfoil Glenn Martin 2 . . . 46

4.5 CLα evaluated in AVL compared with experimental data

ob-tained by Fink and Lastinger . . . 46

5.1 Maximum allowable angle in the proximity of the runway . . . 48

5.2 Dimensioned drawings of the PARSIFAL-MS1 . . . 49

5.3 Model of the PARSIFAL-MS1 employed in AVL . . . 50

5.4 Map related to the parameter CLα for the PARSIFAL-MS1,

and enlargement of the portion related to low heights from the ground . . . 51

5.5 Geometry employed for the analyses carried out in JAVAFOIL 52

5.6 Geometry employed in JAVAFOIL at angle of attack equal to zero . . . 52

5.7 Disposition of the wings for α equal to 10o _{for different}

posi-tions of CR . . . 54

5.8 Map related to CL for the PARSIFAL-MS1 . . . 54

5.9 Map related to the parameter CMα for the PARSIFAL-MS1,

5.10 Map related to the parameter CM for the PARSIFAL-MS1 . . 56

5.11 Map related to the parameter CLδ e for the PARSIFAL-MS1,

5.12 Map related to the parameter CMδ e for the PARSIFAL-MS1,

5.13 Variations of CLδ e and CMδ e resulting from the differential

rotation of the elevators . . . 59

5.14 Actual maximum variations of CL and CM acting on the

dif-ferential rotation of the elevators . . . 60

5.15 Map related to the parameter CLq for the PARSIFAL-MS1,

5.16 Map related to the parameter CMq for the PARSIFAL-MS1,

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LIST OF FIGURES

5.17 Map related to the Oswald factor e for the PARSIFAL-MS1, and enlargement of the portion related to low heights from the

ground . . . 63

5.18 CL e CM at angle of attack equal to zero for different values of the ratio between the deflection of the rear and the front flaps 64 5.19 Dimensioned drawings of the A320-200 . . . 66

5.20 Model of the A320-200 employed in AVL . . . 67

5.21 Map related to the parameter CLα of the A320-200 . . . 68

5.22 Map related to the parameter CMα of the A320-200 . . . 68

5.23 Map related to the parameter CLδ e of the A320-200 . . . 69

5.24 Map related to the parameter CMδ e of the A320-200 . . . 69

5.25 Map related to the parameter CLq of A320-200 . . . 70

5.26 Map related to the parameter CMq of the A320-200 . . . 70

5.27 Map related to the Oswald factor e of the A320-200 . . . 71

6.1 TODR as a function of VR and δeR . . . 75

6.2 Constraints on rotation speed . . . 76

6.3 Distance traveled at take-off for each pair of critical values of VR and δeR . . . 77

6.4 Graphical determination of BFL and V1 . . . 78

6.5 BFL and V1 for different runway conditions . . . 79

6.6 TODR as a function of the aircraft weight . . . 81

6.7 V1 and VR adimensionalized with stall speed as a function of the aircraft weight . . . 81

6.8 V1 and VR as a function of the aircraft weight . . . 82

6.9 TODR as a function of the horizontal distance between the center of gravity and the main landing gear . . . 83

6.10 Adimensionalized V-speeds V1 and VR as functions of the hor-izontal distance between the center of gravity and the main landing gear . . . 83

6.11 V-speeds V1 and VR as functions of the horizontal distance between the center of gravity and the main landing gear . . . 84

6.12 TODR as a function of runway altitude . . . 85

6.13 Adimensionalized V-speeds V1 and VR as a function of runway altitude . . . 85

6.14 V-speeds V1 and VR as a function of runway altitude . . . 86

6.15 TODR for different flap settings . . . 87

6.16 Adimensionalized V-speeds V1 and VR for different flap settings 87 6.17 V-speeds V1 and VR for different flap settings . . . 88

6.18 Flight path in and out of ground effect . . . 89

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6.20 Incidence in and out of ground effect . . . 91

6.21 Vertical load factor in and out of ground effect . . . 92

6.22 Normal reaction in and out of ground effect . . . 92

6.23 Angular speed in and out of ground effect . . . 93

6.24 Vertical speed and climb gradient in and out of ground effect . 94 6.25 Speed in and out of ground effect . . . 94

6.26 Elevator deflection in and out of ground effect . . . 95

6.27 Flight path for the A320-200 . . . 96

6.28 Vertical load factor time history during rotation and transition-to-climb for the A320-200 . . . 97

6.29 Lift coefficient time history during rotation and transition-to-climb for the A320-200 . . . 97

6.30 Aerodynamic incidence time history during rotation and transition-to-climb for the A320-200 . . . 98

6.31 Normal reaction time history during rotation and transition-to-climb for the A320-200 . . . 98

6.32 Vertical speed and climb gradient time history during rotation and transition-to-climb for the A320-200 . . . 99

6.33 Speed time history during rotation and transition-to-climb for the A320-200 . . . 99

7.1 Determination of the critical shape of the fuselage . . . 102

7.2 Fuselage silhouette for the reference Prandtlplane configuration 104 7.3 Trajectory traveled by the rear fuselage in the case of the reference Prandtlplane configuration and the A320-200 . . . . 105

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List of Tables

5.1 CLα evaluated with JAVAFOIL . . . 53

5.2 CLα evaluated with JAVAFOIL in the second analysis . . . 53

5.3 CMα evaluated with JAVAFOIL . . . 56

6.1 Data related to the PARSIFAL-MS1 . . . 73

6.2 Data related to the A320-200 . . . 74

6.3 Friction coefficients for different runway conditions . . . 79

6.4 BFL and V1 values for different runway conditions . . . 80

6.5 Distances traveled for each segment of take-off in ground (IGE) and out of ground effect (OGE) . . . 89

6.6 V-speeds in ground (IGE) and out of ground effect (OGE) . . 90

6.7 Distances traveled during each segment of the take-off phase for the A320-200 . . . 95

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The subject of the present thesis is the numerical evaluation of take-off per-formance of an innovative aircraft configuration in the framework of the PARSIFAL project. Specifically, the main objectives are the determination of the required take-off distance for a 300 seat Prandtlplane civil transport aircraft and the evaluation of its tail-clearance requirements. In order to achieve the aforementioned objectives, a series of numerical simulations have been carried out in MATLAB, based on a non linear dynamic model which describes in detail each segment of the take-off phase. The aerodynamic effects due to the close proximity of the aircraft to the ground during take-off have been kept into account, and their influence on take-take-off performance is thoroughly discussed. Subsequently, a method for the design of the rear fuselage in compliance of the tail-clearance requirements is presented, based on the data related to the aircraft dynamics obtained in the aforementioned take-off numerical simulations. Finally, in order to highlight the advantages offered by a Prandtlplane configuration during take-off, a comparison is per-formed with the performance and tail-clearance requirements displayed by a reference conventional aircraft.

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Acknowledgements

I wish to thank all the people that have supported me throughout my thesis work. In particular, Prof. Vittorio Cipolla and Prof. Aldo Frediani for their critical revision of the present thesis and, last but not least, Karim Abu Salem, whose constant support and helpful advice have always been vital and greatly appreciated. My special thanks go to my family, my friends and whoever has supported me during my academic career.

