Pulsed Fusion Space Propulsion: Computational Ideal Magneto-Hydro Dynamics of a Magnetic Flux Compression Reaction Chamber

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G. Romanelli Master of Science Thesis

Space Systems Engineering

PULSED FUSION SPACE PROPULSION: COMPUTATIONAL IDEAL MAGNETO-HYDRO DYNAMICS OFA MAGNETIC FLUX COMPRESSION REACTION CHAMBER

Gherardo ROMANELLI

to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Friday February 26, 2016 at 10:00 AM.

Student number: 4299876 Thesis committee: Dr. A. Cervone, TU Delft, supervisor Prof. Dr. E. K. A. Gill, TU Delft Dr. Ir. E. Mooij, TU Delft Prof. A. Mignone, Politecnico di Torino

An electronic version of this thesis is available at http://repository.tudelft.nl/. To boldly go where no one has gone before. James T. Kirk ACKNOWLEDGEMENTS

First of all I would like to thank my supervisor Dr. A. Cervone who has always supported me despite my “quite exotic” interests. He left me completely autonomous in shaping my thesis project, and still, was always there every time I needed help. Then, I would of course like to thank Prof. A. Mignone who decided to give his contribute to this seemingly crazy project of mine. His advice arrived just in time to give an happy ending to this story. Il ringraziamento più grande, però, va di certo alla mia famiglia. Alla mia mamma e a mio babbo, perché hanno sempre avuto ﬁducia in me e non hanno mai chiesto ragioni o spiegazioni alle mie scelte. Ai miei nonni, perché se di punto in bianco, un giorno di novembre ho deciso di intraprendere questa lunga strada verso l’Olanda, l’ho potuto fare anche per merito loro. A tutti gli altri, perché erano sempre li a fare il tifo per me. Poi ci sono i miei cari vecchi amici di Arezzo. Con alcuni ci siamo incontrati “solo” 12 anni fa, con altri anche un po’ prima, e nonostante io sia scappato lontano, loro sono ancora li ogni volta che ne ho bisogno. Un altro saluto va ai colleghi/amici di Pisa, perché anche se abbiamo condiviso un periodo breve, è stato comunque un capitolo fondamentale. Finally, there is Delft where I met new people, new chal- lenges, new feelings, and a new life. Lucky me that I also had some good friends to share the adventure with.

iii

ABSTARACT

Verifying the working principle of a magnetic flux compression reaction chamber might be crucial for the development of pulsed fusion propulsion: a system that has been projected to possibly revolutionise manned space exploration. For that purpose, an exhaustive computational Magneto-Hydrodynamics (MHD) analysis is a necessary step. This master thesis investigated the possibility of using PLUTO1 to estimate the ideal-MHD of a multi-coil parabolic reaction chamber. PLUTO is a freely-distributed and modular code for computational astrophysics that, although not originally programmed for engineering applications, has demonstrated great adaptation capabilities: implementing the boundary conditions to effectively emulate a magnetic flux compression reaction chamber has eventually been possible. Besides, the attained results are in accordance with theoretical projections and previous numerical analyses. However, the outcomes pointed out that ideal-MHD could be an over-simplified model: relativistic conditions, that are not properly reproduced by the ideal-MHD equations, have been identified in several locations of the computational domain. In addition, some aspects of the real system physics have yet to be thoroughly investigated as well as mathematically described. Therefore, further investigations are required. According to this research, no other computational analyses of a multi-coil parabolic reaction chamber (i.e. the latest and most promising magnetic flux compression reaction chamber concept) have been found in the literature. Therefore, the results hereby reported contribute to the body of knowledge of plasma physics and nuclear fusion applied to space propulsion. In particular:

The objective of this master’s thesis project is to contribute to the development of a magnetic ﬂux compression reaction chamber for space propulsion applications, by completing the ﬁrst computational ideal-MHD analysis of the plasma expansion in a multi-coil parabolic chamber

Besides, completing such a project has answered to the following research questions:

• How is the thrust generated in a magnetic ﬂux compression reaction chamber?

– What is the theoretical background at support of the projected working principle of a reaction chamber? – What is the latest and most promising reaction chamber concept?

• How have the so far projected performance been derived?

1http://plutocode.ph.unito.it/

v viA BSTRACT

– Which were the assumptions taken to estimate the performance of the rocket? – Can the same assumptions be conﬁrmed by a more detailed plasma physics? – How much do the simpliﬁcations and assumptions taken affect the results of the estimation?

• Can the same performance be reproduced by a more extensive computational analysis?

– What is the plasma physics model that better reproduces the dynamics in a magnetic flux compression reaction chamber? – What are the available codes that can be used to perform the computational analysis? – What are the assumptions/simplification that can be taken to attain an efficient (i.e. in a reasonable amount of time) and still effective analysis? – Can the computational analysis be performed on a commercial laptop? CONTENTS

List of Figures xi

List of Tables xv

Abbreviations xvii

Physical Constants xix

List of Symbols xxi

1 Introduction1 1.1 Magnetic Flux Compression Reaction Chamber...... 4 1.2 Research Contributions...... 7 1.3 Thesis Outline...... 8

2 Plasma Dynamics: Physical Description 11 2.1 Plasma: the 4th state of matter...... 11 2.2 Classical Electromagnetism...... 12 2.2.1 Maxwell’s equations...... 14 2.2.2 Single-Particle Dynamics...... 15 2.3 Plasma Fluid Theory...... 15 2.3.1 Zeroth Moment...... 16 2.3.2 First Moment...... 16 2.3.3 Second Moment...... 16 2.3.4 Single-Fluid Theory...... 17 2.4 Ideal-MHD...... 20 2.4.1 Collisionless Plasma...... 21 2.4.2 High-Conductivity Plasma...... 21 2.4.3 Non-Relativistic Plasma Wave Velocity...... 22

3 Magnetic Flux Compression Reaction Chamber: Operation 25 3.1 Working principle...... 25 3.1.1 Seed Magnetic Field Generation...... 26 3.1.2 Magnetic Field Compression and Momentum Transfer...... 29 3.2 Previous Concept Designs...... 32 3.2.1 Project Daedalus...... 32 3.2.2 Vehicle for Interplanetary Space Transport Applications (VISTA) 33 3.2.3 Human Outer Planet Exploration (HOPE)...... 34

vii viii CONTENTS

4 Magnetic Flux Compression Reaction Chamber: Elementary Analysis 39 4.1 System Energy Balance...... 40 4.1.1 Estimated Performance...... 45 4.2 Simpliﬁed Analytical Model...... 46 4.2.1 Initial Magnetic Flux Derivation...... 53 4.3 Analytical Model Validation...... 57 4.3.1 Initial Conditions...... 57 4.3.2 Integration Problem...... 59 4.3.3 Results...... 60 4.3.4 Final Remarks...... 60

5 Magnetic Flux Compression Reaction Chamber: Computational Code Selection 63 5.1 Selection of the Computational Code...... 63 5.2 PLUTO...... 65 5.2.1 Computational Domain and Solving Strategy...... 66 5.2.2 Deﬁning Initial Conditions...... 69 5.3 Previous Numerical Analysis...... 70 5.3.1 Computational Problem Set-Up...... 71

6 Magnetic Flux Compression Reaction Chamber: Computational Analysis 75 6.1 Final Set-Up...... 75 6.1.1 Computational Domain...... 76 6.1.2 Plasma Pellet and Seed Magnetic Field Definition...... 76 6.1.3 Boundary Conditions...... 78 6.1.4 Ambient Conditions...... 83 6.1.5 Entropy Switch...... 86 6.1.6 Runtime Analysis...... 87 6.2 Validation...... 89 6.3 Results...... 91 6.3.1 Simplified analytical model verification...... 97

7 Conclusions 101 7.1 Research Contributions...... 104 7.2 Recommended Future Work...... 105

A MATLAB script 107 A.1 Arrays Deﬁnition...... 107 A.2 Integration Problem...... 108

B PLUTO 111 B.1 pluto.ini...... 111 B.2 deﬁnitions.c...... 112 B.3 init.c...... 113 B.4 userdefoutput.c...... 119 CONTENTS ix

B.5 ct.c...... 120 Bibliography 123

LISTOF FIGURES

1.1 Thrust, specific impulse (Isp ), and input power of nowadays available space propulsion systems [1]...... 2 1.2 Comparison between the section of a de Laval nozzle a), and of a magnetic nozzle b) [2]...... 3 1.3 Artistic impression of a Fusion Driven Rocket (FDR)[3]...... 4 1.4 Artistic impression of how the magnetic field works as a shield in a reaction chamber [4]...... 6 1.5 Successive stages of plasma expansion in a magnetic flux compression reaction chamber [5]...... 7

2.1 Resultant force applied to two charged particles according to Coulomb’s Law [6]...... 13

3.1 Right-hand-rule to identify the magnetic field generated by a current cable [7]...... 27 3.2 Magnetic field generated by the current through an electric coil.... 28 3.3 Cross section of a reaction chamber that visualises the geometry of the field and the location of the electric coils [4]...... 28 3.4 Reaction chamber cross section parallel to the ρφ-plane. The initial seed magnetic field (expressed by the crossed circles) fills the whole chamber (green circle) [5]...... 29 3.5 Visualisation of the eddy currents (red and green arrows) in the chamber wall (green circle), and the expanding plasma (red circle). [5].... 30 3.6 Net effect on the magnetic field that has been compressed between the chamber wall (green circle) and the expanding plasma (red circle) [5]. 31 3.7 Section of a Winterberg/Daedalus class nozzle [8]...... 33 3.8 Schematic of the Vehicle for Interplanetary Space Transport Applica- tions (VISTA) engine [9]...... 34 3.9 3D model of the reaction chamber for the Human Outer Planet Explo- ration (HOPE) mission [5]...... 35 3.10 Cross section of an electric coil [5]...... 36 3.11 Variation of the induced current density (j) and magnetic field magnitude (B) through the thickness of a conductor (x)[5]...... 36

4.1 Energy flow diagram of the system [10]...... 40 4.2 Spherical coordinate system...... 43 4.3 Initial conditions of the elementary magnetic flux compression reaction chamber analysis [4]...... 47 4.4 Exemplification of how the magnetic flux has been derived [4]..... 48

xi xii LISTOF FIGURES

4.5 Area of the conical annulus at an angle θ [5]...... 49 4.6 Illustration of the area of plasma shell section considered [4]...... 50 4.7 Decomposition of the plasma shell section area along the coordinate axis [4]...... 51 4.8 Angular position of the coils (a), and plasma shell portion associated to the each coil (b) [5]...... 52 4.10 Enlarged view of one plasma section trajectory (dashed line) to show that its reﬂection happens before reaching the chamber wall (black solid line)...... 60 4.9 Plot of the plasma section trajectories according to the MATLABTM script results...... 61

5.1 Example of a 2-dimensional computational grid as implemented in PLUTO [11]...... 67 5.2 Primitive variable labelling strategy for the solution of the Riemann problem...... 67 5.3 Flow diagram of PLUTO’s reconstruct-solve-average (RSA) strategy [12] 68 5.4 Geometry of the seed magnetic ﬁeld in the reaction chamber of Project Daedalus [8]...... 72 5.5 Integration domain of the computational problem [13]...... 73

6.1 Initial seed magnetic field magnitude [Gauss] and streamlines (t 0). 80 = 6.2 Coloured plot of the magnetic field magnitude [Gauss], and visualisation of the imposed geometric condition to attain the field compression (i.e. dashed line) (t 0)...... 81 = 6.3 Compressed magnetic field lines, and magnitude [Gauss](t 2.85µs) 82 ' 6.4 Plasma velocity profile at t 7.83µs and different ambient densities: 9 3 ' 11 3 ρ 2.0 10− g/cm (left), and ρ 2.0 10− g/cm (right)...... 84 1 = · 2 = · 6.5 Thermo-magnetic pressure profile ptm at t 7.83µs and different am- ' 5 bient pressures: p 30.0mbar (left), and p 1.0 10− mbar (right). 85 1 = 2 = · 6.6 Evolution of the TRC (t 0 left, t 2.85µs right)...... 89 = ' 6.7 Density profile at t 2.85µs (left and right). Magnetic field lines in- ' cluded (left)...... 90 6.8 Density profile at t 0 (bottom), and t 1.42µs (top). Selective entropy = ' 5 switch (bottom and top). Ambient conditions p2 1.0 10− mbar and 11 3 = · ρ 2.0 10− g/cm (bottom and top)...... 92 2 = · 6.9 Density profile at t 3.56µs (bottom), and t 6.41µs (top). Selec- ' ' tive entropy switch (bottom and top). Ambient conditions p2 1.0 5 11 3 = · 10− mbar and ρ 2.0 10− g/cm (bottom and top)...... 93 2 = · 6.10 Density profile at t 9.26µs (bottom), and t 14.24µs (top). Selec- ' ' tive entropy switch (bottom and top). Ambient conditions p2 1.0 5 11 3 = · 10− mbar and ρ 2.0 10− g/cm (bottom and top)...... 94 2 = · 6.11 Variation of plasma axial momentum due to different initial ambient 5 9 3 conditions (p1 30.0mbar, p2 1.0 10− mbar and ρ1 2.0 10− g/cm , ρ2 11 =3 = · = · = 2.0 10− g/cm )...... 96 · LISTOF FIGURES xiii

6.12 Variation of plasma axial momentum due to different ENTROPY_SWITCH 5 11 3 states (ambient conditions p 1.0 10− mbar, ρ 2.0 10− g/cm ) 97 2 = · 2 = · 6.13 Variation of plasma axial momentum due to different ENTROPY_SWITCH 11 3 states (ambient conditions p 30.0mbar, ρ 2.0 10− g/cm )... 98 1 = 2 = ·

LISTOF TABLES

4.1 Energy balance symbols explained [10, pag. 3]...... 41 4.2 Input parameters for the Magnetised Target Fusion (MTF) rocket performance estimation [10]...... 45 4.3 Estimated performance of the MTF rocket [10]...... 46 4.4 MATLABTM problem initial conditions...... 58 4.5 Electric coil speciﬁcations [5]...... 58

5.1 Brand-new Vs. Dedicated code to numerically solve ideal-MHD.... 64

6.1 Computational domain characteristics...... 76 6.2 Computational analysis initial conditions...... 77 6.3 Electric coil specifications [5]...... 77 6.4 Electric coil geometry in adimensionalised units [5]...... 79 6.5 Different ambient conditions used for the computational analysis... 84 6.6 Computational domain of the validation problem...... 90 6.7 Icarus Project electric coil specifications [13]...... 91 6.8 Results of the validation problem...... 91 6.9 Verification of Thio et al.’s results...... 98

ABBREVIATIONS

BIS British Interplanetary Society.5, 32, 71 EMF electromotive force. xxi, 81 ETO Earth To Orbit.4 FDR Fusion Driven Rocket. xi,4 HD Hydrodynamics. 65, 71 HOPE Human Outer Planet Exploration. xi,32,34, 35, 39, 102 HRSC High Resolution Shock-Capturing. 65 Isp speciﬁc impulse. xi,1,2 IMLEO Initial Mass in Low Earth Orbit.1 LFR Laser Fusion Rocket.37 MHD Magneto-Hydrodynamics.v, xv,8, 11, 17, 19–22, 34, 62–65,69–72,74,75, 83, 84, 86,88,91, 101, 103–105 MPD Magneto-Plasma-Dynamic.2,3 MTF Magnetised Target Fusion. xv,40,45,46 NASA National Aeronautics and Space Administration.1,34,37,39,46,53,57,76,77, 85, 98, 102, 104 RHD Relativistic Hydrodynamics. 65 RMHD Relativistic Magneto-Hydrodynamics.65,99, 104, 105 RSA reconstruct-solve-average. xii,67,68, 86 SPH Smoothed Particle Hydrodynamics.70 SPMHD Smoothed Particle Magneto-Hydrodynamics. 70 SSVC Space Simulation Vacuum Chamber. 85 VASIMR VAriable Speciﬁc Impulse Magnetoplasma Rocket.3,71 VISTA Vehicle for Interplanetary Space Transport Applications. xi,33,34,70

xvii

PHYSICAL CONSTANTS [14]

speed of Light c 2.998 108 m/s ' · 19 elementary charge e 1.602 10− C ' · 2 standard Gravity g0 9.80665m/s ' 23 2 2 1 Boltzmann constant k 1.3807 10− m kg s− K − B ' ·

