dart · 2002. 10. 24. · normal coefficient cn in continuum folw for a sferical blunted cone with...

STABILITY ANALYSIS BASED ON SIMULATION OF THE DELFT AEROSPACE RE-ENTRY TEST

DEMONSTRATOR

DART

11 April 2001

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Cover:

The REVolution configuration of DART

Stability Analysis based on simulation of the Delft Aerospace Re-entry Test demonstrator DART

Andrea Guidi

Printed at:

Division of Control and Simulation Faculty of Aerospace Engineering Delft Univeristy of Technology Kluyverweg 1, 2629 HS - Delft The Netherlands

I

STABILITY ANALYSIS BASED ON SIMULATION OF THE DELFT AEROSPACE RE-ENTRY TEST

DEMONSTRATOR

DART

Master Thesis by

Andrea Guidi Under the supervision of:

Prof. J.A. Mulder Dr. Q.P. Chu

Division of Control and Simulation

Faculty of Aerospace Engineering Delft Univeristy of Technology Kluyverweg 1, 2629 HS - Delft

The Netherlands

The DART Project is a Collaboration of:

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a Fara, Annamaria e Guglielmo…

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V

Preface

In spite of its name a Preface is often one of the last parts to be written. Furthermore in spite of the value that is often give to it is one of the most difficult parts to write down. In few lines the author should write something that represents his work and at the same time thank all the people and organization that contribute to develop the work.

I will leave the former task to the next abstract and herein I will thank the people that made this work feasible and that supported me in this last year.

First of all I wish to thank the whole Control & Simulation division and its friendly environment that was a determinant component of the success of the present work. Within the ‘Division’ two people were most involved in the assignment and they are my mentors Bob Mulder, chairman of the division, and Ping Chu my direct supervisor and leader of our small group within the division: the ‘Planet Entry Group’, which I also thank a lot.

Then the people that lead the DART project and that gave me the opportunity to take part in it also needs to be thanked and are Tom van Baten, project leader of the DART project, and Jeroen Buursink. Tom van Baten supported me a lot also in other questions during this year of Erasmus.

To the Erasmus exchange program also goes a special thanks of mine, wishing that it would improve more and more the diffusion of the European Culture.

I need then to thank also my home University “Federico II di Napoli” that formed me in my first years of study and especially two professors that were very close to me in this last year: Francesco Marulo and Fabrizio Nicolosi.

I need also to thank ESA and DASA for the windtunnel Database of the IRDT mission that was used for the validation of the CFD code and especially the person of Lionel Marraffa from ESA-ESTEC.

Finally, last but not least my friend Biagio that helped me in the layout and my sister Fara that has always watched my work… from the desktop of my laptop…

DART Stability Analysis

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Contents

Preface V

Contents VII

List of Symbols XI

List of Abreviation XII

List of Figure XIII

Abstract XV

C h a p t e r I

Introduction 1

I.1 Background 1

I.2 Dart Capsule Characteristics 2

I.3 State Space System Model 4

I.4 Re-entry Computational Fluid Dynamics toolbox 5

I.5 Stability 6

I.6 Outline of the Report 8

C h a p t e r I I

Re-entry Flight Dynamic of an Axisimmetric Body 11

II.1 Flight Symulator 11

II.2 GESARED 12

II.3 Reference Frames 14

II.4 Euler Angles 16

II.5 Total angle of attck 20

II.6 Equation of Motion 21


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C h a p t e r I I I

Re-entry Computational Fluid Dynamic 27

III.1 Hypersonic Rarefied Flow 27

III.2 Hypersonic Transitional Flow 29

III.3 Hypersonic Continuum Flow 29

III.4 Viscous influence in the Hypersonic Continuum Flow 33

III.5 High Temperature Effect 35

III.6 Integration of the coefficients with the panel method 36

III.7 Supersonic, Transonic and Subsonic Aerodynamics 39

III.8 Derivatives wit respect to the Pitching moment or Damping Coefficient 46

C h a p t e r I V

Validation 51

IV.1 GESARED Implementation validation 51

IV.2 CFD software validation 54

C h a p t e r V

Stability Analysis 59

V.1 Static and Dynamic Stability 59

V.2 Static Stability 60

V.3 Location Center of Gravity and Center of Pressure 64

V.4 Dynamic Stability and Angle of Attack convergence. 67

V.5 Coning Motion and Initial Re-entry condition 76

V.6 Impact dispersion due to Lift Nonaveraging, Lunar Motion 84 V.6.1 Lunar motion at Constant Roll Rate 84 V.6.2 Lunar motion at a Variable Roll Rate 85

V.7 Other phenomena affecting re-entry 89 V.7.1 Magnus Effect 89 V.7.2 Meteorological Characteristic 90 V.7.3 Heat Shield Ablation 90

C h a p t e r V I

Conclusion and Remarks 91

VI.1 Aerothermodynamics 91

VI.2 Stability 92

VI.3 Other investigations 95


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References 97

APPENDIXES 99

Appendix A: GEARED User Manual 101

Appendix B: RCDF implementation 105

Appendix C: R-CDF User Manual 107

Appendix D: Aerodynamic Database Plot of DART 109

REVolution configuration 109

VOLNA configutation 111

Appendix E: Aerodynamic Heating and Airload 113

Appendix F: 6 d.o.f. Simulations 115

REVolution Configuration with a spin rate of 30 deg./sec. 117

VOLNA Configuration with a spin rate of 30 deg./sec. 123

REVolution Configuration with a spin rate of 90 deg./sec. 129

VOLNA Configuration with a spin rate of 90 deg./sec. 136

Analytic Index 147


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List of Symbols

α angle of attack [deg] αTRIM trim angle of attack [deg] π pi (3.14159) [-] ρ air density [kg/m3] θ cone semi-apex angle [-] θ theta local deflection angle (Euler angle) [deg] φ roll orientation relative to wind (Euler angle) [deg] ψ precession angle (Euler angle) [deg] Ω exoatmospheric coning rate [deg/sec] ω natural pitch frequency [deg/sec] ρ∞ free stream air density [kg/m3] CA axial force coefficient [-] CAb axial force coefficient of the base pressure [-] CD drag coefficient [-] CDb drag coefficient of the base pressure [-] CL lift coefficient [-] CLb lift coefficient of the base pressure [-] CN normal force coefficient [-] CNα aerodyn. normal force derivative [-] CNb normal force coefficient of the base pressure [-] Cmq aerodyn. Pitch damping derivative [-] Cp pressure coefficient [-] Cp max pressure coefficient in stagnation point [-] Cp, b pressure coefficient of the base pressure [-] D aerodynamic reference diameter [m] D drag [N] F force [N] FA aerodynamic force [N] Fx force in x direction [N] Fy force in y direction [N] H reference altitude for exponential altitude [m] I moment of inertia Ix roll moment of inertia L lift [N] L reference length [m] m vehicle mass [kg]


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M moment [Nm] MY moment in y direction [Nm] M Mach number [-] M∞ free stram Mach number [-] Mx moment in point x [Nm] p roll rate [deg/sec] p0 initial roll rate [deg/sec] p pressure [Pa] p∞ free stream pressure [Pa] pstag stagnation point pressure [Pa] Pstagnation max maximum stagnation point pressure [Pa] R, r radius [m] RB base radius [m] RN nose radius [m] S aerodynamic reference area [m2] d nosa diameter [m] D base diameter [m]

List of Abreviation

c.g. center of gravity c.p. center of pressure d.o.f. degree of freedom AOA Angle of Attack CFD Computational Fluid Dynamics CRV Crew Re-entry Vehicle DART Delft Aerospace Re-entry Test demonstrator ESA European Space Agency ESA-ESTEC European Space Research and Technology GNC Guidance Navigation and Control NLR National Aerospace Laboratory (The Netherlands) TPS Termal Protection System


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List of Figure

Chapter I Figure I.1 REVolution configuration, RN=0.250 m, RB=0.670 m, L=1.360 m ϑ1=15o, ϑ2=30o 3 Figure I.2 VOLNA configuration, RN=0.392 m, RB=0.6 m, L=1.2 m ϑ1=0o, ϑ2=15o 3 Figure I.3 Main geometrical characteristics of the blunted biconical shape 6 Chapter II Figure II.1 Simulink model of GESARED 13 Figure II.2 Inertia Frame and Body-centered Frame 14 Figure II.3 Generalized Body axis in symmetric flight 16 Figure II.4 Euler Angle Rotations 17 Figure II.5 Angulare sequence 19 Figure II.6 Total angle of attack 20 Figure II.7: Forces Acting on a Re-entry Body 22 Figure II.8: The relation between the wind frame and the vertical frame 23 Figure II.9: Rotation Coordinate System 23 Figure II.10 Definition of state variables with respect to the earth-fixed reference frame. 26 Chapter III Figure III.1 Re-entry Flow Regimes 28 Figure III.2: Variation of foredrag coefficient with Mach number for various cones (R/rb=0.4) 40 Figure III.3: Variation of Normal Coefficient slope with Mach number for various cone (R/rb=0.4) 41 Figure III.4: Variation of Normal force coefficient slope half angle cone and bluntness at M=0.68 42 Figure III.5: Variation of Normal force coefficient slope half angle cone and bluntness at M=1.05 42 Figure III.6:Variation of centre of pressure with cone half angle and bluntness at M=0.68 42 Figure III.7:Variation of centre of pressure with cone half angle and bluntness at M=1.0 42 Figure III.8: Variation of centre of pressure with Mach number of various cones (R/rb=0.4) 43 Figure III.9: Diagrammatic representation of transonic flow configurations for cones 44 Figure III.10 Cp/D variation as function of alpha 45 Figure III.11 The pure q-motion 47 Figure III.12 Harmonic q-motion, α = constant, θ = γ 48 Figure III.13 Harmonic α motion, q=0 49 Figure III.14 Harmonic combined q- and α-motion 49 Chapter IV Figure IV.1 My against α 51 Figure IV.2 Fz against α 52 Figure IV.3 Mz against β 53 Figure IV.4 Fy against β 54 Figure IV.5 Comparison between CFD toolbox and windtunnel data of the axial coefficient CA and the normal coefficient CN in continuum folw for a sferical blunted cone with inclination angle of 45o 55 igure IV.6 Comparison between CFD toolbox and windtunnel data of the axial coefficient CA and the normal coefficient CN in free molecular folw for a sferical blunted cone with inclination angle of 45o 56


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Chapter V Figure V.1 Statically and Dynamically Stable (left) and Statically unstable and Dynamically Stable (right) 59 Figure V.2 REV-shape - Cm versus alpha 60 Figure V.3 Volna-shape – Cm versus alpha 61 Figure V.4: Total angle of attack (in degrees) 62 Figure V.5: Trim angle of attack (in degree) 63 Figure V.6 Angle of attack and angle of side slip for a 66 Figure V.7 total angle of attack and alpha trim for a REV configuration with a Xc.g. in 0.57 m from the base 66 Figure V.8Total angle of attack related to alpha and beta for the REVolution configuration at spin 30 deg/sec 67 Figure V.9 Total Angle of attack (deg.) convergence of the REVolution shape with a spin of 30 deg../sec. and 90 deg./sec 68 Figure V.10 Total Angle of attack (deg.) convergence of the Volna shape with a spin of 30 deg../sec. and 90 deg./sec 69 Figure V.11Comparison of Amplitude and Frequency of the damping oscillation 70 Figure V.12 Angle of attack convergence at spin rate of 150 deg/sec 71 Figure V.13 spin rate variation REVolution shape with 30 deg/sec 71 Figure V.14 Comparasion of the variation in spin rate of the REV and Volna shapes 72 Figure V.15 Total angle of attack and trim angle of attack 73 Figure V.16 Rolling trim response 73 Figure V.17 Spinning top gyro 74 Figure V.18 Euler Angle of the REVolution shape at 30 deg/sec spin 75 Figure V.19 Precession 77 Figure V.20 Side-moment coefficient due to coning motion; M=1.4 77 Figure V.21 Vortex displacements during planar and coning motions; θ =26° , M=1.4 78 Figure V.22 Precession rate for REVolution shape with spin 30 deg/sec 79 Figure V.23 Precession rate for Volna shape with spin 30 deg/sec 80 Figure V.24 Precession rate for REVolution shape with spin 150 deg/sec 80 Figure V.25 Precession mode 81 Figure V.26Projection of the path of the Nose.(REVolution shape betwwen sec 35-50) - Tridimensional projection of the nose 82 Figure V.27 Projection of the path of the Nose.(Volna shape between sec. 20-33) - Tridimensional projection of the nose 83 Figure V.28 Re-entry vehicle “Lunar” motion with lift and constant roll rate 85 Figure V.29 Re-entry vehicle Lunar motion with lift and roll rate variation 86 Figure V.30 Magnus Effect 89 Chapter VI Figure VI.1 Blunted Multi conic sonic lines 92 Figure VI.2 Propagation of the error ellipsoid 93 Figure VI.3 Center of gravity offset because of moving mass 94 Figure VI.4 Moving mass Controller 95


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Abstract

Delft Aerospace Re-entry Test demonstrator (DART) is a small ballistic re-entry vehicle. Re-entry is a young and very challenging field of study because it covers almost all the possible conditions of an aerodynamic flow. This work is a Stability Analysis of the DART. The stability characteristics of an aircraft have their roots in the aerodynamics of the vehicle.

During the preliminary design phase, a strong interaction between different disciplines is demanded for an effective design. The vehicle equations of motion are a unifying framework for studying many important aspects of a design. One important factor among the several fields is the aerodynamics of the vehicle body. The Flight Simulator needs to simulate the vehicle dynamics, and in order to do this, the aerodynamic coefficients for the vehicle are needed. In order to achieve this goal the implementation of a Computational Fluid Dynamic (CFD) toolbox is presented

Although the simpleness of an axisymmetric shape, the re-entry of such a vehicle is characterized by several complex phenomenologies that were analyzed with the aid of the flight simulator.

The results of the simulations were in agreement with theoretical prediction and shown a statically and dynamically stable behavior of the vehicle.

Validation of both the used software is presented as well as the all information about the executed simulations are supplied in order to help the reader to understand and eventually repeat the experience with the same or different shapes.


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C h a p t e r I Introduction I.1 Background

Delft Aerospace of Delft University of Technology is proposing to build a small reusable ballistic re-entry vehicle together with the following European partners: ESA-ESTEC (European Space Agency), Fokker Space, NLR (National Aerospace Laboratory, The Netherlands), ITAM ( Institute of Theoretical and Applied Mechanics - Siberian Branch of the Russian Academy of Sciences; Novosibirsk) , and SRC Makeyev (State Rocket Center Makeyev, Miass, Russia).

Re-entry is a young and very challenge field of study because it covers almost all the possible conditions of an aerodynamic flow. A re-entry flight starts from an altitude of around 150 km and a velocity higher than 7000 m/s (this is tipical of a body caming from a low Earth orbit). During the re-entry the vehicle is slowed down and goes from a hypersonic velocity characterized by a Mach number higher than 20-30, to a supersonic velocity in a lower part of the atmosphere. From the order of magnitude is already clear the severity of the condition supported from the body. If all the kinetic energy is transformed in heat it is easy to calculate that no structure can stand these environmental conditions.

Main goal of the mission are:

new concept of structure new material separation and reattachment of the boundary layer

This work is as a Stability Analysis of the Delft Aerospace Re-entry Test demonstrator (DART). An aircraft in flight is constantly subjected to forces that disturb it from its flight path. Rising columns of hot air, down drafts, gusty winds, etc., make the air bumpy and the aircraft is thrown off its course. How the aircraft reacts to such a disturbance from its flight attitude depends on its stability characteristics. The stability characteristics of an aircraft have their roots in the aerodynamics of the vehicle. Furthermore, the "classical theory" of stability is expressed in the language of the aerodynamicists. During the initial design phase, a strong interaction between different disciplines is demanded for an effective design. The vehicle equations of motion are a unifying framework for studying many important aspects of a design and for evaluating


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problems that arise during the analysis phase of the design. From these equations a ‘State Space Model’ can be built to simulate all the aspects of the design. The software implementation of the equations of motion is the ‘Flight Simulator’. Aerothermodynamics, flight dynamics, structural analysis, thermal protection system, and GNC (guidance, navigation and control) are all based upon the output of the ‘Flight Simulator’. In stability analysis a ‘Flight Simulator’ can be used. Recently, the Control and Simulation Division of Delft Aerospace developed such tools on demand of ESTEC for the design of the CRV (Crew Re-entry Vehicle) for the International Space Station.

