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Bachelor Thesis May 29 2009 Aerodynamic Performance and Stability Simulation of Different Flying Wing Model Airplane Configurations Author: Flavio Gohl Supervisor: Stefan Leutenegger Dario Schafroth Professor: Prof. Dr. Roland Siegwart
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Page 1: Nuri ETH Calculation

Bachelor ThesisMay 29 2009

Aerodynamic Performance andStability Simulation of Di!erent

Flying Wing Model AirplaneConfigurations

Author:Flavio Gohl

Supervisor:Stefan Leutenegger

Dario Schafroth

Professor:Prof. Dr. Roland Siegwart

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Abstract

In flying wing design, the stability criteria often decrease the aircraft’sperformance and vice versa. Therefore a pitch unstable wing can have ahigher performance. The question to be answered in this thesis is, howmuch performance can be won by a pitch unstable wing, stabilised witha PID controller.

An application is implemented for studying flying wing dynamic andstability behaviour. The simulation is based on a vortex lattice methodintegrated in a rigid body simulation. The vortex lattice method is mod-elled with singularity elements in chord and spanwise direction on a three-dimensional wing. With this method, three-dimensional wing geometries,asymmetrical flap deflection and asymmetrical inflow can be simulated.The application is qualitatively evaluated with real flight tests of a flyingwing with measurements of the period time of a phygoid oscillation.

Three di!erent wings, amongst them a pitch unstable wing with highperformance, are analysed in their dynamic stability behaviour and per-formance. The pitch unstable wing has a slightly higher performance thanthe optimised stable wing.

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AcknowledgementI would like to express my sincere appreciation to all those who have contributed,directly or indirectly, to this Bachelor’s thesis in form of technical or othersupport. I give my special thanks to Stefan Leutenegger and Dario Schafroth,Michael Möller, Jens Walther and Philippe Chatelain and Ursina Gysi.

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Contents1 Introduction 1

1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Work structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 State of the Art 32.1 State of the Art of Aerodynamical Force Calculation . . . . . . . 32.2 State of the Art of Flight Simulators . . . . . . . . . . . . . . . . 3

3 Methods for Aerodynamic Force Calculation 53.1 Vortex Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Theoretical Background on Vortex Methods . . . . . . . . 73.1.2 Lifting Line Method and Nonlinear Lifting Line Method

in General . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.3 Lifting Line Method . . . . . . . . . . . . . . . . . . . . . 113.1.4 Nonlinear Lifting Line Method . . . . . . . . . . . . . . . 123.1.5 Vortex Lattice Method . . . . . . . . . . . . . . . . . . . . 14

4 Implementation of a Suitable Method 184.1 Nonlinear Lifting Line Method . . . . . . . . . . . . . . . . . . . 19

4.1.1 Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.4 Influence Coe!cients . . . . . . . . . . . . . . . . . . . . . 214.1.5 Initial Guess . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.6 Iterative Process / Coupling the Profile Information . . . 224.1.7 Force and Moment Calculation . . . . . . . . . . . . . . . 224.1.8 Evaluation of the Nonlinenar Lifting Line Method . . . . 23

4.2 Vortex Lattice Method . . . . . . . . . . . . . . . . . . . . . . . 254.2.1 Inclusion of the Profile . . . . . . . . . . . . . . . . . . . . 264.2.2 Mesh Generating . . . . . . . . . . . . . . . . . . . . . . . 264.2.3 Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.4 Influence Coe!cients . . . . . . . . . . . . . . . . . . . . . 284.2.5 Flap deflections . . . . . . . . . . . . . . . . . . . . . . . . 304.2.6 Force and Moment Calculation . . . . . . . . . . . . . . . 304.2.7 Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.8 Evaluation of the Vortex Lattice Method . . . . . . . . . 31

4.3 Integrating the Methods in the Simulation . . . . . . . . . . . . . 34

5 Evaluation of the Simulation 365.1 Flight Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Results of the Simulation 396.1 Flying Wing, FG-WingX-02 . . . . . . . . . . . . . . . . . . . . . 39

6.1.1 Stability Results . . . . . . . . . . . . . . . . . . . . . . . 416.2 Optimised Stable Wing . . . . . . . . . . . . . . . . . . . . . . . 43

6.2.1 Stability Results . . . . . . . . . . . . . . . . . . . . . . . 456.3 Dutch Roll Mode (germ. Taumelschwingung) and Spiral Mode . 476.4 Unstable Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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6.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.4.2 Static Results . . . . . . . . . . . . . . . . . . . . . . . . . 486.4.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . 50

7 Discussion 55

8 Conclusion 56

9 Future Work 57

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1 IntroductionAn everlasting problem in aircraft design, especially in flying wing design is therequirement of stability in pitch roll and yaw axis which is strongly coupled withthe aircraft performance and the manoeuvrability. The stability is coupled witha loss of the aircraft’s performance and vice versa. A demonstrative exampleis the elevator of a classic aircraft configuration, it produces a negative lift(excluded canard configurations).

An aircraft with an optimal performance and optimal stability is a require-ment, which probably is never reached, it is always the challenge to find anoptimum. A main step to improve the aircraft performance is, to integrate thefin, elevator and fuselage in one wing. No fuselage, elevator and fin would beneeded, therefore weight can be saved and drag reduced. Even the civil avia-tion recognises, that such flying wing combinations are forward looking, due tointegrating the big fuselage in the wing and transforming it into a lift creatingelement.

Due to this trade-o" between the stability and the performance, the questionis: What happens with the performance and the stability if the wing is unstableand a controller garantees the stability? With a close look at flying wing modelairplanes combined with a controller, the question can be advanced to: Howgood is the flight performance of a pitch unstable flying wing without takingregard on pitch stability? Is it possible to reach better glide ratios and sinkrates if the unstable wing is guided by a pitch controller?

In order to optimise the power consumption of a new prototype of conven-tional solar UAV airplane in higher altitude, it might be advantageous to switchto a flying wing if a significant di"erence in performance is found.

To answer this question, a fast dynamic simulation application is built forstudying aircraft stability and performance. The simulation is based on a com-plex aerodynamic model for complex, three-dimensional wing configurationswith twist, tapering, sweep and dihedral integrated in a six degree of freedomrigid body motion simulation. Influences, such as flap deflections, sideslip angleand angular rates are considered in the calculation of the aerodynamic forcesand moments. The aerodynamic force calculation method is a Vortex Latticemethod, modified for short computing time. For each time step the methodcalculates a force distribution in spanwise and chordwise direction, for a threedimensional wing.

To answer the question about the unstable wing’s gain of performance, apitch unstable wing is designed and compared with self stable flying wings.For testing the flight characteristics and the dynamic stability, the flying wingsare simulated in the dynamic simulation. The unstable wing is stabilised by acontroller.

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1 INTRODUCTION

1.1 ObjectivesThe objectives of this work are:

• Literature review about existing software solutions

• Establishing a Matlab/Simulink nonlinear dynamic model in order to sim-ulate the dynamic behaviour of di"erent flying wing model airplane con-figurations

• Verification of the model with flight experiments using an existing RCflying wing model

• Simulation of di"erent flying wing model airplane solutions

– Self stable flying wing with high flight performance– Low-sweep pitch unstable wing with high performance pitch sta-

bilised by a controller.

• Comparison of the performances

1.2 Work structureThis thesis is structured in three parts. In the first part, a theoretical back-ground in vortex methods is given. The Nonlinear Lifting Line Method andthe Vortex Lattice Method are described in more details. Information aboutexisting software solutions are given. A list of existing simulations and theircapacities is provided.

The second part describes how the methods are implemented and what as-sumptions with regard to a dynamic simulation are made.

In the third part results of the evaluation are presented and results are shownand discussed in the section results of the simulation.

In the Appendix some important code fragments, a manual of the code andthe most important information about designing flying wings are shown.

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2 STATE OF THE ART

2 State of the ArtIn the section 2.1 it is presented what methods are commonly used for aerody-namic force calculation. In the section 2.2 di"erent existing simulators, whichtry to integrate an aerodynamical model, are presented.

2.1 State of the Art of Aerodynamical Force CalculationToday high order panel codes are commonly used. For dynamical simulation,an unsteady Kutta-Joukovsky boundary condition is made, as well as unsteadywake arrangements by modelling the wake with taking regard to the flight stateof earlier simulated time steps. First panel codes were developed in 1962. Thestrongest e"orts in these codes were made in pre- and post-processing and alsoin the automatical coupling of profile information, where inner viscous e"ectsare separately analysed. With vortex methods, e"ects of propulsion and internalflows trough turbines etc. can be modelled. In addition flow separation can bemodelled with vortex methods and give the wing a nonlinear behaviour. Thereare even methods which can couple the inner viscous e"ects in the boundarylayer with the potential flow, so that also turbulent boundary layers can bemodelled. Detailed information can be found in [18, 8, 12, 22]

For real time simulations many simplifications are commonly made ore hugelook up tables are generated from measurement or calculations. See in thesection ’Simulation of the dynamic’.

2.2 State of the Art of Flight SimulatorsThere are various aircraft simulators which try to simulate the behaviour ofaircrafts. A simulation of aircraft with a su!ciently complex aerodynamic forcecalculator for 3D wings was not found. Most simulators are interested in bestgraphical visualisation, but not in physical aircraft behaviour. Some simulatorswhich try to include a more complex integration of aerodynamic forces are listedbelow. A very advanced simulator is X-plane. Not only is the graphical engineoutstanding, but also the aerodynamic forces calculation is at a high level.

JSBSim An Open Source project. It calculates aerodynamic forces from lookup tables. Polar data are included for a range of angles from 0...180°.Propulsion and ground e"ects are implemented as well, and even a con-troller for auto piloting is included. All stability derivates are modelledlinearly. [2]

Flight Gear An Open Source project. The flight gear uses flight dynamicmodels from JSBsim. [15]

X-Plane A well known simulator. The user can define his own airplane and letit fly in X-plane. X-plane tries to approximate the wing as a finite wing.E"ects as aspect ratio, taper ratio, and sweep of the wing are influencingthe aerodynamics. Even compressible e"ects are implemented. A 2Dairfoil polar maker is included. [14]

André Noth The simulator simulates an infinite wing and neglects the e"ectsof a finite wing on the lift distribution. For induced drag simple approx-imations are done to integrate the e"ects of taper ratio and aspect ratio.

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2 STATE OF THE ART

Xfoil is generating 2D profile coe!cients which are integrated over thewingspan. A dynamic of six degrees of freedom is implemented in mat-lab simulink. The simulator is made for controller design.

The simulators listed above are not suited to analyse di"erent wing geometriesand to study stability and performance behaviour of flying wings. For thisreason a simulator is established and will be introduced in this thesis.

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3 METHODS FOR AERODYNAMIC FORCE CALCULATION

3 Methods for Aerodynamic Force Calculation

The aim is, to design an algorithm which can calculate the aerodynamic forcesintroduced by the wing. Then these forces are given into the dynamic simulationand must be calculated at each time step.

There are many methods to calculate the forces over a wing. The programshould in principle calculate the forces for a flying wing configuration.

The specifications for an implementation with regard to a dynamic simula-tion are:

• very short calculation time for real time simulation

• no elevator and fin

• complex wing (dihedral, winglets, swept wings , etc.)

• asymmetrical incident flow

• rotation speed of the wing which must be regarded for damping the move-ment about pitch, roll and yaw axis

• asymmetrical flap deflection

• sideslip angle which must be regarded

• uncomplicated implementation

• consideration of the profile information

Methods which achieve these specifications:

• lifting line method (described in section 3.1.3)

• nonlinear lifting line method (described in section 3.1.4 )

• lifting surface method or even more complex panel methods (described insection 3.1.5 )

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3 METHODS FOR AERODYNAMIC FORCE CALCULATION

3.1 Vortex MethodsInner e!ects Inner e"ects are e"ects which are acting in the boundary layerdue to viscous motion. The boundary layer is extremely small in comparisonwith other dimensions. The results shown in Table 3.1 give an impression of theboundary layer’s dimensions.

Table 1: Boundary layer, laminar and turbulent [19]

These inner e"ects can not be neglected, they comprise the informationabout the viscous drag. This problem can be split up and separately be calcu-lated, because its influence on the outer boundary is small. So the calculationof the viscous drag is carried out separately. However, there are many programswhich can calculate viscous drag, for example:

• Xfoil

• Wineppler

Outer e!ects Pressure forces are e"ects which act outside of the boundarylayer. They are calculated with vortex methods. Details are described in [8],page 18. Examples of programs, which can calculate potential flow, are listedbelow.

