TESI DI LAUREA MAGISTRALE
Analysis and Simulation of an Intake for
Air-Breathing Electric Propulsion Systems
Relatori:
Prof. Luca d’Agostino
Universit`
a di Pisa
Prof. Thierry Magin
von Karman Institute for Fluid Dynamics
Candidato:
Pietro Parodi
The Air-Breathing Electric Propulsion (ABEP) concept employs an air-intake to col-lect the residual atmospheric gas and feed it to an ecol-lectric thruster. In this way, these systems could compensate for the drag force affecting spacecraft in Very Low Earth Orbit and enable low altitude missions with acceptable lifetimes.
The objective of this thesis is to investigate the physical phenomena involved in the operation of the intake and evaluate their impact on its performance through models of increasing complexity.
First, the orbital environment experienced by a spacecraft in VLEO is simulated through the use of an orbital propagator coupled with the available atmospheric mod-els. The results highlight the large variation of the flow conditions due to latitudinal, seasonal, and solar cycle related effects.
Then, the intake is analyzed using a simple lumped parameter model, which enables to estimate the performance of the intake and derive scaling laws for the device. The model shows that, to provide acceptable flow rate and compression, the intake must be matched to the transmissivity characteristics of the thruster. The condition of drag compensation is very stringent and requires a high collection efficiency to be satisfied. Next, the Direct Simulation Monte Carlo (DSMC) method is used to investigate complex flow effects. The degree of rarefaction of the gas has an effect on the per-formance. For realistically low densities, increasing the amount of collisions obstructs the particles incoming from the freestream, causing a decrease of the performance. Moreover, the misalignment of the incoming flow causes a performance degradation. It can reach 20% for a 5° misalignment angle and exceed 40% for a 10° misalignment. Increasing the amount of specular reflections increases the collection efficiency and decreases the density ratio, by as much as 5% for an accommodation coefficient α of 0.9.
Finally, a method based on view-factors for the simulation of free molecular flows is developed and verified. Tests on the intake geometry show good agreement with DSMC. The capability of changing boundary conditions at very low computational cost makes it a promising tool for the multipoint optimization of the geometry.
I would like to thank Prof. d’Agostino for being available to supervise this work and for his precious life and career advice.
I would also like to thank Prof. Thierry Magin, my supervisor at the von Karman Institute, for giving me the chance to work with his research group in the Aeronau-tics and Aerospace Department, for sharing with me a little part of his impressive knowledge, and for always keeping my research topics interesting.
I am really grateful to my advisors at the VKI for all the precious time they spent guiding and reviewing my work, and also for the occasional laughter.
Thanks to all my friends, those that I have known since I can remember, and those that I’ve met more recently in Pisa and in Belgium. I have learned a lot from you and you have made these intense years so much more enjoyable.
Il rigraziamento pi`u grande `e riservato come sempre alla mia famiglia: ai miei genitori, a mio fratello e ai miei nonni, per il loro costante supporto.
1 Introduction 1
1.1 Motivation . . . 1
1.2 State of the Art of Air Breathing Electric Propulsion Systems . . . 3
1.3 Objectives of the Thesis . . . 6
1.4 Outline of the Thesis . . . 6
2 Physical Models and Numerical Methods for Rarefied Gases 7 2.1 Dilute Gases . . . 8
2.2 Kinetic Theory . . . 10
2.2.1 The Distribution Function . . . 10
2.2.2 The Boltzmann Equation . . . 11
2.2.3 The Equilibrium State . . . 12
2.3 Numerical Simulation of Rarefied Flows . . . 15
2.3.1 Methods Based on the Solution of the Boltzmann Equation . . . 15
2.3.2 Methods Based on the Physics of the Flow . . . 15
2.4 The SPARTA DSMC Software . . . 18
2.4.1 Simulation Grid and Solid Objects . . . 18
2.4.2 Boundary Conditions . . . 19
2.4.3 Particle-Surface Collisions . . . 20
2.4.4 Elastic Gas Phase Collisions . . . 20
2.4.5 Inelastic Gas Phase Collisions and Chemical Reactions . . . 22
2.4.6 The Ambipolar Approximation . . . 22
3 The Very Low Earth Orbit Environment 23 3.1 Atmospheric Models . . . 23
3.1.1 Models for Density and Composition . . . 25
3.1.2 The HWM14 Model for the Horizontal Wind . . . 26
3.2 Development of a Software to Calculate the Flow Conditions for a Space-craft in VLEO . . . 26
3.2.1 Calculation Procedure . . . 27
3.2.2 Verification . . . 29
3.3 Calculation of Flow Conditions . . . 29
3.4 Concluding Remarks . . . 35 iii
4.2 Transmission Probabilities of Ducts . . . 38
4.3 Lumped Parameter Model of the Intake . . . 41
4.3.1 Description of the Model . . . 41
4.3.2 Performance Parameters of the Intake . . . 43
4.4 Concluding Remarks . . . 47
5 Direct Simulation Monte Carlo Analysis of the Intake 49 5.1 Description of the 2D and 3D Geometries Employed . . . 49
5.2 Effect of the Degree of Rarefaction on the Performance . . . 52
5.3 Influence of Chemical Reactions in the Gas Phase . . . 55
5.4 Effect of the Flow Misalignment on the Performance . . . 58
5.5 Effect of the Accommodation Coefficient α on the Performance . . . 59
5.6 Concluding Remarks . . . 61
6 Development of a View-Factor Based Method for Free Molecular Flows 63 6.1 Description of the Method . . . 63
6.1.1 Working Hypotheses . . . 64
6.1.2 Flow of Particles Between two Surfaces . . . 64
6.1.3 Solution . . . 67
6.1.4 Calculation of Surface Pressures . . . 67
6.2 Implementation . . . 69
6.3 Verification . . . 69
6.4 Results for the Intake Geometry . . . 71
6.5 Concluding Remarks . . . 71
7 Conclusion 73 7.1 Future Work . . . 75
A Measurement Units and Frames of Reference for Orbital Mechanics Calculations 81 A.1 The Geocentric Equatorial Frame . . . 81
A.2 The Geocentric Right Ascension-Declination Frame . . . 82
A.3 The Earth-Centered, Earth-Fixed Reference Frame and Geodetic Coor-dinates . . . 83
A.4 The East-North-Radial Reference Frame . . . 84
B Transmission Probabilities for Useful Geometries 87 B.1 Transmission Probabilities for 2D Parallel-Plate Ducts . . . 87
B.2 2D and 3D Transmission Probability Equivalence . . . 88
B.3 Transmission Probabilities for a Cylindrical Annulus . . . 88
Acronyms
ABEP Air-Breathing Electric Propulsion ABIE Air-Breathing Ion Engine
BL Borgnakke-Larsen
CFD Computational Fluid Dynamics
CIRA COSPAR (Committee on Space Research) International Reference Atmosphere DSMC Direct Simulation Monte Carlo
ECEF Earth-Centered, Earth-Fixed
ECSS European Cooperation for Space Standardization EP Electric Propulsion
EQU Equatorial Orbit
ESA European Space Agency EUV Extreme ultraviolet FMF Free Molecular Flow FUV Far-ultraviolet
GMAT General Mission Analysis Tool
GOCE Gravity field and steady-state Ocean Circulation Explorer HET Hall-Effect Thruster
HS Hard-Sphere
HWM Horizontal Wind Model
IADC Inter-Agency Space Debris Coordination Committee IJK From the unit vectors of the Earth-centered, inertial frame IPG Inductive Plasma Generator
IPT Inductive Plasma Thruster
ITER International Thermonuclear Experimental Reactor JB Jacchia-Bowman
LTAN Local Time at the Ascending Node MD Molecular Dynamics
MPI Message-Passing Interface
MSIS Mass-Spectrometer-Incoherent-Scatter
NASA National Aeronautics and Space Administration NRL Naval Research Laboratory
NS Navier-Stokes NTC No-Time-Counter PFG Particle Flow Generator RAM-EP Ram Electric Propulsion
TCE Total Collision Energy VHS Variable Hard-Sphere VKI von Karman Institute VLEO Very Low Earth Orbit VSS Variable Soft-Sphere VTK The Visualization Toolkit Sub- and Superscripts
1, i From the freestream to the thermalization chamber 1, o From the thermalization chamber to the freestream
2 From the thermalization chamber to the thruster’s outflow ∞ Freestream quantity ? Reduced quantity c In the chamber w Wall Symbols α Accommodation coefficient δ Mean molecular distance
˙