In the severity of this flight, stability assumes a new important role that is connected to the prediction of the heat load distribution and to its minimization. Selection of the shape of a re-entry vehicle is usually driven by the need for high aerodynamic drag, low aerothermal heating, and sufficient aerodynamic stability. The flight of DART from the re-entry into atmosphere to the touch-down must be aerodynamically stable, without tumbling, within the wide range from Hypersonic to subsonic. Therefore, the shape was established on the premise that it should be a simple combination of spherical part and conical part, so that the aerodynamics could be figured out by simple investigation for such a wide flight velocity range. To minimize complexity, other critical choices in the design of DART were made. Passive flight control and a ballistic trajectory were chosen (in order to contain aerodynamic heating a semi-ballistic trajectory can also be considered). Spin-up of the vehicle is required in order to maintain its trajectory attitude and to eliminate deviations from the desired initial attitude and errors caused by misalignment of the center of gravity. DART requires sufficient aerodynamic stability in order to overcome the spin stiffness and minimize any angle of attack variation throughout all speed regimes, maintaining a controlled attitude until parachute deployment.

I.2 Dart Capsule Characteristics

At the moment two configurations are under investigation for DART. Thereinafter we will refer to them as REVolution configuration and VOLNA configuration.

In the geometry of the REVolution shape the forebody is a 15° half-angle sphere-cone with nose radius RN = 0.250 m and base radius RB = 0.4555 m. The afterbody, also called afterwards “flare”, is a 30° half-angle cone which terminates with an overall diameter of 1.34 m (Figure I.1) .

In the geometry of the VOLNA shape the forebody is a wide half sphere of radius RN = 0.392 m followed by a cylinder of 0.388 m length. The afterbody (or flare) is a 15° half-angle cone (Figure I.2).


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Figure I.1 REVolution configuration, RN=0.250 m, RB=0.670 m, L=1.360 m ϑ1=15o, ϑ2=30o

Figure I.2 VOLNA configuration, RN=0.392 m, RB=0.6 m, L=1.2 m ϑ1=0o, ϑ2=15o


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The flare is for stability as well as to invoke the aerodynamic phenomenon of separation and reattachment. This phenomenon is one of the main goals of the re-entry mission of DART. The reason is because the geometrical detail is found in aerodynamic control surfaces and the phenomenon is therefore of importance for future hypersonic aircraft and aircraft-like reusable launch vehicles.

The Structure is a hot-metal structure of PM-1000 superalloy, an Oxide-Dispersion Strengthened material. This alloy can withstand temperatures up to 1200 ° C and the aggressive atomic oxygen environment of the re-entry by means of a strong oxide layer protecting the underlying material [Buursink et al. ]. This concept of the structure and the test of this new material are the other main goals of the mission.

Last consideration on the characteristics of DART, that will be further discussed later in the study of the center of gravity location, is its Thermal Protection System (TPS). The nose of the vehicle will encounter the highest heat load. It is prevented from melting by a water cooling system. The entire nose is part of a water tank. The amount of water is estimated to be less then 8 kg for a 200 mm nose. This variation of mass during the flight represents nearly the 5% of the weight of the vehicle which are estimated to be 137 Kg for the REVolution and 127 Kg for the VOLNA. It is obvious that this variation will induce a backward motion of the c.g. location.

I.3 State Space System Model

In order to describe a space vehicle returning to the surface of a rotating Earth

through the atmosphere the following model assumption are made:

The vehicle's mass is constant and negligible with respect to the Earth.

The Earth is an oblate sphere, rotating with constant angular velocity and having a gravity field including +J4 harmonic unaffected by the vehicle's mass.

The forces acting on the vehicle are Gravity and Aerodynamics forces.

Earth winds when compared to the horizontal velocity of the vehicle can be ignored such that the vehicle's ground and air speed are identical.

The atmosphere is described by the US Standard Atmosphere 1962.

Once made the above hypothesis the motion of a rigid body can be divided into the translational motion of the c.g. and the rotational motion about the c.g. . The vectorial reppresentation of the loads acting on it are


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F m a

M I

= ⋅

= ⋅ &ω

From these equations the 6 degree of freedom equation of motion are extrapolated. These last give information about position, velocity, attitude and angular rates in three directions, therefore requiring 12 state variables.

In this report the Aerodynamic forces acting on the vehicle are modeled (the lefthand of the newtonian equation) through Computational Fluid Dynamic (CFD) toolboxes implemented in MATLAB. The equations of motion implementation is the Flight Simulator software GESARED implemented in the Control and simulation division. Use of it is made in order to study the stability of DART.

I.4 Re-entry Computational Fluid Dynamics toolbox

During the preliminary design, some needed parameters are not yet available (e.g. aerodynamic coefficients, moment of inertia). One crucial factor among the several fields of a preliminary design of an aircraft is the aerodynamics of the vehicle body. The estimation and measurement of the Aerodynamic of the vehicle give its rapresentation in the flight simulator. This simulation give an essential contribution to a complete understand of the dynamic behavior of the vehicle. The simulation, in the case of the Aerodynamic analysis, is performed with CFD. Computational Aerodynamics is a relatively young field of study. These kinds of calculations, are rather complex and require a large amount of computer power. However, another necessity during the preliminary phase is to have a relatively simple way of evaluation for different solutions of the design. This is necessary so that a quick and iterative process will lead to a final design that meets all the requirements and objectives. In order to achieve it a study in this area was carried out and tools for RCFD were implemented in GESARED. These tools, with the equations implemented in it and presented in the third Chapter of this report, calculate the aerodynamic coefficients (e.g. CL, CD, CM) from the main geometrical parameters of the vehicle. A blunted bi-conical shape is implemented and only 5 parameter are needed in order to model it: nose radius, base radius, total length of the vehicle, and inclination angle of the two cone parts. The aerodynamic regimes implemented in the RCFD toolboxes are: hypersonic rarefied (also known as free molecular flow), hypersonic transitional, and hypersonic continuum, all with the influence of viscosity. Thess tools easily allow the designer to tradeoff and analyze different shapes within the flight simulator GESARED.


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Figure I.3 Main geometrical characteristics of the blunted

biconical shape

I.5 Stability

Generally a body is stable if, perturbed in any of its degrees of freedom, it returns to its initial attitude, without corrective action by any other device. Static stability is the initial tendency of an aircraft, when disturbed, to return to its original position. Dynamic stability is the overall tendency of an aircraft to return to its original position, following a series of damped out oscillations. Stability may be positive, meaning the aircraft will develop forces or moments which tend to restore it to its original position; neutral, meaning the restoring forces are absent and the aircraft will neither converge from its disturbed position, nor diverge; negative, meaning it will develop forces or moments


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which tend to promote divergence. Negative stability is, in other words, the condition of instability.

Stability may be longitudinal, lateral, or directional, depending on whether the disturbance has affected the airframe within the pitching, rolling, or yawing plane.

The stability analysis of an aircraft can be carried out analytically by studying the aerodynamic properties of the aircraft in the limit of the small perturbation approximation. In this approximation the aerodynamic characteristic can be linearized (approach followed by Bryan in one of the first studies in stability in 1911).

A complete Non-linear analysis of the stability of an aircraft cannot be performed without the aid of the computational power of the computer. The role played by the computer in modern flight dynamics is crucial.

A flight simulator provides to the flight dynamicist the kinematics and dynamic characteristics related to position and attitude. Damping of the vehicle from external disturbances can be extrapolated and conclusions on the static and dynamic behavior of the vehicle can be drawn. The non-linear 6 d.o.f. re-entry flight simulator GESARED was used in order to study the stability and controllability of the test vehicle DART.

In the above statements the word “aircraft” was mentioned. “Aircraft” is intended to represent any kind of vehicle that is capable of “flight”. However, “aircraft” can be divided into two main categories: Lifting Body, and Non-Lifting Body. In the former category belongs the aeroplane, with a typical characteristic being the “symmetry” of the vehicle about a central plane. The latter category usually represents all types of capsule. Often, the main characteristic of a capsule is for it to be axisymmetrical. It is easy to understand that the stability of the two different categories will have quite different characteristics.

In the present work the stability of a axisymmetric body is analyzed. It can be immediately noted that, in an axisymmetric vehicle, there is no difference in the approach taken with longitudinal and directional stability. For the two, a “total angle of attack” will be introduced and the stability analysis will be carried out in an “In plane” flight with the introduction of the “Windward-Meridian” plane. The directional stability of an axisymmetric body does not have a direct influence on the flight path, but it has an indirect influence through the coupling with the pitching moment. The roll-rate behavior was one of the “big surprises” in the development of current high performance ballistic missiles. (Platus D.H. AIAA 82-4019) Above all, a careful analysis of the frequencies of pitch and roll should be carried out in order to avoid undesired resonance. Undesired resonance could be the cause of extremely high amplification of the angle of attack. Another important aspect of the roll moment is in the spinning of the vehicle. Often, for various purposes, a spin is required which should be of as constant a value as is possible.


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I.6 Outline of the Report

This report aims to execute a simulator study of the preliminary solutions of DART.

In Chapter II an overview of the re-entry flight dynamics is given, with a short description of the main topics of reference frames, equations of motion and Euler angles. In the same Chapter the mentioned software GEASERD (General Simulator for Atmospheric Re-Entry Dynamics), used to perform the stability analysis, and is adaptation to the study of an axisymmetric body is generally described.

Objective of Chapter II is to describe the aerodynamics of the DART and asses if the requirement of stability are met. The description must be constructed with sufficient breadth and detail to populate an aerodynamic database suitable for 6 degree of freedom trajectory simulations. Data from various sources are compiled. These include Impact Theory analysis to describe the rarefied flow in transitional regime that take place in the beginning of the simulated re-entry flight, as can be seen in Figure III.1, Modified Newtonian law for hypersonic continuum regime and exixting windtunnel data in supersonic, transonic and subsonic regimes.

A crucial point in modeling is its validation. Each model is subjected to simplifications and the correctness of each simplification must be validated. Furthermore when a set of equation is implemented in a software its implementation also must be validated too. Error in the implementation may occur. In Chapters II and III two softwares used in the accomplishment of this report are presented. In Chapter IV a validation of them is gaven. The flight simulator GESARED was already validated in its structure from the Control and Simulation division [Costa R. R. Et al.]. However, in this study a new vehicle was implemented and a validation of this implementation is needed. In the first section a device of validation is carried out through the output of the force acting on the vehicle. In the second section a validation of the CFD software is presented through comparison with other sources.

Although the simpleness of an axisymmetric shape, the re-entry of such a vehicle is characterized by several complex phenomenolog as Lunar motion, Conig motion, Roll-Pitch coupling, Magnus effect and so on. The main phenomenology are shortly overviewed in Chapter V. In the same Chapter a comment on each of the previously mentioned phenomenologies are parallely developed in the specific of DART.

Finally Chapter VI summarizes the ‘Conclusion and Remarks’ of the work, underlines the research results, points out conclusions based on the content of the report, and sudjests further potential researches.


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Additionally same Appendixes are added at the end. In Appendix A and C an user manual of the Flight Simulator GESARED and the CFD code is reported.

In Appendix B a functional description of the main file of the CFD is given.

In Appendix D are reported the complete table of the Aerodynamic coefficient of the 2 concept.

Appendix E give a description of how were obtained all the plot of the report and summarize some additional plot.

Finally Appendix F summarize a tools integrated in GESARED in order to model the heat load and the airload during the re-entry flight


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11

C h a p t e r I I Re-entry Flight Dynamic of an

Axisimmetric Body

II.1 Flight Symulator Afther the establishment of one or more design proposal, analysis of these

preliminary solutions of a design problem must give an interaction of the expected properties of the respective design proposal. Analysis of the concept design is based on simulation: “Imitating the behavior of a system by means of another system that resamble the first in certain aspect and therefore is model of the first system” [Reozenburg, N.F.M. ].

All branches of technical sciences provide mathematical models that the designer can use to predict certain properties of his design concept. Many of these models are based on systems of differential equations. Sometimes, these systems of equations can be simplified to equations fit for hand calculations. In this case the designer has a powerful tool to vary design properties in search for an optimal design. However, very often the mathematical models are so complicated that the solution is only possible with advanced numerical tools and powerful computer. This is the case of a complete non-linear stability analysis and the numerical tools is the flight simulator.

A way in which several aspects of an aerospace design can be related is summarized as shown in the following “Test Triangle”:

Wind Tunnel

Flight Simulator Flight Test

Purely Computational Purely Physical

Before any engineering system can be optimized, it goes through the above process several times. It can be called the “testing phase” of the design and can be


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globally defined as “simulation”. This global simulation uses both computational and physical tests to evaluate and refine all aspects of an aerospace system. A flight simulator is a fully computational “simulation” of the system, a flight test is a fully physical “simulation” of the system. In the middle the windtunnel is a partially physical simulation of the system that is then analyzed through computational simulation. Engineering test and simulation plays an extremely important role in aerospace engineering. In a larger sense DART itself can be seen as a flight test of a larger re-entry study. It is in fact reusable and it will allow Delft Aerospace to iterate the “test triangle” several times and acquire and refine knowledge in re-entry. The opportunity to use real data to improve Computational Simulations both in the field of Aerodynamics (e.g. Computational Fluid Dynamics or CFD) and Flight Dynamics (e.g. a Flight Simulator) is one of the mainsprings of a Flight test.

II.2 GESARED

In order to provide an environment to design control laws for re-entry vehicles, Control and Simulation division of Delft Aerospace developed a simulation tools of atmospheric re-entry. This simulation tools is called GESARED (GEneral Simulation for Atmospheric Re-Entry Dynamics) and was implemented in MATLAB/SIMULINK.

GESARED is a nonlinear Newtonian re-entry dynamic model of six degree of freedom.

The purpose of GESARED is to simulate the Earth’s atmospheric re-entry in six degree of freedom. It was earlier implemented for the Crew Re-entry Vehicle of the International Space Station. One of the task of this stability analysis was to implement the GESARED flight simulator tools for an axisymmetric re-entry vehicle. Several of the differences between a lifting body and a ballistic vehicle are pointed out through this report. Some of them also requir modification in the software of the simulator in order to adapt it to these different characteristics. The most important is mentioned and describe more in detail in the section II.5 and regard the concept of the “total angle of attack”.


____________________________________________________________________________________

13

STORE

Store Results

PLOT

Plot Results

INIT

Initialization

Position

F Grav

Position alt

Gravity and Planet shape

F Aerody n

M Aerody n

F Grav

Position

Vel v ector

Euler ang

Aerody n ang

p, q, r

GESARED

Position alt

Speed

Euler angles

Aerody n ang

p, q, r

M aerody n

F aerody n

Force and Moment computationwith RCFD on-line

Figure II.1 Simulink model of GESARED

Here a short overview of some of the original characteristic of GESARED will be gave (R.R. Costa, Q.P. Chu 2000).

The simulator is designed with modularized software scheme and open system structure, so as to have the generality and versatility for multiple purpose applications.

The ‘State Space System Modeling’ is the one described in section I.3

To reduce the limits and singularities in simulating the re-entry flight dynamics, the state vector for the dynamic equations to be implemented in GESARED is carefully determined. For transnational motion, the vehicle position is expressed in Spherical coordinates, while the velocity is expressed in Cartesian coordinates. For the angular motion, the dynamic equation is described with the Euler angle rates, while the attitude (kinetic equations) is expressed with the quaternions, which does not have any singularity.

The simulator is implemented in MATLAB/SIMULINK environment, mainly due to is powerful abilities and potentials in both the simulation and the computation.


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14

The choice of the atmospheric model is very important, which affects the aerodynamic computation directly. The re-entry flight phase is assumed to be within the altitude of 120 km. Two different model for the atmosphere are implemented the standard 1962 and the GRAM 99.

Because of the importance of the topics in the next two section will be illustrate the reference frame and the equation of motion used in GESARED.

II.3 Reference Frames Although all of this reference frames are well known from aviation and from space

application its useful to report them in the beginning of this report. A clear memory of each of them especially of the Body frame and the Wind frame will be absolutely necessary in order to understand complex phenomenology like lunar motion, coning motion, roll-pitch coupling described in the following chapters.

Figure II.2 Inertia Frame and Body-centered Frame


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15

All the reference frame are right handed and cartesian. The first two are Eart centered frame, the other four are body center frame. In order to give an immediate clarity a Tabular Shape is chosen:

Reference Frames Index

Origin X axes Z axes Y axes

Eart Centerd Inertial

I The center of the Eart

Points to the Greenwich meridian at time t0

Points North Completes the right-hended system

Eart Centered Rotational

R The center of the Earth

Points to the Greenwich meridian

Points North Completes the right-hended system

Vertical V Vehicle’s center of mass

Points North Points Down to Earth’s center

Completes the right-hended system

Body B Vehicle’s center of mass

Point forward in the plane of symmetry of the vehicle

Point Down in the plane of symmetry of the vehicle


Wind W Vehicle’s center of mass

Points in the direction of the airspeed

Points Down collinear with the lift force


Trajectory T Vehicle’s center of mass

Point in the direction of the airspeed

Points Down in the vertical plane


For completeness the three rotation matrix are gave:

−=

εεεε

cos)sin(0)sin()cos(0

001

XR

−=

εε

εε

cos0)sin(010

)sin(0)cos(

YR

−=

1000)cos()sin(0)sin()cos(

εεεε

ZR


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16

The ε meaning a rotation of an angle ε about the axis X, Y and Z respectively.