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Chapter 1 Introduction

The enormous progress in the design and development of civil transport aircraft has made increasingly harder to devise new solutions capable of sig-nificantly improving the performance of a conventional configuration. As a matter of fact, the technological development of conventional aircraft has already reached a certain degree of maturity, and thus current aircraft are very close to an optimum. On the other hand, as time progresses, more in-terest is shown for better and better aircraft in terms of performance, fuel consumption, safety and environmental impact, and hence the only viable option for improvement in one or more of these regards is the development of novel configurations.

The innovative biplane configuration known as Prandtlplane is of particu-lar importance in the non conventional aircraft architectures landscape, as it offers potential benefits in all the aforementioned aspects. The Prandtlplane configuration is characterized by two wings with opposite sweep angles, joined by a pair of vertical surfaces. This configuration is based on the concept of “ Best wing system ”, first introduced by Prandtl, and it is notable for the higher aerodynamic efficiency compared to that of conventional aircraft of equal span in the same flight conditions. This configuration has been thor-oughly investigated by University of Pisa, and it has been proven to be suitable for the application in ULM (IDINTOS project) and medium-sized civil transport aircraft as well.

As of today, University of Pisa coordinates a project called PARSIFAL (Prandtlplane ARchitecture for the Sustainable Improvement of Future Air-pLanes), which is in the framework of the larger Horizon 2020 project fi-nanced and promoted by the European Union. PARSIFAL aims to develop new design tools to be used for Prandtlplane civil transport aircraft and assess their performance compared with that of equivalent conventional aircraft.

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The European partners collaborating with University of Pisa within PARSI-FAL are ONERA (Meudon, France), ENSAM (Bordeaux, France), Technical University of Delft (Delft, The Netherlands), DLR (Hamburg, Germany).

The configuration proposed within PARSIFAL, which is also the config-uration analyzed in the present thesis, is the size of B737 and A320, but it can boast a capacity of 300 passengers, which is typical of upper category aircraft [7].

A 3D rendering illustrating how the Prandtlplane configuration proposed within PARSIFAL would appear is shown in Fig. 1.1.

Figure 1.1: 3D rendering of the Prandtlplane configuration proposed within PARSIFAL

Several characteristics of the Prandtlplane configuration could prove to be beneficial during take-off:

- The reduction in induced drag compared to that of conventional aircraft is especially crucial during low speed flight, which is the case of take-off and landing. As a matter of fact, during those situations induced drag is the preponderant component of the overall aerodynamic drag acting on the aircraft. Besides, in the case of the Prandtlplane configuration, the front wing height with respect to the runway is not limited by clearance requirements due to the presence of the engines, as in the case of current conventional aircraft. As a result, the closer proximity to the

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ground of the front wing in the Prandtlplane configuration results in a larger influence of ground effect on its aerodynamics, whose foremost implication is a further decrease in induced drag.

- Another aspect distinguishing the Prandtlplane configuration from a conventional aircraft is that pitch is controlled by means of a pair of contra-rotating elevators, one for each wing, as shown in Fig. 1.2.

Figure 1.2: Elevators in a Prandtlplane configuration

The pair of contra-rotating elevators allows for pitch control with a pure couple, and thus in the case of the Prandtlplane configuration the deflection of the elevators does not result in any significant effect on the overall lift. Furthermore, the differential deflection of the two surfaces makes for a better degree of versatility, even allowing for small incre-ments in lift resulting from nose-up pitching moincre-ments. This peculiarity of the Prandtlplane configuration is of particular interest during take-off, as it is possible to perform maneuvers in such a way as to avoid near-stall conditions better, even when an engine failure has occurred. The main objective of the present thesis is to assess and preliminarily eval-uate the benefits of all the aforementioned characteristics of the Prandtlplane configuration on its take-off performance. The three-view drawing related to the Prandtlplane configuration proposed within PARSIFAL is shown in Fig. 1.3.

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Figure 1.3: Three-view drawing of the Prandtlplane configuration proposed within PARSIFAL

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Chapter 2 Take-off dynamics description

and modelization

2.1 Description of take-off

The take-off maneuver can be broken down into three main segments, each of them described by a specific set of ordinary differential equations:

- Ground roll: During this phase, the aircraft speed progressively increases because of the thrust generated by the propulsion system, which also has to overcome both aerodynamic drag and the friction due to the

contact with the ground. Then, when a specific speed called VR is

reached, the pilot starts the rotation by acting on the cloche, deflecting the elevators. The only degree of freedom necessary to the description of this segment is the distance traveled along the runway.

- Rotation: As soon as VR is reached, the pilot starts the rotation phase,

which consists in the rotation of the aircraft on the main landing gear. The progressive increase in incidence results in an increase of the gen-erated lift, until the aircraft leaves the ground completely and solely balanced by the aerodynamic forces. At the same time the normal re-action exerted by the runway, as well as friction, progressively decrease to the point that they vanish when the aircraft becomes airborne. - Transition to climb: Following rotation, the aircraft follows a roughly

circular path, substantially reminiscent of an ideal pull-up maneuver, which is then followed by the subsequent climb phase. According to the FAR §25.113 paragraphs (a) and (b), at the end of the runway the aircraft must reach a height at least equal to 35 ft, called screen height,

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and at that point its speed cannot be lower than 1,2 times the stall speed with flaps extracted.

The diagram presented in Fig. 2.1 illustrates the different phases of the take-off maneuver described above.

SRull SRot SM an

ho

Figure 2.1: Take-off phases and related distances

2.2 Ground roll

As mentioned in the previous section, the only “ active ” degree of freedom during this phase is the distance traveled by the aircraft along the runway. Hence, this phase is described by a single differential equation, whose expres-sion can be easily understood by referring to the diagram shown in Fig. 2.2.

L D W T MA RN RT

Figure 2.2: Forces acting on the aircraft during ground roll

The forces acting along the direction of the aircraft velocity on the runway are the thrust produced by the propulsive system T , aerodynamic drag D

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2.2 Ground roll

and the friction RT due to the contact between the landing gear tires and

the ground. Therefore, the resulting equation of motion is: W

g dV

dt = T − D − RT . (2.1)

Assuming the validity of the Coulomb friction model, friction RT and normal

reaction RN are directly proportional. As a result, introducing the rolling

friction coefficient µ, the relationship between RT and RN can be written as

follows:

RT = µRN. (2.2)

The friction coefficient, however, is a function of both speed and tire pressure, but for the purpose of this thesis the assumption that this value remains constant throughout ground-roll does not lead to gross errors. Consequently,

in order to solve equation 2.1, RN needs to be computed, and that can be

easily accomplished by considering the vertical equilibrium equation:

RN = W − L . (2.3)

The last step to be done in order to close the mathematical problem is the characterization of the aerodynamic forces L e D in terms of their respective

aerodynamic coefficients CL and CD:

L = 1 2ρV 2_SC L, (2.4) D = 1 2ρV 2_SC D. (2.5)

Additionally, the overall aerodynamic drag acting on the aircraft can be decomposed in two terms, which are parasitic and induced drag. Hence, assuming that the aircraft drag polar is parabolic, the following expression can be written:

CD = CD0 + kCL

2_. _(2.6)

It is worth noticing that, during ground roll, CL and CD remain constant

throughout this phase, as there are no variations in terms of incidence or height with respect to the ground. Thus, given all the considerations made so far and remembering equation 2.3, equation 2.1 can be rearranged in the definitive form presented below:

dV dt = g T W − ρV2(CD0 + kCL 2 ) 2W/S − µ 1 −ρV 2_C L 2W/S . (2.7)

On a final note, the value of the thrust-to-weight ratio to be inserted in equation 2.7 is variable with time, given the fact that it is a function of the

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aircraft speed. Nevertheless, it is still possible to employ an equivalent mean value, in a manner such as to obtain sufficiently accurate results. Specifically, [1] suggests using the following expression, with BPR being the bypass ratio of the engines installed on the aircraft:

T W = 0.75 5 + BPR 4 + BPR T W max . (2.8)

Therefore, equation 2.7 can be integrated either analytically or numerically, thus making it possible to calculate the distance traveled in ground-roll.