N Molecular flow rate η Collection efficiency λ Mean free path h i Average of a quantity ˆ
n Surface normal unit vector r Position vector
u Bulk velocity v Velocity vector
O Bachmann–Landau “big O notation X Property transported by the molecules µ Viscosity coefficient
ν Density ratio
Ω Right ascension of the ascending node ω Argument of perigee
ωE Angular velocity of the Earth in inertial space
Φ Flux of a molecular property Ψ Yaw angle
ρ Mass density
σ Collision cross section τ Transmission probability A Surface area
B Molecular flux received by a surface D Drag force
d Molecular diameter
f Velocity distribution function g Relative molecular speed
J2 Second zonal harmonic of the gravitational field k Boltzmann constant L Characteristic length m Molecular mass n Number density S Speed ratio
s Distance between elemental surfaces T Temperature
T Thrust force
t Time
ue Effective exhaust velocity
v Speed
vmp Most probable thermal speed
we East-wise wind velocity component
wn North-wise wind velocity component
Wp Ratio of real to simulated particles
X Reducing variable for the transmission probability dS Surface element
Kn Knudsen number Ma Mach number
Introduction
The Gravity field and steady-state Ocean Circulation Explorer (GOCE) spacecraft was launched in 2009 by the European Space Agency (ESA). It provided scientific data of exceptional quality and value on the Earth’s gravitational field and on the upper layers of the atmosphere, products that have broad application in the fields of geodesy, oceanography, solid-earth physics and glaciology. The scientific objectives required the mission to be operated in orbits with altitudes as low as 224 km. Earth orbits with a mean altitude below 250 km are referred to as Very Low Earth Orbits (VLEOs).
At these altitudes spacecraft are subject to an aerodynamic drag that, although small, must be compensated with some means of propulsion if the orbit needs to be maintained for a time comparable to ordinary mission lifetimes (in the order of 10 years). GOCE was equipped with an electric propulsion (EP) system, specifically with a QinetiQ T5 gridded ion thruster, able to continuously compensate for the drag with a thrust in the range of 1 to 20 mN, thus providing a so-called drag-free mode of operation.
The lifetime of the GOCE mission was limited by the ≈40 kg of Xe propellant on board. Once this was exhausted, the spacecraft re-entered the Earth’s atmosphere in less than three weeks.
1.1
Motivation
As proved by the GOCE mission, there are many advantages to fly spacecraft in VLEOs [47], including:
• Increased resolution for optical payloads: Optical payloads for spacecraft are usually equipped with high quality optics, which are limited in angular res-olution by the Rayleigh criterion, which states that the maximum theoretical resolution of an optical system X is limited by diffraction phenomena, as fol-lows:
X = 6.71 × 10−9 h
D (1.1)
where h is the spacecraft’s altitude and D the aperture diameter of the instru-ment. Therefore, for the same optical payload, decreasing the altitude improves the ground resolution.
• Increased radiometric performance: The power density of an electromag-netic signal P is proportional to the inverse square distance r from its source, P ∝ r−2. The power of the signal can be improved by reducing the operating altitude, which allows to use less sensitive instruments.
• Increased geospatial position accuracy: A shorter path length to the target increases the geospatial accuracy of the imagery. The spacecraft position and attitude uncertainties have a smaller arm length to propagate and hence the images taken can be geolocated with a greater accuracy.
• No end-of-life maneuver is required for disposal: ESA guidelines, follow-ing the InterAgency Debris Coordination Committee (IADC) recommendations, state that inactive spacecraft should be de-orbited within 25 years. Spacecraft in VLEO naturally re-enter the atmosphere in a short time frame because of aerodynamic drag.
• Lower risk of collision with space debris: Because of the higher atmospheric density, space debris have a lower residence time in the VLEO region than in higher orbits. For this reason, VLEO satellites may face a lower risk associated with space debris impact.
• Lower radiation levels: Radiation in Earth orbit has its origin in particles trapped by the Earth’s magnetic field (Van Allen belts), solar particles, and cosmic rays. The atmospheric layer in VLEO partially protects the spacecraft from solar activity effects, such as solar flares and coronal mass ejections. • Increased available launch mass: Because of the lower target altitude, flying
at VLEO would increase the available mass to be launched with the same amount of propellant.
The GOCE mission proved that it is possible to fully compensate for the aerody-namic drag with an EP system. Unfortunately for common mission lifetimes, which are in the order of 10 years, this is not possible, because an excessive propellant mass fraction would be required [20]. A possible solution would be to collect the residual atmospheric gas and use it as a propellant for the electric thruster, thus partially or completely eliminating the need for propellant storage on board of the spacecraft. This concept is referred to as Air-Breathing Electric Propulsion (ABEP) or RAM-EP.
The air-intake is one of the main components of an ABEP system. This device collects the atmospheric gas and feeds it to one or more electric thrusters in the appropriate conditions of density and flow rate.
Air-intakes can be classified into two categories: i) passive intakes, which are devices without any moving part and that rely on gasdynamic phenomena to collect the gas and increase its density, ii) active intakes, which would use ground proven vacuum pumping technologies, such as cryogenic, turbomolecular or diffusion pumps, to achieve the same goal. While the latter, possibly preceded by a passive stage, would allow to reach much higher pressures, more compatible with traditional electric thrusters, its feasibility has been judged to be low [9], with a significant impact in terms of mass
and/or power that would most likely make it less competitive than a conventional stored propellant solution. For this reason recent research has focused on the study and development of passive intake systems. A brief survey of the state of the art of these devices is shown in Section 1.2.
1.2
State of the Art of Air Breathing Electric
Propul-sion Systems
In this section, some of the most advanced concepts for ABEP systems are shown and their peculiar characteristics pointed out.
The reader will notice that, while the proposed designs differ under many points of view, such as the degree of integration with the payload, the type and the location of the electric thuster, the intakes all present features in common. This is because they all operate based on the same working principle, that will be illustrated in detail in Chapter 4.
Concept #1 from the Institute of Space and Astronautical
Sci-ence (ISAS), JAXA, Japan
Figure 1.1: ABEP system concept from the Institute of Space and Astronautical Sci-ence, Japan [34].
One of the first proposals for an ABEP system was presented in the early 2000s by K. Nishiyama [34] at the Institute of Space and Astronautical Science, Japan. Named Air Breathing Ion Engine (ABIE), it may be considered as an engine module which could be installed outside of a satellite similar to GOCE.
A microwave discharge ion engine was chosen, in virtue of its ability to efficiently achieve a moderate density plasma at low pressure. The reported pressure range suitable for the discharge is from 0.005 to 0.5 Pa. This rather large engine would be located in the aft side of the payload, and atmospheric gas would be able to reach
it by passing through a series of ducts (or “straws”) surrounding the payload and by being scattered via a reflector towards the chamber and ionization zone.
The backflow of particles from the chamber to the inlet, which degrades the per-formance, is limited by the narrow inlet ducts with a high length-to-radius aspect ratio.