In Figure II.3, is shown the realtion between Euler angle, Aerodynamic angle and the flight path angle in a symmetric flight.

Figure II.3 Generalized Body axis in symmetric flight

II.4 Euler Angles

Because of the importance of this set of angle some more details are gave on them. The Euler angles θ, ψ, φ (Figure II.4), are defined by the right-handed rotation about the 3 axes of a right handed system of axes. The sense of rotation and the order in which the rotations are conduced about the 3 axes in turn are very important since the angles do not obey the commutative law.


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17

Figure II.4 Euler Angle Rotations

Therefore, there are 12 possible sequences. Sometimes each particular sequence is designated as an Euler angle scheme. The attitude of an aircraft is defined as the angular orientation of the body-frame with respect to the Earth axes. Attitude angles therefore are a particulare application of Euler angles.


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18

The question arises as to which sequence is the ‘best’. The answer is problem dependent, as might be the solution of coordinate system (e.g. spherical or Cartesian).

An Euler sequence convinient for relating a re-entry vehicle frame, i.e. a b-frame, to a wind frame, i.e. a v-frame, is the sequence C1C2C1 shown in Figure II.4 . In this sequence 1v-axis is along the velocity vector, which is not necessarly horizontal; the 2v-axis is horizontal and, of course, normal to the 1v-axis (and the velocity vector); the 3v-axis completed the right-handed triad. The b-frame is assumed to be aligned initially with the velocity frame. The body is first rotated about the 1v-axis through the angle θ3 = ψ; next, the body is rotated about the new 2’-axis through the angle θ = α. Thus, the new 1’’-axis along the vehicle’s axis of symmetry makes an angle α with the 1v-axis (which, of course, is in the direction of the positive velocity vector). Since the angle α measures the angular separation between the vehicle’s axis of symmetry (zero lift line) and the velocity vector, it is called the “total angle of attack”. The angle ψ is a measure of the angle between the vertical and the angle of attack plane. If ψ is zero, then the angle of attack plane is vertical. (We would not expect the aerodynamic loads under any circumstance to be function of the angle ψ). The final rotation is about the 1’’-axis (which is also the 1b-axis) through the roll angle θ = φ. This angle is a measure of the rotation of the vehicle relative to the angle of attack plane. If the vehicle is a body of revolution (i.e. if the 1b-axis is an axis of symmetry), the aerodynamic load will not depend upon φ.

The above illustration of Regan on the rotational sequence of Euler angle already

partially illustrate the meaning of a “total angle of attack” becouse of its crucial importance in the understanding of the whole phenomenology of an axialsymmetric body flight dynamic it will be further discussed in the next section.


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19

Figure II.5 Angulare sequence


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20

II.5 Total angle of attck

In an axial-symmetric body, operating at combined attitudes of pitch and yaw, the wind vector approaches the body at some equivalent wind plane at an angle of attack that is a combination of the two angles of pitch and yaw. The equivalent plane will be between the XZ and YZ planes of the body axis and will be called windward-meridian plane. This problem can be solved in two different ways: Geometrically or Analytically. In the first method some considerations on the geometry of an axial-symmetric body, in a flow with the wind vector inclined with the axis of symmetry, are made. The axis of symmetry of the body is inclined at an angle α in the XZ plane and angle β in the XY plane.

Figure II.6 Total angle of attack

Using trigonometry the following relations are derived. The equivalent angle of attack is equal to )cosarccos(cos βαε = and the plane of action is located

at an angle:

=

αβφ

sintanarctan' from the Y axes.

Disadvantage of this method is that in the analysis it is assumed that both of the angles are small because in the modeling of the equations the arc of the angle is assumed to be its chord. This solution can be considered accurate up to an angle of about 15 degrees. In the study of the stability of Dart this assumption can be made considering that it is intended to design a ballistic re-entry configuration. However, in order to achieve more accurate results, and considering that in the angle of attack convergence analysis will be useful start simulation for angle also bigger then 15 degree, this problem can be carried out analytically. This also considering that maybe in the high atmosphere some wide oscillations can be present. It is of importance during the analysis not to confuse the source of error of a too restrictive approximation with the


____________________________________________________________________________________

21

true oscillation of the vehicle. This means that with a series of transformations between different reference frames, that the resulting final reference frame in which there acts only one angle of attack is determined. The concept is that a rotation from a reference frame to an other can be obtained with several angular rotation and with a different order, applying it to a transformation from the body frame to a body frame two rotation matrix can be obtained. The first use the angle of attack and the angle of side slip

)()(),( βαβα RRT yWB ⋅=←

A second transformation use angle the “total angle of attack” and the windward-meridian plane rotation. The problem it is solved simply in an equality between matrix.

With this different method the value of the angle of attack is unvaried from the geometrical value. What is slightly different is the inclination of the windward-meridian plane.

( )2coscos1

sincossin'βα

βαφ⋅−

⋅= a

Finally, the force acting in this new plane is calculated and then projected in the X and Y directions.

II.6 Equation of Motion

The Equations of Motion implemented in the simulator can have different form depending on the choice of axis reference frame. In GESARED the Velocity is expressed in Cartesian coordinates using the Vertical frame (north, east and down). The position is expressed in spherical coordinates of latitude, longitude and distance towards the Earth’s Center. It is assumed through the report a basic knowledge of the subject. However, because of the importance of the argument and because the wide variety of litterature that treat the argument in different way, for completeness of the report a summar of the subject is gave.

The main governing equation in flight mechanics is Newton's Second Law:


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22

maF =

In order to start modelling the system, the external influence on the system should be described. The total external force acting on the vehicle is the result of the sum of two forces: the aerodynamic force, and the gravitational force.

gAE FFF +=

L

W

D

V

γ

Figure II.7: Forces Acting on a Re-entry Body

The aerodynamic force is generated by the air flow over the vehicle's surface,

depending on the velocity and angle of attack.

The motion of the vehicle can be divided into two different parts in order to simplify the model. The first is a 3 degree of freedom translational motion and the second is a 3 degree of freedom angular motion. Due to the 3 degree of freedom angular motion the mentioned forces can generate a moment. This generated moment entirely depends on the co-ordinate system that is chosen. This includes the force that generates the moment as well as the magnitude and direction of this moment and of the forces. The decision for an appropriate co-ordinate system is therefore a very important decision to be made.

The most convenient system to use is a system that uses wind axes co-ordinates. This is a rotating co-ordinate system with respect to Earth-fixed axes. The origin of the wind reference frame is located at the centre of mass of the body. The Xw axis is collinear with the wind-velocity vector. For a northern wind the Xw axis has a positive direction. For a wind in the local horizontal plane, the Zw axis is positive pointing downward. The Yw axis is the right-handed supplement of the other axes (Figure II.8).


____________________________________________________________________________________

23

Figure II.8: The relation between the wind frame and the vertical frame

In the calculus of the aerodynamic coefficient for a revolution body operating at

combined pitch and yaw angles, the wind vector approaches the vector of a body in a equivalent wind plane at an angle )coscos(cos βαε a= . This makes the problem a bi-dimensional one. Using this simplification, Figure II.8 can be simplified into Figure II.9. This figure gives a clear view of the coordinate system.

Figure II.9: Rotation Coordinate System


____________________________________________________________________________________

24

Using al this information, the following relationships can be derived:

( )scoordinate

rotating

fixed

VVdtdVa ×+=

= ω&

k=k’

( )γθω && −= 'k

and :

VjV

kjiV

iVV

ωωω '00

00'''

'

==×

=

The components of the external force are:

gAE W

WLD

F

+

−−

=γγ

sincos

Combining these two relations we obtain:

=

VmVm

FE &ω

It should be noted that

γθ

γ

cos

sin

VR

VR

=

=&

&

These auxiliary relationships will be useful in locating the vehicle with respect to the Earth axes.


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25

The following trajectory equations can be generated:

hs

D

L

e

VSCD

VSCL

VRVhR

WD

gV

gRV

WL

gV

αρρ

ρ

ρ

γθ

γ

γ

γγ

=

=

=

=

==

+

=−

−−=

2

2

2

21

21

cossin

sin

cos1

&

&&

&

&

The trajectory path can be determined when the parametres W/S, CL, CD, and the

initial conditions iiii hV θγ ,,, are provided .

The aerodynamic force can also produce a moment because the centre of pressure (the point where the aerodynamic force is applied) and the centre of gravity shall not be coincident. To calculate this moment, the vector between the centre of gravity and the centre of pressure should be know. This vector is expressed in the body axis system.

This moment is given by:

( )ABWacABacFB FTrFrM ×=×=

where BWT is the transformation matrix from the wind frame to the body frame.

From this equation we can obtain the other 3 equations of motion for the angular motion.

An important consideration on this equation is that the motion of a re-entry vehicle is highly non-linear.

To solve this equation (both the 6 Degree of freedom and the 3 Degree of freedom) a trajectory simulation is needed.


____________________________________________________________________________________

26

Figure II.10 Definition of state variables with respect to the earth-fixed reference frame.

27

C h a p t e r I I I Re-entry Computational Fluid

Dynamic

III.1 Hypersonic Rarefied Flow

Dart Re-entry start released from the Volna launcher at an altitude of 100 Km with a flight path angle of -5° and a designed spin. At this altitude the atmosphere is composed of widely spaced molecules. Surface impacts of these molecules will exert the first aerodynamic forces on the entry vehicle. Knudsen number, Kn is defined as the ratio of the quiescent gas's mean free path to the vehicle maximum diameter. It represents a normalization of the mean free path at each altitude. As long as the Kn > 10, the associated aerodynamic forces can be accurately computed by a free molecular method.

For the free molecular flow the kinematics model of interaction particle-surface postulated by Maxwell assumes there are no collisions between gas molecules in flow field. The surface is impacted by free stream particles, which are diffusely reflected after full thermal accommodation. Unlike hypersonic continuum aerodynamic, where forces exerted on the blunt body are primarily the integrated effect of surface pressures, free molecular flow aerodynamics contain a significant contribution from shear stress.

An inclination method derived from this theory (Regan, 1999) was implemented. The following equations give the values of the pressure and friction coefficients.

θθσ

θσ

sincos2

sin)2(2 2

Tf

Np

C

C

=

−=

where σN is the normal momentum accommodation and σT the tangential momentum accommodation.


____________________________________________________________________________________

28

i

riT

wi

riN PP

PP

τττσ

σ

−=

−−

=

From the pressure and friction coefficients once projected on the body frame the axial coefficient CA, the normal coefficient CN and the moment coefficient CM can be calculated.

( )[ ]

( ) ( )( )[ ] dsnVnCnCrScq

M

dsnVnCnCSq

F

m

iiiiiipi

m

iiiP iii

∑

∑

=∞

∞

=∞

∞

××+−×=

××+−=

1

1

ˆˆˆ

ˆˆ)(

τ

τ)

Force and moment have then to be integrated on the surface of the body in order to obtain the value of the total force and moment acting on the body. This integration for the Free Molecular flow will be made through a panel method explained in the following chapters.

Figure III.1 Re-entry Flow Regimes


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29

III.2 Hypersonic Transitional Flow

The two different flow regime must be linked so that they cover any specific flight condition.

A bridge function is needed. The choice of this bridge function is very important and it should also be pointed out that from its initial condition DART start its re-entry flight in this regime. The dominance of one flow on the other is dependent on the logarithmic of the Knudsen Number. According to Regan, 1999 Mrar can be computed using an error function correlation as show below:

( ) acKnberfaM rar ++⋅⋅= )log(

With:

a = 0.5

b = 0.30709257318569

c = 0.80628539465167

and obtained statistically, Kn is the Knudsen number.

Mrar will represent the weight of the rarefied flow, and 1-Mrar will be the weight of the continuum.

III.3 Hypersonic Continuum Flow For the continuum flow the modified Newtonian law θ2

max sinpp CC = was integrated for the biconical blunted cone shape. The modified Newtonian law is a so called surface inclination method. It is a non-linear extension of the result that was obtained for inviscid flow over body in supersonic and subsonic flow where with the

well known formula 1

22 −

=∞M

Cpϑ simply from the inclination of the surface was

possible to determine the pressure.


____________________________________________________________________________________

30

maxpC is the maximum value of the pressure coefficient. This equals exactly the Cp in the stagnation point behind a normal shock wave. The total pressure behind a normal shock wave, at the free-stream Mach number, can be calculated using the exact shock-wave theory (Anderson, 1989)

Yields: 2

21

2

22

2max2

121

)1(24)1(2

∞

∞−

∞

∞

∞

−

++−

−−

+=

MM

MM

MCp γγ

γγγγ

γγ

γγ

The Modified Newtonian Law is Mach number dependent.

The force coefficients for a sphere and for a cone need to be determined.

For simplicity, an expression for the forces in a body-fixed co-ordinate system is developed, for instance the axial and the normal force. The axial force and the normal force are given by the following formulas (Bertin, 1994)

( )( )( ) idSnpp

RUC

Sb

Aˆˆ

21

122

⋅−

= ∫∫ ∞

∞∞ πρ

( )( )( ) )ˆ(ˆ

21

122

jdSnppRU

CS

b

N −⋅−

= ∫∫ ∞

∞∞ πρ

Where AC is the axial force coefficient and NC is the normal force coefficient. i is the unit vector in the x direction, j is the unit vector in the y direction, n is the vector perpendicular to surface S, Rb is the base radius and ∞U is the freestream velocity in the x direction.

The integral has to be calculated for the three different parts of the blunt biconical shape. For the continuum flow a direct numerical integration was used. As a further explanation of this, two different method will be given.

( )

⋅+−⋅⋅

⋅= 1

421

422

max cossin81sin1cos

412 ϑαϑα

B

PAsphere R

CC

⋅+⋅⋅

⋅⋅+⋅⋅⋅⋅

⋅= 1

221

221

21112

max1 cossin

21sincostan

21costan

2ϑαϑαϑϑϑ hRh

RC

C NB

PAcone


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31

⋅+⋅⋅

⋅⋅+⋅⋅⋅

⋅= 2

222

222

21222

max2 cossin

21sincostan

21tan2 ϑαϑαϑϑ hRh

RCC bcone

B

PAcone

142

2max coscossin

412 ϑαα ⋅⋅⋅⋅⋅

⋅= N

B

PNsphere R

RCC

⋅⋅+⋅⋅⋅⋅⋅⋅⋅

⋅= 1

221112

max1 tan

21coscossincossin2

1ϑϑϑϑαα hRh

RCC N

B

PNcone

⋅⋅+⋅⋅⋅⋅⋅⋅⋅

⋅= 2

221222

max1 tan

21coscossincossin2

1ϑϑϑϑαα hRh

RCC N

B

PNcone

where the subscript 1and 2 indicate respectively the first and the second cone, RB is the base radius of the vehicle, h is the length of each truncated cone, ϑ the inclination angle of the cones and α the angle of attack.

Finally an approximation of the base pressure can be calculated using the Gaubeaud formula:

( )

−

+

−−

+

= ∞

∞∞

11

1211

22 28.24.1

2 γγγ

γγM

MMC

bp

Where Cpb is the pressure coefficient induced by the base pressure.

The base pressure coefficient can be calculated the Cnb and the Cabwith:

α

α

sin

cos

bb

bb

pA

pN

CC

CC

=

−=

The final value of CA and CN is obtained as the sum of all the above contributions.

For the calculation of CM the following method was used

Calculation of the moment around the sphere-cone apex:

The axial component of the coefficient of moment around the apex of the spherical part is given by:


____________________________________________________________________________________

32

−+⋅⋅⋅⋅⋅⋅

⋅⋅=

1sin152cossin)cos(

2sin

13

152

13

max2

1

3

ϑϑϑϑπ

απ

PtN

NMAsphere

CxR

RC

The normal component of the coefficient of moment around the apex of the spherical part is given by:

C R

R x CMNcone

N

N t P

= ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ − − ⋅ − −FHG

IKJ

π α

π ϑ ϑ ϑ ϑ ϑ

3

12 4

13

1

25

13

1

214

215

1

sin

( cos ) cos sin cos sinmax c h

Therefore the coefficient of moment of a sphere of radius RN and section xt is the following:

MNsphereMAsphereMsphere CCC +=

The formula of the truncated cone is reported for a generic cone and don’t repeated for the two part of the vehicle:

The two axial and normal moment around the apex point are:

C

RR

C

C

RR

RR

C

MAtcone

t

B

P

MNtcone

t

B

t

B

P

=

⋅ −FHGIKJ

FHG

IKJ

⋅

=

⋅

⋅ − +FHGIKJ

FHG

IKJ

⋅

3 1

2

3

1 32 2

2

1 1

3

31

1

3

sin cos

sin

cos

sin

sin

max

max

ϑ ϑ

α

ϑ

ϑ

α

It should be noted that repeating the calculation for the second part of a cone the apex of the two conical part will be in different position.

