Preliminary design and stability analysis of a de-orbiting system for CubeSats In the frame of the QB50 mission

Travail de ﬁn d’étude présenté en vue de l’obtention du grade d’ingénieur civil électromécanicien par Guerric DE CROMBRUGGHE Laurent MICHIELS

Promoters: Pr. Philippe Chatelain, EPL Pr. Thierry Magin, VKI Readers: Pr. Miltiadis Papalexandris, EPL Dr. Cem Ozan Asma, VKI Dr. Vladimir Pletser, ESA

June 2011

Abstract Atmospheric re-entry is a key to further developments in space exploration. The purpose of the CubeSat re-entry vehicle, currently under development in the frame of the QB50 mission initiated by the von Karman Institute, is to oﬀer a cost-eﬃcient and ﬂexible validation tool. The techniques 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 atmospheric re-entry. There are thus a certain number of problems to solve. This report focuses on the 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 drag area. Therefore, three possible geometries were designed, and their aerodynamic characteristics for the lower thermosphere (170 km down to 100 km) were obtained with DSMC steady-ﬂow simulations. Those characteristics were then used to perform a three-degree-of-freedom analysis with a Simulink tool especially developed for this study. It appears that deploying a surface perpendicular to the ﬂow downstream the satellite is the most suitable option to de-orbit and provide passive stabilization. The study was performed for a 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 geometrical parameters. 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 thermal loads. The Simulink program could also be further developed and used as a powerful predictive tool 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 his external vision on the project, Tamas Banyai and Erik Torres from the VKI for all the time they spent in explanations and bugﬁxing, Dr. Raimondo Giammanco from the VKI for his technical support on the cluster and remote connections, Pr. Vladimir Riabov from Rivier College for his explanations on the non-monotonic behaviour of aerodynamic coeﬃcients, and Dr. Mark Schoenenberger from NASA Langley for sharing his knowledge on damping coeﬃcients.

Contents Contents

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1 Introduction 1.1 The QB50 mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scope of the CubeSat re-entry mission . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives of the present study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Survey of de-orbiting techniques 2.1 Mission objectives . . . . . . . . . . . . . 2.2 De-orbiting options . . . . . . . . . . . . . 2.2.1 Propulsion . . . . . . . . . . . . . 2.2.2 Aerodynamic drag increase . . . . 2.2.3 Tethers . . . . . . . . . . . . . . . 2.3 Potential de-orbiting systems comparison 2.4 Selected drag increase geometries . . . . . 2.4.1 Basic . . . . . . . . . . . . . . . . . 2.4.2 Badminton . . . . . . . . . . . . . 2.4.3 Flower . . . . . . . . . . . . . . . . 2.4.4 Plate . . . . . . . . . . . . . . . . .

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3 Rareﬁed ﬂow theory and modelling 3.1 Short introduction to rareﬁed ﬂows . . . . . . . 3.2 Direct Simulation Monte Carlo . . . . . . . . . 3.3 Parameters settings . . . . . . . . . . . . . . . . 3.3.1 Pre-processing parameters . . . . . . . . 3.3.2 Note on the Fnum parameter . . . . . . 3.3.3 Processing parameters . . . . . . . . . . 3.4 Results quality evaluation . . . . . . . . . . . . 3.5 Uncertainties . . . . . . . . . . . . . . . . . . . 3.6 Method validation: Apollo re-entry capsule test 3.7 Transitional regime characteristics . . . . . . .

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4 Aerodynamic coeﬃcients database 4.1 Requirements on the coeﬃcients . . . . 4.1.1 Drag and lift coeﬃcients . . . . . 4.1.2 Pitch moment coeﬃcient . . . . . 4.2 Aerodynamics for Kn 100 . . . . . . . 4.2.1 Drag force . . . . . . . . . . . . . 4.2.2 Pitch moment . . . . . . . . . . . 4.2.3 Preliminary conclusions . . . . . 4.3 Basic geometry in transitional regime . . 4.3.1 Inﬂuence of the spherical section

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Flower geometry in transitional regime . 4.4.1 Inﬂuence of the ﬂaps . . . . . . . Plate geometry in transitional regime . . 4.5.1 Inﬂuence of the length of the link Inﬂuence of key parameters . . . . . . .

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5 Dynamic study and stability analysis 5.1 Re-entry modelling . . . . . . . . . . . . . . . 5.1.1 Equations of motion . . . . . . . . . . 5.1.2 Algorithm . . . . . . . . . . . . . . . . 5.2 Program development and validation . . . . 5.2.1 First step: gravitational force . . . . . 5.2.2 Second step: drag and lift terms . . . 5.2.3 Third step: moment equation . . . . . 5.3 Determination of the damping coeﬃcient with 5.4 Application to the selected geometries . . . . 5.4.1 Basic geometry . . . . . . . . . . . . . 5.4.2 Flower geometry . . . . . . . . . . . . 5.4.3 Plate geometry . . . . . . . . . . . . . 5.5 Inﬂuence of key parameters . . . . . . . . . . 5.5.1 Trigger altitude . . . . . . . . . . . . . 5.5.2 Mass . . . . . . . . . . . . . . . . . . . 5.5.3 Solar activity . . . . . . . . . . . . . .

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6 Geometry selection 6.1 Criteria presentation . . . . . . . 6.2 Selection matrix . . . . . . . . . 6.3 Guidelines for a complete system 6.3.1 Link . . . . . . . . . . . . 6.3.2 Plate . . . . . . . . . . . .

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7 Conclusion 71 7.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.3 Last words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Bibliography

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A Decoupled pressure ﬁeld hypothesis

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B Simulink program constructive details 79 B.1 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.2 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.3 Third step: complete Simulink program . . . . . . . . . . . . . . . . . . . . . . . 81 C Validation of the Simulink program 84 C.1 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 C.2 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 D Calculation of the damping coeﬃcient on the second face

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E InďŹ‚uence of the pitch moment damping and the lift force 88 E.1 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 E.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 concerns the 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 of study. Today, research is mainly conducted through computational methods, ground-based experimentation, and ﬂight data analysis and extrapolation. Only a few spacecraft, such as the European Experimental Re-entry Testbed (Expert) or the Intermediate eXperimental Vehicle (IXV), are developed as validation tools. Although they will provide a considerable amount of information to the research community, they are very costly and are developed with extended timelines. CubeSat is the name given to a standardized format of nano-satellite as deﬁned by the California Polytechnic State University and the University of Stanford. A single CubeSat unit consists in a 10x10x10 cm3 cube, weighing around 1 kg. It is typically built from oﬀ-the-shelf components, and oﬀer thus an inexpensive solution that is easier and faster to develop than ordinary satellites. Since its standardization, in 1999, hundred of academic groups, and an increasing number of private companies, have developed their own CubeSat mission to ﬂy an experiment or a radio transmitter. [1] Those small satellites are currently gaining popularity, to such an extent that the overpopulation of satellites orbiting around the Earth becomes a major concern. The current guidelines stipulate that spacecraft should stay on orbit less than 25 years after the end of their mission. This will be diﬃcult to respect, for CubeSats in particular as they are less under the inﬂuence of solar pressure and atmospheric drag bigger satellites. There is thus a strong need for 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 the bigger spacecraft, and will be able to ﬂy experiments after a shorter development period thanks to its standard platform. Furthermore, the de-orbiting technique conceived for this particular application could also be used as new debris mitigation systems.

1.1 The QB50 mission QB50 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, each separated by a few hundred kilometres. The mission will provide information about the temporal and 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 need for 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 other hand, in situ exploration with stratospheric balloons is not possible higher than 42 km, and ground based lidars and radars can maximally sense up to 105 km. The only measurements of the lower thermosphere are provided by sporadic sounding rocket launches. It is, therefore, the least 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 identical set of sensors. The remaining volume will be available for the research or technological demonstrations of a partner university, allowing the network to carry more than ﬁfty diﬀerent original payloads. On the 29th of April 2011, 69 letters of intent had been received from universities all over the world. The Université Catholique de Louvain is one of them. [3]

1.2 Scope of the CubeSat re-entry mission A 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 atmospheric re-entry. Even if they will most probably not reach the ground, they are designed to survive down to 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 atmospheric re-entry. There are thus a number of challenges, which can be divided into four categories: • De-orbiting. When referring to small satellites, de-orbiting systems are often considered as a way to reduce the lifetime of the spacecraft to less than 25 years. De-orbiting is here envisaged as 2

a way to perform a controlled re-entry over a determined geographical area. The timescale involved 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, which means 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, converted into 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 its initial altitude, and v its initial velocity. The total heat transfer is approximately 95.7 M J. It is usually considered that only a small fraction of that energy, in the order of 103 , will go into the satellite, the rest being transferred to the surrounding air [4]. That leaves 95.7 kJ to dissipate within a small volume, keeping its temperature below 70 °C to stay within the electronics’ operational range. It would be possible to raise that limit up to 150 °C if military components are used. An eﬃcient thermal management is thus necessary. • Communication. Due to the huge amount of heat to dissipate, the satellite is expected to burn in the atmosphere before having reached the ground. All the data have to be sent continuously, as it will be impossible to recover them after the ﬂight. The presence of plasma in front of the satellite’s nose will make it impossible to transmit telemetry directly down to the Earth through the atmosphere. The signal will have to be sent upward to telecommunication satellites such as the Iridium constellation, able to relay it back to mission control. Therefore, the satellite should not pass over the poles, where the telecommunication network coverage is poor. • Stability As a direct conclusion of the two last points, a stability system is needed to keep the frontal heat shield facing the incoming ﬂow and the antenna pointing towards outer space.

1.3 Objectives of the present study The 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 analysis of a de-orbiting system for the re-entry satellites of the QB50 mission. This report presents the progression that led from the systems requirements to an aerodynamic drag increase concept able to de-orbit and provide passive stabilization. The second chapter consists in a survey of the techniques that could be used to de-orbit small satellites. After a presentation of the requirements and constraints, the diﬀerent solutions are introduced and their feasibility is assessed. All of them are then compared, and the most suitable one 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 that the 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-orbiting geometries that were proposed in the second chapter, both for rareﬁed regime and for the region between rareﬁed and continuum regimes. The inﬂuence of some special design features is also discussed. In the ﬁfth chapter, a dynamic model that solves the equations of motion governing the re-entry in high altitudes is elaborated, using Simulink and Matlab. The program is built in diﬀerent steps, in order to understand the inﬂuence of each force and moment acting on the satellite. 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 or velocity, is studied. Using diﬀerent criterion from both static and dynamic analyses, the geometries are ﬁnally compared in the sixth chapter, in order to select the most eﬃcient de-orbiting structure. The results 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 CubeSat re-entry mission, in order to select the most suitable one. The objectives of the de-orbiting systems are ﬁrst deﬁned in terms of performance and constraints. The main options available are then brieﬂy examined. Focus is set on systems able to perform the 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 such as solar sails, magnetic navigation, MEMS thrusters arrays, etc., were not considered due to their obviously poor performance for this kind of manoeuvre or due to their low technology readiness level (TRL). Indeed, the QB50 mission is supposed to be launched in 2014, and one cannot aﬀord the risk to wait for technological improvements or developments. A qualitative comparison is then made between all the options, with comments on the criteria and an interpretation of the results. The best option is then selected. Finally, the three drag increase geometries that were further studied are presented, together with the geometrical characteristics of the re-entry satellite itself.