2.3 Rotation

As soon as the aircraft speed reaches VR, the pilot deflects the elevators in

order to produce an increase in the overall nose-up pitching moment. As a consequence, after a short lapse of time during which the load on the front landing gear progressively goes to zero, the aircraft begins to rotate on the main landing gear. As such, the consequent increase in incidence determines

a proportional increase in CL.

The aircraft pitch-attitude is the second degree of freedom to be added to the dynamic description of this segment of take-off, and therefore a second equation needs to be introduced in order to describe the rotational dynamics of the aircraft. The resulting set of equations of motion can be derived referring to the diagram shown in Fig. 2.3.

L D W T MA RN RT θ xB xV h d

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2.3 Rotation

The equation related to the aircraft rotational dynamics can be expressed setting the center of gravity as the reference point, simplifying the final ex-pression which is:

Iy d2_θ

dt2 = MA− hRT − dRN. (2.9)

The dimensions h and d, which describe where the main landing gear is placed with respect to the aircraft center of gravity, can be expressed in terms of the pitch angle θ as shown in Fig. 2.4.

dc d hc h θ0 θ+ θ0 PC

Figure 2.4: Center of gravity position with respect to the point of contact with the runway

Thus, from simple trigonometry considerations the following set of expres-sions can be derived:

h = phc2+ dc2sin (θ + θ0) , (2.10) d = phc2+ dc2cos (θ + θ0) , (2.11) θ0 = arctan hc dc . (2.12)

Furthermore, these expressions can be expanded using the properties of trigonometric functions, obtaining as a result the following equivalent

expres-sions in terms of hc and dc: Thus, from simple trigonometry considerations

the following set of expressions can be derived:

h = hccos θ + dcsin θ , (2.13)

d = −hcsin θ + dccos θ . (2.14)

The expressions exposed above have been obtained neglecting the ulterior vertical displacement resulting from the load acting on the main landing gear.

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The normal reaction forces due to the contact between main landing gear and runway can be derived from the following vertical equilibrium equation:

RN = W − L − T sin θ . (2.15)

Likewise, still adopting Coulomb’s model of friction, the forces due to friction can be expressed as:

RT = µ (W − L − T sin θ) . (2.16)

Finally, forces and moments acting on the aircraft can be expressed in terms of the aircraft longitudinal aerodynamic derivatives, thus obtaining the fol-lowing expressions: L = 1 2ρV 2_S(C L|δe=0+ CLq ¯ c 2V dθ dt + CLδeδe) , (2.17) MA = 1 2ρV 2_S_c_¯_(C M|δe=0+ CMq ¯ c 2V dθ dt + CMδeδe) . (2.18)

It is worth mentioning that, in these expressions, neither CLα nor CMα

ap-pear, as the values of CL and CM in absence of elevator deflection depend

also on the height from the runway because of the effects arising from the close proximity of the aircraft to the ground.

The equation of motion related to the aircraft displacement along the runway is fairly similar to equation 2.1, the only difference being the presence of an additional term related to thrust, which has been assumed to be aligned

to the xB axis of the body axes reference frame:

W g

dV

dt = T cos θ − D − RT . (2.19)

The initial conditions to be imposed in order to solve the set of differen-tial equations consisting of equation 2.9 and equation 2.19 are pitch-attitude

equal to zero and speed equal to VR. In a similar fashion to what has been

done in section 2.2, aerodynamic drag can be expressed assuming that the aircraft drag polar is parabolic, although the deflection of the elevator sur-faces might result in an appreciable increase of the coefficient k. Rotation ends as soon as the aircraft becomes airborne, which is a condition that can be translated in the following expression:

W − L − T sin θ = 0 . (2.20)

It is necessary to mention that equation 2.15 neglects the fact that the aircraft center of gravity is subjected to a roto-translational motion, and thus its

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2.3 Rotation

acceleration is different from zero. Nevertheless, the application of Rivals’ theorem allows to evaluate said acceleration from that of the point of contact with the ground, and thus the following vector equation can be written:

~aCG = ~aP C + ~Ω ∧ (~Ω ∧ (CG − P C)) +

d~Ω

dt ∧ (CG − P C) . (2.21)

The last two terms in equation 2.21, for a two-dimensional motion as the one in the case at hand, are, respectively, the normal and tangential component of the acceleration of the center of gravity with respect to the straight line joining the center of gravity to the point of contact with the runway. There-fore, according to D’Alembert’s principle, the determination of the vertical dynamics of the aircraft can be formulated in terms of an equivalent equi-librium problem, by introducing the inertial forces shown in the diagram of Fig. 2.5. d h θ+ θ0 dΩ dt FN FT

Figure 2.5: Inertial forces due to the rotational motion of the aircraft center of gravity

The expressions of the moduli of FN and FT, referring to Rivals’ theorem,

are: FT = W g dΩ dt √ d2_{+ h}2_, _(2.22) FN = W g Ω 2√_d2_{+ h}2_. _(2.23)

Consequently, the vertical equilibrium with the addition of the inertial terms equation becomes:

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Expanding the inertial terms further, the resulting definitive form is: RN = W + W d g dΩ dt − L − T sin θ − W h g Ω 2 . (2.25)

For the sake of completeness, Equation 2.19 with the addition of the inertial terms becomes: W g dV dt = T cos θ + W h g dΩ dt + W d g Ω 2_{− D − R} T . (2.26)

Thus, the presence of inertial forces slightly alters both the expression of the nose-down pitching moment related to the reaction forces and the condition under which the aircraft leaves the ground. However, in the vast majority of cases, the influence of these additional terms is rather limited.

2.4 Transition to climb

Following the rotation phase, the aircraft has gained enough lift to become airborne and thus its vertical speed, or “ rate of climb ”, becomes the third degree of freedom necessary in order to describe the transition-to-climb phase. As a consequence, a third equation related to the aircraft vertical speed needs to be introduced in the set of equations of motions necessary for the description of this phase. The equations of motion can be written by referring to the diagram in Fig. 2.6.

α xB x γ xV θ L W T D MA

Figure 2.6: Forces acting on the aircraft during transition-to-climb

The absence of the terms linked to the contact with the runway results in the following expression for the equation related to the rotational dynamics of the aircraft:

Iy d2_θ

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2.4 Transition to climb

Conversely, the equation of the horizontal dynamics is slightly different from the form employed for rotation, as a non zero climb gradient implies the rotation of the aerodynamic forces L and D with respect to the local vertical axes reference frame. Hence, the resulting equation can be expressed as:

W g

dV

dt = T cos θ − D cos γ − L sin γ . (2.28)

Finally, the last differential equation needed in order to describe the air-craft dynamics during transition is related to the vertical speed, and it can be expressed as follows:

W g

dVz

dt = T sin θ + L cos γ − W − D sin γ . (2.29)

The aircraft pitch-attitude and the angle of attack can be expressed in terms of the following equations:

γ = arctan Vz

V

, (2.30)

α = θ − γ . (2.31)

On a final note, the evaluation of the horizontal and vertical distances

XT T C and ZT T C traveled by the center of gravity of the aircraft during the

transition-to-climb can be carried out by integrating V and Vz with respect

to time, leading to the following expressions:

XT T C = Z T T C V dt , (2.32) ZT T C = Z T T C Vzdt . (2.33)

The summary of all the equations of motion discussed in this chapter is shown in the box below.

