Preliminary design andstability analysis of a

de-orbiting system for CubeSats

In the frame of the QB50 mission

Travail de fin d’étude présenté en vuede l’obtention du grade d’ingénieur

civil électromécanicien par

Guerric DE CROMBRUGGHELaurent MICHIELS

Promoters:Pr. Philippe Chatelain, EPLPr. Thierry Magin, VKI

Readers:Pr. Miltiadis Papalexandris, EPLDr. Cem Ozan Asma, VKIDr. Vladimir Pletser, ESA

June 2011

Abstract

Atmospheric re-entry is a key to further developments in space exploration. The purpose ofthe CubeSat re-entry vehicle, currently under development in the frame of the QB50 missioninitiated by the von Karman Institute, is to offer a cost-efficient and flexible validation tool. Thetechniques used could also be applied as de-orbiting systems specifically designed for CubeSats,for which there is a strong need.

Up to now, no spacecraft with such reduced dimensions has performed a controlled atmo-spheric re-entry. There are thus a certain number of problems to solve. This report focuses onthe preliminary design and stability analysis of a de-orbiting system.

After a survey of the available techniques, it was decided to increase the satellite’s dragarea. Therefore, three possible geometries were designed, and their aerodynamic characteristicsfor the lower thermosphere (170 km down to 100 km) were obtained with DSMC steady-flowsimulations. Those characteristics were then used to perform a three-degree-of-freedom analysiswith a Simulink tool especially developed for this study.

It appears that deploying a surface perpendicular to the flow downstream the satellite is themost suitable option to de-orbit and provide passive stabilization. The study was performed fora 0.3x0.3 m2 square plate connected with a 1 m long flexible link to the satellite’s rear face.Results show that the system’s efficiency can easily be improved by varying the geometricalparameters.

Now that the efficiency and stability of such a de-orbiting concept have been demonstrated,a more detailed design could be done, taking into account other effects such as the thermalloads. The Simulink program could also be further developed and used as a powerful predictivetool for the dynamics of re-entry vehicles at high altitudes.

Keywords: de-orbiting, re-entry, stability, DSMC, CubeSat, QB50

Acknowledgements

The authors would like to thank Pr. Philippe Chatelain from the UCL, Dr. Cem Ozan Asma,Pr. Thierry Magin, and Zsolt Varhegyi from the VKI, Dr. Vladimir Pletser from the ESA for hisexternal vision on the project, Tamas Banyai and Erik Torres from the VKI for all the time theyspent in explanations and bugfixing, Dr. Raimondo Giammanco from the VKI for his technicalsupport on the cluster and remote connections, Pr. Vladimir Riabov from Rivier College forhis explanations on the non-monotonic behaviour of aerodynamic coefficients, and Dr. MarkSchoenenberger from NASA Langley for sharing his knowledge on damping coefficients.

Contents

Contents i

1 Introduction 11.1 The QB50 mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of the CubeSat re-entry mission . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Objectives of the present study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Survey of de-orbiting techniques 52.1 Mission objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 De-orbiting options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Aerodynamic drag increase . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Tethers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Potential de-orbiting systems comparison . . . . . . . . . . . . . . . . . . . . . . 132.4 Selected drag increase geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Badminton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Flower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.4 Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Rarefied flow theory and modelling 173.1 Short introduction to rarefied flows . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Direct Simulation Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Parameters settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Pre-processing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.2 Note on the Fnum parameter . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.3 Processing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Results quality evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 Method validation: Apollo re-entry capsule test case . . . . . . . . . . . . . . . . 243.7 Transitional regime characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Aerodynamic coefficients database 274.1 Requirements on the coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Drag and lift coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.2 Pitch moment coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Aerodynamics for Kn � 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Pitch moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 Preliminary conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Basic geometry in transitional regime . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Influence of the spherical section . . . . . . . . . . . . . . . . . . . . . . . 31

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4.4 Flower geometry in transitional regime . . . . . . . . . . . . . . . . . . . . . . . . 344.4.1 Influence of the flaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Plate geometry in transitional regime . . . . . . . . . . . . . . . . . . . . . . . . . 364.5.1 Influence of the length of the link and the size of the plate . . . . . . . . . 39

4.6 Influence of key parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Dynamic study and stability analysis 425.1 Re-entry modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Program development and validation . . . . . . . . . . . . . . . . . . . . . . . . 445.2.1 First step: gravitational force . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.2 Second step: drag and lift terms . . . . . . . . . . . . . . . . . . . . . . . 465.2.3 Third step: moment equation . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Determination of the damping coefficient with the modified-Newtonian method . 495.4 Application to the selected geometries . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.1 Basic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.2 Flower geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.3 Plate geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5 Influence of key parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5.1 Trigger altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5.2 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5.3 Solar activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Geometry selection 656.1 Criteria presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Selection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3 Guidelines for a complete system . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3.1 Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.2 Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7 Conclusion 717.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.3 Last words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Bibliography 74

A Decoupled pressure field hypothesis 77

B Simulink program constructive details 79B.1 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.2 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.3 Third step: complete Simulink program . . . . . . . . . . . . . . . . . . . . . . . 81

C Validation of the Simulink program 84C.1 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84C.2 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

D Calculation of the damping coefficient on the second face 87

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E Influence of the pitch moment damping and the lift force 88E.1 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88E.2 Lift force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Chapter 1

Introduction

Atmospheric re-entry is a key to further developments in space exploration, whether it concernsthe safe return of astronauts on Earth or the landing of robots on Mars, Venus, or even Titan.Since the very beginning of the space age, it has been considered as an important area ofstudy. Today, research is mainly conducted through computational methods, ground-basedexperimentation, and flight data analysis and extrapolation. Only a few spacecraft, such as theEuropean Experimental Re-entry Testbed (Expert) or the Intermediate eXperimental Vehicle(IXV), are developed as validation tools. Although they will provide a considerable amount ofinformation to the research community, they are very costly and are developed with extendedtimelines.

CubeSat is the name given to a standardized format of nano-satellite as defined by the Califor-nia Polytechnic State University and the University of Stanford. A single CubeSat unit consistsin a 10x10x10 cm3 cube, weighing around 1 kg. It is typically built from off-the-shelf compo-nents, and offer thus an inexpensive solution that is easier and faster to develop than ordinarysatellites. Since its standardization, in 1999, hundred of academic groups, and an increasingnumber of private companies, have developed their own CubeSat mission to fly an experimentor a radio transmitter. [1]

Those small satellites are currently gaining popularity, to such an extent that the over-population of satellites orbiting around the Earth becomes a major concern. The current guide-lines stipulate that spacecraft should stay on orbit less than 25 years after the end of theirmission. This will be difficult to respect, for CubeSats in particular as they are less under theinfluence of solar pressure and atmospheric drag bigger satellites. There is thus a strong needfor de-orbiting systems specifically designed for CubeSats.

Those considerations inspired the idea of a CubeSat re-entry vehicle. When demonstrated,it will provide an alternative solution, maybe less complete but more cost-efficient than thebigger spacecraft, and will be able to fly experiments after a shorter development period thanksto its standard platform. Furthermore, the de-orbiting technique conceived for this particularapplication could also be used as new debris mitigation systems.

1.1 The QB50 missionQB50 is a space mission initiated by the von Karman Institute, dedicated to an in situ

study of the lower thermosphere. It will consist in a network of over fifty 2-unit CubeSats, eachseparated by a few hundred kilometres. The mission will provide information about the temporaland spatial variations of the atmospheric parameters. Due to the drag force, the satellites will

1

naturally decay from the altitude of 320 km down to 90 km in about 3 months, without needfor propulsion. The launch is planned for 2014. [2]

The lower thermosphere is the layer of the atmosphere located between � 90 and � 200 km(Figure 1.1). It is too rarefied for remote sensing by Earth observation satellites. On the otherhand, in situ exploration with stratospheric balloons is not possible higher than 42 km, andground based lidars and radars can maximally sense up to 105 km. The only measurements ofthe lower thermosphere are provided by sporadic sounding rocket launches. It is, therefore, theleast explored layer of the atmosphere.

Figure 1.1: Atmospheric layers, the QB50 mission will evolve in the lower thermosphere

The satellites will most probably be divided in two groups, each of them carrying an identicalset of sensors. The remaining volume will be available for the research or technological demon-strations of a partner university, allowing the network to carry more than fifty different originalpayloads. On the 29th of April 2011, 69 letters of intent had been received from universities allover the world. The Université Catholique de Louvain is one of them. [3]

1.2 Scope of the CubeSat re-entry missionA few satellites in the QB50 network will consist in 3-unit CubeSats. Instead of burning in the

atmosphere, as will the other satellites, their mission will be to perform a controlled atmosphericre-entry. Even if they will most probably not reach the ground, they are designed to survive downto 50 � � � 70 km and provide key information about their trajectory and surrounding environment.

Up to now, no spacecraft with such reduced dimensions has performed a controlled atmo-spheric re-entry. There are thus a number of challenges, which can be divided into four cate-gories:

• De-orbiting.When referring to small satellites, de-orbiting systems are often considered as a way toreduce the lifetime of the spacecraft to less than 25 years. De-orbiting is here envisaged as

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a way to perform a controlled re-entry over a determined geographical area. The timescaleinvolved is thus a few orbits rather than a few years.

• Thermal loads.The satellite should start its re-entry at orbital speed at an altitude of � 150 km, whichmeans over 7, 800 m/s. For an ideal re-entry, it would reach the ground at zero velocity.All that energy, both kinetic and potential, has to be dissipated - in this case, convertedinto heat. That heat can be estimated by equaling it to the satellite’s energy:

Q = m � (g � h+v2

2)

Where Q is the dissipated heat, m the satellite’s mass, g the gravitational constant, h itsinitial altitude, and v its initial velocity. The total heat transfer is approximately � 95.7MJ . It is usually considered that only a small fraction of that energy, in the order of10�3, will go into the satellite, the rest being transferred to the surrounding air [4]. Thatleaves � 95.7 kJ to dissipate within a small volume, keeping its temperature below 70 °Cto stay within the electronics’ operational range. It would be possible to raise that limitup to 150 °C if military components are used. An efficient thermal management is thusnecessary.

• Communication.Due to the huge amount of heat to dissipate, the satellite is expected to burn in the atmo-sphere before having reached the ground. All the data have to be sent continuously, as itwill be impossible to recover them after the flight. The presence of plasma in front of thesatellite’s nose will make it impossible to transmit telemetry directly down to the Earththrough the atmosphere. The signal will have to be sent upward to telecommunicationsatellites such as the Iridium constellation, able to relay it back to mission control. There-fore, the satellite should not pass over the poles, where the telecommunication networkcoverage is poor.

• StabilityAs a direct conclusion of the two last points, a stability system is needed to keep the frontalheat shield facing the incoming flow and the antenna pointing towards outer space.

1.3 Objectives of the present studyThe first and the last challenges, de-orbiting and stability, were unified under one single

problem. The objective of this study is to conduct a preliminary design and stability analysisof a de-orbiting system for the re-entry satellites of the QB50 mission. This report presents theprogression that led from the systems requirements to an aerodynamic drag increase conceptable to de-orbit and provide passive stabilization.

The second chapter consists in a survey of the techniques that could be used to de-orbit smallsatellites. After a presentation of the requirements and constraints, the different solutions areintroduced and their feasibility is assessed. All of them are then compared, and the most suitableone is presented.

The third chapter presents the theory and modelling techniques for hypersonic rarefied flows.The theory of rarefied flow is introduced, followed by a description of the modelling code used,the parameter settings, and their validation with the Apollo module test case. The regimes thatthe re-entry satellite will experience during the beginning of its flight are described.

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The fourth chapter presents and discusses the aerodynamics of the different de-orbitinggeometries that were proposed in the second chapter, both for rarefied regime and for the regionbetween rarefied and continuum regimes. The influence of some special design features is alsodiscussed.

In the fifth chapter, a dynamic model that solves the equations of motion governing there-entry in high altitudes is elaborated, using Simulink and Matlab. The program is built indifferent steps, in order to understand the influence of each force and moment acting on thesatellite. It is then validated and applied to the de-orbiting geometries described previously.The evolution of the flight parameters during re-entry, such as altitude, angle of attack orvelocity, is studied.

Using different criterion from both static and dynamic analyses, the geometries are finallycompared in the sixth chapter, in order to select the most efficient de-orbiting structure. Theresults are discussed, and a few guidelines for a more evolved system are given.

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Chapter 2

Survey of de-orbiting techniques

This chapter consists in a survey of the different de-orbiting options available for the CubeSatre-entry mission, in order to select the most suitable one.

The objectives of the de-orbiting systems are first defined in terms of performance andconstraints.

The main options available are then briefly examined. Focus is set on systems able to performthe de-orbiting on their own, or at least as primary systems. Cold gas, chemical propulsion,electric propulsion, aerodynamic drag increase and tethers were considered. Other options suchas solar sails, magnetic navigation, MEMS thrusters arrays, etc., were not considered due totheir obviously poor performance for this kind of manoeuvre or due to their low technologyreadiness level (TRL). Indeed, the QB50 mission is supposed to be launched in 2014, and onecannot afford the risk to wait for technological improvements or developments.

A qualitative comparison is then made between all the options, with comments on the criteriaand an interpretation of the results. The best option is then selected.

Finally, the three drag increase geometries that were further studied are presented, togetherwith the geometrical characteristics of the re-entry satellite itself.

2.1 Mission objectivesStrictly speaking, the re-entry satellite considered for this study is a 2-unit CubeSat with an

ablative heat shield on its top side, approximately corresponding to one additional unit (seesection 2.4.1 for detailed information). The geometrical constraints are thus the same as for a3-unit CubeSat. Its total mass is expected to be around 3 kg [1]. Although the mass budget isnot clearly defined yet, it is foreseen that the mass of the de-orbiting system should not exceed500 g, the rest being reserved for the other systems and the payloads.

The external volume of a 3-unit CubeSat is 10x10x32.75 cm3 [1]. Again, even if the volumebudget is not clearly defined yet, it is foreseen that the volume of the de-orbiting system shouldnot exceed half a unit, 5x10x10 cm3: its shape should also be taken into account, as it maycause integration issues (e.g. spherical tank, pipes and cylindrical thruster).

The average power directly available from the solar cells while the satellite is on orbit isexpected to be only in the order of 4 W [1]. Furthermore, those solar cells will most probablybe lost during the re-entry. This is a very limiting factor. Indeed, most of the valves usedon the market for propulsion systems already require at least several watts to open and bekept open. Fortunately, the power requirement of the re-entry satellite while it is on orbit isquite low: minimum telemetry and possibly some periodical measurements. All the power canthus be used to charge a set of batteries with high capacity, and discharge them during the re-entry phase. The battery system has to be chosen and dimensioned carefully, as it needs to be

5

able to survive a power peak during the de-orbiting manoeuvre (e.g. valve opening, propellantignition, mechanism deployment, attitude control, etc.) and then provide enough power to runthe experiments while using the telemetry.

Numerous battery packages are available for small satellites. The NanoPower BP-4 fromGomspace will be used as reference for this preliminary study [5]. It simply consists in fourPanasonic CGR18650HG cells. Even if it is quite massive and volumic, 213 g and 23x90x96mm3, it is the only one able to provide a maximum power discharge of 12 W in one single unit.It has a capacity of over 3.6 Ah, with a nominal voltage ranging from 7.4 to 8.4 V . The totalenergy available is thus approximately 30 Wh. This battery set is not the final system, but justa representative reference to allow numeric investigation.

The satellite must complete its trajectory in less than half an orbit, in order to avoid thepoles were the telecommunication network coverage is poor. This objective has to be translatedin terms of speed reduction ∆V in order to allow for quantification of the mass of propellantneeded for de-orbiting techniques based on propulsion. A first estimate shows that an impulsivespeed reduction of �50 m/s at an altitude of 170 km is enough to reduce the distance coveredfrom more than 15 orbits (Figure 2.1(a)) to approximately a quarter of an orbit (Figure 2.1(b)).Those figures were obtained with the model developed in chapter 5. The Earth’s radius hasbeen divided by 40 in the figure for more visibility. Although it is based on an estimate, thisscenario will be taken as reference for the preliminary survey.

Since that scenario is based on an impulsive burn, the time ∆t needed to deliver that speedreduction should be as short as possible, with a lower limit set to avoid overly important ac-celerations on the satellites. A ∆t greater than 0.5 s guarantees accelerations lower than 30g.

The satellites could stay up to 3 months on orbit before de-orbiting, starting at an altitude of320 km possibly down to 170 � � � 120 km. During that phase, the de-orbiting system will mostprobably be in sleep mode.

satellite’s trajectoryEarth

(a) (b)

Figure 2.1: Possible de-orbiting manoeuvre; a �50 m/s impulsive speed increment at analtitude of 170 km (b) compared with the natural decay (a). The figures were obtained with

the model developed in chapter 5, assuming the atmospheric density is exponential

The requirements are summarized in Table 2.1.

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Table 2.1: Requirements for the de-orbiting system

Sleep phase (months) Trigger altitude (km) ∆V (m/s) Acceleration (g)

< 3 120 ... 300 > 50 < 30

Mass (g) Volume (cm3) Power (W ) Energy (Wh)

< 500 < 500 < 12 < 29

2.2 De-orbiting optionsThe main de-orbiting options are discussed in this section. A few practical examples of existing

models are also given for each technology.