12 vacuum Permittivity ε 8.854 10− F/m 0 ' · pi π 3.14159 ' 7 2 vacuum Permeability µ 4π 10− N/A 0 ' ·

xix

LISTOF SYMBOLS

ROMAN SYMBOLS

~a acceleration vector ~A vector potential B~; B magnetic field vector Bx1; Bx2; Bx3 magnetic field components B magnetic field magnitude Ca Courant number eˆ axis versor eth thermal energy etot total energy E~; E electric potential vector Ecap electrical input energy Ecp,f charged particle energy from the fusion reaction Ecp,n neutron energy fraction recovered by the moderator E jet jet propulsion energy Eliner energy of the imploding plasma liners En neutron energy from the fusion reaction En,cap neutron energy fraction absorbed by the moderator En,esc neutron energy fraction lost to deep space Er c recharging energy Etar get initial energy of the plasma target Eφ azimuthal component of the electromotive force (EMF) f magnetic field variation frequency fα plasma species distribution function frep fusion pulse frequency ~fB magnetic force ~fC Coulomb force ~fL Lorentz force F flux vector G fusion reaction energy gain I electric current Isp specific impulse ~j; j current density k thermal conductivity l f focal length L electric inductance LD Debye length

xxi xxii LISTOF SYMBOLS

Lc System characteristic dimension m mass M j momentum n plasma particle density ND ; Λ number of particles contained in a Debye sphere p hydrostatic pressure pB magnetic pressure pm magnetic pressure ptm thermo-magnetic pressure pz,exp expanding plasma axial momentum pz,re f reﬂected plasma axial momentum pz ; ∆px3 change of plasma axial momentum q value of the electric charge Qασ momentum change due to particle collision Qdr i ve waste heat from the plasma pellet generator and ignition system Qnozzle dissipated heat Qpl asma dissipated heat due to non-isentropic transformation Qr c heat dissipated by the recharging system ~r ; r vector position rx1; rx2; rx3 vector position components S source vector t time T thrust force T temperature U conservative variable vector ~v; v velocity vector vex rocket exhaust velocity vx1; vx2; vx3 velocity vector components V~ general vector ﬁeld V primitive variable vector W energy loss due to particle collision z cylindrical axial coordinate

GREEK SYMBOLS

α plasma species δ; δs; δc skin depth θ colatitude λm f p mean free path ν collision frequency ρ cylindrical coordinate: radius ρchar ge charge density ρm mass density σ; σ0 plasma species σ electrical conductivity xxiii

τ viscus stress tensor τ magnetic ﬁeld variation time τ collision time φ azimuthal coordinate Φ magnetic ﬂux

VECTOR CALCULUS SYMBOLS

gradient operator ∇ divergence operator ∇∇· curl operator ∇∇× tensor (outer) product ⊗

1 INTRODUCTION

Due to the lack of knowledge of prolonged deep space travel effects on astronauts, the transfer time of a manned interplanetary mission should be as short as possible (i.e. less than 6 months, based on past experience) [15,16]. Also, limiting the total cost of the mission plays a decisive role when discussing its feasibility. Besides, the propulsion system has been recognised as playing a critical role in affecting the two just mentioned parameters. Thus, to allow for a relatively low cost and fast mission, a high specific power system, a low Initial Mass in Low Earth Orbit (IMLEO), and a high Isp have been identified as primary requirements [17, 18]. The low IMLEO would in fact limit the total number of launches necessary to bring the spacecraft in orbit, hence, the total cost of the mission, as well [17]. On the other hand, the high Isp would assure an efficient propulsion system, and the high specific power (i.e. high thrust) a fast transit time. Figure 1.1 shows a comparison between some different propulsion systems in terms of Isp and thrust level; the same figure shows the required power (i.e. efficien- cies included) to run the system. Among them, chemical rockets show the highest thrust capabilities. Their Isp is however quite low compared to other solutions. Thus, that being an expression of the rocket efficiency, its final mass is still not competitive for a relatively low-cost mission. In fact, to attain a higher Isp , it is the exhaust speed of the propellant that has to increase, and that is proportional to its energy content: part of the thermal energy of the propellant is converted to kinetic energy. In chemical rockets, the energy of the propellant (i.e. fuel + oxidizer) comes form the com- bustion of the fuel, thus, it is limited by the molecular binding energy content of the reactant [19]. An option to overcome the just mentioned limit, is to transfer thermal energy to the propellant through an external power source. Electrothermal and nucler thermal rockets belong to that category, for example. The first ones heat up the propellant thanks to an electrical resistance or an applied current arc (i.e. electrothermal, in the picture), while the latter extract the energy from a nuclear reaction (i.e. either fission, or fusion). Despite the quite critical subject that nuclear power is, this type of thermal rocket shows appealing capabilities for some proposed manned missions: National Aeronautics and Space Administration (NASA) has selected one of them for

1 2 1.I NTRODUCTION 1 the Human Exploration of Mars Design Reference Architecture 5.0 [17]. However, there are other types of rockets that might show better capabilities, still. In fact, the highest exhaust speed, and hence Isp , is attained when accelerating ions (which are the propellant, here) thanks to an electric or magnetic field (i.e. electrostatic and electro-magnetic thrusters, in the picture). Ion thrusters and hall effect thrusters apply to the group of electrostatic thrusters, and have already been employed for different unmanned space missions (deep space travel, as well) [20]. However, they show a thrust that is still too little for crewed interplanetary flights (i.e. flight times would be too long) [3].

FIGURE 1.1: Thrust, Isp , and input power of nowadays available space propulsion systems [1]

On the other hand, although no engine has been completed and employed, yet, electro-magnetic propulsion might finally fill the gap between thermal rockets and electrostatic thrusters by providing an efficient and high specific power option [1]. Among the others, one of the assets such rockets have resides in the way the high kinetic energy of the propellant is attained: the propellant is not heated up by an external source as in thermal rockets, but as in chemical rockets the energy comes again from the bonds of its constituent elements. However, while in a chemical rocket the energy derives from the split of molecular bonds, in an electro-magnetic thruster it comes from the separation of nuclear constituents; and that can thus be orders of magnitude higher [21]. In fact, the propellant used for electro-magnetic thrusters is an ionised gas (i.e. plasma) that is accelerated through the interaction with an electro-magnetic field. As a result, by expanding (i.e. accelerating) the plasma out of the rocket, they directly convert its thermal energy to kinetic energy, and hence, they avoid the Carnot cycle limitation (i.e. efficiency limitation) that is otherwise inevitable when the gas is externally heated [22–24]. For example, the Magneto- 3 1 Plasma-Dynamic (MPD)[25] thruster, and the VAriable Specific Impulse Magneto- plasma Rocket (VASIMR)[26] belong to this category. However, such propulsion systems are still in their development phase. There is in fact a quite exotic and yet common to all of them aspect that is required for such thrusters to work as planned: that is the magnetic nozzle. A magnetic nozzle is in fact what is responsible for the expansion (i.e. the acceleration) of the hot plasma out of the rocket. It would thus have the same task a de Laval nozzle (i.e. solid nozzle) has in some reaction engines (e.g. chemical rockets, jet engines, etc.). Indeed, both kinds of nozzle aim to convert thermal energy of the propellant into kinetic energy. Nevertheless, a magnetic nozzle differs from an ordinary de Laval nozzle, for it is not a solid wall, but the generated magnetic field, that confines and drives the plasma out of the spacecraft: the plasma being composed of charged particles (i.e. ions) reacts to any magnetic field it interacts with. Also, the magnetic field prevents the plasma from touching the structure of the spacecraft. Hence, all the possible damages that might otherwise be caused by the incredibly hot plasma are avoided. Figure 1.2 shows a comparison between a de Laval nozzle and a magnetic nozzle. The magnetic field is generated in the latter by the current flowing through electric coils sustained by a structural frame. The configuration of the coils and the intensity of the current determine the shape and strength of the field. Hence, the fluid is confined in and driven out of a de Laval nozzle due to the hydrostatic pressure that comes from the contact between the fluid itself and the nozzle’s walls (see the arrows in part a) of the picture). On the other hand, it is the so called magnetic pressure that performs the same task in a magnetic nozzle. However, despite the different approach, the general working principle of a solid and a magnetic nozzle are quite similar, if not basically the same [25, 27].

FIGURE 1.2: Comparison between the section of a de Laval nozzle a), and of a magnetic nozzle b) [2]

Even though electro-magnetic thrusters such as the MPD and the VASIMR are projected to have great capabilities, there is still another option that might surpass them both: especially, in terms of cost and mass [28]. That concept is also projected to exploit the acceleration of plasma to generate thrust; however, the propellant thermal energy would derive from the fusion reaction of the ions. Nuclear fusion has in fact been projected as the most promising alternative to power a spacecraft for manned interplanetary flights [29, 30]. In addition, the plasma expansion is thereby 4 1.I NTRODUCTION 1 assured by a device that, while still similar to a magnetic nozzle, is quite different in shape and structure. That is the so called magnetic flux compression reaction chamber, and is meant to take the most out of the fusion reaction. An example of a rocket engine that is meant to use such a reaction chamber would be the FDR under development at MSNW1. That propulsion system has been thoroughly investigated during the literature review that preceded the thesis project hereby reported. That study discussed the FDR as capable of sending a crewed spacecraft (as the one showed in Figure 1.3) to Mars in 90 days. A specific impulse I sp = 2435s, and a thrust level T 1.5 kN [3] have in fact been projected achievable. Be- ∼ sides, the estimated total wet mass of the spacecraft would be within the payload capacity of the nowadays available launch vehicles [3]. That shall thus limit the total cost of the mission (all the previously proposed manned missions to Mars did require multiple Earth To Orbit (ETO) launches [17]). In addition, even though still in its earliest development phase, were all the expectations met, the FDR might be flight ready by 2023 [3].

FIGURE 1.3: Artistic impression of a FDR[3]

Thus, the working principle of a magnetic ﬂux compression reaction chamber, which has been computationally analysed for this thesis project, is introduced in the next section.

1.1. MAGNETIC FLUX COMPRESSION REACTION CHAMBER The idea of using magnetic ﬁelds for propulsion purposes was ﬁrst introduced by Everett and Ulan in one of their studies [31]. However, the concept of magnetic

1http://msnwllc.com/ 1.1.M AGNETIC FLUX COMPRESSION REACTION CHAMBER 5 1 flux compression reaction chamber owes its birth to Professor Friedwardt Winter- berg [32]. The reaction chamber was soon implemented in the design of an advanced spacecraft. That was the project Daedalus, and was first published in 1978 by the British Interplanetary Society (BIS) as a supplement to Volume 31 of their journal [8]. Project Daedalus concerned the design of a spacecraft for manned interplanetary travel, and included a quite detailed study of all the vehicle’s subsystems. The same issue contained a description of the aforementioned chamber, as well. For that reason, that very new category of magnetic nozzle got the name of Winterberg/Daedalus Class Magnetic Flux Compression Reaction Chamber [8]. The theoretical working principle of a magnetic flux compression reaction chamber is quite straight forward. The chamber can be imagined as a bowl-shaped structure that is placed outside of the ship, and have the concave part facing deep space. The plasma is brought to fusion conditions somewhere along the axis of the “bowl”, and inside of it. As a result of the fusion reaction, the energy content of the plasma greatly increases: its thermal energy, and hence its internal pressure arise. Moreover, since the propulsion system is meant to be employed in open space (i.e. the plasma is surrounded by vacuum, and hence by a hydrostatic pressure close to zero), the pressure difference between the inside and the outside of the plasma is quite remarkable. Thus, given the shape of the reacting pellet (i.e. almost spherical), the plasma starts expanding radially in all the directions, and the dynamics of the system can be rightfully imagined as similar to what of a bomb that explodes inside a bowl. For example, if the bomb is placed on the ground, and the bowl is used to cover it, when the bomb explodes the bowl is pushed by the force of the explosion and moves away from the ground. However, one could argue that the explosion might crash the bowl, beside moving it, and that is exactly the same issue a magnetic flux reaction chamber (i.e. the “bowl”) has when interacting with an expanding fusion plasma (i.e. the “bomb”). The energy that comes from a fusion reaction (as well as its temperature) is so incredibly high that no nowadays available material is capable of withstanding a direct contact without reporting serious damages [10]. Nevertheless, magnetic fields can interact with the moving plasma. In particular, if properly designed, a magnetic field can work as a shield between the expanding plasma and the the chamber. In addition, the magnetic field can counteract the plasma expansion. In fact, a magnetic pressure is associated to any magnetic field, and its action is analogous to an ordinary hydrostatic pressure. Thus, were a magnetic field generated within the bowl and to surround the plasma pellet, the pressure outside the plasma would not be the one associated with vacuum, but a somewhat higher one. A pressure higher than zero would thus act against the motion of the plasma, decreasing its expansion acceleration. Moreover, in the case the pressure outside the plasma gets high enough (i.e. higher than the plasma internal pressure), the motion of the latter could even be stopped. There comes the idea of a magnetic flux compression reaction chamber. The chamber is a bowl-shaped structure designed to generate a magnetic field, and the field is such that fills the whole inside of the chamber. However, under fusion reaction conditions (i.e. high temperature, and high reaction rate), the field is swept 6 1.I NTRODUCTION 1 by the expanding plasma. Thus, when the plasma starts its expansion, the field has to modify its shape accordingly. Also, the same field cannot pass through the structure of the bowl itself (this constraint comes from the way the field is generated), and hence the field gets confined and compressed between the expanding plasma and the chamber structure. Moreover, due to the specific properties of a magnetic field, when that gets compressed between two conductive surfaces (i.e. the plasma pellet and the bowl), its magnitude (i.e. intensity) increases. In addition, since the magnetic pressure is proportional to the magnetic field magnitude, one can derive that the pressure outside the plasma grows as well. Thus, if the magnetic field is properly selected, the dynamics of the system can evolve in such a way that the expansion of the plasma is stopped before it can reach the structure of the chamber: initially, the plasma internal pressure is higher than the magnetic pressure, and the plasma starts to expand; later on, the magnetic pressure gets higher than the one inside the plasma, and the motion of the latter is slowed down to a halt. That is a consequence of the magnetic flux compression, and is the way a magnetic field acts as a shield for the structure of the chamber. Figure 1.4 pictures what just explained: how the field inside the chamber is generated (i.e. magnetic coils in the picture) is explained in the coming chapters.

FIGURE 1.4: Artistic impression of how the magnetic ﬁeld works as a shield in a reaction chamber [4]

There is however another interesting property of magnetic fields that comes handy in such a situation, and that is responsible for the addition of the word reaction to the chamber’s name: when the force that has caused the deformation of the magnetic field ceases, the field gets back to its original shape. That derives from the energy balance of the system, and can be explained by electromagnetic principles: the details are better described in the coming chapters. However, the consequence this behaviour has can already be introduced. When the plasma stops its expansion, so does the force that is deforming the field. Therefore, when the plasma expansion comes to an end, the field starts getting back to its original shape in the exact same way a spring would do (see Fig. 1.5). Thus, if one remembers that the field cannot penetrate neither the plasma nor the chamber structure, it is easy to deduce that, as a consequence of the field getting back to its original shape (i.e. to occupy the 1.2.R ESEARCH CONTRIBUTIONS 7 1 whole chamber), the chamber and the plasma starts moving apart. Specifically, if the plasma is expelled from one side, the chamber is in reaction pushed towards the other. Thus, since the chamber is firmly connected to the spacecraft, if one moves, the other follows. Moreover, expelled the plasma, the same process is repeated at a certain frequency to attain a pulsed thrust: that is in fact the reason why this concept is named pulsed fusion propulsion.

FIGURE 1.5: Successive stages of plasma expansion in a magnetic ﬂux compression reaction chamber [5]

Therefore, that is how a magnetic flux compression reaction chamber propels a spacecraft along its interplanetary trajectory. However, the detailed physics that describes its working principle is not that trivial as it may sound from the previous description. In fact, a thorough investigation has been required to clearly understand it. From the outcome of that research, a specific and complete computational analysis has been pointed out as necessary to finally verify the working principle of the chamber. From that comes the purpose of the thesis project hereby reported, the foreseen research contribution of which are pointed out in the next section.

1.2. RESEARCH CONTRIBUTIONS Several solutions that differ in the geometry of the chamber and the magnetic field generation have been proposed. The latest design, i.e. a multi-coil parabolic chamber, is extensively described in the coming chapters of this report. Nevertheless, despite some literature has been found that brings some theory at support of the working principle of both hemispherical [8] and parabolic [4,5, 10] reaction chambers, that very same literature pointed out that a thorough demonstration is still missing. In fact, some very strong assumptions have been so far been taken to reproduce the performance of such a nozzle. Moreover, the plasma interaction with the reaction chamber has yet to be extensively confirmed by the solution of the equations that describe the system dynamics. As it is proved in the coming report, solving the system of differential equations that reproduce the plasma dynamics in reaction to pressure, external loads, and electromagnetic forces, is not that trivial. It requires indeed a numerical integration, and hence, due to the great amount of computational steps involved, a dedicated software or code to complete the procedure. The aim of this thesis project is thus to 8 1.I NTRODUCTION 1 provide a numerical solution of the plasma dynamics in a magnetic flux compression reaction chamber. That is meant to complete a preliminary verification of the chamber’s working principle: i.e. validating the assumptions so far taken at support of the analytical models of the chamber, and verifying the estimated performance. The simulated dynamics starts from the instant right after the fusion reaction has been completed, hence, it reproduces the plasma expansion, only. Thus, no clarifications and verifications concerning neither the fusion reaction ignition, nor the generation of the plasma pellet are included in this report; the literature study that preceded this thesis was in fact more focused on those aspects. As a result:

For that purpose, research has been done about the nowadays available software that may complete such a task. The selection process used to ﬁnally choose PLUTO2, a freely-distributed and modular code for computational astrophysics, is also discussed in the next. Therefore, this research brings a contribute to the development process of new space propulsion systems that shall advance manned space exploration capabilities. Besides, according to the results found in the literature, computational analysis of magnetic ﬂux compression reaction chambers are still very scarce and limited to old design. Hence, the thesis outline is detailed below.