Concept #2 from the Institute of Space Systems (IRS),
Ger-many
This concept is shown schematically in Figure 1.2. It uses an Inductive Plasma Thrusters (IPT), an electrode-less device based on inductively heated plasma gen-erators (IPG). In IPTs no critical components have direct contact with the plasma, therefore all issues concerning erosion are eliminated. A wide range of propellants can be used without the risk of performance degradation over time typical of other devices. Moreover, the IPT’s functional principle enables ignition at very low densities and removes the need for a neutralizer [43].
In this concept, the thruster is fed by a passive intake which is composed of small inlet ducts through which air can reach a thermalization chamber where it is trapped and led to the thruster. The authors of the concept did not exclude the possibility to use a device called “Knudsen Compressor” which, exploiting the phenomena of thermal creep flow, would increase the pressure of the feed gas [42].
Figure 1.2: ABEP system concept from the Institute of Space Systems, Germany [43].
Concept #3 from SITAEL, Italy
SITAEL has recently undertaken numerical and experimental research activities to test if Hall effect thrusters (HET) could represent a suitable system for ABEP. The resulting system, called RAM-EP, is shown in Figure 1.3. It is composed of a passive intake and a double stage HET, comprising of an ionization stage followed by an acceleration stage.
The preliminary geometry of the system is better defined with respect to the other concepts because a model, with a frontal area of 0.126 m2, was tested in a ground facility in 2017. The inlet ducts are arranged in a split-ring configuration, with constant radial increments and a number of subdivisions per ring aimed at keeping the aspect ratio of the ducts as uniform as possible.
Figure 1.3: The RAM-EP system concept by SITAEL, Italy [8].
The experimental setup, shown in Figure 1.4, consists of a particle flow generator (PFG), to generate a flow of particles as similar as possible to the freestream flow in orbital conditions, and the RAM-EP assembly. Both devices are placed on separate thrust balances in a vacuum chamber.
The tests proved the successful ignition and steady-state operation of the RAM-HET with the gas mixture 1.27 N2+ O2, representative of atmospheric conditions. It was, however, unable to produce a positive net thrust [8]. Andreussi reports that the intake component showed the expected behavior, with a performance that, although not directly measured, was sufficient for the ignition and acceleration of the propellant. The RAM-HET instead, while showing good ionization capability in the first stage, did not provide the expected acceleration in the second stage. This was identified as the main responsible for the lack of thrust.
Figure 1.4: SITAEL’s experimental setup for the test of the RAM-EP system [8].
A generic model of the intake, which can be applied to all the previous concepts, will be used in this thesis. Where the specification of a detailed geometry will be required, SITAEL’s concept will be used as a reference.
1.3
Objectives of the Thesis
The objective of this work is to investigate, through models of increasing complexity, the physical phenomena involved in the intake’s operation and their effect on its per-formance. In order to establish the operating conditions for the intake, a software will be developed to collect information on the atmospheric conditions along a determined orbit.
The investigation will then proceed with the analysis of the device through the simplest model, a lumped parameter model. This model will establish the performance parameters for the intake and its scaling laws.
The intake will then be analyzed through the Direct Simulation Monte Carlo (DSMC) method, a numerical technique for rarefied flows based on molecular sim-ulation. The DSMC method will allow to simulate the flow on a realistic geometry and with complex gas phase and gas surface effects.
Finally, a simulation method based on the view-factor technique will be developed, implemented, and tested on the same realistic geometry. While requiring the use of simplifying assumptions, this method could have an advantage in terms of computa-tional cost with respect to DSMC. Moreover, it could represent a means of verification for the DSMC calculations.
1.4
Outline of the Thesis
The thesis is structured as follows:
• In Chapter 2, we will introduce the fundamental concepts regarding rarefied gases that will be functional to the development of Chapters 4, 5, and 6. We will also introduce the Direct Simulation Monte Carlo (DSMC) numerical method. • In Chapter 3, we will investigate the flow conditions in VLEO, the operational
environment for ABEP systems, and we will develop a software to compute atmospheric conditions representative of a given mission scenario.
• In Chapter 4, the intake will be analyzed in the simplest level of complexity, by means of a lumped parameter model. From this model the two fundamental performance parameters of the intake will emerge: the density ratio ν and the collection efficiency η; together with the scaling laws which govern the device. • In Chapter 5, DSMC will be employed for a more detailed and accurate analysis,
providing precious insight on many flow phenomena.
• Chapter 6 will be devoted to the development and verification of a simulation method for free molecular flows, based on view-factors between discrete surfaces. The method will be tested on the intake’s geometry and the results compared to DSMC.
• Finally, Chapter 7 will draw the conclusions of the thesis work and will illustrate some possible future developments.
Physical Models and Numerical
Methods for Rarefied Gases
This chapter will provide the essential information regarding rarefied gases that will be used in the following chapters to investigate the intake device. Detailed information regarding rarefied gases can be found for instance in [14] and, with more focus on numerical simulation, in [13] and [16].
A gas flow may be modeled at the macroscopic or the microscopic level. At the macroscopic level, the gas is regarded as a continuous medium and it is described by familiar flow properties, such as the velocity, density, temperature, and pressure. The accepted mathematical model for a continuous gas is provided by the Navier-Stokes (NS) equations. At the microscopic level instead, the gas must be described considering the properties of each individual molecule it is composed of: position, velocity, and internal state. The accepted mathematical model for gases at the microscopic level is the Boltzmann equation.
In the Boltzmann equation, the fraction of molecules that occupy a given “position” in the physical and velocity space is the only dependent variable. This combined space is called phase space. In the most general case, it has six dimensions. Due to this high dimensionality of the microscopic model, the macroscopic description is much more amenable to analytic and numerical solutions. Unfortunately, the transport terms in the NS equations fail when the gradients of the macroscopic variables extend on length scales of the order of the so-called mean free path.
The mean free path λ is the average distance traveled by the molecules between collisions. It is therefore natural to express the degree of rarefaction of a gas through the ratio of the mean free path to a characteristic dimension L. This ratio is called Knudsen number (Kn).
Kn = λ
L (2.1)
Taking L as the length scale of the macroscopic gradients, it is generally accepted that the error in the continuum formulation becomes significant for Kn > 0.01.
Two limits for the degree of rarefaction are identifiable: For Kn 1, the transport terms in the NS equations vanish and the Euler equations for inviscid flow are obtained. Conversely, if Kn 1, a collisionless, or free molecular flow is obtained, in which the
gas molecules only interact with solid objects and not between themselves.
2.1
Dilute Gases
Consider a gas consisting of a single species. Such a gas is referred to as a simple gas. The number of molecules per unit volume is called number density n. On average, the available volume for each particle is equal to 1/n. Therefore, the mean molecular distance will be δ = n−1/3.
A simplistic model for the molecules regards them as hard elastic spheres of diam-eter d. This is the so-called hard-sphere (HS) model. Clearly, these spheres collide whenever the distance between their centers decreases to d. More realistically, the molecules exert a force on one another which is essentially function of their distance. A typical intermolecular force is shown in Figure 2.1. The force is effectively zero at large distances, it becomes weakly attractive as the distance is diminished, and de-creases again to become strongly repulsive for even shorter distances. This class of potentials takes the name of Lennard-Jones potentials.
F (r)
r
A
ttraction
Repulsion
Figure 2.1: Typical intermolecular force F as a function of the distance r between the centers of the molecules.
For sufficiently low densities, the molecular distance δ is large compared to the molecular diameter d. In this condition, each molecule will mostly be moving outside of the range of influence of the other molecules. The small number of interactions therefore concern two molecules at a time. In this sense, binary collisions are over-whelmingly more probable than three-body collisions. This situation defines a dilute gas.