• Tornado (MATLAB code)

• AVL

• XFLR

• Miarex (MATLAB code)

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3.1.1 Theoretical Background on Vortex Methods

With the vortex methods listed above, the flow around the wing gets modelledas a potential flow. This means that only linear aerodynamics can be studied,e"ects of stall and other e"ects at high angles of attack are negligible. The flowbehaviour at high mach numbers is also negligible.

The process of modelling the potential flow happens with an intelligent vor-tex arrangement, which models the flow around the wing. The mathematicalbackground for these methods are given in this section.

Biot Savart The induced velocity from a vortex line is calculated with formula1. This is the velocity field, which generates a vortex line with a constantcirculation strength.

u =!4!

ˆ

ds! r

| r |3 (1)

10 Drehungsbehaftete Stromungen 81

Gesetz von Biot-Savart

Bisher haben wir das Wirbelstarkefeld ! eines gegebenen Geschwindigkeitsfeldes u betrachtet, welchesman einfach durch Rotationsbildung erhalt. Jetzt fragen wir, welches Geschwindigkeitsfeld zu einemgegebenen Wirbelstarkefeld gehort. Zur Vereinfachung betrachten wir den freien Raum ohne Begren-zungen durch Wande.Analog zum Magnetfeld eines stromdurchflossenen Leiters in der Elektrodynamik ist die azimutaleGeschwindigkeit u! um einen unendlich langen geraden Wirbelfaden (Potentialwirbel) mit Zirkulation! im Abstand r gegeben durch

u! =!

2"r.

Allgemeiner “induziert” ein der Raumkurve C folgendes Wirbelfadenstuck mit konstanter Zirkulation! das Geschwindigkeitsfeld

u(x) =!4"

!

C

ds! r

|r|3 .

C

x

y

z

x

r

du

ds !

Fur den Beitrag eines endlich langen, geraden Wirbelfadenstucks ergibt sich damit in einem durch denAbstand r und die Winkel #1 und #2 bestimmten Punkt die induzierte azimutale Geschwindigkeit

u" =!

4"r(cos #1 " cos #2) .

1

2

!

Wirbelfadenstück

u senkrecht zur Zeichenebene

!

"r

Beim Grenzubergang zu einem unendlich langen geraden Wirbelfaden (#1 # 0, #2 # ") ergibt sichdaraus wieder die Geschwindigkeit um einen Potentialwirbel. Der Fall eines halbunendlichen Wirbelfa-dens ist ebenfalls erfasst.Ist die Wirbelstarke !(x) im Raum verteilt, so erhalt man das von ! induzierte Geschwindigkeitsfelddurch ein Volumenintegral, das die Beitrage aller Wirbelelemente zur Geschwindigkeit u(x) im Punktx aufsummiert:

Stand 30. September 2008

Figure 1: Vortex filament and the induced velocity

If the vortex filament is a line, the integral is transformed to:

u! =!

4!r(cos "1 " cos "2) (2)

10 Drehungsbehaftete Stromungen 81

Gesetz von Biot-Savart

Bisher haben wir das Wirbelstarkefeld ! eines gegebenen Geschwindigkeitsfeldes u betrachtet, welchesman einfach durch Rotationsbildung erhalt. Jetzt fragen wir, welches Geschwindigkeitsfeld zu einemgegebenen Wirbelstarkefeld gehort. Zur Vereinfachung betrachten wir den freien Raum ohne Begren-zungen durch Wande.Analog zum Magnetfeld eines stromdurchflossenen Leiters in der Elektrodynamik ist die azimutaleGeschwindigkeit u! um einen unendlich langen geraden Wirbelfaden (Potentialwirbel) mit Zirkulation! im Abstand r gegeben durch

u! =!

2"r.

Allgemeiner “induziert” ein der Raumkurve C folgendes Wirbelfadenstuck mit konstanter Zirkulation! das Geschwindigkeitsfeld

u(x) =!4"

!

C

ds! r

|r|3 .

C

x

y

z

x

r

du

ds !

Fur den Beitrag eines endlich langen, geraden Wirbelfadenstucks ergibt sich damit in einem durch denAbstand r und die Winkel #1 und #2 bestimmten Punkt die induzierte azimutale Geschwindigkeit

u" =!

4"r(cos #1 " cos #2) .

1

2

!

Wirbelfadenstück

u senkrecht zur Zeichenebene

!

"r

Beim Grenzubergang zu einem unendlich langen geraden Wirbelfaden (#1 # 0, #2 # ") ergibt sichdaraus wieder die Geschwindigkeit um einen Potentialwirbel. Der Fall eines halbunendlichen Wirbelfa-dens ist ebenfalls erfasst.Ist die Wirbelstarke !(x) im Raum verteilt, so erhalt man das von ! induzierte Geschwindigkeitsfelddurch ein Volumenintegral, das die Beitrage aller Wirbelelemente zur Geschwindigkeit u(x) im Punktx aufsummiert:

Stand 30. September 2008

Figure 2: Vortex filament as a line

If the vortex filament is infinitely long, the formula is:

u! =!

4!r(3)

Force on a vortex line According to the Kutta-Joukovsky theorem, avortex #!moving with the velocity #v experiences a force #F .

"#F = $("#V !"#! ) · l (4)

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3.1.2 Lifting Line Method and Nonlinear Lifting Line Method inGeneral

Vortex arrangement / Singularity Element The flow is modelled as apotential flow. As singularity element, for the Lifting Line Method as well asfor the Nonlinear Lifting Line Method, a horseshoe vortex is chosen.

The singularity element is placed, in order that the side vortex lines areleaving the wing in flow direction, and the bound vortex is laying on the wing inthe direction of the leading edge. Assuming that the angle are small the trailingvortices can be laid in the x-y plane of the body co-ordinate system.

A better physical arrangement is, if the trailing vortices are leaving the wingin flow direction. These di"erent arrangements are shown in figure 5. Thetrailing vortices in the wake (behind the wing) have to be aligned in x directionor better in flow direction, because in this case there will not be any force actingon the trailing vortices.

"#v !""""#!Wake = 0 (5)

The arrangement is shown in Figure 3.1.2.

Figure 3: Horseshoe vortex arrangementa) with elementary wingsb) Prandtl methodfigure copied from [24], page 24

The bound vortex is laid on the 1/4 line of the wing, and the collocationpoints are lying on the 3/4 line of the wing in the middle of the trailing vortices.On the collocation points the no slip condition is made. The detailed vortexarrangement of the Lifting Line method is shown in Figure 4.

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Figure 4: Vortex arrangement and normal vectors in the collocation points withthe starting vortex in infinity, copied from[8]

Figure 5: Trailing vortex arrangementleft: trailing vortex leaving the wing in flow direction but following the profileright: trailing vortex leaving the wing in flow directionCopied from [8]

Vortex Theorems A vortex is always closed. This means that vortices areclosed filaments, or vortex rings. The horseshoe vortex is also closed, with thestarting vortex in infinity. Thus its influence is negligible. A second importantassumption is, that the circulation strength along the vortex ring is constant.

Forces The lift force of a single element is calculated (Kutta JoukovskyTheorem):

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Lift(i) = $! · V!(i) · !(i) · "y

V! speed of the flow

$! density of air (1.225 kg/m3)

"y width of a horseshoe vortex

!(i) circulation strength of element i

The lift force is generated by the infinite flow speed on the bound vortex. Thelift force is aligned vertically to the flow speed (see formula 4 on page 7).

The induced drag is calculated with the induced flow speed of the trailingvortices. They generate a downwash, which induces a velocity on the boundvortex. The induced velocity on the bound vortex generates a force. This forceis the induced drag. Only the trailing vortices have an influence on the induceddrag.

Draginduced = $! · Vind(i) · !(i) · "y

The forces are placed on the 1/4 line on each bound vortex.

Tre!tz Analysis The induced drag can also be calculated in the so calledTre"tz plane, far behind the wing. The induced velocity is much easier tocalculate in the Tre"tz plane than over the wing, because the trailing vorticescan be modelled as infinitely long in both directions.

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3.1.3 Lifting Line Method

Boundary Condition In the classic Lifting Line Method, the circulationstrength is calculated with a no slip condition. No flow can penetrate the profile.The sum of induced velocity, infinite velocity and induced velocity of the wake,projected on normal direction, is zero. This equation is valid at the collocationpoints.

#winduced · #nsolid + #wi · #nsolid + #V! · #nsolid = 0 (6)

#winduced velocity of the bound vortex

#wi induced velocity of the wake (trailing vortices)

#V! incident flow velocity

The normal vectors of the wing #nsolid are vertical to the chord line, so theboundary condition takes only the chord as profile information, cambering isnegligible. To have better profile information in the boundary condition, thefollowing methods integrate the profile better.

Equation 6 gives for each collocation point a linear equation with the un-known circulation strength !(i). The enormous advantage of the vortex methodis, that the circulation ! is linear in equation 1, 2 and 3. So equation 6 can bewritten as a matrix and vector equation:

K · ! = RHS (7)

K nxn matrix

! vector with length n

RHS incident flow ( #V!) projected on normal vector #n

In order to solve this equation for the circulation vector !, only a matrix inver-sion has to be done.

Closed analytical solutions of equation 6 are given in [24], page 7-10. Thenumerical solution of this equation is described in section Implementation.

It is more or less arbitrary to evaluate this equation in a point, which islaying on the 3/4 line. However this alignment is commonly used and providesgood results. It might be useful to analyse, where that point has to lie withdi"erent profiles. A sample calculation, why the collocation point is laying onthe 3/4 line is shown in [12], page 23

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3.1.4 Nonlinear Lifting Line Method

The Nonlinear Lifting Line Method was developed in 1946, see [25].

Circulation Strength In Weissinger’s Nonlinear Lifting Line Method, thecirculation strength is calculated by iteration. At first the induced angle mustbe calculated:

%ind = arctan(Vind/V!) (8)

The downwash is calculated on the collocation points. See Figure 6 depictinga geometrical interpretation of equation 8.

Figure 6: Induced angle and angle of attack

V

induced angle

effective angle

geometrical angle

V

Vind

induced angle

Profile data With the induced angle, the angle of attack can be calculated:

%effective = %geometric " %induced (9)

With the e"ective angle, the local lift force can be calculated. This is donewith known airfoil polars (measurements, Xfoil, Wineppler). With this informa-tion, nonlinear profile behaviour can be coupled with the circulation distributionand the lift force. Especially at high angles of attack this information is essen-tial, as the profile then shows a highly nonlinear behaviour and can not bemodelled as a perfect flat plate with a linear behaviour.

Iterative Process

1. An initial circulation distribution is estimated.

2. The induced angle of attack %ind is calculated.

3. The local angle of attack (angle between infinite velocity and chord minusthe induced angle) is calculated.

4. The lift distribution with known airfoil polar data is calculated.

5. With the Kutta Joukovsky theorem, a new circulation distribution can becalculated.

6. The old and the new circulation distributions are compared. A new circu-lation distribution with both of them is iteratively generated. The iterativeprocess begins with point 1.

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The new circulation is calculated with the following formula: [18], page 277

!n+1(i) = (1"D)!n(i) + D!new(i) (10)D is the damping factor, D $ 0.05. The iteration is made, until the maximal

di"erence between new and old circulation is small enough.References: [8, 18, 17]

Advantages, Disadvantages and Limitations of the Nonlinear LiftingLine Method

It is important to know the limitations of the Nonlinear Lifting Line Method.The main advantage is the inclusion of the nonlinear profil data. Even experi-mental profil data can be coupled with the potential flow.

Advantages

• Only few calculations are necessary.

• Nonlinear e"ects of the profile can be studied.

• Flap deflections are not integrated in the K matrix.

Disadvantages

• Momentum distribution: The fact is, that the lift force is not always placedin the quarter of the chord, as it is for non swept wings (the sweep angleis measured at the chord quarter line). For a flying wing the momentumequilibrium is essential. It has an influence on the centre of pressure, thuson the positioning of the centre of gravity. It is of great importance toknow the centre of gravity, not only for the construction, but also forthe mass of stability. The measure of stability is defined as the distancebetween the centre of gravity and the neutral point of the aircraft1. Thismeans that the profile data of CM is not exactly the value which can beextracted from the local angle of attack. This problem can only be solvedwith a panel method with several panels in the chord direction, see section3.1.5.