2.2.1 PropulsionThere is a large variety of propulsion systems used for attitude control of bigger spacecraft

that could be used as de-orbiting systems for medium to small satellite. However, most of themare too volumic, massive, or power-consuming for the particular case of CubeSats.

When needed, the mass of propellant mp will be roughly estimated using the following rela-tion [6]:

mp = m0 � [1� exp( �∆VIsp � g )] (2.1)

Where m0 is the total mass of the satellite before the burn, which is expected to be 3 kg,Isp the specific impulse of the thruster, and g the acceleration due to the gravity at the surfaceof the Earth, 9.81 m/s2. The duration of the burn ∆t will be estimated using the followingrelation [6]:

∆t =mp � g � Isp

F(2.2)

Where F is the mean thrust. When the amount of propellant available is fixed in advance,the speed increment ∆V will be estimated using another form of equation 2.1:

∆V = g � Isp � ln m0

m0 �mp(2.3)

The subscript thruster is for the thruster itself without taking the propellant and the rest ofthe dry mass into account, and the subscript tot is for the entire system.

Cold gas propulsion

The cold gas propulsion technology has numerous advantages: it offers the greatest degreeof simplicity among all the propulsion systems, it benefits from an important heritage in spaceapplications, and it uses contamination-free non-toxic propellant, e.g. N2. It has been usedmainly for attitude control of medium to big spacecraft, and not for propulsion of small satellites,hence the reduced mass and volume of the thrusters but high power requirement to handle thevalves. [7, 8, 9]

However, the characteristically low specific impulse of cold gas thrusters results in hugeamounts of propellant needed to achieve the required speed increment. Furthermore, theirrelatively low thrust results in a very long time to reach that speed increment. Considering thehigh power needed to hold the valve open, it also results in important energy consumption.

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The Moog 58X125A is a typical example of the smallest products available on the market [10].It has already flown on four missions. Its characteristics are summarized in Table 2.2. It appearsthat the burn duration is way too long for a rapid de-orbiting. Plus, the required energy exceedsby far what is available in the batteries.

Table 2.2: The Moog 58X125A characteristics

mthruster (g) mp (g) Vthruster (cm3) F (mN) Isp (s) ∆t (h) Power (W ) Energy (Wh)

9 226 4.9 4.4 65 9 10 91

Nevertheless, Marotta has developed a low-power micro-thruster which fits the requirementsand is already space qualified [11]. Its characteristics are summarized in Table 2.3.

Table 2.3: The Marotta low-power micro-thruster characteristics

mthruster (g) mp (g) F (mN) Isp (s) ∆t (min) Power (W ) Energy (Wh)

70 226 445 65 5   1   0.083

Propellants such as butane and ammonia can be stored in their liquid form and will phasechange into gas upon expansion. It allows for lighter and less volumic tanks, and storage at lowerpressure, which reduces leakage concerns and yet maintains the simplicity of cold gas thrusters.Up to now, the piezoactuated butane propulsion system from Vacco is the only one existing inbreadboard model [8, 9]. Its characteristics are summarized in Table 2.4. It is designed as anattitude control system, with five thrusters in total. Since the tank is integrated, its maximalspeed increment is limited to a certain value, which is below the objective. [8, 9]

Table 2.4: The Vacco piezoactuated butane propulsion system characteristics

mtot (g) Vtot (cm3) ∆V (m/s)

456 250 26

Chemical propulsion: monopropellant

Hydrazine thrusters are commonly used for attitude control and to produce small speed incre-ments. Unfortunately, none of those available commercially fulfils the requirements of CubeSatapplications. A few research and development models are getting close to the requirement en-velope, but none of them passed flight qualification. Furthermore, hydrazine requires to behandled by experienced personnel as it is toxic and flammable. [7, 8, 9]

Among the research and development models, the JPL Hydrazine Milli-Newton Thruster(HmNT), which was originally developed for precision pointing and formation flying, is themost suitable one [8, 9]. Its characteristics are summarized in Table 2.5.

Hydrogen peroxide has a lower degree of toxicity than hydrazine and is thus more suitablefor university student groups, hence the development of hydrogen peroxide micro-thrusters forCubeSats. On the other hand, it is usually less efficient than hydrazine, and subject to slowdecomposition, which can lead to tank over-pressurization over time. Furthermore, none of thethrusters in development are ready for flight.

8

Table 2.5: The JPL HmNT characteristics

mthruster (g) mp (g) Vthruster (cm3) F (mN) Isp (s) ∆t (min) Power (W ) Energy (Wh)

40 100 8 129 150 19 8 2.45

Chemical propulsion: bipropellant

Even if the bipropellant thrusters offer greater specific impulse compared to their monopro-pellant counterparts, the cost of added complexity and dry mass is too important for CubeSatapplications. [7, 8, 9]

Chemical propulsion: solid

Typical values of the specific impulse for small solid rocket motors reach above 250 s. Theamount of propellant needed for a certain speed increment is thus lower than for any other kindof chemical propulsion system. The propellant being solid, less volume is necessary for storage.Since there is no valve, the only power requirement is less than 1 W for the igniter.

Solid rocket motors are only able to give one single burn, which is not that much of aninconvenience for a de-orbiting manoeuvre. Plus, there is usually a notable uncertainty onthe thrust and total specific impulse delivered. If that uncertainty is too high, an attitudecontrol system may be needed to correct the trajectory and attitude after or during the burn.Plus, solid rocket motors deliver large thrusts on short duration, which may lead to excessiveacceleration. [7, 8, 9]

The STAR 3A, manufactured by ATK, is among the smallest thrusters available on the mar-ket [12]. It has already flown on two missions. Its characteristics are summarized in Table 2.6.The thruster is cylindrical and its length nearly entirely fills a 3-unit CubeSat, leaving only littleroom for the systems and payloads. Also, the mass of the thruster exceeds the constraints, butit includes all the systems needed - unlike the mass estimate for the other chemical propulsionsystems where the mass of the tank and piping were not taken into account.

Table 2.6: The ATK STAR 3A characteristics

mtot (g) Vtot (cm3) F (N) Isp (s) ∆V (m/s) ∆t (s) Acceleration (g)

890 984 613.85 241.2 98.24 0.5 30

The STAR 4G, also manufactured by ATK, is a so-called ’slow burner’ [12] (Figure 2.2). Itscharacteristics are summarized in Table 2.7. The achieved speed increment is much greater thanwhat is needed.

Table 2.7: The ATK STAR 4G characteristics

mtot (kg) Vtot (cm3) F (N) Isp (s) ∆V (m/s) ∆t (s) Acceleration (g)

1.5 755.4 307 275.6 ¡ 1, 000 10 10

Electric propulsion: electromagnetic and electrostatic thrusters

Electric thrusters have been used for many years for fine attitude control and station keeping.They are characterized by a very high specific impulse, ranging from 500 s to 3, 400 s, whichallows for less propellant. However, this advantage is largely countered by the small thrust

9

Figure 2.2: The STAR 4G solid rocket motor, Credits: ATK

delivered and the amount of power needed to generate the magnetic and electric fields used toaccelerate the propellant. [7, 8, 9]

If mass is saved on the propellant, it is at the cost of very long de-orbiting times, to becounted in tens of days, and thus huge amounts of energy.

The Miniature Xenon Ion (MiXI) thruster developed by the JPL is a representative exampleof the status of the technology [8, 9]. Its characteristics are summarized in Table 2.8. The firingtime exceeds by far the de-orbiting window, and the power required is too important for thebatteries.

Table 2.8: The JPL MiXI thruster characteristics

mp (g) F (mN) Isp (s) ∆t (days) Power (W ) Energy (Wh)4.8 1.5 3, 200 ¡ 1 13 � � � 50 312

The same results would be obtained with other kinds of electric micro-thrusters. Hall effectthrusters, for example, are capable of higher thrusts, up to 15 mN , at the cost of lower specificimpulse, from 1, 000 s to 1, 500 s, and higher power requirements, ranging from 100 to 300 W .

Electric propulsion: resistojets

Resistojets are similar to classic gaseous or liquid propulsion techniques to which a heatexchanger would be added. The increased temperature of the propellant results in higher specificimpulse and higher thrust. The exchanger adds complexity, mass and power requirements,especially for small thrusters. Despite their simplicity, they are thus not applicable yet toCubeSats. [7, 8, 9]

2.2.2 Aerodynamic drag increaseThe initial altitude of the re-entry satellite being particularly low, the remaining atmosphere

could be used to de-orbit it through by increasing its aerodynamic drag. It could be done bydeploying surfaces in order to increase the satellite’s cross-sectional area. Using the definitionof the ballistic coefficient, which describes the sensitivity of a flying object to the aerodynamicbrake, one can see that increasing the area drag Adrag leads to a decreased ballistic coefficientCb, and finally to an increased deceleration. [13]

Cb =m

CD �Adrag

10

The drag coefficient CD is generally assumed to be equal to 2.2 at altitudes at which spacecraftorbit, although experiments and analytic results have shown that it can vary widely [14] [15].The mass of the satellite is 3 kg. Assuming for now that there is a permanent attitude controlto keep angle of attack, the drag area of a 3-unit CubeSat is 10x10 cm2. Thereby, the ballisticcoefficient of the satellite before the deployment of the drag increase system is Cb = 136.36kg/m2.

The effect of the drag area on the lifetime of the satellites was estimated using Satellite ToolKit (STK) with the astrogator propagator. STK is a software able, among other things, togive an approximation of a spacecraft’s trajectory through the atmosphere, knowing its initialposition, mass and drag coefficient. The results were calculated for an initial orbit at an altitudeof 170 km on the 1st of June 2014.

Drag area

The speed reduction ∆V used previously is defined as an impulsive burst. The case of theaerodynamic drag increase is different as the speed reduction occurs now continuously. Therefore,the difference between the orbital velocity and the satellite’s velocity at a certain altitude, noted∆valtitude, will be considered. It is representative of the de-orbiting system’s efficiency from itsdeployment down to a certain altitude.

Table 2.9 shows that the ballistic coefficient increases and the lifetime of the satellite decreaseswhen the drag area is increased. That information is also shown in Figure 2.3. The lifetime isthe duration of the satellite’s trajectory from its initial orbit till it reaches the ground.

A drag area of 5 m2 is already enough to reduce the de-orbiting time from the initial valueof more than 14 hours to only 22 minutes. This value is expected to be sufficient for a quickde-orbiting. The speed difference ∆v is measured at 100 km.

Table 2.9: Influence of the drag area on the ballistic coefficient, lifetime and speed reductionprovided, based on simulations performed with STK

Drag area (m2) 0.01 0.15 1 5Ballistic coefficient (kg/m2) 136.36 9.1 1.36 0.27

Lifetime (min) 860 90 41 22∆v100km (m/s) 23 244 407 434

Although the altitude is globally decreasing, there is repeated signal with a period of one orbit.Those periodical variations present two minima and two maxima. The satellite’s trajectorydescribes thus a decreasing elliptical orbit, most probably due to the introduction of a dragforce.

Trigger altitude

Table 2.10 compares the efficiency of a 1 m2 drag area for different deployment altitudes.The ∆t is the time needed to reach the altitude of 70 km and 50 km respectively. The distancecovered in terms of latitude ∆Lat is also indicated for both altitudes. The lowest trigger altitude,120 km, corresponds to the expected beginning of the intense aerodynamic heating, and thusthe formation of plasma in front of the ablative heat shield.

For example, it will take 10 minutes for a satellite to decay from an initial orbit of 120 kmdown to 70 km. During that period, it will cover a total latitude difference of 37.9°.

11

Figure 2.3: Comparison of the lifetime of the satellite with different drag areas, based onsimulations performed with STK

Table 2.10: Efficiency of a 1 m2 drag area for different trigger altitudes for an orbit inclinationof 79°, obtained with STK

Initial altitude (km) ∆t70 (min) ∆Lat70 (°) ∆t50 (min) ∆Lat50 (°)200 79 279 80 281150 29 96 29.5 98130 16 60.6 16.5 61.7120 10 37.9 11 39.2

Conclusion

From a theoretical point of view, de-orbiting techniques based on the aerodynamic drag in-crease seem to be really efficient, even with moderated drag areas.

Drag increase systems would consist of thin membranes deployed and maintained by a struc-ture, which could be inflatable or mechanical. A drag area of 0.15 m2 will weigh around 30 g,and have a stowed volume of 79 cm3 [16]. This fulfils the requirements of mass and volume.

2.2.3 TethersElectromagnetic tethers are increasingly being considered as a light, compact, cheap and

reliable way to de-orbit small satellites. They offer the advantage of operating without propellantor input power. Their mass and volume requirements are thus also smaller compared to othersystems.

On the other hand, the behaviour of tethers in space is extremely difficult to predict and tocontrol. Indeed, with a length ranging from a few meters to a few kilometres, tethers experiencevery different conditions along their length. Their discretization in a series of small bodies,needed for trajectory prediction, has to be done with a high degree of precision, resulting in ex-pensive computing time. Plus, experiments showed that unpredicted instabilities might appear.The case of electrodynamic tethers is even more complicated, since they involve electromagneticinteractions with the atmospheric plasma and the magnetosphere. [17]

12

Furthermore, the goal of that de-orbiting is to shorten the lifetime of the satellite, in orderto reduce the amount of small debris in orbit. Electrodynamic tethers are thus dimensioned forde-orbiting time in the order of years. [18]

The nanoTerminator developed by Tethers Unlimited is the only commercially available elec-trodynamic tether for small satellites [19]. The mass of its package is less than 80 g, with avolume of 0.5x8.3x10 cm3, what is small enough to fit on the face of a CubeSat. It will be testedon low orbit for the first time on one of the QB50 satellites. Unfortunately, it would require atleast more than one day to de-orbit and is thus not suited for the re-entry satellites [20]. Plus,its length, 30 m, is considerable compared to the size of the satellite and may lead to importantinstabilities.

Another concept, the electrostatic tether, is still under investigation. A prototype should flyin a few years onboard a CubeSat developed by Estonian university students. Estimations show,however, that it is less efficient than electrodynamic tethers to perform a de-orbiting. [21]

2.3 Potential de-orbiting systems comparisonThe de-orbiting systems where compared based on how they meet the following criteria:

• Mass: (+) if the mass of the system is considerably smaller than the requirement, (0) ifit is more or less equal to the requirement, (�) if it is larger, and (��) if it is equal orlarger than the mass of the entire satellite.

• Volume: same points as for the mass.

• Power: same points as for the mass, the energy is also taken into account.

• ∆t: (+) if it is in the order of seconds, (0) if it is in the order of minutes, (�) if it is inthe order of hours, and (��) if it is in the order of days.

• TRL: (+) if the corresponding technology is space qualified, (0) if it is ready for flight,and (�) if it is still a prototype.

• Stability: (+) if the system may provide stabilization to the satellite, (0) if its influenceon the stability can be neglected, and (�) if it may cause instabilities.

• Applicability: this criterion defines the ease with which the technology is applied to de-orbiting. The tether receives therefore a (+), as it was especially conceived for thatpurpose. Inversely, all the propulsion techniques receive a (�). Indeed, the thrust vectorneeds to be aligned with the velocity vector of the satellite. It requires post-burn attitudemodification, e.g. 180° rotation if the thruster is on the opposite side of the heat shield, orcomplex integration, e.g. two thrusters right after the heat shield producing symmetric andsynchronized thrusts under opposite angles with respect to the velocity vector. Both casesresult in mass increase and more complex operations during the de-orbiting manoeuvre.

The total points is negative or null for every options but the aerodynamic drag. Furthermore,some options have a (��) and could thus not be used at all. This confirms that the rapidde-orbiting of CubeSats is something new, for which a new tool has to be developed.

13

Table 2.11: Comparison of the different de-orbiting options

Criteria

Options Mas

s

Volu

me

Powe

r

∆t

TR

L

Stab

ility

App

licab

ility

TotalCold gas 0 0 0 0 + 0 - 0Monopropellant 0 0 0 0 - 0 - 2 -Solid - - - + + + - - 2 -Electric 0 0 - - - - + 0 - 4 -Aerodynamic drag increase 0 0 + 0 0 + 0 2 +Tethers + + 0 - - 0 - + 0

According to the total points, aerodynamic drag increase is the best option. Furthermore, itpresents the following advantages:

• It is an efficient solution, able to de-orbit in less than one orbit with appropriate dimen-sioning.

• It fits in the requirements envelope in terms of mass and volume, and will require onlylittle power during deployment.

• A well-dimensioned system with a specific shape will ensure passive stabilization of thesatellite.

• It allows for several degrees of freedom during its conception (shape, dimensions, type ofstructure, materials, etc.) as well as during the flight itself (altitude of deployment and ofjettison).

2.4 Selected drag increase geometriesChanging geometries and deployment mechanisms are not something new to the CubeSat

community. Their reduced size and volume forced engineers to find solution to deploy antennas.Mechanisms are being designed in order to deploy solar panels that would significantly increasethe power available onboard (Figure 2.4). Recently, spectacular advances such as the NanoSail-Dsatellite, a solar sail demonstrator launched by a team of researchers from NASA Ames, provedthat even the most ambitious concepts have a chance of success. However, major drawbackshave also been identified, as the first satellite failed and the second had issues to deploy its sail.