The macroscopic quantities in a continuum model are defined at a point in space. However, values at a “point” must be sampled from the properties of the molecules within a small volume V enclosing the point. If the number of molecules in the volume nV is not high enough, macroscopic properties such as bulk velocity, density, and temperature, may noticeably fluctuate from their average value.
Figure 2.2 shows the different flow regimes and approximations for a gas. The horizontal axis shows the density in logarithmic scale, normalized with the atmospheric density at sea level ρ0, and the corresponding altitude in the Earth’s atmosphere.
Higher altitudes and lower densities occur towards the left. The vertical axis shows the length scale L considered, with L = 1 m marked with a dashed line. This plot is valid for a nitrogen gas.
The gas can be considered dilute in the region to the left of the line δ ≈ d. For nitrogen gas or air, the limit density for a dilute gas is about 1.5 ρ0.
In the region Kn < 0.01 (right side of the limit line), methods from continuum gas dynamics provide an accurate model. For Kn > 10, conversely, very few collisions occur in the volume of interest and the flow is approximately free molecular. The intermediate region of 0.01 < Kn < 10 is often referred to as the transition regime. Here, collisions in the gas phase cannot be neglected, but the continuum model leads to inaccurate results. It follows that, in the transition and free molecular regimes, a microscopic level approach must be adopted.
In the region below the line nL3 = 1 × 106, the number of molecules in the charac-teristic volume is less than 106. Therefore, statistical fluctuations of the macroscopic
properties may be encountered.
10−15 10−12 10−9 10−6 10−3 100 ρ/ρ0 = n/n0 [−] 10−9 10−7 10−5 10−3 10−1 101 103 Characteristic Length L [m] -25 0 25 50 75 100 125 150 175 200 225 250 Approximate Altitude [km] Kn= 0.01 Kn= 10 Kn= 1000 Molecular flow Con tinuum flow nL3 = 1 ×106 Fluctuationsimp ortan t δ ≈ d
2.2
Kinetic Theory
2.2.1
The Distribution Function
At a particular instant, the state of a gas at the microscopic level is defined by the position r = (x, y, z), velocity v = (vx, vy, vz), and internal state of each molecule.
The number of molecules in a real gas is so large that the adoption of a statistical description in terms of probability distributions is required.
It is convenient to consider the position space and the velocity space as a single, six-dimensional phase space. In the phase space, the dynamic state of each molecule is appropriately described as a point.
Let dN (r, v, t) denote the number of points inside the volume element dr dv around the phase space coordinates (r, v), at the instant t. The distribution function f (r, v, t) is defined as the density of the representative points in the phase space, as follows:
f (r, v, t) = dN (r, v, t)
dr dv (2.2)
The number density n(r, t) is a macroscopic variable, which can be defined in the phase space as the number of points in dr around the coordinates r at the instant t, per unit volume, irrespective of velocity. It can be obtained from the integral:
n(r, t) = 1 dr ∞ Z −∞ dN (r, v, t) = ∞ Z −∞ f (r, v, t) dv (2.3)
In general, if some property X (r, v, t) is associated to each molecule, the distribution function allows to calculate its average value, according to the following equation:
hX (r, v, t)i = 1 n(r, t) ∞ Z −∞ X (r, v, t) f (r, v, t) dv (2.4)
Note that the average value is only a function of position and time, since the integration is over the whole velocity space.
When the particle property is an integer “power” of velocity, this process establishes a moment of the distribution function. The first moment of the distribution function is the average or bulk velocity. Through higher order moments one can obtain the momentum flow dyad and the total energy flow triad, which are analogous to the macroscopic quantities of momentum and energy fluxes.
We can also compute the flux of property X transported by the molecules through a surface immersed in the gas. The particles with velocity between v and v + dv that will cross the surface dS in the interval between t and t + dt must be initially in the prism of base dS and side v dt, as indicated in Figure 2.3. The volume of the prism is: dr = v · dS dt = v · ˆn dS dt (2.5) where ˆn is the unit vector normal to the surface.
v dt ˆ n · v dt v dS ˆ n
Figure 2.3: Prism of volume dr = v · dS dt = v · ˆn dS dt containing the molecules that will cross dS in the time interval dt.
From the definition of f , the number of molecules in the volume of the prism that have velocity between v and v + dv is:
f dr dv = f v · ˆn dS dt dv (2.6)
The total amount of property X that crosses dS is obtained by multiplying this number of particles by X and integrating the result over all possible velocities. If the flux Φ is desired, which is the amount of property transported per unit area and per unit time, we must then divide by dS dt. The final expression is:
Φ(X ) = Z
v · ˆn > 0
X f v · ˆn dv (2.7)
2.2.2
The Boltzmann Equation
Consider an element of phase space dr dv that does not change in shape or size with time. The rate of change of the number of particles inside this volume will be:
∂N ∂t =
∂
∂t(f dr dv) (2.8)
There are three physical processes affecting the rate of change of the number of particles inside the volume:
• Convection across dr due to molecule motion at velocity v.
• Convection across dv due to molecule acceleration a caused by an external force. • Intermolecular collisions that change particle velocities, hence scatter particles
Accounting for all these effects, we can obtain the Boltzmann equation: ∂f ∂t + v · ∇f + a · ∇vf = δf δt coll (2.9) where ∇ is the traditional del operator and ∇v is the del operator in the velocity space:
∇v = ˆ x ∂ ∂vx + ˆy ∂ ∂vy + ˆz ∂ ∂vz .
This is the basic governing equation for all dilute gas dynamics. The right hand side is called the collision term and, even when only binary collisions are considered, is a complex integral quantity. The integro-differential nature of the equation makes it challenging to solve analytically or with conventional numerical methods. The possible solution methods will be briefly discussed in Section 2.3.
A significant contribution to the theory of rarefied gas dynamics was the devel-opment of the Chapman-Enskog analysis: a mathematical framework connecting the Boltzmann to the Navier-Stokes equations. It is possible to find the precise distribu-tion funcdistribu-tion that reduces the Boltzmann to the Navier-Stokes equadistribu-tions, therefore providing a theory linking interatomic forces to macroscopic transport properties: the coefficients of viscosity, thermal conductivity, and diffusivity.
2.2.3
The Equilibrium State
The equilibrium distribution function is the time-independent solution of the Boltz-mann equation in the absence of external forces. In the equilibrium state, the inter-actions between molecules do not cause any change in the distribution function with time and there are no spatial gradients in the number density.
If we assume that there are no external forces acting on the system (a = 0) and that the particles are uniformly distributed in space, the distribution function is ho-mogeneous (∇f = 0). Since we are looking for a steady-state solution, it is also time-independent (∂f /∂t = 0). According to Equation 2.9, the equilibrium distribu-tion funcdistribu-tion satisfies:
δf δt
coll
= 0. (2.10)
It can be shown that the distribution satisfying Equation 2.10 is the Maxwell-Boltzmann velocity distribution function:
f (v) = n m 2πkT 3/2 exp m |v − u| 2 2kT ! , (2.11)
where m is the molecular mass of the gas, k is the Boltzmann constant and u is the bulk velocity of the gas. Quantity T follows the thermodynamic definition of the kinetic temperature as average kinetic energy of the molecules:
3 2nkT = 1 2nmhv 2i = 1 2m ∞ Z −∞ f v2dv (2.12)
Next we can consider the distribution of speeds F (v), where v = |v|. This is obtained by integrating Equation 2.11 over all directions for the velocity v. We consider a gas at rest (u = 0), or a reference frame that is moving with the bulk velocity of the gas. The result is:
F (v) = 4πn m 2πkT 3/2 v2 exp −mv 2 kT (2.13) This results in a distribution such as the one of Figure 2.4.