MNtconeMAtconeMtcone CCC +=

CC R x C R

R Lx C R

R LMMsphere N t Mtcone B

B

t Ncone B

B

=⋅ ⋅ ⋅ + ⋅

⋅− ⋅ ⋅

⋅( cos )ϑ 1

2 3

2

2

2

d i


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33

III.4 Viscous influence in the Hypersonic Continuum Flow

The viscosity plays a relevant role in the hypersonic regime. Viscous drag is due to the stress on the aerodynamic surface and in the boundary layer. The decreased momentum in the flowfield results in a corresponding loss of momentum of the aerodynamic system. Some of the physicals aspects involved in the viscous drag loss are the presence of shear layers, turbulent transition and boundary layer separation.

An other effect due to the presence of the boundary layer is that the flow sees the body shape modified by the presence of a tick hypersonic boundary layer. In hypersonic theory this phenomenology is often called viscous interaction. This phenomenon consists of the fact that the viscous boundary layer changes the nature of the outer flow, and in turn these inviscid flow changes feed back as change in the boundary layer. In hypersonic flow there are two kinds of interaction. One is due to the particularly thick boundary layer interaction and is known as “Pressure Interaction”. The other interaction is due to the impingement of the strong shock wave on the boundary layer, and is called Shock wave-Boundary Layer interaction. This last phenomenon was not analysed in the present study.

Numerical methods for CFD are mostly concerned with the solution of the system of partial differential equations of Navier-Stokes. But as already mentioned this will be a basic piece of work for further improvement of the various regimes, and the main goal is to implement only an as complete as possible toolbox of CFD methods in the Flight Simulator GESARED. Furthermore, the Navier Stokes integration requests a lot of computer power and on a state-of-art computer still a lot of run time. One of the goals here is an acceptable run time in a standard desk-top computer. Therefore an engineering approach was used.

The viscous effect generates two forces acting on the body. One is tangential to the surface of the body and comes from the explained phenomenon of the shear stresses in the boundary layer. The second is normal to the shape and is due to the presence of the boundary layer itself. The undisturbed flow sees a body modified by the presence of the boundary layer. The tangential force was implemented by calculating the friction coefficient with an engineering approximation called “Reference Temperature method”. This method is based on the assumption of a dependency of the value of Cf from a reference temperature somewhere inside the boundary layer.

Without going deep into this explanation of this method (Zoby, E.V., J 1981, and Anderson, 1990), the formulas implemented for the laminar boundary layer are:

21

'Re

664.0ω−

∞

=

TT

C

x

f


____________________________________________________________________________________

34

and for the turbulent boundary layer:

51

5 'Re

0592.0ω−

∞

⋅

=

TT

C

x

f

In both case the T’ is a function of Mach and T∞ in the undisturbed flow:

2

211'

∞∞

⋅−

+= MTT γ

Both these formulas are for a flat plate. To apply the above results to cones simply multiply the right hand side of the equation by the Mangler fraction, 3 . In the design of DART for the first shape under investigation (Figure I.1) the friction was determined only on the two conical part of the vehicle. This choice came from consideration on the shape of DART in which the spherical surface of the nose is small enough to be neglected. In the second shape (Figure I.2) the above formula for the flat plate was implemented in the toolbox of panels as made for the coefficient of the Free molecular flow.

As mentioned, in the study of the viscosity an important phenomenon is involved, the transition between laminar and turbulent flow. For the accurate prediction of skin friction the knowledge of the transitional Reynold number is critical. However no theory exists for accurate prediction of ReT . Any knowledge concerning it comes from experimental data (Reshotko, Eli 1976)

For the transition between the two different regimes, turbulent and laminar, also an engineering approximation is used. In this approximation the transition is related to the value of the Reynolds number (Bowcutt, K. G. 1986)

The value of the ReT is assumed to be function of the Mach number through:

)10209.1exp(421.6)log(Re 641.24eT M⋅×= −

e

TeeT

xVµ

ρ=Re

Once the value of Re for transition is known it is easy to determine the distance xt where transition occurs.


____________________________________________________________________________________

35

The main effect of the viscous interaction is the variation of the CP due to the presence of the Boundary Layer.

In that case also an engineering method related to experimental data is applied. Within the viscous interaction two other phenomena can be identified, and as often in Aerodynamics, they can be related to a similarity parameter: the Viscous Interaction Parameter χ (chi bar).

ee

wwMµρµρχ

Re

3∞=

χ is used to determine the transition between the two different kinds of interaction: “strong” and “weak”. A large value of χ corresponds to the strong interaction and vice versa. The value associated with transition is for practical purposes assumed to be 3. Also it is directly related to the value of the pressure through:

χ5.01+=∞P

P

in case of strong interaction and

χ078.01+=∞P

P

in case of weak interaction.

It should be noted that the discussed viscous interaction is in case of a laminar boundary layer. In case of a turbulent boundary layer the same adjustment would have to be made. However, most viscous interaction theory is based on laminar flow. This is because viscous interaction occurs at large Mach numbers and small Re numbers and this condition promote laminar boundary layer. Hence in this basic study, turbulent interaction will not be considered.

III.5 High Temperature Effect

Finally in order to complete an overview of all the problems connected with the hypersonic flow regime the high temperature effect is briefly introduced. In classical compressible aerodynamics the flow is considered to be a continuum, monophase,


____________________________________________________________________________________

36

chemically inert, at constant composition and without electric charge. Basically the flow is considered thermodynamically as a system of 3 degrees of freedom. If ∞M is high enough, viscous dissipation within the boundary layer causes high temperatures, which in turns causes chemical reactions within the boundary layer, and the above statement is no longer applicable.

In such a case the classical system of equations for compressible aerodynamics (Eulerian and Navier-Stokes) are not totally applicable; diffusion of chemical species and energy changes due to chemical reactions must be included.

Nevertheless, the application of the classical equations to relatively moderate hypersonic conditions yield useful results. Considering that DART is a small re-entry vehicle, and that its design constraints limit the maximum skin temperature at 1200°C its flight can be considered a moderate hypersonic flight.

Furthermore because this study starts basically from a stability analysis an other effect of the high temperature has to be underlined. Flight experience with the space shuttle has indicated a much higher pitching moment at hypersonic speeds then predicted. Maus et al. (Maus et al 1984) argue that this discrepancy is due to the effects of a chemically reacting shock layer. At first glance there is little difference between the calculation for a non-reacting shock layer with 4.1=γ , and the reacting shock layer assuming local chemical equilibrium. Hence pressure distributions are always somewhat insensitive to chemical reaction effects. However for a long moment arm this difference in pressure distribution can result in non-negligible variation in the pitching moment.

This is not the case for our design considering that the maximum length of DART is around 1.5 meter. Indeed this is an other proof that the high temperature reacting gas effects will not have high influence in the stability analysis of DART .

In order to have an idea of the error magnitude it should be noted that the modified Newtonian law itself gives an error estimated to be about 5% of the real value. Consequently the correction of the viscosity effect already generates an output with an error value of less then 5%. It would be useless in this basic study to introduce an effect in which the uncertainty due to the approximation of the method would possibly be bigger than the improvement itself.

III.6 Integration of the coefficients with the panel method

All the methods discussed above are local methods that then need to be integrated over the shape of the body. The numerical integration is feasible for a relative simple


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37

shape, but can be difficult in the study of complex structure. In order to maintain the toolbox as general as possible, two different methods were chosen for the free molecular and for the continuum flow integration. The continuum flow is more sensitive to small variations of the shape than the free molecular flow and that is also the part of the flight that has more influence on the stability. For this flow regime, a direct integration on the blunted double cone is used. The reason is that in case of calculations on different shapes this method uses less computer time.

For the Free Molecular integration a panels method is used. Basically the shape was discretized in the way of a classic Finite Element Method, and then the pressure coefficients are applied to all the panels, summed and divided into the different components.

The method is structured in the following way. A primary tool calculates all the necessary parameters from a minimum number of input parameters. This minimum input is 5 parameters; nose radius, base radius, inclination angle of the first cone, inclination angle of the second cone and the total length of the vehicle.

Once assigned the value of RB,RN,L,ϑ1 ,ϑ2 (Figure I.3) the other geometry parameter are calculated.

The distance along the x-axis (the axis of symmetry of the body) where the first conical part attaches to the spherical part (a cylinder can be considered a cone with angle of inclination zero, this statement also clarifies which angle is assumed to define the cone)

( )1sin1 ϑ−⋅= Natt Rx

Then the radius of the cone at the attachment is calculated as: 1cosϑ⋅= Natt RR

The length of the two conical part is found to be

( )( )12

11112 tantan

tantansin1cosϑϑ

ϑϑϑϑ−

⋅−⋅−−⋅−=

LRRh NB

21 hxLh att −−=

Finally the value of the base radius of the fist cone is calculated

111 costan ϑϑ ⋅+⋅= NBcone RhR

Using this data the structure is divided in a number of nodes that can be varied according to the accuracy wanted. The structure is divided in “n” layers in the longitudinal direction and then each resulting circle in “m” parts.


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38

Also, the x coordinate in the conical part has a fixed step increment that depends from the number n that is chosen. For the spherical part the x coordinate is calculated keeping a constant increment in the angle from the symmetry axis along the sphere in order to have the same size on the lateral sides of each panel.

( )( )( )ϑdiRix N ⋅−−⋅= 1cos1)(

where dϑ is a constant angular step .

The y and z coordinate are found through the calculation of the Radius of each circle at the different x coordinates and its projection on the two axes. From geometrical considerations the radii are:

( ) ϑtan)()( ⋅−+= attatt xixRir

for the spherical part, and

ϑtan))(()( 1 ⋅−−+= hxixRir attB

for the two conical part.

The resultant matrix of the node is:

N(i,j,1)=x(i); N(i,j,2)=y(i,j); N(i,j,3)=z(i,j);

The matrix of the panels were built with the following “For” cycles:

for i=2:1:n for j=1:1:m panel(i,j,1,:)=N(i,j,:); panel(i,j,2,:)=N(i+1,j,:); panel(i,j,3,:)=N(i+1,j+1,:); panel(i,j,4,:)=N(i,j+1,:); end end It should be noted that the first level is implemented separately because it is

composed of just 3 panels.

In the panel matrix the coordinates of the k-corner of each panel are

xp(k)=panel(i,j,k,1); yp(k)=panel(i,j,k,2); zp(k)=panel(i,j,k,3);


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39

Plotting all the panels for the two shapes under study for DART at the moment

results in Figure I.1and Figure I.2.

III.7 Supersonic, Transonic and Subsonic Aerodynamics In a supersonic flow for thin bodies at small angles of attack the inviscid

compressible flow can be described by a linear partial differential equation, leading to the supersonic expression for pressure coefficient on a surface with local deflection angle θ :

12

2 −=

∞MC p

θ

With the same procedure of the hypersonic regime it would be possible to calculate from the value of Cp the value of CA and CN and then Cd and Cl.

However in this case it is prefered to take the data directly from one of the many aerodynamic tables in the literature. (i.e. Nicholas O., 1964; Owens V.R. e.o. 1966)

It should be noted that the same consideration on the linearity of the normal force coefficient and pitching moment coefficient with angle of attack will be made.The linearity was maintained between approximately 4± degrees. No significant changes in CN slope were observed here as the bluntness increased from 0 to 0.5. (Figure III.4 & Figure III.5)

The same consideration which was made in the hypersonic regime on the stability can also be made in the other speed regimes. In fact the next sketch shows that the variation of the centre of pressure with the R/rb ratio is very small (Figure III.6 &

Figure III.7)


____________________________________________________________________________________

40

Figure III.2: Variation of foredrag coefficient with Mach

number for various cones (R/rb=0.4)


____________________________________________________________________________________

41

Figure III.3: Variation of Normal Coefficient slope with

Mach number for various cone (R/rb=0.4)


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42

Figure III.4: Variation of Normal force

coefficient slope half angle cone and bluntness at M=0.68

Figure III.5: Variation of Normal force

coefficient slope half angle cone and bluntness at M=1.05

Figure III.6:Variation of centre of pressure with cone half angle and bluntness at M=0.68

Figure III.7:Variation of centre of pressure with cone half angle and bluntness at

M=1.0


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43

Figure III.8: Variation of centre of pressure with Mach number of various cones (R/rb=0.4)


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44

Figure III.9: Diagrammatic representation of transonic

flow configurations for cones

Only one consideration has to be made on the position of the centre of pressure in transonic regime. Looking at the sketch of Figure III.8, a not negligible downstream and upstream movement of the centre of pressure was observed in the sonic Mach number region. After that, a subsequent stabilisation beginning at the transition of local flow from transonic to supersonic speeds can be seen. In order to understand this movement it is interesting to analyse the behaviour of the shock wave in the transonic region in Figure III.9.


_____________________________________________________________________________________

45

During the hypersonic flight the location of the center of pressure was assumed constant and equal to the value extrapolated from the tables form a value of Mach equal to 5. In the supersonic flight the location of the center of pressure is fully extrapolated from aerodynamic tables.

In this calculations as well as in the population of the Supersonic database the two shape was approximated with the two blunted cone envelope a nearly 20° half-angle cone with a ratio d/D = 0.35 (d = RN , D = RB ) for the REVolution and a nearly 10° half-angle cone with a ratio d/D = 0.326 for the VOLNA shape.

The table at our disposal don’t indicate directly the value of our shape, the one reported is for a value of d/D = 0.4. However, consulting the table in Figure III.6 and linearizing it in Excel in Figure III.10 it can be seen that the two line are very closed. The variation of the location of the family of bluntness d/D = 0.33, 10 degree half-angle and 0.d/D = 35, 20 degree half-angle, can be assumed coincident with the respective variation of the family of d/D = 0.4. Where in table of other coefficient the distance was not negligible the two equation generated in Excel were used for interpolate the values.

y = -0,0492x + 1,1904

y = -0,0552x + 1,3937

00,10,20,30,40,50,60,70,80,9

1

0 10 20 30

d/D=0.4

d/D=0

Lineare(d/D=0.4)

Lineare(d/D=0)

Figure III.10 Cp/D variation as function of alpha


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46

The two location are estimated to be

Xcp = 0.362 m from the base for the REVolution shape

And

Xcp = 0.7825 m from the base for the VOLNA shape

The extrapolated value of the VOLNA shape it seems too big, it is already ahead the c.g. location of 0.095 m. However, looking at Figure III.6 is immediately recognizable that the variation between the 15° and the 10° cones is quite big, the 15° cone will have a c.p. at 0.54 m. A more conservative averaged value of 0.661 m was used in the simulations.

The reason because was reported this calculations, was not especially for the interest in the calculations itself, but rather for focus the attention on this uncertainty of the VOLNA shape that will needs of further investigation because of its particular cylindrical-conical shape.

III.8 Derivatives wit respect to the Pitching moment or

Damping Coefficient

At hypersonic speeds, ballistic range tests do not discern significant changes in the values on the dynamic derivatives with Mach number. In addiction past expreience with 6-DOF simulations have also indicated that the long variation in the dynamic derivatives at the higher Mach number have little effect on the hypersonic flight dynamics of the vehicle. Often this coefficient, because of limited budget and the high priority to acquire static stability and drag performance, have been estimated rather then measured.The derivative with respect to the pitching moment is indicated by Cmq and afterwards will be called damping coefficient. In the simulation executed for this report an estimation of Cmq=-0.5 was assumed. However, in order to understand the meaning of this derivative short consideration on the angular motion are given.


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47

Defining q-motion the motion indicated in Figure III.11, the center of rotation lies on

the top axis through the aircraft’ center of gravity. The distance to the c.g. is qVR = . At

a positive pitching velocity the center of rotation lies on the negative ZS-axis, i.e. above the aircraft

Figure III.11 The pure q-motion

In all points of the aircraft, with exception of the c.g. and all the points on the Zs-axis, the geometric angle of attack is proportional to the angular velocity and to the distance in Xs-direction to the c.g. ,


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48

VqL

Lxx

Rxx ref

ref

gcgc ⋅−

=−

=∆ ....α

Then in principle the local airspeed varies with the angular velocity, because in a given point velocity varies proportional with the distance from the center of rotation. Only in points at distance R of the center of rotation, i.e. the c.g. , the velocity is not influenced by the pitch rate.

The c.g. is the only point of the aircraft where during the q-motion neither the magnitude of the velocity, nor the direction (AOA) vary.