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Ground-roll: W g dV dt = T − D − RT Rotation: W g dV dt = T cos θ − D − RT Iy d2_θ dt2 = MA− hRT − dRN Transition-to-climb: W g dV

dt = T cos θ − D cos γ − L sin γ

W g

dVz

dt = T sin θ + L cos γ − W − D sin γ

Iy d2_θ

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Chapter 3 Numerical simulation of the

take-off phase

In the previous chapter, the derivation of the set of equations that allows to analyze the dynamics of the take-off maneuver in every segment has been described. The model complexity, mainly ascribable to the high level of non linearity and the strong coupling between the degrees of freedom of the aircraft, requires an approximate numerical approach in order to solve the dy-namic problem. As such, a variety of different scripts has been implemented in MATLAB, whose purpose is the main focus of this chapter.

3.1 Numerical discretization of differential

problems

Mathematical problems based on a set of ordinary differential equations, such as the one discussed in the present thesis, can be solved performing a dis-cretization process that involves the subdivision of the time domain in very small intervals. As a consequence, all the differential operators appearing in the dynamic model must be redefined accordingly to the selected discretiza-tion process. In fact, the discretizadiscretiza-tion process can be carried out in multiple ways, and a considerable amount of options are available depending on the number of different points considered within the selected time step for the approximate evaluation of time derivatives. In the simplest and most known form of time discretization, time derivatives are approximated as follows:

du

dt ≈

u(t + ∆t) − u(t)

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Another important decision to be made, in order to obtain an approxi-mate solution of an ordinary differential equation, is how to carry out the integration of the right hand side. In that regard all possible methods can be divided in two categories:

- Explicit methods: In these schemes the right hand side is evaluated at the initial point in the considered time step. The simplest first order explicit scheme is the explicit Euler method, whose form is:

u(t + ∆t) − u(t)

∆t = f (t) . (3.2)

This scheme allows to obtain an approximate numerical solution for any given ordinary differential equation, by solving a series of linear equations. Hence, these methods are very fast, but they can display numerical instability in some particular instances.

- Implicit methods: In these schemes, on the contrary, the right hand side is evaluated at the final point in the considered time step. The simplest first order implicit scheme is the implicit Euler method, whose form is:

u(t + ∆t) − u(t)

∆t = f (t + ∆t) . (3.3)

This scheme has the advantage of being numerically stable whatever the time step, but the equation that has to be solved in order to ob-tain the approximate solution at each step is non linear. Therefore, these methods are highly computationally demanding and considerably slower than their explicit counterpart.

The convergence of the approximate solution can be made faster either by reducing the time step, or by employing higher order schemes.

Explicit Euler schemes have been chosen for solving the set of differential equations describing take-off dynamics, setting a time step equal to 0,001 s.

3.2 Numerical procedure for the simulation

of take-off dynamics

On the basis of the aforementioned considerations, it is possible to implement a numerical procedure for the simulation of the dynamics of each segment of take-off.

In the case of ground roll, the only equation needed for the dynamic simulation of this phase can be solved by computing at each time step the

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3.2 Numerical procedure for the simulation of take-off dynamics aircraft speed from the value obtained at the previous step. Specifically, the application of the explicit Euler scheme yields the following approximate equation: Vn+1= Vn+ g ∆t T W − Dn W − µ 1 − Ln W . (3.4)

The numerical procedure comes to a halt as soon as VR has been reached.

The whole procedure is illustrated in the block diagram shown in Fig. 3.1.

Figure 3.1: Numerical procedure for the simulation of ground roll

The simulation of rotation is described by one more equation, related to pitch. The application of the explicit Euler scheme to these set of equations yields the following approximate expressions:

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Vn+1 = Vn+ g ∆t T W cos θn− Dn W − RT n W , (3.5) ωn+1 = ωn+ ∆t MAn Iy − hn RT n Iy − dn RN n Iy , (3.6) θn+1 = θn+ ∆t ωn. (3.7)

In this case, the procedure comes to an end when the reaction force exerted by the runway goes to zero. The procedure is illustrated in the block diagram shown in Fig. 3.2. Equazioni Rullaggio START RN<0 END YES NO V_R, 0 , 0 , δ_R V_n+1, θ_n+1, ω_n+1 V_n, θ_n, ω_n Rotation Equations

Figure 3.2: Numerical procedure for the simulation of rotation

The elevator deflection δe time history can be modeled as a short ramp,

simulating the time needed to achieve the desired deflection angle, followed by a steady state.

Last, the simulation of the transition-to-climb is based on the following set of expressions, while the resulting procedure is illustrated in the block diagram shown in Fig. 3.3.

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3.2 Numerical procedure for the simulation of take-off dynamics Vn+1 = Vn+ g ∆t T W cos θn− Dn W cos γn− Ln W sin γn , (3.8) Vz n+1 = Vz n+ g ∆t T W sin θn+ Ln W cos γn− 1 − Dn W sin γn , (3.9) ωn+1 = ωn+ ∆t MAn Iy , (3.10) θn+1 = θn+ ∆t ωn, (3.11) γn+1 = arctan Vz n+1 Vn+1 , (3.12) αn+1 = θn+1− γn+1. (3.13) Equazioni Rullaggio START h h_o END YES NO V_M, θ_M, ω_M, 0 , δ_e(t) V_n+1, θ_n+1, ω_n+1, γ_n+1 Vn, θn, ωn, γn Transition-to-climb Equations <

Figure 3.3: Numerical procedure for the simulation of transition-to-climb

The procedure continues until the screen height of 35 ft is reached, as estab-lished by FAR §25.113 paragraphs (a) and (b). It is important to mention

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that the block called “ Transition-to-climb Equations ” also includes the de-termination of the aerodynamic incidence and the evaluation of the distance traveled along and above the runway.

3.3 Modeling of the elevator deflection time

history

The set of differential equations related to the transition-to-climb contains

the elevator deflection δe as an input. Consequently, it is necessary to

as-sume an elevator deflection time history to ensure that the simulated ma-neuver satisfies the requirements of comfort of passengers and minimal fuel consumption. For instance, a simple unchecked maneuver is not satisfactory enough in that regard, given the fact that the resulting simulated flight path suffers from the presence of very large oscillations, as shown in Fig. 3.4.

Figure 3.4: Flight path and rate of climb for an unchecked maneuver

The main reason for this fact is that the elevator deflection required to per-form the rotation is not the same needed to trim the aircraft during the subsequent climb phase.