Figure 2.4: Concepts of deployable solar panels, Credits: Pumpkin and AAUSAT

14

Some of these ideas could be used to increase the satellite’s drag, but the aerodynamic forceacting on the fins at lower altitudes might be too high. Although they can be used as source ofinspiration, more robust geometries need to be designed for re-entry vehicles.

Due to the time needed to run the simulations, it was decided to study only four geometries,similar in terms of volume and dimensions. They are directly inspired by existing or planneddeployable mechanisms for small satellites, keeping in mind that their goal is both to increasethe drag area and to provide passive stabilization. The design is kept at a conceptual level,regardless of the materials or other implementation details.

2.4.1 BasicThe basic geometry represents the current design of the satellite on its own, without any

drag increase system deployed (Figure 2.5). The satellite is a 3-unit CubeSat. Two units arededicated to the systems, instrumentation and payloads, while the third one is an ablative heatshield.

From a geometrical point of view, it is approximated by a box whose front part, the heatshield, is shaped as a section of a 0.15 m radius sphere. The cross-sectional area of the boxis a square with 0.1 m long sides. The total length of the box, from its bottom to the top ofthe spherical section, is 0.3 m. For this geometry as for the next ones, the centre of gravity isassumed to be right in the middle of this basic satellite.

Figure 2.5: The meshed basic geometry in the Gambit environment

2.4.2 BadmintonThe first concept is directly inspired by the legacy of past re-entry vehicle designs. The side-

panels are deployed with a certain angle, resulting in a geometry close to a typical sphere-frustumconfiguration (Figure 2.6). Past missions and experiments show that the dynamic stability ofsuch a configuration is better than the spherical section configuration.

Four panels with dimensions of 0.005x0.1x0.3 m are stuck to the sides of the basic satellite,0.05 m from the top of the sphere, with an angle of 20°.

Figure 2.6: The meshed badminton geometry in the Gambit environment

15

2.4.3 FlowerThe flower geometry consists in a rotation of the side panels around the edge of the satellite’s

rear face until they are perpendicular to their initial position. A slightly inclined flap wasadded to each panel in order to determine under which conditions it could provide a significantstabilizing spin to the satellite (Figure 2.7).

Four panels with dimensions of 0.005x0.1x0.3 m are fixed to the edges of the back of thebasic satellite. The flaps are fastened to the basis of each of those panels, with an inclination of10°. Their dimensions are 0.005x0.05x0.1 m.

Figure 2.7: The meshed flower geometry in the Gambit environment

2.4.4 PlateThe plate geometry is a very conceptual representation of a parachute. It consists in a square

plate, attached with a string to the back of the satellite (Figure 2.8).The dimensions of the plate are 0.005x0.3x0.3 m. In order to facilitate the meshing, the

string is represented by a 1 m long box whose cross-sectional area is a square with a side of0.0025 m. This geometry will be studied for both a rigid link, with the plate parallel to thesatellite’s rear face, and a flexible link, with the plate perpendicular to the direction of the flow.

Figure 2.8: The meshed plate geometry in the Gambit environment

16

Chapter 3

Rarefied flow theory and modelling

Since aerodynamic drag increase is selected as de-orbiting technique, it is necessary to modeland understand the aerodynamics of the different geometries.

Rarefied flow is first introduced, followed by the theory and governing principles of thededicated modelling technique.

The parameter settings that were used for this particular study are then reviewed and jus-tified. This section is strongly recommended for a deeper understanding of the modelling tech-nique. Both the code and the parameter settings were validated with a practical test case, theApollo capsule.

The last section, finally, discusses the evolution of a few characteristic dimensionless numbersthroughout the transitional regime.

3.1 Short introduction to rarefied flowsThe Navier-Stokes equations can be derived from the Boltzmann equation as long as the devi-

ation from equilibrium of the Maxwellian distribution function is small, which is the underlyingassumption for continuum fluid dynamics. This is no longer the case for rarefied flows, for whichdeviation from equilibrium is significant. [22]

The degree of rarefaction of a gas flow, and thereby its degree of non-equilibrium, canbe determined with the Knudsen number. It is defined as the ratio between the mean freepath λ, which is the mean distance covered by a particle between successive collisions, and acharacteristic length scale L.

Kn =λ

L(3.1)

Three flow regimes can be identified from the Knudsen number. For Kn   0.01, the flow iscloser to collisional equilibrium: continuum fluid dynamics equations can be used. For Kn ¡ 10,the flow is in free-molecular regime: collisions occur scarcely, even for particles reflected awayafter hitting a surface. Between those extremes, the flow is in transitional regime.

Those limits are not clearly defined, as they rely on the definition of the characteristic lengthscale and other parameters not included in the definition of the Knudsen number, such as thefree-stream velocity. The upper limit on the Knudsen number, for example, is not sufficient toensure free-molecular regime for objects moving at very high speed [14]. However, they can beused as indications.

During its atmospheric re-entry, the satellite will experience the three types of regime: fromfree-molecular, while on orbit, till continuum as it approaches the ground (Figure 3.1). The

17

values of its aerodynamic coefficients have to be determined for those three regimes. Exper-imentation of low-density flows being both complex and expensive, a numerical approach ispreferred.

Y

Z

X

Mach

1412108642

(a)

X Y

Z

Mach

2218141062

(b)

X Y

Z

Mach

25211713951

(c)

Figure 3.1: Effect of the rarefaction on the Mach number flow field for the flower geometry,respectively at Kn = 30.2 (a), Kn = 2.14 (b) and Kn = 0.345 (c). Rarefied flows are clearlydifferent from continuous flows, in this case the bow shock is thicker and the wake is barrely

present

3.2 Direct Simulation Monte CarloSeveral approaches exist for numerical modelling of rarefied gas flows: direct Boltzmann

equation solution methodology, gas kinetic Navier-Stokes schemes, moment methods, and directsimulation Monte Carlo (DSMC). Among those, DSMC is the most widely used technique. It hasa high degree of accuracy, is conceptually simple, and is easily applicable to complex geometries.

The method was developed in the early 1960’s by Prof. Graeme Bird, from the Universityof Sydney, and has been continuously developed and improved since then. It is valid for any gasflow for which the collision cross section is smaller than the distance between atoms or molecules.Unfortunately, the memory and computing performance requirements increase much faster thanthe number of simulated particles. Therefore, it is only used for rarefied and transitional flowssimulation. Computational fluid dynamics (CFD), which is not applicable to flows with a certaindegree of non-equilibrium, is preferred for continuum flow simulation, where DSMC methodswould require too important computational power. [22]

Basically, the DSMC performs a probabilistic simulation on a limited number of simulatedparticles, each of them representing a large number of molecules or particles, in order to repro-duce the physics of the Boltzmann equation. Operations are performed in two sequences.

The particles are first moved through the computational domain, according to their velocityvector, and assigned to a new cell if necessary. Particles leaving the computational domain areremoved.

The particles are then organized into pairs and collisions are performed. Collisions betweenparticles and between particles and a surface are calculated using specified probabilistic, phe-nomenological models.

The simulation begins in vacuum, and ends when steady state conditions have been reached.When the simulation is finished, the field and surface quantities are sampled. [23]

The simulations presented in this study were performed with the Rarefied Gas DynamicsAnalysis System (RGDAS), developed by the Khristianovich Institute of Theoretical and Ap-plied Mechanics of the Siberian Branch of the Russian Academy of Sciences. It is an extended

18

version of another computational system, the Statistical Modelling In Low-Density Environment(SMILE). It is developed to solve advanced problems of high-altitude aerothermodynamics. Thecore code of the system is written in FORTRAN90, and the user interface in C++. [24]

3.3 Parameters settingsThe parameters taken by RGDAS as input are divided into two categories: pre-processing, and

processing. The pre-processing settings are themselves divided into six subsystems: chemistry,flow data, geometry and domain, starting surface, parameters of the numerical method, andparameters of remote run. The starting surface subsystem is used for cases such as thrusterplume analysis, and is thus not necessary for this study. The parameters of remote run are notused either.

3.3.1 Pre-processing parametersChemistry (kinetic models)

The variable hard sphere (VHS) molecular model is used to simulate the molecular collisions,and the Larsen-Borgnakke statistical model for the energy exchange between kinetic and internalmodes [25]. Chemical reactions were not considered. It will be demonstrated in section 4.6 thattheir influence is negligible anyway.

Flow data

The first type of flow data is the global flow parameters: angle of attack and slip angle,temperature, a speed parameter (velocity, Mach number or speed ratio) and a density parameter(mean free path, density, number density or Knudsen number).

It was decided to assume an orbital re-entry. The free-stream velocity u8 at a certain altitudeh is thus equal to the orbital velocity at that altitude:

u8 =

cG �MEarth

REarth + h(3.2)

Where G is the gravitational constant 6.673 � 10�11 m3/kg � s2, MEarth the Earth’s mass5.9722 � 1024 kg, and REarth the Earth’s mean radius 6, 378.1 km.

In reality, the atmosphere will slow the spacecraft down. However, as it will be demonstratedin section 4.6, the influence of a small variation in the free-stream velocity on a spacecraft’saerodynamic behaviour is negligible. For an accurate result, though, iteration on the speedwould be necessary.

The free-stream atmospheric parameters were set according to the Jacchia Reference Atmo-sphere for an exospheric temperature of 1, 200 K (Table 3.1). It is valid from 90 up to 2, 500 km,and includes thus the lower thermosphere. Furthermore, it is based on spacecraft drag data,which matches the field of this study. [26] As it is demonstrated in section 5.5.3, the thermo-spheric parameters vary considerably with the solar activity. The model chosen for this studyis close to the medium activity. The number density n8 and the temperature T8 are directlyused as parameters.

The second type of flow data is the flow parameters for every species: mole fraction, rotationaltemperature and vibrational temperature. The mole fractions were set according to the sameatmospheric model (Table 3.2), while the rotational and vibrational temperatures of diatomicspecies are assumed to be the same as the free-stream global temperature.

19

Table 3.1: Free-stream conditions according to [26]

Altitude (km) n8 (m�3) ρ8 (kg/m3) T8 (K)

170 2.2702 � 1016 8.7777 � 10�10 892150 5.3055 � 1016 2.1383 � 10�9 733140 9.3528 � 1016 3.8548 � 10�9 625130 1.9429 � 1017 8.2075 � 10�9 500120 5.2128 � 1017 2.2642 � 10�8 368115 9.8562 � 1017 4.3575 � 10�8 304110 2.1246 � 1018 9.6068 � 10�8 247105 5.0947 � 1018 2.3640 � 10�7 208100 1.1898 � 1019 5.5824 � 10�7 194

Table 3.2: Atmospheric composition according to [26]

Altitude (km) XN2 XO XO2

170 0.54820 0.40826 0.04354150 0.61557 0.32982 0.05461140 0.65173 0.28646 0.06181130 0.69113 0.23799 0.07089120 0.73271 0.18278 0.08451115 0.75386 0.14835 0.09779110 0.77042 0.10635 0.12323105 0.78319 0.05873 0.15808100 0.78440 0.03877 0.17683

Geometry and domain

Even if a 3D geometry editor is available in RGDAS, Gambit was preferred to draw thedifferent geometries and generate their surface mesh. The surface mesh was made out of trianglesand generated with the generic solver. Complex surfaces, such as the frontal spherical section,were refined for a better resolution. The geometries were fully described in section 2.4.

The gas-surface interactions are assumed to be completely diffusive, without sticking, andwith full energy accommodation. The surface is assumed to be non-catalytic with the cold-wallapproximation imposing a temperature of Tw = 300 K.

The optimal size of the domain depends on the Mach number at its boundaries. It reliesmainly on the size and shape of the spacecraft, and on the flow regime.

Two numerical criteria must be respected in order to ensure an accurate simulation [22]. First,the mean distance between collisions, often referred to as mean collision separation, must besmaller than the local mean free path. Since RGDAS performs grid adaptation, this criterion ismet by setting a correct initial cell size and by allowing a large enough maximum cell numberand maximum cell division if needed. Depending on the size of the domain, the initial numberof cells will be chosen so that the initial cell dimension in the direction of the free-stream ∆x isless than the mean free path in the free-stream λ8:

∆x   λ8 (3.3)

A rough estimate of the mean free path is performed to set the initial cell size. The volumicdensity ρ8 is used to estimate the pressure p8 (equation 3.4), using the ideal gas approximationand considering the specific gas constant of air (equation 3.5).

20

p8 = Rair � ρ8 � T8 (3.4)

Rair =R

Mair(3.5)

Mair =¸i

Xi �Mi (3.6)

Where R is the ideal gas constant and Mair the molar mass of the mixture, defined accordingto equation 3.6 as the molar mass of each of the constituent Mi weighed according to its molefraction Xi.

Assuming a Maxwell distribution for the velocities of the particles, the mean free path isthen calculated:

λ8 =kB � T8?

2 � π � d2N2� p8

(3.7)

Where the collision diameter dN2 is that of di-nitrogen, which is the major constituent inthe lower thermosphere (Table 3.2), and kB is the Boltzmann constant.

After a few pre-processing computation steps, RGDAS will give as output the exact value ofthe mean free path. If needed, the estimate made previously on the initial cell size upper limitis then corrected. As shown in Table 3.3, the estimate was close enough to the output to beused with a conservative security factor of 0.6.

Table 3.3: Approximated mean free path verification

Altitude (km) λ8 (m) λ8RGDAS (m)

170 94.0717 99.9150150 38.6163 38.6920140 21.4209 20.5210130 10.0607 9.0623120 3.6469 3.0239115 1.8950 1.4926110 0.8595 0.6410105 0.3493 0.2488100 0.1479 0.1034

Numerical parameters

The second numerical criterion is the time-step interval, which should be smaller than themean collision time. This is because the operations of particle movement and particle collisionare decoupled during each time-step. Since RGDAS does not perform local time-step adaptation,a global time-step interval must be carefully defined.

The global time-step is equal to the time it would take for a free-stream particle to cover thedistance equal to the initial cell dimension in the direction of the free-stream ∆x, consideringits thermal speed in addition to the free-stream velocity (equation 3.8). This ensures that aparticle never crosses more than one cell in a single time-step.

Such a definition of the time-step is enough for the entire domain. Indeed, the cell dimensionbecomes smaller when grid adaptation is performed, what only occurs in regions where the localKnudsen number is smaller. In those regions, the velocity of the flow is also significantly reducedanyway. Their ratio remains thus more or less the same.

21

∆t   ∆x

u8 + 3 � ?2 �Rair � T8(3.8)

The number of processors and the maximum number of particles per processor also have to befixed. The simulations were performed at the von Karman Institute, most of them on a publiccomputer and a few test cases on the Beowulf cluster. The public machine used is equipped withfour Dual Core AMD Opteron Processor 280. Maximum one million particles per processorswere allowed for every simulation, in order to limit memory usage.

The number of particles per collision cell was set equal to 8, or 16 for cases close to free-molecular regimes. Those numbers are large enough to keep a reasonable ratio between simula-tion particles and real particles.

The other parameters concern the grid adaptation. Preliminary results showed that gridadaptation is only necessary for altitudes lower than 120 km. For those cases, the possible cellnumber growth was set equal to 5, and the maximum level of collision cell division was set equalto 64. This means that the length of a cell can be reduced by a factor of 4 in every direction ifneeded.

It became necessary, at an altitude of 100 km, to force cell division in order to respectthe first numerical criterion (equation 3.3). This was done by setting the minimum number ofparticles allowed to be in a newly split cell equal to 4, in the processing parameters.

3.3.2 Note on the Fnum parameterThe number of real particles represented by each simulated particle Fnum (equation 3.9) is an

important parameter for DSMC as it is representative of how close the simulated flow is to thereal flow. It is very difficult to estimate it in advance, as it depends on many other parameterssuch as the cells dimension, the number of particles per cell, the allowed level of grid adaptation,and the maximum number of particles per processor.

Fnum =Nreal

Nsimulated(3.9)

The number of simulated particles is usually in the order of � 105 � � � 106, which is severalorders of magnitude below the number of real particles. The lower limit of the number of realparticles Nrealmin

is the product between particle number density and domain volume (equation3.10). Denser regions will cause the number of real particles to increase.

Nrealmin= Vtot � n8 (3.10)

3.3.3 Processing parametersThe processing part is divided into two main phases. The first phase will let the flow establish

itself, without sampling the macro-parameters. The number of steps of the first phase MACSshould thus be at least larger than the number of steps needed for a particle to cross the domain’slength L in free-stream conditions (equation 3.11). Ideally, the number of steps of the first phaseshould be one or two orders of magnitude greater (� 500 � � � 5, 000 steps).

MACS ¡ L/u8∆t

(3.11)

22

The second phase is the sampling of the macro-parameters. The number of sampling steps canbe very different depending on the goal of the simulation - more steps are performed to obtainsmall quantities, such as the pitch moment on a small satellite at very high altitude (� 5, 000, 000steps), than for a general picture of the Mach number flow field (� 300, 000 steps). Figure 3.2illustrates the convergence of the drag and pitch moment coefficients on the basic geometry foran angle of attack of 15° at an altitude of 150 km. The sampling phase is stopped when theresults have clearly converged.