• For the lift calculation, all profile data is needed. So a look up table ofprofile data is necessary, which is also needed for viscous drag calculation.

• The wake is modelled in a simple way.

Limitations

• According to NACA technical note [26], ’The calculations are subject tothe limitations of lifting line theory and should not be expected to giveaccurate results for wings of low aspect ratio and large amounts of sweep’

1There are some di!erent definitions of the neutral point, there is a geometrical and anaerodynamical neutral point, mostly they are nearly the same points. Commonly the measureof stability is given in percent of the mean aerodynamical chord length. More details ofgeometrical and aerodynamical neutral points see [19], page 104

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3.1.5 Vortex Lattice Method

The Vortex Lattice Method, implemented in this thesis, is a Lifting SurfaceMethod. Therefore the theoretical information which is given in this section islimited to the Lifting Surface Method. The method is similar to the Lifting LineMethod, it is only enriched with more singularity elements in chord direction.

The boundary condition includes profile information, so the cambering hasan influence on the lift coe!cient. However, the influence of the whole profileis not as strong as in the Nonlinear Lifting Line Method. Viscous drag can notbe calculated with these methods either, this is done with experimental profilepolars or with software solutions.

More complex panel methods are mathematically similar to the low orderpanel method, but there are other boundary conditions and other singularityelements. They use a Dirichlet boundary condition for a thick body, and formodelling the potential flow they use doublet panels (vortex rings) or constant-strength sources.

The method, implemented in this thesis, is similar to the method presentedin [6], this method is also known as ”A Multi-lifting Line Method and its Ap-plication on Design and Analysis of Nonplanar Wing Configuration”.

Singularity element Basically a vortex ring or a combination of a ringand a horseshoe vortex may be chosen as a singularity element. The choice isdepending on the e"ects which would be studied and on the provided calculationtime. For the Vortex Lattice Method implemented in this thesis, the horseshoevortex is chosen as singularity element.

Mesh and Vortex Arrangement The wing is divided into several el-ements in wingspan direction as well as in chord direction. The result is arectangular mesh over the whole wing surface. In each of these rectangularpanels a horseshoe vortex is laid. The arrangement is shown in Figure 7. It isimportant for the trailing vortices, that they are laid on the surface of the wing.If they are not laid on the surface, e"ects such as twist do not have a su!cientlystrong influence. The following Figures 8 and 9 show more geometrical detailsof the horseshoe vortex placement.

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3 METHODS FOR AERODYNAMIC FORCE CALCULATION

Collocation Point

Trailing Vortex

Bound Vortex

c/2

Wake

Figure 7: Vortex arrangement of the Vortex Lattice Method

Figure 8: Detailed horseshoe vortex and alignment in a panel, copied from [8]

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3 METHODS FOR AERODYNAMIC FORCE CALCULATION

Tomas Melin. KTH, Department of Aeronautics. Page 22 (45)

A Vortex Lattice MATLAB Implementation for Linear Aerodynamic Wing Applications.

4.3.2 Twist and the skewed vortex loop.

When adding twist to the layout, it implies that the geometric angle of attack

varies with span, the design is no longer a flat plate but a mildly skewed surface.

The twist will cause the two outgoing vortex legs from a panel are no longer

parallel, see figure 11. This is the source of the vortex-sling arrangement in

Tornado, which is used instead of the more commonplace horseshoe vortex.

4.3.3 Camber and thin airfoil boundary application.

To extend the geometry even more, the wing could also be cambered. In

Tornado the wing is still regarded as flat with a thin wing approximation where

the boundary conditions are shifted. That is, the normal of the cambered surface

is calculated and the non-flow-through boundary condition is employed at the

chord line (see figure 12). This approximation is common and used in a variety

of methods.

Fig 11: Twisted vortex sling.Figure 9: Trailing vortices and twist, the trailing vortices are not parallel, copiedfrom [13]

Wake The wake modelling in this simple model does not take the wakeroll up and the unsteady flow into account. For more precise results, the wakewould be modelled by vortex panels with time variant circulation strength. Thewake in this implementation contains all trailing vortices which are followingthe local chord direction.

Boundary Condition and Profile Information To take care of profileinformation, the boundary condition must be improved. The boundary condi-tion is made on the skeleton line, so that the cambering has an influence onthe circulation distribution. So the normal direction on the skeleton line is cal-culated, and in this direction the total flow must be zero. This simply means,that no flow can go through the skeleton line. This is an approximation of theprofile, it is assumed that the profile is thin (see Figure 10). This approximationis commonly used. See in [6, 18, 8, 12, 21, 11]

Tomas Melin. KTH, Department of Aeronautics. Page 23 (45)

A Vortex Lattice MATLAB Implementation for Linear Aerodynamic Wing Applications.

4.3.4 The polyhedral wing.

The geometry may be even more intricate when we allow cranked wings, i.e.

wings that are polyhedral, like the F-16 main wing, see figure 13. However from

the geometric layout and meshing point of view, this is not a big problem as

every polyhedral wing may be broken down into quadrilateral partitions. In

Tornado, this partitioning takes place early in the user input of geometry

definitions.

Fig 12: Camber and shifted boundary condition.

Fig 13: Cranked wing on a F16 type of aircraft.

Figure 10: Influence of the cambering in the boundary condition.The Figure is copied from [13]

System of Equations The resulting equation for the calculation of thecirculation strength in a collocation point is the same as equation 6. Thisequation is solved for each panel and the resulting system of equations can be

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3 METHODS FOR AERODYNAMIC FORCE CALCULATION

solved for the circulation strength in each panel. The system of equations canbe written in a matrix form, similar to equation 7. To solve the system ofequations, only a matrix inversion must be done.

Advantages and Disadvantages of the VL-Method in general

Advantages

• A force distribution over the chord and the span is made. This providesa better momentum distribution.

• The centre of gravity can be calculated more exactly.

Disadvantages

• The wake arrangement with infinite trailing vortices is an enormous sim-plification.

• The Kutta condition is a steady Kutta condition and neglects dynamicflow behaviour.

• E"ects of flow separation and transition are neglected in the potentialflow, but they are not negligible in the airfoil viscous drag.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4 Implementation of a Suitable MethodThere are various existing programs, which can calculate aerodynamic forces.

• Tornado (MATLAB code) [21]

• AVL [11]

• XFLR [10]

• Miarex (MATLAB code) [23]

The best results can be obtained with panel methods. There is an Open Sourcetool called Tornado [21]. It was tested, if this software could be used for adynamic simulation. Unfortunately the calculation times are too long and avast look up table would be necessary, because the program has to calculate anew mesh and a new influence matrix for each flap deflection.

There is another Open Source tool from Mark Drela [11], which is a VortexLattice Method as well, but the calculation time is also too long, similarly toTornado. Flap deflections must always be meshed again.

Another problem is, that these two programs do not care about the frictionof the profile. Therefore the friction force is only the induced drag. For thewhole drag force it is necessary to integrate the lift distribution with profiledata. This has to be calculated separately and again increases the calculationtime.

XFLR is another program with great potential which calculates with di"er-ent methods. Often comparisons with XFLR are made in this thesis. IntegratingXFLR in a dynamic simulation would not pe possible either, the geometry foreach flap deflection would have to be changed. In XFLR flap deflections aredefined as a new profile. Therefore for each change of the profile, the profilecoe!cients have to be recalculated for the interpolation. Miarex is a kind ofNonlinear Lifting Line Method combined with Xfoil, but not e!cient enougheither for dynamical solutions. Miarex is also limited in the wing geometries.

Some requirements, which do not allow to generate a look up table, are:

• asymmetrical flap deflections

• sideslip angles

• angular rates for damping the movement in yaw-, pitch- and roll-axes

These variables generate too many flight states.After checking the qualities and limitations of the di"erent programs, a Lift-

ing Line Method was chosen, which has the great advantage that flap deflectionsare not cared about in the influence matrix of the vortices. The flap deflectionsare only considered in the iteration. A reduced look up table would be gener-ated with the only parameter &, the sideslip angle. Unfortunately the resultswere insu!cient, because the method does not care about a force distributionin chord direction.

Finally a Vortex Lattice Method was implemented. The method was opti-mised for a fast calculation, so that the method can be used for dynamical simu-lation. Some simplifications with flap deflections were made. In both methods,simplifications are made in the wake modelling. Otherwise, the simulation timewould be much longer.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.1 Nonlinear Lifting Line MethodThe main idea of this implementation is, to split up the calculations. Themeaning is to calculate as much as possible before the simulation starts so thatduring the simulation only the most important calculations must be carried out.

A simple structure of the program is shown in Figure 11, listing the mostimportant functions and procedures.

load all input data

geometry

definition

initial state

definition

profile polar

definitionXfoil

polars

generate geometry and vortex mesh

calculate influence coefficient

Nonlinear Lifting Line MethodProcess Diagramm

make initial guess for

gamma start distribution

iterate calculate

forces and moments

geometry

mesh influence matrix K

gamma start andflap deflection

gamma

Manual input

Figure 11:

The following chapter gives more detailed information about the most im-portant functions.

4.1.1 Inputs

The geometry can be a 3D wing geometry with flaps, dihedral, twist and sweepangle. Even discontinuous functions of the twist angle and the geometry arepossible. As inflow information, the flow velocity must be defined in threecomponents, and the angular rates have to be initialised.

4.1.2 Mesh Generation

The coordinate system is defined as commonly used in aircraft design. They-axis is in the spanwise direction, the x-axis goes backwards of the wing and

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4 IMPLEMENTATION OF A SUITABLE METHOD

the z-axis is looking away from the earth. This definition is di"erent than thedefinition of the dynamics.

As singularity element, the horseshoe vortex is chosen. The trailing vorticesare aligned in x-direction in this implementation .

The wing geometry can be devided into several partitions. Each partition isa trapezoid with geometry data listed below. The geometry of the wing is onlydefined for a wing half, then the geometry is mirrored.

b_root root chord, skalar

b_tip tip chord skalar

alpha_g geometrical angle to the body x-axis, this is a vector and containsthe root angle and the tip angle

s span of the partition

n number of single panels

phi sweep angle at leading edge

x0 reference point, where the partition has to be placed, it is defined atthe tip of the root chord for each partition and therefore is a matrix.

With this parameters the function generate_Mesh.m gives as output the meshdata for the vortex placements and some other geometry data, which are usedfor the force calculation. The most important outputs are:

T_left all left horseshoe points

T_right all right horseshoe points

A all collocation points

The trailing vortices are leaving the wing in x-direction. This is a small angleapproximation of the wake. Physically they have to leave the wing in the flowdirection behind the wing. Therefore twist e"ects are neglected in the vortexarrangement. The e"ect of the wing’s geometrical angles are taken into accountin the calculation of the new Gamma distribution (!), see Algorithm 1 in Ap-pendix A . The new Gamma circulation is calculated with the geometrical andthe induced angle and then damped with the old circulation.

4.1.3 Sideslip Angle

The whole geometry is rotated with an angle & around the z-axis. The sideslipangle is calculated:

1 betha_in=atan(global_flow_speed (2)/global_flow_speed (1));

However, the trailing vortices are still pointing into the x-direction.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.1.4 Influence Coe"cients

The Kutta condition is fulfilled at each collocation point. The induced velocitiesof all horseshoe vortices must be calculated at the collocation point i, and then becompared with the incident flow V! projected on the local geometrical normalvector. For all collocation points the system has the following form:

Coll. Points Matrix K Gamma Boundary Cond.1 a1,1 a1,2 a1,3 ... a1,N !1 RHS1

2 a2,1 a2,2 a2,3 ... a2,N !2 RHS2

3 a3,1 a3,2 a3,3 ... a3,N · !3 = RHS3

4 a4,1 a4,2 a4,3 ... a4,N !4 RHS4

... ... ... ... ... ... ... ...N ... ... ... ... ... !N RHSN

Table 2: System of linear equations of the Nonlinear Lifting Line Method

The influence coe!cients ai,j are calculated with the function vortex.m, vor-tex_left.m, vortex_right.m. They are principally vectors, projected on eachnormal vector in the collocation point. The function is plotted in Algorithm 2Appendix A .

The circulation Gamma can be excluded from the calculation of the influencecoe!cients, because it is linear in the equation.

The right hand side of the equation is the free stream flow projected on thenormal vector.