F (v)
v vmp hvi phv2i
Figure 2.4: Distribution of speed for a Maxwell-Boltzmann distribution function. Three characteristic speeds can be identified: the most probable speed vmp,
corre-sponding to the speed for which F (v) is maximum, the average speed hvi, and the root mean square speed phv2i. They have the following values:
vmp= r 2kT m (2.14) hvi = r 8kT πm = 2 √ π vmp (2.15) phv2i = r 3kT m = r 3 2vmp (2.16)
A number of useful properties for a gas at equilibrium can be derived from the distribution function. Two of them, used extensively for this thesis, are i) the mean free path λ, which is the mean distance traveled by molecules between consecutive collisions, and ii) the flux Φ of gas molecules through a surface. These quantities will now be briefly introduced.
It is possible to show that the mean free path for a simple gas at equilibrium is: λ = √1 2nσ = 1 √ 2nπd2 (2.17)
where σ is the collision cross section, which is πd2 for hard-sphere gas molecules of diameter d. If multicomponent mixtures or more complex collision models are involved, the value of λ can be computed using the expressions in Appendix D of [16].
Using Equation 2.7, we can compute the flux of molecules that crosses a surface immersed in the gas at equilibrium. The gas has a component vxof the bulk velocity in
the direction of the surface normal, therefore v · ˆn = vx. The flux is obtained by setting
the transported property X = 1 in Equation 2.7 and integrating in the appropriate velocity space: Φ = ∞ Z vz=−∞ ∞ Z vy=−∞ ∞ Z vx=0 f (vx, vy, vz) vxdvxdvydvz (2.18)
Note that the lower integration limit for vx is zero because backward moving molecules
will never reach the surface. Carrying out the integration, the final result is: Φ = 1 4n r 8kT πm n e−Sx2 +√πS x[1 + erf(Sx)] o (2.19) where Sx = vx/vmp is the speed ratio of the gas in the direction normal to the surface.
Notice that the speed ratio can be put in relation to the Mach number Ma as follows: S = V vmp = V r 2kT m =r γ 2Ma (2.20)
where V is the bulk velocity and γ is the ratio of specific heats of the gas. Two important limits can be considered for Equation 2.19:
• Sx → 0: No stream velocity in the x direction. The expression for the flux
reduces to: Φ = 1 4n r 8kT πm = 1 4n hvi (2.21)
Therefore, even if there is no average velocity towards the surface, there is a finite mass flux due to the thermal velocity of the molecules. This flux is therefore called thermal flux.
• Sx 1: The stream in the x direction is hypersonic. In this case, the expresison
reduces to:
Φ = n vx (2.22)
In this case, the flux of particles due to the random motion is insignificant with respect to the flux due to the very high bulk velocity. This flux is ususally called hypersonic flux.
2.3
Numerical Simulation of Rarefied Flows
Numerical methods for the solution of the Boltzmann equation can be divided into two different classes: some are based on the direct discretization of the equation, while others are based directly on the simulation of the physics of the flow. For these, an effort must be later made to link the methodology to the Boltzmann equation with sufficient rigor to claim that they are indeed providing its solution.
2.3.1
Methods Based on the Solution of the Boltzmann
Equa-tion
The most direct approach to solving the Boltzmann equation numerically is to apply the conventional methods of computational fluid dynamics (CFD). The velocity distri-bution function would be the only dependent variable for an atomic gas and it could be discretized in the phase space. However, a major problem arises because of the large number of degrees of freedom of the phase space, and consequently the excessively high number of points or elements required for the representation. In addition, the velocity space domain is in theory of infinite extent, requiring restricted bounds to be chosen which are not so easy to predict in some cases [13]. This method goes by the name of direct Boltzmann CFD and, despite the difficulties, it has been applied successfully to simple problems.
2.3.2
Methods Based on the Physics of the Flow
The difficulties in the solution of the Boltzmann equation fundamentally reside in the particle nature of the gas, but this feature can be used to one’s advantage by developing physically based simulation methods. The gas in this case is modeled by a large number of simulated molecules in a computer. The machine stores the position vector, velocity vector and internal state of each of the simulated molecules, evolving them in time.
The first of such approaches was the molecular dynamics (MD) method. Here, the molecular motion, the “collisions”, and the interactions with the boundaries are treated deterministically. The motion of the molecules in particular can involve the effect of intermolecular forces, extending the validity of this method to dense gases (while the Boltzmann equation is restricted to dilute gases).
Following the reasoning of Bird [13], the simulation of a flow field extending 30 mean free paths in each direction would require a number of molecules nλ3 ∝ (n0/n)2,
corresponding approximately to 108 molecules at standard density. The same problem
at 100 times the standard density would require instead just 104 molecules. For this
reason, MD calculations are usually restricted to dense gases (or liquids). This also implies that, for rarefied gas problems, involving densities in the order of 10−6 standard densities, the required number of simulated molecules would become exceedingly high, in the order of 1020.
The Direct Simulation Monte Carlo Method
The Direct Simulation Monte Carlo (DSMC) method was first introduced by G. A. Bird in 1963 [13]. DSMC is part of the physics based methods but, contrarily to MD, it relies on the assumption that the gas is dilute. This allows the uncoupling, over a small time interval, of the molecular motion and the intermolacular collisions. As long as this time step is small in comparison with the mean collision time, the results have been proven to be independent on its actual value.
An enormous reduction in computational cost is achieved by regarding each sim-ulated particle as a fixed number Wp of identical real particles. This is typically a
very large number, in the order of 1010, bringing the number of molecules that would
have to be simulated by MD to an acceptable value. The stochastic or probabilistic nature of this method lies in many aspects of the implementation: from the generation of particles and their properties at the boundaries to the calculation of post-collision conditions and chemical reactions. The diagram of Figure 2.5 shows the fundamental algorithm for DSMC, which is summarized in the following steps:
• The first operation consists in the initialization of the computational grid and of the surfaces representing the physical objects. The grid has a very different function from the grid of classic CFD solvers using finite difference or finite volumes schemes. In DSMC, the particles are sorted by grid cell according to their position, and pairs of particles are chosen randomly within each cell as candidates for a collision. It is important that the grid resolves the local mean free path λ, otherwise the algorithm would consider non realistic interactions between distant particles that wouldn’t interact in reality.
• The algorithm proceeds by seeding the domain with an initial population of particles, if desired. This can usually reduce the time needed to reach a steady-state.
• Here the main loop of the algorithm begins, first step of which is to insert in the domain the particles coming from outside the domain, initializing them with velocities and positions, compatible with probability distribution functions pre-dicted by kinetic theory.
• The particles are then moved according to simple Newtonian laws of motion, accounting for body forces and checking for interactions with boundaries and surfaces. The particles must then be sorted in the respective grid cell.
• Once the particles are sorted, intermolecular collisions are performed between pairs of particles within each cell, computing the post-collision properties for the particles according to the chosen models for scattering, chemical reactions and internal energy transfer.
• Finally, the macroscopic flow quantities of interest (number density, velocity, temperature, etc.) are sampled, computing them as averages of the properties of the single particles within each cell. Another requirement for the size of the
Start
Set up simulation grid and surfaces Distribute particles into cells with their initial position and velocity
Insert particles at boundaries
Move individual particles and compute interactions with boundaries
Sort particles into cells
Select collision pairs and perform intermolecular collisions
Sample flow properties
Output results
N > Niter Stop
True False
Figure 2.5: Flowchart for the typical DSMC algorithm. Adapted from [40].
cells therefore is that they must be compatible with the desired resolution for the flow field. These quantities can be output and saved to a file in the desired format.