It should be noted that already in “normal” flight condition this effect is scarcely relevant. In high atmosphere re-entry where the velocity is several order of magnitude bigger then the one of the angular velocity this effect is highly negligible. It can also be noted that the dumping effect is negligible in the high part of the atmosphere where is dominant the damping due to change in the density of the atmosphere.

If the angular velocity during a q-motion is not constant, but varies harmonically with time, the aircraft describes due to this q-motion a sinusoidal trajectory, as indicated in Figure III.12.

Figure III.12 Harmonic q-motion, α = constant, θ = γ

The most characteristic feature of this trajectory is that the angle of attack at the aircraft’s c.g. remains constant during the motion. The center of gravity moves along an


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49

undulating trajectory, such that, γαθ &&& +== q . Both the angle of pitch and of flight path angle will vary harmonically.

In Figure III.13 the case for harmonic plunge is presented, i.e. α varies harmonically, however, now q=0.

Figure III.13 Harmonic α motion, q=0

A different motion to be distinguished from the q-motion, results if the center of rotation lies in the aircraft’s c.g. . The angle of attack varies also in the center of gravity and αθ && == q and by consequence, 0=−= αθγ &&& . The resulting motion is along a stright line and is shown if Figure III.14.

Figure III.14 Harmonic combined q- and α-motion

This motion may be considered as a superposition of a q-motion as defined above

and a variable α-motion, such that the condition αθ && == q is satisfied.

The Cmq derivative is the most important damping coefficient and is as:

CVSL

Mq

CqLV

mqmq= ⋅ ∂

∂=

∂

∂

112

2ρ


_____________________________________________________________________________________

50

It is clear from this short overview on the dynamic of the angular motion, that is a quite complex task to determine the value of the Cmq .

51

C h a p t e r I V Validation

IV.1 GESARED Implementation validation

An interesting verification of the elementary dynamics of the system can be made through the value of the forces and moments acting on the system and the attitude of the system. The aerodynamic forces in flight change in magnitude, direction and location, however their behavior can be predicted from flight dynamics. It is well known that for a positive variation of the angle of attack α in the plane XY, a statically stable vehicle sustains a negative moment that tend to restore its αTRIM.

5 10 15 20 25 30

-30

-20

-10

0

10

20

30

t

My

.... ;

Alp

ha _

___

Figure IV.1 My against α


_____________________________________________________________________________________

52

The respect of this condition during the simulation can be verified in Figure IV.1 can be verified that this condition is respected during the simulation.

The value of the nornal force induced by a positive α should be also negative and this condition is also respected.

0 5 10 15 20 25 30 35

-20

-15

-10

-5

0

5

10

15

20

25

t

Fz ..

.. ; A

lpha

___

_

Figure IV.2 Fz against α

It should be here noted that the component of the considered forces are expressed in the body reference frame and, while in the aerodynamic frame the lift force is positive, it results negative in the body reference frame because of the difference in the orientation of the axis. Therefore the negative value of Fz was expected (Figure IV.2).

The same can be said about the angle of side slip β. In a body reference frame a positive β induce a positive moment, the z-axis points down and consequently a nose-left angle (that is defined positive) induce a positive moment. The value of the normal force as for the angle of attack is negative. The next picture shows that also in this case the elementary dynamics of the problem is respected in the simulation.


_____________________________________________________________________________________

53

Figure IV.3 Mz against β

The order of magnitude of the forces acting on the body have also been compared with the ones acting on similar vehicles and have shown high conformity.

This section not only aims of validation but also to a preliminary understanding of the dynamic of the re-entry flight of DART.

0 5 10 15 20 25 30 35-60

-40

-20

0

20

40

60

t

Mz

.... ;

Bet

a __

__


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54

0 5 10 15 20 25 30 35

-20

-15

-10

-5

0

5

10

15

20

t

Fy ..

.. ; B

eta

____

Figure IV.4 Fy against β

IV.2 CFD software validation For the validation of the toolbox the windtunnel data obtained from the ESA-

DASA, IRDT mission (for a 45o blunted cone) is compared with the output of the software for the same shape. Figure IV.5 and igure IV.6 shows this comparison.


_____________________________________________________________________________________

55

Figure IV.5 Comparison between CFD toolbox and windtunnel data of the IRDT mission

of the axial coefficient CA and the normal coefficient CN in continuum folw for a sferical blunted cone with inclination angle of 45o


_____________________________________________________________________________________

56

igure IV.6 Comparison between CFD toolbox and windtunnel data of the IRDT mission

of the axial coefficient CA and the normal coefficient CN in free molecular folw for a sferical blunted cone with inclination angle of 45o


_____________________________________________________________________________________

57

The errors in the axial coefficient and normal coefficient is increasing with the angle of attack. Particularly good results are obtained with the CA in the continuum and rarefied flow where the two curves match up to an angle of about 50° for the continuum and for all the values for the rarefied. In the case of the CN in continuum flow the error increases significantly beyond a value of alpha of about 20° and up to 35° in the rarefied.

This work is preliminary aimed to the stability analysis of DART therefore approximated methods were implemented. However, it can be a preliminary work for further study on application of CFD tools in flight dynamics opening new challenging fields of research and new concepts of Guidance Navigation and Control (GNC).

"...As so often the case in many field of engineering, simplicity is a most desirable virtue" (M.V.Cook)

Further validation of the simulation software shall be achieved with comparison to flight test data from the actual DART flight. In light of this, as already mentioned, DART is a flight test itself for Delft Aerospace in order to acquire knowledge on re-entry hypersonic aerodynamics and on validation of different codes developed in the faculty (e.g. the present work and the flight simulator GESARED).


_____________________________________________________________________________________

58


59

C h a p t e r V Stability Analysis

V.1 Static and Dynamic Stability

An aircraft which, following a disturbance, oscillates with increasing up and down movements until it eventually tumbles or enters a dangerous dive would be said to be unstable, or to have negative dynamic stability.

An aircraft that has positive dynamic stability does not automatically have positive static stability. Designers may have elected to build in, for example, negative static stability and positive dynamic stability in order to achieve their objective in maneuverability. In other words, negative and positive dynamic and static stability may be incorporated in any combination in any particular design of aircraft. Figure V.1 better emphasize this characteristic.

Figure V.1 Statically and Dynamically Stable (left) and

Statically unstable and Dynamically Stable (right)

An aircraft may be inherently stable, that is, stable due to features incorporated in the design, but may become unstable due to changes in the position of the center of gravity caused by consumption of fuel, improper disposition of the disposable load, and so fuort.


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60

In order to achieve a highly accurate prediction of the landing site, and a high “comfort” for the on-board instrumentation DART has to be statically and dynamically stable.

V.2 Static Stability

As mentioned in the introduction, “the stability characteristics of an aircraft has is roots in the aerodynamics of the vehicle”, this because the response of the aircraft at the forces acting on its surface during the flight is given by its “Aerodynamic Coefficients”. But more important for the stability analysis are the Derivatives of these Aerodynamic Coefficients.

Figure V.2 REV-shape - Cm versus alpha

The derivatives express in fact the behavior of the coefficients (and consequently of the vehicle) at variation of flight condition. The aerodynamic coefficients are function of several parameters but in simplified aerodynamic tables two main parameters are


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61

identified: Angle of Attack (AOA) and Mach number (ρ and V are intrinsically expressed through Mach). Variation of this two parameters in the flight condition cause variation in the value of the aerodynamic coefficients. “An aircraft is statically stable in a given steady flight condition, if a change in angle of attack or in angle of side slip, causes a change of the aerodynamic moment about the center of gravity, i.e. a change in Cm or a change in Cn respectively. Which tend to rotate the aircraft back to its original attitude relative to the airflow.”

Since the condition of static stability is ∂∂

<Cm

α0 , the above mentioned statement is

quite obvious for the static stability. In Figure V.2and Figure V.3 it can be clearly noticed that both the two shapes of DART respect this condition and are therefore statically stable.

Figure V.3 Volna-shape – Cm versus alpha

However, static stability is strictly connected to the location of the center of gravity. The above coefficients are all referred to a fix point, in this case coincident with the


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62

center of gravity of the vehicle, therefore they are not the only sources of moment on the vehicle. A component deriving from the aerodynamic normal forces should be added and this component depends on the location of the center of pressure through:

( )ααα d

dCxxd

dC

ddC N

pcgcmm gc

...... −+= .

The position of the center of pressure can be a crucial point in the static stability of the vehicle. When the location of the center of pressure falls between the center of gravity and the nose of the vehicle static instability may occur during the flight.

A statically stable vehicle always assumes ‘Trim’ condition during the flight. Trim is the condition for which the summation of all the moments acting on the vehicle is identically zero. It is characterized by an angle of attack called ‘αTRIM’.

In the next section further considerations on the Figure V.4 and Figure V.5 are given. From the is only underlined that both the configuration reach an ‘αTRIM’ after a certain dumping time .

Figure V.4: Total angle of attack (in degrees)

0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

RE

V

0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

Vol

na

Altitude


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63

Figure V.5: Trim angle of attack (in degree)

It should be noted that the trim angle of the Volna shape is much smaller then the one of the REVolution shape, but it is reached after a longer 'damping' phase.

Another consideration can be made on the variability of the trim angle. The REVolution shape does not experience a constant trim angle. As expected the trim angle can change at different altitude. On the other side once reached 13 km value of the trim of the Volna shape becomes very closed to zero and remains almost constant.

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 104

0

0.2

0.4

RE

V

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 104

0

0.1

0.2

0.3

0.4

Vol

na

Altitude


_____________________________________________________________________________________

64

V.3 Location Center of Gravity and Center of Pressure

As first notation through this paragraph and through the whole report the location of center of gravity and center of mass were assumed coincident. This assumption is not restrictive for a re-entry vehicle. As mentioned in the former section for a good stability of a blunted body the center of gravity (c.g.) location has to be closes enough to the nose. At hypersonic continuum condition, blunt shapes exhibits acceptable stability even for a c.g. position behind the maximum diameter location of the vehicle. Conversely at supersonic speed a c.g. not enough close to the nose can result in a dynamic instability that induces oscillatory motions. At transonic speed this motion is further amplified and in subsonic regime the strength of the dynamic instability can overcome the static stability of the shape and causing the vehicle’s oscillations to diverge into tumbling motion. Therefore the c.g. requirement for a re-entry vehicle whose entry profile includes subsonic flight is driven by the subsonic dynamic stability of the vehicle. Designers sometimes avoid this further study in the subsonic flight regime by deploying parachutes at supersonic speeds. Further investigation will have to be made for DART in order to trade off the better solution in its design.

In the paragraph of the characteristics of DART an influence of the TPS on the center of gravity location was mentioned. The variation of mass induced by the TPS can be mostly neglected for the hypothesis of constant mass made in the modeling of the system but it cannot be neglected for what concerns the stability characteristics related to the center of gravity location. A variation of mass in the nose part of the vehicle result in a backward motion of the c.g. with destabilizing effect on the vehicle. In turn with the decreasing of the Mach number, especially in the supersonic flight, the vehicle experiences a forward movement of the center of pressure. The center of pressure is identified for our purposes as that longitudinal station at which the components of the aerodynamic moment vector normal to the vehicular plane of symmetry is zero. The center of pressure would move longitudinally with changes in the angle of attack. This movement of the center of pressure with angle of attack means that either the vehicle could operate at only one trimmed angle of attack or the center of gravity would have to be moved to change the trim angle of attack. The importance of the simultaneous movement of this two point is clear for what said in the previous section. The moment due to the aerodynamic forces is proportional to the arm (Xc.g. – Xc.p.) and will be a stabilizing moment if the c.g. is located between the c.p. and the nose, destabilizing in the opposite case, for this last characteristic is also often defined as the 'static margin'. A backward motion of the c.g., and a forward motion of the c.p., can give arise destabilizing condition.

An estimation of this phenomenology during the flight of DART is done.

Starting from the data reported in Buursink at al., the variation is of about 8 kg and the vehicle weight is known. Assuming the weight of the water in the center of the tank


_____________________________________________________________________________________

65

located at a distance RN from the nose, the center of gravity location of the complete body is gaven by the formula:

REV

waterREVystemrestofthescgwaterwatercgcg M

MMXMXX

)(** −+= −−

The location of the c.g. of the complete body for the REVolution configuration is known to be at 0.603 m from the base, the resulting c.g. of the resto of the sistem, without water, is evaluated at 0.57156 m from the base with a consequent shift of 0.0314 m from the initial calculation.

The location of the c.g. of the complete body for the Volna configuration is known to be at 0.689 m from the base, the resulting c.g. of the resto of the sistem, without water, is evaluated at 0.6549 m from the base with a consequent shift of 0.0341 m from the initial calculation.

An estimation of the c.p. location during the flight was found in litterature [NASA TN D-3088] . The two locations are estimated to be

Xcp = 0.362 m from the base for the REVolution shape

And

Xcp = 0.661 m from the base for the Volna shape.

Finally the two arm can be calculated:

REVolution ; Xcp – Xcg = 0.362 – 0.57156 = -0.21 m

Volna ; Xcp – Xcg = 0.661 – 0.6549 = 0.0061 m

The equation of the Cm derivatives is reported here again with the inverted sign considering that the distance was expressed from the base and not from the nose.

( )ααα d

dCxxd

dC

ddC N

gcpcmm gc

...... −+=

It is clear that the REVolution shape does not suffer from static instability due to consumption of water. The Volna shape can be subjected to such instability in fact the final position of the c.g. and c.p. originate a destabilizing moment.A Simulation was executed with a shifted c.g in the location xc.g. = 0.57 for the REVolution configuration. In it is depicted a high frequency oscillation in alpha and in Figure V.7 total angle of attack and alpha trim for a REV configuration with a Xc.g. in 0.57 m from the base a very different αTrim from the previous of Figure V.5.


_____________________________________________________________________________________

66

Figure V.6 Angle of attack and angle of side slip for a REV configuration with a Xc.g. in 0.57m

Figure V.7 total angle of attack and alpha trim for a REV configuration with a Xc.g. in 0.57 m from the base

80 100 120 140 160 180 200 220 240 260

-5

0

5

alph

a

80 100 120 140 160 180 200 220 240 260

-5

0

5

beta

time

0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

Tau

0.5 1 1.5 2 2.5 3 3.5

x 104

0.2

0.4

0.6

0.8

1

1.2

Tau

trim

- d

eg.

altitude - m


_____________________________________________________________________________________

67

V.4 Dynamic Stability and Angle of Attack convergence.

The dynamic stability evaluation of an axisymmetric re-entry vehicle is especially concerned on the behavior of its angle of attack during the flight through the atmosphere. The variation in the angle of attack is essential for prediction of the trajectory of the vehicle and for heating requirement of the structure of the vehicle. The concept of the total angle of attack was already explained in Chapter I. In the following lines the meaning of the total angle of attack related to the output of the flight simulator is practically shown (Figure V.8).

Figure V.8Total angle of attack related to alpha and beta for the REVolution configuration at spin 30 deg/sec

GESARED is a complete 6 degree of freedom (d.o.f.) simulation tool, its output in the attitude is in the three minensional field. Figure V.8 shows the angle of attack and the angle of side slip with the resulting total angle of attack.

0 50 100 150 200 250

-100

1020

alph

a

0 50 100 150 200 250

-10

0

10

beta

50 100 150 200 2500

10

20

Tau


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68

Following the behavior of the wind-ward meridian plane, its rotation and its rotation rate, is carried out from the output of GESARED. Its rotation rate is an important parameter because can generate asymmetry in the wear down of the material of the vehicle and consequence of general asymmetry in the dynamic motion of the vehicle.

In order to have a dynamically stable vehicle its angle of attack has to converge. A lot of study in literature can be found on the angle of attack convergence behavior of axisymmetric vehicles. Most of them with spin condition. In the present section only the behavior of spin re-entry vehicle is treated. The importance of the spin is analyzed in the next section.

Figure V.9 Total Angle of attack (deg.) convergence of the REVolution shape with a spin of 30 deg../sec. and 90

deg./sec

For what concerns the study of the dynamic stability during a re-entry flight the aid of a flight simulator is strongly requested. It is in fact practically impossible to execute analytically a complete non-linear analysis of the attitude of a re-entry flight.

0 50 100 150 200 250 3000

10

20

30

RE

V s

pin

30

0 50 100 150 200 2500

10

20

30

RE

V s

pin

90

time


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69

Figure V.10 Total Angle of attack (deg.) convergence of the Volna shape with a spin of 30 deg../sec. and 90 deg./sec

The behavior of both the shapes was tested in the flight simulator GESARED. Simulations were performed with a 25° AOA initial condition in order to simulate the response of the vehicle to a disturbance that may occur during the flight causing a variation in attitude from its αTrim. Figure V.9 and Figure V.10 depict that both the configurations reach their αTrim after a damping. The REV shape is characterized by a longer dumping time and a smaller amplitude in the oscillations. However it also experience a higher frequency of oscillation (Figure V.11).