0 1 2 3 4 5

x 106

−0.05

0

0.05

Processing steps

Pitc

h m

omen

t coe

ffici

ent

0 1 2 3 4 5

x 106

3.8

3.85

3.9

Dra

g co

effic

ient

CdCm

Figure 3.2: Practical example of the convergence of the drag and pitch moment coefficient -basic geometry for an angle of attack of 15° at an altitude of 150 km. More than 5 � 106 steps

are necessary for the drag coefficient to converge

3.4 Results quality evaluationRGDAS also allows for the visualization of the flow fields for several parameters, some of

which can be useful when checking the quality of the computations.

The Mach number flow field is used to check if the domain is well defined. The Mach numberupstream should be very close to the free-stream Mach number, so that most of the frontalbow shock is included. This condition, however, is difficult to fulfil at higher altitudes, closeto the free-molecular regime, for which the thickness of the bow shock is much bigger than thedimension of the satellite itself (Figure 3.1(a)).

A supersonic flow downstream is a sufficient condition for hypersonic flows. The entire wakeis thus not included.

The initial cell size, and maximum cell splitting, is verified with a flow field representing thelocal mean free path which is compared to the local cell dimension. A rephrasing of equation3.3 indicates that this parameter should be greater than one on the entire domain:

λ

∆x¥ 1 (3.12)

23

The number of particles in each cell Ncell can be used to check if the cell splitting was correctlyperformed.

The time-step, finally, is checked by comparing the local speed of the flow to the speed neededto cover the local cell dimension in one time-step. Again, a rephrasing of equation 3.8 indicatesthat that parameter should be smaller than one on the entire domain:

(|ux|+ c) � ∆t∆x

¤ 1 (3.13)

Where c is the thermal speed.

3.5 UncertaintiesWhen all the parameters are set correctly and the verifications have been performed, three

remaining sources of uncertainties on the output can be identified:

• Deviations from the models and hypotheses used. The free-stream velocity, for example,will be different from the orbital velocity as discussed in section 3.3.1.

• Errors inherent to the DSMC method and its implementation in RGDAS.

• Statistical scatter on the results due to the Monte Carlo method. The DSMC methodallows for its quantification, and RGDAS returns it as output. This uncertainty is usuallyquite small, and only becomes significant for small values, such as the pitch moment on asmall satellite at very high altitude. It will be graphically represented when needed.

3.6 Method validation: Apollo re-entry capsule test caseMoss et al. at NASA Langley Research Center, conducted a study on the aerodynamics of

the Apollo capsule in rarefied conditions [27]. Their results were used to validate the methodexposed in the previous sections. The atmospheric model, molecular collision model, energyexchange model, and surface properties assumptions used for the simulations presented herewere inspired from their research. Their study was performed using DSMC simulation with theDS3V program of Bird [28]. No specific information was provided about the domain’s size. Thecode and the domain are thus the only differences between their study and the validation of themethod presented here.

The results are shown in Figures 3.3. The considered points correspond to altitudes of 100,110, 120, 130, 140, 150, and 170 km. The agreement with the results from Moss et al. is good,and could probably be even better if more sampling steps were performed.

The evolution of the coefficients along the transitional regime corresponds to a pattern oftenencountered for simple shaped bodies [29] [30] [31] and other re-entry vehicles such as the Expertmodule [32]: a plateau for continuum regime, another plateau for the free-molecular regime, anda smooth transition between both, that can be approximated with a bridging function [32] [33].However, in this case the continuum regime plateau is not reached yet.

3.7 Transitional regime characteristicsWhen studying the aerothermodynamics of an object, the two crucial parameters are free-

stream density ρ8 and velocity u8. Indeed, the most important properties of the incoming fluid

24

10−4

10−2

100

102

104

1.4

1.5

1.6

1.7

1.8

1.9

2

Knudsen number

Dra

g co

effic

ient

Moss et al.validation

(a)

10−4

10−2

100

102

104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Knudsen number

Lift

coef

ficie

nt

Moss et al.validation

(b)

Figure 3.3: Validation of the drag (a) and lift (b) coefficients with the Apollo test case

can be estimated based on those parameters [34]: its mass flux ρ8 � u8, its momentum fluxρ8 � u28, and its energy flux 1

2 � ρ8 � u38. Two dimensionless numbers describe those parameters:the Knudsen number for density, and the Mach number for velocity. Their evolution throughthe upper part of the atmosphere for this study is represented in Figure 3.4, together with theReynolds number.

100 110 120 130 140 150 160 17010

−2

10−1

100

101

102

103

Tra

nsiti

onal

reg

ime

char

acte

ristic

s

Altitude (km)

Knudsen numberReynolds numberMach numberMach number Apollo

Figure 3.4: Characteristic dimensionless numbers evolution through the transitional regime forthe QB50 re-entry satellite. The Mach number of the Apollo test case is also represented

There is no generally accepted guideline on how the characteristic length scale should bechosen. Some authors recommend to consider a local length scale related to the physics of theflow field, such as the shock thickness [22]. However, in that case, it would vary continually withthe altitude. It was thus decided to choose a length scale based on the geometry, which is moregeneral. When choosing the characteristic length to study the Reynolds number of a flow overan aircraft, the length of the vehicle itself is often used. The reference is thus the length of the

25

satellite Lref = 0.3 m.

The flow evolves from free-molecular at high altitudes down the transitional regime withoutclearly reaching continuum. This observation will be confirmed by the simulations, although thefree-molecular regime limit seems to be rather around Kn = 100.

A vehicle is considered to be evolving at hypersonic speed when it passes Mach 5. AboveMach 10, one speaks of high-hypersonic speed. With the parameters of the simulation, the flowregime is high-hypersonic for both the Apollo module, simulated at a constant speed of 9, 600m/s, and for the re-entry satellite, simulated at orbital speed. At those velocities, thermalconsiderations become the driving design requirement, hence the need for a thermal protectionsystem (TPS), and an effective stabilization system to keep it facing the flow. The heat shield’sshape and materials of the QB50 re-entry satellite were chosen for their ability to reduce theheat dissipated in the satellite, by allowing the ablation of the shield and increasing the heattransfer to the atmosphere [35].

The Reynolds number also varies widely. Even if the regime is high-hypersonic, the verylow atmospheric density gives only little inertia to the fluid at high altitudes, hence the smallReynolds number. The Knudsen, Mach, and Reynolds numbers are directly linked through thevon Karman relation:

Re =Ma

Kn�cγπ

2(3.14)

26

Chapter 4

Aerodynamic coefficients database

The objective of this chapter is to present and discuss the aerodynamic coefficients databasesthat were obtained with the method described in chapter 3 for the geometries presented insection 2.4. Those databases will be used in chapter 5 to compute the dynamic evolution of thesatellite’s flight parameters between those altitudes.

Explanations regarding the requirements on the coefficients are first given. This study beinglimited to three-degree-of-freedom, only the drag and lift coefficient and the pitch momentcoefficient are considered.

An overview of the main aerodynamic characteristics of all the geometries for a regime ofKn � 100 is presented. That first step is conducted in terms of force and moments rather thancoefficients, allowing for direct comparison between the geometries. It is not the case in thenext sections, where it is the understanding of the aerodynamic behaviour of the geometry thatmatters.

A complete analysis in terms of aerodynamic coefficients of the three remaining geometriesis then presented, for several angles of attack and altitudes ranging from 170 km down to 100km. More investigations are conducted on special features, such as the frontal spherical sectionof the basic satellite, the flaps of the flower geometry, and some geometrical parameters of theplate geometry.

4.1 Requirements on the coefficientsThe aerodynamic forces and moment coefficient considered for this study are represented in

Figure 4.1. They are taken in the same referential as the flow: the positive drag force is alignedwith the velocity vector but pointing in the opposite direction. The positive pitch moment andangle of attack are counter-clockwise.

Figure 4.1: Aerodynamic forces and moment acting on the satellite

27

4.1.1 Drag and lift coefficientsThe drag and lift coefficients CD and CL are respectively defined according to equations 4.1

and 4.2.

CD =D

12ρ8u

28Aref

(4.1)

CL =L

12ρ8u

28Aref

(4.2)

Where D and L are the drag and the lift forces, and Aref a reference area. The reference areais defined as the satellite’s cross-sectional area when it has an angle of attack of 0°. Therefore,Aref is defined respectively for the basic, the badminton, the flower, and the plate geometry as0.010 m2, 0.051 m2, 0.150 m2 and 0.090 m2.

As specified in section 2.1, one of the objective for the de-orbiting system is to provide a rapidde-orbiting. The drag coefficient should be as high as possible, in order to increase to drag forceand thereby provide a large reduction of the speed and thus short re-entry duration.

The satellite is not designed as a glider, a waverider, or anything similar, and there are thusno special requirements on the lift coefficient. However, a negative lift coefficient may reducethe de-orbiting time by pulling the satellite down to the Earth.

4.1.2 Pitch moment coefficientThe pitch moment coefficient CM is defined according to equation 4.3.

CM =Mz

12ρ8u

28LrefAref

(4.3)

Where Mz is the pitch moment.

The pitch of a flying object is at equilibrium for an angle of attack at which its pitch momentcoefficient is equal to zero CM = 0. This equilibrium is stable if the partial derivative of thepitch moment coefficient with respect to the angle of attack is negative around that point, alsocalled trim point:

CMα =BCM

Bα   0 (4.4)

While trying to stabilize the satellite during its re-entry, the goal is to keep it aligned withits velocity vector. This ensures that the heat shield is facing the incoming flow and the antennapointing towards outer space. There should thus be a stable trim point for an angle of attack of0°. The ideal magnitude of the slope around that point will depend on dynamic considerations.

4.2 Aerodynamics for Kn � 100

The aerodynamics of the four geometries were compared at an altitude of 150 km, whichcorresponds to a regime of Kn = 129. That altitude was chosen for two reasons. First, it isincluded in the system’s operation range. Second, it corresponds to the very beginning of thetransitional regime, and therefore computation time is short enough to allow simulations formultiple angles of attack.

28

The simulated angles of attack were 0, 5, 15, 30, 45, 65, 90, 135, and 180°. The range wasextended to an angle of attack of 180° in order to have an idea of the aerodynamics of the reversegeometries (e.g. what if the side-panels were deployed in front of the satellite instead of at itsrear for the flower geometry?). More angles of attack were simulated close to 0°, providing abetter precision around the satellite’s nominal position. Due to the symmetry of the geometries,there is no need to simulate negative angles of attack.

Although the output of RGDAS is available in terms of coefficients, it was preferred in thissection to translate it in terms of force and moment to allow for direct comparison. It is notthe case of the next section, where it is the understanding of the aerodynamic behaviour of thegeometry that matters.

The results for the drag force and pitch moment are presented in Figures 4.2. Due to thesymmetry of the geometries, the drag force from 0° to �180° is obtained with an orthogonalsymmetry of all the points with respect to the 0° axis. Continuity requires thus a zero slope at0° and 180° for the drag force. The pitch moment from 0° to �180° is obtained with a centralsymmetry of all the points with respect to the origin.

0 50 100 1500

0.004

0.008

0.012

0.016

0.02

Angle of attack (deg)

Dra

g fo

rce

(N)

basicflowerplate (rigid link)badminton

(a)

0 50 100 150

−0.008

−0.006

−0.004

−0.002

−0.00

0.001

Angle of attack (deg)

Pitc

h m

omen

t (N

m)

basicflowerplate (rigid link)badminton

(b)

Figure 4.2: Drag force (a) and pitch moment (b) acting on the satellite for various geometriesand angles of attack for Kn = 129

4.2.1 Drag forceAs it can be concluded from Figure 4.2(a), the flower geometry is clearly the one providing the

most important drag force. This is not surprising: it is also the one having the most importantdrag area, followed by the plate. However, if the system’s efficiency is defined as the dragcoefficient for a certain drag area, a quick similarity study shows that the plate may be moreefficient than the flower for all the angles of attack (equation 4.5). The same conclusion will bedrawn in section 4.5.1 for small angles of attack in the entire transitional regime.

Cdplate

Arefflower

Arefplate

¡ Cdflower(4.5)

Furthermore, the present data were obtained for a rigid link (e.g. a boom) between thesatellite and the plate. Under certain conditions, a flexible link (e.g. a cable) would allow theplate to remain perpendicular to the flow, always providing the same considerable drag area. In

29

that case, though, all the drag force acting on the plate would not directly result in drag forceacting on the satellite. Part of it would reinforce the pitch moment with a magnitude dependingon the angle of attack.

It is interesting to note how the drag force acting on the plate geometry first increases withthe angle of attack, and then decreases after having reached a maximum around 20°. The plateis thus still in the wake generated by the satellite, and becomes visible to the flow only after acertain angle of attack. As it will be confirmed in section 4.5.1, a longer rope would maximizethe drag effect of the plate at this altitude.

The similar values of the drag force for an angle of attack of 0° or 180° indicates that theflow regime is de facto close to free-molecular. Indeed, the cross-sectional area in this case isthe most important parameter, regardless of the shape of the satellite.

4.2.2 Pitch momentAs it can be seen in Figure 4.2(b), all the geometries have two trim points: one stable at 0°,

which means that they all meet the requirement on the pitch moment, and one unstable at 180°.The badminton and the basic are clearly the less interesting geometries.

For every angle of attack, the plate is the geometry generating the most important pitchmoment. Again, this result has to be interpreted carefully, as a rigid link was considered. Thelever arm for the resulting force applying on the plate is thus the entire distance between theplate and the satellite’s centre of gravity.

For a flexible link, that lever arm would be reduced to the distance between the satellite’scentre of gravity and the point of attachment of the flexible link, which is always the same nomatter what the length of the link is. The generated moment is thus much smaller than for arigid link. If questions of mass and volume are not taken into account, the optimal length for aflexible link is just long enough to ensure that the plate is out of the satellite’s wake.

4.2.3 Preliminary conclusionsThe basic and the badminton geometries are obviously not interesting for this objective.

Furthermore, all the geometries present an unstable trim point for an angle of attack of 180°:the reverse geometries are thus not interesting either. Nevertheless, the basic geometry providesa reference from which to judge the efficiency of the different geometries. Only the basic, flowerand plate geometries will thus be further studied.

The practical feasibility of a rigid link is questionable. It was thus decided to pursue theanalysis with a flexible link. Their aerodynamic characteristics should not be too different forsmall angles of attack anyway.

It is not necessary to study the selected geometries for every angles of attack, as the satelliteis expected to oscillate around its nominal orientation, at an angle of attack of 0°. Linearityis assumed around the nominal position for small angles of attack for all the aerodynamiccoefficients. The different geometries will thus be studied for 0° and 15°, at altitudes of 170,which correspond to the free-molecular regime, 150, 140, 130, 120, 115, 110, 105 and 100 km,which is the limit of DSMC. It corresponds to a major section of the transitional regime, fromKn = 333 down to Kn = 0.35.

Besides, the altitude of 100 km is known as the von Karman line, which marks the frontierbetween the Earth’s atmosphere and outer space according to the Fédération AéronautiqueInternationale (FAI). Below 100 km, the distance covered by the satellite in terms of latitude isnegligible anyway, as it can be seen in Figure 2.1(b).

30

4.3 Basic geometry in transitional regimeThe evolution of the aerodynamic coefficients along the transitional regime for the basic

geometry can be found in Figures 4.3 and 4.4. The lift and pitch moment coefficients are onlyshown for an angle of attack of 15°, as they are null for 0°.

On both the drag and the pitch moment coefficients, a peak is present around Kn = 1, whichcorresponds to an altitude around 110 km.

That non-monotonic behaviour may be surprising. Indeed, most of the cases studied in theliterature present the same profile as for the Apollo module: a plateau for continuum regime,another plateau for the free-molecular regime, and a smooth transition between both (see section3.6).

Up to now, experimental and numerical investigations on the evolution of the aerodynamiccoefficients in the transition flow regimes are related to simple shape bodies such as spheres,blunt plates, disks, wedges, cylinders, cones, etc. However, the non-monotonic behaviour of theaerodynamic characteristics was found in experimental [36] [37] and numerical DSMC investi-gations [38] [39] [40] for some cases. The general behaviour of certain results obtained for thedrag coefficient of cylinders and plates in particular present the same kind of peak as the onefound for the drag coefficient basic geometry (see for example Figure 4.7).

The peak on the pitch moment coefficient causes the satellite to be unstable in the lowerpart of the transitional regime, as it will be shown in section 5.4.1. The stabilization systemshould thus imperatively be operating at those altitudes as a small perturbation may result ina dramatic divergence of the satellite’s angle of attack.

The irregularities on the curves describing the lift and pitch moment coefficients, whose mag-nitudes are smaller, give an idea of the typical uncertainties on the results that were discussedin section 3.5.

4.3.1 Influence of the spherical sectionA comparative study of the satellite without and with its frontal spherical section (Figure 4.8)

is performed in order to understand its role in the aerodynamics of the basic geometry. Theresults of that study, conducted for an altitude of 150 km, which corresponds to a regime ofKn = 129, are shown in Figures 4.5 and 4.6.

As shown in Figures 4.5 , the drag and lift coefficients are slightly more important withoutthe frontal spherical section, although they are very close to the one with the frontal sphericalsection, especially for values around 0° and 180°.