• At this point, the loop will continue until the desired number of steps Niter has
been reached. This number has to be set before starting the simulation.
DSMC is in principle capable of both transient and steady state simulations. In both cases, in fact, a transient simulation is performed. In terms of macroscopic properties the flow will eventually reach a steady-state or a quasi-steady-state. When a steady-state solution is desired, sampling of the flow properties is started only after
a sufficient initial transient, and the samples are averaged over multiple time steps to reduce the inevitable statistical scatter.
Clearly, the DSMC algorithm does not recourse to a mathematical model of the flow, rather its foundation is deeply physical and based on gas kinetics. It has been demonstrated, however, that the Boltzmann equation can be derived through a sim-plified version of the DSMC procedures [12]. Even if the relationship has not been demonstrated for modern algorithms, the experimental validation that the method has received since its introduction constitutes proof of its validity.
For the intake, where the gas is certainly dilute and the flow regime is expected to be free molecular or, in the worst case, transitional, DSMC has been identified as the optimal method for simulation. The following chapter will present the chosen DSMC software, SPARTA, and briefly illustrate its features.
2.4
The SPARTA DSMC Software
The DSMC simulator SPARTA (Stochastic PArallel Rarefied-gas Time-accurate An-alyzer) [25, 1] is an open source software written by M. A. Gallis and S. J. Plimipton at Sandia National Laboratories, a US Department of Energy laboratory. It is written in C++ and parallelized with the MPI message-passing library for excellent scaling on distributed- or shared-memory parallel machines and clusters. SPARTA can perform 2D, 2D-axisymmetric, or 3D simulations.
2.4.1
Simulation Grid and Solid Objects
The grid, internally generated by the software, is a Cartesian, hierarchical grid with binary refinement. Cartesian means that the faces of a grid cell are aligned with the Cartesian (x, y, z) axes. Hierarchical means that it follows a quadtree (2D) or octree (3D) organization, where cells can be recursively refined by a factor of 2 in each coordinate direction. An illustration of the octree structure is shown in Figure 2.6. This is a very important feature for the simulation of flows with large non-uniformity of the density. Moreover, it enables to have a more uniform number of particles per cell accross the domain.
For the solid objects, SPARTA accepts triangulated (3D) or line-segmented (2D) surfaces. Surface elements are assigned to the grid cells they intersect with, so that particle/surface collisions can be efficiently computed. Since the Cartesian grid is not fitted to the body, the software internally computes the intersection between the surface and each grid cell. If a cell is intersected by the surface, it can become one of two types: a cut cell, which has only one flow region, or a split cell, which has more than one disjoint flow regions or sub cells. An illustration is provided in Figure 2.7.
Figure 2.6: Illustration of a Cartesian octree grid and the associated data structure in three successive levels of refinement.
cut cell split cell
sub cell
Figure 2.7: Illustration of cut and split cells, defining flow regions and solid regions. Adapted from [1].
2.4.2
Boundary Conditions
A variety of boundary conditions are provided in SPARTA, including subsonic bound-ary conditions. Two types of boundbound-ary conditions have been used for this work: the hypersonic “outflow” (which also works as an inflow) and a vanish condition. The latter is a very simple condition, consisting of the elimination of whatever particle reaches the boundary. This simulates the entrance of a particle into a region from which it cannot come back.
For hypersonic outflow boundary conditions the number density n, velocity ¯V, and translational temperature Ttr of the flow are specified. A standard procedure is
implemented in SPARTA, where the number of particles that should enter the domain from a given boundary surface of area A and normal ˆn during the timestep ∆t is computed using Equation 2.23, which is derived from Equation 2.19.
FDSMC= A ∆t Wp n vmp 2√π n e−Sn2 +√πS n[1 + erf(Sn)] o (2.23) where Sn = ( ¯V · ˆn)/vmp is the speed ratio normal to the surface.
Each of the FDSMCparticles is created with properties that are randomly generated
using an acceptance-rejection algorithm. The properties are sampled from the appro-priate Maxwellian distribution at the macroscopic temperature and bulk velocity. The particles that reach the outflow boundary with an outward velocity simply vanish. A detailed description of the procedure can be found in [16].
2.4.3
Particle-Surface Collisions
SPARTA implements Maxwell’s gas-surface interaction model. Maxwell’s model con-siders two kinds of interactions: specular and diffuse interactions.
In a specular interaction, occurring with probability (1 − α) the particle rebounds elastically as if it was hitting a flat surface. The collision results in an inversion of the component of the molecule’s velocity normal to the surface and no change in its tangential component. Thus, the angle of reflection is the same as the angle of incidence. Meanwhile, the molecule’s thermal energy is assumed to remain the same. In a diffuse interaction, occurring with a probability (α), the molecule attains thermal equilibrium with the surface and then rebounds from the surface according to the Maxwellian velocity distribution at the local surface temperature. The resulting distribution function of the rebound angle is proportional to the cosine of the angle between the velocity and the surface normal. For this reason it is said that the diffuse reflection follows a cosine law.
For this model, therefore, two parameters should be specified: the surface tem-perature and the accommodation coefficient α. A representation of the two types of interaction is shown in Figure 2.8.
α = 0 α = 1
Figure 2.8: Illustration of the extreme cases of Maxwell’s gas-surface interaction model: fully specular reflection (α = 0) and fully diffuse reflection (α = 1).
2.4.4
Elastic Gas Phase Collisions
For the gas phase collisions, models of increasing level of realism and complexity are available: the simplest is the Hard-Sphere (HS) model, in which the collision cross section σ for a specific species-pair is a constant and the scattering is isotropic. This leads to a viscosity dependence with temperature that does not agree with the experiments. An improvement is obtained in the Variable Hard-Sphere (VHS) model, by allowing the cross section to have a dependence on the relative velocity of the
collision pair g, according to Equation 2.24. σTVHS = πd2ref gref
g 2ν
(2.24) In this case, the simulated viscosity under equilibrium conditions follows itself a power law, as follows: µVHS = µVHSref T Tref ω (2.25) where ω ≡ ν + 1/2.
The parameters of the VHS model can be set to obtain the desired viscosity coef-ficient µ or self diffusion coefcoef-ficient D, but not both independently.
The Variable Soft-Sphere (VSS) model remediates by using the same cross section of VHS, but with anisotropic scattering. Here the scattering angle depends on the impact parameter through the parameter α. The VSS scattering model is particularly useful in simulating mixtures in which mass diffusion must be accurately modeled. The VSS model has been chosen for the simulations of the intake as it is the most realistic model available in SPARTA.
DSMC simulations involving air typically include 11 chemical species. Such a mixture is referred to as “NASA air 11”. The values of the parameters used by the VSS model for these species are shown in Table 2.1.
Species d [m] ω [−] Tref [K] α [−] O2, O2+ 3.96 × 10−10 0.77 273.15 1.4 N2, N2+ 4.07 × 10−10 0.74 273.15 1.6 O, O+ 3.00 × 10−10 0.8 273.15 1 N, N+ 3.00 × 10−10 0.8 273.15 1 NO, NO+ 4.00 × 10−10 0.8 273.15 1 e 7.00 × 10−13 0.5 273.15 1
Table 2.1: Parameters of the VSS model for the 11 air species.
The values in Table 2.1 refer to the collision of molecules of the same species. The parameters for the collision of unlike-species are considered in SPARTA as the arithmetic average of the parameters for the two species.
In DSMC, the probability that a pair of particles in a given cell collide is computed as:
PDSMC= σ(d, g) g Wp∆tDSMC/VDSMC (2.26)
where σ is the collision cross section, generally dependent on the relative velocity of the collision pair g, and VDSMC is the volume of the DSMC cell. Since the probability
of collision is applied to each particle individually, DSMC can reproduce correctly the nonequilibrium collision rate and can be accurate for arbitrary (non-Maxwellian) velocity distribution functions.