1.3. THESIS OUTLINE The results of a detailed survey of plasma physics, as well as the different mathematical models that can be used to describe plasma dynamics are first reported in Chap- ter2. There, the principles that explain the operation of a magnetic flux compression reaction chamber are introduced. In the same chapter, the ideal-MHD equations are derived, and why those have been specifically selected for this project is also explained. Then, Chapter3 discusses the details of the operation of a reaction chamber. In addition, a brief history about the evolution of the various proposed concepts is included. Particular emphasis is given to the differences and advantages of the latest design in respect to the original one. Following that, the simplified analytical solution that describes the plasma dynamics in a reaction chamber, and that has been found in the literature, is reviewed in Chapter4. There, the Matlab ® script that has been realised to reproduce the simplified model’s estimations is also detailed. Managing to reproduce the same results helped to clarify all the assumptions and simplifications that have been so far taken to complete the preliminary design of the latest reaction chamber. Furthermore, from that analysis, the knowledge that has been necessary to accurately reproduce the dynamics of the chamber (i.e. for the computational analysis that followed) has been acquired.

2http://plutocode.ph.unito.it/ 1.3.T HESIS OUTLINE 9 1 Thus, an exhaustive investigation, followed by a thorough selection process, has been used to choose the more suitable and easily implementable code to perform the numerical analysis. That is reported in Chapter5, where previous and similar studies are also reviewed. Finally, Chapter6 discusses the results of the computational analyses that have been completed. The aim was to reproduce the projected performance found in the literature (i.e. attained through the simplified analytical models). Thus, the functionalities of the selected code have been extensively probed to find out the most efficient way to simulate and reproduce the reaction chamber dynamics. Hence, the final configuration used is there clearly explained. Eventu- ally, all the outcomes and research contributions of this project are summarised in Chapter7, which concludes this report.

2 PLASMA DYNAMICS: PHYSICAL DESCRIPTION

This chapter clarifies the fundamental theory that describes plasma dynamics. First, the definition and nature of plasma are introduced in Section 2.1. There, why a specific physical model is necessary to describe its temporal evolution is pointed out. Then, Section 2.2 and 2.3 discuss how a simplified single-fluid model is derived from the description of the separate constituents of a plasma. There ideal-MHD equations are also introduced: those have in fact been used to model the plasma dynamics within a magnetic flux compression reaction chamber. Thus, Section 2.4 eventually justifies the fairness of using the most simplified MHD model for the purpose of this research.

2.1. PLASMA: THE 4th STATE OF MATTER Matter can be found in the universe in several states. Which state shows in a specific place and time depends on the ratio between the environmental thermal energy and the binding energy of the elements that compose the matter. The most common and known states are solid, liquid, and gas. When atomic and molecular binding energy are both higher than the environmental thermal energy, the matter is found in the solid state. However, when the temperature of the environment increases, the molecules, first, then the atoms, start to separate from each other. Hence, the other two states of matter successively appear [14]. If the temperature increases even more, the atomic binding energies are eventually also exceeded. At that point, the singular positive and negative ions (i.e. protons and electrons) start to separate, as well. A gas that undergo such a process becomes an ionised gas, and the ratio between its charged ions and total number of particles (i.e. ions/(neutr ons ions) is its degree of ionisation. When the degree of ionisa- + tion is sufficient for the gas to start showing plasma-like behaviours (i.e. 1%), the ∼ gas becomes plasma [14]. That is indeed the 4th state of matter. Plasma is claimed to account for the majority of the known matter in the universe.

11 2.P LASMA DYNAMICS: 12 PHYSICAL DESCRIPTION

In fact, it can be found in stars, nebulae, and interstellar space (e.g. solar wind) [14]. Nevertheless, plasma can also appear in terrestrial environment (e.g. lightings, and 2 the Aurora Borealis). Moreover, beside natural cases, plasma can be artificially attained in fluorescent lamps and plasma torches, for example, or in laboratory experiments. However, apart from laboratory experiments where a specific non-neutral condition may be sought, plasma resulting from the ionisation of neutral gas can be considered as quasi-neutral [14]. That implies that positive and negative ions can be assumed equally numbered. Hence, the ion density (i.e. number of ions per unit of volume) is the same for protons and electrons [1]. An important consequence of this quasi-neutrality is a phenomenon known as Debye shielding: the effects of applied external potentials, or of concentration of charges internally to the plasma, are shielded in a length (i.e. Debye length LD ) that is much smaller than the system characteristic dimension L (i.e. L L )[1]. That is also directly related to the c D ¿ c high conductivity of plasmas: since electric currents are free enough to flow inside the plasma, any internal electric field is shorted out. In addition, Debye shielding is related to (as well as due to [1]) collective behaviours that are found in plasmas. In fact, collective behaviours imply that there are enough particles (i.e. N 1) for D À the Debye shielding to happen, and that long range electromagnetic forces have only strong non-local effects [1]. These characteristics of plasmas turned out to be quite useful for the modelling of its dynamics [1]. In fact, due to the nature of a plasma, its dynamics is eventually based on the motion of its constituents particles (i.e. the ions) [14]. However, if one thinks to plasma as to an ionised gas and includes collective behaviours, it might be argued that its dynamics can be described as one of a fluid. That is in fact correct. Nevertheless, its constituent particles being charged, plasma is reactive to electromagnetic forces as well. Therefore, specific models are needed to accurately describe its dynamics. These models combine fluid dynamics with classical electromagnetism. The next sections are to derive the plasma theory that has been used to model the operation of a magnetic flux compression reaction chamber. For the sake of the discussion, fluid dynamics principles are not further treated in this report. Electro- magnetic interactions are in fact dominant in a reaction chamber.

2.2. CLASSICAL ELECTROMAGNETISM Classical electromagnetism is a branch of theoretical physics that describes the dynamics of charged particles under the effect of electric and magnetic ﬁelds combined. At the basis of this theory are protons, and electrons. These are the fundamental charged particles. The charge of an individual particle is an universal physical 19 constant (i.e. e 1.602 10− C), and the difference between protons and electrons is = · that the former is positively charged, while the latter is negatively. As a consequence, protons and electrons attract each other, whereas alike particles do repel. The charge of particles (i.e. q) that results from the combination of protons and electrons is the result of the algebraic sum of the single charges. Thus, a force is always mutually ap- 2.2.C LASSICAL ELECTROMAGNETISM 13 plied between two or multiple charged particles. That can be expressed through the Coulomb’s Law [6]. Figure 2.1 shows an example of two interacting charges. Hence, Equation 2.1[6] deﬁnes the force ~f2 applied to a charge q2 due to the interaction with 12 2 another charge q , according to Coulomb’s Law. There, ε 8.854 10− F/m is the 1 0 ≈ · vacuum permittivity, and~r1 and~r2 are the vector position of each charge.

FIGURE 2.1: Resultant force applied to two charged particles according to Coulomb’s Law [6]

1 q1q2 ~f2 (~r2 ~r1) (2.1) = 4πε ~r ~r 3 − 0 || 2 − 1|| A direct consequence of this force is the deﬁnition of electric ﬁeld:

1 q1 E~1 (~r ~r1) (2.2) = 4πε ~r ~r 3 − 0 || − 1|| Thus, every charged particle has an associated electric ﬁeld, that has effect in every point of the space around the particle itself (i.e. identiﬁed by ~r ), and is directed according to ~r ~r . Whether another particle is attracted or repelled depends on the − 1 relation between the respective signs of the charges. Therefore, Coulomb’s Law can also be expressed as:

~f Eq~ (2.3) C = In addition, there is another fundamental force that is described by electromagnetism. That is the one applied to charged particles when they interact with a magnetic ﬁeld: if a particle moves in the space at a certain speed (~v), and the trajectory of the particle intercepts some magnetic ﬁeld lines, a force is applied to the charge [6].

~f q~v x B~ (2.4) B = Hence, adding Equation 2.3 and Equation 2.4, the resultant force is found. That is deﬁned as the Lorentz Force (fL)[6]. 2.P LASMA DYNAMICS: 14 PHYSICAL DESCRIPTION

~fL ~fC ~fB 2 = + (2.5) q(E~ ~v x B~) = + Explaining how electric and magnetic ﬁelds evolve in space and time is then necessary to complete the classical electromagnetism model. Thus, a system of differential equations, named after the mathematician and physician who ﬁrst published them, James Clerk Maxwell, is introduced.

2.2.1. MAXWELL’S EQUATIONS Maxwell’sdifferential equations that describe the evolution in space and time of electromagnetic ﬁelds’ mutual interaction are reported from Equation 2.6 to 2.9[6].

ρchar ge E~ (2.6) ∇ · = ε0

B~ 0 (2.7) ∇ · = ∂B~ E~ (2.8) ∇ × = − ∂t

1 ∂E~ B~ µ0~J (2.9) ∇ × = + c2 ∂t

7 2 2 Where, µ 4π 10− N/A is the vacuum permeability, and 1/c µ ε , and c 2.998 0 = · = 0 0 ' · 108 m/s is the speed of light in vacuum. Out of all Maxwell’s, Equation 2.7 turned out to be fundamental when modelling the dynamics of a magnetic flux compression reaction chamber: it applies an important constraint to the analytical definition of the magnetic field vectorial components. In fact, Equation 2.7 implies that every magnetic field must be solenoidal: a field is defined solenoidal when its net flux through any closed surface (i.e. H B~ d~S) s · is constantly equal to zero [6]. That is a consequence of the divergence theorem expressed in Equation 2.10, and means that there are no identifiable sources of a magnetic field (i.e. magnetic fields vectors are distributed in closed lines, only) [6].

I I B~ d~S B~ dV (2.10) S · = V ∇ · In Chapter3, where the operation of a magnetic flux compression reaction chamber is better described, it is proven why this particular property of magnetic fields is the key of the chamber’s operation. Therefore, combining Maxwell’s equations with the Lorentz force’s allows to model the motion of charged particles under the effect of electric and magnetic fields. This introduces the first way of modelling plasma physics, i.e. describing the motion of each ion. That takes the name of Single-Particle Dynamics [14]. 2.3.P LASMA FLUID THEORY 15

2.2.2. SINGLE-PARTICLE DYNAMICS The same Equation 2.5 can be used to describe the acceleration of a single charged particle [14]: 2

d~v m ~fL dt = (2.11) q(E~ ~v x B~) = + This, combined with the already mentioned Maxwell’s equations, describes the motion of a single charged particle. However, to thoroughly describe the dynamics of one particle, the interaction with all the other plasma constituents has to be included as well [14]. Hence, since ion densities of fusion plasmas can be of the order of 1020, it is clear how extensive an integration of each particle motion can get. Neverthe- less, the description of plasma dynamics can be greatly simplified at the expense of the accuracy. However, the next sections prove how the simplified model can still be reliable under certain circumstances. In fact, due to its already mentioned collective behaviours and quasi-neutrality, plasma can be described as a fluid, and hence modelled as such.

2.3. PLASMA FLUID THEORY To describe plasma as the result of collective behaviour of its ions, the average properties of the ionised gas are included. Thus, a statistical physics approach is necessary. For that purpose, one can make use of the Boltzmann equation (see Eq. 2.12[1]). That describes the microscopic average distribution of charged particles in a plasma that is not in thermal equilibrium [33]. There, the variation in time of the distribution function (fα) of a species of particles (α) is expressed. The function is described in the six-dimensional space ((r , r , r ), (v , v , v )), where is the gra- x y z x y z ∇ dient of the function in the spatial domain, the gradient of the function in the ∇v velocity domain, v the velocity expressed as a column vector, and a the acceleration. In fact, notations in bold font are used in the next to express column vectors and tensors. Thus: µ ¶ dfα ∂fα ∂fα v fα a v fα (2.12) dt = ∂t + · ∇ + · ∇ = ∂t coll Where, the term on the right hand side describes the effect of particle collisions. However, to derive a plasma-ﬂuid theory, a macroscopic description is rather necessary, and that can actually be attained by expressing the velocity moments of Equation 2.12[33]. To derive a more general solution, the collision term is for the moment included [33]; however, plasma particle collisions are constrained in such a way that [1]:

• The total number of particles is not changed by collisions;

• The total momentum and energy of one species is not changed by the collision between particles of the same species; 2.P LASMA DYNAMICS: 16 PHYSICAL DESCRIPTION

• The total momentum and energy of the system is not changed after the collision between particles of different species. 2 Under these conditions, the velocity moments of Equation 2.12 can be attained as in the next. Their derivation is not within the purpose of this thesis, hence they are reported directly from the literature [1].

2.3.1. ZEROTH MOMENT First, the Boltzmann equation is integrated in the velocity space, and, by taking into account the constraints for the collision term, the zeroth moment of the Boltzmann equation can be expressed as in Equation 2.13[1]. That is in fact the equation of continuity for the ion particle density (nα); hence, vα is the average velocity of species-α particles.

∂nα (nαvα) 0 (2.13) ∂t + ∇ · =

2.3.2. FIRST MOMENT Then, Boltzmann equation is multiplied by mαv (where mα is the species mass) before being integrated again in the velocity domain. That gives us the momentum transport equation [1].

dvα nαmα nαqα (E vα B) (pαI) Rασ (2.14) dt = + × − ∇ −

Where (pαI) is the pressure multiplied by the identity dyadic tensor I, and Rασ includes the effect of collisions (σ is a different plasma species) [1].

2.3.3. SECOND MOMENT Finally, the second velocity moment of the Boltzmann equation allows to express heat transport equation as in 2.15[1]:

µ 2 ¶ µ 2 ¶ ∂ 3pα mαnαvα 5 mαnαvα Qα pαvα vα qαnαvα E ∂t 2 + 2 + ∇∇· + 2 + 2 − · = (2.15) µ ∂W ¶

= − ∂t Eασ

³ ∂W ´ Where Qα is the heat flux, and is again to include the effects of colli- − ∂t Eασ sions [1].The heat flux might be expressed including the third moment, however, it is usually either neglected or expressed by replacing the energy equation with an equation of state [33]. Thus, the three velocity moments represent the equations that macroscopically describe the average behaviour of different plasma species (α, σ, etc.). In fact, beside protons and electrons, the species that composes a plasma can be much more: e.g. different and heavier ions, electrons of different average speed and/or temperature, 2.3.P LASMA FLUID THEORY 17 etc. However, the system can be simplified by considering all the electrons and protons respectively belonging a single category. Such a description of plasma dynamics takes the name of two-fluid model [33]. Nonetheless, this description can be further 2 simplified. In the two-fluid model, electrons and protons do interact with each other through collisions (i.e. electromagnetic interaction), and that is indeed still extensive to model. Thus, a single fluid theory would be much more accessible.

2.3.4. SINGLE-FLUID THEORY The most commonly applied way to describe the plasma as a single fluid takes the name of MHD, where the dynamics of the plasma is described in its centre-of mass frame (CM) [33]. First, the MHD model requires to define single-fluid variables:

mass density X ρm(r , t) nαmα = α

macroscopic velocity P α nαmαvα v(r , t) P = α nαmα

current density X J(r , t) nαqαvα = α

total pressure X p(r , t) pα(r , t) = α Thus, including the just deﬁned variables, and the collision’s constraints, the following expressions are derived from the three moment equations (i.e. Eq. 2.13, 2.14, and 2.15)[1, 33]:

The continuity equation

∂ρm (ρm v) 0 (2.16) ∂t + ∇ · =

The momentum transport equation

∂(ρm v) £ ¤ ρm v v τ pI J B (2.17) ∂t + ∇ · ⊗ + + = ×

The energy transport equation · µ 2 ¶ ¸ ∂etot ρm v E B v eth p k T v τ × 0 (2.18) ∂t + ∇ · 2 + + − ∇ + · + µ0 = 2.P LASMA DYNAMICS: 18 PHYSICAL DESCRIPTION

Where, τ is the viscous stress tensor, and k T is the Fourier’s Law to express the con- ∇ ductive heat transfer. However, since plasma can ideally be considered an inviscid 2 fluid, and its conductive heat transfer negligible, both the terms are hereby assumed 0 [1]. In addition, the total energy per unit of volume e has been introduced [1]: = tot 2 2 ρm v B etot eth (2.19) = 2 + + 2µ0 Where, the first two terms on the right hand side are the specific kinetic energy (i.e. per unit of volume), and the specific thermal energy, respectively. Those two together represent the specific hydrodynamic energy, whereas the last term is the specific magnetic energy. The specific magnetic energy is indeed equivalent to the magnetic pressure, and, as better explained in Chapter3, is also what assures the operation of a magnetic flux compression chamber. As a matter of fact, the analogy between plasma single-fluid dynamics and the Navier-Stokes equations of classical fluid dynamics is now clear: indeed, the only difference resides in the terms that include the influence of magnetic fields. Then, a relation to express J as a function of the plasma velocity is added to the system [1]. For that purpose, the Ohm’s Law is introduced. That can be a complex relation for plasma where the current is the result of the combined motion of all the particles; nonetheless, further simplifications can be implemented. That includes neglecting all the second order terms, as well as expressing the collision integral by an average collision frequency. Also, slow temporal changes of the plasma current (i.e. ∂J 1), and large spatial scales are assumed [33]. As a result, the generalised ∂t ¿ Ohm’s Law becomes [33]:

J σ(E v B) (2.20) = + × Which, assuming a perfectly conductive plasma (i.e. σ ), can be further reduced ∼ ∞ to [33]:

E v B 0 (2.21) + × = Hence, the term E B of Equation 2.18 can be written as [34]: ×

E B (v B) B × = − × × (2.22) B 2v (v B)B = − · In addition, recalling Equation 2.9, and assuming non-relativistic plasma velocities (i.e. v c), the term 1 ∂E is negligible in respect to v B [14]. Thus, Equation 2.9 ¿ c2 ∂t × becomes:

B µ J ∇ × = 0 2.3.P LASMA FLUID THEORY 19

As a results, applying the identity [34]:

1 2 ( B) B (B )B (B B) ∇ × × = · ∇ − 2∇ · µB 2 ¶ (B )B = · ∇ − ∇ 2 the term J B that si included in Equation 2.17 can be written as: ×

1 J B ( B) B × = µ0 ∇ × × (B )B µ B 2 ¶ · ∇ (2.23) = µ0 − ∇ 2µ0 (B )B µ B 2 ¶ · ∇ I = µ0 − ∇ · 2µ0

Therefore, simplifying Equations from 2.16 to 2.18 by adding Equation 2.21, 2.22, and 2.23, MHD can be expressed as:

∂ρm (ρm v) 0 ∂t + ∇ · = µ 2 ¶ ∂(ρm v) B (B )B ρm v v pI I · ∇ ∂t + ∇ · ⊗ + + 2µ0 = µ0 ∂B ( v B) 0 ∂t + ∇ × − × = · µ 2 ¶ ¸ ∂etot ρm v 1 2 (v B)B v eth p B v · 0 ∂t + ∇ · 2 + + + µ0 − µ0 =

Then, implementing the identities (i.e. including Eq. 2.7)[34]:

( v B) B( v v ) v( B B ) ∇ × − × = ∇ · + · ∇ − ∇ · + · ∇ (v B B v) = ∇ · ⊗ − ⊗ (B )B (B B) B( B) · ∇ = ∇ · ⊗ − ∇ · (B B) = ∇ · ⊗ and adding the term [11]:

B 2 ptm p = + 2µ0 to express the thermo-magnetic pressure (thermal+magnetic), the MHD equations can be ﬁnally expressed as [11]: 2.P LASMA DYNAMICS: 20 PHYSICAL DESCRIPTION

∂ρm (ρm v) 0 (2.24) ∂t + ∇ · = 2 µ ¶ ∂(ρm v) B B ρm v v ptm I ⊗ 0 (2.25) ∂t + ∇ · ⊗ + − µ0 = ∂B (v B B v) 0 (2.26) ∂t + ∇ · ⊗ − ⊗ = · ¸ ∂etot ¡ ¢ (v B)B v etot ptm · 0 (2.27) ∂t + ∇ · + − µ0 = Furthermore, to close the system, an equation of state has to be included. Thus, depending on the assumptions concerning the nature of the reaction, that equation changes. For an ideal gas, the equation of state takes the form: p eth (2.28) = γ 1 − where γ 5 in a 3-dimensional plasma. = 3 As a result, due to the simplifications applied, this single-fluid model takes the name of ideal-MHD[14]. The derivation and discussion of the other MHD models (e.g. resistive-MHD) is not within the purpose of this report. What is discussed in the next section, however, are the reasons why, for the case hereby examined (i.e. fusion plasma dynamics in a magnetic flux compression reaction chamber), ideal- MHD can be an accurate model.

2.4. IDEAL-MHD Previous studies (i.e. both theoretical and experimental) showed that fusion plasma can effectively be modelled through ideal-MHD[35, 36]. In fact, as already mentioned in the previous sections of this chapter, the conditions that are required for the validity of ideal-MHD are:

• large temporal variation scale of the plasma currents (i.e. ∂J 1); ∂t ¿ • large spatial scale of the system

• collisionless plasma;

• high-conductivity plasma

• non-relativistic velocities

A plasma that shows such characteristics is referred to as strongly-magnetised collisionless plasma. Besides, the same conditions have been discovered sufficient to confirm the applicability of an ideal gas law as the equation of state of fusion plasma [14]. The first and the second condition are directly related to the presence of De- bye shielding effects [14], and experiments involving fusion plasmas have indeed 2.4.I DEAL-MHD 21 confirmed that they are met [35]. However, further explanation is needed to prove that the other conditions are also verified. Thus, the next sections show how fusion plasma, as what is involved with the operation of a magnetic flux reaction chamber, 2 can be effectively modelled according to ideal-MHD.

2.4.1. COLLISIONLESS PLASMA To justify the hypothesis of collisionless plasma, the collision frequency (ν ) is in- ασ0 troduced. One particular species of particles is identified with α, while σ0 identifies one of the other species. Hence, the total collision frequency (να) of the species α with all the other plasma species is defined as the sum of the single frequencies [14].

X νs ν (2.29) ≈ ασ0 σ0 Specifically, any angular deviation of a particle trajectory due to Coulomb interactions with other particles is identified as a collision. Nevertheless, the collision frequency is not the inverse of the time between single collisions. In fact, it refers to the interval in which the number of collisions is sufficient to attain an angular deviation of more than 90° [14]. The distance a particle travels in that interval is defined as mean free path (λm f p ), and when that is much greater than the system’s characteristic dimension (i.e. λ L), the hypothesis of collisionless plasma is con- m f p À firmed [14]. Obviously, lower the collision frequency is, longer the mean free path. Indeed, the collision frequency can be expressed as a function of plasma temperature (T )[14]:

e4 lnΛ n ν (2.30) 2 1/2 3/2 ∼ 4πε0m T Where Λ is equivalent to the number of particles contained in the so called Debye sphere: a Debye sphere, that is centred on a particle, defines the range within which Coulomb interactions influence the particle itself [14]. Hence, for high enough tem- peratures, the collision frequency becomes negligible. As a consequence, fusion and space plasmas can be confidently assumed being collisionless [14]. Besides, external applied magnetic fields are what primarily drive the dynamics of a collisionless plasma (i.e. strongly magnetised plasma) [14].

2.4.2. HIGH-CONDUCTIVITY PLASMA The electrical conductivity of plasmas (σ) can be deﬁned as [14]:

ne2τ σ 1.96 = m Hence, it is proportional to the number of particles (n), and the collision time (τ). To quantify the high-conductivity of fusion plasma, the collision time (τ) is thus fundamental. In fact, the collision time is related to the collision frequency (ν) through the relation [14]: 2.P LASMA DYNAMICS: 22 PHYSICAL DESCRIPTION

1 τ 2 ∼ nσ2ν 2 3/2 (2.31) ε0pmT ∼ e4n

where a similar relation to the one of Equation 2.30 has been used to express the collision frequency. Therefore, one can see that the conductivity of plasma is again proportional to its temperature. In particular, a high temperature causes a high conductivity (i.e. σ T 3/2). As a result, the conductivity of fusion plasmas can be conﬁ- ∝ dentially assumed high enough to conﬁrm the hypotheses of ideal-MHD[35].

2.4.3. NON-RELATIVISTIC PLASMA WAVE VELOCITY At the beginning of this thesis project, the assumption of non-relativistic conditions was not questioned due to what found in the literature [35, 36], where they confidently assure that non-relativistic plasma velocities have been confirmed by experiments. That may in fact be true for the plasma velocity itself (i.e. v in the ideal-MHD model). Nevertheless, at the end of the project, and following a discussion with Pro- fessor A. Mignone from the Politecnico of Torino, how the same constraint should also be extended to plasma wave velocities, has been pointed out. If one considers again the motion of single ions, their dynamics is eventually found of an oscillatory kind. Hence, that can be described as the motion of a wave. Specifically, when the motion is parallel to the magnetic field lines, the ion dynamics is associated to what of sound waves. On the other hand, when the motion is perpendicular, the dynamics is identified with the so called Alfvén waves [14,35,36]. Without further discussing the nature of these waves, what is hereby important is the related velocity (i.e. Alfvén speed). In fact, while it can be rightfully assumed that the speed of sound waves is consistently non-relativistic, that is not always the case for the Alfvén speed. As a matter of fact, the Alfvén speed (v A) is expressed as [14]: s B 2 v A (2.32) = µ0ρm

where B is the intensity of the magnetic ﬁeld, and ρm the mass density of plasma. Hence, the constraint of a ﬁnite Alvén speed prevents the possibility of implementing a density 0 in the computational domain of any MHD numerical solver; fur- = thermore, due to numerical stability issues, the density cannot even be 1. ¿ However, the next chapters show how implementing vacuum conditions (i.e. extremely low density ambient conditions) has in fact been necessary for the computational analysis completed for this project. Consequently, relativistic Alfvén velocities have been attained in some locations of the computational domain. Besides, since ideal-MHD does not implement a rigorous constraint on the upper limit of the estimated velocities, some simulations showed values higher than the speed of light. That has been eventually mitigated through some adaptations of the problem set- up (see Chap.6). Nevertheless, despite ideal-MHD has been found suitable under 2.4.I DEAL-MHD 23 certain simulation conditions, the highlighted issue raised some interest for further investigations. 2

3 MAGNETIC FLUX COMPRESSION REACTION CHAMBER: OPERATION

The description of a magnetic ﬂux compression reaction chamber operation can now be expanded and further discussed with the aid of the plasma physics concepts pointed out in Chapter2. Thus, Section 3.1 gives a much more detailed analysis of the working principle of a reaction chamber. In addition, some of the subsystems, that are fundamental for the chamber to work as such, are also introduced. By the end of the same section it shall be clear how the reaction chamber generates thrust. Moreover, a brief review of some reaction chamber concepts, that have so far been proposed, is included in Section 3.2. Besides, the latest and most promising of the designs (i.e. a multi-coil parabolic chamber) is there pointed out. That is in fact the model that has been the subject of this research.

3.1. WORKINGPRINCIPLE The idea of what then became a magnetic flux compression reaction chamber was first introduced by Friedwardt Winterberg for his proposed deuterium micro-bomb space propulsion concept in 1955 [32, 37]. That was his solution for an efficient and affordable access to interplanetary destinations (i.e. including manned missions). In fact, as he mentioned again in 2010 [37], Winterberg still believes that nuclear pulse propulsion is the most viable option for that purpose. Besides, as it has already been discussed in the introduction of this report, all other propulsion options show some drawbacks that are remarked by Winterberg in his article [37]. In particular, in the propulsion concept proposed by Winterberg, the energy to drive the rocket is extracted from the fusion explosion of small Deuterium bombs (i.e. Deuterium microbombs rocket propulsion). The explosion, which is ignited behind the ship, generates a plasma expansion that is reflected towards the back of the spacecraft in order to attain a positive thrust due to the momentum conservation of the system. To reflect the plasma, and thus attain thrust, some kind of device is necessary. For the Project Orion [38, 39] a similar concept had been already introduced:

25 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 26 OPERATION

there, nuclear fission bombs would repeatedly explode and expand against a pusher plate behind the spacecraft, and the ablation of the plate itself would thus generate a thrust in reaction. The project was never realised due to the excessive cost and mass, as well as the public opinion against such big nuclear explosions [39]. However, Win- terberg derived from that project what he referred to as “magnetic mirror” [37], and that eventually became the magnetic flux compression reaction chamber. The resul- 3 tant system was projected to be much lighter and efficient, as well as safer, than what proposed for the Project Orion [32, 37, 39]. Specifically, the reaction chamber takes the shape of an open, concave, and axisymmetric structure that surrounds the fusion explosion which is ignited somewhere along its axis. However, the generation of thrust is not attained as in Project Orion, but accomplished through the electromagnetic interactions between the fusion plasma and the magnetic field that is generated within the chamber. In fact, the field stops the incredibly hot fusion plasma from touching the chamber wall, and, eventually, reflects it out of the chamber. However, to explain the details of how the reflection actually happens, it is first useful to introduce how the initial magnetic field (i.e. seed magnetic field) is generated.

3.1.1. SEED MAGNETIC FIELD GENERATION In fact, when an electric current (I) flows along any line (e.g. a cable), a resultant magnetic field is generated in the whole space around the line itself. The amplitude of the field, evaluated at a point identified by a vector ~r , depends on the linear integral of the vectorial product between dl~ (i.e. infinitesimal part of the line (d~l)) and ~r rˆ ~r . Such a relation comes from the Biot-Savart Law [6], and is expressed as: = || || Z µ0 Id~l rˆ B~ × (3.1) = 4π r 2 Thus, when a current flows through a conductive cable (i.e. a line), a magnetic field is generated in the space around the cable itself. Figure 3.1 is useful to visualise what the distribution of the generated magnetic field is like. The field can be visualised as composed by multiple and concentric circular lines, centred around the cable itself, and having the direction in accordance with the right-hand-rule: the thumb of the right hand points the same direction of the current’s, while the magnetic field lines follow the other fingers. Moreover, depending on the geometry of the cable (i.e. the direction it follows), the resultant magnetic field changes accordingly. Thus, solving the Biot-Savart integral allows to find the direction and the magnitude of the magnetic field in every point of the space. As a consequence, the solution of the Biot-Savart integral depends on the geometry of the cable. Besides, despite the integral can be analytically solved for some basic configurations, an exact solution of more complex geometries may require an extensive numerical integration. What is of interest for the analysis of a magnetic flux compression reaction chamber is the field generated by a current coil: that is a linear cable closed in a circle. 3.1.W ORKINGPRINCIPLE 27

The Biot-Savart integral for a single electric coil configuration has been approximately solved, and an analytic expression of the field attained [40, 41]. Specifically, it is the expression of the magnetic vector potential that has been evaluated; from that, the field can then directly be derived. In fact, a vector potential ~A is as- 3 sociated to every vector field V~ , so that the relation V~ ~A is satisfied. Also, including the = ∇ × mathematical identity that states that the curl of any vector field is always a solenoidal field (i.e. ¡ ~A¢ 0 [34]), it is clear how a vector po- ∇ · ∇ × = tential is related to solenoidal fields [40]. As a re- FIGURE 3.1: Right-hand-rule to sult, a vector potential ~A can be defined for ev- identify the magnetic field generated ery magnetic field B~. Furthermore, the magnetic by a current cable [7] vector potential is an expression of the potential energy that is needed to maintain a generated magnetic field in the specific configuration [40]. Extensive discussions about the magnetic vector potential associated with a current coil have been found in the literature [40–42]. The final relation that has been used is reported in Equation 3.4. However, its full derivation is not included for the sake of simplicity. Hence, Equation 3.2[40] shows the magnetic vector potential (i.e. expressed in polar coordinates (ρ,φ,z)) associated with an infinitesimal section of the loop. For symmetry reasons, the azimuthal component of the potential is the only non-zero one. There, a is the radius of the loop, I the current, ρ¯ and z¯ identify the location of the point where the potential is evaluated, and φ is the azimuthal coordinate of the infinitesimal section of cable.

µ I a Z 2π cos(φ)dφ ~ 0 Aφ(ρ¯,z¯) p (3.2) = 4π 0 a2 ρ¯2 z¯2 2aρ¯ cosφ + + − The integral of Equation 3.2 can be solved by including the complete elliptic integrals K (k2) and E(k2), where k2 is [40, 41]:

4aρ k2 (3.3) = a2 ρ2 z2 2aρ + + − Thus:

· 2 ¸ µ0I a (2 k )K (k) 2E(k) ~A ( ,z) − − (3.4) φ ρ p 2 = π a2 ρ2 z2 2aρ k + + − From this last relation, the components of the magnetic field can be derived from the curl of the vector potential (i.e. B~ ~A). Hence, the resultant magnetic field of a = ∇ × circular current loop, the geometry of which is visualised in Figure 3.2, can finally be approximately expressed. 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 28 OPERATION

However, the seed magnetic field of a reaction chamber is in fact the result of the combination of multiple and coaxial electric coils. Re- calling that the chamber has an open, concave, and axisymmetric shape (i.e. bowl-shape), several coils, that share the same main axis of the 3 bowl, are fixed to its structure. Also, depending on their number and geometry, the resultant magnetic field changes in shape and character- FIGURE 3.2: Magnetic field gener- istics. Figure 3.3 shows a possible example of a ated by the current through an elec- chamber cross section. There, each pair of light 1 tric coil blue dots, that are symmetric with respect to the chamber axis, is indeed the cross section of each coaxial coil. The same figure shows the geometry of the generated magnetic field. Thus, since the section lies on the ρz-plane (i.e. in polar coordinates), due to symmetry reasons, the magnetic field has no azimuthal component.

FIGURE 3.3: Cross section of a reaction chamber that visualises the geometry of the ﬁeld and the location of the electric coils [4]

As a result, the seed magnetic field in a reaction chamber can actually be expressed as the algebraic summation of the vector potential components that are associated with every single coil. That is in fact what has been done for this project, and is better explained in Chapter6. Therefore, now that the seed field generation has been clarified, how thrust is consequently attained can finally be discussed. The key here is the compression of the magnetic field trapped between fast-moving conductive armatures.