RHSi = "(U!, V!, W!) · #ni (11)

The influence coe!cients are only depending on the geometry and the sideslipangle &. The assumption is, that the wake is stationary, so that the vortex ar-rangement, despite a variation in the angle of attack, rests always the same.

This influence matrix is saved for several sideslip angles and later interpo-lated at the desired sideslip angle.

4.1.5 Initial Guess

The initial function for the circulation strength is chosen in this implementationas:

1 Gamma_start=-inv(K)*RHS

This is a linear approach which later is iteratet with nonlinear profile infor-maion.

The RHS is:1 RHS=v_abs.*sin(alpha_g)

v_abs norm of the velocity, v_abs is a vector

alpha_g geometrical angle, a vector

The twist e"ect is neglected in the RHS, because the influence of twist on thenormal vector in the collocation points is neglibible.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.1.6 Iterative Process / Coupling the Profile Information

The iteration is started with the following commonly used parameters.1 %Damping Factor

D=0.05;3 %Stop criteria

min =0.001;

The iteration function is the core of the total program, it is plotted in Algo-rithm 1 on page 60 Appendix A .

A short description of what the function does:The induced velocity w_i here is calculated with the Tre"tz analysis. It is

calculated with the following formula:w_i =1/2*( K_far*Gamma_alt);

K_far is the influence matrix for the Tre"tz analysis. The e"ective angle ofattack is calculated:

1 alpha_i=-atan(w_i ./(v_abs ’));

The induced angle alpha_i could also be simplified to: alpha_i=-w_i./v_abs.The new circulation Gamma_new is calculated with the following two for-

mulas:

Kutta-JoukovskyL = $ · ! · v · "y (12)

Profile_CL

L = CL · $ · v2

2· "y · b (13)

Therefore the circulation Gamma is:

! = CL · v

2· b (14)

The CL value is calculated with an interpolation function between severalpolars. In the interpolation, the flap angle and the angle of attack are the inputvalues. With the CL distribution, the viscous drag CD is interpolated from theprofile polars.

The convergence process of the iteration is not a robust process. So thedamping factor and the stop criteria have to be chosen carefully. The practicehas shown, that the values given here mostly provide good results.

4.1.7 Force and Moment Calculation

The lift force is calculated with the CL distribution. It is projected on thespeed normal direction, vertical to the flow speed vector and vertical to thebound vortex. The induced drag force is calculated with the formula:

Dinduced(i) = "$ · !(i) · winduced · "y (15)

The viscous drag is calculated with the integration of the local viscous dragover the span of the wing which is calculated with profile polar information.The drag force is projected in flow speed direction. The point of attack of the

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4 IMPLEMENTATION OF A SUITABLE METHOD

forces is on the bound vortex. The resulting moment is evaluated in the originof the wing. To the resulting moment the local momentum coe!cient of theprofile integrated over the span of the wing is added.

4.1.8 Evaluation of the Nonlinenar Lifting Line Method

Unfortunately in the Tre"tz analysis sweep angles are neglected. So a variationof the sweep angle does not change the induced velocities in the Tre"tz plane.For swept wings, it is important to include the influence of the sweep angle, soa solution is searched to include the sweep angle. This is done by calculatingthe induced velocitities on the bound vortex. The Tre"tz plane only recognisesthe infinite vortex lines, the di"erence of the starting point’s x-component isneglected because it is too far away. On the bound vortex, the influence ofdi"erent x-components of the trailing vortex starting point is considered. Thiscauses an influence on the sweep. The implementation results showed, that theinfluence is too strong and gives a qualitatively incorrect distribution of lift.

To illustrate the e"ect of Tre"tz and bound analysis, a Figure with thedi"erence of the two analysis types is added, see Figure 12. The wing is a sweptwing without dihedral. The geometry definition is listed below.

Wing definition of the test wing:

1 p=3; %number of Partitionsny=[0,0,0]*pi /180; %Dihedral of the Partitions

3n=[10 ,10 ,7]; %Number of collocation points

5 s=[0.5;0.3;0.2]; %Span of partitionb_root =[0.2;0.2;0.2]; %root chord

7 b_tip =[0.2;0.2;0.2]; %tip chordphi =[20 ,20 ,20]*pi /180; %Sweep angle , back is ""

positive9

%geometrical twist angles11 alpha =[0 -2 -2 -3 -3 -3.8]*pi /180;

!! !"#$ !"#% !"#& !"#' " "#' "#& "#% "#$ !"#!

"#'

"#(

"#&

"#)

"#%

*

+,--,

.

.

!! !"#$ !"#% !"#& !"#' " "#' "#& "#% "#$ !"#!

"#'

"#(

"#&

"#)

"#%

*

/0

.

.

+,--,.

+,--,.1234.5678839.:;,<*=2=

/0.>2=362?@32A;

/0.>2=362?@32A;.1234.5678839.:;,<*=2=

Figure 12: Di"erences between alternative calculations of the induced angleAngle of attack: 6 degreesCalculations made with Nonlinear Lifting Line Method Gohl

The test wing from above is compared with other results. Here comparedare the CL distributions, the circulation distribution curve is qualitatively the

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4 IMPLEMENTATION OF A SUITABLE METHOD

same because the chord is constant and therefore not plotted, see Figure 13.

!! !"#$ !"#% !"#& !"#' " "#' "#& "#% "#$ !"#!

"#'

"#(

"#&

"#)

"#%

"#*

"#$

+

,-

.

.

/01231456.-378319.-314.:48;0<=.>[email protected];2

/01231456.-378319.-314.:48;0<.CD-E

F5142.:48;0<.CD-E

/01231456.-378319.-314.:48;0<=.G0H1<[email protected];2

Figure 13: Comparison of the Lifting Line Method with a Panel calculation forthe swept wing with twistangle of attack: 5°

Advantages of the Nonlinear Lifting Line Implementation

• Nonlinear behaviour of profile information influences the circulation dis-tribution.

• Flap deflections do not need to be regarded in mesh generation. The flapinfluence is regarded in the iterative function where the local lift coe!cientis interpolated.

• The calculation is very fast.

Disadvantages of the Nonlinear Lifting Line Implementation

• The twist e"ect is not modelled well, because all trailing vortices are leav-ing the wing and not following the local chord direction.

• This method does not provide accurate results for wings with low aspectratios.

• The results of swept wings are not usable, because sweep angle e"ects areneglected.

• An interpolation between Reynolds numbers is not done in this imple-mentation. When profile polars are made, the Reynolds number must beestimated.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.2 Vortex Lattice MethodThe Vortex Lattice Method is implemented as a consequence of the unsatisfac-tory results provided by the Nonlinear Lifting Line Method and the limitationsof the method (see section 3.1.4 on page 12 and section 3.1.4 on page 13).

The Vortex Lattice implementation is similar to the Nonlinear Lifting LineMethod. However there are some significant changes in the mesh compositionand in the structure of the linear system of equation, and there are other mod-ifications, which are shown in this section. The basic problem of the VortexLattice method is, that the dimension of the influence matrix is much higher.If a vortex lattice method should be implemented in a dynamic simulation, thecalculation time is a severe problem. To reduce the computing time, some im-portant modifications in saving the data and in calculating the influence matrixare done. The most important steps in the program are shown in a flow diagram,see Figure 14

load all input data

geometry

definition

inflow

information

profile polar

definitionXfoil

polars

generate geometry and vortex mesh

calculate influence coefficient

Vortex Lattice MethodProcess Diagramm

generate RHS, insert flap deflection

inverting Matrix K and calculate

Gamma

calculate forces and moments

geometry

mesh Influence matrix K

Gamma

Manual input

Figure 14: Process diagram of the Vortex Lattice Implementation

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.2.1 Inclusion of the Profile

The profile coordinates can directly be included in the implementation. Thenthe mean line is calculated from the profile data, as an example see the profileMH45 in Figure 15:

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #

!!"%

!!"$

!!"#

!

!"#

!"$

!"%

,!-,./

0!-,./

1

1

Figure 15: Mean line of the profile MH45 calculated with profile.m

4.2.2 Mesh Generating

The wing is divided into rectangular elements in spanwise direction and in chorddirection. In each of the panel a horseshoe vortex is laid. The trailing vortices inthis implementation are following the chord direction, otherwise the twist e"ectwould be neglected. The arrangement in top view is shown in the theoreticalpart (see Figure 7 on page 15). For details about di"erent mesh arrangementsand results, see in section 4.2.8 on page 31. A detailed description of a panel isshown in Figure 16

T_right

T_left

P

bound

vortex Trailing

vortex

!

!

!

Panel

V"

r1

r2

n

T_left

behind

T_right

behind

P

Figure 16: Detailed description of a single panel

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4 IMPLEMENTATION OF A SUITABLE METHOD

The mesh points are saved in matrices:

TleftX =

TleftX1,1 TleftX1,2 TleftX1,3 ...TleftX2,1 TleftX2,2 ... ...TleftX3,1 ... ... ...

... ... ... TleftXm,n

The first index is the number of the element in chord direction, the secondindices is the number of element in spanwise direction. The same is applied forall other coordinates and points. The organisation of the elements is shown inthe following Figure:

1,1

1,2

1,3

2,1

3,13,2

2,2

2,3

2,4

3,3

3,4

3,5

. . .

. . .

. . .

Figure 17: Organisation of the indices of the mesh points

With this matrix structure, MATLAB can calculate faster, because thebasic vector calculations can be calculated much faster.

The normal vector is now calculated at the mean line of the profile, seeFigure 16. It is calculated with the cross product:

r1 ! r2

% r1 ! r2 %= "#n (16)

For the calculation of the normal direction, the collocation point must lie onthe profile mean line.

4.2.3 Sideslip Angle

If the flow has a y-component, the mesh must be changed. In this implemen-tation, the trailing vortices only are rotated around the z-axis. The influencecoe!cients are saved for a range of sideslip angles and then the desired sideslipangle is interpolated between this data.

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4.2.4 Influence Coe"cients

The influence coe!cients are organised as the following Table shows:

Coll. Points Matrix Kp1,1 : a1,1 a1,2 a1,3 ... a2,1 a2,2 a2,3 ... !1,1 RHS1,1

p2,1 : a1,1 a1,2 a1,3 ... a2,1 a2,2 a2,3 ... !1,2 RHS2,1

p3,1 : a1,1 a1,2 a1,3 ... a2,1 a2,2 a2,3 ... !1,3 RHS3,1

... a1,1 a1,2 a1,3 ... a2,1 a2,2 a2,3 ... ... RHS1,2

p1,2 : ... ... ... ... ... ... ... ... · !2,1 = RHS2,2

p2,2 : ... ... ... ... ... ... ... ... !2,2 RHS3,2

p3,2 : ... ... ... ... ... ... ... ... !2,3 RHS1,3

... ... ... ... ... ... ... ... ... ... RHS2,3

p1,3 : ... ... ... ... ... ... ... ... ... RHS3,3

... ... ... ... ... ... ... ... ... ... ...

Table 3: System of linear equations

The boundary condition is evaluated at each collocation point, like for ex-ample, the equation for the collocation point 1,1:

a1,1 ·!1,1+a1,2 ·!1,2+a1,3 ·!1,3+...+a2,1 ·!2,1+...+am,n ·!m,n = ""#V! ·"#n (17)

Attention, many errors can occur if the numbering of the circulation vec-tor and the numbering of the RHS vector are exchanged. There are writ-ten functions, which can rewrite the numbering of these vectors (rewrite.m,rewrite_RHS.m).

The calculation of influence coe!cients is basically similar to the calcula-tion in the nonlinear implementation. The principle to calculate the influencecoe!cients of a vortex line is described below.

The basic formula to calculate the induced velocity of a straight vortexfilament is given in the theoretical part. So the cosines of the angles betweenR_0 and R_1, R_1 and R_0 must be found. These angles are calculated withthe help of the dot product. The geometrical illustration is shown in Figure 18.

cos(#) =R0 · R1

% R0 % ·% R1 % (18)

The distance r is calculated with the help of the cross product:

r =R0!R1% R0 % (19)

The resulting induced velocity is calculated with the principle of equation18 and 19:

a =R1!R2

% R1!R2 %214!

(R0 · R1% R1 % "

R0 · R1% R2 % ) (20)

The value % R0 % in equation 20 can be cancelled.

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4 IMPLEMENTATION OF A SUITABLE METHOD

!