0 50 100 150 200 2500

10

20

30

spin

30

deg/

sec

0 20 40 60 80 100 120 140 160 1800

10

20

30

spin

90

deg/

sec

time


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Figure V.11Comparison of Amplitude and Frequency of the damping oscillation

0 5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

RE

V

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

Vol

na

time

70 75 80 85 90 95 100

012345

RE

V

70 75 80 85 90 95 100

2

4

6

8

Vol

na

time


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71

Figure V.12 Angle of attack convergence at spin rate of

150 deg/sec

A simulation at high spin rate was executed in the REVolution shape and, as expected, a very high spin rate induces a smaller oscillation due to the stiffness of the spin (Figure V.12). However, the convergence is lightly delayed.

Most of the study treats the case of constant roll rate. However, because of the susceptibility of re-entry vehicles to roll rate excursions it is necessary to investigate the influence of roll acceleration on the angle of attack convergence and on the windward meridian rotation rate.

Figure V.13 spin rate variation REVolution shape with 30 deg/sec

0 50 100 150 200 250 3000

5

10

15

20

25

30

time

Tau (deg)

0 50 100 150 200 250 3000

10

20

30

time

spin

rate

5.5 6 6.5 7 7.5 8 8.5 9 9.5

x 104

29.2

29.4

29.6

29.8

30

altitude

spin

rate


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72

From the can be seen that the spin rate is not constant for all the flight (as expected) but it can be considered constant in the beginning of the re-entry until an altitude of about 70 km.

Figure V.14 Comparasion of the variation in spin rate of the REV and Volna shapes

During atmospheric re-entry a spinning body with a trim angle of attack configuration, such as the once inducible by aerodynamic configuration asymmetries, encounters a trim amplification which is strongly dependent on the magnitude of the ratio of the spin frequency to the body natural or critical frequency. In the vicinity of equal spin and critical frequencies, the body angle of attack can become very large and result in a significant perturbation in the subsequent trajectory history. Although flight simulators are generally used to predict angle of attack divergence in the region of transient amplification, a simplified analytical description of the phenomenology is interesting.

Several studies have shown that spin rate changes arise primarily from coupling between lateral mass asymmetries and steady trim forces. This principle is illustrated in Figure V.15.

0 20 40 60 80 100 120 140 160 180 200

10

15

20

25

30

time

spin

rate

, (R

EV

- ,

Voln

a :

)


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Figure V.15 Total angle of attack and trim angle of attack

The resultant angle of attack is described by two time varying damped vectors, R1 and R2, rotating at different rates, ω0 - ∆ω and ω0 + ∆ω, respectively, about the trim component αTRIM. For positive spin rate p, R2 always rotates faster than R1 in a counterclockwise direction, whereas R1 changes its direction of rotation depending on the relative magnitude of ω0 and ∆ω. The αTRIM orientation and magnitude vary with

proximity to resonance

−=II

p x1ω ( where ω α= −FHG

IKJ

C qSdI

m ). With a c.g. displaced

from the geometric (or aerodynamic) centerline, a roll torque is generated and its magnitude and direction depend on the orientation and magnitude of the total angle of attack. The variation of αTRIM magnitude and orientation with proximity to resonance p/ω is depicted in Figure V.16 for a constant environment ω.

Figure V.16 Rolling trim response


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74

With increasing in p/ω, αTRIM rotates counterclockwise, to a ψ = - 90° at resonance and grows to a peak magnitude αT limited only by the aerodynamic damping factor. A further increase in p/ω above resonance rotates and decreases αTRIM to 0. It should be noted that large αTRIM is maintained about resonance. With changing in the environment, only the magnitude of the amplification αT/ αST is affected by the changing in the damping factor, the orientation is only slightly affected.

Thus, if significant undamping occurs due to unsteady aerodynamic and ablation effects, the trim exhibits a growth similar to the usual dynamic motion instability. (Price D.A. Jr, Ericsson E.L. 1971)

The motion of a spinning re-entry body, is analogous to the motion of a spinning top or gyroscope under the action of gravity in which the static moment ω2Isinθ, which tends to decrease θ for a statically stable missile, is equivalent to the top turning moment -Wlsinθ, which tend to increase θ (Figure V.17). Such a motion is described in literature in terms of the classical Euler angle coordinates. A great contribution of D.H. Platus in this field was the application and development of the classical Euler coordinate system to describe modern ballistic re-entry vehicle motion. Certain aspects of re-entry vehicle motion are conveniently described in the terms of Euler angles. One of the examples is the roll-rate behavior that was one of the “big surprises” in the development of the current high performance ballistic missiles. Some vehicles spun into resonance (in which the roll rate is approximately equal to the vehicle natural frequency) with catastrophic effects of large angle of attack amplification.

Figure V.17 Spinning top gyro


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75

One Euler angle is for practical purposes, the precession angle of the lift vector in space. The coupled behavior of this Euler angle with the total angle of attack Euler angle, analogous to the nutation angle of a spinning top or gyro determines cross-range dispersion associated with lift non-averaging. The vehicle motion during re-entry, described in terms of time-dependent Euler angle [ψ(t),φ(t),θ(t)], is assumed to be of the form

[ψ(t),φ(t),θ(t)] = [ψqs(t),φqs(t),θqs(t)] + [ψ+(t),φ+(t),θ+(t)]

where [ψqs(t),φqs(t),θqs(t)] represents a quasi-steady component that varies relatively slowly with time (of the order of the dynamic pressure) and [ψ+(t),φ+(t),θ+(t)] represents an oscillation of higher frequency about the average ( quasi-steady ) values.

Figure V.18 Euler Angle of the REVolution shape at 30 deg/sec spin

0 50 100 150 200 250

-80-60-40-20

020

thet

a

0 50 100 150 2006080

100120140

psi

0 50 100 150 200 2500

2000

4000

phi

time


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76

Neglecting the oscillating component Platus carried out an analytic analysis of the angle of attack convergence and solution for the quasi-steady behavior was obtained. From this analysis it comes out that for values of the angle of attack sufficiently greater then trim the precession rate ψ& can oscillate either in a positive (Figure V.19 c) or negative (Figure V.19 a) mode, and it depend on the initial re-entry condition. Once ψ& is determined, the windward-meridian rotation φ& , for small θ, is simply the difference

between the roll rate p and ψ& . The roll acceleration term pp& is a dumping term that adds

to or subtract from the dumping coefficient, depending on the direction of the rolling acceleration. Negative value of p& (deceleration) produce a momentary angle of attack divergence.

The analytical expression found from Platus for the angle of attack convergence ratio reduce to the simple result :

ρ

ωθθ bx e

IpI −⋅

=2

0

0

, where

+−

= 2

2 2sin8 md

ICC

IHSdb N

mqα

γ

The result indicate that for roll rates that are not excessively supercritical (i.e. (1-2ωI/pIx) << 1 ), the angle of attack convergence depends only on the re-entry rate and is independent on the roll acceleration.

V.5 Coning Motion and Initial Re-entry condition

Before re-entry, the vehicle is in a state of moment-free motion, and the relation between the various angular rates is uniquely determined. In general the vehicle has an arbitrary motion around the center of gravity compounded by the sum of 4 simple motions. Three of these are well known and are an initial roll rate p0, pitch rate q0 , yaw rate r0 . The fourth motion the “coning motion” is a steady precession with a precession rate Ω and a cone half-angle α(Figure V.19 a). This last motion should be further analysed. From experimental investigation (Tobak M. , 1969) vortexes induced by this motion take asymmetrical disposition (Figure V.21) with respect to the angle of attack plane during coning motion and become great source of side moment (Figure V.20).


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Figure V.19 Precession

Figure V.20 Side-moment coefficient due to coning motion; M=1.4


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78

Figure V.21 Vortex displacements during planar and coning motions; θ =26° , M=1.4

In Figure V.21 are shown photographs of the vortex cores at successive station proceeding rearward from the nose; the left column for φ& = 0 , the right column for


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79

φ&=600 rpm . The plane of θ is indicated on each photo by a dashed line. It is clear that at φ& = 0 the vortices are of equal intensity and symmetrically displaced with respect to the θ plane, whereas at φ& = 600 rpm the vortices become asymmetrically displaced with respect to the θ plane and may be of unequal intensity. A measure of the side moment coefficient due to coning motion is shown in Figure V.20.

The coning axis is, in general, inclined with respect to the flight path, and the velocity vector may lie inside the coning circle (Figure V.19 b) or outside the coning circle (Figure V.19 a). Limiting cases of exoatmospheric motion, where the vehicle is at an angle of attack with no coning or is coning symmetrically about the flight path, are shown in Figure V.19 c and Figure V.19 d respectively. The exoatmospheric roll rate, precessin rate, and cone half-angle are related by the expression:

αcos0

IpIx=Ω .

Figure V.22 Precession rate for REVolution shape with spin 30 deg/sec

0 50 100 150 200 250 3000

2

4

6

8

omeg

a (d

eg/s

ec)

0 10 20 30 40 50 60 70 80

6

6.5

7

time

omeg

a (d

eg/s

ec)


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Figure V.23 Precession rate for Volna shape with spin 30 deg/sec

Figure V.24 Precession rate for REVolution shape with spin 150 deg/sec

0 50 100 150 200 2502

3

4

5

6

7

omeg

a (d

eg/s

ec)

0 10 20 30 40 50 60 70 80

5.45.65.8

66.26.4

time

omeg

a (d

eg/s

ec)

0 50 100 150 200 250 3000

10

20

30

40

omeg

a (d

eg/s

ec)

10 20 30 40 50 60 70 8032

33

34

35

time

omeg

a (d

eg/s

ec)


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81

In the above Figure V.22, Figure V.23 and Figure V.24 the precession behavior of the two shapes and different spin rates can be observed. It is interesting to notice that the Volna Shape experiences a precession smaller than the REVolution shape and with a smaller value of the variation of the rate. Another interesting note is that increasing the value of the spin rate, increases also the variation of the spin acceleration. This will be interesting in the study of the Lunar motion that is strongly dependent from that rate.

After entering the atmosphere, the statically stable vehicle is subjected to an aerodynamic pitch moment that tends to align the vehicle with the flight path. However, because of the angular momentum comprised of the roll and the precession rates, which, on the average, is directed along the coning axis, the pitch moment is resisted by gyroscopic forces that induce a precession of the angular momentum vector about the average flight path. This precession is opposite in direction to the angular momentum (retrograde) for the case of Figure V.19 a and c. . A projection of the path described by the nose of the vehicle on a plane perpendicular to the flight path would be as shown in Figure V.25 a and b when the path is viewed along the direction of the flight.

Figure V.25 Precession mode

The residual coning motion, in the same direction as the angular momentum vector (clockwise), has been called nutation (Figure V.25 c) by Nicolaides (Nicolaides, J.D. 1953), and the retrograde precession in the opposite direction has been simply called precession (Figure V.25 a). In general the two motion exists simultaneously. The Euler angles were defined with respect to an inertial frame of reference that moves along the flight path (Figure II.3), so that ψ& represents the precession about the velocity vector. Therefore, the two exoatmospheric conditions represented in Figure V.19 c and d correspond to initial conditions on ψ& of ψ& = 0 and ψ& = Ω , respectively.


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Figure V.26Projection of the path of the Nose.(REVolution shape

betwwen sec 35-50) - Tridimensional projection of the nose

-14 -12 -10 -8 -6 -4 -2 0 2 488

90

92

94

96

98

100

102

104

106

108

psi

thet

a

050

100150

200

-100

-50

0

5070

80

90

100

110

120


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Figure V.27 Projection of the path of the Nose.(Volna shape between sec. 20-33) - Tridimensional projection of the nose

-40 -30 -20 -10 0 10 20 3070

75

80

85

90

95

100

105

110

115

120

050

100150

200250

-100

-50

0

5040

60

80

100

120

140


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V.6 Impact dispersion due to Lift Nonaveraging, Lunar

Motion

Assuming that eventual asimmetries relative to the vehicle are fixed in time, the Lift force derived in the trim configuration, and necessary to balance the torque caused by asymmetry, is also fixed. Consequentely there is an unbalanced force acting on the vehicle in a constant direction and generating an acceleration away from the zero-lift flight path. In order to average out this lifting effects most of the statically stable missiles and re-entry vehicles are provided with a nominal roll rate.

The response frequency of the vehicle to angle of attack is in general much greater then the roll rate, at list for altitude where the variation in roll rate are determinant for impact prediction. As consequence the resulting angle of attack does not converge and the vehicle assumes, in contrast, an initial, or re-entry, angle of attack. When the initial angle of attack is damped, and the vehicle assumes the αTRIM , the vehicle respons in pitch and yaw will prevent the roll rate from rotating the vehicle meridian plane contains αTRIM , out of the wind (with respect to the body reference frame). It means that the windward meridian plane is always the meridian that contains αTRIM , just as it would be if the vehicle were not rolling. However, the vehicle is rolling and the consequence is that the vehicle-fixed lift force is rotating about the mean flight path at the rate of the roll, and the resulting vehicle acceleration is similarly distributed. The result is a constant lift at constant roll that generates a spiral trajectory about the mean flight path without significant deviation from the zero lift trajectory. However, if the roll varies, the lift is not distributed for an equal amount of time in all directions about the zero-lift flight path and deviation occurs. The previously described motion of a re-entry vehicle rolling with a vehicle-fixed windward meridian is called “Lunar motion” (Crenshaw J.P. 1971).

Two main kinds of Lunar motion are possible: with constant roll rate and with variable roll rate. Of majour importnace on the re-entry vehicle impact is the second. The first is a very particular case but has been reported in order to better understand the more general case of a roll rate variation.

V.6.1 Lunar motion at Constant Roll Rate When the αTRIM is small, after that the oscillatory portion of the angle of attack is

dampened to the trim value, the lift and roll can be assumed both constant, and the roll is much less than the response frequency of the vehicle in pitch and yaw but not close to zero. The resulting motion is a rotation of the c.g. about the mean flight path (as shown in Figure V.28) and translation down the main flightpath.


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85

Figure V.28 Re-entry vehicle “Lunar” motion with lift and constant roll rate

The direction of the lift is toward the mean flightpath and is balanced by the centrifugal force

L C qA Wg

P rL= =FHGIKJα

2

where P is the roll rate, and r is the distance from the vehicle c.g. to the mean flightpath. The increasing in drag, caused by lift is a negligible fraction of the zero-lift drag for the supposed case with a small αTRIM . Since L is constant, the resulting acceleration is averaged about the mean flight path.

V.6.2 Lunar motion at a Variable Roll Rate

Define the mean flightpath as place of the instantaneus center of curvature of the motion produced by lift as the vehicle proceeds through the atmosphere. Suppose that although L remain constant the roll variates. Since the lift is no longer avaraged in time, there will be a lateral motion of the center of curvature and a motion about the center of curvature as shown in Figure V.29.


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Figure V.29 Re-entry vehicle Lunar motion with lift and roll rate variation

In Figure V.29 the angles δx and δy represent the deflections of the mean flight path resulting from lift. The xb-yb plane contains the lift vector and the motion about the center fo curvature. The velocity V is the velocity of the istantaneus center of curvature as the vehicle proceeds. The magnitude of the tangential velocity is several order bigger then the one of the other components, following that the lift serves only to change the diraction of the velocity.

For stright or nearly stright flightpaths, the differential distance at ground level normal to the intended flight path resulting from a differential deflection of the flightpath is approximately equal to the arch length:

∆ R t

td VdtIM= zδ

Where tIM the impact time (Crenshaw J.P. 1971).


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Distribution of the Lift around the body for REV at 30 spin

-100 -50 0 50 100 150-150

-100

-50

0

50

100

150

Liftx

Lifty


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88

Figure dimensional distribution of the Lift forces for REV at 30 spin

2025

3035

40

-100

0

100

200-150

-100

-50

0

50

100

150

t

Lunar Motion

Fx

Fz


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89

V.7 Other phenomena affecting re-entry

Many phenomena affect the motion of a re-entry vehicle. The main phenomenology of an axisymmetric re-entry vehicle were analyzed through the present chapter especially with respect to the characteristic of DART, some other less predictable or less determinant are shortly mentioned in this last section.

V.7.1 Magnus Effect Magnus dynamic effect is important while determining targeting accuracy of

missiles, and re-entry vehicles. A spinning object creates a sort of vortex of rotating air around itself. On the side where the motion of the vortex is in the same direction of the windstream to which the object is exposed, the velocity is enhanced. On the opposite side, the velocity decreases. According to Bernoulli’s principle, the pressure is lower on the side where the velocity is greater, and consequently there is an unbalanced force at right angles to the wind. This is the Magnus force.