Despite the fact that the oscillations are clearly visible only long after the end of take-off, the effects on the evaluation of take-off dynamics and performance are still not negligible. Therefore, it is necessary to assess how the elevator deflection needed to perform a given maneuver varies with the

vertical load factor nz, in order to understand how transition-to-climb and

the subsequent climb phase differ in that regard. To that end, if the terms related to ω are neglected, the linear system describing the generic trim

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3.3 Modeling of the elevator deflection time history problem is: CLα CLδe CMα CMδe α δe = nz 2W/S ρV2 −CM0 . (3.14)

The solution of the above linear system related to the elevator deflection

angle δe can be written as follows:

δe= − CM0CLα+ nz 2W/S ρV2 CMα CLαCMδe− CLδeCMα . (3.15)

Hence, if the aircraft is stable in pitch, which is the case of most conven-tional aircraft and the Prandtlplane configuration as well, the larger the load factor associated with the maneuver, the larger the elevator deflection angle required to perform it. As a result, during the climb the elevator deflection required is lower than that needed during the transition-to-climb, which is in turn lower than that needed to perform the rotation.

The simplest maneuver that satisfies all the conditions outlined above is a two-stage partially checked maneuver, whose generic time history is illus-trated in the diagram shown Fig. 3.5.

t δe

δeR

δeF

tR τd

Figure 3.5: Two-stage maneuver

The maneuver is described by the parameter τd, which is the time interval

between the end of rotation and the beginning of the reduction in the elevator

deflection applied, and δef, which is the final value of the elevator deflection

angle. The transition from zero to δeR is assumed to last 1 s, and the resulting

“ slope ” in the diagram of Fig. 3.5 is considered to be the same for the transition from δeR to δef.

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3.4 Optimization procedure for a two-stage

maneuver

The choice of the values to be used for the two parameters describing the two-stage maneuver defined earlier needs to be based on the amplitude of the oscillations observed in the resulting maneuver. To that end, a measure of the amplitude of these undesired oscillations needs to be rigorously defined. Specifically, the following expression has been found to be suitable for that purpose: A= Z t2 t1 dVz dt 2 dt . (3.16)

This mathematical expression, which is reminiscent of an integral norm,

mea-sures how large the variations of the vertical speed Vz are within the time

interval considered. As a matter of fact, if it is assumed that the vertical speed at steady state is the sum of a constant and a purely oscillatory com-ponent, the expression above filters only the oscillatory one and evaluates its amplitude in absolute value. Thus, this expression constitutes, in fact, a well-defined means of measuring the oscillatory behavior of the vertical

speed. Nevertheless, the time interval comprised between t1 and t2 has to

be sufficiently extended with respect to the period of the oscillations, and placed in such a way as not to include the initial response.

On these bases, a code written in MATLAB has been used to evaluate, given rotation speed and the initial elevator deflection angle, the values of

the two parameters τd and δef that minimize the control parameter defined

above, thus minimizing the oscillatory response. In order to make the code execution faster the following features have been implemented:

- First, set a pre-optimized value of τd given rotation speed and elevator

deflection, the code searches for the value of δef that minimizes the

control parameter. Subsequently, the optimization procedure is carried

out on τd with the value of δef determined before, in order to find the

final pair of values that minimizes the control parameter. The method exposed above has been made possible by the fact that the search for the minimum in the problem at hand can be carried out on each of the two parameters involved separately.

- The range step of the parameters for the problem at hand is adaptive, meaning that its sign and absolute value change dynamically as the control parameter approaches the minimum. Specifically, the control

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3.5 Simulation of take-off dynamics given a predetermined pitch-attitude time history

minimum, and thus it is possible to first perform analyses with large range steps as long as the control parameter decreases at each step, in order to rapidly scan the range within which the minimum is searched. Subsequently, as soon as the control parameter starts increasing, the range step is reduced and its sign changed, in order to scan the range within which the minimum is searched backwards with a higher degree of accuracy. As soon as the control parameter starts increasing again, the range step is reduced and its sign changed a second time, and thus the same set of operations are carried out in a loop until the minimum has been found with the desired precision.

The outcome of the optimization procedure discussed above is shown in Fig. 3.6 .

Figure 3.6: Flight path and rate of climb for a two-stage maneuver

3.5 Simulation of take-off dynamics given a

predetermined pitch-attitude time

history

The method for the simulation of the transition-to-climb phase exposed ear-lier, while effective, still requires an optimization procedure to be carried out before obtaining the desired results, and thus it might be excessively time consuming when several simulations in a row have to be made in order to assess take-off performance. Fortunately, it is also possible to simulate the transition-to-climb assuming that the aircraft is constrained to follow a spec-ified time history set a priori for one of the dynamic variables involved in the problem at hand [9].

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Specifically, pitch-attitude θ is the most suitable choice for that purpose, and therefore an adequate time history has to be selected such as to be sufficiently representative of an actual maneuver. Moreover, the order of continuity of the function selected as the time history must be sufficiently high, in order to prevent the simulation from being affected by the presence of unnatural peaks. On these bases, a possible choice for the function describing

the pitch-attitude time history is the following one, with θ0 and ω0 being,

respectively, incidence and angular speed at the end of rotation and T the time needed to reach the steady state condition:

θ(t) = ω0 t − 2 T π 1 − cos π t 2 T + θ0. (3.17)

The generic appearance of the time history defined in equation 3.17 is shown in Fig. 3.7.

θ

t

θ0 T ω0

Figure 3.7: Assumed pitch-attitude time history

The selected time history satisfies both the incidence and angular speed ini-tial conditions, its order of continuity is sufficiently high and the resulting maneuver is plausible. Nevertheless, multiple choices are possible for the function to be used as a pitch-attitude time history. For instance, a polyno-mial expression suitable for that purpose can be found by imposing a series of conditions that ensures the order of continuity needed in order not to have a solution affected by spikes. As a result, the conditions listed above can be translated in a set of linear equations which yields the coefficients of the polynomial, whose order is equal to the number of conditions imposed. The

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solution of the linear system shown below yields the coefficient of the fourth

order polynomial of the form A t4 _{+ B t}3 _{+ C t}2 _{+ D t + E that ensures the}

continuity up to the second order for a pitch-attitude time history assumed to achieve a steady state after a time period equal to T:

      0 0 0 0 1 0 0 0 1 0 0 0 2 0 0 4 T3 _{3 T}2 _{2 T 1 0} 12 T2 _{6 T} ₂ _{0 0}             A B C D E       =       θ0 ω0 ˙ω0 0 0       (3.18)

The comparison between the angular speed time histories resulting from the assumption of the two different pitch-attitude time histories is shown in Fig. 3.8. 0 1 2 3 4 5 6 t [s] 0 1 2 3 4 5 6 7 8 ! [deg/s] Polynomial expression Trigonometric expression

Figure 3.8: Angular speeds time histories resulting from different assumed pitch-attitude time histories

The angular speed time history resulting from the trigonometric assumed time history displays a corner, as opposed to the one resulting from the polynomial assumed time history

On these bases, a second MATLAB code has been written, whose de-scription is provided in the following points:

- in order to obtain the elevator deflection angle needed at the consid-ered time step, the equation related to the rotational dynamics can be

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rearranged as follows: δe = 2 Iy ρ V2_S d2_θ¯ d t2 − CM|δe=0− CMq ¯ c 2 V d¯θ d t CMδe , (3.19)

where ¯θ is the assumed pitch-attitude time history;

- the elevator deflection angle determined at the previous point is in-putted in the remaining two differential equations, allowing to deter-mine the values of the vertical and horizontal speed at the time step considered.

The block diagram in Fig. 3.9 illustrates the procedure discussed above.