L(x,y,z)

R(x,y,z)

P(x,y,z)

R_0

R_1

R_2

r

Figure 18: Vortex influence coe!cient

A numerical problem in calculating the influence coe!cients can occur, if acollocation point lies too close to the vortex line. The induced velocity then risesto infinite. To avoid this problem, a region is defined in which the velocities areset to zero. The problem occurs if:

% R1 %< ' (21)

% R2 %< ' (22)

% R1!R2 %2< ' (23)

If one of these three equations is fulfilled, then the value of a is set to zero.So the equation 20 for all values of the denumerators is always defined.

The idea is, to calculate the induced velocity from all bound vortices, lefttrailing vortices and right trailing vortices for one collocation point i at once.This is much faster. All the operations of the equations 18, 19 and 20 canbe calculated with matrices. The function vortex_panel.m needs as input allpoints in matrix structures and gives out the coe!cients. The structure of a(see equation 20) is therefore a matrix. The indices are organised in the way,that the coe!cient am,n is the influence of the panel m,n on the collocationpoint i. These coe!cients are directly rewritten as vectors, so that they are inthe structure of a row of the influence matrix, see the system of equation inTable 3

The function is added in Algorithm 3, Appendix AThe calculation of the influence matrix for one collocation point is made in

a loop for each collocation point. See the function in Algorithm 4, Appendix A.

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For lift forces, the calculation of the influence coe!cients is done with thecollocation points on the 3/4 line of the panel. For induced drag forces, thecalculation of influence coe!cients is done on collocation points lying on thebound vortex. Then the K matrix is saved as the inverted matrix, so thatduring the simulation no matrix inversion is needed.

4.2.5 Flap deflections

Flap deflections in this implementation are made with the boundary condition.All surface normals which lie in the flap region are rotated around the y-axiswith the angle of the flap deflection.

4.2.6 Force and Moment Calculation

The lift forces act on the bound vortices. They are calculated for each panel:

Liftforce =n!

i=1

m!

j=1

$ · !j,i ·&yj,i (24)

The induced drag is calculated with the following formula:

Draginduced =n!

i=1

m!

j=1

"$ · sign(!j,i) · !j,i · windj,i ·&yj,i (25)

The term wind is the induced velocity from the trailing vortices. It is calcu-lated with the following formula

wind = Kbound · ! (26)

Kbound is the influence matrix of the trailing vortices on the bound vortices.It is calculated with the following formula:

Kbound = KboundX 'Unormalvel + KboundY ' Vnormalvel + KboundZ 'Wnormalvel

(27)

KboundX, KboundY , KboundZ are the influence coe!cient matrices in thedirections X, Y, Z.

Unormalvel , Vnormalvel , Wnormalvel are the components of the vectors ver-tical to the flow speed and the bound vortex vector.

The viscous drag is interpolated with the profile polars.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.2.7 Stall

In the simulation high e"ective angles might occur. It could therefore be, thatCL values of 2.0 or more appear (as a consequence of the linear behaviour of themethod). Values of CL over 1.5 for normal profiles do not exist physically. Inthis case the interpolation function for calculating the CD values has a problem,because in the polar table no CD value at this too high CL value exists. In orderto enable the simulation to calculate stall situations, CL values which are notfound in the table are assumed as the last element in the table. The results ofthis method are not signigicant, therefore the Vortex Lattice Method does notgive accurate results for stall behaviour.

4.2.8 Evaluation of the Vortex Lattice Method

Advantages

• Any desired wing geometry with sweep, dihedral, twist, winglets and sev-eral wing parts can be calculated. Even elevator and fin can be added tothe geometry, but the mesh must be looked at carefully due to singulari-ties.

• Airfoil cambering is taken in account.

Disadvantages

• Non-linearities of the air’s viscousity and dependencies of speed are ne-glected in the lift force and the induced drag force.

• In this implementation a strong wake simplification is made, see in sec-tion 4.2.8.

• An interpolation between Reynolds numbers is not done. Influences ofspeed and the chord distribution of the wing are approximated to thesame Reynolds numbers.

Influence of Di!erent Mesh Types

In this part the di"erences between a flat mesh and a mesh on skeleton line of thewing are shown. It elucidates again the simplification of the mesh arrangementof the implemented method. A comparison of di"erent mesh types is done andanalysed qualitatively and quantitatively.

The boundary conditions are still evaluated at the skeleton line. The greatdi"erence is in direction of the trailing vortex. It follows the skeleton directionor the chord direction.

31

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4 IMPLEMENTATION OF A SUITABLE METHOD

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Figure 19: Di"erent mesh typessquares: collocation points and trailing vortices are liyng on the skeleton line ofthe wing,circles: collocation points and trailing vortices are laying on the chord line ofthe wingtriangles: a calculation from XFLR with VLM classic method, induced dragwith Tre"tz analysis

The greatest di"erence is visible in the induced drag. This is also the valuewhich causes most of the problems in this method. The wake modeling isnot physical. The trailing vortices are not leaving the wing in the flow speeddirection. In this implementation, the twist has a too strong influence on thedirection of the trailing vortices. But for exact lift distribution and center oflift calculations, the vortices must follow the profile chord or the local skeletonline.

For calculating exact glide numbers, the method provides too high valuescaused by too small induced drag values (see Figure 19). The di"erence ofthe induced drag is shown in Figure 20, where the wing is the model gliderFG-WingX-02.

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4 IMPLEMENTATION OF A SUITABLE METHOD

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Why not let the trailing vortices follow the flow speed direction? For a staticcalculation, there would not be any problems, but for a dynamic simulation thereare obstacles. The vortex arrangement for dynamic analysis should not change.If it changes, the following steps have to be calculated:

1. The vortex arrangement must be corrected at each time step.

2. All the influence coe!cients in the Biot-Savart influence matrix K changeand must be recalculated.

3. The matrix inversion must be recalculated. The matrix can have dimen-sions of 500x500.

4. An interpolation with the sideslip angle is not possible, because the meshis not always the same. So the arrangement has also to be corrected withsideslip angle. But this step 4 could be directly done in step 1.

If all these calculations had to be done at each time step, this would cost muchmore simulation time.

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4 IMPLEMENTATION OF A SUITABLE METHOD

4.3 Integrating the Methods in the SimulationNow that the aerodynamic forces may be calculated as a function of incidentflow direction, flap deflection and angular rates, the airplane dynamics maybe calculated by applying rigid body dynamics. For the rigid body motionsimulation, the existing MATLAB/Simulink model by André Noth is used.

Therefore, only the mass, centre of gravity and the inertia must be added.The inertia is calculated in the function inertia_calc.m. Some simplification

are made:

• homogeneous mass distribution

• constant thickness of the wing, with thickness c defined in wingdef.m

• wing, modelled with many rectangular boxes

• neglected dihedral

Vortex Lattice Method

The input data from the Euler-Lagrange dynamics and the output data of theforce calculating block is shown in the following flow Figure 21:

Aerodynamic Forces

Fx, Fy, Fz

Mx, My, Mz

flap deflections

speed in body system

angluar rates

in earth system

phi, theta, psi

Figure 21: Flow diagram of the aerodynamic force block

The data must be transformed into the wing’s body system. Moreover itis essential to calculate the angular rates p, q, r around the body axis. Theinterpolation of the influence matrices must be done. Then the velocity vectoris calculated. The angular rates are taken in account in the velocity vector. Itis calculated with following principle:

#vlocal = " #Omega!"#R + #V! (28)

Omega is a vector with the angular rates in the body system, R is thedistance between the center of gravity and the local panel.

All the vectors have to be reorganised for the RHS calculation, even thesurface normal vectors. The flap deflection is calculated with changing theRHS. The mesh arrangement is not changed for the flap deflection. This is asimplification in favour of faster calculation, otherwise the whole mesh wouldhave to be recalculated. Finally the circulation distribution is calculated withthe interpolated influence matrices at the desired sideslip angle.

Then the forces and moments are calculated and delivered to the output.

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4 IMPLEMENTATION OF A SUITABLE METHOD

Nonlinear Lifting Line Method

The inclusion of the Nonlinear Lifting Line Method in the dynamics is similar tothe Vortex Lattice Method. Some important di"erences occur in the calculationof the circulation. The circulation is calculated with the initial circulation atthe first time step. For the next time step, the circulation distribution from theold time step is used for starting the iteration. The changes of the circulationdistributions are always small, so that only a few iterations are necessary.

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5 EVALUATION OF THE SIMULATION

5 Evaluation of the SimulationThe simulation is evaluated with real flight tests. In order to at least qualita-tively compare the behaviour of the simulation to real flight, flight test are madeand taken on video. The experimental data are collected with a self built modelflying wing. More information about the wing can be found in section 6.1 onpage 39 .

5.1 Flight Test ResultsFor the flight test, a well defined flight attitude must be found, as the simulationneeds the same initial condition as in the flight test.

This attitude can be reached with the following scenario. At first the wing isaccelerated and then, at high speed, the aileron are set to the initial condition.The wing slows down until the velocity is zero and then the wing tilts forward.It begins to oscillate periodically as it stabilises on its own. The period timecan be measured by a clock. Then the period times can be compared withsimulated results. For visualisation, the course of the flight is shown in thefollowing Figure:

acceleration

with flap

deflection

flaps set to

the initial state

velocity: zero angle theta: -90°

first period second period

comparison with simulation

no action by the pilot

Figure 22: Path for flight tests

Measured times are written in Table 4. The measurements were made ingood weather conditions, early in the morning, so that weather e"ects could beneglected. Two flights were evaluated. Flight one: Only one period time wasmeasurable. Flight two: Two period times were measurable.

Phygoid Nr. 1 2 ToleranceFlight 1 [s] 5.8 not measurable +/- 0.3Flight 2 [s] 5.5 5.4 +/- 0.3

Table 4: Measured times of the phygoids

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5 EVALUATION OF THE SIMULATION

5.2 Simulation ResultsThe following attitude is chosen for the evaluation:

# -85° real flight dataspeed ! 1 [m/s] a realistic value is chosen

centre of gravity 0.21 m data from VLM Gohlflap deflection ! "4° (+/- 0.5 degrees) real flight dataangle of attack 2° a realistic value is chosen

Table 5: Initial Condition

The simulation results with the initial condition in Table 5 are shown inFigure 23. In Table 6 the measured period times are shown. The polars in Xfoilare made with a turbulence model of Ncrit=9, details about the turbulencemodel see in Xfoil manual, in the section Viscous Formulation [4].

Period Nr. 1 2 3Period time [s] 5.76 5.88 5.76

Table 6: Simulation Results, times of the first three periods of the phygoidoszillation

There are several reasons for the slight variation of the simulated and mea-sured values:

• The model is not absolutely perfect, it is a simplification of the reality.Values such as the induced drag are inexact and can cause small errors.The calculation of the inertia is another simplification. There could alsooccur 3D flow e"ects, which are neglected.

• The airfoil polars are generated with Xfoil. The turbulence model ofN crit = 9 in Xfoil might be incorrect.

• The model wing is not perfect. The trailing edge is not as sharp as thesimulated profile is. In addition the leading edge is less rounded than theoriginal airfoil, due to the manufacturing method.

• The initial condition is not absolutely correct as it is estimated. The initialcondition of #init = "#

2 causes some problems with the Tait-Bryan angles(more details see in [16]) so that the angle must be approximated.

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5 EVALUATION OF THE SIMULATION

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38

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6 RESULTS OF THE SIMULATION

6 Results of the Simulation

6.1 Flying Wing, FG-WingX-02In the next three sections three di"erent wings are presented, simulated andanalysed. The flying wings are tuned for high flight performance and are com-pared in their stability and performance. The most important design rules aboutflying wings are written in Appendix C.

Before this thesis was written, a model glider, FG-WingX-02, was built.The glider was calculated with [20]. The design contains nine di"erent chordwidths and furthermore the twist is not only linear. The chord increases withthe spanwise coordinate, which is unusual for a conventional wing. This e"ectimproves the stability and increases the Reynolds numbers at the wing tips.The wing is designed for best glide ratio with flap deflection and for good fastflight without flap deflection.

Originally the wing was designed without winglets. The side stability shouldhave been generated with the high sweep angle. However practical tests showed,that the side stability was not adequate. An improved design with winglets isdescribed in 6.2. Basically it is possible to attain side stability only with a sweepangle. For more details see the Horten wings.