2.1 Mission objectives Strictly 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 (see section 2.4.1 for detailed information). The geometrical constraints are thus the same as for a 3-unit CubeSat. Its total mass is expected to be around 3 kg [1]. Although the mass budget is not clearly deﬁned yet, it is foreseen that the mass of the de-orbiting system should not exceed 500 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 volume budget is not clearly deﬁned yet, it is foreseen that the volume of the de-orbiting system should not exceed half a unit, 5x10x10 cm3 : its shape should also be taken into account, as it may cause 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 is expected to be only in the order of 4 W [1]. Furthermore, those solar cells will most probably be lost during the re-entry. This is a very limiting factor. Indeed, most of the valves used on the market for propulsion systems already require at least several watts to open and be kept open. Fortunately, the power requirement of the re-entry satellite while it is on orbit is quite low: minimum telemetry and possibly some periodical measurements. All the power can thus be used to charge a set of batteries with high capacity, and discharge them during the reentry 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, propellant ignition, mechanism deployment, attitude control, etc.) and then provide enough power to run the experiments while using the telemetry. Numerous battery packages are available for small satellites. The NanoPower BP-4 from Gomspace will be used as reference for this preliminary study [5]. It simply consists in four Panasonic CGR18650HG cells. Even if it is quite massive and volumic, 213 g and 23x90x96 mm3 , 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 total energy available is thus approximately 30 W h. This battery set is not the ﬁnal system, but just a representative reference to allow numeric investigation. The satellite must complete its trajectory in less than half an orbit, in order to avoid the poles were the telecommunication network coverage is poor. This objective has to be translated in terms of speed reduction ∆V in order to allow for quantiﬁcation of the mass of propellant needed for de-orbiting techniques based on propulsion. A ﬁrst estimate shows that an impulsive speed reduction of 50 m/s at an altitude of 170 km is enough to reduce the distance covered from 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 has been divided by 40 in the ﬁgure for more visibility. Although it is based on an estimate, this scenario 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 speed reduction should be as short as possible, with a lower limit set to avoid overly important accelerations on the satellites. A ∆t greater than 0.5 s guarantees accelerations lower than 30 g. The satellites could stay up to 3 months on orbit before de-orbiting, starting at an altitude of 320 km possibly down to 170 120 km. During that phase, the de-orbiting system will most probably be in sleep mode. satellite’s trajectory Earth

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(b)

Figure 2.1: Possible de-orbiting manoeuvre; a 50 m/s impulsive speed increment at an altitude 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. 6

Table 2.1: Requirements for the de-orbiting system Sleep phase (months) <3 Mass (g) < 500

Trigger altitude (km) 120 ... 300 Volume (cm3 ) < 500

∆V (m/s) > 50

Power (W ) < 12

Acceleration (g) < 30

Energy (W h) < 29

2.2 De-orbiting options The 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 Propulsion There 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 them are 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 relation [6]: mp = m0 [1 exp(

∆V )] Isp 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 surface of the Earth, 9.81 m/s2 . The duration of the burn ∆t will be estimated using the following relation [6]: mp g Isp (2.2) F 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: ∆t =

∆V = g Isp ln

m0 m0 mp

(2.3)

The subscript thruster is for the thruster itself without taking the propellant and the rest of the 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 degree of simplicity among all the propulsion systems, it beneﬁts from an important heritage in space applications, and it uses contamination-free non-toxic propellant, e.g. N2 . It has been used mainly 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 the valves. [7, 8, 9] However, the characteristically low speciﬁc impulse of cold gas thrusters results in huge amounts of propellant needed to achieve the required speed increment. Furthermore, their relatively low thrust results in a very long time to reach that speed increment. Considering the high 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 appears that the burn duration is way too long for a rapid de-orbiting. Plus, the required energy exceeds by far what is available in the batteries. Table 2.2: The Moog 58X125A characteristics mthruster (g) 9

mp (g) 226

Vthruster (cm3 ) 4.9

F (mN ) 4.4

Isp (s) 65

∆t (h) 9

Power (W ) 10

Energy (W h) 91

Nevertheless, Marotta has developed a low-power micro-thruster which ﬁts the requirements and 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) 70

mp (g) 226

F (mN ) 445

Isp (s) 65

∆t (min) 5

Power (W ) 1

Energy (W h) 0.083

Propellants such as butane and ammonia can be stored in their liquid form and will phase change into gas upon expansion. It allows for lighter and less volumic tanks, and storage at lower pressure, 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 in breadboard model [8, 9]. Its characteristics are summarized in Table 2.4. It is designed as an attitude control system, with ﬁve thrusters in total. Since the tank is integrated, its maximal speed 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) 456

Vtot (cm3 ) 250

∆V (m/s) 26

Chemical propulsion: monopropellant Hydrazine thrusters are commonly used for attitude control and to produce small speed increments. Unfortunately, none of those available commercially fulﬁls the requirements of CubeSat applications. A few research and development models are getting close to the requirement envelope, but none of them passed ﬂight qualiﬁcation. Furthermore, hydrazine requires to be handled 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 the most 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 suitable for university student groups, hence the development of hydrogen peroxide micro-thrusters for CubeSats. On the other hand, it is usually less eﬃcient than hydrazine, and subject to slow decomposition, which can lead to tank over-pressurization over time. Furthermore, none of the thrusters in development are ready for ﬂight.

8

Table 2.5: The JPL HmNT characteristics mthruster (g) 40

mp (g) 100

Vthruster (cm3 ) 8

F (mN ) 129

Isp (s) 150

∆t (min) 19

Power (W ) 8

Energy (W h) 2.45

Chemical propulsion: bipropellant Even if the bipropellant thrusters oﬀer greater speciﬁc impulse compared to their monopropellant counterparts, the cost of added complexity and dry mass is too important for CubeSat applications. [7, 8, 9] Chemical propulsion: solid Typical values of the speciﬁc impulse for small solid rocket motors reach above 250 s. The amount of propellant needed for a certain speed increment is thus lower than for any other kind of 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 an inconvenience for a de-orbiting manoeuvre. Plus, there is usually a notable uncertainty on the thrust and total speciﬁc impulse delivered. If that uncertainty is too high, an attitude control 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 excessive acceleration. [7, 8, 9] The STAR 3A, manufactured by ATK, is among the smallest thrusters available on the market [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 little room for the systems and payloads. Also, the mass of the thruster exceeds the constraints, but it includes all the systems needed - unlike the mass estimate for the other chemical propulsion systems where the mass of the tank and piping were not taken into account. Table 2.6: The ATK STAR 3A characteristics mtot (g) 890

Vtot (cm3 ) 984

F (N ) 613.85

Isp (s) 241.2

∆V (m/s) 98.24

∆t (s) 0.5

Acceleration (g) 30

The STAR 4G, also manufactured by ATK, is a so-called ’slow burner’ [12] (Figure 2.2). Its characteristics are summarized in Table 2.7. The achieved speed increment is much greater than what is needed. Table 2.7: The ATK STAR 4G characteristics mtot (kg) 1.5

Vtot (cm3 ) 755.4

F (N ) 307

Isp (s) 275.6

∆V (m/s) ¡ 1, 000

∆t (s) 10

Acceleration (g) 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, which allows 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 to accelerate the propellant. [7, 8, 9] If mass is saved on the propellant, it is at the cost of very long de-orbiting times, to be counted in tens of days, and thus huge amounts of energy. The Miniature Xenon Ion (MiXI) thruster developed by the JPL is a representative example of the status of the technology [8, 9]. Its characteristics are summarized in Table 2.8. The ﬁring time exceeds by far the de-orbiting window, and the power required is too important for the batteries. Table 2.8: The JPL MiXI thruster characteristics mp (g) 4.8

F (mN ) 1.5

Isp (s) 3, 200

∆t (days) ¡1

Power (W ) 13 50

Energy (Wh) 312

The same results would be obtained with other kinds of electric micro-thrusters. Hall eﬀect thrusters, for example, are capable of higher thrusts, up to 15 mN , at the cost of lower speciﬁc impulse, 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 heat exchanger would be added. The increased temperature of the propellant results in higher speciﬁc impulse and higher thrust. The exchanger adds complexity, mass and power requirements, especially for small thrusters. Despite their simplicity, they are thus not applicable yet to CubeSats. [7, 8, 9]

2.2.2 Aerodynamic drag increase The 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 by deploying surfaces in order to increase the satellite’s cross-sectional area. Using the deﬁnition of the ballistic coeﬃcient, which describes the sensitivity of a ﬂying object to the aerodynamic brake, one can see that increasing the area drag Adrag leads to a decreased ballistic coeﬃcient Cb , 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 spacecraft orbit, 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 control to keep angle of attack, the drag area of a 3-unit CubeSat is 10x10 cm2 . Thereby, the ballistic coeﬃcient of the satellite before the deployment of the drag increase system is Cb = 136.36 kg/m2 . The eﬀect of the drag area on the lifetime of the satellites was estimated using Satellite Tool Kit (STK) with the astrogator propagator. STK is a software able, among other things, to give an approximation of a spacecraft’s trajectory through the atmosphere, knowing its initial position, mass and drag coeﬃcient. The results were calculated for an initial orbit at an altitude of 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 the aerodynamic 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 its deployment down to a certain altitude. Table 2.9 shows that the ballistic coeﬃcient increases and the lifetime of the satellite decreases when the drag area is increased. That information is also shown in Figure 2.3. The lifetime is the 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 value of more than 14 hours to only 22 minutes. This value is expected to be suﬃcient for a quick de-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 reduction provided, based on simulations performed with STK Drag area (m2 ) Ballistic coeﬃcient (kg/m2 ) Lifetime (min) ∆v100km (m/s)

0.01 136.36 860 23

0.15 9.1 90 244

1 1.36 41 407

5 0.27 22 434

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 trajectory describes thus a decreasing elliptical orbit, most probably due to the introduction of a drag force. 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 distance covered 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 thus the 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 km down 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 on simulations performed with STK Table 2.10: Eﬃciency of a 1 m2 drag area for diﬀerent trigger altitudes for an orbit inclination of 79°, obtained with STK Initial altitude (km) 200 150 130 120

∆t70 (min) 79 29 16 10

∆Lat70 (°) 279 96 60.6 37.9

∆t50 (min) 80 29.5 16.5 11

∆Lat50 (°) 281 98 61.7 39.2

Conclusion From a theoretical point of view, de-orbiting techniques based on the aerodynamic drag increase seem to be really eﬃcient, even with moderated drag areas. Drag increase systems would consist of thin membranes deployed and maintained by a structure, 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 Tethers Electromagnetic 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 propellant or input power. Their mass and volume requirements are thus also smaller compared to other systems. On the other hand, the behaviour of tethers in space is extremely diﬃcult to predict and to control. Indeed, with a length ranging from a few meters to a few kilometres, tethers experience very 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 expensive computing time. Plus, experiments showed that unpredicted instabilities might appear. The case of electrodynamic tethers is even more complicated, since they involve electromagnetic interactions 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 order to reduce the amount of small debris in orbit. Electrodynamic tethers are thus dimensioned for de-orbiting time in the order of years. [18] The nanoTerminator developed by Tethers Unlimited is the only commercially available electrodynamic tether for small satellites [19]. The mass of its package is less than 80 g, with a volume of 0.5x8.3x10 cm3 , what is small enough to ﬁt on the face of a CubeSat. It will be tested on low orbit for the ﬁrst time on one of the QB50 satellites. Unfortunately, it would require at least 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 important instabilities. Another concept, the electrostatic tether, is still under investigation. A prototype should ﬂy in 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 comparison The 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) if it is more or less equal to the requirement, () if it is larger, and () if it is equal or larger 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 in the 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 inﬂuence on the stability can be neglected, and () if it may cause instabilities. • Applicability: this criterion deﬁnes the ease with which the technology is applied to deorbiting. The tether receives therefore a (+), as it was especially conceived for that purpose. Inversely, all the propulsion techniques receive a (). Indeed, the thrust vector needs to be aligned with the velocity vector of the satellite. It requires post-burn attitude modiﬁcation, e.g. 180° rotation if the thruster is on the opposite side of the heat shield, or complex integration, e.g. two thrusters right after the heat shield producing symmetric and synchronized thrusts under opposite angles with respect to the velocity vector. Both cases result 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 rapid de-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

Options

Mass

Volume

Power

∆t

TRL

Stability

Applicability

Criteria

Total

Cold gas Monopropellant Solid Electric Aerodynamic drag increase Tethers

0 0 0 0 +

0 0 -0 0 +

0 0 + -+ 0

0 0 + -0 --

+ + + 0 0

0 0 0 + -

0 +

0 2242+ 0

According to the total points, aerodynamic drag increase is the best option. Furthermore, it presents the following advantages: • It is an eﬃcient solution, able to de-orbit in less than one orbit with appropriate dimensioning. • It ﬁts in the requirements envelope in terms of mass and volume, and will require only little power during deployment. • A well-dimensioned system with a speciﬁc shape will ensure passive stabilization of the satellite. • It allows for several degrees of freedom during its conception (shape, dimensions, type of structure, materials, etc.) as well as during the ﬂight itself (altitude of deployment and of jettison).