On the contrary, the effect on the pitch moment coefficient is significant. As shown in Figure4.6(a), the pitch moment coefficient of the basic geometry is negative for every angle of attackexcept 0° for which it is null. The satellite’s nominal position is thus a stable trim point. Theexact behaviour of the pitch moment coefficient without the frontal spherical section, zoomed inFigure 4.6(b), cannot be determined, as the statistical scatter is too important on such a smallvalue. However, one can easily see that the satellite is unstable for small values of the angleof attack and conclude with a certain degree of confidence that the frontal spherical sectionprovides a passive stabilization effect for small values of the angle of attack in rarefied regime.However, that stabilizing effect is negligible when it is compared to the one provided by theother geometries, as shown in Figure 4.2(b).

31

10−2

100

102

104

106

2.4

2.7

3

3.3

3.6

3.9

4.2

4.4

Knudsen number

Dra

g co

effic

ient

angle of attack: 0°

angle of attack: 15°

(a)

10−2

100

102

104

106

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Knudsen number

Lift

coef

ficie

nt

(b)

Figure 4.3: Drag (a) and lift (b) coefficients of the basic geometry in the transitional regime

10−2

100

102

104

106

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Knudsen number

Pitc

h m

omen

t coe

ffici

ent

angle of attack: 15°Cm = 0

Figure 4.4: Pitch moment coefficient of the basic geometry in the transitional regime

32

0 50 100 1502.5

3.5

4.5

5.5

6.5

7

Angle of attack (°)

Dra

g co

effic

ient

basic (with spherical section)box (without spherical section)

(a)

0 50 100 150−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Angle of attack (°)

Lift

coef

ficie

nt

basic (with spherical section)box (without spherical section)

(b)

Figure 4.5: Influence of the spherical section on the drag (a) and lift (b) coefficients of thebasic geometry at an altitude of 150 km for various angles of attack

0 50 100 150

−0.16

−0.12

−0.08

−0.04

0

0.02

Angle of attack (°)

Pitc

h m

omen

t coe

ffici

ent

basic (with spherical section)box (without spherical section)Cm = 0

(a)

0 50 100 150−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Angle of attack (°)

Pitc

h m

omen

t coe

ffici

ent

box (without spherical section)Cm = 0

(b)

Figure 4.6: Influence of the spherical section on the pitch moment coefficient for the basicgeometry at an altitude of 150 km for various angles of attack (a), and zoom on pitch moment

acting on the box (b)

33

Figure 4.7: Drag coefficient of a plate in the transitional regime in air at M8 = 10 and α = 0,Figure from [31], with experimental data from [36] and free-molecular data from [41]

Figure 4.8: The satellite without (box) and with (basic) its frontal spherical section have thesame external dimensions, the frontal spherical section is cut in the body of the satellite

In free-molecular regime, the cross-sectional area is the governing factor for the aerodynamicsof a spacecraft. This explains why the drag coefficient without the frontal spherical section isslightly more important, with a peak around 90°. Since this spherical section is cut into thebody of the satellite, it slightly reduces its cross-sectional area, especially for angles of attackaround 90°.

Without the frontal spherical section, the satellite is completely symmetrical. As the centreof gravity is in its middle, the fractions of cross-sectional resulting in positive and negativepitch moment are the same for every angle of attack. Therefore, there should be no pitchmoment in free-molecular regime. It appears that the modification of the flow in nearly free-molecular regime, for Kn = 129, is enough to provide a very small pitch moment, hence the smallpitch moment coefficient. The addition of the frontal spherical section breaks that symmetry,significantly enough to provide a stabilizing pitch moment for the basic geometry.

4.4 Flower geometry in transitional regimeThe evolution of the aerodynamic coefficients along the transitional regime for the flower

geometry can be found in Figures 4.9 and 4.10. The lift and pitch moment coefficients are onlyshown for an angle of attack of 15°, as they are null for 0°.

Again, a peak is clearly visible for the drag and the pitch moment coefficients around Kn = 10.This time, however, the peak on the pitch moment coefficient is negative, while it was positivefor the basic geometry. The nominal angle of attack of the flower geometry is thus a stable trimpoint for the entire transitional regime.

In this particular case, the concave geometry of the satellite may reinforce the non-monotonicbehaviour found for the basic geometry. Indeed, significant and unpredictable interference effectsmay exist between the body of the satellite and the deployed panels. [42] Those effects were

34

10−2

100

102

104

106

1.8

2

2.2

2.4

2.6

2.8

Knudsen number

Dra

g co

effic

ient

angle of attack: 0°angle of attack: 15°

(a)

10−2

100

102

104

106

−0.1

−0.08

−0.06

−0.04

−0.02

Knudsen number

Lift

coef

ficie

nt

(b)

Figure 4.9: Drag (a) and lift (b) coefficients of the flower geometry in the transitional regime

10−2

100

102

104

106

−0.32

−0.3

−0.28

−0.26

−0.24

−0.22

Knudsen number

Pitc

h m

omen

t coe

ffici

ent

(a)

10−2

100

102

104

106

−2

0

2

4

6

8

10x 10

−3

Knudsen number

Rol

l mom

ent c

oeffi

cien

t

angle of attack: 0°angle of attack: 15°

(b)

Figure 4.10: Pitch (a) and roll (b) moment coefficients of the flower geometry in thetransitional regime

35

already identified for specific cases such as side-by-side plates or torus (see for example Figure4.11).

Figure 4.11: Aerodynamic of side-by-side plates in the transitional regime at M8 = 10, Figurefrom [31]. The filled symbols are for the drag coefficient and the empty symbols for the liftcoefficient, l are for H = 0.25L, ♢ for H = 0.5L, △ for H = 0.75L, and l for H = 1.25L,

where H is the distance between the plane of symmetry and each plate’s inside surface and Ltheir length

4.4.1 Influence of the flapsDespite their small dimensions (see section 2.4.3 for the entire geometrical description), the

flaps create a certain roll moment coefficient. That coefficient increases as the altitude decreases.Again, a peak is clearly visible around Kn = 10 for an angle of attack of 15°. Nevertheless, itremains two orders of magnitude below the pitch moment coefficient.

Spinning the spacecraft around an axis will force it to hold that axis by gyroscopic effect. Inthis case, the flaps will create a torque on the satellite, forcing it to spin around the axis parallelto its velocity vector for its nominal position. No matter how small it is, that spin-stabilizationwill have a beneficial effect, enhancing the aerodynamic stabilization.

4.5 Plate geometry in transitional regimeThe evolution of the aerodynamic coefficients along the transitional regime for the plate

geometry can be found in Figures 4.13(a) and 4.14. The simulations were performed for aflexible link between the satellite and the plate. The plate remains thus perpendicular to theflow, no matter what the angle of attack is, as illustrated in Figure 4.12. Once more, the pitchmoment coefficient is only shown for an angle of attack of 15°, as it is null for 0°.

Figure 4.12: Schematic view of the contribution of the plate on the pitch moment

36

Assuming the body of the satellite itself and the plate do not interfere in each other’s pressurefield, it is possible to decouple the total drag coefficient CD in two terms: a term CB

D due to thebody of the satellite , and a term CP

D due to the plate itself. The first term corresponds to thedrag coefficient obtained for the basic geometry in section 4.3. Therefore:

CPD = CD � CB

D (4.6)

The magnitude of the drag force acting on the plate is assumed to be greater than the oneacting on the body of the satellite. That assumption will be verified in section 6.3. The flexiblelink is thus tight. The drag force acting on the plate is therefore directly applying on the pointof attachment of the link on the body of the satellite while the lift force, which is negligibleanyway, is not transmitted. The point of attachment is assumed to be a single point in themiddle of the satellite’s rear face. The drag force has the same orientation with respect to thesatellite’s main axis as the link. For the satellite’s nominal position, the drag force due to theplate is thus perpendicular to the surface. When there is a certain angle of attack, the drag forcemakes a certain angle with the surface and creates a certain pitch moment. The force coefficientCPN corresponding to its fraction which is parallel to the surface, and thereby perpendicular to

the vector drawn between the centre of gravity and the point of attachment, is:

CPN = sinα � (CD � CB

D) (4.7)

The pitch moment coefficient CM is then obtained with equation 4.8, where 0.15 m is thelever arm: the distance between the centre of gravity and the point of attachment.

CM =0.15

Lref� sinα � (CD � CB

D) (4.8)

The hypothesis made on the decoupling between the two pressure fields is valid for continuumhypersonic regime if the link is long enough, but its validity has to be assessed for the rarefiedregime. Therefore, the pressure field of the entire plate geometry (plate and satellite) is comparedto the pressure field of the entire geometry minus the pressure field of the basic geometry (platealone). From the results, available in appendix A, it can be concluded that the hypothesis isvalid for the lowest part of the transitional regime, but is questionable for highly rarefied regime.

Another method would have been to run simulations for the plate on its own. That approachwould not have allowed for direct visualisation of the pressure fields and would have requiredmore DSMC simulations.

The behaviour of the drag coefficient is similar to the one of the flower geometry. The peakmay be caused by interferences between the satellite’s rear face and the plate.

Even if the values of the drag coefficient are smaller than for the flower, a quick similaritystudy shows that the plate geometry might generate more drag than the flower geometry. Indeed,as for the study for a regime of Kn � 100, equation 4.5 is verified for both angles of attack of0° and 15° at every altitude. That assumption will be partially verified in the next section.

With the hypothesis of decoupled pressure fields, the lift coefficient of the plate geometry isexactly the same as for the basic geometry. Indeed, the flexible link is not transmitting the forcetangential to its length. The lift force acting on the plate would have been negligible anyway,as it is perpendicular to the flow direction.

The behaviour of the pitch moment coefficient is also similar to the one of the flower geometry.The nominal angle of attack is thus a stable trim point throughout the entire transitional regime.

37

10−2

100

102

104

106

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Knudsen number

Dra

g co

effic

ient

angle of attack: 0°angle of attack: 15°

(a)

10−2

100

102

104

106

1.5

2.5

3.5

4.5

5.5

Knudsen number

Dra

g co

effic

ient

plate geometrywith a 2m long linkwith a 0.15m2 plate

(b)

Figure 4.13: Drag coefficient of the plate geometry in the transitional regime (a), andcomparison with different geometrical parameters with the nominal angle of attack (b)

10−2

100

102

104

106

−0.31

−0.28

−0.25

−0.22

−0.19

−0.16

Knudsen number

Pitc

h m

omen

t coe

ffici

ent

Figure 4.14: Pitch moment coefficient of the plate geometry in the transitional regime

38

4.5.1 Influence of the length of the link and the size of the plateNew simulations were conducted on particular points of the transitional regime (Kn = 0.345,

Kn = 2.14, Kn = 30.2, and Kn = 129) for a flexible link of 2 m instead of 1 m and a platewith a square area of 0.150 m2, the same as the cross-sectional area of the flower geometry. Thisallows not only for the study of the influence of those geometrical parameters, but also for thecomparison of the flower and the plate geometries with respect to the same reference area.

The results are shown in Figure 4.13(b). The simulations were only performed for the nominalangle of attack. The comparison is thus only based on the drag coefficient, although it can givea qualitative idea on the pitch moment coefficient. The same reference area Aref = 0.09 m2 isused for the three cases.

The behaviour of the drag coefficient for the plate with a longer link seems to get closer towhat is observed for simple shape bodies, although a small peak remains, similar to what wasobserved for the basic geometry. The longer distance between the satellite and the plate itselfis thus attenuating the interference effect between the plate and the satellite’s rear face.

Moreover, the value of the drag coefficient is greater with a longer link, except for the peak.This is easily explained, as the satellite’s wake has less influence at a longer distance. It isconfirmed by the smaller difference for the near-continuum region, where the wake’s length isreduced anyway.

Both the bigger plate and the longer link are interesting, although the enhancement obtainedwith the bigger plate is more visible. Before comparing it with the flower geometry, though,the results obtained for the bigger plate need to be normalized with the same reference areaAref = 0.150 m2, what is not done in Figure 4.13(b).

Table 4.1: Compared drag coefficients of the flower and the plate geometries with the samedrag area Adrag = 0.150 m2

Kn CDplateCDflower

0.34 1.7194 1.93992.14 3.0949 2.755830.2 2.1828 2.1668129 2.2123 2.1239

From the results, summarized in Table 4.1, it can be concluded that the plate geometryconcept is slightly more interesting than the flower geometry for the highly rarefied regime.This is not the case anymore when the regime evolves towards continuum. At Kn = 0.345, theflower geometry is more interesting.

Nevertheless, the considerable drag coefficient increase obtained with only a slightly greaterplate area offers the promise of even better results for bigger areas.

The different values of the drag coefficient for the peak may be explained with Figure 4.15,where the number of collisions per particle per second for the plate geometry in the peak is repre-sented for the three geometrical variations. The bigger plate generates visibly more interactionsbetween particles.

4.6 Influence of key parametersTwo strong hypotheses were made in regards to the parameter settings: that the satellite

would be evolving at orbital velocity, and that no chemistry would be involved. This may

39

Z

X

Y

# collisions / particle / s250022001900160013001000700

(a)

Z

X

Y

# collisions / particle / s

26002400200016001200800

(b)

Z

X

Y

# collisions / particle / s250022001900160013001000700

(c)

Figure 4.15: Effect of the geometrical variations on the number of collisions per particle persecond for the plate geometry in the peak (Kn = 2.14): nominal case (a), 0.150 m2 plate (b),

and 2 m long link (c)

introduce an error in the aerodynamic behaviour of the satellite, especially at low altitudeswhere speed reduction is the greatest and chemical reactions most likely to occur due to theimportant aerodynamic heating. The plate geometry with a drag area of 0.150m2 was simulatedat an altitude of 100 km with an angle of attack of 0°, which corresponds to Kn = 0.345, forthree different free-stream velocities and with chemistry.

6000 6500 7000 7500 80002.4

2.42

2.44

2.46

2.48

2.5

Free−stream velocity (m/s)

Dra

g co

effic

ient

without chemistrywith chemistry

Figure 4.16: Drag coefficient of the plate geometry with a 0.150 m2 drag area at Kn = 0.345for various free-stream velocities

As shown in Figure 4.16, the drag coefficient slowly increases with the free-stream velocity.However, this speed reduction during re-entry will only be in the order of 150 m/s, and so theeffect on the final result is not significant.

These results confirm what studies on the effect of free-stream velocity performed on theApollo module [27] and on the Shuttle Orbiter [43] have shown - that the aerodynamic coeffi-cients’ behaviour is similar for increasing velocity and increasing rarefaction.

40

There is a visible difference between the results when chemistry is considered and when it isnot. That confirms the hypothesis that the aerodynamic heating at 100 km is already importantenough to cause chemical reaction which can have an effect on the drag coefficient. Nevertheless,that difference is relatively small.

41

Chapter 5

Dynamic study and stability analysis

Because of the dimensions of the satellite, the altitudes considered, and the high-hypersonicregime, this mission is a very particular case that has not been analysed yet in the literature.Therefore, a specific dynamic model has to be elaborated in order to further study the perfor-mance of the different geometries. The model developed in this chapter allows for the analysisof the evolution of the satellite’s flight parameters, such as altitude, angle of attack, velocity orangle of incidence. It will also allow for a deeper understanding of the real influence of eachvariable used in the equations of motion that govern the re-entry at high altitudes.

The first section describes the dynamic model that will be used. The second section explainsits development and the steps that led to its application. The effects of damping are then studied,before discussing the results that were obtained when applying the model to the aerodynamicdatabases generated for the basic, the flower and the plate geometries, in the fourth section.The chapter ends with an analysis of the influence of some key parameters.

5.1 Re-entry modellingThis dynamic model uses Matlab and Simulink, coupled with the aerodynamic coefficients

databases, to solve numerically a three-degree-of-freedom system describing the satellite’s mo-tion.

5.1.1 Equations of motionThe vectorial form of the sum of forces and moment on the satellite gives the following

equations of motion [44] (illustrated in Figures 5.1 and 5.2):$'&'%mBvBt = �Fg er +D eD + L eL

BωBt =

Mz

Jzzez

(5.1)

Where the forces and moment are defined as:

Fg =G �MEarth

|r|2 m (5.2)

D = ρ(h) � CD(h, α) � v2

2�Aref (5.3)

L = ρ(h) � CL(h, α) � v2

2�Aref (5.4)

Mz = ρ(h) � v2

2�Aref � Lref � (Cα

M (h, α) + CMω(h, α) � ω) (5.5)

42

Figure 5.1: Graphic definition of the variables used in the equations of motion, referential andforces

Figure 5.2: Graphic definition of the variables used in the moment equation, moment andangles

The first equation of the system 5.1 is the sum of forces. It describes the temporal variationof the satellite’s velocity, m being its mass. The first term of the right hand side represents thegravitational force Fg. The Earth is assumed to be stationary and perfectly spherical and therelative motion of the atmosphere is neglected. The second and third terms are respectively thedrag and lift forces described in equations 5.3 and 5.4 with the conventions of Figure 5.1, asdescribed in section 4.1. This time, though, it is not the flow speed u8 that is considered butthe satellite’s v.

The second equation of system 5.1 describes the variation of the angular velocity due to themoment Mz acting on the satellite. Jzz is the inertia term following the z-axis and ω the angularvelocity. The integration of this angular velocity gives the angle of attack α, which is the anglebetween the direction of the satellite’s velocity and its main axis as shown in Figure 5.2.