Figure 24: My WingVECTORWORKS EDUCATIONAL VERSION

VECTORWORKS EDUCATIONAL VERSION

Some data of the wing:

Wing span: 2 mWing span with winglets: 2.32 mWing load: 25.945 g/dm2

Taper Ratio: 2.0With flap deflections of -3 degrees the XFLR polar calculation generates the

following results (see Figure 6.1):

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6 RESULTS OF THE SIMULATION

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The best glide ratio is about 23 at an angle of attack of 9°. The best sinkrate is at an angle above 10°. Angles above 10° are in a critical region of stall.

40

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6 RESULTS OF THE SIMULATION

6.1.1 Stability Results

The wing is simulated for the the initial condition shown in Table 7. In AppendixD is shown, how the equilibrium point is calculated. The wing is dynamicallystable, which shows Figure 26.

v_initial 7 m/scenter of gravity 0.21 mflap deflection -4°

sideslip 0°angle of attack 6°

#init -85°

Table 7: Initial point for simulation

The result is plotted in Figure 26.

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6 RESULTS OF THE SIMULATION

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The simulation of the wing FG-WingX-02 for longer times has shown, thatthe wing is spiral mode unstable (similar behaviour as shoen in Figure 29). Thisbehaviour is noticeable in practical flight test. If the wing flees once into a curvethen he would tilt more and more into the curve. This dynamic behaviour isslow and can be stabilised by the pilot

42

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6 RESULTS OF THE SIMULATION

6.2 Optimised Stable WingOriginally the wing FG-WingX-02 was designed without winglets. As the wingletswere added later, the circulation distribution is not elliptic anymore.

The optimised wing has the same outline but a di"erent twist. The twist wasincreased the most at the tip of the wing, where the winglet’s influence is thestrongest. The wing is optimised at best glide ratio, not at best sink rate for abetter stall behaviour because the angle of attack at best sink rate is very high.This modification is done, to create a self stable wing with high performance.

The polars calculated in XFLR are shown in Figure 28.

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43

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6 RESULTS OF THE SIMULATION

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The main di"erence between the optimised stable wing and the original wingis the angle of attack, when the best glide ratio occurs. The optimised stablewing is fundamentally better manoeuvrable at best glide ratio. The angle ofattack at best glide ratio is further from stall and even flying at best sink ratewould be possible. In the middle of the wing, the CL value is increased forbetter stall behaviour, but the performance hardly improved.

The lift distribution at the operating point is shown in Figure 27. For com-parison only the lift distribution of the real model glider and the model gliderwithout winglets are plotted in the same Figure. The spanwidth coordinate isfolded on the y-axis for better visualisation.

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6 RESULTS OF THE SIMULATION

6.2.1 Stability Results

Simulations are done, to analyse the stability behaviour of the modified wing.The operating point is defined, as shown in Table 8.

equilibrium point initial pointv_initial 8.4437 m/s data from VLM Gohl 8.4437

angle of attack 8° 5°%Glide 2.16°

centre of gravity 0.2461 m data from VLM Gohl 0.2461flap deflection -2° -2°

sideslip 3° 3°

Table 8: Operating point

Then the wing is simulated for stability analysis. The wing should be stable.The simsulation results with the configuration listed in Table 8 are shown inFigure 29

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6 RESULTS OF THE SIMULATION

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The time the wing needs to balance itself is about 50 seconds. The dimensionof the momentum in y-axis after 30 seconds is 1 · 10#4Nm. The dynamicalbehaviour is similar to the wing FG-WingX-02.

After 100 seconds the wing begins to instabilise itself around yaw and rollaxes caused by small numerical noise. These e"ects can only be canceled with acomplex optimisation of the geometry (dihedral and winglet, see section 6.3 onthe next page). These modifications are not done here, they are too complex.The modified wing does not garantee the spiral stability and dutch roll modestability, but the dynamic of these instabilities are slow and can be stabilisedby the pilot.

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6.3 Dutch Roll Mode (germ. Taumelschwingung) andSpiral Mode

A problem with flying wing, which is di!cult to handle, is the dutch roll andspiral mode stability. Here some of information about yaw stability coupledwith dutch roll and spiral mode is added. For flying wings these problems oftenappear.

E"ects of dihedral and sweep angle are qualitatively explained. This sectiondoes not give detailed theoretical information, it only helps to understand thesimulated results and enlightens the problem of yaw stability coupled with dutchroll mode and spiral mode. More detailed theoretical information can be foundin [3], page 180 and in [9]

To reach yaw stability the most simple solution is to add large enoughwinglets. Another way is to add a high sweep angle, which stabilises the wing bydrag. The dihedral is basically added for better roll stability, but if it was addedon the entire wing it would not stabilise the wing on the yaw axis. Dihedral andsweep angle are quasi similar roll angle stabilising elements.

Dutch roll means, that the wing oscillates in yaw axis coupled with an oscil-lation in roll axis. This is caused by too much dihedral and not enough influenceof the fin (If the winglets are attached behind the centre of gravity, they act asfins.). The opposite problem which occurs with too strong influence of the finand too little dihedral, is the spiral mode. The wing tends to turn more andmore into the curve until its flight path is a spiral.

To design a wing which fulfils both stability criteria (roll and yaw) is chal-lenging. Winglets are often a good solution, they do not have a dihedral e"ect.Another solution could be to add a negative dihedral. It eliminates the e"ectof dihedral on the sweep, so that dutch roll and spiral stability is su!cient. Onthe other hand, negative dihedral would also cause negative e"ects in fast flight.

An important design rule for flying wing is, that dihedral should not beadded to the wing. If dihedral is added for roll stability, dutch roll and spiralstability must be studied carefully. For the two stable built wings (see above),the spiral mode is slow enough to be controlled by the pilot and therefore is nota hazardous instability.

6.4 Unstable WingA wing is designed which is unstable and has a high performance. The designcriteria is to reach a high glide ratio. This is done by a wing without sweep angleand twist angle, as these elements create a loss of performance. To minimisethe induced drag an elliptical circulation distribution is aimed. Therefore theresulting ideal geometry would be a perfect ellipse. Due to the low Reynoldsnumbers at the wing tip, the chord at the wing tip is increased and a smalltwist is added. Therefore the shape of the wing is not a perfect ellipse, but thecirculation distribution is a perfect ellipse. Winglets are added to garantee theside stability. The are not added to create an influence on the flow on the wing,but only to garantee the side stability of the wing. They act similarly to fins.Nevertheless supplementary winglets could be added in order to improve theperformance of the wing, but this is not done here.

The centre of lift is not lying in front of the neutral point, so the wing isabsolutely pitch unstable. As the wing is unstable in the pitch axis, a controller

47

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6 RESULTS OF THE SIMULATION

must be designed to garantee the pitch stability. Otherwise, without a controller,a simulation would not be possible.

Consequently the yaw stability is tested in simulation. With the controllerthe wing should be stable on the pitch axis and the stability in yaw and rollaxes must be fulfilled by the self-stability of the wing. It must be tested as well,if the wing is manoeuvrable. In other words, can the side stability of the wingcompensate the negative momentum from rudder deflections?

6.4.1 Geometry

The geometry definition is shown in Appendix A, Algorithm 5 on page 63.For visualisation a mesh plot is shown in Figure 30.

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6.4.2 Static Results

The polars are calculated with a Vortex Lattice Method in XFLR. A calculationwith a panel method is added. In XFLR, the panel method is a 3D method whichplaces the singularity elements on the top and the down surface of the wing.Detailed informations about the XFLR calculation methods can be found in [1].

The best glide ratio can be reached at an angle of attack of 5° to 6°, themaximum value with the VLM calculation is 26.

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6 RESULTS OF THE SIMULATION

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6.4.3 Controller

The controller is a PID controller. The controller parameters are calculatedwith the Ziegler Nichols Method [5]. The parameters are listed in Appendix A.

An actuator saturation was added, because the polars are only made for flapdeflections of -15° ... 15°. The di"erence of #init " #out (pitch angle is #out)must be multiplied with -1, because the flap deflection is positive if the flap goesdown (Xfoil definition). The controller is shown in Figure 32.

49

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6 RESULTS OF THE SIMULATION

Figure 32: Simulator with controller

Attention: This controller is only designed for controlling the pitchaxis. The control design in this way is not meant to be realised. Butfor studying the instabilities of the other axes it is only essential, thatthe pitch axis is stable.

6.4.4 Simulation results

The simulation is started with the following initial condition, see Table 9.

equilibrum point initial pointv_initial 7.25 m/s data from VLM Gohl 7.25 m/s

angle of attack 6° at best glide ratio 4°%Glide (glide angle) 2.72°centre of gravity 0.0883 m data from VLM Gohl 0.0883 mflap deflection 0° 0°

sideslip 3°

Table 9: Initial point

The simulation results are shown in Figure 33.

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Figure 33: Simulation Results

In the consequence it is tested, what happens with di"erent flap deflections inorder to fly curves, with a closer look at the angle of sideslip. This is important,because rudder deflections are coupled with an unpleasant e"ect, the momentumin the yaw axis. It causes a sideslip angle in the opposite direction.

First simulation results showed, that with this wing it is not possible to fly

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6 RESULTS OF THE SIMULATION

a curve! The sideslip increases too much. Therefore the wing must be stabilisedmore in the yaw axis.

A positive dihedral or larger winglets cause the best e"ect on the side stabil-ity. It is important to place the winglets far enough behind the physical centreof gravity. To improve the side stability, the winglets are placed 50 cm behindthe wing (see Figure 30). After that, the simulation results are much better andflying curves has become possible. Simulation results are shown in Figure 34with the aileron deflection shown in Table 11. The simulation is started withthe following initial point (see Table 10). In Figure 35 the relative sideslip angleis plotted versus the time.

The characteristic of &in shows clearly that the wing is stable, even with flapdeflections.

The speed is plotted in Figure 6.4.4

Initial pointv_initial 7 m/s arbitrary

angle of attack 5° arbitrary%Glide 2.72° calculated from glide ratio

centre of gravity 0.0883 m data from VLM Gohlflap deflection 0°

Table 10: Initial point

time [s] 3 - 3.5 3.5 - 6 6 - 9s 9 - 60left aileron 1° 0 -1° 0 + / - controller deflection added

right aileron -1° 0 1° 0 + / - controller deflection added

Table 11: Flap deflectioins

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6 RESULTS OF THE SIMULATION

!"!

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Figure 34: Simulation results with flap deflections. At first a left curve, then aright curve is flown.

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6 RESULTS OF THE SIMULATION

! "! #! $! %! &! '!!!(!'

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Figure 35: Sideslip angle of the simulation with flap deflections, as shown inFigure 34

0 10 20 30 40 50 606.4

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

Time

Speed

Figure 36: Speed of the flying wing

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7 DiscussionThe simulation results show that the performance of an accurately optimisedstable wing can be topped by a pitch controlled unstable wing. However, thefact is, that even a well designed self stable wing with an elliptic circulationdistribution can have an excellent performance and fulfil the stability criteria.Therefore, the question whether it is worth to design a pitch unstable wing witha controller can not be answered here.

The simulated unstable wing has calculated glide number of about 26 in thearea of Reynolds numbers of 100000, whereas comparison the optimised stablewing has the best glide ratio at 24, with the same Reynolds number. The selfstable flying wing FG-WingX-02 has best glide ratios about 23. These glideratios are calculated with a vortex lattice method with the panels lying on theprofile mean line. For this comparison only glide ratios are considered, thereare many other criterias to compare (as example: stall behaviour or curve flyingbehaviour). The statement, that the controlled pitch unstable wing has the bestglide ratio does not necessarily mean, that it is the best wing in every respect .

A comparison of the calculated glide ratios with real measured glide ratiosof contest wing airplanes is di!cult to find, and most gliders are not tuned forbest glide ratio. Some flight data of succeeded F3B competition models arepresented in [19], their glide ratios are around 15 - 17.

For more accurate statements wind tunnel tests or exact measured flightpaths would have to be done.

The simulation results show clearly, that the time until the wing stabilisesitself from a disturbance is long (duration in the tests: about 50 seconds at aninitial angle " of -90°). The reason is the low damping around the pitch axis.The damping basically depends on the induced velocity of the angular rate pon the wing. The further the wing elements are away from the pitch-axis, thehigher the damping velocity is, as written in the following formula: v = (!"R.

All the simulated wings tend to fly a curve. If the wing once has slightlytilted around the roll axis, the wing does not act with a moment against thistendency. Therefore, the wing tilts into a curve, until an equilibrium point inthe curve is reached. The reason for this behaviour is, that a numerical noisealways acts on the calculated moments. Caused by these small disturbances,the wing passes into a dutch roll instability.