Since typically many particle pairs are present in each cell, algorithms have been developed that drastically improve the numerical efficiency without loss of accuracy.
The algorithm implemented in SPARTA is the No-Time-Counter (NTC) algorithm of Bird, the details of which can be found in [13]. The fundamental idea of the NTC algorithm is that, instead of applying a small collision probability to many particle pairs, one can obtain identical results by applying a larger probability to fewer particle pairs.
2.4.5
Inelastic Gas Phase Collisions and Chemical Reactions
Once a pair of simulated particles has been selected for a collision, it is further consid-ered for chemical reactions and translational-rotational-vibrational energy transfer.
SPARTA must calculate the probability that a given chemical reaction occurs. The model employed for our simulations is the Total Collision Energy (TCE) model. The TCE model allows for reaction probabilities to be defined based on known, measured, reaction rates in thermal equilibrium. The model is described in detail in [13]. The re-actions considered for the NASA air 11 mixture are those of Park’s reaction mechanism [35].
For inelastic collisions, the rotational and vibrational energy transfer probabilities are computed from species-specific constants called collision numbers. If a molecule is selected for energy transfer, post-collision energies are computed according to the redistribution method of Borgnakke and Larsen (BL) [15]. With this method, the post-collision energy for each degree of freedom is sampled from an equilibrium distribution that corresponds to the collision energy of the individual collision. This method allows to recover the correct behavior where, over successive collisions, molecular energies and distribution functions relax toward a state that satisfies the equipartition of energy.
2.4.6
The Ambipolar Approximation
The ambipolar approximation is a computationally efficient way to model low-density plasmas which contain ions and electrons. If the ambipolar approximation is activated in SPARTA, electrons are not free particles which move independently. This would require a simulation with a very small time step due to the small mass and high speed of the electrons. Instead each ambipolar electron is assumed to stay “close” to its parent ion, so that the plasma gas appears macroscopically neutral. Each pair of particles thus moves together through the simulation domain, as if they were a single particle, which is how they are stored within SPARTA [1].
The Very Low Earth Orbit
Environment
A Very Low Earth Orbit (VLEO) is an Earth orbit with an altitude below 250 km. Spacecraft in these orbits experience lifetime-limiting levels of aerodynamic drag. The performance of an Air-Breathing Electric Propulsion system operating in VLEO will be influenced by the characteristics of the atmospheric gas, including its composition, density, temperature, and velocity. Unfortunately, these parameters are highly vari-able, depending not only on the altitude, but also on other conditions related to the Earth’s climate and the solar activity.
In many ABEP studies, the authors estimate their boundary conditions for the freestream gas through the use of space- and time-averaged data, usually for one or more fixed levels of solar activity. Representative values from averaged data are certainly a good starting point for the design of an ABEP system, but the variability of the conditions is not considered. This could become important when the margins for the design of the operational conditions has to be analyzed. For this reason, one of the objectives of this thesis is to develop a software that is able to evaluate the atmospheric conditions encountered by a spacecraft during a given mission scenario, using the available models for the atmosphere.
In the following sections, the models chosen to obtain the atmospheric properties will be presented and the details of the calculations performed by the software will be illustrated.
3.1
Atmospheric Models
The Earth’s atmosphere can be classified into different regions based on temperature, composition, or molecular collision rate.
The region of interest for ABEP spacecrafts is entirely contained in a region called thermosphere. This layer of the atmosphere extends from a minimum in the tem-perature profile called mesopause (≈90 km) to the altitude where the vertical scale height is approximately equal to the mean free path (≈500 km, depending on solar and geomagnetic activity levels). Below the mesopause, the atmosphere is well mixed and the mixture of gases is distributed vertically in a hydrostatic equilibrium that
de-pends on the mean molecular mass. In the thermosphere, instead, diffusive equilibrium dominates and each individual species is distributed according to its own molecular mass. Lighter gases tend to spread out over a greater vertical extent and at high levels predominate over the heavier ones [2]. In addition, solar radiation in the extreme ultraviolet (EUV) band is almost completely absorbed within this region, causing a sharp temperature increase to values in the order of 1000 K and the photochemical dissociation of O2.
Figure 3.1 shows a typical composition and temperature profile in the thermo-sphere. From an altitude of 100 km the composition departs from the composition of the lower, well mixed layers and N2 and O become the prevalent components. Typical
number densities in the region of interest from 180 to 250 km are between 1015 and
1017m−3. 100 200 300 400 500 Altitude [km] 1011 1013 1015 1017 1019 Num b er densit y [m − 3 ] N2 O O2 Ar N He H Total density
Composition and Temperature, NRLMSISE-00
200 400 600 800 1000 T emp erature [K] Temperature
Figure 3.1: Altitude composition and temperature profile from NRLMSISE–00 at the equator on 21 March 2015, 18:00 Local Time.
Two different models have been used as a source for the thermochemical atmo-spheric data: NRLMSISE-00 (shortened NRLMSIS) and JB2008 (shortened JB08). The standards of the European Space Agency (ECSS–E–ST–10–04C) [4] recommend the use of both NRLMSIS and JB2006 (the predecessor of JB2008) for calculations regarding the neutral atmosphere. In particular, the use of NRLMSIS is suggested for temperatures and composition calculations, while JB06 is suggested for total atmo-spheric densities. These models have a long history of development and refinement dating back to the early years of spaceflight, as illustrated by Figure 3.2.
The third model that has been used, HWM14, is a model of the horizontal wind in the atmosphere. It is also recommended by the ECSS standards [4]. The horizontal wind influences the direction of the flow arriving at the spacecraft. This could cause a misalignment of the incoming flow with respect to the axis of the intake system.
Figure 3.2: Development of the models of the Earth’s atmosphere. The models em-ployed in this work are shown underlined in the bottom part of the figure. They are all derived from past models and are some of the most recent available.
3.1.1
Models for Density and Composition
NRLMSISE–00 thermospheric model
The MSIS-class models for composition, total mass density, and temperature from which NRLMSIS is derived, have been the reference empirical models for the upper atmosphere for decades. A. E. Hedin based past MSIS-class models on over two decades of data [26, 27]. In addition, orbital decay data on which the Jacchia models are primarily based, have been included in the newer version [36].
One can interpret the NRLMSISE–00 (Mass Spectrometer and Incoherent Scat-ter Radar Extended) model as a semi-empirical “view” of its extensive underlying database. The model takes statistical variability into account and interpolates, or extrapolates, to estimate composition and temperature for conditions not covered by the database. Extending from the ground to the exobase, NRLMSIS provides temper-ature, number densities of atmospheric species (N2, O2, O, N, He, Ar, H), total mass
Jacchia-Bowman 2008 (JB2008) empirical thermospheric density model The Jacchia-Bowman Empirical Thermospheric Density Model JB2008 [5] was devel-oped starting from the JB2006 model which was in turn based on the CIRA72 model of the atmosphere. The CIRA72 model integrates the diffusion equations using the Jacchia 71 temperature formulation to compute density values for a given geographical location and solar conditions.
Driving solar indices are computed from on-orbit sensor data, which are used for the solar irradiances from the extreme (EUV) to far ultraviolet (FUV). The indices drive exospheric temperature equations, developed to represent the thermospheric EUV and FUV heating. The model accounts for the semiannual density cycle that results from EUV heating. Geomagnetic storm effects are modeled using the Dst index as the driver of global density changes.
The model is validated through comparisons with accurate daily density drag data previously computed for numerous satellites in the altitude range from 175 to 1000 km.