Figure 3.9: Numerical procedure for the simulation of transition-to-climb by assuming a predetermined pitch-attitude time history

The flight path resulting form this alternative method for the simulation of the transition-to-climb phase is shown in Fig. 3.10, with the time interval

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Figure 3.10: Flight path and rate of climb resulting from the selected pitch-attitude time history

The distance traveled during the transition-to-climb estimated with this method is not significantly dissimilar with that obtained through the appli-cation of the first method. As a matter of fact, the distance evaluated with the first method is equal to 199 m, while the second method yields a value equal to 215 m, which is conservative.

Fig. 3.11 shows the comparison between the elevator deflection time his-tories obtained by applying to the case study the different methods discussed in this thesis for the simulation of the transition-to-climb phase.

0 1 2 3 4 5 6 t [s] -5 0 5 10 15 20 /e [deg] Two-stage 4th_{order polynomial}

Figure 3.11: Elevator deflection time histories resulting from different simu-lation methods

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While in the two-stage maneuver the elevator deflection simply decreases reaching the steady state value, the assumption of a fourth order polyno-mial pitch-attitude time history results in a compensation that precedes the steady state. Nevertheless, the deflection angle at steady state is substan-tially identical for the two maneuvers.

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Chapter 4 Ground effect aerodynamics

During take-off, the close proximity of the aircraft lifting surfaces with the ground affects their aerodynamic characteristics, both in terms of aerody-namic drag and lift-generating capabilities. This section covers the theo-retical fundamentals of ground effect aerodynamics, as well as its numer-ical evaluation performed with the aerodynamic VLM solver AVL. Subse-quently, a comparison between the results of a conventional configuration and a Prandtlplane configuration is made.

4.1 Theoretical and experimental models for

the description of ground effect

The first tangible effect of the aircraft proximity to the ground is the sub-stantial reduction of induced drag, which can be estimated with a theoretical model devised by C. Wieselsberger [13]. This model, based on the hypothe-sis of potential flow, schematizes the wake of each lifting surface as a vortex sheet composed of “ horseshoe ” vortices, and allows to evaluate the reduction in drag caused by ground effect by adding a mirrored copy of the aircraft ge-ometry with respect to the ground plane, as schematically shown in Fig. 4.1.

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Γ

Figure 4.1: Mirroring procedure for ground effect evaluation

The resulting system of vortices is symmetric with respect to the ground plane, and thus the resulting flow field is symmetric as well. Therefore, the vertical component of the velocity field on the ground plane is equal to zero, meaning that the flow field obtained satisfies the boundary condition imposed by the presence of the ground.

This problem can be solved with the application of Prandtl’s theory of multiplanes, yielding the following expression:

∆k = −σ 1

πAR, (4.1)

where ∆k is the reduction in the coefficient k employed in the calculation of the induced drag coefficient, AR is the wing aspect ratio and σ is a coefficient that quantifies the effects produced by the mutual interaction between the wing and its mirrored copy. The approximate analytical expression of the influence coefficient σ in Equation 4.1 found by Prandtl is:

σ = 1 − 1.32hCA b 1.05 + 7.4hCA b , (4.2)

where hCA is the wing height from the ground. This theoretical model has

been experimentally tested several times, and it has been proved that it re-mains valid for any value of the wing height from the ground but those typical of ekranoplanes, which however are not of interest for the cases investigated in this thesis.

Another effect correlated to the proximity to the ground is the increase

of the aerodynamic coefficient CLα, which enhances the lift-generating

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4.2 Numerical evaluation of ground effect with AVL

regarding this phenomenon, but there is still a large amount of experimen-tal data on the matter [16]. In particular, Fink, Lastinger [12] and Carter [11] indisputably confirmed with their experiments the existence of a definite effect for a large variety of airfoils differing in aspect ratio and thickness.

4.2 Numerical evaluation of ground effect

with AVL

The program employed for the numerical evaluation of the aerodynamic char-acteristics of the different configurations here analyzed is AVL, developed by M. Drela and H. Youngren [17]. This code is a VLM, short for “ Vortex

Lattice Method”, and is structured as follows:

- each lifting surface, approximated with their mean surface, is subdi-vided by a grid in cells that contain a control point, where the boundary conditions on the velocity are imposed;

- a “ horseshoe ” vortex is assigned to each cell, which presents a constant circulation value along its length, as shown in Fig. 4.2;

- the application of the Biot-Savart law allows for the evaluation of the velocities induced by the vortices assigned to each cell, and the imposi-tion the condiimposi-tion of tangency of the flow field to the mean line where the control points are located defines a linear problem in terms of the circulation values of each vortex;

- the resulting solution, employing the Kutta-Jukowski theorem, allows for the computation of the spanwise lift distribution, total lift and aerodynamic moment, induced drag.

Because of the hypothesis of potential flow, this method is neither suitable for the investigation of viscous effects nor near-stall conditions. Furthermore, the geometry of the selected airfoils is relevant only as long as their mean line is concerned. Nevertheless, AVL allows for the rapid evaluation of a large number of case studies related to a single configuration, whether conventional or not, in a variety of different operative conditions and, as such, it is an extremely useful tool in the first stage of the designing process and for a first investigation of the performances of a given aircraft.

AVL builds the aircraft model on the basis of a text file identified by

the extension .avl, where the geometry of each surface and the mesh to be applied on them for defining the control points are specified. In the input

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z x y 1 2 3 Γ1 Γ2 Γ3

Figure 4.2: Lifting surface as modeled in a VLM

file it is possible to define slender body elements such as fuselages, whose interaction with the flow field is modeled in terms of “ source-doublet ” pairs, but it is also possible modeling fuselages by inserting their top-view profile, without losing in accuracy. AVL allows to define control surfaces such as ailerons, elevators and flaps on a main surface, specifying their type, spanwise extension and chordwise hinge axis position. Moreover, it is possible to define a correlation between the deflection of two different control surfaces placed anywhere on the aircraft, by identifying them with the same denomination and subsequently specifying the gain with respect to the deflection angle considered in defining the entity of the control applied. However, in this way the deflection of a control surface is modeled as a rotation of the mean line of the airfoils composing the surface where the control surface is defined, and thus any kind of high-lift device is modeled as a simple plain flap.

In AVL two approaches are possible for the evaluation of ground effect: 1. Specifying in the input file the symmetry of the aircraft model with

respect to the ground plane, in a similar fashion to Wieselsberger’s model.

2. Inserting an additional surface simulating the ground plane presence, specifying that this surface is unaffected by any rotation performed by the aircraft and that the forces and moments computed on it are not to be included in the final results.

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4.3 Validation of the numerical results obtained in AVL

The first approach is preferable, as in the second approach the boundary conditions to be imposed because of the presence of the ground are satisfied in a limited number of points, which are the control points of the additional surface that simulates the presence of the ground.

4.3 Validation of the numerical results

obtained in AVL

A validation procedure has been carried out in order to assess the accuracy of AVL in simulating ground effect. Specifically, the results obtained in AVL in the case of a single rectangular wing with aspect ratio equal to 7,2 have been compared with those predicted by the models and the experimental data presented in Section 4.1.

The comparison between the results obtained numerically and those pre-dicted by the Wieselsberger’s model regarding induced drag is shown in Fig. 4.3, with σ being the influence coefficient appearing in Equation 4.1.