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8 ConclusionThe most important tasks and limitations for a dynamical method were defined,and consequently numerical methods which fulfil the tasks were presented. Ex-isting statical and dynamical applications were introduced and discussed. The-oretical information about vortex methods was given.

A complex method for calculating aerodynamic forces with regard to com-plex three-dimensional wing geometries, angular rates and asymmetrical flapdeflections was implemented in MATLAB. It turned out, that a model, whereall flow e"ects would have been included consumed a lot of computer processtime. Therefore the implemented vortex lattice method was modified for shortcalculation times and some simplifications in the wake arrangement were made.The wake was modelled statically with vortex filaments. Horseshoe vortices assingularity elements in chord and spanwise direction were chosen. This VortexLattice Method was integrated in a rigid body motion simulation implementedin Simulink.

With the simulation tool, it is possible to study dynamical and staticalresults for arbitrary three-dimensional flying wing configurations. Statementsabout stability in all axis and manoeuvrability with flap deflections can be made.Complex dynamical e"ect such as spiral mode, dutch mode and phygoid modecan be simulated and analysed.

To answer the question about the unstable wing’s gain of performance, apitch unstable wing was designed and compared with two self stable flying wings.For testing the flight characteristics and the dynamic stability, the flying wingswere simulated in the dynamic simulation. The unstable wing was stabilised bya pitch controller to garantee the pitch stability.

The results show, that flight performance with an unstable wing can be wonand high calculated glide ratios can be reached. The unstable wing, simulatedin this thesis reaches glide ratios about 26. For comparison, the stable modifiedflying wing reaches a glide ratio about 24 at Reynolds numbers of 100000. Theinteresting result is, that the pitch unstable wing does not have a significantlybetter glide ratio in comparison to a well designed self stable flying wing. Butthe unstable wing has a much better stall behaviour, because the angle of attackat best glide ratio is smaller than than the angle of attack at best glide ratioof the stable wings . So the question whether it is worth to design a controllerand achieve better results of the glide ratio must be answered in considerationof other criteria needed.

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9 Future WorkFor a continuative work there are many modifications to do. In general threeoptimisation strategies might be useful:

Increasing the simulation speed to improve the vortex model. A high im-provement of the calculation speed could allow automatic geometry variationto find best geometry parameters. An improved model would be necessary tocombine more aerodynamic e"ects and therefore give more accurate results. Abetter evaluation of the simulation would elucidate, if even a improvement ofthe model is necessary.

Increasing the Simulation Speed

With a modified 6DOF simulator the computing time could be improved. Up-grading the computing time of the calculation methods would also be necessaryfor a faster simulation.

Improving the Model

Many improvements in wake modelling and flow separation could be done. Asimple improvement in this implementation would be, that Xfoil is automati-cally coupled with the vortex methods. An interpolation for di"erent Reynoldsnumbers could also improve the physical model without expending a lot of com-puting time. To achieve better results for induced, a better wake modellingwould be necessary. For even better results, a vortex lattice method with alattice of vortices over the top and bottom surface of the wing could be laid.

Evaluating the Simulation

The simulation results could be evaluated more precisely with flight path mea-surements or even with wind tunnel measurements.

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References[1] Author unknown, guidelines for xflr5 v4.16. Analysis of foils and wings

operating at low Reynolds numbers, April, 2009.

[2] Jon S. Berndt. jsbsim.sourceforge.net/. Manual.

[3] Michael Cook. Flight Dynamic Principles. Number ISBN-13: 978-0-7506-6927-6 in second edition. BUTTERWORTH HEINEMANN, 2007.

[4] Mark Drela and Harold Youngren. Xfoil 6.9 user primer. 30 Nov 2001.

[5] Lino Guzzella. Analysis and Synthesis of Single-Input Single-Output Con-trol Systems. vdf, 2007.

[6] Karl-Heinz Horstmann. Ein Mehrfach - Traglinienverfahren und seine Ver-wendung für Entwurf und Nachrechnung nichtplanarer Flügelanordnungen.1988.

[7] Michael Wohlfahrt Karl Nickel. Schwanzlose Flugzeuge. Birkhäuser Verlag(Flugtechnische Reihe Band 3), 1990.

[8] Joseph Katz and Allen Plotkin. Low-Speed Aerodynamics. Cambridge Uni-versity Press, 2001.

[9] Bill Kuhlman and Bunny Kuhlman. Swept wings and e"ective dihedral.RC Soaring Digest, March 2000.

[10] GNU General Public License. Xflr5 is an analysis tool for airfoils, wingsand planes operating at low reynolds numbers.

[11] Harold Youngren Mark Drela. Avl,http://web.mit.edu/drela/public/web/avl/.

[12] W. H. Mason. Applied computational aerodynamics text/notes aerodynam-ics of 3d lifting surfaces http://www.aoe.vt.edu/. Department of Aerospaceand Ocean Engineering Virginia Polytechnic Institute and State University,Mar 11, 1998.

[13] Tomas Melin. A vortex lattice matlab implementation for linear aerody-namic wing applications. Master’s thesis, Royal Institute of Technology(KTH), 2000.

[14] Austin Meyer. www.x-plane.com/.

[15] Stuart Buchanan Jon Berndt Bernhard Buckel Cameron Moore Curt OlsonDave Perry Michael Selig Darrell Walisser Michael Basler, Martin Spottet al. www.flightgear.org/. The Flight Gear Manual 1.9.0.

[16] A. Noth, S. Bouabdallah, and R. Siegwart. Dynamic Modelling of Fixed-Wing UAVs. ETH Zürich, 2008, Version 2.0.

[17] D. Bruce Owens. Weissinger’s model of the nonlinear lifting-line methodfor aircraft design. AIAA 98-0597, 1979.

[18] Ion Paraschivoiu. Aerodynamique Subsonique. 1998.

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[19] Helmut Quabeck. Design, Leistung und Dynamik von Segelflugmodellen.HQ Modellflugliteratur, 1994.

[20] Frank Ranis and Herbert Stammler. Nurflügel v2.17 (freeware).

[21] redhammer project. Tornado 1.0. http://www.redhammer.se/tornado/,RELEASE 2.3 2001-01-31.

[22] P. SANTINI and P. GASBARRI. Lifting surface in subsonic unsteadyregime. Universit ‘ a di Roma “La Sapienza”, Dipartimento di Aerospaziale,via Eudossiana 18; 00184 Roma, Italy, 5 May 1998).

[23] Matthieu Scherrer. Méthode d’intégration sur une aile de résultat expéri-mentaux et xfoil.

[24] Herrmann Schlichting and Erich Truckenbrodt. Aerodynamik desFlugzeuges, Band II. 1969.

[25] James C. Sivellis and Rober H. Neely. Method for calculating wing char-acteristics by lifting-line theory using nonlinear section lift data. NationalAdvisory Commitee for Aeronautics Langley Field Report No. 865, 1946.

[26] James C. Sivellis and Robert H. Neely. Method for calculating wing char-acteristics by lifting-line theory using nonlinear section lift data. TechnicalNote no. 1269, April 1947.

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Appendix

Appendix A) Most Important Code FragmentsImplementation Nonlinear Lifting Line Method

Algorithm 1 Iteration function (Gammait_nonlin.m) of the Nonlinear LiftingLine Methodfunction[Gamma_alt ,alpha_i ,w_i]= Gammait_nonlin(K,K_far ,Gamma_start ,D,min ,b,""

v_abs ,angle_of_attack ,y,alpha_p ,CL_p ,flap_angle ,poldef)2 Gamma_alt=Gamma_start;

Max =1;4 while Max >min

w_i =1/2*( K_far*Gamma_alt); %here: Trefftz Analysis , with factor""1/2, else 1!

6 alpha_i=-atan(w_i ./(v_abs ’));Gamma_new =(1/2* CL(alpha_p ,CL_p ,flap_angle ,alpha_g ’-alpha_i ,""

poldef) ’.*b’).*v_abs ’;8 Gamma =(1-D)*Gamma_alt+D*Gamma_new;

Max=abs(max(Gamma -Gamma_new));10 Gamma_alt=Gamma;

12 endend

Algorithm 2 Generate Vortices (generate_vorices.m)of the Nonlinear LiftingLine Method

1 function[K]= generate_Vortices_VarIII(A,T_left ,T_right ,n,N)for i=1:2* sum(n)

3 for j=1:2* sum(n)%Collocation Point , where the Velocity will be calculated: i

5 %this velocity gets summated from element 0 till n: j%Velocity of the upper Vortex

7 w_up=vortex(T_left(:,j),T_right(:,j),A(:,i));%Velocity of the left Vortex

9 w_left=vortex_left(T_left(:,j),A(:,i));%Velocity of the right Vortex

11 w_right=vortex_right(T_right(:,j),A(:,i));

13 w=w_up+w_left+w_right;%Projection in the normal direction

15 K(i,j)=w(1)*N(1,i)+w(2)*N(2,i)+w(3)*N(3,i);end

17 endend

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Appendix

Implementation: VLM Method

Algorithm 3 Function vortex_panel.m of the Vortex Lattice Method

1 %This Program calculates the velocity divided by the Zirculation%written and tested: 20. Februar 2009 by Flavio Gohl

3 %The Code is modified to the code in book low speed Aerodynamics page 255 ""from Joseph Katz.

%Attention , the code is not the same!5 function[a_i_X_vec ,a_i_Y_vec ,a_i_Z_vec ]= vortex_panel(L_X ,L_Y ,L_Z ,R_X ,R_Y ,""

R_Z ,P_X ,P_Y ,P_Z ,m,n,epsilon)einsmatrix=ones(m(1),sum(n)*2);

7 %define the three vectors r_0 , r_1 ,r_2r_0_X=R_X -L_X;

9 r_0_Y=R_Y -L_Y;r_0_Z=R_Z -L_Z;

11r_1_X=P_X*einsmatrix -L_X;

13 r_1_Y=P_Y*einsmatrix -L_Y;r_1_Z=P_Z*einsmatrix -L_Z;

15r_2_X=P_X*einsmatrix -R_X;

17 r_2_Y=P_Y*einsmatrix -R_Y;r_2_Z=P_Z*einsmatrix -R_Z;

19%Calculate r_1xr_2=vprod

21 vprod_X=r_1_Y.*r_2_Z -r_1_Z.* r_2_Y;vprod_Y=r_1_Z.*r_2_X -r_1_X.* r_2_Z;

23 vprod_Z=r_1_X.*r_2_Y -r_1_Y.* r_2_X;

25 %Calculate normnorm_vprodsquare=vprod_X .^2+ vprod_Y .^2+ vprod_Z .^2;

27 R1=sqrt(r_1_X .^2+ r_1_Y .^2+ r_1_Z .^2);R2=sqrt(r_2_X .^2+ r_2_Y .^2+ r_2_Z .^2);

29%!!!!! Gamma is linear with K, so it comes later in the equation !!!!!! a is""

a velocity/Gamma31 a_i =(1/(4* pi)*1./ norm_vprodsquare).*(( r_0_X.* r_1_X+r_0_Y .*r_1_Y+r_0_Z.*""

r_1_Z)./R1 -(r_0_X .* r_2_X+r_0_Y .*r_2_Y+r_0_Z.*r_2_Z)./R2);

33 %Check singularities[zeile ,colonne ]=find(R1 <epsilon);

35 a_i(zeile ,colonne)=0;[zeile ,colonne ]=find(R2 <epsilon);

37 a_i(zeile ,colonne)=0;[zeile ,colonne ]=find(norm_vprodsquare .^2< epsilon);

39 a_i(zeile ,colonne)=0;

41 %influence coefficientsa_i_X=a_i.* vprod_X;

43 a_i_Y=a_i.* vprod_Y;a_i_Z=a_i.* vprod_Z;

45%write a_i in a row , in the structure of K

47 for i=1:m(1)a_i_X_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=a_i_X(i,:);

49 a_i_Y_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=a_i_Y(i,:);a_i_Z_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=a_i_Z(i,:);

51 end

53 end

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Appendix

Algorithm 4 Function generate_vortices_panel.m of the Vortex LatticeMethod

1 function[K]= generate_Vortices_panel(A_X ,A_Y ,A_Z ,T_right_X ,T_right_Y ,""T_right_Z ,T_left_X ,T_left_Y ,T_left_Z ,T_right_behind_X ,T_right_behind_Y"",T_right_behind_Z ,T_left_behind_X ,T_left_behind_Y ,T_left_behind_Z ,n,m,""U,V,W)

epsilon =1*10^ -20;3 lenj =0;