3.1.2. MAGNETIC FIELD COMPRESSIONAND MOMENTUM TRANSFER Every conducting material is characterised by its magnetic diffusivity (Dm). That quantifies how much an external magnetic field can penetrate within the material, and can be expressed as a function of the magnetic permeability (µ), and the electrical conductivity (σ) of the material itself [43]. Specifically, higher the conductivity of the material is, less the field penetration [43]: 3 1 Dm (3.5) = µσ The magnetic diffusivity is eventually responsible for the thrust generation in the reaction chamber. In fact, according to the 3rd and 4th Maxwell’s relations (i.e. Equa- tions 2.8 and 2.9), a time and spatially varying magnetic field is related to, and responsible for, applied electric potentials. Also, in the case the electric potential is applied to conductive materials, eddy currents (i.e. induced currents) are also attained. These currents, as the Biot-Savart law shows (i.e. 3.1), are also associated with the generation of a magnetic field (i.e. induced magnetic field). As a result, when two fast moving armatures move towards each other, any external magnetic field that happens to be between them, starts to penetrate through their thickness. Moreover, due to the eddy currents that are consequently generated, the direction of the induced magnetic field is such that it prevents the external field to penetrate further [9]. As a result, the field not being able to escape, gets compressed between the two armatures. According to the first reaction chamber concept, the two conductive armatures are the chamber wall (see Fig. 3.3), on one side, and the expanding plasma (highly conductive, see Chap.2), on the other. Thus, as a consequence of the plasma expansion, some field lines do penetrate within both, and eddy currents are hence generated [43,44]. Also, due to symmetry reasons, the induced eddy currents have azimuthal component, only. Figure 3.4 shows a cross section of the chamber parallel to the ρφ- plane (i.e. in polar coordinates). That is referred to the very initial conditions, when the seed field has just been generated by the current in the coils. The FIGURE 3.4: Reaction chamber cross section parallel to the ρφ-plane. The initial seed mag- green circle represents the conductive netic field (expressed by the crossed circles) wall, while the crossed-circles express fills the whole chamber (green circle) [5] the direction of the field lines entering the plane of the figure. Also, no eddy currents have been yet generated through the conductive wall. On the other hand, Figure 3.5 shows a more advanced situation when the plasma (i.e. red circle) has al- 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 30 OPERATION

ready started its expansion. Eddy currents (i.e. green and red arrows) have also been generated. Thus, the figure clarifies the resultant effect on the seed field.

FIGURE 3.5: Visualisation of the eddy currents (red and green arrows) in the chamber wall (green circle), and the expanding plasma (red circle). [5]

As a result, the direction of the induced currents is such that the seed field is amplified between the armatures, while its further penetration within the conductors is impeded [5]. The net conditions of the compressed magnetic field are thus reported in Figure 3.6. The compressed magnetic field amplification is also due to its solenoidal nature: the field is amplified to maintain the magnetic flux through the annular section of Figure 3.6 equal to the initial magnetic flux of Figure 3.4. Furthermore, for the chamber to complete its task, the plasma reflection has to be accomplished. Thus, as already mentioned in the introduction of this report, the rebound of the plasma is attained due to the effect of the applied magnetic pressure (pm). Section 2.3.4 pointed out that this pressure is in fact the specific magnetic potential energy. Thus, from Equation 2.19:

B 2 pm (3.6) = 2µ0 As a result, the field amplification caused by the compression is clearly related to an increase of the magnetic pressure. Besides, this pressure results in the only applied force that counteracts the expansion of the plasma: the system is in fact outside of the spacecraft and surrounded by open space (i.e. negligible hydrostatic pressure of vacuum) [4,5]. Hence, magnetic pressure must eventually become greater than the plasma thermal pressure itself, and that condition has to be maintained for a sufficient time for the plasma to completely stop before touching the chamber wall [43]: due to the high temperature of fusion plasmas, a contact with the chamber must 3.1.W ORKINGPRINCIPLE 31 be avoided. Also, the same process can be described from the system energy balance point of view. In fact, the plasma stops when its kinetic energy is 0: during the whole process, plasma kinetic energy, minus the involved heat dissipation and non-isentropic transformation losses, is converted into magnetic potential energy. The intensity of the initial seed field (i.e. the initial magnetic potential energy) de- termines the distance from the wall at which the plasma stops [43]. That is better discussed in Chapter4. 3 Eventually, when the plasma stops, the external forces that have compressed the field cease, as well. Thus, since the condition of compressed magnetic field (i.e. increased magnetic potential energy) is not one of equilibrium, if no external loads are applied, the field tends to get back to its original condition of equilibrium: i.e. to fill the whole inside of the chamber. As a result, the plasma reverses its motion, and is pushed out of the open part of the chamber: the potential energy of the field, diminished by the involved losses, is thus transformed back to plasma kinetic energy. In addition, since the to- FIGURE 3.6: Net effect on the magnetic field tal momentum of the system is con- that has been compressed between the cham- served, the change of plasma momen- ber wall (green circle) and the expanding plasma (red circle) [5] tum caused by its reflection is balanced by a change of the spacecraft’s (i.e. momentum transfer). Due to symmetry reasons, the effective change in momentum is ideally directed along the axis of the chamber itself, hence, the thrust is also generated in the same direction [4,5, 43]. When all the plasma has been expelled from the chamber, a new fusion reaction is ignited in the same spot, and the whole process repeats all over again: from that the name of pulsed fusion propulsion [32,37,39]. To summarise, for the concept to work as such, an initial magnetic field (i.e. seed magnetic field) has to first be generated to fill the inside of the chamber. Then, the motion of the plasma has to be reversed in order to generate thrust in reaction. That is accomplished thanks to the rise of magnetic pressure due to the field compression between the two conductive armatures (i.e. the chamber wall, and the expanding plasma) [4,5, 43, 44]. Of course, a perfectly symmetric reflection of the plasma shall never be achieved in a real system. Hence, a parameter to express such a possible loss should be implemented to complete a valid projection of the chamber performance. Besides, an accurate and extensive analysis of the system dynamics would be necessary to properly define that. Nevertheless, all the so far discussed chamber models, either analytical or numerical, do not implement such a parameter: in fact, they all include the perfect axial symmetry of the system among the simplifications. 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 32 OPERATION

However, the reaction chamber proposed by Winterberg has stimulated the interest of many researchers since it was ﬁrst published in 1955 [37]. As a result, many concept studies have been proposed and discussed. Some of them have been published as complete proposals for interplanetary missions: beside the design of the whole propulsion system, they added a full spacecraft design as well as mission sce- narios. Therefore, the most prominent of such studies are reported in the next. Among 3 them, the last project described (i.e. project HOPE) included a multi-coil parabolic reaction chamber. That is indeed the concept that has been investigated for the purpose of this thesis: it is the latest update of the reaction chamber design, as well as the most promising and extensively treated in the literature.

3.2. PREVIOUS CONCEPT DESIGNS The concept designs that followed Winterberg’s ﬁrst publications were what really tried to prove the capabilities of his idea. There, whole mission architectures have been discussed: that included orbit trajectories, as well as spacecraft system designs. In addition, each of the studies proposed a different implementation (i.e. number of coils and chamber geometry) of the same Winterberg’s concept. In the next sections, the three most detailed projects that have been found in the literature are reported. There, the respective reaction chamber design only is described. In fact, highlighting the differences between different concepts has helped to clarify why it is a parabolic multi-coil solution that is nowadays considered the most promising.

3.2.1. PROJECT DAEDALUS Project Daedalus was an interstellar concept mission published by the BIS in 1978. The spacecraft would be powered by nuclear fusion and use a reaction chamber (i.e. Winterberg/Daedalus class nozzle) in order to redirect the plasma expansion and generate thrust. The chamber would be of a hemispherical shape, with the seed magnetic field generated by a series of 4 coils, and a superconducting wall to assure the field compression. Also, the reaction would be ignited at the geometrical centre of the chamber [8]. Figure 3.7 shows the section of the same nozzle. The three excitation field coils would be responsible of generating the seed field, while the fourth, labelled as induc- tion generator coil, would generate electricity power from the current induced by the magnetic field compression. The cusp shape in the magnetic field lines is due to ignition system requirements [8]. According to their report, that coil configuration had been based on some computer aided simulations. Nevertheless, the detailed results of such simulations have not been published in the journal, and they only mention that the 4-coil configuration was eventually selected to assure a proper shape (i.e. proper shielding property) of the magnetic field [8]. In addition, the sizing of the chamber itself (i.e. geometry and electric coils description) was just based on preliminary calculations and some very strong assumptions [8]. However, it was the first of its kind, and inspirational for all that came after. 3.2.P REVIOUS CONCEPT DESIGNS 33

FIGURE 3.7: Section of a Winterberg/Daedalus class nozzle [8]

3.2.2. VEHICLEFOR INTERPLANETARY SPACE TRANSPORT APPLICATIONS (VISTA)

This project, led by C. D. Orth, included the design of the vehicle mentioned in the title, and was carried out at the Lorentz Livermore National Laboratory2. Despite its name, VISTA was primarily meant to investigate the feasibility of a manned transport vehicle to Mars. However, some preliminary estimations for missions beyond Mars were also included. The need to implement a magnetic reaction chamber came from the necessity to employ nuclear fusion power for propulsion purposes: this option was selected to keep the transfer time as short as possible [9]. In fact, they estimated a round trip transfer time to Mars of 145 days, and to Titan of 500 days [9].

Figure 3.8 (where the laser modules are for plasma ignition purposes) shows a schematic of the VISTA engine. One of the major differences from the Winterberg-Daedalus class nozzle, is the location of the fusion reaction in respect to the reaction chamber itself. In fact, in the VISTA vehicle the fusion reaction would be ignited outside of the reaction chamber, whereas, in a Winteberg-Daedalus class nozzle the reaction would take place in proximity of the hemisphere centre. The result is that the former interacts with only a limited volume of plasma (i.e. most of it expands directly into space), while the latter redirects a bigger portion of plasma [9]. However, not many details have been found concerning the dynamics and the spe- ciﬁc design of the VISTA engine [4]. Thus, this concept has been discarded, and no further descriptions are included in this report.

2https://www.llnl.gov 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 34 OPERATION

FIGURE 3.8: Schematic of the VISTA engine [9]

3.2.3. HUMAN OUTER PLANET EXPLORATION (HOPE) In 2003 NASA included a derivation of the Winterberg/Daedalus class reaction chamber in one of their mission design concept: i.e. the HOPE mission [4]. The reaction chamber model they selected had first been investigated and analysed by Thio et al. in a previous work of theirs [10], and the main differences from other chamber concepts are its parabolic shape, and the fusion reaction ignited at its focus. Theo- retically, that grants a more effective and efficient use of the fusion plasma energy. In fact, due to the well known geometrical properties of a parabola, anything that comes radially from its focal point (i.e. the expanding plasma) is eventually reflected along the parabola axis. Therefore, the generated thrust being proportional to the change of plasma momentum parallel to the chamber axis, reflecting all the plasma towards that very direction assures the maximum possible efficiency of the system [13]. Be- sides, in his publication, Orth mentioned that the efficiency of the VISTA propulsion system (estimated as high as 64%) could be increased up to 75%, had the plasma assumed a parabolic shape before being reflected [9]. As a result, the radius of the chamber (i.e. 6.5m [4]) is much smaller than what proposed for the Daedalus project (i.e. 50m (first stage), and 20m (second stage) [8]). Nevertheless, the theory of MHD being scale invariant [13], the working principle of the reaction chamber remains the same. Hence, Figure 3.9 shows the 3D model of the complete chamber structure. There, the parts in yellow are indeed the 8 coaxial coils that generate the seed magnetic field, while those in grey are the carbon com- posite frame structure (i.e. non conductive) of the chamber. In fact, to increase the view factor of the fusion reaction (i.e. the probability that any neutron released by the reaction does not collide with the chamber structure), no solid wall is included [4]. 3.2.P REVIOUS CONCEPT DESIGNS 35

Furthermore, the frame of the chamber is widely spaced, and has limited cross sections directed towards the focus of the parabola [4]. As a matter of fact, the lack of a solid conductive wall is the other main difference of this very concept from all the others: the exact same role of the 3 wall is hereby fulfilled by the 8 coils. In fact, as already mentioned in Section 3.1.2, a current is induced in the chamber wall in reaction to the magnetic field compression. How- ever, a wall being missing here, the induced current is in the coils themselves. Nevertheless, the coils have to apply the seed FIGURE 3.9: 3D model of the reaction chamber for the magnetic field, as well, and are HOPE mission [5] thus connected to a generator to complete that task: the potential applied by the generator would hamper any other current induced in the same coil [5]. As a consequence, the electric coils of a reaction chamber have to be designed to simultaneously generate a seed magnetic field, and allow for an induced current to flow through them. Also, the operation of the latter should not affect that of the former.

ELECTRIC COILS DESIGN Thus, according to the latest studies [5] every ring is to be composed of two concentric coils. The internal ones are referred to as seed-coils, and made of superconducting material. They are in fact what actually generates the initial seed magnetic field that fills the inside of the chamber. On the other hand, the external coils are called thrust coils and made of normally-conducting material: the current is in fact induced through them, hence, they are responsible for the thrust generation [5]. Figure 3.10 clarifies what just explained. Since it is the internal ring that generates the field that fills the inside of the chamber, that field has to pass as less affected as possible through the thickness of the thrust coil. In fact, in reaction to the seed field generation, a current, which aims to counteract the generation of the seed field itself, is induced through the thrust ring. Therefore, if not designed accordingly, the thrust ring might also completely block the field from filling the chamber. To overcome this possibility, what happens inside a conductor when hit by a magnetic field comes to aid. The intensity of the induced current that is generated inside the conductor decreases along the thickness of this, and so does any external magnetic field that penetrates. The trend of the induced current density, and the penetration of the magnetic field is illustrated in Figure 3.11. 1 The depth at which the current density is reduced by a factor of e is defined as skin region [5]. 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 36 OPERATION

FIGURE 3.10: Cross section of an electric coil [5]

As a result, if the thrust ring is too thick, the seed magnetic field might be completely blocked and not fulfil its task. However, if a proper thickness is chosen (i.e. much thinner than the skin region) the seed field can propagate to fill the chamber. Nevertheless, not blocking the seed field from filling the chamber is not the only requirement the thrust rings should comply. Actually, they should also prevent the amplified compressed field from FIGURE 3.11: Variation of the induced current density reaching the seed coils. That (j) and magnetic field magnitude (B) through the thick- is to not affect the operation of ness of a conductor (x)[5] the last. Indeed, the equation of the skin region depth shows how it is possible to fulfil these two seemingly conflicting requirements. Equation 3.7 shows that the skin depth (δ) is inversely proportional to the square root of the magnetic field frequency (ν)[5]. s ρ δ (3.7) = π νµ There, substituting the frequency with the inverse of the variation period (τ) Equa- tion 3.7 becomes [5]: s ρτ δ (3.8) = πµ Therefore, it is clear how for the same conductive material properties (i.e. resistivity 3.2.P REVIOUS CONCEPT DESIGNS 37

(ρ), and magnetic permeability (µ)) the skin depth changes according to how fast the variation of the magnetic field is. Now, the generation of the seed field, despite being dependent on the selected pulse frequency, is completed in a time between 10 and 50ms, whereas the field compression between 1 and 10µs [5]. Thus, the ratio between the skin depth of the seed field (δs), and that of the compressed one (δc ) is [5]: 3 r 50 10 3 r  · − 223.6 δs τs  1 10 6 = r · − δc = τc =  10 10 3  · − 31.6  10 10 6 · − = Hence, in the worst case scenario, the skin depth of the seed field is still 31.6 times the other. In addition, NASA argues that selecting a conductor thickness only 5 times greater than δc would be sufficient to grant an effective operation of the chamber. As a result, according to the proposed design, a thrust ring made of titanium diboride and molybdenum (Ti B2/Mo), with a thickness between 0.04 and 0.35cm, would be feasible and adapt for the specific application [4,5]. In fact, a similar ploy had been already proposed by Hyde in his concept design of a Laser Fusion Rocket (LFR) for interplanetary missions [45]. According to what found in the literature, his study has been the first to propose the use of conducting coils to attain the magnetic field compression. In addition, he argued that a metal skin around the seed-field coils is necessary to complete the chamber’s task. Thus, the just described thrust rings can be considered as a derivation of that “simple”metal skin concept. However, the quality of the papers found being pretty poor, no extensive material has been gathered about his project.

PROJECTED CAPABILITIES To show the possible capabilities of this engine, NASA have completed some preliminary investigations about its possible application for interplanetary missions: particularly interesting is the option of a manned round trip to Mars. For the specific purpose, since the resultant rocket is projected to employ prolonged burn times (i.e. medium-thrust rocket), a numerical analysis has been implemented to compute the trajectory of the spacecraft [5]. As a result, despite implementing numerous simplifications, the study showed that a 90-day transfer time might be feasible. Besides, were all the expectations of the estimated performance met, the same destination could be reached in 30 days [3,5]. In conclusion, since it nowadays is the most advanced, as well as most thoroughly investigated concept, a multi-coil parabolic chamber has been selected as the subject of this thesis. Besides, there is an ongoing project that aims to complete the first flight-ready system by 2023, and employs the same kind of reaction chamber [3, 46]. That is the project under development at MSNW3, and has confirmed similar performance and mission capabilities to what from NASA’s study. Moreover, MSNW’s rocket has been the subject of the literature review that preceded this thesis research [47].