The pitch moment coefficient CM described in chapter 4 is only valid when studying thestatic stability. In this chapter, it will include a damping term. To avoid any confusion thisstatic pitch moment coefficient is now noted Cα

M . The expression of the total moment includesthus now two terms. The first one implies Cα

M , which is the linear part, given by the DSMCstatic study. The second one implies the damping coefficient CMω, and is therefore proportionalto the angular velocity. It will be calculated analytically in section 5.3.

43

The angle of incidence i, also defined in Figure 5.2, is the angle between absolute velocity andthe line parallel to the ground. It corresponds to the variation in the satellite’s direction withthe azimuthal direction, and can be found knowing that i = ϕ+ψ. Those angles are defined as:

ϕ = arctan yx

ψ = arctan vxvy

(5.6)

The components of the absolute velocity are determined using the angle of incidence (equations5.7). The radial velocity vradial points to the Earth’s centre and is responsible for the diminutionof the altitude, while the azimuthal velocity vazimuthal is always parallel to the ground.

vradial = v � sin i vazimuthal = v � cos i (5.7)

For circular orbits around the Earth, the angle of incidence is null and the velocity is onlyazimuthal. When the drag influence is considered, the satellite begins to decay and its velocitybecomes partially radial, while its angle of incidence takes a positive value.

5.1.2 AlgorithmThe main steps of the algorithm are illustrated in Figure 5.3. The aim of the dynamic model

is to couple the system 5.1 to the aerodynamic coefficients database, in order to be able toconsider their variation with the altitude and angle of attack.

A loop that calculates the altitude and the angle of attack for every time-step is created.Given the variables of position, velocities, angle of attack and angular velocity at a certainmoment, it computes the altitude and scrolls through the aerodynamic coefficients databases tofind CL, CD and Cα

M , and calculates CMω by means of a Matlab routine. These coefficients arethen used to integrate the equations of motion to the next time-step, using the Simulink blocks,and obtain new values of the position, velocity, angle of attack and angular velocity.

Figure 5.3: Block scheme of the algorithm used

5.2 Program development and validationThe program is here presented according to the chronological phases that led to its final

development. It was built in three steps. The reasoning was always the same: first write thevectorial equations in their scalar form, then construct the Simulink program related to theequations, and finally validate the program using STK or test cases.

At first, the program considers only the gravitational force. Drag and lift forces are thenadded to the model. Finally the moment equation is taken into account.

Some of the Simulink block schemes are so complex that their presence in the text was notjustified. However, the reader can refer to appendix B, for more detailed views.

44

5.2.1 First step: gravitational forceOnly the gravitational force is taken into account. The goal is thus to write a program able

to simulate circular orbits around the Earth, generating sinusoid functions for the position andvelocities of the satellite.

Equations

The sum of the forces acting on the satellite is summarized as:

m � BvBt = �G �MEarth

|r|2 m er (5.8)

Rewriting this equation in a scalar form, with respect to the conventions established in Figure5.1, with:

er =

x?x2+y2y?

x2+y2

(5.9)

Gives: $'''&'''%m � BvxBt = � xa

x2 + y2� G �MEarth

x2 + y2m

m � BvyBt = � yax2 + y2

� G �MEarth

x2 + y2m

(5.10)

Construction

Those equations are now translated in Simulink. The loop created is shown in Figure 5.4.Equations 5.10 are integrated twice to get the coordinates, using a fixed-step integration method(ode8, Dormand-Prince), with a step size of 1 s.

Figure 5.4: First Simulink program, only the gravitational force is considered. Doubleintegration of the equations of motion

The centre of the program is the ”Equations of Motion” block detailed in Figure B.1. Ittranscribes equations 5.10 using basic mathematical operations defined in the Simulink Library.

45

The altitude is calculated at every time-step according to equation 5.11, converted in Simulinkin the ”Alt Calculation” block, as shown in Figure B.2:

h =ax2 + y2 �REarth (5.11)

The validation of this first step is achieved using two simple cases and STK. A circular orbitat 150 km is first simulated, and then a speed reduction at 170 km. The results are presentedin appendix C.1.

5.2.2 Second step: drag and lift termsThe drag and lift forces are now added to the model. The satellite is slowed down by the

aerodynamic forces and its altitude decays.

Equations

The first equation of system 5.1 can be written in a scalar form, using the definitions of thevectors eD and eL. The right hand side of this equation is now the sum of three forces: thegravitational force, the drag force, and the lift force:

mBvBt = �Fg er +D eD + L eL

With:

eD =

� vx?v2x+v2y

� vy?v2x+v2y

eL =

� vy?v2x+v2yvx?v2x+v2y

(5.12)

Becomes, after simplification:

$'''''&'''''%m � BvxBt = � xa

x2 + y2� G �MEarth

x2 + y2m�

bv2x + v2y

2� ρ(h) �Aref � (vx � CD(h) + vy � CL(h))

m � BvyBt = � yax2 + y2

� G �MEarth

x2 + y2m�

bv2x + v2y

2� ρ(h) �Aref � (vy � CD(h)� vx � CL(h))

(5.13)

Construction

The Matlab script is now added to the loop. It takes the altitude as input and returns thecorresponding atmospheric density, the drag, and the lift coefficients. The atmospheric modelis an interpolation of Jacchia’s model, the same as the one chosen in section 3.3.1 to generatethe aerodynamic coefficients databases. The drag and lift coefficients are stored in the script fordifferent altitudes and interpolated linearly between them.

The new elements added to the program are shown in red frames in Figure B.3. The ”Equa-tions of Motion” block needs a return of the velocities vx and vy, as well as the three coefficientsreturned by the Matlab program to be able to calculate the drag and lift terms of equations5.13.

46

Again, the equations are transcribed in Simulink using basic mathematical operations definedin the library, as illustrated in Figures B.4 and B.5.

The validation of this second step is given in appendix C.2. It consists in a free-fall from analtitude of 1 km, taking into account the gravitational and drag forces.

5.2.3 Third step: moment equationThe model is now completed in order to analyse the attitude of the satellite during the re-entry.

Equations

The equations developed until here remain the same, but the moment equation is added tothe model. Now that attitude is considered, the aerodynamic coefficients do not rely only onaltitude as previously but also on the angle of attack.

$''''''''''&''''''''''%

m � BvxBt = � xax2 + y2

� G �MEarth

x2 + y2m�

bv2x + v2y

2� ρ(h) �Aref � (vx � CD(h, α) + vy � CL(h, α))

m � BvyBt = � yax2 + y2

� G �MEarth

x2 + y2m�

bv2x + v2y

2� ρ(h) �Aref � (vy � CD(h, α)� vx � CL(h, α))

BωBt =

ρ(h)

Jzz� vx

2 + v2y2

�Aref � Lref � (CαM (h, α) + ω � CMω)

(5.14)

Construction

This second equation of system 5.1 is also integrated twice in order to obtain the value of theangle of attack, as it is shown in Figure 5.5.

Figure 5.5: Third step, the moment equation. The variables on the dashed arrows are alreadygiven by the previous version of the program

The complete Simulink model is shown in Figure B.8, and the latest additions are highlightedin red frames. The subsystems are described in Figures B.6 and B.7.

47

The Matlab program is modified, so as to take as supplementary input the angle of attackand to return the three aerodynamic coefficients, the damping coefficient and the density. Thesevalues are now stored for different altitudes and angles of attack, and interpolated linearlybetween them.

Global validation

The validation of the complete program is achieved with a free-fall from an altitude of 100km. It already provides information on the behaviour of some of the satellite’s flight parameters,such as the velocity, altitude, or angle of attack, and is thus presented in the text.

The satellite falls from the altitude of 100 km with an initial angle of attack of �80° andno initial velocity (Figure 5.6). The atmospheric model at those altitudes is assumed to beexponential: ρ(h) = 1.225 � e� h

7,600 kg/m3. The drag coefficient is constant and equal to 2, whilethe moment coefficient is abritrary chosen as proportional to the opposite of the angle of attack:CαM = �0.001 � α. There is thus a stable trim point for an angle of attack of 0°. The expression

of the moment does not include the damping term yet, CMω = 0.

Figure 5.6: Free-fall from an altitude of 100 km, with an initial angle of attack of �80°

At first, both altitude and absolute velocity, pointing in the negative y direction, quicklydecreases, (Figure 5.7(a)). However, at an altitude of 44.9 km, the atmospheric density becomestoo important and the drag term is then greater than the gravity term. The velocity begins todecrease and seems to reach an asymptote, which can be calculated analytically. Indeed, at theend of its fall, the satellite’s acceleration is assumed to be null, and:

¸F = 0 ÝÑ m � g(h) = D(h)

= ρ(h) � v2(h)

2�Aref � CD

v(h) =

d2m � g(h)

ρ(h) �Aref � CD(5.15)

As foreseen, the curve obtained in equation 5.15 is the asymptote of the absolute velocity forthe lower altitudes (Figure 5.7(b)).

The angle of attack oscillates with an increasing frequency around 0°, its stable trim point.The evolution of the attenuation of this angle is inversely proportional to the dynamic pressure:pdyn = ρ(h) � v2(h)/2. From the altitude of 100 km down to 29 km, the important increase indensity causes the dynamic pressure to increase too. As a result, the oscillations are attenuated.

48

Nevertheless, after an altitude of 29 km, the decrease in velocity becomes greater than theincrease in density, and the dynamic pressure decreases. Therefore the angle of attack diverges- at first quickly, and then more slowly (Figure 5.8).

5.3 Determination of the damping coefficient with the modified-Newtonian method

In order to get closer to reality, the pitch moment coefficient has to be completed with itsdynamic term, the damping coefficient CMω. That damping term describes the effects that theangular velocity has on the pitch moment coefficient. It could not be computed with RGDAS,which can only model steady flows. It is here obtained applying the dynamical method of themodified-Newtonian theory.

Basic geometry

The total pitch moment coefficient, including both the static and the damping components,can be obtained analytically for simple geometries by integrating the pressure coefficient over thesurface intersected by the flow (face 1 and 2 in Figure 5.9). The basic geometry is approximatedhere with a box having the external dimensions of a 3-unit CubeSat (Figure 4.8). A newreferential attached to the satellite is defined.

The primary step will be to calculate the pressure coefficient. The force acting on a surfaceincident to a hypersonic flow can be simplified by considering that the normal component ofthe momentum flux is completely transferred as a force to the surface, while the componenttangential to the surface remains unchanged [45] [46]. The development is first made for theface 1:

FN = muN = (ρ8u8 �A � cosα) � (u8 �A � cosα) = ρ8u2NA (5.16)

And the pressure coefficient is defined and simplified knowing that in hypersonic regime theincident pressure is much greater than free-stream pressure:

Cp =p� p812ρ8u

28

� FN/A12ρ8u

28

(5.17)

Replacing equation 5.16 in equation 5.17:

Cp =2u2Nu28

(5.18)

If damping is neglected, only the linear velocity is taken into account and uN = u8,N =u8 cosα is the normal velocity seen by the faces. If damping is considered, this expression ismodified with the angular velocity as depicted in Figure 5.10.

uN = u8,N + ((ω � r) � n) (5.19)

49

0 50 100 150 200 250 300 3500

100

20

40

60

80

Time (s)

Alti

tude

(km

)

0 50 100 150 200 250 300 3500

200

1000

400

600

800

Abs

olut

e V

eloc

ity (

m/s

)

velocityaltitude

957 m/s

44,9 km

(a)

051015200

50

100

150

200

Altitude (km)

Vel

ocity

(m

/s)

simulationanalytical calculation

(b)

Figure 5.7: Altitude and velocity evolution during free-fall with drag considered (a), and zoomon the last altitude with asymptotic validation (b)

020406080100−100

−80

−60

−40

−20

0

20

40

Ang

le o

f atta

ck (

°)

0204060801000

20

40

60

Altitude (km)

Dyn

amic

pre

ssur

e (1

0kg/

ms²

)

dynamic pressureangle of attack

altitude: 29 km

Figure 5.8: Dynamic pressure influencing the angle of attack’s evolution during free-fall

50

Figure 5.9: Notations used for the damping coefficient determination

Figure 5.10: Contribution of the angular velocity to the total normal velocity, face 1

With ω the angular velocity of the satellite, n the normal to the surface, and r the vectorfrom the centre of gravity to the surface element:

u8 = u8

cosαsinα0

ω =

00ω

n =

�100

r =

xy0

Equation 5.19 becomes after calculation:

uN = u8 cosα+ ωy

And is replaced in equation 5.18:

Cp =2u2Nu28

= 2 � cos2α+ 4 � cosα � y ωu8

+ 2 � y2 ω2

u28

Now that the pressure coefficient is determined for the first face, it has to be multiplied bythe lever arm and integrated over the whole face to obtain the pitch moment coefficient.

CM,1ez =1

ArefLref

»y ey � Cp ex dS

=�1

ArefLref

» l

0b Cp y ez dy

51

Where b is the width of the satellite, 0.01 m. In this case:

CM,1ez =�b

ArefLref

» l

02 cos2α y ez

+4 cosα y2 ω

u8ez

+2 y3ω2

u28

ez dy

The first term of this integral does not depend on the angular velocity ω. It corresponds to thepitch moment coefficient when damping is neglected, and was already calculated with DSMC.The second and third terms are respectively the first and second order damping coefficients, anddepend respectively on ω and ω2.

After calculation, it appears that the first and third terms are null, due to the approximationon the geometry. The difference with the result obtained with DSMC in section 4.3.1 is due tothe underlying hypotheses of modified-Newtonian method, which stipulates that the momentumflux tangential to the surface remains unchanged and that the pressure field is constant all alongthe surfaces facing the flow and null on the others.

Only the first order damping term remains:

CM,1 = � b

ArefLref� cosα � l

3

3

ω

u8(5.20)

The same development is applied to the second face intercepted by the flow, and available inappendix D. The normal velocity is now:

uN = u8 sinα� ωx

Giving again a pitch moment coefficient that only contains a damping part:

CM,2 = � b

ArefLref� sinα � L

3

3

ω

u8(5.21)

Finally, the total moment coefficient is the sum of the contributions of the two intersectedfaces:

CM = CM,1 + CM,2

=�b

3 �ArefLref� (l3 cosα+ L3 sinα) ω

u8

(5.22)

And the damping coefficient is defined as:

CMω =BCM

Bω =�b

3 �ArefLrefu8� (l3 cosα+ L3 sinα) (5.23)

For example, if α = 15°, this expression can be evaluated (u8 = 7, 806.32 m/s is taken at 170km, where the orbital velocity is the smallest):

CMω = �1.132 � 10�5 s

52

The evolution of this coefficient with regards to the angle of attack is shown in Figure 5.11for an altitude of 170 km.

0 20 40 60 80−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−5

Angle of attack (°)

Dam

ping

coe

ffici

ent (

s)(15° ; −1.32e−5 s)

Figure 5.11: Evolution of the damping coefficient with the angle of attack for the basicgeometry at an altitude of 170 km (equation 5.23)

Flower geometry

In the same way, the damping coefficient was calculated for the flower geometry, still ne-glecting the frontal spherical section. The pressure coefficient is now integrated on the six facesintersected by the flow. The reference area is now the one of the flower geometry: 0.150 m2.When the angle of attack is not null, the shadow due to the satellite itself on the upper deployedpanel has to be taken into account.

CMω = �5.4 � 10�5 s

This coefficient is almost five times greater than it was in the basic geometry, but remainsrelatively small in comparison with the other aerodynamic coefficients described in section 4.4.

Plate geometry

Because of the flexible link, only the satellite has an angular velocity while the plate remainsperpendicular to the flow. For this reason, the damping coefficient is the same as for the basicgeometry.

The damping coefficient greatly depends on the geometry of the satellite, with l and L to thecube, and on the flow velocity, with the inverse of u8. Because of the very small size of thesatellite and the importance of its velocity, this coefficient remains small in these three cases,and should not have a visible influence on the evolution of the angle of attack, as it is shown inappendix E. However, it was included in the final model.

53

It is calculated for every time-step in the Matlab program, using the angle of attack andabsolute velocity, and returned into Simulink together with the drag, lift, and atmosphericdensity coefficients.

5.4 Application to the selected geometriesNow that the dynamic model is complete and validated, the dynamic behaviour of the basic

geometry and two selected de-orbiting geometries is studied in order to assess their performances.The simulations are done starting at an altitude of 170 km, and ending at 100 km. The rangefor which the aerodynamic coefficients have been determined.

5.4.1 Basic geometryThe model developed up to here is now applied to the basic geometry. The aerodynamic

coefficients are taken in section 4.3. The inertia term for a 3-unit CubeSat around the z-axis isJzz = 0.0075 kg �m2. The same value will be considered for the next geometries.

Figure 5.12(a) proves that the angle of incidence i remains small during the re-entry, havinga maximum value of 0.2379° at 100 km. The satellite’s velocity remains almost parallel to theground. Equations 5.7 gives a radial velocity of maximum 32.46 m/s at 100 km, while azimuthalvelocity is nearly equal to the absolute velocity: 7, 813 m/s.