%Rewrite the surface normals5 for i=1:m(1)

U_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=U(i,:);7 V_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=V(i,:);

W_vec (1,[(i-1)*sum(n)*2+1:1:(i)*sum(n)*2])=W(i,:);9 end

11 for i=1:2* sum(n)for j=1:m(1)

13 %velocity of the bound vortex[a_i_X ,a_i_Y ,a_i_Z] = vortex_panel(T_left_X ,""

T_left_Y ,T_left_Z ,T_right_X ,T_right_Y ,T_right_Z ,A_X(j,i),""A_Y(j,i), A_Z(j,i),m,n,epsilon);

15%Velocity of the left Vortex

17 [La_i_X ,La_i_Y ,La_i_Z] = vortex_panel_l(""T_left_behind_X ,T_left_behind_Y ,T_left_behind_Z ,T_left_X ,""T_left_Y ,T_left_Z ,A_X(j,i), A_Y(j,i), A_Z(j,i),m,n,epsilon"");

19 %Velocity of the right Vortex[Ra_i_X ,Ra_i_Y ,Ra_i_Z] = vortex_panel_r(T_right_X ,""

T_right_Y ,T_right_Z ,T_right_behind_X ,T_right_behind_Y ,""T_right_behind_Z ,A_X(j,i), A_Y(j,i), A_Z(j,i),m,n,epsilon)"";

21wx=a_i_X+La_i_X+Ra_i_X;

23 wy=a_i_Y+La_i_Y+Ra_i_Y;wz=a_i_Z+La_i_Z+Ra_i_Z;

25%Projection in the normal direction

27 K(j+lenj ,:)=wx.*U_vec+wy.*V_vec+wz.*W_vec;

29 endlenj=j+lenj;

31 endend

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Appendix

Simulations

Algorithm 5 Wing geometry of the unstable flying wing

%===================2 %== Wing Geometry

%===================4 betha_end =1;%betha_end /10 is the real angle!

6 p=5;%number of partitionsny=[0 0 0 0 90]*pi /180;%Dihedral of the Partitions

8 n=[6 5 4 3 3];%number of collocation points in y-Directionm=[4 4 4 4 4];%number of collocation points in x-Direction

10 s=[0.5 0.25 0.9 -0.75 0.1 0.15];%Spanwidth of partitionb_root =[0.24 0.215 0.18 0.155 0.14];

12 b_tip =[0.215 0.18 0.155 0.125 0.06];phi =[2 6 8 9 10]/360*2* pi; %sweep angle(back is positiv !)

14 alfa =[0 0 0 0 0 0 0 0 0 0]*pi /180; %Twist anglex_0_rel =[0 0 0

16 tan(phi (1))*s(1) s(1) sin(ny(1))*s(1)tan(phi (2))*s(2) s(2) sin(ny(2))*s(2)

18 tan(phi (3))*s(3) s(3) sin(ny(3))*s(3)tan(phi (4))*s(4) +0.5 s(4) sin(ny(4))*s(4)

20 tan(phi (5))*s(5) s(5) sin(ny(5))*s(5)]’;

22 x_0(:,1)=[0 0 0]’;for i=2: length(x_0_rel)

24 x_0(:,i)=x_0(:,i-1)+x_0_rel(:,i);end

26 cog = [0.0883 0 0]; % Correct centre of gravityc=0.012 ; %Thickness of the Wing for Inertia

28 flap_hinge =0.75;

Algorithm 6 controller.m, parameters used for simulating the unstable flyingwing

1 %Controller Datakp_crit =50;

3 T_crit =0.14;kp=0.6* kp_crit;

5 Ti=0.5* T_crit;Td =0.125* T_crit;

7 Num=kp*[Ti*Td Ti 1];Denom =[0 Ti 0]; P=kp;

9 TI=kp/Ti;TD=kp*Td;

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Appendix

Appendix B) Manual

Manual of the Vortex Lattice MethodStatical analysis

Copy the m-files (on CD-Rom) and the simulink model into a folder, in thecurrent directory of the m-files should be made a folder named ’polars’, whereXfoil polars are saved.

1. Calculate in Xfoil the profile polar for di"erent flap angles.User Guide for creating polars:

(a) open Xfoil(b) define the point, where the momentum is calculated(c) load mh45.dat or any other profile(d) oper(e) visc tipe Re(f) vpar (enter viscous parameter menu) N 9.0 (set new lower turbulence

level) <return> (back to oper)(g) optionally type iter for more iterations, so that the calculation con-

verges(h) write the polars:

i. set the initial Value of alfa bsp: alfa -5ii. pacc enter the file name as ex.: ’mh45.pol’iii. enteriv. aseqv. if it doesn’t work, increase the iterations (for example iter 400 )

(i) Flap deflection:

i. gdesii. flapiii. then set the valuesiv. exec

(j) if not converged, change the panelling! ppar and pane, then save theprofil and try again

2. Save all polars in *.dat files in the folder ’polars’.

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Appendix

3. In the function create_polar_table.m, define and include the polar:poldef =[15 7.5 0 -7.5 -15]*2*pi /360;

2[alpha_p (:,1,:),CL_p(:,1,:),CD_p(:,1,:),CM_p(:,1,:)]= import_polar(’""

mh45f15.dat’);4 [alpha_p (:,2,:),CL_p(:,2,:),CD_p(:,2,:),CM_p(:,2,:)]= import_polar(’""

mh45f7_5.dat’);[alpha_p (:,3,:),CL_p(:,3,:),CD_p(:,3,:),CM_p(:,3,:)]= import_polar(’""

mh45pol.dat’);6 [alpha_p (:,4,:),CL_p(:,4,:),CD_p(:,4,:),CM_p(:,4,:)]= import_polar(’""

mh45fm7_5.dat’);[alpha_p (:,5,:),CL_p(:,5,:),CD_p(:,5,:),CM_p(:,5,:)]= import_polar(’""

mh45fm15.dat’);

For example, five flap deflections are saved at the angles defined in poldef.

4. Define your wing. Attention, all partitions must have the same number ofpanels in chord direction! Otherwise, the mesh matrices can not be built.Take as example:

1 betha_end =40;

3 p=2; %number of Partitionsny=[0 90]*pi /180; %Neigungswinkel of the Partitions

5n=[4 2]; %anzahl Aufpunkte fr die ""

einzelne Partition / Flgelhlfte7 m=[5 5];

s=[1 0.6]; %Spannweite jeder ""Parition

9 b_root =[0.2 0.2]; %Breite , Symmetrische Eingabeb_tip =[0.2 0.2];

11 phi =[20 29]/360*2* pi; %Pfeilungswinkel(gegen hinten ""positiv !)

alpha =[0 0 0 0]*pi /180;13 x_0 =[0 0 0

tan(phi (1))*s(1) s(1) sin(ny(1))*s(1)]’;15

x_0_Trefftz =[0 0 0;tan(phi (1))*s(1) s(1) sin(ny(1))*s(1)+sin(alpha (1)"")*100] ’;

17

19 cog = [0.212690364338495 0 0]; % Correct center of ""gravity

21 c=0.012 ; %Thickness of the Wing for ""Inertia

23 flap_hinge =0.75;

If results with sideslip are analysed, check that the value of betha_incontains the desired sideslip angle. The range of sideslip angles are defined:[-betha_end/10 : 0.1 : betha_end/10]

5. Create an initial.m file and write the initial values of the flow. In thisfile, the flap vector is initialized. You must define for each partition if itis flapped or not, and if it is the left or right aileron. For the wing above,the flap vector would have the following form:

1 flap_angle_part =-[0 left_ail right_ail 0];

In this vector, the winglets do not have flaps.

6. Include your wingdef.m file in the code main_panel.m the name is free,however you could define it as wingdef_myfirstwing.m. The initial.m file

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Appendix

should not be unnamed, otherwise the name has to be changed in otherfunctions.

7. type main_panel.m in MATLAB workspace for running the application.

(a) Type CA, CW, CW_ind, CW_visc, CM, CL_distribution, CD_distribution,Lift_Matrix, xg_X (Centre of pressure), and many others

(b) Mesh plots and force distributions over the wing are shown in figures.Check in the mesh plot if the geometry is panelled correctly.

Numerical Problems It might occur, that some collocation points are badlypositioned. So change the geometry or change the panel number in chordor spanwise direction, then try again. The value of epsilon can also bechanged in the function generate_vortice_panel.m.

Dynamic analysis

1. First make a static analysis for initial values (see above).

2. Calculate with inertia_calc.m the inertia Ixx, Iyy, Izz or put the inertiasfrom CAD data into the file initial.m.

3. Write your correct centre of gravity in the initial.m file, take it from thestatic analysis or from experimental data.

4. Run the main_panel.m file for a range of betha_end=50 (defined inwingdef.m). Then, tables for sideslip angles from -5°... 5° are made in0.1° steps. This may take some time (one or two minutes).

5. In the function calculate_force_and_moment_flavio_panel.m write thecorrect flap vector.

6. Start the simulator by André Noth modified by Flavio Gohl for VLMcalculations. In the solver preferences of the simulator, set the step time.

7. Start

8. The values can be visualised with running show_uav_final.m. The sideslipangles are saved in a vector betha_in, it can be plotted versus the time ifdesired.

Errors which occur during simulation:

• The sideslip angle is too high, so that no interpolation is possible

– increase the range of betha_end in wingdef.m and try again, or mod-ify your geometry.

• The flap vector is not defined correctly in the file calculate_force_and_moment_flavio_panel.m

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Appendix

Appendix C) Design CriteriaThe most important design rules for flying wings, recommended by the author.

1. Choice of a profile type. For flying wings it is recommended to choose aprofile with a constant pressure point. This means that the momentumcoe!cient at the 1/4 line should be zero or slightly positiv. Good profileswith a high CLmax at low reynolds numbers are the MH45 and MH60.

2. For a pitch stable wing, at first it is important to have the wing’s centreof gravity in front of the aerodynamic neutral point.

3. With a simple flapped wing, it must be decided at which CL the airplanehas to fly in trimmed flight. It is recommended to choose a high CL valuewith a flap deflection, so that the wing without flap deflection is in fastflight. The reason is, that at slow speed the induced drag is very high(Cdind = const ' CL

2) and the viscous drag small (because the dynamicpressure is small), and therefore a flap deflection is not too expensive. Infast flight, the viscous drag must be extremely small, and consequently itis best to fly without flap deflection!

4. At the chosen CL it must be decided, what the shape of the circulationdistribution should look like. For minimal induced drag, it is elliptic. Thiscould cause a problem in combination with point 1, as the center of gravityis not necessarily in front of the neutral point. In this case it is an iterativeprocess to find a geometry which can fullfill both requirements of point 1and 3. For an e!cient design it is recomended, to have an aspect ratio of0.4 to 1.0. Only in this range both requirements can be fullfilled. [7]page107

5. For good stall behaviour, it is advantageous, to place the center of gravityin front of the the C-Point. The C-Point is the point of lift distributionwhich is proportional to the local chord. This setting is also possible ifthe aspect ratio is between 0.4 and 1.0 [7]page 105

6. Reynolds numbers: It is important to have a closed look at the localreynolds numbers, as they may decrease severely at the tip of the wing.However, even at te wing tip, the reynolds number must be su!cientlyhigh. In case of this problem it is better to twist the profile a little andgive more chord length for better reynolds numbers.

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Appendix

Appendix D) Estimation of Initial ConditionIn this part, a method to find the speed and angle theta in equilibrium at adesired angle of attack is described. The method presented here saves a lot ofsimulation time for finding the equilibrium point. The following points must becalculated, and written in the files initial.m and wingdef.m.

1. A desired angle of attack must be chosen (for example at best glide ratioor at best sink rate)

2. The correct centre of gravity at the desired angle of attack must be cal-culated.

3. With the CL value (which is estimated or calculated) at the desired angleof attack, the velocity can be calculated:

vnorm ="

2 · m · g

CL · $ · A(29)

vinitial =

#

$cos(%) · vnorm

0sin(%) · vnorm

%

& (30)

4. From the glide ratio the glide angle can be calculated:

%Glide = arctan'

CD

CL

((31)

5. The initial angle Theta can be calculated:

# = "(%Glide " %Attack) (32)

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