Figure V.30 Magnus Effect


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90

The more recent studies agree that the Magnus force results from the asymmetric distortion of the boundary layer displacement thickness caused by the combined spinning and flow past the body. In the case of a sphere or a cylinder, the so called “whirlpool”, or more accurately the “circulation”, does not consist of air set into rotation by friction with a spinning object. Actually a spherical or cylindrical object can impart a spinning motion only to a very thin layer next to the surface. The motion imparted to this layer affects the manner in which the flow separates from the surface in the rear. Boundary layer separation is delayed on the side of the spinning object that is moving in the main direction as the free stream flow, while the separation occurs prematurely on the side moving against the free stream flow. The wake then shifts toward the side moving against the free stream flow. As a result, the flow downtream the object is deflected, and the resulting change in momentum flux causes a force in the opposite direction (downwards in the case in Figure V.30).

This phenomenon is influenced by the conditions in the thin layer next to the body, known as the boundary layer, and there certain anomalies may arise in the force if the spin of the body introduces anomalies in the layer, such as making the flow turbulent on one side and not the other.

V.7.2 Meteorological Characteristic For accurate prediction of the location of a long-range ballistic missile impact, the

effect of the atmosphere’s meteorological characteristics (especially density and wind profiles) near the impact point must be considered. Aiming instructions that are given at the firing point are frequently based on a standard density profile and zero wind, with corrections then applied to account for the existing winds and for deviation in density from the standard profile. Such correction are obtained by use of 3 or 6 DOF trajectory simulations. However, the use of a closed-form solution to the equation of motion, if sufficiently simple while still accurate, is more economical, is faster, is more useful in situations where the computing capability is limited, and provides helpful insight to the user. The report of Louis S. Glover in the References presents such a solution, including derivations of the working equations.

V.7.3 Heat Shield Ablation An other factor is heat-shield ablation, which depends on the state of the boundary

layer. During the period when a transitional boundary layer exists, the effects of ablation on the motion are substantially more complex. In this altitude regime, vehicles generally exhibit a momentary angle of attack divergence. Such alterations in motion may be detrimental to the system performance of ballistic and maneuvering re-entry vehicles. This alpha divergence is usually accompanied by large oscillations in the effective pitching moment coefficient slope during the transition progression over the surface. (Martellucci A.)

91

C h a p t e r V I Conclusion and Remarks

The primary objective of this work is to build a background work for further study in stability analysis of DART in order to achieve the best final design. In order to do that the report was layouted supplying all the information about simulations (Appendix F). This information can be necessary especially in case a future reader want to validate or repeat some of the experiences, or want to perform new simulation with different shapes. The main idea related to a simulation report is that the experience is nothing but a numerical experiment. Through the report same theory background was furnished in order to better understand and analyze the numerical results. Here same conclusion and remarks on further study are gave.

VI.1 Aerothermodynamics

Successful study in the stability of DART require, as much as possible an accurate description of the vehicle aerodynamics. The description must span the hypersonic rarefied, hypersonic continuum, supersonic, transonic and subsonic flow regimes. Strong interaction between the flight dynamicists and the aerodynamicists is demanded in further investigation on the Stability of DART.

The Newtonian Flow assumption is reasonably accurate as long as the sonic line remains on the vehicle spherical nose. This is true for an angle of attack of 5 degree or less down to Mach number 10. It means that a disturbance that induces a deviation of more then 5 degrees at less than Mach 10 is not certainly well described by the Newtonian law. Moreover, in the flare-shape of the DART, especially in the case of the REVolution concept, it should be investigate the possibility that the for high alpha the sonic line hit the flare with consequence modification in the aerodynamics of the vehicle and in turn in its stability.

Further investigation must be carried out for the CMq and CLp, through the report was assumed a guess value of 0.05 for the former and 0.005 for the last. In that phase the damping coefficients are not a critical issue. The damp due to the variation in density is several orders of magnitude higher, however during the flight in low atmosphere it is not anymore a negligible parameter.


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Figure VI.1 Blunted Multi conic sonic lines

For what concerns the supersonic-subsonic especially the VOLNA configuration needs further investigation because the particular shape which does not present the same characteristics of the most common tables where the aerodynamic database was founds. It was already underlined in section VII.7 .

VI.2 Stability

Both the configurations show good static and dynamic stability. However the “static margin” in both the configuration and particularly in the VOLNA configuration, is very small. During flow regime from the supersonic to the subsonic, instability can be due to forward movement of the center of pressure. Furthermore there are uncertainties in the location of the center of gravity due to the presence of a TPS that originates a variation in the mass in the vehicle.

Further investigation is certainly needed in the center of gravity and center of pressure location. A possible solution that needs further investigation is the deployment of the parachutes supersonic speeds.

For what concerns dynamic stability as already pointed out in the Aerothermodynamic section further investigation is certainly needed on damping coefficient. Especially in the case of the roll, the damping coefficient it result to be the major source of damp to the roll. Because the great importance of that movement of the body, particularly its rate as underlined in the report, the importance of this parameter become clear.


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93

The VOLNA shape presents a more slender shape with respect to the REVolution. Slender shapes suffer more in lateral disturbances. As already underilined in the previous section its needs of more investigations.

Moreover simulation with better atmospheric model can be carried out. In all the simulations the Atmospheric Model 1962 was used because of the high time consuming of other models (e.g. GRAM '99) and the need to perform several simulations.

Further investigation is certainly needed in the “trajectory error analysis”. Uncertainties in the initial condition of the trajectory, along with uncertainties in the physical properties of the atmosphere through which the Re-entry vehicle traverses, contributes to uncertainties in the impact point. Several studies are available on the subject in order to handle that uncertainty of great importance for the prediction of the landing point. Another important phenomenology that needs to be further analyzed in this context is the Lunar Motion that as seen in section V.6 can strongly influence the trajectory error of a spin re-entry vehicle.

Figure VI.2 Propagation of the error ellipsoid

Other source of uncertainty is the inertia of the vehicle especially in the component of the product of inertia. They are quite hard to predict with very high accuracy. In the simulations they were assumed all zero, but some of the simulation were performed with


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94

the original asymmetric Inertia matrix furnished by the CAD software in which was implemented the complete structure of the shapes from Ir. Buursink. The asymmetric Inertia of the vehicle induces variations in the roll of the vehicle. Follows that the value of the products of inertia should be carefully predict, and the behavior of the roll rate at product of inertia variation needs to be further investigated.

Possible solution for handle with all this uncertainties is a slightly controlled vehicle with the implementation of a “moving mass” subsystem(Figure VI.3). Such a system can be a very small controller that with small effort in analysis and costs can give higher performance to the vehicle in the solution of several problems (e.g. the control of the trim angle at different altitudes, the control of eventual asymmetry of the system due to imperfection in the construction or in the wear-down of the materials of the skin).

Figure VI.3 Center of gravity offset because of moving mass

Moreover a moving mass controller as depicted in Figure VI.4 can also improve the desired spin rate quality. In fact, it can in a certain range also control the value of the velocity of rotation of the vehicle. A level of passive roll control is also achievable by predicting the eventual wear-down of the skin.


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Figure VI.4 Moving mass Controller

VI.3 Other investigations

Another important investigation is the eventual slight ablation of the nose as well. It was seen in the report that very small asymmetry in the nose part can be cause of dangerous asymmetry in the motion.

Investigation on Natural Frequency and Vibration of the structure are also strongly recommended. This field influences a lot in fact the interaction between different characteristic motions of the re-entry, e.g. Roll-Pitch coupling.


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96

Analysis of the possible effects due to wind and atmospheric variations are another important field of investigation. In the state space model, depicted in section I.3 the effect of the wind in the environment was assumed negligible.

97

References Andreson, J.D. Jr , Hypersonic and high temperature gas dynamics, AIAA 2000 Bertin, J.J. , Hypersonic aerothermodynamics AIAA, 1994 Buursink J., Van Baten T.J. et al. “The Delft Aerospace Re-entry Test demonstrator”

IAF, 2000 Bowcutt, K. G., Anderson J.D. , and Capriotti D. : “Viscous Optimized Hypersonic

Waveriders”, AIAA Paper no. 87-1257, 1986 Cook, M. V. “Flight Dynamic Principles” Arnold, London, 1997 Costa, R.R. , et al , “Atmospheric Re-entry Modeling and Simulation: Application to a

Lifting Body Re-Entry Vehicle”, AIAA Paper 2000-4086 De Leon Pablo et al. Research in Atmospheric Re-entry Vehicles Using High Attitude

Balloons IAF-00-V.2.08 , 2000 Ericson L., “Effect of boundary layer transition on vehicle dynamics”, Journal of

spacecraft, vol. 6 pp. 1404

Glover Luis S., “Approximate Equations for Impact Dispersion Resulting from Winds and Deviations in Density”, Journal of Spacecraft and Roket, VOL. 9 NO 7, pp. 483484

Levy L. L. an Tobak M. , “Nonlinear Aerodynamic s of Bodies of Revolution in Free Flight”, AIAA Journal , VOL. 8 NO 12 , pp 2168, 2171, New York 1970

Maus, J. R., et al. , Hypersonic Mach number and Real gas Effect on Space Shuttle

Orbiter Aerodynamics , Journal of Spacecraft and Rockets, vol. 21, n0.2, March-April 1984, pp 136-141

Nicolaids, J.D. , “On the Free Flight Motion of Missiles Having Slightly Configurational

Asymmetry” , Rept. BRL-858, 1953 Ballistic Research Laboratory. Platus D.H., “Angle-of-Attack Convergence and Windward-Meridian Rotation Rate of

Rolling Re-entry Vehicles”, AIAA Journal , Vol. 7 NO. 12 pp 2324-2330, New York 1969 Platus D.H. “Dispersion of Spinning Missiles due to Lift Nonaveraging”, AIAA

Journal , VOL. 15 NO 7, pp 909, 915 , Arlington 1976


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Platus D.H. , “Ballistic Re-entry Vehicle Flight Dynamics” AIAA 82-4019; AIAA Journal vol 5 No 1, pp 4,16, 1982

Prince D.A. and Eircsson L.E. , “A New Treatment of Roll-Pitch Coupling for Ballistic

Re-Entry Vehicles” , AIAA Journal Vol. 8 NO 9 pp 16081615 Regan, Frank J. , Dynamics of atmospheric re-entry, AIAA 1994 Reshotko, Eli : “Boundary Layer Stability and Transition” in Annual Review of fluid

Mechanics, vol. 8, 1976, pp 311-349 Tobak M. et al. “Aerodynamics of Body of Revolution in Coning Motion” AIAA

Journal VOL. 7 NO 1, pp 95, 99 , New York 1968 Zoby, E.V., J.N. Moss, and K. Sutton: “Approximate Convective-Heating Equations

for Hypersonic Flows” Journal of Spacecraft and Rockets, vol. 18, no. 1, January-February 1981, pp64-70


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APPENDIXES


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100

[THIS PAGE WAS INTENTIANLLY LEFT BLANCK]


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101

Appendix A: GEARED User Manual Once in Matlab environment and after changed the directory of the workspace with

the one where are stored the file of GESARED, type “dartx” that recall the simulink file of GESARED for the implementation of DART.

SIMULINK will run the following model:

STORE

Store Results

PLOT

Plot Results

INIT

Initialization

Position

F Grav

Position alt


F Aerody n

M Aerody n

F Grav

Position

Vel v ector

Euler ang

Aerody n ang

p, q, r

GESARED

Position alt

Speed

Euler angles

Aerody n ang

p, q, r

M aerody n

F aerody n


If the initial condition have to be tuned open with te MATLAB editor the file init.m, in it are stored the initial condition of the flight in the following way:

function init(); %TUDelft, LR, B&S %Function that intializes the simulator % Andrea Guidi 17/11/2000 %New version adapted for the interaction for the Aerodynamic software clear all close all %definition of global variables global ATmod_aerdb Er DGTORAD RADTODG online cpress


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102

global Rn Rb Rf rt thetac thetafl xt hc hf L sc_Lref sc_Sref global sc_Inertia xcg sc_Mass xcpc global CXc_X CXc_Y CXc_V CYZc_X CYZc_Y CYZc_V CMc_X CMc_Y CMc_V global CXfm_X CXfm_Y CXfm_V CZfm_X CZfm_Y CZfm_V CMYfm_X CMYfm_Y CMYfm_V global CXs_X CXs_Y CXs_V CYZs_X CYZs_Y CYZs_V CMs_V Cpres_V Cpres_Y CMs_X CMs_Y Cpres_X global CMQ CLp%for damping global e_limit e_limitalfa e_limitbeta e_limitsugma sigmaold global i k kk phiold %PARAMETRI start = 0 %load matlab.mat %[n l]=size(t) shape = 1 control = 0; inputalpha = 25; spin = 90 RADTODG = 57.29578; % RAdian TO DeGree transformation factor DGTORAD = 1.0/RADTODG; % DeGree TO RAdian transformation factor Er = 7.292115147*10^(-5); % (rad/s) %Earth rotation velocity % Following is the spacecraft related parameters ------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Geometry REFERENCE PARAMETERS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % shape = input('shape 1 or 2 ?') if shape==1 load revdb xcpc = [0.362*2*Rf 0 0]; elseif shape==2 load volnadb xcpc = [0.65*2*Rf 0 0] end Ixy = 0 Iyx = Ixy; Izx = 0; Ixz = Izx; Iyz = 0; Izy = Iyz; %damping CMQ = -0.5; CLp = -0.005;


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103

%reference measure sc_Sref = (Rf^2)*pi; % (m*m) sc_Lref = 2*Rf; % (m) the radius of DART was take sc_Inertia = [Ixx Ixy Ixz; Iyx Iyy Iyz; Izx Izy Izz];% The inertial tensor (kg*m^2) if start==0 %%%%%%%%%%%%%%%%%%%%%% % Initial Conditions % %%%%%%%%%%%%%%%%%%%%%% %remember: all angle has to be express in Rad R = 6489108.190466388; %Initial distance to the center of Earth alt = 100000.000000 %Initial altitude of the vehicle lat = 7.560000*pi/180 ; %Initial latitude long = -119.500000*pi/180; %Initial longitude %initial Velocity V = 7000.500000 ; %Initial total velocity FPA = -5.000*pi/180 ; %initial flight path angle Heading = 90.5*pi/180;% 180000*pi/180 ; %initial heading angle %initial Euler angle %roll = 0; %pitch = 0; %yaw = 0; %initial Aerodynamic Angle alpha = inputalpha*pi/180; %inputalpha*pi/180; %alphaold = alpha sideslip = 0*pi/180 ; bank = 0; %initial angular velocities p = spin*pi/180; q = 0; r = 0; clear x clear t clear tout else %load simdiscstep001 n=n(1)-2 ninit clear x clear t clear tout alt(1) end % Format of the Initial Condition altformat = 1; % Altitude format % 1 for altitude


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104

% other for distance to the center of Earth attformat = 0; % Attitude format % 1 for pitch, yaw and roll angles % 0 for aerodynamic angles % other for quaternions Velformat = 1; % Velocity format % 1 for total velocity, FPA and heading % other for Velocity North, East and Down ATmod_aerdb = 1; % Atmospheric model to be used in the aerodynamic database. % 1 for Standard Atmosphere 1962 % other for GRAM 99

In the model are already implemented the two shape described in the present report. In case different shape would be implemented reed the Appendix D, build a new database and for a new shape of DART and change in init.m the name of the file that contain the following information about the vehicle:

Once chose the initial condition of the simulation run it with a double click on the yellow button named “INIT” and then with the “play” button of SIMULINK.

To see the result of the simulation, double click on the green button named “PLOT”

The graphic implemented are:

Altitude , Latitude, Longitude, Total Velocity, flight-path angle, heading, roll rate, pitch rate, yaw rate, angle of attack, sideslip angle, bank angle, teta Euler angle, psi Euler angle, fi Euler Angle.


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105

Appendix B: RCDF implementation

The tools can be used in two different ways. In the classic way an aerodynamic database can be built. A matrix of aerodynamic coefficients at different values of Mach and Alpha is built and then interpolated in the flight simulator. An other more challenging way is to calculate the exact value of the coefficient in real time connecting directly the RCFD tools to the Flight Simulator as shown in the next figure.

2F aerodyn

1M aerodyn

Flow Characteristic

Aerodyn Coef. Continuum

Aerodyn Coef. Free Molecular

RCFD

Aerodyn ang

Euler

pqr

Position alt

Speed

Aerodyn Coef. Free Molecular

Aerodyn Coef. Continuum

Aerodyn F

Aerodyn M

Flow Characteristic

Force & Moment Computatuin

5p, q, r

4Aerodyn ang

3Euler angles

2Speed

1Position alt

GESARED is implemented in SIMULIK that is particularly suitable for such an operation.