3http://msnwllc.com/ 3.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 38 OPERATION

However, further investigations are required before the capabilities of this rocket can be confirmed, and its development advanced. Among them, an extensive computational analysis is still missing in order to validate the simplified analytical model of the chamber [5]. Thus, contributing to finally filling that gap has been the aim of this very thesis project, and the next chapters report the actual work that has been completed. First, the review of the simplified analytical model that describes the 3 dynamics of a multi-coil parabolic chamber is detailed. In fact, completing an exhaustive revision of that model has helped to clarify the operation of this reaction chamber. 4 MAGNETIC FLUX COMPRESSION REACTION CHAMBER: ELEMENTARY ANALYSIS

The first extensive attempt of reproducing the plasma dynamics in a multi-coil parabolic reaction chamber is included in a part of NASA’s HOPE conceptual design [4]. Thereby, Adams et al. selected Thio’s concept over Orth’s one, due to a lack of documentation of the latter. In fact, Thio et al. included a quite detailed investigation about the energy balance of the propulsion system [10]. Nevertheless, despite they have managed to replicate the dynamics of the plasma in a parabolic reaction chamber, Adams’ research is supported by some basic calculations, and the assumptions taken are still quite strong [4]. Thus, this chapter presents the review of both Thio’s and Adam’s work. This analysis helped to clearly understand the way the simplified analytical model of the reaction chamber has been derived, as well as which assumptions it is based on. In addition, from the results of this review, the initial conditions to apply for the rest of the project have been derived: i.e. the performance estimated by Thio et al. [10] are what the computational analysis of this thesis has aimed to verify. First, the same results as from Thio’s work (i.e. system energy balance) are again derived In Section 4.1. Then, Section 4.2 describes the simplified analytical model of the reaction chamber as it has been developed by Adams et al.. Eventually, the chamber performance attained through the energy balance are compared to the results of the simplified analytical model by applying the same initial conditions. The calculations have been completed using a compiled MATLABTM script. Again, such a comparison has helped to verify the understanding of the two different approaches, as well as to really get familiar with the projected operation of a magnetic flux compression reaction chamber. The development of the script, as well as the achieved results, are discussed in Section 4.3. Also, the paper that are reviewed in this chapter use the term magnetic nozzle, instead of reaction chamber. Thus, might that name appear in the next, it is to be assumed equivalent to magnetic flux compression reaction chamber.

39 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 40 ELEMENTARY ANALYSIS

4.1. SYSTEM ENERGY BALANCE Thio et al. discussed the feasibility of a nuclear fusion powered space propulsion system that would make use of a multi-coil magnetic ﬂux compression parabolic reaction chamber [10]. To estimate the operation of that reaction chamber they employed a quite detailed energy balance of the whole propulsion system. That same balance is hereby reviewed.

FIGURE 4.1: Energy ﬂow diagram of the system [10]

Figure 4.1 reports the energy flow diagram of the system discussed by Thio et al., while Table 4.1 the description of all the symbols used in the same study. There, the plasma pellet that is generated at the focus of the parabolic chamber is brought to fusion condition by the compression due to imploding plasma jets (i.e. plasma liners) that converge to the plasma location: the method of bringing a magnetised plasma pellet to fusion conditions by the compression due to impacting plasma jets (i.e. liners) is called MTF. The generation of the plasma pellet, as well as the ignition process, is accomplished with the energy that comes from some capacitors (Ecap ). Out of the fusion reaction, one gets the energy of both charged particles (i.e. electrons and protons, E ) and neutrons (E ). However, since neutral particles (i.e. q 0 in the Maxwell’s cp,f n = equations) cannot be stopped by the magnetic field, their energy would be totally lost, were an appropriate neutron moderator not used. Eventually, the energy that is transferred to the nozzle (i.e. compressed magnetic field plus induced current in the coils) is partially returned to the plasma in order to generate thrust (E jet ); another part is dissipated (Qpl asma and Qnozzle ), and what remains is used to recharge the system (Er c ). Thus, according to the energy flow diagram, the energy that at the end of every fusion pulse is available to produce thrust (E jet ) is [10]: 4.1.S YSTEM ENERGY BALANCE 41

Ecap Electrical input energy to generate the plasma pellet and drive the ignition system Eliner Energy of the imploding plasma liners Etar get Initial energy of the plasma target Qdr i ve Waste heat from the plasma pellet generator and ignition system Ecp,f Charged particle energy from the fusion reaction En Neutron energy from the fusion reaction Ecp,n Neutron energy fraction recovered by the moderator En,esc Neutron energy fraction lost to deep space En,cap Neutron energy fraction absorbed by the moderator 4 E jet Jet propulsion energy Qpl asma Heat dissipated in the plasma due to non-isentropic transformation Qnozzle Heat dissipated in the electric coils during the magnetic ﬁeld compression Er c Energy extracted from the induced currents in the nozzle and used to recharge the system Qr c Heat dissipated by the recharging system

TABLE 4.1: Energy balance symbols explained [10, pag. 3]

E jet Ecp Qpl asma Qnozzle Er c = − − − (4.1) (1 ε )ε E = − c k cp where

E E E cp = cp,f + cp,n Er c εc = E Q Q cp − pl asma − nozzle Qpl asma Qnozzle εk 1 + = − Ecp

Also, the energy that comes from the fusion reaction (E f us) is expressed as a function of the input energy of the reaction itself. That is equal to the initial energy of the plasma pellet (Etar get ) plus the energy that comes from the ignition system (Eliner ). Thus, since the fusion reaction increases the input energy by an ampliﬁcation factor (G), the output energy is [10]:

E G(E E ) (4.2) f us = liner + tar get The same is also equivalent to the sum of the released charged particles energy (Ecp,f ) and that of the neutrons (En)[10].

E E E f us = cp,f + n 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 42 ELEMENTARY ANALYSIS

However, as already mentioned, the magnetic ﬁeld cannot stop the neutrons expansion: their energy would be totally lost if a neutron moderator were not used. In fact, the material of the converging plasma liners includes hydrogenous elements with high neutron scattering capabilities: the neutrons break the bonds of the hydrogenous elements by impacting with them, hence, they are responsible for the release of charged particles that add to the expanding plasma (i.e. a fraction of the neutrons energy is recovered) [10]. Thus, by adding the recovered energy (Ecp,n) to the initial charged particle energy (Ecp,f ), the energy that is available to compress the magnetic ﬁeld (Ecp ) is [10]: 4

E E E cp = cp,f + cp,n E η E (4.3) = cp,f + mod n E (1 η )E = f us − − mod n Then, to evaluate the performance of the reaction chamber, the plasma momentum before and after the reflection are introduced. In fact, as already explained in Chap- ter3, due to the variation of plasma axial momentum after the reflection, a force (i.e. the thrust) is applied to the chamber as a reaction. Thus, to derive an expression of the plasma momentum, its initial velocity is first expressed. However, that is attained from the very energy that is available for thrust generation E jet (i.e. the energy that is converted back to plasma kinetic energy after it is stopped by the magnetic compression), and not from the initial plasma energy Ecp . Hence, it takes into account all the possible losses involved, already [10]:

s 2E jet v = mp s (4.4) 2(1 ε )ε E − c k cp = mp

In addition, the plasma is initially assumed of a spherical shape of radius R0. Thus, its mass is [10]:

4 3 mp πR ρ0 = 3 0 Nevertheless, as introduced in Chapter3, the parabolic chamber intercepts the trajectory of a fraction of the expanding plasma, only. That fraction depends on the aperture angle (θα) of the chamber itself: the aperture angle is in fact the angular position of the ﬁrst electric coil calculated from the chamber axis having its positive direction towards the chamber exit. Figure 4.2 is to picture the spherical reference frame that has been used for all the derivations discussed in the next. Thus, the product of the local axial velocity component (i.e. v cos(θ)) times the initial plasma density (ρ0) is integrated over the initial volume of plasma to attain the plasma momentum before its reﬂection (pz,exp )[10]: 4.1.S YSTEM ENERGY BALANCE 43

FIGURE 4.2: Spherical coordinate system

Z 2π Z π Z R 2 pz,exp ρ0v cos(θ)r sin(θ)dr dθ dφ = 0 θα 0 3 2 R0 sin (θα) (4.5) 2πρ0v = − 3 2 1 2 mp v sin (θα) = −4 The reflection is then assumed as perfectly elastic (i.e. all the losses have been already accounted for in the expression for the plasma initial velocity). Hence, the reflected velocity is taken equal to the initial one. However, a mean angular deviation (δθD ) is also included: the plasma might in fact slightly deviate from the axial direction after the reflection. Therefore, the plasma momentum after the reflection is again calculated for the same mass of plasma (i.e. same density and integration limits) times the reflected velocity (i.e. v cos(δθD )) [10].

Z 2π Z π Z R 2 pz,re f ρ0v cos(δθD )r sin(θ)dr dθ dφ = 0 θα 0 R3 ¯ ¯π 0 ¯ ¯ 2πρ0v cos(δθD ) ¯ cos(θ)¯ = 3 − θα R3 0 (4.6) 2πρ0v cos(δθD ) [1 cos(θα)] = 3 + 3 µ ¶ R0 2 θα 2πρ0v cos(δθD ) 2cos = 3 2 µ ¶ 2 θα mp v cos(δθD )cos = 2 where the trigonometric identity [10]: 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 44 ELEMENTARY ANALYSIS

µ ¶ 2 θα 1 cos(θα) 2cos + = 2 has been used. As a result, the change of plasma axial momentum is attained [10]:

pz pz,re f pz,exp = ·− µ ¶ ¸ 2 θα 1 2 mp v cos(δθD )cos sin (θα) = 2 + 4 4 · µ ¶ µ ¶ µ ¶¸ 2 θα 1 2 θα 2 θα (4.7) mp v cos(δθD )cos 4cos sin = 2 + 4 2 2 · µ ¶¸ µ ¶ 2 θα 2 θα mp v cos(δθD ) sin cos = + 2 2 where the trigonometric identity [10]: µ µ ¶¶ θα θα sin(θα) 2cos sin = 2 2 has been used. As a consequence, the chamber efﬁciency (η j ) can be evaluated as the ratio between the actual plasma momentum change and the total initial plasma momentum: the latter is given by the total mass of plasma (mp ) times the real initial plasma ³ 2E ´ velocity (i.e. no losses included) v cp [10]. 0 = mp

pz η j = mp v0 pz r = 2Ecp (4.8) mp mp · µ ¶¸ µ ¶ p 2 θα 2 θα (1 εc )εk cos(δθD ) sin cos = − + 2 2

Thus, to derive the chamber efficiency, one can estimate a value for both the εk and εc coefficients. The former is linked to the dissipated heat due to non-isentropic transformations and to ohmic heating of the chamber coils. However, due to the nature of fusion plasmas (i.e. high temperature and reaction rate), and the properties of the superconducting coils that are used, the coefficient εk can be assumed as high as 0.85 [10]. On the other hand, εc depends on the energy that is necessary to recharge the capacitors and initialise a new reaction after every pulse. Thus, it varies according to the efficiency of the recharging system, as well as to what of the plasma pellet generation and ignition sequence. Thio et al. discussed 0.042 as a reasonable value for that coefficient [10]. Thus, once the chamber efficiency is quantified, the axial jet momentum (M j ), the exhaust velocity (vex ), the specific impulse (Isp ), and the rocket thrust (T ) can respectively be found as [10]: 4.1.S YSTEM ENERGY BALANCE 45

q M η 2m E (4.9) j = j p cp

M j vex (4.10) = mp

vex Isp (4.11) = g0

T f M (4.12) 4 = rep j where g0 is the standard gravity (i.e. Earth sea-level gravity). As a result, from all the above defined relations, Thio et al. estimated the performance of a specific propulsion system combined with a magnetic flux compression reaction chamber. All the assumptions concerning the energy requirements and ef- ficiencies are based on the knowledge and technological state of the art of the time when the study was completed [10]. However, comparable values have been found in more recent studies, as well [3].

4.1.1. ESTIMATED PERFORMANCE To derive the performance of the projected engine, Thio et al. started from the energy requirements of the fusion reaction. The assumed nuclear reaction is the so called MTF: the literature study that preceded this thesis project highlighted MTF as the most promising alternative for attaining a feasible nuclear fusion reactor. The fuel projected to feed the reaction was selected through a trade off that included neutron production, fuel availability, and feasibility in the short term [10]. Concerning the neutron production, that is a critical issue when discussing mp 2.26g nuclear fusion applications: neutrons Eliner 14MJ escaping the reaction chamber are suf- Etar get 2.5kJ ﬁciently energetic to possibly cause se- En 0.35E f us vere damages to the spacecraft structure θα 5/12π (e.g. neutron embrittlement [4]), and to δθD π/12 the crew, as well. As a result, despite it εk 0.96 is not the least energy demanding, Thio εc 0.031 et al. chose a Deuterium-Deuterium (D- ηmod 0.98 D) fusion reaction as reference for their G 70 project [10]. A Deuterium-Tritium (D- f 200Hz T) would in fact require less energy to rep be ignited, but release a higher quantity TABLE 4.2: Input parameters for the MTF of neutrons [10]. However, even though rocket performance estimation [10] sea water is very rich of Deuterium, Tri- tium is not easily found in nature. On the other hand, a better option might be Deuterium-Helium-3 (D He3) which is a practically aneutronic reaction. Never- − theless, no He3 is available on the Earth. 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 46 ELEMENTARY ANALYSIS

Therefore, Thio et al. derived the en- E f us (from Eq. 4.2) 980MJ ergy requirements for a D-D fusion re- Ecp (from Eq. 4.3) 973MJ action from simulations they had them- η j (from Eq. 4.8) 0.81 selves completed for another study [48]. M j (from Eq. 4.9) 1.7kgm/s The results showed that a fusion gain of vex (from Eq. 4.10) 752km/s 70 is possible with an input energy of 3 ' Isp (from Eq. 4.11) 77 10 s 14MJ (i.e. plasma initial energy plus fu- · T (from Eq. 4.12) 340kN sion ignition energy), and by using 2.26g of fuel [10, 48]. Furthermore, 200Hz is TABLE 4.3: Estimated performance of the MTF 4 pointed out as a reasonable limit for the rocket [10] nuclear pulse repetition rate: reaching this value is discussed as a feasible and good compromise between system requirements and outcome performance [10]. As a result, the input parameters that Thio et al. have employed to estimate the performance of a MTF rocket are summarised in Table 4.2. Thus, from the just deﬁned conditions and through the relations expressed in the previous sections, the resultant performance of the MTF rocket has been derived and reported in Table 4.3[10].

4.2. SIMPLIFIED ANALYTICAL MODEL To complete a simplified analytical model of the plasma dynamics in a reaction chamber, the plasma is considered as an ideal gas that expands under the effect of its internal hydrostatic pressure, and the external magnetic pressure. The dynamics is then evaluated through an iterative process: instantaneous analytical solutions of all the involved physical quantities are attained, and each of them is updated at every time step according to the projected evolution of the system [4]. The same derivations have been reviewed again by NASA’s Advanced Concept Office in another study of theirs, and the same results attained [5]. However, both NASA’s works neglect the whole initial part of the plasma expansion: the plasma is assumed having a parabolic shape (i.e. parabolic plasma shell), and its thermal energy completely converted in kinetic energy [4,5]. Therefore, the iterative process starts with the plasma at its maximum speed, and the magnetic field trapped in the small interstice between the plasma and the chamber wall. As a consequence, the only force that affects the dynamics of the plasma is due to magnetic pressure. Furthermore, for the whole integration process, the plasma and the field are assumed of a parabolic shape, both having the focus coincident with the chamber’s. Figure 4.3 illustrates such initial conditions. Thus, exploiting the axial symmetry of the system, only one section of the nozzle is modelled. As already mentioned in Section 2.2.1 and 3.1.2, due to the solenoidal nature of a magnetic field, its flux (Φ) is uniform and constant in time. Hence, applying the trigonometric properties of a parabolic section, the magnetic flux is expressed in spherical coordinates as [4,5]:

Φ A [B(θ)cos(π/2 θ/2)] A B(θ)sin(θ/2) (4.13) = con − = con Where θ is the colatitude, B is the local magnetic ﬁeld magnitude calculated at θ, and 4.2.S IMPLIFIED ANALYTICAL MODEL 47

FIGURE 4.3: Initial conditions of the elementary magnetic ﬂux compression reaction chamber analysis [4]

Acon is the conical section the magnetic flux is calculated through. Figure 4.4 exem- plifies the used reference frame, and, hence, how the equation has been derived. The magnitude of the field is considered constant along the r-direction. Adams et al. report that such a simplification is due to the already limited (i.e small) volume occupied by the field, however, no more detailed proofs are brought at support [4]. In fact, as the Biot-Savart Law suggests (see Eq. 3.1), it is sound to assume the field would vary with the distance from the coils. The derivation then continues with an expression for Acon(θ). That is attained through the following mathematical simplifications. The final result is shown in Equation 4.14. Figure 4.5 illustrates the geometry considered.

dA (h) 2πhdr con = h r sin(θ) = Z rmax Acon(θ) 2πr sin(θ)dr = rmin πsin(θ)(r 2 r 2 ) = max − min 2a r = 1 cos(θ) −

l 2 l 2 f − f p Acon(θ) 4πsin(θ) (4.14) = (1 cos(θ))2 − Combining Equation 4.13 and 4.14, the magnetic ﬁeld (B(θ)) can thus be expressed as [4,5]: 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 48 ELEMENTARY ANALYSIS

FIGURE 4.4: Exempliﬁcation of how the magnetic ﬂux has been derived [4]

Φ (1 cos(θ))2 B(θ) − (4.15) = 2π(l 2 l 2 ) 2sin(θ)sin(θ/2) f − f p Also, from the trigonometric identities [4,5]:

1 cos(θ) tan(θ/2) − = sin(θ) (1 cos(θ))2 sin(θ)tan(θ/2)2 − = sin(θ) (1 cos(θ))2 2sin(θ/2)cos(θ/2)tan(θ/2)2 − = sin(θ) sin(θ/2)2 (1 cos(θ))2 − cos(θ/2) = 2sin(θ)sin(θ/2)

the B ﬁeld can eventually be expressed as [4,5]

Φ sin(θ)2 B(θ) (4.16) = 2π(l 2 l 2 ) cos(θ/2) f − f p The last equation expresses the magnetic ﬁeld magnitude as a function of θ and the constant magnetic ﬂux (Φ). From that, the study continues with considering the plasma shell divided in multiple parabolic sections: each section reacts to a constant and uniform magnetic pressure [4,5]. Concerning the magnetic pressure, there is a mistake in the equations used in both the papers [4,5]: the value of the pressure 4.2.S IMPLIFIED ANALYTICAL MODEL 49

FIGURE 4.5: Area of the conical annulus at an angle θ [5]

is doubled. Such a mistake has been recognised while trying to reproduce the performance of the rocket proposed by Thio et al.. The correct deﬁnition of magnetic pressure, hereby repeated, is as from Equation 3.6.