2.4 Selected drag increase geometries Changing 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 increase the power available onboard (Figure 2.4). Recently, spectacular advances such as the NanoSail-D satellite, a solar sail demonstrator launched by a team of researchers from NASA Ames, proved that even the most ambitious concepts have a chance of success. However, major drawbacks have 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 force acting on the ﬁns at lower altitudes might be too high. Although they can be used as source of inspiration, 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 planned deployable mechanisms for small satellites, keeping in mind that their goal is both to increase the 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 Basic The 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 are dedicated to the systems, instrumentation and payloads, while the third one is an ablative heat shield. From a geometrical point of view, it is approximated by a box whose front part, the heat shield, is shaped as a section of a 0.15 m radius sphere. The cross-sectional area of the box is a square with 0.1 m long sides. The total length of the box, from its bottom to the top of the spherical section, is 0.3 m. For this geometry as for the next ones, the centre of gravity is assumed to be right in the middle of this basic satellite.

Figure 2.5: The meshed basic geometry in the Gambit environment

2.4.2 Badminton The ﬁrst concept is directly inspired by the legacy of past re-entry vehicle designs. The sidepanels are deployed with a certain angle, resulting in a geometry close to a typical sphere-frustum conﬁguration (Figure 2.6). Past missions and experiments show that the dynamic stability of such 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 Flower The 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 was added to each panel in order to determine under which conditions it could provide a signiﬁcant stabilizing 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 the basic satellite. The ﬂaps are fastened to the basis of each of those panels, with an inclination of 10°. Their dimensions are 0.005x0.05x0.1 m.

Figure 2.7: The meshed ﬂower geometry in the Gambit environment

2.4.4 Plate The 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 of 0.0025 m. This geometry will be studied for both a rigid link, with the plate parallel to the satellite’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 model and understand the aerodynamics of the diﬀerent geometries. Rareﬁed ﬂow is ﬁrst introduced, followed by the theory and governing principles of the dedicated modelling technique. The parameter settings that were used for this particular study are then reviewed and justiﬁed. This section is strongly recommended for a deeper understanding of the modelling technique. Both the code and the parameter settings were validated with a practical test case, the Apollo capsule. The last section, ﬁnally, discusses the evolution of a few characteristic dimensionless numbers throughout the transitional regime.

3.1 Short introduction to rareﬁed ﬂows The Navier-Stokes equations can be derived from the Boltzmann equation as long as the deviation from equilibrium of the Maxwellian distribution function is small, which is the underlying assumption for continuum ﬂuid dynamics. This is no longer the case for rareﬁed ﬂows, for which deviation from equilibrium is signiﬁcant. [22] The degree of rarefaction of a gas ﬂow, and thereby its degree of non-equilibrium, can be determined with the Knudsen number. It is deﬁned as the ratio between the mean free path λ, which is the mean distance covered by a particle between successive collisions, and a characteristic length scale L. Kn =

λ L

(3.1)

Three ﬂow regimes can be identiﬁed from the Knudsen number. For Kn 0.01, the ﬂow is closer 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 away after 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 length scale and other parameters not included in the deﬁnition of the Knudsen number, such as the free-stream velocity. The upper limit on the Knudsen number, for example, is not suﬃcient to ensure free-molecular regime for objects moving at very high speed [14]. However, they can be used as indications. During its atmospheric re-entry, the satellite will experience the three types of regime: from free-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. Experimentation of low-density ﬂows being both complex and expensive, a numerical approach is preferred. Z

X

Z

Y

X

Mach 14 12 10 8 6 4 2

(a)

Z

Y

X

Mach 22 18 14 10 6 2

(b)

Y

Mach 25 21 17 13 9 5 1

(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 clearly diﬀerent from continuous ﬂows, in this case the bow shock is thicker and the wake is barrely present

3.2 Direct Simulation Monte Carlo Several approaches exist for numerical modelling of rareﬁed gas ﬂows: direct Boltzmann equation solution methodology, gas kinetic Navier-Stokes schemes, moment methods, and direct simulation Monte Carlo (DSMC). Among those, DSMC is the most widely used technique. It has a 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 University of Sydney, and has been continuously developed and improved since then. It is valid for any gas ﬂow 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 than the number of simulated particles. Therefore, it is only used for rareﬁed and transitional ﬂows simulation. Computational ﬂuid dynamics (CFD), which is not applicable to ﬂows with a certain degree of non-equilibrium, is preferred for continuum ﬂow simulation, where DSMC methods would require too important computational power. [22] Basically, the DSMC performs a probabilistic simulation on a limited number of simulated particles, each of them representing a large number of molecules or particles, in order to reproduce the physics of the Boltzmann equation. Operations are performed in two sequences. The particles are ﬁrst moved through the computational domain, according to their velocity vector, and assigned to a new cell if necessary. Particles leaving the computational domain are removed. The particles are then organized into pairs and collisions are performed. Collisions between particles and between particles and a surface are calculated using speciﬁed probabilistic, phenomenological 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 Dynamics Analysis System (RGDAS), developed by the Khristianovich Institute of Theoretical and Applied 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. The core code of the system is written in FORTRAN90, and the user interface in C++. [24]

3.3 Parameters settings The 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, and parameters of remote run. The starting surface subsystem is used for cases such as thruster plume analysis, and is thus not necessary for this study. The parameters of remote run are not used either.

3.3.1 Pre-processing parameters Chemistry (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 internal modes [25]. Chemical reactions were not considered. It will be demonstrated in section 4.6 that their 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 altitude h is thus equal to the orbital velocity at that altitude: u8 =

cG M

Earth

REarth + h

(3.2)

Where G is the gravitational constant 6.673 1011 m3 /kg s2 , MEarth the Earth’s mass 5.9722 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 demonstrated in section 4.6, the inﬂuence of a small variation in the free-stream velocity on a spacecraft’s aerodynamic behaviour is negligible. For an accurate result, though, iteration on the speed would be necessary. The free-stream atmospheric parameters were set according to the Jacchia Reference Atmosphere 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 thermospheric parameters vary considerably with the solar activity. The model chosen for this study is close to the medium activity. The number density n8 and the temperature T8 are directly used as parameters. The second type of ﬂow data is the ﬂow parameters for every species: mole fraction, rotational temperature and vibrational temperature. The mole fractions were set according to the same atmospheric model (Table 3.2), while the rotational and vibrational temperatures of diatomic species are assumed to be the same as the free-stream global temperature. 19

Table 3.1: Free-stream conditions according to [26] Altitude (km) 170 150 140 130 120 115 110 105 100

n8 (m3 ) 2.2702 1016 5.3055 1016 9.3528 1016 1.9429 1017 5.2128 1017 9.8562 1017 2.1246 1018 5.0947 1018 1.1898 1019

ρ8 (kg/m3 ) 8.7777 1010 2.1383 109 3.8548 109 8.2075 109 2.2642 108 4.3575 108 9.6068 108 2.3640 107 5.5824 107

T8 (K) 892 733 625 500 368 304 247 208 194

Table 3.2: Atmospheric composition according to [26] Altitude (km) 170 150 140 130 120 115 110 105 100

XN2 0.54820 0.61557 0.65173 0.69113 0.73271 0.75386 0.77042 0.78319 0.78440

XO 0.40826 0.32982 0.28646 0.23799 0.18278 0.14835 0.10635 0.05873 0.03877

XO2 0.04354 0.05461 0.06181 0.07089 0.08451 0.09779 0.12323 0.15808 0.17683

Geometry and domain Even if a 3D geometry editor is available in RGDAS, Gambit was preferred to draw the diﬀerent geometries and generate their surface mesh. The surface mesh was made out of triangles and 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, and with full energy accommodation. The surface is assumed to be non-catalytic with the cold-wall approximation imposing a temperature of Tw = 300 K. The optimal size of the domain depends on the Mach number at its boundaries. It relies mainly 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 be smaller than the local mean free path. Since RGDAS performs grid adaptation, this criterion is met by setting a correct initial cell size and by allowing a large enough maximum cell number and maximum cell division if needed. Depending on the size of the domain, the initial number of cells will be chosen so that the initial cell dimension in the direction of the free-stream ∆x is less 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 volumic density ρ8 is used to estimate the pressure p8 (equation 3.4), using the ideal gas approximation and considering the speciﬁc gas constant of air (equation 3.5). 20

p8 = Rair ρ8 T8 Rair = Mair =

(3.4)

R Mair

(3.5)

Xi M i

(3.6)

¸ i

Where R is the ideal gas constant and Mair the molar mass of the mixture, deﬁned according to equation 3.6 as the molar mass of each of the constituent Mi weighed according to its mole fraction Xi . Assuming a Maxwell distribution for the velocities of the particles, the mean free path is then calculated: λ8 =

?

kB T 8 2 π d2N2 p8

(3.7)

Where the collision diameter dN2 is that of di-nitrogen, which is the major constituent in the 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 of the mean free path. If needed, the estimate made previously on the initial cell size upper limit is then corrected. As shown in Table 3.3, the estimate was close enough to the output to be used with a conservative security factor of 0.6. Table 3.3: Approximated mean free path veriﬁcation Altitude (km) 170 150 140 130 120 115 110 105 100

λ8 (m) 94.0717 38.6163 21.4209 10.0607 3.6469 1.8950 0.8595 0.3493 0.1479

λ8RGDAS (m) 99.9150 38.6920 20.5210 9.0623 3.0239 1.4926 0.6410 0.2488 0.1034

Numerical parameters The second numerical criterion is the time-step interval, which should be smaller than the mean collision time. This is because the operations of particle movement and particle collision are 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 the distance equal to the initial cell dimension in the direction of the free-stream ∆x, considering its thermal speed in addition to the free-stream velocity (equation 3.8). This ensures that a particle 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 dimension becomes smaller when grid adaptation is performed, what only occurs in regions where the local Knudsen number is smaller. In those regions, the velocity of the ﬂow is also signiﬁcantly reduced anyway. 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 be ﬁxed. The simulations were performed at the von Karman Institute, most of them on a public computer and a few test cases on the Beowulf cluster. The public machine used is equipped with four Dual Core AMD Opteron Processor 280. Maximum one million particles per processors were 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 freemolecular regimes. Those numbers are large enough to keep a reasonable ratio between simulation particles and real particles. The other parameters concern the grid adaptation. Preliminary results showed that grid adaptation is only necessary for altitudes lower than 120 km. For those cases, the possible cell number growth was set equal to 5, and the maximum level of collision cell division was set equal to 64. This means that the length of a cell can be reduced by a factor of 4 in every direction if needed. It became necessary, at an altitude of 100 km, to force cell division in order to respect the ﬁrst numerical criterion (equation 3.3). This was done by setting the minimum number of particles allowed to be in a newly split cell equal to 4, in the processing parameters.

3.3.2 Note on the Fnum parameter The 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 the real ﬂow. It is very diﬃcult to estimate it in advance, as it depends on many other parameters such as the cells dimension, the number of particles per cell, the allowed level of grid adaptation, and the maximum number of particles per processor. Nreal

Fnum =

Nsimulated

(3.9)

The number of simulated particles is usually in the order of 105 106 , which is several orders of magnitude below the number of real particles. The lower limit of the number of real particles Nrealmin is the product between particle number density and domain volume (equation 3.10). Denser regions will cause the number of real particles to increase. Nrealmin = Vtot n8

(3.10)

3.3.3 Processing parameters The 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 M ACS should thus be at least larger than the number of steps needed for a particle to cross the domain’s length L in free-stream conditions (equation 3.11). Ideally, the number of steps of the ﬁrst phase should be one or two orders of magnitude greater ( 500 5, 000 steps). M ACS

8 ¡ L/u ∆t

22

(3.11)

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

0.05

3.9

0

−0.05

3.85

0

1

2

3

Processing steps

4

5

Drag coefficient

Pitch moment coefficient

Cd Cm

3.8

6

x 10

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 evaluation RGDAS 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 number upstream should be very close to the free-stream Mach number, so that most of the frontal bow shock is included. This condition, however, is diﬃcult to fulﬁl at higher altitudes, close to the free-molecular regime, for which the thickness of the bow shock is much bigger than the dimension of the satellite itself (Figure 3.1(a)). A supersonic ﬂow downstream is a suﬃcient condition for hypersonic ﬂows. The entire wake is thus not included. The initial cell size, and maximum cell splitting, is veriﬁed with a ﬂow ﬁeld representing the local mean free path which is compared to the local cell dimension. A rephrasing of equation 3.3 indicates that this parameter should be greater than one on the entire domain: λ ∆x

¥1

23

(3.12)

The number of particles in each cell Ncell can be used to check if the cell splitting was correctly performed. The time-step, ﬁnally, is checked by comparing the local speed of the ﬂow to the speed needed to cover the local cell dimension in one time-step. Again, a rephrasing of equation 3.8 indicates that 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 Uncertainties When 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 method allows for its quantiﬁcation, and RGDAS returns it as output. This uncertainty is usually quite small, and only becomes signiﬁcant for small values, such as the pitch moment on a small satellite at very high altitude. It will be graphically represented when needed.