0 0.05 0.1 0.15 0.2 0.25 0.3 h_CA/b 0.2 0.3 0.4 0.5 0.6 0.7 0.8 < AVL Wieselsberger

Figure 4.3: Induce drag evaluated in AVL compared with that predicted by Wieselsberger

The two curves obtained almost overlap and hence, since Wieselsberger’s model has been amply proven to be accurate [15] [16], it can be concluded that AVL predicts the entity of the influence of ground effect on induced drag reasonably well. In fact, this outcome is not completely unexpected, since in both AVL and Wieselsberger’s model the evaluation of ground effect is

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carried out with the addition of a mirrored copy of the aircraft with respect to the ground plane.

The evaluation of CLα has been carried out by employing the same airfoil

used by Fink and Lastinger in their wind tunnel tests, a Glenn Martin 2 airfoil modified in order to provide it with a flat bottom [12]. The geometry of the modified Glenn Martin 2 airfoil employed by Fink and Lastinger is shown in Fig. 4.4. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c -0.1 0 0.1 0.2 y/c

Figure 4.4: Airfoil Glenn Martin 2

The comparison between the experimental data obtained by Fink and Lastinger and the results obtained in AVL is shown in Fig. 4.5.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 h_CA/c 0 0.5 1 1.5 2 2.5 " C L , AVL Fink-Lastinger

Figure 4.5: CLα evaluated in AVL compared with experimental data obtained

by Fink and Lastinger

The two curves are almost coincident in this case as well, and therefore AVL is a suitable tool for a first evaluation of ground effect.

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Chapter 5 Aerodynamic characteristics in

ground effect

In this chapter the numerical results obtained in AVL concerning the main aerodynamic characteristics of the reference Prandtlplane configuration are exposed. Moreover, a comparison is made with the results obtained for a conventional configuration, highlighting even more the peculiarities detected for the Prandtlplane configuration.

5.1 Computational procedure and data

elaboration

In order to investigate the specific aerodynamic behavior in ground effect of a given aircraft, a code capable of interfacing with AVL and then elaborat-ing the resultelaborat-ing data in an easy-to-read format has been implemented in MATLAB. Specifically, the code is structured in the following steps:

1. First of all, the range of values considered in the analysis for pitch angle and height of the aircraft from the ground is defined. At the same time, the code identifies the conditions for which there is contact with the runway in terms of the maximum allowable incidence beyond which a tail-strike occurs. The expression employed in order to compute this angle, whose derivation is made evident in Fig. 5.1, assumes the following form: θmax = arcsin hCG lf − xCG

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lf

x_CG

hCG

θmax CG

Figure 5.1: Maximum allowable angle in the proximity of the runway

2. Subsequently, when the input file indicating the height and position of the center of gravity along the fuselage has been set up, the code instructs AVL to perform an analysis with the pitch angle considered and then save the data obtained in a text file. The analysis is per-formed only for the cases where there is no contact with the ground, i.e. the pitch angle considered is lower than the critical value previously computed.

3. Once all the necessary analyses in AVL have been performed, the pro-gram extracts all the relevant data, which is then stored and saved as matrices ready to be used for the subsequent elaboration and employ-ment in MATLAB. Specifically, these matrices are used for the evalua-tion of the aerodynamic derivatives at any given time in the codes for the simulation of the take-off maneuver.

4. Lastly, the matrices obtained at the previous step are used for the generation of contour lines, which visually represent the trend of a given parameter with respect to height from the ground and pitch angle. Furthermore, in the resulting graph the line representing the values of the critical angle is plotted as well.

The reference Prandtlplane configuration analyzed in the present thesis is the PARSIFAL-MS1, which is the object of the investigations carried out within the PARSIFAL project. The dimensioned drawings of this configura-tion are shown in Fig. 5.2, while the related model employed in the analyses carried out in AVL is shown Fig. 5.3.

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5.1 Computational procedure and data elaboration

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5.2 Numerical evaluation of CL and CLα for the reference

Prandtlplane configuration

5.2 Numerical evaluation of C

L

and C

Lα

for

the reference Prandtlplane configuration

3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] CL , 3.8 3.9 3.9 4 ₄ 4.1 _4.1 4.2 4.2 4.3 4.3 4.4 4.4 4.4 4.5 4.5 4.5 4.6 4.6 4.6 4.7 4.7 4.8 4.8 4.9 5 5.1 5.2

Contact with the ground

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 h [m] 0 1 2 3 4 5 6 7 8 9 10 , [deg] CL , 4.3 4.4 4.4 4.4 4.5 4.5 _4.5 4.6 4.6 4.7 _4.7 4.7 4.8 4.8 4.8 4.9 4.9 4.9 5 5 5 5.1 5.1 5.2 5.3 5.4

Figure 5.4: Map related to the parameter CLα for the PARSIFAL-MS1, and

enlargement of the portion related to low heights from the ground

In the graph shown in Fig. 5.4 it is rather evident how a condition of close

proximity to the ground determines a noticeable increase of CLα, which is not

dissimilar from the behavior displayed by a single isolated wing in ground effect, as exposed in Chapter 4. Nevertheless, this increment becomes smaller as the incidence increases.

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In addition to the analysis carried out in AVL, a simple two dimensional potential solver has been used for investigating a similar situation to the previous one. Specifically, the software JAVAFOIL has been selected, given the fact that it allows to simulate ground effect through the “ mirroring ” operation previously shown in Chapter 4. The test has been conducted on two airfoils NACA0012 properly disposed in order to schematically represent the configuration analyzed in AVL. The diagram in Fig. 5.5 illustrates the layout of the two airfoils representing the wings of the Prandtlplane config-uration adopted in JAVAFOIL, while Fig. 5.6 shows how the configconfig-uration is visualized in JAVAFOIL with angle of attack equal to zero. JAVAFOIL

¯ c 3 ¯c 0.2 ¯c ¯ c 0.5 ¯c 0.4 ¯c CR

Figure 5.5: Geometry employed for the analyses carried out in JAVAFOIL

Figure 5.6: Geometry employed in JAVAFOIL at angle of attack equal to zero

applies rotations with respect to the center of rotation CR indicated in the diagram of Fig. 5.5. The results of this additional analysis are reported in table 5.1.

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5.2 Numerical evaluation of CL and CLα for the reference Prandtlplane configuration α [deg] CL CLα [1/rad] 0 0,161 – 1 0,284 7,05 2 0,394 6,31 3 0,495 5,79 4 0,589 5,36 5 0,677 5,07

Table 5.1: CLα evaluated with JAVAFOIL

The trend of CLα as a function of incidence found in this two-dimensional

analysis agrees with the results obtained in AVL, as there is a progressive

reduction of CLα with incidence.

The influence of the center of rotation position on the parameter CLα has

also been investigated in order to understand if the trend observed might be ascribed to the relative motion of the two wings with respect to the ground plane. Specifically, a second analysis has been conducted in JAVAFOIL

moving CR rearwards of a distance equal to ¯c. The results of this analysis

are shown in Table 5.2.

α [deg] CL CLα [1/rad] 0 0,161 – 1 0,286 7,14 2 0,393 6,13 3 0,490 5,56 4 0,580 5,16 5 0,665 4,87

Table 5.2: CLα evaluated with JAVAFOIL in the second analysis

Despite the fact that, as expected, the values of CLα with the rearmost

CR are smaller than the respective values with CR placed on the front,

the differences detected are not ascribable to the relative motion of the two wings with respect to the ground plane, considering that the position of the center of rotation has a large influence on the entity of the displacements in a rotation, as shown in Fig. 5.7.