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106

A short description of the software is given. The GESARED block in the next figure is the implementation of the equation of motions, in it once gave as input the dynamic of the vehicle the kinematics is derived. The “Force and Moment computation” block calculate the dynamic forces acting on the vehicle using as aerodynamic model the one generated by the RCFD toolbox. This sublevel of this block is the one shown in the above figure. In it can be seen the block of the RCFD toolbox and the block of the forces and moments computation. Third Block in GESARED is the one that simulate the planet of the Re-entry in this case of course the earth was modelled.

STORE

Store Results

PLOT

Plot Results

INIT

Initialization

Position

F Grav

Position alt


F Aerody n

M Aerody n

F Grav

Position

Vel v ector

Euler ang

Aerody n ang

p, q, r

GESARED

Position alt

Speed

Euler angles

Aerody n ang

p, q, r

M aerody n

F aerody n


This real time option is highly time consuming, about 10 times (or more) longer, therefore can be executed after a preliminary analysis in order to obtain more accurate data, especially in the critical part of the bridge of different regimes data table. In this case of implementation more parameter can be used in the calculation of the coefficient, where would be complex implement and interpolate a table of more then two dimensions.

An other aspect of the importance of this multidisciplinary approach in the study of flight dynamic is underlined by the follow sentence of M.V. Cook (M.V. Cook 1997): "Modern flight dynamics is concerned not only with dynamics, stability and control of the basic airframe but also with the sometimes complex interaction between aeroplane and flight control system. Since the flight control system comprises motion sensor, a central computer, central hardware a study of subject becomes a multidisciplinary activity". What came out is that future on-board system would be able to use real time CFD in order to achieve better performances.


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107

Appendix C: R-CDF User Manual

Once in Matlab environment and after changed the directory of the workspace with the one where are stored the file of RCFD toolbox open with the MATLAB editor the file “dbcreator.m” the following file will appear:

%Database creator % fiele to create different database for different shape of Dart % with the Toolbox CFD global Rn Rb Rf rt theta thetafl xt hc hf L Minf alfas %INPUT Nwe Dart shape 1 Rf=1.340/2; %Rb=0.911/2; Rn=0.467/2; L = 1.360; thetac=15*pi/180; thetafl=30*pi/180; % definition of the Mach number Range of tha data base Minf=[4:1:10 11:2:30 35 40]; %Definition of the angle of attack range of the data base %alfas = 0.0001*pi/180:3*pi/180:60.0001*pi/180; %[Rad] alfas = [-75 -60 -50:5:-20 18:2:-4 -2:0.2:2 4:2:18 20:5:50 60 75]*pi/180; %[RAD] geomnlf dbiteration dbiterationfm save Adatabbase CXfm_X CXfm_Y CXfm_V CZfm_X CZfm_Y CZfm_V CMYfm_X CMYfm_Y CMYfm_V Rn Rb Rf rt theta thetafl xt hc hf L CXc_X CXc_Y CXc_V CYZc_X CYZc_Y CYZc_V CMc_X CMc_Y CMc_V

Once defined the geometrical parameter of the vehicle and the parameter of the Table in terms of Mach number and Angle of Attack run the file “dbcreator” in the workspace.

The output willl be a .mat file Adatabase in which are stored all the Aerodynamic and Geometric characteristic of the vehicle.

N.B. : In order to use this file in GESARED the mass and Inertia information have to be add.


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108



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109

Appendix D: Aerodynamic

Database Plot of DART In the following Figures are plotted the Aerodynamic Coefficient of the two shape

of DART presented in Figure 1.1 and 1.2 and obtained with the Copmutational Fluid Dynamic Toolbox described in Chapter II.

The CA, CN, and CM are respectively Axial coefficient, Normal coefficient and Momentum coefficient in the free molecular flow for the first three figures and continuum flow for the second three.

REVolution configuration

Appendix 1Aerodynamic coefficient in Free Molecular Flow

-80 -60 -40 -20 0 20 40 60 80-2

0

2

CN

alpha

-80 -60 -40 -20 0 20 40 60 800

1

2

3

CA

alpha

-80 -60 -40 -20 0 20 40 60 80-5

0

5

CM

alpha


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110

Appendix 2Aerodynamic Coefficient in Continuum flow

Appendix 3 Aerodynamic Coefficient in Supersonic flow

-80 -60 -40 -20 0 20 40 60 80-1

0

1C

N

alpha

-80 -60 -40 -20 0 20 40 60 800.4

0.6

0.8

CA

alpha

-80 -60 -40 -20 0 20 40 60 80-0.5

0

0.5

CM

alpha

0 5 10 15 20 25 30 35 400

0.5

1

1.5

CN

alpha

0 5 10 15 20 25 30 35 400.2

0.3

0.4

0.5

CA

alpha

0 5 10 15 20 25 30 35 40-0.4

-0.2

0

CM

alpha


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111

VOLNA configutation

Appendix 4Aerodynamic coefficient in Free Molecular Flow

-80 -60 -40 -20 0 20 40 60 80-5

0

5

CN

alpha

-80 -60 -40 -20 0 20 40 60 800

1

2

3

CA

alpha

-80 -60 -40 -20 0 20 40 60 80-5

0

5

CM

alpha


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112

Appendix 5Aerodynamic Coefficient in Continuum flow

Appendix 6 Aerodynamic Coefficient in Supersonic flow

-80 -60 -40 -20 0 20 40 60 80-1

0

1C

N

alpha

-80 -60 -40 -20 0 20 40 60 800.4

0.6

0.8

CA

alpha

-80 -60 -40 -20 0 20 40 60 80-0.2

0

0.2

CM

alpha

0 5 10 15 20 25 30 35 400

0.5

1

1.5

CN

alpha

0 5 10 15 20 25 30 35 400

0.2

0.4

CA

alpha

0 5 10 15 20 25 30 35 40-0.4

-0.2

0

CM

alpha


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113

Appendix E: Aerodynamic Heating

and Airload

Other crucial parts of re-entry aerodynamics are the heating and airloading. For this reason the stability analysis was initiated with the trajectory simulation of DART. Trajectory analysis provides important data for these two fields, like the magnitude of the velocity and the magnitude of the accelerations, for the flight envelope of the re-entry. For every aircraft a flight envelope is a closed area in the V-Z diagram that specifies the operating conditions. This parameter is very important in the prediction of the heatflux that the vehicle has to sustain. The Trajectory analysis is executed with a 3 degree of freedom simulator. GESARED implements a nonlinear Newtonian re-entry dynamic model with 6 degrees of freedom. Hence, the software had to be adapted the for a 3 DoF simulation. Basically, in the modeling of the motion in GESARED, the 3 DoF of the translational motion and the 3 DoF of the rotational motion are considered and implemented in two different blocks which are independent of one another. In a 3 DoF simulator only the first block is necessary. However, because of the intrinsic origin of the aerodynamic force, the attitude of the vehicle cannot be completely ignored. Two choices are possible. In the first choice the attitude of the vehicle is given simply by the initial conditions and is assumed to be unvaried until the end of the flight, in relation to the wind reference frame (this means that the angle of attack and side slip are kept constant, but not the eulerian angle). A second solution can be, after some consideration on the behavior of the glide in the re-entry flight, to implement a table of values of the angle of attack at different altitudes through the atmosphere. As mentioned above, once the flight envelope of the vehicle is known, then preliminary estimations of several environmental conditions can be made. For these estimations a toolbox was created in GESARED in order to calculate the heatflux, g-loading, and pressure distributions during the re-entry flight. The theory implemented uses the most common approximations available. For the calculation of the Heatflux the Chapman formula is

implemented: n

c

R

VV

Q

⋅∗

=

3

0

81006584.1ρρ

GESARED can output all of the variables in the above formula for 3 Dof.

For the calculation of the pressure, basic aerodynamic formulas were implemented which determined the value of pressure at the stagnation point. Then, using modified Newtonian Theory, the pressure distribution was calculated as a function of the local inclination angle of the body. Is not necessary describe the g-load. It is simply given by the relation ∆V/∆t.


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114



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115

Appendix F: 6 d.o.f. Simulations All the simulations were executed on PC Pentium II 233 Mhz, in Windows

environment. The Software package used was the flight simulator GESARED, already described in Chapter II. In Appendix XXX is also furnished a “User Manual” of the software for the basic information needed in order to run a simulation.

GESARED is a MATLAB/SIMULINK model, therefore basic knowledge of this software package are needed in order to use it.

The following are the initial condition for all the simulations:

%initial Position

alt = 100000.000000 %Initial altitude of the vehicle lat = 68.967 %Initial latitude long = 33.083 %Initial longitude %initial Velocity V = 7000.500000 %Initial total velocity FPA = -5.000 %initial flight path angle Heading = 90.5 %initial heading angle %initial Aerodynamic Angle alpha = 25 sideslip = 0 bank = 0 %initial angular velocities p = spin q = 0 r = 0

Two different spin condition were analyzed for the two shape.

30 degree per second, and 90 degree per secnd.

The FPA was choose coincident to the one furnished from the Volna launcher. The latitude and longitude are coordinate of the launching site in North Siberia.


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116



117

REVolution Configuration with a spin rate of 30 deg./sec.


118


119


120


121


122


123

VOLNA Configuration with a spin rate of 30 deg./sec.


124


125


126


127


128


129

REVolution Configuration with a spin rate of 90 deg./sec.


131


132


133


134


135


136

VOLNA Configuration with a spin rate of 90 deg./sec.


137


138


139


140


141


142

Finally are reported same relevant code especially the one used for the figure of the report.

% build windmeridian [n l] = size(x) k=0 for i=1:n gamma(i)=x(i,12)+2*k*pi; if x(i+1,12)<0 if x(i,12)>0 k=k+1 end end end w=0 for i=1:n zeta(i)=Tauezeta3(i,2)+2*w*pi; if Tauezeta3(i+1,2)<0 if Tauezeta3(i,2)>0 w=w+1 end end end wwm = zeta-gamma; %plot omega frequency % Revolution clear all load rev30 Ixx = 19.156 Iyy = 81.567 [n l]=size(t) omega =0*t; for i=1:n omega(i)=(Ixx/Iyy)*(x(i,7)*cos(Tauezeta(i,1))); end figure subplot(2,1,1) plot(t,omega*180/pi)


143

ylabel('omega (deg/sec)') subplot(2,1,2) plot(t,omega*180/pi) xlabel time ylabel('omega (deg/sec)') zoom %Volna clear all Ixx = 20.21 Iyy = 98.137 load volna30 [n l]=size(t) omega = 0*t for i=1:n omega(i)=(Ixx/Iyy)*(x(i,7)*cos(Tauezeta(i,1))); end figure subplot(2,1,1) plot(t,omega*180/pi) ylabel('omega (deg/sec)') subplot(2,1,2) plot(t,omega*180/pi) xlabel time ylabel('omega (deg/sec)')


144

% Lunar motion and NNon-avaraging Lift %In this is included the effect of the Rolling %Liftx = Lift.*cos(Tauezeta(:,2)); %Lifty = Lift.*sin(Tauezeta(:,2)); %plot(Liftx,Lifty) % plot(Fx,Fy) %generating of the Lift force Fyz=sqrt(Forze(:,8).^2+Forze(:,9).^2); Lift = -Forze(:,7).*sin(Tauezeta(:,1))+Fyz.*cos(Tauezeta(:,1)); %generating of the rotation of the wind meridian plane [n l] = size(x) % n=n-1 k=0 for i=1:n gamma(i)=x(i,12)+2*k*pi; if x(i+1,12)<0 if x(i,12)>0 k=k+1 end end end w=0 %n=n-1 for i=1:n zeta(i)=Tauezeta(i,2)+2*w*pi; if Tauezeta(i+1,2)<0 if Tauezeta(i,2)>0 w=w+1 end end end wwm = zeta-gamma; rotplane=wwm; %Tauezeta(:,2)-x(:,15); figure plot(t,rotplane) %decomposition of the Lift in the x and y direction Liftx = Lift'.*cos(rotplane); Lifty = Lift'.*sin(rotplane);


145

%booo % figure %plot(Tauezeta(:,2)) % Plot part of lunar motion nini=5000 nfin=8000 t1=t(nini:nfin); Lifty1=Lifty(nini:nfin); Liftx1=Liftx(nini:nfin); %plot of the lunar motion figure plot3(t1,Liftx1,Lifty1) xlabel('t') ylabel('Fx') zlabel('Fz') title('Lunar Motion') %________________ %plot of the lunar motion figure plot3(t,Liftx,Lifty) xlabel('t') ylabel('Fx') zlabel('Fz') title('Lunar Motion')

%_______________________________


146

% Precession %proiezione di tau sul piano %in x e y taux=Tauezeta(:,1).*cos(Tauezeta(:,2))*180/pi; tauy=Tauezeta(:,1).*sin(Tauezeta(:,2))*180/pi; tin=8000 tfin=9690 tpr=t(tin:tfin); tauxpr=taux(tin:tfin); tauypr=tauy(tin:tfin); figure plot(tauxpr,tauypr) figure plot3(tpr,tauxpr,tauypr) % Without influence of rolling rotplane=Tauezeta(:,1)-x(:,15); taux=Tauezeta(tin:tfin,1).*cos(rotplane)*180/pi; tauy=Tauezeta(tin:tfin,1).*sin(rotplane)*180/pi; tpr=t(tin:tfin); tauxpr=taux(tin:tfin); tauypr=tauy(tin:tfin); figure plot(tauxpr,tauypr) figure plot3(tpr,tauxpr,tauypr)


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147

Analytic IndexAblation 50 Aerodynamic angle .....................................35 angle of attack V, XIII, 2, 7, 37, 39, 40, 50,

7, 8, 9, 11, 13, 17, 20, 21, 22, 23, 26, 27, 31, 32, 34, 35, 36, 39, 44, 50, 73

Angle of Attack .................................... 20, 26 atmosphere ......1, 2, 4, 33, 39, 46, 8, 26, 41,

45, 49, 73 axisymmetric .........XIII, 7, 8, 31, 26, 27, 48 ballistic missiles ........................................... 7, 34 boundary layer... XIII, 1, 52, 53, 54, 55, 48,

49, 50, 58 bridge function ............................................48 cartesian. 34 Center of Gravity ........................................23 center of mass ................................ 34, 21, 23 center of pressure................................. 21, 23 CFD XIII, 5, 31, 52, 15, 16, 17, 67 coning motion................... 33, 36, 37, 39, 41 convergence..................................................26 critical frequency .........................................32 CRV VI, 2 Damping Coefficient....................................6 DART XIII, 1, 2, 7, 31, 53, 55, 57, 17,

73, 74 Delft University of Technology.................1 diameter V, VI, 2, 46, 23 directional........................................................7 Dynamic Stability ................................. 18, 26 Earth 1, 4, 31, 34, 37, 40 Equation of Motion....................................40 ESA 1, 14 Euler angle....................V, 32, 35, 37, 33, 34 exoatmospheric ...............................V, 39, 41 flare 2, 4 flight path...XIII, 1, 7, 35, 46, 9, 39, 41, 44,

45, 46 Fluid Dynamics ............................... VI, 5, 31 Free Molecular ...................................... 47, 56 GESARED.........5, 7, 31, 40, 52, 17, 66, 73 GRAM 99 .....................................................33

gyro 34 Heat Shield ...................................................50 High Temperature Effect..........................54 Hypersonic Continuum Flow ..................48 Hypersonic Rarefied Flow ........................46 Hypersonic Transitional Flow..................48 Impact Theory ...............................................8 IRDT 14 Knudsen Number.......................................48 lateral 7, 57, 32, 45 lift nonaveraging..........................................34 Lifting Body............................................. 7, 58 longitudinal ........................................7, 56, 23 Lunar Motion...............................................44 Mach VI, 1, 49, 53, 54, 6, 20, 23, 58,

66 Magnus Effect.......................................48, 49 MATLAB...........................................5, 31, 32 Meteorological .............................................49 Modified Newtonian Law.........................49 Non-linear analysis........................................7 nutation angle...............................................34 panel method ...............................................55 Pitching moment...........................................6 PM-1000 4 precession ........................... V, 34, 35, 36, 41 q-motion 7, 8, 9 Re-entry XIII, 1, 2, 30, 46, 36, 58, 67 Reference Frames.................................33, 34 side slip 40, 12, 20, 73 spin rate 30, 31, 32, 33, 41 State Space Model .........................................2 Static stability..................................................6 TEST i, VI, XIII, 1, 30, 58 the Boundary Layer. ...................................54 Total angle of attck .....................................39 trim angle ...................................V, 22, 23, 32 Viscous influence ........................................52 vortex 36, 38, 48 Wind Tunnel ................................................30 Windward-Meridian............................... 7, 58

dart · 2002. 10. 24. · normal coefficient cn in continuum folw for a sferical blunted cone with...

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