3.6 Method validation: Apollo re-entry capsule test case Moss 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 method exposed in the previous sections. The atmospheric model, molecular collision model, energy exchange model, and surface properties assumptions used for the simulations presented here were inspired from their research. Their study was performed using DSMC simulation with the DS3V program of Bird [28]. No speciﬁc information was provided about the domain’s size. The code and the domain are thus the only diﬀerences between their study and the validation of the method 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 often encountered for simple shaped bodies [29] [30] [31] and other re-entry vehicles such as the Expert module [32]: a plateau for continuum regime, another plateau for the free-molecular regime, and a 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 characteristics When studying the aerothermodynamics of an object, the two crucial parameters are freestream density ρ8 and velocity u8 . Indeed, the most important properties of the incoming ﬂuid 24

0.35

2

1.9

Moss et al. validation

Moss et al. validation

0.3

Lift coefficient

Drag coefficient

0.25

1.8

1.7

0.2

0.15

1.6 0.1

1.5

1.4 −4 10

0.05

−2

0

10

2

10

0 −4 10

4

10

10

−2

0

10

10

2

10

4

10

Knudsen number

Knudsen number

(a)

(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 ﬂux ρ8 u28 , and its energy ﬂux 12 ρ8 u38 . Two dimensionless numbers describe those parameters: the Knudsen number for density, and the Mach number for velocity. Their evolution through the upper part of the atmosphere for this study is represented in Figure 3.4, together with the Reynolds number. 3

Transitional regime characteristics

10

2

10

1

10

0

10

−1

10

Knudsen number Reynolds number Mach number Mach number Apollo

−2

10

100

110

120

130

140

150

160

170

Altitude (km)

Figure 3.4: Characteristic dimensionless numbers evolution through the transitional regime for the 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 be chosen. Some authors recommend to consider a local length scale related to the physics of the ﬂow ﬁeld, such as the shock thickness [22]. However, in that case, it would vary continually with the altitude. It was thus decided to choose a length scale based on the geometry, which is more general. When choosing the characteristic length to study the Reynolds number of a ﬂow over an 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 without clearly reaching continuum. This observation will be conﬁrmed by the simulations, although the free-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. Above Mach 10, one speaks of high-hypersonic speed. With the parameters of the simulation, the ﬂow regime is high-hypersonic for both the Apollo module, simulated at a constant speed of 9, 600 m/s, and for the re-entry satellite, simulated at orbital speed. At those velocities, thermal considerations become the driving design requirement, hence the need for a thermal protection system (TPS), and an eﬀective stabilization system to keep it facing the ﬂow. The heat shield’s shape and materials of the QB50 re-entry satellite were chosen for their ability to reduce the heat dissipated in the satellite, by allowing the ablation of the shield and increasing the heat transfer to the atmosphere [35]. The Reynolds number also varies widely. Even if the regime is high-hypersonic, the very low atmospheric density gives only little inertia to the ﬂuid at high altitudes, hence the small Reynolds number. The Knudsen, Mach, and Reynolds numbers are directly linked through the von Karman relation: Ma Re = Kn

26

c γπ 2

(3.14)

Chapter 4

Aerodynamic coeﬃcients database The objective of this chapter is to present and discuss the aerodynamic coeﬃcients databases that were obtained with the method described in chapter 3 for the geometries presented in section 2.4. Those databases will be used in chapter 5 to compute the dynamic evolution of the satellite’s ﬂight parameters between those altitudes. Explanations regarding the requirements on the coeﬃcients are ﬁrst given. This study being limited to three-degree-of-freedom, only the drag and lift coeﬃcient and the pitch moment coeﬃcient are considered. An overview of the main aerodynamic characteristics of all the geometries for a regime of Kn 100 is presented. That ﬁrst step is conducted in terms of force and moments rather than coeﬃcients, allowing for direct comparison between the geometries. It is not the case in the next sections, where it is the understanding of the aerodynamic behaviour of the geometry that matters. A complete analysis in terms of aerodynamic coeﬃcients of the three remaining geometries is then presented, for several angles of attack and altitudes ranging from 170 km down to 100 km. More investigations are conducted on special features, such as the frontal spherical section of the basic satellite, the ﬂaps of the ﬂower geometry, and some geometrical parameters of the plate geometry.

4.1 Requirements on the coeﬃcients The 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 aligned with the velocity vector but pointing in the opposite direction. The positive pitch moment and angle of attack are counter-clockwise.

Figure 4.1: Aerodynamic forces and moment acting on the satellite

27

4.1.1 Drag and lift coeﬃcients The drag and lift coeﬃcients CD and CL are respectively deﬁned according to equations 4.1 and 4.2. CD = CL =

D 8 u28 Aref

(4.1)

L 8 u28 Aref

(4.2)

1 2ρ

1 2ρ

Where D and L are the drag and the lift forces, and Aref a reference area. The reference area is 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 as 0.010 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 rapid de-orbiting. The drag coeﬃcient should be as high as possible, in order to increase to drag force and 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 thus no special requirements on the lift coeﬃcient. However, a negative lift coeﬃcient may reduce the de-orbiting time by pulling the satellite down to the Earth.

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

1 2ρ

8

Mz 8 Lref Aref

u2

(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 moment coeﬃcient is equal to zero CM = 0. This equilibrium is stable if the partial derivative of the pitch moment coeﬃcient with respect to the angle of attack is negative around that point, also called trim point: CM α =

B CM 0 Bα

(4.4)

While trying to stabilize the satellite during its re-entry, the goal is to keep it aligned with its velocity vector. This ensures that the heat shield is facing the incoming ﬂow and the antenna pointing towards outer space. There should thus be a stable trim point for an angle of attack of 0°. 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, which corresponds to a regime of Kn = 129. That altitude was chosen for two reasons. First, it is included in the system’s operation range. Second, it corresponds to the very beginning of the transitional regime, and therefore computation time is short enough to allow simulations for multiple angles of attack.

28

The simulated angles of attack were 0, 5, 15, 30, 45, 65, 90, 135, and 180°. The range was extended to an angle of attack of 180° in order to have an idea of the aerodynamics of the reverse geometries (e.g. what if the side-panels were deployed in front of the satellite instead of at its rear for the ﬂower geometry?). More angles of attack were simulated close to 0°, providing a better 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 this section to translate it in terms of force and moment to allow for direct comparison. It is not the case of the next section, where it is the understanding of the aerodynamic behaviour of the geometry that matters. The results for the drag force and pitch moment are presented in Figures 4.2. Due to the symmetry of the geometries, the drag force from 0° to 180° is obtained with an orthogonal symmetry of all the points with respect to the 0° axis. Continuity requires thus a zero slope at 0° and 180° for the drag force. The pitch moment from 0° to 180° is obtained with a central symmetry of all the points with respect to the origin. 0.02

0.001 basic flower plate (rigid link) badminton

Pitch moment (Nm)

Drag force (N)

0.016

−0.00

0.012

0.008

−0.002

−0.004

−0.006 basic flower plate (rigid link) badminton

0.004 −0.008

0

0

50

100

150

0

50

Angle of attack (deg)

100

150

Angle of attack (deg)

(a)

(b)

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

4.2.1 Drag force As 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 important drag area, followed by the plate. However, if the system’s eﬃciency is deﬁned as the drag coeﬃcient for a certain drag area, a quick similarity study shows that the plate may be more eﬃcient than the ﬂower for all the angles of attack (equation 4.5). The same conclusion will be drawn in section 4.5.1 for small angles of attack in the entire transitional regime. Cdplate

Areff lower Arefplate

¡ Cd

f lower

(4.5)

Furthermore, the present data were obtained for a rigid link (e.g. a boom) between the satellite and the plate. Under certain conditions, a ﬂexible link (e.g. a cable) would allow the plate 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 force acting on the satellite. Part of it would reinforce the pitch moment with a magnitude depending on the angle of attack. It is interesting to note how the drag force acting on the plate geometry ﬁrst increases with the angle of attack, and then decreases after having reached a maximum around 20°. The plate is thus still in the wake generated by the satellite, and becomes visible to the ﬂow only after a certain angle of attack. As it will be conﬁrmed in section 4.5.1, a longer rope would maximize the 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 the ﬂow regime is de facto close to free-molecular. Indeed, the cross-sectional area in this case is the most important parameter, regardless of the shape of the satellite.

4.2.2 Pitch moment As 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 pitch moment. Again, this result has to be interpreted carefully, as a rigid link was considered. The lever arm for the resulting force applying on the plate is thus the entire distance between the plate and the satellite’s centre of gravity. For a ﬂexible link, that lever arm would be reduced to the distance between the satellite’s centre of gravity and the point of attachment of the ﬂexible link, which is always the same no matter what the length of the link is. The generated moment is thus much smaller than for a rigid link. If questions of mass and volume are not taken into account, the optimal length for a ﬂexible link is just long enough to ensure that the plate is out of the satellite’s wake.

4.2.3 Preliminary conclusions The 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 provides a reference from which to judge the eﬃciency of the diﬀerent geometries. Only the basic, ﬂower and plate geometries will thus be further studied. The practical feasibility of a rigid link is questionable. It was thus decided to pursue the analysis with a ﬂexible link. Their aerodynamic characteristics should not be too diﬀerent for small angles of attack anyway. It is not necessary to study the selected geometries for every angles of attack, as the satellite is expected to oscillate around its nominal orientation, at an angle of attack of 0°. Linearity is assumed around the nominal position for small angles of attack for all the aerodynamic coeﬃcients. 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, from Kn = 333 down to Kn = 0.35. Besides, the altitude of 100 km is known as the von Karman line, which marks the frontier between the Earth’s atmosphere and outer space according to the Fédération Aéronautique Internationale (FAI). Below 100 km, the distance covered by the satellite in terms of latitude is negligible anyway, as it can be seen in Figure 2.1(b). 30

4.3 Basic geometry in transitional regime The 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 only shown 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, which corresponds to an altitude around 110 km. That non-monotonic behaviour may be surprising. Indeed, most of the cases studied in the literature 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 section 3.6). Up to now, experimental and numerical investigations on the evolution of the aerodynamic coeﬃcients 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 the aerodynamic characteristics was found in experimental [36] [37] and numerical DSMC investigations [38] [39] [40] for some cases. The general behaviour of certain results obtained for the drag coeﬃcient of cylinders and plates in particular present the same kind of peak as the one found 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 lower part of the transitional regime, as it will be shown in section 5.4.1. The stabilization system should thus imperatively be operating at those altitudes as a small perturbation may result in a dramatic divergence of the satellite’s angle of attack. The irregularities on the curves describing the lift and pitch moment coeﬃcients, whose magnitudes are smaller, give an idea of the typical uncertainties on the results that were discussed in section 3.5.

4.3.1 Inﬂuence of the spherical section A 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. The results of that study, conducted for an altitude of 150 km, which corresponds to a regime of Kn = 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 without the frontal spherical section, although they are very close to the one with the frontal spherical section, especially for values around 0° and 180°. On the contrary, the eﬀect on the pitch moment coeﬃcient is signiﬁcant. As shown in Figure 4.6(a), the pitch moment coeﬃcient of the basic geometry is negative for every angle of attack except 0° for which it is null. The satellite’s nominal position is thus a stable trim point. The exact behaviour of the pitch moment coeﬃcient without the frontal spherical section, zoomed in Figure 4.6(b), cannot be determined, as the statistical scatter is too important on such a small value. However, one can easily see that the satellite is unstable for small values of the angle of attack and conclude with a certain degree of conﬁdence that the frontal spherical section provides 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 the other geometries, as shown in Figure 4.2(b).