If the initial angle of attack is set to zero, no oscillation occurs and the satellite remains stableduring the entire re-entry. Figure 5.12(b) shows that the basic satellite without any de-orbitingmechanism is decaying down to 100 km in more than 13 hours. That corresponds to more than9 slightly elliptical orbits around the Earth, according to Figure 5.13. Again, the Earth’s radiushas been divided by 40 in the figure, in order to distinguish the different orbits of the trajectory.

If the injection angle of attack is perturbed to a value of 15°, the satellite begins to oscillatearound its nominal position, with a magnitude slowly damped over time (Figure 5.14(a)). Thedrag coefficient being much greater at 15° than at 0° (section 4.3), the satellite decays fasterwhen the angle of attack is perturbed, in only 11 hours (Figure 5.12(b)). Moreover, the positivepeak of the pitch moment coefficient that exists at an altitude of 105 km for this geometrycauses the angle of attack to diverge. At 107 km, both α and ω quickly increase. The satel-lite becomes unstable and the simulation stops. However, the angular velocity remains small,reaching maximum 0.27°/s (Figure 5.14(b)).

As shown in Figure 5.15, it is also possible to determine the difference ∆v between the absolutevelocity of the satellite and the orbital velocity for each altitude. This difference is due to thedrag force acting on the satellite and will characterize the efficiency of the de-orbiting systems.In this case it is equal to 12 m/s.

At first, the velocity increases because the gravity terms dominates the drag and lift in theequations of system 5.13. That force being proportional to the inverse of the square of thedistance to the Earth’s centre, the behaviour of the augmentation of absolute velocity dependson the diminution in altitude. Around an altitude of 115 km, and in agreement with the rapidatmospheric density increase after that point, the drag term becomes greater than the gravityterm, and the satellite slows down brutally.

54

0 2 4 6 8 10 12 140

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Time (hours)A

ltitu

de (

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initial angle of attack: 0°initial angle of attack: 15°

107.7 km

(b)

Figure 5.12: Basic geometry, evolution the angle of incidence (a) and the altitude (b) with time

satellite’s trajectoryEarth

Figure 5.13: Basic geometry, representation of the satellite’s trajectory (more than 9 orbits),the Earth’s radius has been divided by 40

55

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ngul

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ity (

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(b)

Figure 5.14: Basic geometry, evolution of angle of attack (a) and angular velocity (b)(αinit = 15°) with altitude

1001101201301401501601707805

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7840

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Altitude (km)

Abs

olut

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loci

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orbital velocitysatellite’s velocity

12 m/s

Figure 5.15: Basic geometry, evolution of ∆v with altitude (αinit = 15°)

56

5.4.2 Flower geometryCoefficients are now taken in section 4.4, and the reference area is 0.150 m2. If the initial

angle is 0°, it takes 92 min reach the altitude of 100 km (Figure 5.16), a bit more than oneorbit (Figure 5.17). Because the drag coefficient is smaller for an angle of attack of 15°, it willbe one minute longer. Thus, initial angle of attack has no major influence on the de-orbitingtime, unlike for the basic geometry.

Attenuation is much more important here than it was for the basic geometry. The perturbedinjection angle of attack is divided by six in 90 min. It oscillates between �3° and 3° at thealtitude of 100 km (Figure 5.18(a)). Moreover, the flower satellite remains stable during theentire simulation. It oscillates more than 10 times faster than the basic, ω reaching almost 4°/s(Figure 5.18(b)).

It can be observed that the rapid diminution of the magnitude of the pitch moment coefficientafter the peak at an altitude of 110 km (see section 4.4) only leads to a small divergence of pitchangle. This effect appears mostly in the angular velocity.

The ∆v is also much greater, 164 m/s, mostly due to the drag area which is now equal to0.150 m2. Figure 5.19 clearly shows that the influence of this de-orbiting mechanism becomesimportant after an altitude of 115 km.

5.4.3 Plate geometryThe reference area is now equal to the surface of the plate, 0.09 m2. The link between the

plate and the satellite being flexible, the reference length is still equal to 0.3 m. With an initialangle of attack of 0°, and because of the smaller value of both the drag area and drag coefficient,the distance covered is about one orbit and a half (5.21) in 134 min (Figure 5.20). Again,because the drag and lift coefficients are very similar at 0° and 15°, the satellite will be fallingin only 40 s less if the insertion angle of attack is set to 15°.

The angle is attenuated as shown in Figure 5.22(a), and oscillates again around 3° and �3°at the altitude of 100 km. The angular velocity is in the same order of magnitude as for theflower geometry, although it is slightly smaller (Figure 5.22(b)).

The ∆v provided here is around 115 m/s, which is lower than for the flower geometry, due tothe smaller value of the pitch moment coefficient (Figure 5.23).

57

0 20 40 60 80 100100

110

120

130

140

150

160

170

Time (minutes)

Alti

tude

(km

)

Figure 5.16: Flower geometry, evolution of the altitude with time

satellite’s trajectoryEarth

Figure 5.17: Flower geometry, representation of the satellite’s trajectory, the radius of theEarth has been divided by 40

58

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1

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3

4

Altitude (km)

Ang

ular

vel

ocity

(°/

s)

(b)

Figure 5.18: Flower geometry, evolution of angle of attack (a) and angular velocity (b)(αinit = 15°) with altitude

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satellite’s velocityorbital velocity

164 m/s

Figure 5.19: Flower geometry, evolution of ∆v with altitude

59

0 20 40 60 80 100 120 140100

110

120

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140

150

160

170

Time (minutes)

Alti

tude

(km

)

Figure 5.20: Plate geometry, evolution of the altitude with time

satellite’s trajectoryEarth

Figure 5.21: Plate geometry, representation of the satellite’s trajectory, the radius of the Earthhas been divided by 40

60

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(b)

Figure 5.22: Plate geometry, evolution of angle of attack (a) and angular velocity (b)(αinit = 15°) with altitude

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Figure 5.23: Plate geometry, evolution of ∆v with altitude

61

5.5 Influence of key parametersQuite a few parameters are not yet fixed, but only approximated. Among these is the trigger

altitude, which will probably be decided during the flight itself depending on the trajectory ofthe satellites, the mass of the satellite, which depends on its final design, and the solar activity,which depends on the exact launch date. Their influence is studied in this section.

Other parameters, such as pitch moment damping and lift, have less influence in this partic-ular case. They are discussed in appendix E, in order to understand their role.

5.5.1 Trigger altitudeAn important parameter of influence on the re-entry time is the trigger altitude. It is impor-

tant to know when the de-orbiting system must be deployed in order to keep the satellite visibleduring the end of its course. Regarding the plate geometry, Table 5.1 shows the effect that thevariation of trigger altitude has on the ∆v at 100 km, and on the time needed to cover the 50km between 150 and 100 km. The satellite evolves at orbital velocity before deployment. Theplateau of free-molecular regime is assumed to be reached at 170 km.

Table 5.1: Effects of trigger altitude

Trigger altitude 150 km 170 km 200 km

∆v100km (m/s) 114 115 116∆t150�100km (s) 3, 505 3, 250 3, 030

The first line of Table 5.1 does not vary much with the trigger altitude. This means that, inthese three cases, the satellite reaches an altitude of 100 km with more or less the same absolutevelocity.

Figure 5.24 shows that the trigger altitude has a significant effect on the radial velocityonly until 120 km, which explains the small variations in the time needed to cover the 50 lastkilometres. Nevertheless, beyond 120 km, the re-entry will occur in the same way for the threecases, having the same vradial and ∆v100km. The altitude of deployment will thus have a verylimited effect on the re-entry duration after 120 km. A higher trigger altitude will not reallyhelp to reduce the time needed to cover the last kilometres of the re-entry. It should thus ratherbe decided in function of deployment feasability and stability questions.

5.5.2 MassThe total mass of the satellite is not fixed yet. In the case of the plate geometry, it will even

vary during the flight due to the deployment of the structure.

If the first equation of system 5.1 is divided by m:

BvBt = �GM|r|2 er +

D

meD +

L

meL

It appears that the acceleration depends on the inverse of the mass through the drag and liftterms. A smaller mass will thus increase the influence of those two forces, and the satellite willdecay much faster (Figure 5.25).

The mass also plays a role in the second equation of 5.1 through calculation of the inertiaterm in:

BωBt =

Mz

Jzzez

62

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50

60

70

80

Altitude (km)

Rad

ial v

eloc

ity (

m/s

)

trigger altitude: 150kmtrigger altitude: 200kmtrigger altitude: 170km

Figure 5.24: Influence of the trigger altitude on the radial velocity of the satellite. Thevariations which look like angular points on the curve of the trigger altitude are in fact rapid

changes in the radial velocity

0 50 100 150 200100

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130

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160

170

Time (minutes)

Alti

tude

(km

)

mass: 3 kgmass: 2 kgmass: 4 kg

Figure 5.25: Influence of satellite’s mass on the de-orbiting time

Again, a smaller mass, and thereby a lower Jzz, increases the importance of the stabilizingmoment. It also means that the satellite has less time to attenuate the perturbation of the angleof attack (see Figure 5.25). The effect of the mass on the attenuation is therefore almost null.

63

5.5.3 Solar activityThe atmospheric parameters, including its density, vary considerably with the solar activity.

Figure 5.26 shows the variation observed on the evolution of the satellite’s altitude over time fordifferent choices of atmospheric models. The simulations were conducted for the plate geometry.The atmospheric model is only changed in the Simulink dynamic analysis tool, while all theaerodynamic coefficients are still the one computed with Jacchia’s atmospheric model.

As expected, the decrease of altitude is strongly influenced by solar activity, and the timenecessary to de-orbit can be almost twice as important between periods of low and high activity.However, the real effect of the solar activity will most probably be less important. Indeed,during periods of low activity the atmosphere is less dense. The flow regime is thus rarefied tilllower altitudes, and the drag coefficient remains greater.

Although the QB50 mission is supposed to be launched during a period of low solar activity,the present study was conducted with a model close to the medium activity. The Sun’s activitybeing regulated by cycles with a mean period of 11.5 years, it is more often in medium activityand the present study is thus more general. Furthermore, there are more reference points inJacchia’s atmospheric model, allowing for an important gain of precision.

0 0.5 1 1.5 2 2.5 3100

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150

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170

Time (hours)

Alti

tude

(km

)

JacchiaMSISE90 − medium activityMSISE90 − high activityMSISE90 − low activity

Figure 5.26: Influence of solar activity, through the atmospheric model chosen, on thede-orbiting time

64

Chapter 6

Geometry selection

Based on the results presented and discussed in chapters 4 and 5, it is now possible to deter-mine which among the flower and the plate geometries is the most suitable concept.

6.1 Criteria presentationThe criteria used are representative of a geometry’s ability to de-orbit and stabilize. It is good

to remind the reader that these geometries are only concepts. The final de-orbiting structurewill obviously be different in its dimensions and shape. Therefore, a descriptive parameterwhich allows for objective comparison is needed. Having no information on the complexity ofthe different geometries, or on the mass or volume they represent, or on their reliability, itwas decided to weigh them according to their drag area. Nevertheless, it should be noted thateven the drag area is not a perfect denominator as it does not result in linear variations of ageometry’s aerodynamic behaviour (see section 4.5.1).

• De-orbit:

– Time to reach an altitude of 100 km with the nominal insertion angle of attack of 0°multiplied by the drag area: to minimize,

– Time to reach an altitude of 100 km with a perturbed insertion angle of attack of 15°multiplied by the drag area: to minimize,

• Stabilize:

– Altitude at which the maximum magnitude of the angle of attack drops below 5° witha perturbed insertion angle of attack of 15° divided by the drag area: to maximize,

The system is supposed to be deployed at an altitude of 170 km.

6.2 Selection matrix

Table 6.1: Comparison of the different geometries

Criterion Basic Flower PlateAdrag �∆tnominal (s �m2) 481.3 821.4 722.3Adrag �∆tperturbed (s �m2) 392.8 830.4 717.5

(Altitude for α   5°)/Adrag (km/m2) Unstable 12.74 10.2

65

The first surprising observation is that the basic geometry seems to be the most efficient onefor what concerns de-orbiting. This is easily explained. Indeed, the entire drag area of thebasic geometry, which consists solely of its frontal spherical section, is directly exposed to thefree-stream. This is not the case for the flower geometry: the flow hitting the side panels hasalready been slowed down by the body of the satellite itself. In the same way, the plate of theplate geometry is exposed to the body of the satellite’s wake rather than to the free-stream.

From a theoretical point of view, and if the only goal is to de-orbit without stabilizing(e.g. if stabilization is ensured with another active system), the most efficient geometry is thussomething similar to the basic geometry, with an important frontal area. That frontal area,however, will be exposed to important temperatures. Therefore it needs to be designed as anheat shield, which is most probably impossible to deploy and will cause numerous additionalconstraints in terms of mass, volume, and integration.

Furthermore, in the particular case of the basic geometry it is unstable. The reverse plateand flower geometries, which are similar concepts, are also unstable as it was concluded in section4.2.3.

The de-orbiting performances of the plate and flower geometries are very similar, althoughthe plate geometry is more efficient for de-orbiting and the flower geometry the most efficientfor stabilizing.

These results of the dynamic study must not be the only criteria considered. In this regard,other advantages of the plate geometry must be highlighted:

• More degrees of freedom:De-orbiting devices designed with the plate configuration allow for a choice in regardsto the area of the plate and the length of the link. Furthermore, section 4.5.1 showedthat they both can increase the performance in both stabilization and rapid de-orbiting.Moreover, greater drag areas are more likely to be deployed with the plate geometry ratherthan with the flower geometry, which is limited by the external surface of the satellite.

• More likely to resist:The side panels of the flower are attached to the satellite by only one of their extremities,on which they create an important moment (Figure 6.1). Inversely, the link can haveseveral points of attachment on the plate, depending on the resistance needed.

• Less points of failure:The flower device is made out of four distinct mechanical parts, while the plate consists ofonly one piece. The loss of a side panel is thus a point of failure, which would cause thesatellite to be dramatically unstable. The loss of the plate would just cause the satelliteto return to its basic configuration, which is stable for small angles of attack.

For all these reasons and due to its efficiency, the plate geometry should be selected as themost suitable structure for the QB50 mission.

6.3 Guidelines for a complete systemNow the plate geometry has been selected as the most suitable drag increase concept, a few

guidelines are given for its practical implementation. They should not be considered as definitiverequirements, but as leads for a further development.

66

X Y

Z

Pressure (Pa)

9.586.553.520.5

Z

X

Y

Pressure (Pa)

7.56.55.54.53.52.51.50.5

Figure 6.1: Pressure flow field around the flower and the plate geometry for Kn = 0.345, whichcorresponds to an altitude of 100 km. The pressure on the plate is lower and more uniformly

distributed than on the panels of the flower

6.3.1 LinkThe tension force in the link with regards to the altitude is represented in Figure 6.2. The

velocities computed in section 5.4 were used instead of the orbital velocity for the conversionfrom aerodynamic coefficient to force, although their difference does not significantly influencethe general behaviour. The bump observed around 110 km corresponds to the peak in the dragcoefficient.

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1.2

1.5

1.8

Altitude (km)

Ten

stio

n fo

rce

in th

e lin

k (N

)

Figure 6.2: Tension force in the link (cable) between the plate and the satellite

The first observation is that the tension force is always positive, which means that the link isalways tight. The plate being in the satellite’s wake, this was not an obvious result. If the linkwas not tight, the effect of the plate would have been null.

67

The tension force is rather small in highly rarefied regime: only 3.8 mN at 170 km. Theforce acting on the plate itself is then 4.5 mN . The magnitude of that force on the undeployedsystem will be even smaller. This is most probably not enough to deploy it. Therefore, the plateshould be deployed at lower altitudes or using another system than just the drag force: e.g. aloaded spring could deploy the folded plane, which could be unfolded and maintained with aprestressed structure.

Although the tension force quickly increases, it barely reaches 1.7 N at an altitude of 100km. Even with a security coefficient, this is obviously within the range of most materials thatcould be used as link: typical sport ropes can approximately bear 0.1 � � � 1 kN/mm2, while steelcables used in the entertainment industry can even go up to 20 kN/mm2. Furthermore, multipleropes could be used, at least as back-up solutions.

It should be noted, though, that the tension in the link will certainly increase even fasterbelow 100 km: a gross extrapolation shows that it reaches � 10 N at 90 km, � 30 N at 80 km,and already � 80 N at 70 km. Even with those levels of tension force, and even if the surface ofthe plate is bigger, one can confidently say that pressure is not a problematic issue at the verybeginning of the re-entry.

The real concern is the temperatures through which the link will evolve: from the coldnessof space to the important heating of the re-entry. Therefore, the link should be designed for itsthermal tolerance rather than its resistance. It could be covered with a protective material chosenfor its low thermal conductivity and high melting temperature, such as fluorinated ethylenepropylene (FEP) or polytetrafluoroethylene (PTFE).

6.3.2 PlateFrom Figure 5.21, it appears that a 0.09 m2 plate is not big enough to ensure a rapid de-

orbiting. Two solutions exist: a longer link, in order to decrease the influence of the satellite’swake, or a bigger plate, in order to increase the satellite’s drag area. As shown in Figure 4.13(b),slightly increasing the drag area to 0.150 m2 is considerably more effective than doubling thelink’s length to 2 m.