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Figure 5.7: Disposition of the wings for α equal to 10o _{for different positions} of CR

In fact, the same trend of CLα with incidence in ground effect has been

observed also for an isolated wing both in AVL and JAVAFOIL.

On a final note, considering that during the take-off simulation the

pa-rameter that is actually used is the lift coefficient CL, the map shown in

Fig. 5.8 is related to the values of the parameter CL obtained in AVL.

3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] CL 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 1 1.1 1.1 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.7 1.8 Contact with the ground

Figure 5.8: Map related to CL for the PARSIFAL-MS1

The values obtained for high values of pitch angle are to be considered wrong, as AVL is not capable of predicting the aerodynamic behavior of a given con-figuration in near-stall conditions.

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5.3 Numerical evaluation of CM and CMα for the reference

5.3 Numerical evaluation of C

M

and C

Mα

for

the reference Prandtlplane configuration

3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] Cm , -1.85 -1.85-1.8_-1.75 -1.7 -1.7 -1.65 -1.65 -1.6 -1.6 -1.55 -1.55 -1.5 -1.5 -1.45 -1.45 -1.4 -1.4 -1.4 -1.35 -1.35 -1.35 -1.3 -1.3 -1.3 -1.25 -1.25 -1.25 -1.2 -1.2 -1.2 -1.15 -1.15 -1.1 -1.1 -1.05 -1.05 -1 -1 -1 -0.95 -0.95 -0.9 -0.9 -0.85 -0.85 -0.8 -0.8 -0.75 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 h [m] 0 1 2 3 4 5 6 7 8 9 10 , [deg] Cm , -2.05-1.95-1.9-2 -1.85-1.8-1.75 -1.7 -1.7 -1.65 -1.65 -1.6 -1.6 -1.55 -1.55 -1.5 -1.5 -1.45 -1.45 -1.4 -1.4 -1.35 -1.35 -1.3 -1.3 -1.25 -1.25 -1.2 -1.2 -1.15 -1.15 -1.1 -1.1 -1.05 -1.05 -1 -1 -1 -0.95 -0.95 -0.9 -0.9 -0.85 -0.85 -0.8 -0.8 -0.75 -0.75 -0.7 -0.7 -0.65 -0.65 -0.6 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 0

Figure 5.9: Map related to the parameter CMα for the PARSIFAL-MS1, and

enlargement of the portion related to low heights from the ground

The map presented in Fig. 5.9 highlights how the proximity with the ground, for low values of incidence, results in the reduction of the nose-down pitching moments due to aerodynamic effects. On the other end, the aircraft pitch stiffness progressively increases with incidence. The main reason for

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closer proximity to the ground results in a larger influence of ground effect compared to that acting on the rear wing.

An additional analysis in JAVAFOIL has been carried out in this case as well, employing the same geometry illustrated in Fig. 5.5. The results of this analysis are shown in Table 5.3.

α [deg] CM CMα [1/rad] 0 -0,054 – 1 -0,048 0,34 2 -0,052 -0,23 3 -0,063 -0,66 4 -0,079 -0,92 5 -0,099 -1,12

Table 5.3: CMα evaluated with JAVAFOIL

Hence, JAVAFOIL predicts a similar behavior to the one evaluated with AVL.

Finally, the map related to the values of CM obtained numerically with

AVL is shown in Fig. 5.10.

3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] Cm -0.55 -0.5 -0.5 -0.45 -0.45 -0.4 -0.4 -0.35 -0.35 -0.3 -0.3 -0.25 -0.25 -0.2 -0.2 -0.15 -0.15 Contact with the ground

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5.4 Numerical evaluation of CLδ e and CMδ e for the reference

5.4 Numerical evaluation of C

L_{δ e}

and C

M_{δ e}

for the reference Prandtlplane

configuration

3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] CL /e -0.1 -0.1 -0.09 -0.09 -0.08 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 h [m] 0 1 2 3 4 5 6 7 8 9 10 , [deg] CL /e -0.06 -0.06 -0.05 -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 0 0

Figure 5.11: Map related to the parameter CLδ e for the PARSIFAL-MS1,

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3 4 5 6 7 8 9 10 h [m] 0 2 4 6 8 10 12 14 16 18 20 , [deg] Cm /e 1.77 1.78 1.79 1.8 1.8 1.81 1.82 1.83 1.84 1.85 1.85 1.86 1.86 1.87 _1.87 1.88 _1.88 1.89 _1.89 1.9 _1.9 1.91 1.91 1.92 1.92 1.93 1.93 1.94 1.94 1.95 1.95 1.96 1.96 1.97 1.97 1.98 1.98 1.99 1.99 2 2 2.01 2.01 2.02 2.02 2.03 2.03 2.04 2.04 2.05 2.05 2.06 2.06 2.07 2.07 2.08 2.09 2.1 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.2

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 h [m] 0 1 2 3 4 5 6 7 8 9 10 , [deg] Cm_/_e 2 2.01 2.02 2.02 2.03 2.03 2.04 2.04 2.05 2.05 2.06 2.06 2.07 2.07 2.08 2.08 2.08 2.09 2.09 2.09 2.1 2.1 2.1 2.11 2.11 2.11 2.12 2.12 2.12 2.13 2.13 2.14 2.14 2.15 2.15 2.16 2.16 2.17 2.17 2.18 2.19 2.2

Figure 5.12: Map related to the parameter CMδ e for the PARSIFAL-MS1,

and enlargement of the portion related to low heights from the ground

The values of the control derivatives are shown in Fig. 5.11 and in Fig. 5.12.

The parameter CLδ e, which is rather small in magnitude even in cruise

con-ditions, is subjected to a further decrease to the point of becoming positive for very small values of the height of the aircraft from the ground. These

values of CLδ e are typical of a Prandtlplane configuration, given the fact that

the pitch control is realized with a pair of contra-rotating elevators placed on the front and the rear wing, whose differential rotation can be tuned in order to grant a pure couple control.

Analysis of take-off performance of a Prandtlplane 300 seat civil transport aircraft

DIPARTIMENTO DI INGEGNERIA CIVILE E

INDUSTRIALE

Corso di Laurea Magistrale in Ingegneria Aerospaziale

Analysis of Take-Off Performance

of a PrandtlPlane 300 seat

civil transport aircraft

Supervisors:

Prof. Vittorio Cipolla

Prof. Aldo Frediani

Karim Abu Salem

Candidate:

Mario Bianchi

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Take-off dynamics description

and modelization

2.1

Description of take-off

2.2

Ground roll

2.3

Rotation

2.4

Transition to climb

Chapter 3

Numerical simulation of the

take-off phase

3.1

Numerical discretization of differential

problems

3.2

Numerical procedure for the simulation

of take-off dynamics

3.3

Modeling of the elevator deflection time

history

3.4

Optimization procedure for a two-stage

maneuver

3.5

Simulation of take-off dynamics given a

predetermined pitch-attitude time

history

θ

t

Chapter 4

Ground effect aerodynamics

4.1

Theoretical and experimental models for

the description of ground effect

4.2

Numerical evaluation of ground effect

with AVL

4.3

Validation of the numerical results

obtained in AVL

Chapter 5

Aerodynamic characteristics in

ground effect

5.1

Computational procedure and data

elaboration

5.2

Numerical evaluation of C

and C

for

the reference Prandtlplane configuration

5.3

Numerical evaluation of C

and C

for

the reference Prandtlplane configuration

5.4

Numerical evaluation of C

and C