31

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

4.2

0.18

0.16

Lift coefficient

Drag coefficient

3.9

3.6

3.3

0.14

0.12

3

0.1

2.7

0.08

2.4 −2 10

0

10

2

4

10

0.06 −2 10

6

10

10

0

2

10

Knudsen number

10

4

10

6

10

Knudsen number

(a)

(b)

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

0.03 angle of attack: 15° Cm = 0

Pitch moment coefficient

0.02

0.01

0

−0.01

−0.02

−0.03 −2 10

0

10

2

10

4

10

6

10

Knudsen number

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

32

7

0.2

6.5

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

0.15

Lift coefficient

Drag coefficient

0.1 5.5

4.5

0.05 0 −0.05 −0.1

3.5 basic (with spherical section) box (without spherical section) 2.5

0

50

100

−0.15 −0.2

150

0

50

Angle of attack (°)

100

150

Angle of attack (°)

(a)

(b)

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

0.02

0.015 box (without spherical section) Cm = 0

0

Pitch moment coefficient

Pitch moment coefficient

0.01 −0.04

−0.08

−0.12 basic (with spherical section) box (without spherical section) Cm = 0

−0.16 0

50

100

0.005

0

−0.005

−0.01

−0.015

150

Angle of attack (°)

0

50

100

150

Angle of attack (°)

(a)

(b)

Figure 4.6: Inﬂuence of the spherical section on the pitch moment coeﬃcient for the basic geometry 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 the same 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 aerodynamics of a spacecraft. This explains why the drag coeﬃcient without the frontal spherical section is slightly more important, with a peak around 90°. Since this spherical section is cut into the body of the satellite, it slightly reduces its cross-sectional area, especially for angles of attack around 90°. Without the frontal spherical section, the satellite is completely symmetrical. As the centre of gravity is in its middle, the fractions of cross-sectional resulting in positive and negative pitch moment are the same for every angle of attack. Therefore, there should be no pitch moment in free-molecular regime. It appears that the modiﬁcation of the ﬂow in nearly freemolecular regime, for Kn = 129, is enough to provide a very small pitch moment, hence the small pitch 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 regime The 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 only shown 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 positive for the basic geometry. The nominal angle of attack of the ﬂower geometry is thus a stable trim point for the entire transitional regime. In this particular case, the concave geometry of the satellite may reinforce the non-monotonic behaviour found for the basic geometry. Indeed, signiﬁcant and unpredictable interference eﬀects may exist between the body of the satellite and the deployed panels. [42] Those eﬀects were

34

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

−0.02

−0.04

Lift coefficient

Drag coefficient

2.6

2.4

2.2

−0.08

2

1.8 −2 10

−0.06

0

10

2

10

4

10

−0.1 −2 10

6

10

0

10

Knudsen number

2

4

10

10

6

10

Knudsen number

(a)

(b)

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

−3

−0.22

10

x 10

angle of attack: 0° angle of attack: 15° 8

Roll moment coefficient

Pitch moment coefficient

−0.24

−0.26

−0.28

−0.3

−0.32 −2 10

6

4

2

0

0

10

2

10

4

10

−2 −2 10

6

10

Knudsen number

0

10

2

10

4

10

6

10

Knudsen number

(a)

(b)

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

35

already identiﬁed for speciﬁc cases such as side-by-side plates or torus (see for example Figure 4.11).

Figure 4.11: Aerodynamic of side-by-side plates in the transitional regime at M8 = 10, Figure from [31]. The ﬁlled symbols are for the drag coeﬃcient and the empty symbols for the lift coeﬃcient, 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 L their length

4.4.1 Inﬂuence of the ﬂaps Despite 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, it remains 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. In this case, the ﬂaps will create a torque on the satellite, forcing it to spin around the axis parallel to its velocity vector for its nominal position. No matter how small it is, that spin-stabilization will have a beneﬁcial eﬀect, enhancing the aerodynamic stabilization.

4.5 Plate geometry in transitional regime The 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 a ﬂexible link between the satellite and the plate. The plate remains thus perpendicular to the ﬂow, no matter what the angle of attack is, as illustrated in Figure 4.12. Once more, the pitch moment 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 pressure B due to the ﬁeld, it is possible to decouple the total drag coeﬃcient CD in two terms: a term CD P due to the plate itself. The ﬁrst term corresponds to the body of the satellite , and a term CD drag coeﬃcient obtained for the basic geometry in section 4.3. Therefore: P B CD = CD CD

(4.6)

The magnitude of the drag force acting on the plate is assumed to be greater than the one acting on the body of the satellite. That assumption will be veriﬁed in section 6.3. The ﬂexible link is thus tight. The drag force acting on the plate is therefore directly applying on the point of attachment of the link on the body of the satellite while the lift force, which is negligible anyway, is not transmitted. The point of attachment is assumed to be a single point in the middle of the satellite’s rear face. The drag force has the same orientation with respect to the satellite’s main axis as the link. For the satellite’s nominal position, the drag force due to the plate is thus perpendicular to the surface. When there is a certain angle of attack, the drag force makes a certain angle with the surface and creates a certain pitch moment. The force coeﬃcient P corresponding to its fraction which is parallel to the surface, and thereby perpendicular to CN the vector drawn between the centre of gravity and the point of attachment, is: P B CN = sin α (CD CD )

(4.7)

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

0.15 sin α (CD CDB ) Lref

(4.8)

The hypothesis made on the decoupling between the two pressure ﬁelds is valid for continuum hypersonic regime if the link is long enough, but its validity has to be assessed for the rareﬁed regime. Therefore, the pressure ﬁeld of the entire plate geometry (plate and satellite) is compared to the pressure ﬁeld of the entire geometry minus the pressure ﬁeld of the basic geometry (plate alone). From the results, available in appendix A, it can be concluded that the hypothesis is valid 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 approach would not have allowed for direct visualisation of the pressure ﬁelds and would have required more DSMC simulations. The behaviour of the drag coeﬃcient is similar to the one of the ﬂower geometry. The peak may 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 similarity study 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 of 0° 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 is exactly the same as for the basic geometry. Indeed, the ﬂexible link is not transmitting the force tangential 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

5.5

3

plate geometry with a 2m long link with a 0.15m2 plate

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

2.8

4.5

Drag coefficient

Drag coefficient

2.6

2.4

2.2

2

3.5

2.5

1.8

1.6 −2 10

0

10

2

10

4

1.5 −2 10

6

10

10

0

2

10

10

4

10

6

10

Knudsen number

Knudsen number

(a)

(b)

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

−0.16

Pitch moment coefficient

−0.19

−0.22

−0.25

−0.28

−0.31 −2 10

0

10

2

10

4

10

6

10

Knudsen number

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 plate New 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 plate with a square area of 0.150 m2 , the same as the cross-sectional area of the ﬂower geometry. This allows not only for the study of the inﬂuence of those geometrical parameters, but also for the comparison 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 nominal angle of attack. The comparison is thus only based on the drag coeﬃcient, although it can give a qualitative idea on the pitch moment coeﬃcient. The same reference area Aref = 0.09 m2 is used for the three cases. The behaviour of the drag coeﬃcient for the plate with a longer link seems to get closer to what is observed for simple shape bodies, although a small peak remains, similar to what was observed for the basic geometry. The longer distance between the satellite and the plate itself is 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 is conﬁrmed by the smaller diﬀerence for the near-continuum region, where the wake’s length is reduced anyway. Both the bigger plate and the longer link are interesting, although the enhancement obtained with 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 area Aref = 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 same drag area Adrag = 0.150 m2 Kn 0.34 2.14 30.2 129

CDplate 1.7194 3.0949 2.1828 2.2123

CDf lower 1.9399 2.7558 2.1668 2.1239

From the results, summarized in Table 4.1, it can be concluded that the plate geometry concept 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, the ﬂower geometry is more interesting. Nevertheless, the considerable drag coeﬃcient increase obtained with only a slightly greater plate 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 represented for the three geometrical variations. The bigger plate generates visibly more interactions between particles.

4.6 Inﬂuence of key parameters Two 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

X

Z

X

Y

Z

# collisions / particle / s 2500 2200 1900 1600 1300 1000 700

X

Y

Z

# collisions / particle / s 2500 2200 1900 1600 1300 1000 700

# collisions / particle / s 2600 2400 2000 1600 1200 800

(a)

Y

(b)

(c)

Figure 4.15: Eﬀect of the geometrical variations on the number of collisions per particle per second 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 altitudes where speed reduction is the greatest and chemical reactions most likely to occur due to the important aerodynamic heating. The plate geometry with a drag area of 0.150 m2 was simulated at an altitude of 100 km with an angle of attack of 0°, which corresponds to Kn = 0.345, for three diﬀerent free-stream velocities and with chemistry.

2.5 without chemistry with chemistry

Drag coefficient

2.48

2.46

2.44

2.42

2.4

6000

6500

7000

7500

8000

Free−stream velocity (m/s)

Figure 4.16: Drag coeﬃcient of the plate geometry with a 0.150 m2 drag area at Kn = 0.345 for 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 the eﬀect on the ﬁnal result is not signiﬁcant. These results conﬁrm what studies on the eﬀect of free-stream velocity performed on the Apollo module [27] and on the Shuttle Orbiter [43] have shown - that the aerodynamic coeﬃcients’ 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 is not. That conﬁrms the hypothesis that the aerodynamic heating at 100 km is already important enough 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-hypersonic regime, 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 performance of the diﬀerent geometries. The model developed in this chapter allows for the analysis of the evolution of the satellite’s ﬂight parameters, such as altitude, angle of attack, velocity or angle of incidence. It will also allow for a deeper understanding of the real inﬂuence of each variable 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 explains its 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 aerodynamic databases 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 modelling This 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 motion.

5.1.1 Equations of motion The 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):

$ Bv '& m = Fg er + D eD + L eL Bt '% Bω = Mz ez Bt

(5.1)

Jzz

Where the forces and moment are deﬁned as: Fg =

G MEarth m |r|2

v2 Aref 2 v2 Aref L = ρ(h) CL (h, α) 2 v2 Mz = ρ(h) Aref Lref (CMα (h, α) + CM ω (h, α) ω) 2 D = ρ(h) CD (h, α)

42

(5.2) (5.3) (5.4) (5.5)

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

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

The ﬁrst equation of the system 5.1 is the sum of forces. It describes the temporal variation of the satellite’s velocity, m being its mass. The ﬁrst term of the right hand side represents the gravitational force Fg . The Earth is assumed to be stationary and perfectly spherical and the relative motion of the atmosphere is neglected. The second and third terms are respectively the drag and lift forces described in equations 5.3 and 5.4 with the conventions of Figure 5.1, as described in section 4.1. This time, though, it is not the ﬂow speed u8 that is considered but the satellite’s v. The second equation of system 5.1 describes the variation of the angular velocity due to the moment Mz acting on the satellite. Jzz is the inertia term following the z-axis and ω the angular velocity. The integration of this angular velocity gives the angle of attack α, which is the angle between 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 the static stability. In this chapter, it will include a damping term. To avoid any confusion this α . The expression of the total moment includes static pitch moment coeﬃcient is now noted CM α , which is the linear part, given by the DSMC thus now two terms. The ﬁrst one implies CM static study. The second one implies the damping coeﬃcient CM ω , and is therefore proportional to 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 and the line parallel to the ground. It corresponds to the variation in the satellite’s direction with the azimuthal direction, and can be found knowing that i = ϕ + ψ. Those angles are deﬁned as: ϕ = arctan

y x

ψ = arctan

vx vy

(5.6)

The components of the absolute velocity are determined using the angle of incidence (equations 5.7). The radial velocity vradial points to the Earth’s centre and is responsible for the diminution of 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 only azimuthal. When the drag inﬂuence is considered, the satellite begins to decay and its velocity becomes partially radial, while its angle of incidence takes a positive value.

5.1.2 Algorithm The 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 to consider 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 certain moment, it computes the altitude and scrolls through the aerodynamic coeﬃcients databases to α , and calculates C ﬁnd CL , CD and CM M ω by means of a Matlab routine. These coeﬃcients are then 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 validation The 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 the vectorial equations in their scalar form, then construct the Simulink program related to the equations, and ﬁnally validate the program using STK or test cases. At ﬁrst, the program considers only the gravitational force. Drag and lift forces are then added 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 not justiﬁed. However, the reader can refer to appendix B, for more detailed views.