Simulink simulations were performed varying the reference area, but keeping the same aero-dynamic coefficients as for the 0.09 m2 plate (Figure 6.3). This is a conservative assumption, asit can be concluded from section 4.5.1 that the drag coefficient is actually slightly more impor-tant for a 0.150 m2 than for a 0.09 m2. The ideal drag area seems to be obtained for a 1 m2

plate. Indeed, it is enough to avoid passing over the poles, and the benefit of a bigger plate isnot worth the technical complexity it would most probably cause. Furthermore, the orbit is notpolar but has an inclinaison of 79 °, the latitude covered will thus be smaller. The final resultis similar to what was announced in Figure 2.1(b).

The material used should be manufacturable in the form of foldable sheets. Again, its thermalproperties are more important than its resistance. Due to the important outer surface of theplate, a material with a high emissivity and a high melting point is a justified choice as it wouldallow the plate to radiate most of its heat.

Although the geometry was tested for square plates, another shape could be considered. Acircle, in particular, would be more advantageous as the thermal and mechanical constraintspresent at its corners would be uniformly distributed all around its perimeter. Furthermore,the plate will need a rigid structure to hold it deployed. If that structure is holding the plate’sperimeter, it will be shorter for the circle than for the square.

68

satellite’s trajectoryEarth

(a) (b)

(c)

(d)

Figure 6.3: Influence of the plate surface on the satellite’s trajectory, with Aref = 0.09 m2 (a),Aref = 0.5 m2 (b), Aref = 1 m2 (c) and Aref = 2 m2 (d)

69

Finally, the stability of the plate itself has to be ensured. The viability of solutions suchas multiple points of attachment along the perimeter, as is the case for parachutes, has to beassessed.

70

Chapter 7

Conclusion

7.1 AchievementsThe objective of this study was to conduct a preliminary design and stability analysis of a

de-orbiting system for CubeSats. We are now able to propose a drag increase concept with afew guidelines for its practical implementation. We proved its ability to de-orbit and stabilizethe QB50 re-entry satellite.

That goal was achieved through the following points:

• Chapter 2: survey of the techniques that could be used as de-orbiting systems for smallsatellites followed by a comparative analysis. The drag increase was selected as the mostsuitable technique for the particular case of the QB50 re-entry mission.

• Chapter 3: deeper understanding of the rarefied and transitional hypersonic flows, andpresentation of the DSMC method used to model them.

• Chapter 4: creation with DSMC of aerodynamic coefficients databases describing fourpossible drag increase concepts.

• Chapter 5: development of a three-degree-of-freedom Simulink program to model thedynamics of a re-entry satellite, and application to the drag increase concepts.

• Chapter 6: comparison between the different concepts based on their ability to de-orbitand to stabilize, selection of the plate geometry, and presentation of a few guidelines forits practical implementation.

Current engineering methods used to model the atmospheric re-entry, implemented in softwaresuch as STK, take into account a constant drag coefficient, most of the time approximated asequal to 2.2. Figure 7.1, based on data yielded for the basic geometry, clearly shows thatsuch a method would have led to completely different and most probably erroneous conclusions.Furthermore, those engineering methods do not even take into account questions of dynamicstability.

A semi-engineering approach would have consisted in computing the aerodynamic coeffi-cients for the continuum regime with the modified-Newtonian method and for the highly rar-efied regime with the free-molecular method, and approximate the transitional regime with abridging function. Again, this would have led to incorrect results as the non-monotonic aerody-namic behaviour of the geometries through the transitional regime was highlighted by DSMCsimulations. Furthermore, it was proven during the analytical calculation of the pitch momentcoefficient (section 5.3) that the underlying hypotheses of modified-Newtonian method were toostrong for accurate results.

71

0 200 400 600 800 1000100

110

120

130

140

150

160

170

Time (min)

Alit

ude

(km

)

drag coefficient: 2.2basic geometry

Figure 7.1: Comparison of re-entry time, between the engineering method and thecomputation of the basic geometry

7.2 PerspectivesTo further develop the de-orbiting and passive stabilization system presented in this study,

we would advise to start with the design of the deployment system. Indeed, its size and thetechnology used should not be too dependent on the choices made for the plate itself. In parallel,a survey should be conducted on the materials and manufacturing processes that could be appliedto the plate and to the link, taking into account the guidelines given in section 6.3.

Once the design step is finished, it will be possible to know the volume and weight of thedeployment system, and therefore what is left for the plate and link. Under that constraint,an optimum should then be reached regarding the geometrical parameters and materials used.At this stage, a few DSMC simulations could be run on test cases, either at high altitudes(� 170 km) to assess the system’s ability to deploy, or at lower altitudes (� 100 km) to gatherinformation on the pressures and temperatures involved.

When the design has been sufficiently developed, full characterization in the transitionalregime should be conducted with DSMC and simulations should be performed with the Simulinkprogram developed for this study. In the long term, it would be helpful to use a multi-bodydynamic system approach , which would consider the properties of materials as well as the localpressure and temperature.

Regardless of the objectives that were set for this particular study, improvements could bebrought to the tools developed, and some phenomena could be further investigated.

The results obtained for this study could be improved by calculating the real inertia matrixand damping of the different geometries, which means including the frontal spherical sectionfor the basic geometry. However, the differences with the current values of these parametersshould be small and therefore the influence of that change on the final result would most likelybe negligible.

72

The Simulink program could easily be upgraded from three-degree-of-freedom to six-degree-of-freedom, which would allow for observation of the coupling degrees of freedom, e.g. thegyroscopic effect provided by the flaps of the flower geometry. That option was not consideredfor this preliminary study as it would have been necessary for more DSMC simulations to takeinto account cases for which a certain angle of attack is coupled with a certain slip angle.However, this upgrade would be interesting for the final design of the de-orbiting system, whichwill most probably be tested both experimentally and numerically for various angles of attackand slip angles in most of the transitional regime, or for test cases which need less simulations,e.g. axisymmetrical re-entry vehicles.

Furthermore, the program was developed for a very general case: only the altitude of thesatellite is known, but there is no information about its position in terms of longitude or latitude.Therefore, general hypotheses, such as the perfectly spherical Earth with a uniform gravity field,were used. For an improved flight simulator, effects such as the Earth’s rotation, a non-uniformgravity field, atmospheric parameters’ variation over time, etc., could be included in equations5.1 and in the databases.

With those enhancements, the Simulink program used for this study would become the coreof a very powerful predictive tool for re-entry vehicles, and for the QB50 re-entry satellite inparticular.

The databases obtained regarding the different geometries could be supplemented with thecalculation of the different aerodynamic coefficients with the modified-Newtonian method in con-tinuum regime and the free-molecular method in highly rarefied regime. Those values, althoughobtained using approximate methods, would both validate the numerical results and completethe curves describing the aerodynamic coefficients with respect to the Knudsen number.

The present study was limited to the stability analysis. As it was mentioned in section 6.3,though, the surrounding temperature will be one of the major constraint for the de-orbitingsystem. Therefore, the results of the simulations performed for this study should be kept andfurther analysed.

Finally, the underlying cause of the peak within the transitional regime is an interesting phe-nomenon that should be further studied. The geometrical parameters of the different geometriescould be changed in order to isolate what causes the sudden drag increase. For the basic ge-ometry, one would change the satellite’s length or smooth the edges, and determine to whichextent these factors are influent. The surface parameters could also be analysed to determinewhere the forces are being applied in function of the rarefaction regime.

7.3 Last wordsThe research presented in this report is thus a solid base for the design of a de-orbiting system

for the QB50 re-entry satellite. The same driving ideas could be more generally applied to thedesign of de-orbiting systems for small satellites, whether it is for re-entry missions or for debrismitigation. Other interesting topics, such as the non-monotonic behaviour of the aerodynamiccoefficients in the transitional regime for particular geometries, were approached, even if theygo beyond the initial scope of this study.

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76

Appendix A

Decoupled pressure field hypothesis

The decoupled pressure field hypothesis was tested for three regimes: closest to continuumKn = 0.345, peak Kn = 2.14, and highly rarefied Kn = 333 (respectively altitudes of 100 km,110 km and 170 km), with the nominal angle of attack 0°. The pressure field of the entire plategeometry (plate and satellite) is compared to the pressure field of the entire geometry minus thepressure field of the basic geometry (plate alone).

It appears that the hypothesis is clearly valid when the flow regime is close to continuum(Figure A.1). The pressure fields are slightly coupled when the regime is more rarefied (FigureA.2), but the pressure level drops from a level of 7.5 Pa down to 0.5 Pa. The plate appearsto be the dominating geometry in both regimes. In highly rarefied regime (Figure A.3), thepressure fields are clearly coupled, although the maximum pressure level is only 0.0013 Pa.

Z

X

Y

Pressure (Pa)

7.56.55.54.53.52.51.50.5

Z

X

Y

Pressure (Pa)

7.56.55.54.53.52.51.50.5

Figure A.1: Plate geometry pressure field and difference between the plate geometry and thebasic geometry pressure field for Kn = 0.345

77

Z

X

Y

Pressure (Pa)

0.450.350.250.150.05

Z

X

Y

Pressure (Pa)

0.450.350.250.150.05

Figure A.2: Plate geometry pressure field and difference between the plate geometry and thebasic geometry pressure field for Kn = 2.14

Z

X

Y

Pressure (Pa)

0.00130.00110.00090.00070.00050.00030.0001

Z

X

Y

Pressure (Pa)

0.00130.00110.00090.00070.00050.00030.0001

Figure A.3: Plate geometry pressure field and difference between the plate geometry and thebasic geometry pressure field for Kn = 333

78

Appendix B

Simulink program constructivedetails

B.1 First stepThe Simulink blocks that translate the equation of motion are presented in Figure B.1. For

now, the variation of the satellite’s velocity is only defined by the gravitational force.

Figure B.1: First Simulink program. Inside the ”Equations of Motion” block

The altitude is computed for every time-step using the Simulink code depicted in Figure B.2.

Figure B.2: First Simulink program. Inside the ”Alt Calculation” block

79

B.2 Second stepThe complete Simulink program used for this second step is presented in Figure B.3. A return

of the velocities and a Matlab script that contains the aerodynamic database were added.

Figure B.3: Second step program: drag and lift forces added

The right hand side of the equation of motion is now the sum of three forces: the gravitationalforce, the drag force, and the lift force (B.4).

Figure B.4: Second step program. Inside the ”Equations of Motion” block

The drag and lift forces are translated in Simulink using the code of Figure B.5.

80

Figure B.5: Second step program. Inside ”Drag and Lift Terms” block

B.3 Third step: complete Simulink programThe computation of the angular acceleration is shown in Figure B.6.

Figure B.6: Complete Simulink model. Inside the ”Equations of motion” block

81

The total pitch moment acting on the satellite has now two parts, a linear one and a dampingpart, as shown in Figure B.7.

Figure B.7: Complete Simulink model. Inside the ”Moment/Jzz” block

The final Simulink program is depicted in Figure B.8. The angle of attack’s equation isintegrated twice, and a return of the angle of attack and angular velocity is added.

82

Figu

reB.

8:C

ompl

ete

Sim

ulin

km

odel

83

Appendix C

Validation of the Simulink program

C.1 First stepThe satellite starts its trajectory at x0 = 0 m and y0 = 6, 521, 000 m, which is equal to an

altitude of 150 km plus the Earth’s radius. The initial azimuthal velocity is oriented followingthe positive x axis and calculated using the definition of orbital velocity (equation 3.2): vx0 =7, 818.286 m/s and vy0 = 0 m/s.

The coordinates, velocities and altitude are evaluated over 300 min and the result is shownin Figure C.1.

0 50 100 150 200 250 300−8

−6

−4

−2

0

2

4

6

8x 10

6

Time (minutes)

Coo

rdin

ates

to e

arth

’s c

ente

r (m

)

xy

5242 s

satellite’s trajectoryEarth

Figure C.1: Orbit at 150 km, evolution of satellite’s coordinates, and altitude during 300 min

As foreseen, x, y, vx and vy behave like sinusoids while the altitude remains constant at 150km. The time taken by the satellite to describe a simple orbit is the period of these sinusoids,equal to 5, 242 s. The same period is computed with STK: 5, 241 s. These two results are closeenough to validate this first step.

Another validation is performed with STK by making the satellite fall from an altitude of 170km to the ground (Figure C.2). The initial velocity following y is not the orbital velocity, but isset to 7, 500 m/s. The lower curve is given by STK, the upper one comes from the simulation.The small difference is most probably due to effects such as the Earth’s rotation that are takeninto account in STK and not in the Simulink program, or the atmospheric model.

84

Figure C.2: Evolution of the altitude over time, when only the gravitational force is considered.The lower curve is obtained in STK while the upper one is the result of the simulation

C.2 Second stepThe validation of this second step is achieved by simulating a free-fall, represented in Figure

C.3. The satellite falls from an altitude of 1 km, with no initial velocity, down to the ground.

Figure C.3: Free-fall from an altitude of 1 km, the atmospheric density is constant

The evolution of the satellite’s velocity and position along the y-axis is shown in FigureC.4. The absolute velocity increases quickly until it reaches a plateau, which can be calculatedanalytically. Indeed, after a certain time, the satellite’s velocity remains constant and the forcesinvolved are at equilibrium:

¸F = 0 ÝÑ mg = D

= ρ � v2

2�Aref � CD

(C.1)

And considering ρ = 1, 2 kg/m3, CD = 2, m = 3 kg and Aref = 0, 01 m2:

85

v =

d2mg

ρ �Aref � CD= 49.67 m/s

Which corresponds to the plateau found in Figure C.4, and so validates this second step.

0 5 10 15 200

20

40

60

Time (s)

Abs

olut

e V

eloc

ity (

m/s

)

0 5 10 15 200

500

1000

250

750

Alti

tude

(m

)

altitudevelocity

49,6739

Figure C.4: Evolution of the absolute velocity and the altitude during a simple free-fall

86

Appendix D

Calculation of the dampingcoefficient on the second face

A new pointing vector is defined for the second face:

n =

0�10

(D.1)

And the expression of normal velocity can be determined as previously:

uN = u8 sinα� ωx

And, when replaced in equation 5.18, leads to the following expression of the pressure coeffi-cient:

Cp =2u2Nu28

= 2 � sin2α � �4 � sinα � x ω

u8�+2 � x2 ω

2

u28

� (D.2)

After replacing in the expression of the coefficient:

CM,2ez =b

ArefLref�» L

02 � sin2α � x � ez

�4 � sinα � x2 ωu8

ez

+2 � x3 ω2

u28

ez � dx

(D.3)

After calculation:

CM,2 = � b

ArefLref� sinα � L

3

3

ω

u8

With L = 0.3 m, this expression can be evaluated:

CM,2 = �9.946 � 10�6ω

87

Appendix E

Influence of the pitch momentdamping and the lift force

Those two coefficients also depend on the final design, but their influence on the dynamicbehaviour of the two selected geometries is very limited. In order to observe the effects thatcould appear with other geometries, their influence is analysed here for hypothetical cases.

E.1 DampingBy comparing the results obtained for the basic geometry in the previous section with a test

where CM,ω = 0 (Figure E.1), it appears that, in this case, damping does not have a significanteffect on the evolution of angular velocity, or angle of attack. Its value is not sufficient to preventthe instability caused by the peak at an altitude of 107 km. The Figure shows only the lastoscillations, because the effect is not even observable on the graphic before that point in time.

An observation of what the damping’s effect on the oscillations could be in a different scenariois performed for an arbitrary constant coefficient: CMω = �0.005. This value would correspondto a bigger satellite travelling at a lower velocity.

In this hypothetical case, amplitude of the oscillations as well as the angular velocity arerapidly attenuated in comparison to the results shown in Figures 5.14(a) and 5.14(b). Evenbetter, at the point of instability previously observed, the damping keeps the angle of attackclose to zero and prevents the exponential increase of angular velocity. The satellite wouldremain stable.

E.2 Lift forceThe satellite is launched at an altitude of 150 km at orbital velocity. The drag coefficient is

constant and equal to 2. The lift coefficient is first set to 0, and then to 1 in a second test.

As shown in Figure E.3, for a null lift coefficient the altitude decreases from 150 km to 80km in 14, 510 seconds (around 4 hours, in less than 3 orbit). It appears that a lift coefficient of1 makes the satellite decay a bit more slowly (in 12 more minutes). This happens as a resultof the positive orientation of the lift force, which has a tendency to send the satellite to higheraltitudes (see Figure 4.1). Because the lift term also depends on the atmospheric density, it hasa more significant effect at lower altitudes.

88

3.75 3.8 3.85 3.9 3.95

x 104

−10

−5

0

5

10

15

Time (s)

Ang

le o

f atta

ck (

°)

with dampingwithout damping

Figure E.1: Influence of damping on angle of attack, last oscillations of basic geometry.CMω = �1, 132 � 10�5 (real case)

100110120130140150160170−15

−10

−5

0

5

10

15

Altitude (km)

Ang

le o

f atta

ck (

°)

Figure E.2: Influence of damping on angle of attack, last oscillations of basic geometry.CMω = �0, 005 (hypothetical case)

89

0 5000 10000 1500080

90

100

110

120

130

140

150

160

Time (s)

Alti

tude

(km

)

lift coefficient: 0lift coefficient: 1

14.510 s15.210 s

Figure E.3: Effect of the lift force on the evolution of altitude, CD = 2

90