B 2 pB (4.17) = 2µ0

To continue, an expression for the area of the plasma shell section is attained. Figure 4.6 helps to visualise the problem. This area is thereby considered as a vector, which can then be decomposed along the coordinate axis [4,5]. Figure 4.7 exempliﬁes the result.

d~A dA eˆ dA eˆ = θ θ + r r dA r 2sin(θ)dθdφ r = dA r sin(θ)dr dφ θ = − ⇓ ~ 2 1 dr dA r sin(θ)dθdφ( eˆθ eˆr ) = −r dθ +

and applying some other manipulations [4,5]: 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 50 ELEMENTARY ANALYSIS

2a r = 1 cos(θ) − dr r sin(θ) dθ = −1 cos(θ) − ~ 2 sin(θ) dA r sin(θ)dθdφ( eˆθ eˆr ) = 1 cos(θ) + − r 2sin(θ)dθdφ(cot(θ/2)eˆ eˆ ) = θ + r 4

FIGURE 4.6: Illustration of the area of plasma shell section considered [4]

Then, from the deﬁnition of solid angle (Ω)[4,5]:

dΩ sin(θ)dθdφ = the section area can ﬁnally be written as [4,5]:

d~A r 2dΩ(cot(θ/2)eˆ eˆ ) (4.18) = θ + r Therefore, the force that is applied to each plasma section due to the magnetic pressure is [4,5]:

d~F pB d~A = − (4.19) p r 2dΩ(cot(θ/2)eˆ eˆ ) = − B θ + r where the negative sign indicates the force acting in the opposite direction to the vector normal to the expanding plasma surface [4,5]. Then, it is assumed that different portions of the plasma shell that subtends the same solid angle (dΩ), has the 4.2.S IMPLIFIED ANALYTICAL MODEL 51

FIGURE 4.7: Decomposition of the plasma shell section area along the coordinate axis [4] same mass. Thus, if M is the total mass of the shell, the mass of one of its sections (dM) is [4,5]:

dΩ dM M = 4π As a consequence, the equation for the applied force can be manipulated to attain dv an expression of the time derivative of the plasma velocity ( dt ).

dM dΩ 4π = M

⇓ 2 dM d~F 4πpB r (cot(θ/2)eˆ eˆr ) = − M θ + d~v d~F dM = dt

As a result [4,5]:

2 d~v 4πpB r (cot(θ/2)eˆ eˆr ) (4.20) dt = − M θ + dv From the last derivation one can see that, as expected, dt is always negative: the cot(θ/2) is in fact always positive for θ between 0 and 2π [4,5]. The thrust that is applied to the rocket can thus be modelled as a reaction to the plasma deceleration [4,5]:

d~F d~F R = − Therefore: 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 52 ELEMENTARY ANALYSIS

sin(θ)dθdφ d~v d~FR M (4.21) = − 4π dt A force is thus applied to each reaction chamber section that covers the same solid angle (i.e. coordinate and extension wise) of each plasma section. Specifically, the total number of sections that is considered is equal to the number of coils (i.e. 8): in fact the force is hereby applied to each thrust ring (see Sec. 3.2.3). As a result, Equation 4.21 is first estimated for each section by an integration in dφ and dθ, then all the separate contributes are added to attain the total thrust. The extremes of in- 4 tegration for φ are 0 and 2π, due to the axial symmetry of the system. On the other hand, concerning the extremes of θ another simplification is employed: i.e. the force applied on each thrust ring is due to the plasma shell that has co-latitudinal coordinates closer to that very coil’s than to any other’s. Figure 4.8 illustrates how the extremes for θ are chosen.

FIGURE 4.8: Angular position of the coils (a), and plasma shell portion associated to the each coil (b) [5]

Therefore, from the integration of dF~ R , and assuming that the force’s dependence dv on the value of θ is not too strong (i.e. for each integration dt is considered constant and equal to its value calculated at θi [4,5]:

θi θi 1 M dv~ ¯ Z 2π Z +2 + ~ ¯ FR dφ θ θ sin(θ) dθ ' −4π dt ¯θ i i 1 i 0 +2 − (4.22) M d~v ¯ · µθ θ ¶ µθ θ ¶¸ ¯ i 1 i i 1 i ¯ cos + + cos − + ' 2 dt θi 2 − 2 Finally, the quantities are expressed in polar coordinates (i.e. (ρ,φ,z)) having the z- axis coincident with the nozzle axis. That way, the z-component of the force is in fact the generated thrust. Equation 4.23 and 4.24 give the transformation from spherical to polar coordinates., hence, the positive direction of the z-axis points the fore of the rocket. 4.2.S IMPLIFIED ANALYTICAL MODEL 53

eˆ eˆ sin(θ) eˆ cos(θ) (4.23) r = ρ + z

eˆ eˆ cos(θ) eˆ sin(θ) (4.24) θ = ρ − z As a result, the ﬁnal equation of the acceleration that is then used to estimate the thrust is reported in Equation 4.25, that is also preceded by its derivation [4,5]: ( d~v 4πp r 2 cot(θ/2)eˆ B θ 4 dt = − M eˆr

~ 2 ( sin(θ) dv 4πpB r 1 cos(θ) eˆθ − dt = − M eˆr ³ ´ ~ 2 sin(θ) dv 4πpB r  1 cos(θ) cos(θ) sin(θ) eˆρ ³ − + ´ dt = − M sin(θ)  1 cos(θ) sin(θ) cos(θ) eˆz − − + ³ ´ ~ 2 sin(θ)cos(θ) sin(θ) sin(θ)cos(θ) dv 4πpB r  +1 cos(θ−) eˆρ ³ 2 − 2 ´ dt = − M sin(θ) cos(θ) cos(θ)  − 1+ cos(θ)− eˆz − ³ ´ ~ 2 sin(θ) dv 4πpB r  1 cos(θ) eˆρ ³ − ´ dt = − M cos(θ) 1  1 cos(−θ) eˆz − 2 ( d~v 4πpB r cot(θ/2)eˆρ dt = − M eˆ − z 2 d~v 4πpB r (cot(θ/2)eˆρ eˆz ) (4.25) dt = − M − As expected, the z-component of the acceleration is always positive, and a function of the magnetic pressure. Thus, to determine the magnetic field, hence the related pressure, Equation 4.16 shows that it is first necessary to evaluate the magnetic flux. The derivation of the flux as discussed by the NASA’s teams is summarised in the next section.

4.2.1. INITIAL MAGNETIC FLUX DERIVATION Since the field is generated by electric coils, an expression that links the magnetic flux to the inductance of and the current flowing through each coil can be defined. Specifically, the review of NASA’s papers [4] and [5] has pointed out the latter as having a more reliable derivation (i.e. less typos and more thorough explanation). There, the derivation starts from the equation of the magnetic self-inductance related to the current (I), and the inductance (L) of each coil:

Φ LI = 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 54 ELEMENTARY ANALYSIS

The same equation can be used to express the magnetic flux contained within each j-ring of the reaction chamber. Figure 3.4 is again useful as a reference here: the magnetic flux through each j-ring is in fact what through the green circle of the figure.

Φ L I (4.26) j = j s,j However, this is valid for the initial conditions, only: when the seed field is generated by each seed current Is,j . Later on, when the plasma expansion starts, the mutual inductance between each ring and each plasma section has to be included, as well. In fact, a current is induced on both the thrust rings and the surface of the plasma due 4 to the resultant Lorentz Force (see Sec. 3.1.2). Thus, during the plasma expansion, the generated magnetic flux is affected by the induced current in the plasma (Ip,j ), and the current in the coils (Ic j ): i.e. induced current plus seed current (see Fig 3.6). As a result, at the point of maximum compression (i.e. when the plasma expansion stops), the magnetic flux related to each j-ring is [5]:

Φ L I M I (4.27) j = j c j + j p,j In a similar way, the ﬂux through each plasma section ring (i.e. red ring of Figure 3.6) is:

Φ L I M I (4.28) p,j = p,j p,j + j c,j However, as Section 3.1.2 has already pointed out, the ﬂux through that section is indeed zero. Thus [5]:

Φ 0 p,j = ⇓ Ip,j M j Lp,j = − Ic,j and, as a consequence [5]:

Ip,j Φj L j Ic j Lp,j Ip,j = − Ic,j Ã 2! (4.29) µ Ip,j ¶ L j Lp,j Ic,j = − Ic,j

From the last relation, an equivalent inductance for the current carrying ring (Lc,j ) is deﬁned. That being a function of the instantaneous electric current is in fact a time varying inductance [5]:

Ã 2! µ Ip,j ¶ Lc,j L j Lp,j = − Ic,j

⇓ Φ L I j = c,j c,j 4.2.S IMPLIFIED ANALYTICAL MODEL 55

Next, the magnetic potential energy related to each coil is introduced as a function of their relative inductance and current. In particular, the initial energy (Es,j ), and at maximum ﬁeld compression (Ec,j ) are estimated [5].

1 2 Es.j L j I = 2 s,j 1 2 Ec.j Lc,j I = 2 c,j Ic.j 4 Then, introducing the current ampliﬁcation factor A j , the magnetic ﬂux and = Is,j the coil energy at maximum compression can each be written as [5]:

Φ A L I j = j c,j s,j 1 2 2 Ec.j A Lc,j I = 2 j s,j Hence, combining the last two equations:

1 2 2 Ec.j A Lc,j I = 2 j s,j Φj Lc,j = A j Is,j

⇓ 1 2 Φj 2 Ec.j A j Is,j = 2 A j Is,j 1 A j Φj Is,j = 2 Finally, from Equation 4.26 the energy at maximum compression can be expressed as a function of the magnetic ﬂux and the coil inductance only [5].

1 Φj Ec.j A j Φj = 2 L j 2 1 A j Φj

= 2 L j

Consequently, the initial energy becomes [5]:

2 1 Φj Es,j = 2 L j and, as a result [5]:

E A E c,j = j s,j 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 56 ELEMENTARY ANALYSIS

Thus, the energy that is transferred to the magnetic field can be defined as the difference between the final and the initial energy of the coils [5].

∆E E E (A 1)E j = c,j − s,j = j − s,j Eventually, assuming an ampliﬁcation factor that is the same for all the rings [5]

X ∆E ∆E j = j 4 Φ2 1 X j (A 1) = − 2 j L j

and, the magnetic ﬂux being uniform in space (i.e. the same for every ring: Φ Φ), j = the equation [5]:

2 Φ X 1 ∆E A 1) = − 2 j L j is ﬁnally attained. From this last relation, an expression that can be used to estimate the magnetic ﬁeld from the design parameters (i.e. A, ∆E, L j ) is derived [5].

Φ2 ∆E (A 1) P 1 − 2 = j L j v (4.30) u 2∆E Φ u t P 1 = (A 1) j − L j

Hence, an analysis of this last relation highlights that:

• the inductance L j of each ring is dependent on the same very ring geometry, only, and hence is a design parameter;

• ∆E is equal to the remaining kinetic energy of the plasma that has to be transferred to the magnetic ﬁeld (i.e. energy of the fusion reaction products minus the losses).

Therefore, one can either select an ampliﬁcation factor and derive a required magnetic ﬂux as a consequence, or do the other way around. The system is affected by the choice in such a way that:

• higher the amplification factor is, more the magnetic field gets compressed before the plasma is reflected;

• higher the initial magnetic ﬁeld is, lower the necessary current ampliﬁcation factor. 4.3.A NALYTICAL MODEL VALIDATION 57

As a result, optimising the combination of the two parameters allows to define the minimum distance from the chamber structure that is reached by the plasma, as well as how much the current induced in the thrust rings is: a compromise between performance and safety of the system has thus to be sought. A higher amplification factor would in fact increase the generated thrust. However, the currents induced in the coils, as well as the loads applied to the structure of the chamber may become excessive and hence, compromise the integrity of the system. NASA discussed an amplification factor of 25 as a reasonable and feasible compromise [4,5]. That is thus the value that has been included in the verification process reported in the next section. That was aimed to verify the estimated perfor- 4 mance by Thio et al. [10] with the application of the just discussed simplified analytical model [4].

4.3. ANALYTICAL MODEL VALIDATION To thoroughly review the simplified model developed by the NASA’s teams [4,5], a MATLABTM script has been implemented to complete the calculations of the iterative process. The aim was to verify whether the model could reproduce the performance estimated by Thio et al. [10]: the same initial conditions as from Table 4.2 have thus been used. In addition, a graphical output in which the trajectory of every plasma section is plotted has been included to highlight the point at which the plasma reflection happens. The aim was to verify that, according to the dynamics modelled, each plasma section would be reflected before reaching the reaction chamber structure. What is discussed in the next sections has thus been the product of self-reflections and interpretation of the model introduced in the read papers [4,5]: the general it- eration procedure is already mentioned in the papers, however, the hereby applied integration strategy has been specifically developed for this thesis. Hence, the final result has been achieved with the aid of MATLABTM R2015a1.

4.3.1. INITIAL CONDITIONS Since Adams et al. derived their chamber concept from Thio’s, the initial conditions for the problem have been selected combining the data from the three reports (i.e. [4,5, 10]). Thio’s data have been used to assign the initial energy conditions of the system, while Adams’ for the general geometry of the chamber and the electric coil details. The magnetic flux compression reaction chamber was thus taken of a parabolic shape with a focal length l f , and equipped with 8 electric coils equally spaced along the θ direction. In addition, the initial magnetic flux is necessary to then evaluate the magnetic field at every step and angular position according to Equation 4.16. Thus, the flux has first been evaluated applying Equation 4.30, where ∆E has been taken equal to E jet , and the amplification factor A to 25 as proposed in [4,5]. As a result, all the initial conditions and the coils specifications (i.e. respective inductance and angular position) are summarised in Table 4.4 and 4.5.