44

5.2.1 First step: gravitational force Only 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 and velocities of the satellite. Equations The sum of the forces acting on the satellite is summarized as: m

Bv = G MEarth m e r Bt |r|2

(5.8)

Rewriting this equation in a scalar form, with respect to the conventions established in Figure 5.1, with:

?xx+y er = ? y x +y

Gives:

2

2

2

2

$ B vx '' m = a x G MEarth m & Bt x2 + y 2 x2 + y 2 '' m Bvy = a y G MEarth m % Bt 2 2 x2 + y 2

(5.9)

(5.10)

x +y

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. Double integration of the equations of motion

The centre of the program is the ”Equations of Motion” block detailed in Figure B.1. It transcribes 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 Simulink in the ”Alt Calculation” block, as shown in Figure B.2: h=

a

x2 + y 2 REarth

(5.11)

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

5.2.2 Second step: drag and lift terms The 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 the vectors eD and eL . The right hand side of this equation is now the sum of three forces: the gravitational force, the drag force, and the lift force: m

Bv = F e + D e + L e g r D L Bt

With:

?vv +v eD = ? v v +v x 2 x y 2 x

2 y

?vv +v eL = ? v v +v

2 y

y 2 2 x y x 2 2 x y

(5.12)

Becomes, after simpliﬁcation:

b2 2 $ ' Bvx = a x G MEarth m vx + vy ρ(h) A (v C (h) + v C (h)) ' m ' x y D L ref & Bt x2 + y 2 2 x2 + y 2 b ' vx2 + vy2 ' ' % m BBvty = ax2y+ y2 Gx2M+Earth m ρ(h) Aref (vy CD (h) vx CL(h)) y2 2

(5.13)

Construction The Matlab script is now added to the loop. It takes the altitude as input and returns the corresponding atmospheric density, the drag, and the lift coeﬃcients. The atmospheric model is an interpolation of Jacchia’s model, the same as the one chosen in section 3.3.1 to generate the aerodynamic coeﬃcients databases. The drag and lift coeﬃcients are stored in the script for diﬀerent altitudes and interpolated linearly between them. The new elements added to the program are shown in red frames in Figure B.3. The ”Equations of Motion” block needs a return of the velocities vx and vy , as well as the three coeﬃcients returned by the Matlab program to be able to calculate the drag and lift terms of equations 5.13.

46

Again, the equations are transcribed in Simulink using basic mathematical operations deﬁned in 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 an altitude of 1 km, taking into account the gravitational and drag forces.

5.2.3 Third step: moment equation The 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 to the model. Now that attitude is considered, the aerodynamic coeﬃcients do not rely only on altitude as previously but also on the angle of attack.

b2 2 $ ' Bvx = a x G MEarth m vx + vy ρ(h) A (v C (h, α) + v C (h, α)) ' m ' x y D L ref ' Bt x2 + y 2 2 x2 + y 2 ' b2 2 & Bvy = a y G MEarth m vx + vy ρ(h) A (v C (h, α) v C (h, α)) ' m y x D L ref ' Bt x2 + y 2 2 x2 + y 2 ' ' ' vx 2 + vy2 % BBωt = ρ(h) 2 Aref Lref (CMα (h, α) + ω CM ω ) J zz

(5.14)

Construction This second equation of system 5.1 is also integrated twice in order to obtain the value of the angle 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 already given by the previous version of the program

The complete Simulink model is shown in Figure B.8, and the latest additions are highlighted in 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 attack and to return the three aerodynamic coeﬃcients, the damping coeﬃcient and the density. These values are now stored for diﬀerent altitudes and angles of attack, and interpolated linearly between them. Global validation The validation of the complete program is achieved with a free-fall from an altitude of 100 km. 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° and no initial velocity (Figure 5.6). The atmospheric model at those altitudes is assumed to be

exponential: ρ(h) = 1.225 e 7,600 kg/m3 . The drag coeﬃcient is constant and equal to 2, while the moment coeﬃcient is abritrary chosen as proportional to the opposite of the angle of attack: α = 0.001 α. There is thus a stable trim point for an angle of attack of 0°. The expression CM of the moment does not include the damping term yet, CM ω = 0. h

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, quickly decreases, (Figure 5.7(a)). However, at an altitude of 44.9 km, the atmospheric density becomes too important and the drag term is then greater than the gravity term. The velocity begins to decrease and seems to reach an asymptote, which can be calculated analytically. Indeed, at the end of its fall, the satellite’s acceleration is assumed to be null, and:

¸

F = 0 ÝÑ m g(h) = D(h)

d v(h) =

= ρ(h)

v 2 (h) Aref CD 2

2m g(h) ρ(h) Aref CD

(5.15)

As foreseen, the curve obtained in equation 5.15 is the asymptote of the absolute velocity for the 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) v 2 (h)/2. From the altitude of 100 km down to 29 km, the important increase in density 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 the increase 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 modiﬁedNewtonian method In order to get closer to reality, the pitch moment coeﬃcient has to be completed with its dynamic term, the damping coeﬃcient CM ω . That damping term describes the eﬀects that the angular 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 the modiﬁed-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 the surface intersected by the ﬂow (face 1 and 2 in Figure 5.9). The basic geometry is approximated here with a box having the external dimensions of a 3-unit CubeSat (Figure 4.8). A new referential attached to the satellite is deﬁned. The primary step will be to calculate the pressure coeﬃcient. The force acting on a surface incident to a hypersonic ﬂow can be simpliﬁed by considering that the normal component of the momentum ﬂux is completely transferred as a force to the surface, while the component tangential to the surface remains unchanged [45] [46]. The development is ﬁrst made for the face 1: FN = mu ˙ N = (ρ8 u8 A cos α) (u8 A cos α) = ρ8 u2N A

(5.16)

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

p p8 1 2 2 ρ8 u8

FN /A 1 2 2ρ u

8 8

(5.17)

Replacing equation 5.16 in equation 5.17: Cp =

2u2N u28

(5.18)

If damping is neglected, only the linear velocity is taken into account and uN = u8,N = u8 cos α is the normal velocity seen by the faces. If damping is considered, this expression is modiﬁed with the angular velocity as depicted in Figure 5.10. uN = u8,N + ((ω r) n)

49

(5.19)

100

1000

800

60

600

40

400 44,9 km

20

Velocity (m/s)

200

80

0 0

simulation analytical calculation

velocity altitude

Absolute Velocity (m/s)

Altitude(km)

957 m/s

100

150

200

250

300

350

100

50

200

50

150

0

0

20

15

10

5

0

Altitude (km)

Time (s)

(a)

(b)

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

60 40

Angle of attack (°)

20 0

40

−20 −40 20 −60 −80 −100 100

Dynamic pressure (10kg/ms²)

altitude: 29 km

dynamic pressure angle of attack 80

60

40

20

0 0

Altitude (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 vector from the centre of gravity to the surface element: cos α 0 1 x u8 = u8 sin α ω= 0 n= 0 r = y 0 ω 0 0 Equation 5.19 becomes after calculation: uN = u8 cos α + ωy And is replaced in equation 5.18: Cp =

2 2u2N ω 2ω 2 + 2 y = 2 cos α + 4 cos α y u28 u8 u28

Now that the pressure coeﬃcient is determined for the ﬁrst face, it has to be multiplied by the lever arm and integrated over the whole face to obtain the pitch moment coeﬃcient. 1 CM,1 ez = Aref Lref =

1

Aref Lref

51

»

y ey Cp ex dS

»l 0

b Cp y ez dy

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

CM,1 ez =

»l

b Aref Lref

2 cos2 α y ez

0

+4 cos α y 2 +2 y 3

ω ez u8

ω2 ez dy u28

The ﬁrst term of this integral does not depend on the angular velocity ω. It corresponds to the pitch 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, and depend respectively on ω and ω 2 . After calculation, it appears that the ﬁrst and third terms are null, due to the approximation on the geometry. The diﬀerence with the result obtained with DSMC in section 4.3.1 is due to the underlying hypotheses of modiﬁed-Newtonian method, which stipulates that the momentum ﬂux tangential to the surface remains unchanged and that the pressure ﬁeld is constant all along the surfaces facing the ﬂow and null on the others. Only the ﬁrst order damping term remains: CM,1 =

l3 ω b cos α Aref Lref 3 u8

(5.20)

The same development is applied to the second face intercepted by the ﬂow, and available in appendix D. The normal velocity is now: uN = u8 sin α ωx Giving again a pitch moment coeﬃcient that only contains a damping part: CM,2 =

b L3 ω sin α Aref Lref 3 u8

(5.21)

Finally, the total moment coeﬃcient is the sum of the contributions of the two intersected faces: CM = CM,1 + CM,2 b (l3 cos α + L3 sin α) ω = 3 Aref Lref u8

(5.22)

And the damping coeﬃcient is deﬁned as: CM ω =

B CM Bω

=

b

3 Aref Lref u8

(l3 cos α + L3 sin α)

(5.23)

For example, if α = 15°, this expression can be evaluated (u8 = 7, 806.32 m/s is taken at 170 km, where the orbital velocity is the smallest): CM ω = 1.132 105 s

52

The evolution of this coeﬃcient with regards to the angle of attack is shown in Figure 5.11 for an altitude of 170 km. −5

0

x 10

−0.5

(15° ; −1.32e−5 s)

Damping coefficient (s)

−1 −1.5 −2 −2.5 −3 −3.5 −4 0

20

40

60

80

Angle of attack (°)

Figure 5.11: Evolution of the damping coeﬃcient with the angle of attack for the basic geometry 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 neglecting the frontal spherical section. The pressure coeﬃcient is now integrated on the six faces intersected 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 deployed panel has to be taken into account. CM ω = 5.4 105 s This coeﬃcient is almost ﬁve times greater than it was in the basic geometry, but remains relatively 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 remains perpendicular to the ﬂow. For this reason, the damping coeﬃcient is the same as for the basic geometry. The damping coeﬃcient greatly depends on the geometry of the satellite, with l and L to the cube, and on the ﬂow velocity, with the inverse of u8 . Because of the very small size of the satellite 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 in appendix 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 and absolute velocity, and returned into Simulink together with the drag, lift, and atmospheric density coeﬃcients.

5.4 Application to the selected geometries Now 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 range for which the aerodynamic coeﬃcients have been determined.

5.4.1 Basic geometry The 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 is Jzz = 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, having a maximum value of 0.2379° at 100 km. The satellite’s velocity remains almost parallel to the ground. Equations 5.7 gives a radial velocity of maximum 32.46 m/s at 100 km, while azimuthal velocity 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 stable during the entire re-entry. Figure 5.12(b) shows that the basic satellite without any de-orbiting mechanism is decaying down to 100 km in more than 13 hours. That corresponds to more than 9 slightly elliptical orbits around the Earth, according to Figure 5.13. Again, the Earth’s radius has 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 oscillate around its nominal position, with a magnitude slowly damped over time (Figure 5.14(a)). The drag coeﬃcient being much greater at 15° than at 0° (section 4.3), the satellite decays faster when the angle of attack is perturbed, in only 11 hours (Figure 5.12(b)). Moreover, the positive peak of the pitch moment coeﬃcient that exists at an altitude of 105 km for this geometry causes the angle of attack to diverge. At 107 km, both α and ω quickly increase. The satellite 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 absolute velocity of the satellite and the orbital velocity for each altitude. This diﬀerence is due to the drag 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 the equations of system 5.13. That force being proportional to the inverse of the square of the distance to the Earth’s centre, the behaviour of the augmentation of absolute velocity depends on the diminution in altitude. Around an altitude of 115 km, and in agreement with the rapid atmospheric density increase after that point, the drag term becomes greater than the gravity term, and the satellite slows down brutally.

54

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

satellite’s trajectory Earth

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

7845 12 m/s 7840

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

56

5.4.2 Flower geometry Coeﬃcients 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 one orbit (Figure 5.17). Because the drag coeﬃcient is smaller for an angle of attack of 15°, it will be one minute longer. Thus, initial angle of attack has no major inﬂuence on the de-orbiting time, unlike for the basic geometry. Attenuation is much more important here than it was for the basic geometry. The perturbed injection angle of attack is divided by six in 90 min. It oscillates between 3° and 3° at the altitude of 100 km (Figure 5.18(a)). Moreover, the ﬂower satellite remains stable during the entire 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 coeﬃcient after the peak at an altitude of 110 km (see section 4.4) only leads to a small divergence of pitch angle. 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 to 0.150 m2 . Figure 5.19 clearly shows that the inﬂuence of this de-orbiting mechanism becomes important after an altitude of 115 km.

5.4.3 Plate geometry The 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 initial angle 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 falling in 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 the ﬂower 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 to the smaller value of the pitch moment coeﬃcient (Figure 5.23).

57

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

satelliteâ€™s trajectory Earth

Figure 5.17: Flower geometry, representation of the satelliteâ€™s trajectory, the radius of the Earth has been divided by 40

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

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

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satelliteâ€™s trajectory Earth

Figure 5.21: Plate geometry, representation of the satelliteâ€™s trajectory, the radius of the Earth has been divided by 40

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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 parameters Quite 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 of the 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 particular case. They are discussed in appendix E, in order to understand their role.