3.1.2
The HWM14 Model for the Horizontal Wind
This empirical model describes the atmospheric wind fields from the surface to the exobase (≈450 km) as a function of latitude, longitude, altitude, day of the year, and hour of the day. The empirical formulation of the model is able to represent the predominant variations of the middle and upper atmosphere to an acceptable degree. These oscillations are in fact periodic, unlike the quasi-random weather in the troposphere, driven by in situ solar heating under the cyclical influence of the Earth’s rotation, tilt, and orbit around the Sun.
An appropriate mathematical basis set is chosen to represent the features. The unknown model parameters are determined via an optimal estimation procedure over the data collected by 73 × 106 measurements from 44 different instruments, both
ground-based and space-based, spanning over 60 years [22].
3.2
Development of a Software to Calculate the
Flow Conditions for a Spacecraft in VLEO
We have written a code in Python 3 that allows to obtain high fidelity statistics on the flow conditions encountered by a spacecraft in VLEO, mainly in terms of species density, temperature, and velocity. The procedure for the computation consists of five consecutive steps:
1. In the first step, the orbit, defined by its initial conditions, is propagated for the desired time interval. The state vector (position and velocity vectors of the spacecraft) is computed at every time step and stored.
2. In the second step, the position from the state vector is expressed in geodetic coordinates, which will be used as input for the atmospheric models.
3. The third step computes and stores the atmospheric data from the three models for each date/time and geodetic coordinates sample.
4. The following step is to post-process this information, which includes the calcu-lation of non-dimensional and derived quantities such as the yaw angle or the mean free path of the gas molecules.
5. Finally, the last step computes and outputs the statistics on the data (mean, standard deviation and minimum/maximum value), and produces the plots of the selected quantities.
The details of the implementation will be presented in the following paragraphs.
3.2.1
Calculation Procedure
Step 1: Orbital Propagation
For the assessment of the atmospheric conditions along an orbital trajectory, it is necessary to compute the position and velocity of the spacecraft at discrete instants in time. This operation is called orbital propagation. A set of initial conditions (in this case the set of standard orbital elements at a certain time) and boundary conditions (the disturbance forces) are given. The initial conditions represent an input to the software because they are part of the mission definition. Boundary conditions, on the other hand, deserve some discussion.
If we assume that the drag force is balanced by the thrust of the ABEP system, the principal disturbance force is caused by the Earth’s oblateness. Earth’s oblateness causes a zonal variation (with latitude) of the gravitational force. The major effects of oblateness on orbits is due to J2, the second zonal harmonic of the gravitaional field.
This disturbance causes a change of all the orbital parameters with time, which would otherwise be constant. It is possible to show that the only parameters that undergo a secular variation are the right ascension of the ascending node Ω and the argument of perigee ω. From [6], the average rate of change of the two parameters is given by Equations 3.1 and 3.2. ˙ Ω = − 3 2 √ µJ2R2 (1 − e2)2a7/2 cos i (3.1) ˙ ω = − 3 2 √ µJ2R2 (1 − e2)2a7/2 5 2 sin 2i − 2 (3.2) For orbits of the right inclination (from 95° to 100°), the orbital plane will rotate, because of J2, at the correct rate such that it will keep a constant angle with the
radial from the Sun. Orbits with this property are called Sun-synchronous orbits (SSO). Simulating SS orbits represents an important feature of the software, since they are a possible target orbit for a future ABEP powered mission.
The propagation of the orbit has been implemented using two different methods: a simple analytic method and a more advanced numerical propagator.
The analytic method consists in propagating the initial orbital elements. Ω and ω are approximated as linear functions of time, obtained through Equations 3.1 and 3.2. The true anomaly ν is propagated by solving Kepler’s equation. All the other parameters are assumed to remain constant. Finally, the state vector in the IJK frame can be computed from the propagated value of the orbital elements. All of the previous procedures can be found in any text on orbital mechanics, such as [6].
As a second method a numerical propagator, the poliastro package for Python, was integrated in the software. Poliastro is an open source library for astrodynamics and orbital mechanics, released under the MIT license [30]. For the propagation it uses Cowell’s method [28] with the perturbative force of J2. The integration is performed
with an explicit Runge-Kutta method. Poliastro directly returns the state vector at the desired time.
Step 2: Calculation of Geodetic Coordinates
Once the position is known in the IJK frame, it undergoes a series of frame trans-formations to ultimately obtain the geodetic coordinates. It is first transformed to the ascension-declination frame using equations A.1 and A.2, then transformed to the ECEF frame using equation A.5. Finally, the geodetic coordinates can be obtained using the WGS84 reference ellipsoid, using the procedure described in Section A.3.
Step 3: Query of the Atmospheric Data
The software then proceeds to call the functions for the three models NRLMSIS, JB08 and HWM14. The required input for the three models are the geodetic coordinates, the geodetic altitude, the date, and the time. In the current implementation, every model obtains values for solar and geomagnetic indices from tabulated historic data. The indices in the simulation results will then correspond to the condition during the simulated time span. This allows to capture the overall variability of the conditions more realistically than a simulation in which the indices are fixed.
Step 4: Post Processing of the Atmospheric Data
The raw atmospheric and orbital data must be post-processed to obtain the quantities of interest. These mainly concern the gas and were introduced in Chapter 2.
An exception is represented by the calculation of the yaw angle of the spacecraft, which is the angle between the velocity of the incoming gas and the direction of the orbital velocity. The calculation is made in the IJK frame: here, the orbital velocity is obtained from the state vector. The velocity of the incoming gas has a contribution from the atmosphere’s rigid rotation with the Earth and a contribution from the wind (relative to the rigidly rotating atmosphere). Using the definition of Equation A.6, it can be expressed in the IJK frame, as shown by Equation 3.3.
where r is the position vector, ωE the angular velocity of the Earth in inertial space,
wn and we the north and east components of the wind computed by HWM14 and ˆn
and ˆe the north and east unit vectors obtained from Equation A.6.
Step 5: Output and Statistics
The software gives the possibility to save the computed data to a file, and enables the user to review it at a later moment or to perform statistical analysis. Statistics on the various parameters (mean, standard deviation and minimum/maximum value) can be computed and saved to a file. Box and whisker plots of the data can be generated.
3.2.2
Verification
A verification of the orbital propagation and of the calculation of the geodetic coordi-nates has been performed by comparing the results with the General Mission Analysis Tool (GMAT), a well established and validated tool for space trajectory optimiza-tion and mission analysis [7]. The direct comparison of the geodetic coordinates with GMAT showed an excellent agreement over a simulation period of 30 days.
3.3
Calculation of Flow Conditions
In this section, we present the results of the calculations for some selected test cases. The parameters shown here will be useful for later considerations on the operation of the intake.
Based on target mission profiles selected in the available literature, three altitudes have been chosen for the calculations: 250, 200, and 180 km. Since the solar cycle has a periodicity of 11 years, an ABEP powered spacecraft would probably experience an entire period of the solar cycle. For this reason, the mission profiles last 11 years, from Jan 1, 2000 to Jan 1, 2011. For each altitude, two test cases are considered: one with an equatorial (EQU) circular orbit and one with a sun-synchronous orbit (SSO) with LT at the ascending node (LTAN) of 18:00. The latter orbit has some advantages from the mission design point of view, such as the constant availability of solar power, and it is often selected as a target orbit in ABEP mission studies.
The following figures show “box and whisker” plots for the six test cases. In these plots, the ends of the box are the upper and lower quartiles, the median is marked by a line inside the box, and the whiskers are the two lines outside the box that extend to the maximum and minimum observations.
Figures 3.3 and 3.4 show the mass density calculated respectively by NRLMSIS and by JB08. This is in fact a common output to both models. The models are in good agreement, giving confidence that the models are being used correctly.