5.5.1 Trigger altitude An important parameter of inﬂuence on the re-entry time is the trigger altitude. It is important to know when the de-orbiting system must be deployed in order to keep the satellite visible during the end of its course. Regarding the plate geometry, Table 5.1 shows the eﬀect that the variation of trigger altitude has on the ∆v at 100 km, and on the time needed to cover the 50 km between 150 and 100 km. The satellite evolves at orbital velocity before deployment. The plateau of free-molecular regime is assumed to be reached at 170 km. Table 5.1: Eﬀects of trigger altitude Trigger altitude ∆v100km (m/s) ∆t150100km (s)

150 km 114 3, 505

170 km 115 3, 250

200 km 116 3, 030

The ﬁrst line of Table 5.1 does not vary much with the trigger altitude. This means that, in these three cases, the satellite reaches an altitude of 100 km with more or less the same absolute velocity. Figure 5.24 shows that the trigger altitude has a signiﬁcant eﬀect on the radial velocity only until 120 km, which explains the small variations in the time needed to cover the 50 last kilometres. Nevertheless, beyond 120 km, the re-entry will occur in the same way for the three cases, having the same vradial and ∆v100km . The altitude of deployment will thus have a very limited eﬀect on the re-entry duration after 120 km. A higher trigger altitude will not really help to reduce the time needed to cover the last kilometres of the re-entry. It should thus rather be decided in function of deployment feasability and stability questions.

5.5.2 Mass The 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:

Bv = GM Bt |r|2

er +

L D eD + eL m m

It appears that the acceleration depends on the inverse of the mass through the drag and lift terms. A smaller mass will thus increase the inﬂuence of those two forces, and the satellite will decay much faster (Figure 5.25). The mass also plays a role in the second equation of 5.1 through calculation of the inertia term in:

Bω = Mz e Bt Jzz z 62

80 70

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

170 mass: 3 kg mass: 2 kg mass: 4 kg

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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 stabilizing moment. It also means that the satellite has less time to attenuate the perturbation of the angle of attack (see Figure 5.25). The eﬀect of the mass on the attenuation is therefore almost null.

63

5.5.3 Solar activity The 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 for diﬀerent 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 the aerodynamic 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 time necessary 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 till lower 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 activity being regulated by cycles with a mean period of 11.5 years, it is more often in medium activity and the present study is thus more general. Furthermore, there are more reference points in Jacchia’s atmospheric model, allowing for an important gain of precision. Jacchia MSISE90 − medium activity MSISE90 − high activity MSISE90 − low activity

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64

Chapter 6

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

6.1 Criteria presentation The 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 structure will obviously be diﬀerent in its dimensions and shape. Therefore, a descriptive parameter which allows for objective comparison is needed. Having no information on the complexity of the diﬀerent geometries, or on the mass or volume they represent, or on their reliability, it was decided to weigh them according to their drag area. Nevertheless, it should be noted that even the drag area is not a perfect denominator as it does not result in linear variations of a geometry’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° with a 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 Adrag ∆tnominal (s m2 ) Adrag ∆tperturbed (s m2 ) (Altitude for α 5°)/Adrag (km/m2 )

65

Basic 481.3 392.8 Unstable

Flower 821.4 830.4 12.74

Plate 722.3 717.5 10.2

The ﬁrst surprising observation is that the basic geometry seems to be the most eﬃcient one for what concerns de-orbiting. This is easily explained. Indeed, the entire drag area of the basic geometry, which consists solely of its frontal spherical section, is directly exposed to the free-stream. This is not the case for the ﬂower geometry: the ﬂow hitting the side panels has already been slowed down by the body of the satellite itself. In the same way, the plate of the plate 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 thus something 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 an heat shield, which is most probably impossible to deploy and will cause numerous additional constraints in terms of mass, volume, and integration. Furthermore, in the particular case of the basic geometry it is unstable. The reverse plate and ﬂower geometries, which are similar concepts, are also unstable as it was concluded in section 4.2.3. The de-orbiting performances of the plate and ﬂower geometries are very similar, although the plate geometry is more eﬃcient for de-orbiting and the ﬂower geometry the most eﬃcient for 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 regards to the area of the plate and the length of the link. Furthermore, section 4.5.1 showed that 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 rather than 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 have several 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 of only one piece. The loss of a side panel is thus a point of failure, which would cause the satellite to be dramatically unstable. The loss of the plate would just cause the satellite to 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 the most suitable structure for the QB50 mission.

6.3 Guidelines for a complete system Now 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 deﬁnitive requirements, but as leads for a further development.

66

Z

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Figure 6.1: Pressure ﬂow ﬁeld around the ﬂower and the plate geometry for Kn = 0.345, which corresponds 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 Link The 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 conversion from aerodynamic coeﬃcient to force, although their diﬀerence does not signiﬁcantly inﬂuence the general behaviour. The bump observed around 110 km corresponds to the peak in the drag coeﬃcient. 1.8

Tenstion force in the link (N)

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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 is always tight. The plate being in the satellite’s wake, this was not an obvious result. If the link was 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. The force acting on the plate itself is then 4.5 mN . The magnitude of that force on the undeployed system will be even smaller. This is most probably not enough to deploy it. Therefore, the plate should be deployed at lower altitudes or using another system than just the drag force: e.g. a loaded spring could deploy the folded plane, which could be unfolded and maintained with a prestressed structure. Although the tension force quickly increases, it barely reaches 1.7 N at an altitude of 100 km. Even with a security coeﬃcient, this is obviously within the range of most materials that could be used as link: typical sport ropes can approximately bear 0.1 1 kN /mm2 , while steel cables used in the entertainment industry can even go up to 20 kN /mm2 . Furthermore, multiple ropes could be used, at least as back-up solutions. It should be noted, though, that the tension in the link will certainly increase even faster below 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 of the plate is bigger, one can conﬁdently say that pressure is not a problematic issue at the very beginning of the re-entry. The real concern is the temperatures through which the link will evolve: from the coldness of space to the important heating of the re-entry. Therefore, the link should be designed for its thermal tolerance rather than its resistance. It could be covered with a protective material chosen for its low thermal conductivity and high melting temperature, such as ﬂuorinated ethylene propylene (FEP) or polytetraﬂuoroethylene (PTFE).

6.3.2 Plate From Figure 5.21, it appears that a 0.09 m2 plate is not big enough to ensure a rapid deorbiting. Two solutions exist: a longer link, in order to decrease the inﬂuence of the satellite’s wake, 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 the link’s length to 2 m. Simulink simulations were performed varying the reference area, but keeping the same aerodynamic coeﬃcients as for the 0.09 m2 plate (Figure 6.3). This is a conservative assumption, as it can be concluded from section 4.5.1 that the drag coeﬃcient is actually slightly more important 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 is not worth the technical complexity it would most probably cause. Furthermore, the orbit is not polar but has an inclinaison of 79 °, the latitude covered will thus be smaller. The ﬁnal result is similar to what was announced in Figure 2.1(b). The material used should be manufacturable in the form of foldable sheets. Again, its thermal properties are more important than its resistance. Due to the important outer surface of the plate, a material with a high emissivity and a high melting point is a justiﬁed choice as it would allow the plate to radiate most of its heat. Although the geometry was tested for square plates, another shape could be considered. A circle, in particular, would be more advantageous as the thermal and mechanical constraints present 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’s perimeter, it will be shorter for the circle than for the square.

68

satellite’s trajectory Earth

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(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 such as multiple points of attachment along the perimeter, as is the case for parachutes, has to be assessed.

70

Chapter 7

Conclusion 7.1 Achievements The 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 a few guidelines for its practical implementation. We proved its ability to de-orbit and stabilize the 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 small satellites followed by a comparative analysis. The drag increase was selected as the most suitable technique for the particular case of the QB50 re-entry mission. • Chapter 3: deeper understanding of the rareﬁed and transitional hypersonic ﬂows, and presentation of the DSMC method used to model them. • Chapter 4: creation with DSMC of aerodynamic coeﬃcients databases describing four possible drag increase concepts. • Chapter 5: development of a three-degree-of-freedom Simulink program to model the dynamics 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-orbit and to stabilize, selection of the plate geometry, and presentation of a few guidelines for its practical implementation. Current engineering methods used to model the atmospheric re-entry, implemented in software such as STK, take into account a constant drag coeﬃcient, most of the time approximated as equal to 2.2. Figure 7.1, based on data yielded for the basic geometry, clearly shows that such 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 dynamic stability. A semi-engineering approach would have consisted in computing the aerodynamic coeﬃcients for the continuum regime with the modiﬁed-Newtonian method and for the highly rareﬁed regime with the free-molecular method, and approximate the transitional regime with a bridging function. Again, this would have led to incorrect results as the non-monotonic aerodynamic behaviour of the geometries through the transitional regime was highlighted by DSMC simulations. Furthermore, it was proven during the analytical calculation of the pitch moment coeﬃcient (section 5.3) that the underlying hypotheses of modiﬁed-Newtonian method were too strong for accurate results. 71

170 drag coefficient: 2.2 basic geometry 160

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

7.2 Perspectives To 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 the technology 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 applied to 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 the deployment 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 gather information on the pressures and temperatures involved. When the design has been suﬃciently developed, full characterization in the transitional regime should be conducted with DSMC and simulations should be performed with the Simulink program developed for this study. In the long term, it would be helpful to use a multi-body dynamic system approach , which would consider the properties of materials as well as the local pressure and temperature. Regardless of the objectives that were set for this particular study, improvements could be brought 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 matrix and damping of the diﬀerent geometries, which means including the frontal spherical section for the basic geometry. However, the diﬀerences with the current values of these parameters should be small and therefore the inﬂuence of that change on the ﬁnal result would most likely be negligible.

72

The Simulink program could easily be upgraded from three-degree-of-freedom to six-degreeof-freedom, which would allow for observation of the coupling degrees of freedom, e.g. the gyroscopic eﬀect provided by the ﬂaps of the ﬂower geometry. That option was not considered for this preliminary study as it would have been necessary for more DSMC simulations to take into 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, which will most probably be tested both experimentally and numerically for various angles of attack and 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 the satellite 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-uniform gravity ﬁeld, atmospheric parameters’ variation over time, etc., could be included in equations 5.1 and in the databases. With those enhancements, the Simulink program used for this study would become the core of a very powerful predictive tool for re-entry vehicles, and for the QB50 re-entry satellite in particular. The databases obtained regarding the diﬀerent geometries could be supplemented with the calculation of the diﬀerent aerodynamic coeﬃcients with the modiﬁed-Newtonian method in continuum regime and the free-molecular method in highly rareﬁed regime. Those values, although obtained using approximate methods, would both validate the numerical results and complete the 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-orbiting system. Therefore, the results of the simulations performed for this study should be kept and further analysed. Finally, the underlying cause of the peak within the transitional regime is an interesting phenomenon that should be further studied. The geometrical parameters of the diﬀerent geometries could be changed in order to isolate what causes the sudden drag increase. For the basic geometry, one would change the satellite’s length or smooth the edges, and determine to which extent these factors are inﬂuent. The surface parameters could also be analysed to determine where the forces are being applied in function of the rarefaction regime.

7.3 Last words The 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 the design of de-orbiting systems for small satellites, whether it is for re-entry missions or for debris mitigation. Other interesting topics, such as the non-monotonic behaviour of the aerodynamic coeﬃcients in the transitional regime for particular geometries, were approached, even if they go beyond the initial scope of this study.

73

Bibliography [1] C. Venturini, L. Abramowitz, J. Johansen, and J. Gee. Cubesat development programs working with the community. In AIAA SPACE 2009, editor, AIAA Paper 2009-6501, 2009. [2] Jean Muylaert. Qb50: An international network of 50 cubesats for multi-point, in-situ measurements in the lower thermosphere and for re-entry research. In ESA Atmospheric Science Conference, 2009. [3] von Karman Institute. http://www.vki.ac.be/qb50/project.php. QB50 Project Description, January 2011. [4] Wilbur L. Hankey. Re-Entry Aerodynamics. AIAA Education Series, 1988. [5] GomSpace. NanoPower BP-4 Datasheet, 2009. [6] James R. Wertz and Wiley J. Larson. Space Mission Analysis and Design. Space Technology Library, 2010. [7] Olivia Billett. Micropropulsion for nanosatellites. In Space and Stanford Systems Development Laboratory, editors, http://ssdl.stanford.edu/ssdl/images/stories/AA236/A06Fall/Lectures/lecture-17.pdf, November 2006. [8] Juergen Mueller, Richard Hofer, and John Ziemer. Survey of propulsion technologies applicable to cubesats. In 57th JANNAF Propulsion Meeting, May 2010. [9] Juergen Mueller, John Ziemer, Richard Hofer, Richard Wirz, and Timothy O’Donnell. A survey of micro-thrust propulsion options for microspacecraft and formation ﬂying missions. In 5th Annual CubeSat Developers Workshop, 2008. [10] Space Product Division. Demonstrated Performance for Fluid and Motion Control Catalogue, section 4 Cold Gas Systems And Components. Moog, 2000. [11] Jean L. Broge. Marotta’s low-power microthruster. Aerospace Engineering, page 7, December 2003. [12] ATK. http://www.atk.com/products/documents/starATK Space Propulsion Products Catalog, May 2008. [13] Bryce Carpenter David Beckett. Rapid de-orbit of leo space vehicles using towed rigidizable inﬂatable structure technology: Concept and feasibility assessment. In AIAA Small Satellite Conference, August 2004. [14] G. E. Cook. Satellite drag coeﬃcients. Technical report, Royal Aircraft Establishment, Farnborough, Hants., UK Farnborough, Hants., UK Farnborough, Hants., UK, 1965. [15] E. M. Gaposchkin. Calculation of satellite drag coeﬃcients. Technical report, Lincoln Laboratory, MIT, 1994. 74

[16] D.C. Maessen, E.D. van Breukelen, B.T.C. Zandbergen, and O.K. Bergsma. Development of a generic inﬂatable de-orbit device for cubesats. In International Astronautical Federation, editor, 58th International Astronautical Congress 2007, volume 3, pages 1860–1870, 2007. [17] Robert P. Hoyt. Stabilization of electrodynamic tethers. In American Institute of Physics Mohamed S. El-Genk, editor, Space Technology and Applications International Forum 2002, Melville, NY, 2002. [18] Robert P. Hoyt. http://www.tethers.com/papers/DeorbitTechnologies.pdf. Tethers Unlimited, Inc. white paper, 2011. [19] nanoTerminator Datasheet, editor. http://www.tethers.com/SpecSheets/nanoTerminator.pdf. Tethers Unlimited, 2010. [20] Robert P. Hoyt. private e-mail, January 2011. [21] Pekka Janhunen. Electrostatic plasma brake for deorbiting a satellite. Journal of Propulsion and Power, 26(2):370–372, March-April 2010. [22] E. Josyula and J. Burt. Review of rareﬁed gas eﬀects in hypersonic applications. In von Karman Institute, editor, Lecture series on models and computational methods for rareﬁed ﬂows, Brussels, Belgium, January 2011. [23] G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994. [24] M. S. Ivanov. User manual for Rareﬁed Gas Dynamics Analysis System (RGDAS). Russian Academy of Sciences, Siberian Branch, Khristianovich Institute of Theoretical and Applied Mechanics, Computational Aerodynamics Laboratory, Novosibirsk. [25] C. Borgnakke and P. S. Larsen. Statistical collision model for monte carlo simulation of polyatomic gas mixture. Journal of Computational Physics, 18:405–420, 1975. [26] L. G. Jacchia. Thermospheric temperature, density and composition : New models. Special Rept. 375, Smithsonian Astrophysical Observatory, Cambridge, MA, March 1977. [27] James N. Moss, Christopher E. Glass, and Francis A. Greene. Dsmc simulations of apollo capsule aerodynamics for hypersonic rareﬁed conditions. In 9th AIAA/ASM Thermophysics and Heat Transfer Conference, 2006. [28] G. A. Bird. The ds2v/3v program suite for dsmc calculations. In American Institute of Physics M. Capitelli, editor, 24th International Symposium on Rareﬁed Gas Dynamics, volume 762, pages 541–546, 2005. [29] Vladimir Riabov. Aerodynamic applications of underexpanded hypersonic viscous jets. Journal of Aircraft, 32(3):471–479, May-June 1995. [30] Vladimir Riabov. Comparative analysis of hypersonic rareﬁed gas ﬂows near simple-shape bodies. Journal of Spacecraft and Rockets, 35(4):424–433, 1998. [31] Vladimir Riabov. Numerical simulation of kinetic eﬀects on low-density hypersonic aerodynamics. In Proceedings of the 25th ICAS Congress, September 2006. [32] Pavel Vashchenkov, Alexandr Kashkovsky, and Mikhail Ivanov. Numerical analysis of highaltitude aerodynamics of the expert capsule. In West-East High Speed Flow Field Conference, November 2007.

75

[33] R. G. Wilmoth, R. C. Blanchard, and J. N. Moss. Rareﬁed transitional bridging of blunt body aerodynamics. In 21st International Symposium on Rareﬁed Gas Dynamics, Marseille, France, July 1998. [34] Bob Reid. Lecture 12 : Aerothermodynamics of the shuttle. Aircraft System Engineering course (MIT), 2005. [35] Gilles Bailet, Cem Ozan Asma, Jean Muylaert, and Thierry Magin. Feasability analysis and preliminary design of an atmospheric re-entry cubesat demontrator. In 7th European Symposium on Aerothermodynamics for Space Vehicles, 2011. [36] V. N. Gusev, A.I. Erofeev, T.V. Klimova, V.A. Perepukhov, V. Riabov, and A.I. Tolstykh. Theoretical and experimental investigations of ﬂow over simple shape bodies by a hypersonic stream of rareﬁed gas. Trudy TsAFI, (1855):3–43, 1977. [37] G. Koppenwallner and H. Legge. Drag of bodies in rareﬁed hypersonic ﬂow. In Progress in Astronautics and Aeronautics, editors, Thermophysical Aspects of Re-Entry Flows, volume 103, 1985. [38] Vladimir Riabov. Numerical study of hypersonic rareﬁed-gas ﬂows about a torus. Journal of Spacecraft and Rockets, 36(2):293–296, 1999. [39] Vladimir Riabov. Aerodynamics of two side-by-side plates in hypersonic rareﬁed-gas ﬂows. Journal of Spacecraft and Rockets, 39(6):910–916, 2002. [40] Vladimir Riabov. Numerical study of interference between simple-shape bodies in hypersonic ﬂows. Computers and Structures, 87:651–663, 2009. [41] Plenum, editor. Rareﬁed Gas Dynamics, pages 345–390. M. N. Kogan, New York, 1969. [42] Vladimir Riabov. private e-mail, May 2011. [43] A. W. Wilhite, J. P. Arrington, and R. S. McCandless. Performance aerodynamics of aeroassisted orbital transfer vehicles. Progress in Astronautics and Aeronautics, 96:165–185, 1985. [44] Jack A. White and Katherine G. Johnson. Approximate solutions for ﬂight-path angle of a reentry vehicle in the upper atmosphere. Technical Report TN D-2379, NASA Langley research centre, 1964. [45] Mark Schoenenberger. Newtonian method of calculating pitch damping with low speed modiﬁcations. Research report, 2006. [46] Murray Tobak and William R. Wehrend. Stability derivatives of cones at supersonic speeds. Technical Report 3788, National Advisory Commitee for Aeronautics, 1956.

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Appendix A

Decoupled pressure ﬁeld hypothesis The decoupled pressure ﬁeld hypothesis was tested for three regimes: closest to continuum Kn = 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 plate geometry (plate and satellite) is compared to the pressure ﬁeld of the entire geometry minus the pressure 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 (Figure A.2), but the pressure level drops from a level of 7.5 P a down to 0.5 P a. The plate appears to be the dominating geometry in both regimes. In highly rareﬁed regime (Figure A.3), the pressure ﬁelds are clearly coupled, although the maximum pressure level is only 0.0013 P a. X

Z

X

Y

Z

Pressure (Pa) 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.5

Y

Pressure (Pa) 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.5

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

77

X

Z

X

Y

Z

Pressure (Pa)

Pressure (Pa)

0.45 0.35 0.25 0.15 0.05

0.45 0.35 0.25 0.15 0.05

Y

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

X

Z

X

Y

Z

Pressure (Pa) 0.0013 0.0011 0.0009 0.0007 0.0005 0.0003 0.0001

Y

Pressure (Pa) 0.0013 0.0011 0.0009 0.0007 0.0005 0.0003 0.0001

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

78

Appendix B

Simulink program constructive details B.1 First step The 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 step The 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 gravitational force, the drag force, and the lift force (B.4).

Figure B.4: Second step program. Inside the â€?Equations of Motionâ€? 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 program The 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 damping part, 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 is integrated twice, and a return of the angle of attack and angular velocity is added.

82

83 Figure B.8: Complete Simulink model

Appendix C

Validation of the Simulink program C.1 First step The 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 following the 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 shown in Figure C.1. 6

8

x 10

5242 s

6

Coordinates to earth’s center (m)

satellite’s trajectory Earth

x y

4 2 0 −2 −4 −6 −8 0

50

100

150

200

250

300

Time (minutes)

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 150 km. 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 close enough to validate this ﬁrst step. Another validation is performed with STK by making the satellite fall from an altitude of 170 km to the ground (Figure C.2). The initial velocity following y is not the orbital velocity, but is set 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 taken into 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 step The 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 Figure C.4. The absolute velocity increases quickly until it reaches a plateau, which can be calculated analytically. Indeed, after a certain time, the satellite’s velocity remains constant and the forces involved are at equilibrium:

¸

F = 0 ÝÑ mg = D =ρ

v2 Aref CD 2

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

85

(C.1)

d v=

2mg = 49.67 m/s Ď Aref CD

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

1000

750 49,6739 40

500

Altitude (m)

Absolute Velocity (m/s)

60

20 250 altitude velocity 0 0

5

10

15

20

0

Time (s)

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

86

Appendix D

Calculation of the damping coeﬃcient on the second face A new pointing vector is deﬁned for the second face: 0 n = 1 0

(D.1)

And the expression of normal velocity can be determined as previously: uN = u8 sin α ωx And, when replaced in equation 5.18, leads to the following expression of the pressure coeﬃcient: Cp =

2 2u2N ω 2 2ω = 2 sin α 4 sin α x +2 x u28 u8 u28

(D.2)

After replacing in the expression of the coeﬃcient: b CM,2 ez = Aref Lref

»L 0

2 sin2 α x ez

4 sin α x2 uω ez +2 x3

ω2 ez dx u28

After calculation: CM,2 =

L3 ω b sin α Aref Lref 3 u8

With L = 0.3 m, this expression can be evaluated: CM,2 = 9.946 106 ω

87

8

(D.3)

Appendix E

Inﬂuence of the pitch moment damping and the lift force Those two coeﬃcients also depend on the ﬁnal design, but their inﬂuence on the dynamic behaviour of the two selected geometries is very limited. In order to observe the eﬀects that could appear with other geometries, their inﬂuence is analysed here for hypothetical cases.

E.1 Damping By 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 signiﬁcant eﬀect on the evolution of angular velocity, or angle of attack. Its value is not suﬃcient to prevent the instability caused by the peak at an altitude of 107 km. The Figure shows only the last oscillations, 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 scenario is performed for an arbitrary constant coeﬃcient: CM ω = 0.005. This value would correspond to a bigger satellite travelling at a lower velocity. In this hypothetical case, amplitude of the oscillations as well as the angular velocity are rapidly attenuated in comparison to the results shown in Figures 5.14(a) and 5.14(b). Even better, at the point of instability previously observed, the damping keeps the angle of attack close to zero and prevents the exponential increase of angular velocity. The satellite would remain stable.

E.2 Lift force The 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 80 km in 14, 510 seconds (around 4 hours, in less than 3 orbit). It appears that a lift coeﬃcient of 1 makes the satellite decay a bit more slowly (in 12 more minutes). This happens as a result of the positive orientation of the lift force, which has a tendency to send the satellite to higher altitudes (see Figure 4.1). Because the lift term also depends on the atmospheric density, it has a more signiﬁcant eﬀect at lower altitudes.

88

15

with damping without damping

Angle of attack (°)

10

5

0

−5

−10

3.75

3.8

3.85

3.9

3.95

Time (s)

4

x 10

Figure E.1: Inﬂuence of damping on angle of attack, last oscillations of basic geometry. CM ω = 1, 132 105 (real case) 15

Angle of attack (°)

10

5

0

−5

−10

−15 170

160

150

140

130

120

110

100

Altitude (km)

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

89

160 lift coefficient: 0 lift coefficient: 1

150

Altitude (km)

140 130 120 110 100 14.510 s

90 80 0

5000

10000

15.210 s

15000

Time (s)

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

90