1(http://nl.mathworks.com/products/matlab/) 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 58 ELEMENTARY ANALYSIS

Chamber geometry 5 θα 12 π a 2.0m Subsystem efﬁciencies

εk 0.96 εc 0.031

ηmod 0.98 4 Initial energies

E f us (from Eq. 4.2) 980MJ Eliner 14MJ Etar get 2.5kJ

En 0.35E f us Ecp (from Eq. 4.3) 973MJ E jet (from Eq. 4.1) 905MJ Fusion reaction G 70

frep 200Hz Plasma initial conditions

mp 2.26g

l f p0 (initial plasma shell focal length) 1.0m vex (from Eq. 4.10) 752km/s v0 (from Eq. 4.8) 928km/s Initial magnetic ﬂux Φ (from Eq. 4.16) 4.61Tm A 25 ∆E E = jet

TABLE 4.4: MATLABTM problem initial conditions

1 2 3 4 7 6 6 6 L [Henry] 3.85 10− 2.35 10− 4.99 10− 8.29 10− · · · · θn [deg] 75 89 103 117 5 6 7 8 5 5 5 5 L [Henr y] 1.24 10− 1.79 10− 2.54 10− 3.68 10− · · · · θn [deg] 131 145 159 173

TABLE 4.5: Electric coil speciﬁcations [5] 4.3.A NALYTICAL MODEL VALIDATION 59

4.3.2. INTEGRATION PROBLEM Once assigned the initial conditions, the actual integration problem had to be defined. According to the approach proposed in the reviewed papers [4,5], all the physical quantities are updated at each new time step. Hence, at every step, the position of each plasma section is first updated according to the linear and angular velocities calculated at the step before. Then, both magnetic field strength and pressure are estimated according to Equation 4.16 and 4.17, respectively. From that, the accelerations are evaluated, and those are eventually used to estimate the thrust. Therefore, to complete the calculations, every physical quantity of the problem has been initially preallocated to an array having a number of rows equal to the num- 4 ber of time steps, and a number of columns equal to the number of electric coils (i.e. i 8)). In fact, as already explained in Section 4.2), the employed model estimates = the dynamics of each plasma section related to each coil (i.e. the plasma section θ θ θ θ i + i 1 i + i 1 extending from 2 − to 2 + ). Thus, preallocating the size of each array has helped to save computational time. The lines of the code that have been used to define all the arrays introduced in the problem are reported in Appendix A.1 (note that the label of each physical quantity might change from what showed in Table 4.4). In addition, due to the simplifying assumptions included, the model hereby introduced is not suitable for reproducing the after-rebound part of the system dynamics, which must thus be completely omitted [4,5]. As a consequence, a way to stop the integration process after the rebound has to be implemented. Besides, despite such an aspect is mentioned in the published discussion of the model, no specific strategy to stop the integration process in included [4,5]. The idea of simply limiting the total integration time was soon discarded. In fact, from the very 2nd time step, and for the whole rest of the integration, every plasma shell section acts as indepen- dent of the dynamics of all the others. Hence, the time step at which the plasma is reflected is different for every section. Hence, another approach not dependent on the total integration time has been adopted: the sign of the velocity along the radial direction (in polar coordinates) has been used to identify whether the plasma had been reflected or not. In fact, an analysis of the calculated results (i.e. plot, and numerical values) showed that the value of plasma radial velocity gets equal to 0, first, and then changes the sign (i.e. it becomes negative). Thus, the completed script checks for each plasma section whether that very condition is met. When a radial velocity 0 is identified, it stops updating ≤ all the physical quantities related to that section, and set its acceleration equal to 0. As a result, the specific plasma section is excluded from the integration to estimate the thrust. The lines of the code that concern the integration process are included in Appendix A.2. Among them, the ploy that has been used to stop updating the thrust estimation goes from line 21 to line 60 of the code. This strategy has been proven to be successful by progressively increasing the total time of integration: for a total time greater than the time of reflection, the calculated thrust does not change further. In addition, the average total thrust that is produced throughout the simulated plasma expansion, is derived as expressed in line 82 of the code: every instantaneous force that is applied to each plasma section is first multiplied by the time step length, and then summed to the others (i.e. total impulse); the result of the sum is then 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 60 ELEMENTARY ANALYSIS

multiplied by the pulse frequency to attain the average thrust. Despite being quite an obvious one, this derivation is also not included in [4,5]

4.3.3. RESULTS Thus, in accordance with Adams’ model [4], the aforementioned script has been used to evaluate the thrust generated by the reaction chamber. The attained results have then been compared to what previously estimated by Thio et al. [10]. As it is obvious (see line 73 of the code in Appendix A.2), the accuracy of the thrust calculation is greatly affected by the time step length: it increases with shorter time steps. 4 Nonetheless, an average thrust of T 340kN has eventually been attained: that is in ' accordance with what estimated by Thio et al. (see Tab. 4.3). Also, due to the implemented strategy, once the reflection is completed, any longer integration time does not affect the result (i.e. the estimated thrust does not change). The plot that shows the trajectory of each plasma section is also hereby included in Figure 4.9. However, the lower half of the chamber section in the picture is in fact just a mere reflection of the upper. Indeed, the model has been formulated to integrate the dynamics of half section, only: the plasma acceleration has been integrated from φ 0 to 2π. Further- = more, the plot is in spherical coordinates and includes a red parabola, that visualises the initial plasma position, as well as a black one for the chamber wall. Every dashed line, that starts from the red parabola, represents the trajectory of each plasma section. The plot shows that all the sections basically follow a radial direction for all the pre-rebound part of the computation. The successive part (i.e. the curvilinear one) has in fact no meaning, however, the different length of each line exem- plifies the fact that every section is reflected at a different time step. Moreover, despite that is not clear due to the low resolution of Figure 4.9, an enlarged view of the same plot shows that the reflection of each section does happen before touching the chamber wall. That can be seen in Figure 4.10: although one trajectory only is pictured, the same behaviour has been recognised FIGURE 4.10: Enlarged view of one for all the sections. The dashed line is again the plasma section trajectory (dashed trajectory of the plasma section, while the one in line) to show that its reflection hap- black represents part of the chamber wall. pens before reaching the chamber wall (black solid line)

4.3.4. FINAL REMARKS Completing this review has helped to clarify the strong assumptions the analytical model is based on. All the simpliﬁcations have been already mentioned in the previous sections of this chapter. Nonetheless, they are hereby summarised again, for the sake of clarity:

• The after-rebound part of the plasma motion is not modelled; 4.3.A NALYTICAL MODEL VALIDATION 61

FIGURE 4.9: Plot of the plasma section trajectories according to the MATLABTM script results

• The magnetic ﬁeld amplitude varies with the θ-coordinate, but is not dependent on the distance from the coils (i.e. r );

• The plasma is assumed starting from a parabolic shape that is then divided into multiple separate sections;

• Each plasma section is kept of a parabolic shape at every time step;

• Each plasma section evolves separately from the others;

• The thermal energy of the plasma is assumed as completely transformed in kinetic energy long before the magnetic ﬁeld starts slowing down the plasma.

Moreover, the considerations that have been derived from this review, and that have actually inspired the work completed for the thesis project, are now summarised. In fact, assuming an initial parabolic shape of the plasma means that it is considered as a single body until that point. Then, each section is considered as it were sepa- rated from all the others, and in a completely different system. As a consequence, it is as the motion of each section does not affect that of any other. Nonetheless, the 4.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 62 ELEMENTARY ANALYSIS

magnetic field being one and interconnected for all the sections, the reliability of the proposed model has been doubted. In addition, the effective feasibility of replacing a conductive wall (as in Project Daedalus) with multiple coils is just qualitatively confirmed by what found in the literature [5, 45]. Magnetic field compression is indeed theoretically flawless when between two conductive armatures (i.e. continuous surfaces). Nonetheless, what the field reaction is when it gets pressed towards the space between two consecutive coils, has not been found extensively discussed in the literature: the magnetic field of the two coils might in fact separate, and let the plasma pass through the chamber structure, as a consequence. That could in fact 4 be confirmed, or disproved, by a more rigorous description of the reaction chamber dynamics (e.g. implementing MHD principles in the model). In conclusion, before the design of any magnetic flux compression reaction chamber can go further, a more accurate modelling of its dynamics is necessary: i.e. the accuracy of the simplified analytical model has to be extensively verified. Therefore, numerical simulations that apply advanced plasma physics models for the description of a reaction chamber dynamics have a remarkable value: great effort has recently been put in developing sophisticated numerical codes that solve the plasma dynamics of complex propulsion systems [1, 49]. A complete numerical simulation would in fact add some valuable assets to the modelling of a reaction chamber:

• Electromagnetic interactions, completely lacking in the analytical model, would be implemented by a more accurate plasma physics description. That would include a more rigorous contribute of the magnetic ﬁeld, which in fact is considered as a mere hydrostatic pressure in the simpliﬁed model;

• The energy evolution of the system would be more thoroughly reproduced. The transformation of plasma thermal energy into kinetic energy, as well as the change of plasma shell shape would be estimated in detail;

• The after-rebound part of the expansion could also be modelled.

Therefore, the implementation of a more rigorous and complete numerical model may allow to completely, or at least partially drop all the simplifications the analytical model is based on. Specifically, computational MHD aims to numerically solve the equations of MHD[1, 13,25, 49–52]. Thus, since Chapter2 discussed how ideal- MHD can describe the dynamics of fusion plasma in a propulsion system, computational MHD has been implemented to complete this thesis project. The next Chapter introduces the aspects and references that brought to select the specific code used. 5 MAGNETIC FLUX COMPRESSION REACTION CHAMBER: COMPUTATIONAL CODE SELECTION

So far, this report has highlighted the need for a more extensive analysis aimed at verifying the operation of a magnetic ﬂux compression reaction chamber. In particular, Chapter2 pointed out ideal-MHD as a viable option that might be used to model plasma dynamics in a reaction chamber, and Chapter4 how such a task does require sophisticated numerical integrations. Thus, this chapter describes the investigation that preceded the computational analysis completed for this thesis project. First Section 5.1 gives the details of the process that eventually brought to the selection of PLUTO as the computational code for this analysis. Hence, Section 5.2 is used to introduce the functions of PLUTO, and its solving strategy. To conclude, Section 5.3 covers the so far achieved results in terms of computational analysis of reaction chamber dynamics. Despite the limited amount of material available in the literature, that has still been a valuable example.

5.1. SELECTIONOFTHE COMPUTATIONAL CODE At ﬁrst, the option of autonomously compiling a code to numerically solve the system of differential equations has been considered: the capabilities of MATLABTM in solving systems of ordinary and partial differential equations has been preliminar- ily investigated [53]. However, due to a quite scarce knowledge of MATLABTM programming and of numerical analysis in general, implementing an already existing dedicated code was recognised as a better alternative. The reasoning that aided that choice is summarised in Table 5.1. There, Brand-New Code identiﬁes the code to be autonomously compiled for the project. Therefore, from the considerations mentioned in the table, compiling a new code from scratch has been understood not being doable within the scheduled time for this thesis project. Also, the successive necessary validation would have become a completely different project on its own. Thus, recalling the objective of this research:

63 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 64 COMPUTATIONAL CODE SELECTION

the seek for a suitable and already distributed code has been recognised as the more seemingly effective and feasible option.

Brand-New Code Dedicated Code Strong knowledge of computer The work left to the user to adapt the programming and numerical code for the speciﬁc project is limited to 5 integration techniques required setting up the problem Extensive debugging might be A dedicated code usually has necessary before the code compiles documentation to support the user with correctly debugging A validation of the code is necessary A distributed code is usually already validated

TABLE 5.1: Brand-new Vs. Dedicated code to numerically solve ideal- MHD

As a consequence, applications of computational-MHD have been ﬁrst investigated in the literature. However, since focus was mainly given to propulsion systems applications, all the results were in fact relative to proprietary codes (e.g. MACH- 21 [25, 49–51] and Nautilus2 [13]). Hence, expanding the query to more general applications of computational MHD (e.g. astrophysics), and giving priority to freely distributed codes, the pool was eventually restricted to two candidates:

• The Pencil Code3;

• PLUTO4.

At that point of the project, no experience in computational-MHD had been achieved, yet. Hence, the process to select one of the two was not based on the spe- ciﬁc capabilities of each: as far as it could be foreseen at that time, the two codes were both capable of achieving the same results. In fact, the choice was mostly based on the gathered impressions after few preliminary exercises completed with the two. In the end, a more user friendly interface, and a more extensive documentation, combined with its open-source nature, were decisive for selecting PLUTO over the other.

1http://www.numerex-llc.com/m2.htm 2https://www.txcorp.com 3http://pencil-code.nordita.org/ 4http://plutocode.ph.unito.it/ 5.2.PLUTO 65

5.2. PLUTO PLUTO is a freely-distributed and modular code for computational astrophysics. It is written in C, and has been developed by a team led by Prof. Andrea Mignone at Politecnico di Torino. It has been validated by solving many astrophysical problems. Examples include applications in stellar and extragalactic jets, magneto-rotational instability, and relativistic Kelvin-Helmholtz instability [12]. Nonetheless, what hereby presented shows that PLUTO may also be suitable for engineering applications, beside the ones it has been programmed for. In fact, PLUTO implements multiple physics modules [11]:

•HD: Newtonian (i.e. classical) hydrodynamics;

• MHD: ideal/resistive magneto-hydrodynamics; 5

• RHD: relativistic hydrodynamics;

• RMHD: relativistic magneto-hydrodynamics;

Among them, ideal-MHD has been the primary module employed for the analysis hereby reported. However, at the end of the project, the possible need to implement the relativistic module has been identiﬁed (see Sec. 2.4.3). Regardless of the physical model employed, the code is programmed to integrate a set of non-linear hyperbolic conservation laws (i.e. Euler Equations [54]), of the form [12]:

∂U F (U) S(U) (5.1) ∂t = −∇ · + where U is the vector containing the conservative variables, F (U) is the flux tensor (i.e. the flux of the conservative variables through the surface limit of the integration domain), and S(U) is the source term. The source term includes the effect of gravity acceleration and potential, which have both not been included for the completed calculations; hence, S(U) has been set 0 for this project. = Ideal-MHD can in fact be expressed through conservative laws, as in Equations from 2.24 to 2.27, or in its primitive form. The first describes the evolution of the conservative variables (ρ,ρ~v,etot ,B~), while the latter of the primitive variables (ρ,~v,p,B~). However, when applying numerical analysis, there are some advantages in expressing the same model in a conservative form. In particular, using conservative laws, the solution of the code converges even though shock-waves are detected in the integration domain [54]. As a result, PLUTO implements the so called High Resolution Shock-Capturing (HRSC) scheme, which has lately been proven to be a reliable and robust tool to solve strongly supersonic flows [12]. Besides, the presence of shock- waves has in fact eventually been highlighted in the solutions completed for this thesis. Therefore, when expressed in their conservative form, U and F (U) used to model ideal-MHD are [12, 54]: 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 66 COMPUTATIONAL CODE SELECTION

 ρ  ρv   x1 ρv   x2   ρvx3 U   =  etot     Bx1     Bx2  Bx3

 T ρvx1 ρvx2 ρvx3  ρv2 p B 2 ρv v B B ρv v B B   x1 + − x1 x1 x2 − x1 x2 x1 x3 − x1 x3  5  ρv v B B ρv2 p B 2 ρv v B B   x2 x1 x2 x1 x2 x2 x2 x3 x2 x3   − + − 2 − 2   ρvx3vx1 Bx3Bx1 ρvx3vx2 Bx3Bx2 ρvx3 p Bx3  F (U)  − − + −  = (E p)vx1 (vx1Bx1)Bx1 (E p)vx2 (vx2Bx2)Bx2 (E p)vx3 (vx3Bx3)Bx3  + − + − + −   0 vx1Bx2 Bx1vx2 vx1Bx3 Bx1vx3   − −   vx2Bx1 Bx2vx1 0 vx2Bx3 Bx2vx3  − − v B B v v B B v 0 x3 x1 − x3 x1 x3 x2 − x3 x2

where, vx1,vx2,vx3 and Bx1,Bx2,Bx3 have been employed to indicate the 3-dimensional 2 ρv p B2 components of the velocity and magnetic field vector, and etot [11]. 2 γ 1 µ0ρ p = + − + Concerning the expression of the total energy, the relation γ 1 comes from the equation of state of a calorically ideal gas, the applicability of which− has been proven in Section 2.4. Nevertheless, primitive variables (i.e. V (ρ,u,v,w,p,B ,B ,B )) are still used = x1 x2 x3 to define the initial conditions of the problem [11]. In fact, updating the flux tensor through primitive variables, instead than conservative, allows to better enforce some physical constraints such as pressure positivity and non-superluminal speeds (i.e. v c) in the whole domain [12]. Thus, the problem starts with the definition of all < the primitive variables in the predefined domain, as well as the boundary conditions. The domain of the integration problem can either be mono-, bi-, or tri-dimentional; it is divided in a logically rectangular structured mesh, and is delimited by ghost zones. The ghost zones are the cells that surrounds the domain, and share common borders with it [11].

5.2.1. COMPUTATIONAL DOMAINAND SOLVING STRATEGY The structured mesh is thus composed of a number of cells that depends on the assigned resolution of the problem: in each orthogonal coordinate direction, the de- ﬁned number of cells (i.e. NX 1,NX 2,NX 3) divides the domain into a discrete number of points. Those points, having coordinates (i, j,k), are in fact the geometrical centre of each computational cell, and the primitive variables are thus deﬁned as cell-centred values [11]. In addition, IBEG, JBEG,KBEG are the coordinates of the lower limit of the computational domain, while IEND, JEND,KEND are of the upper; NXITOT , NX 2TOT , 5.2.PLUTO 67

and NX 3TOT are the number of cells in each direction, including ghost zones. Figure 5.1 exempliﬁes these deﬁnitions with a 2-dimensional case.

FIGURE 5.1: Example of a 2-dimensional computational grid as implemented in PLUTO [11]

When domain, initial conditions, and boundary conditions are defined, the code starts computing the evo- - + lution of the system. During the process, discontinuities arise between adjacent cells, and solving the Euler equa- V-,L V-,R tions is to achieve continuity of the solution at the interface [11]: any discontinuity between two states (i.e. left, - + and right state) is due to a flux. For the specific purpose, a 1-dimensional (i.e. for V+,L V+,R each orthogonal direction) Riemann problem is computed at the interface of adjacent cells. A Riemann prob- x1,2,3 lem is in fact an initial value problem, in which a single FIGURE 5.2: Primitive vari- discontinuity is found in volume averaged conservative able labelling strategy for quantities (i.e. states) [54]. Specifically, the solving strat- the solution of the Riemann egy implemented in PLUTO can be exemplified by the problem scheme of Figure 5.3. That represents the so called RSA strategy [12]:

• RECONSTRUCT: A piecewise monotonic interpolation is applied at each computational cell to reconstruct the volume averaged primitive variables V from U. Then, according to the coordinate direction, a left (-) and right (+) interface is deﬁned for each computational cell, in respect to its adjacent ones. Thus, depending whether it is on the left (L) or the right (R) side of the interface, each variable is labelled consequently. The label assigned to the same variable changes depending on the speciﬁc pair of cells considered (see Fig. 5.2).

• SOLVE: For each discontinuity that is found between left and right state (i.e. V ,L V ,R or V ,L V ,R ) at the interface of adjacent computational cells, the + 6= + − 6= − 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 68 COMPUTATIONAL CODE SELECTION

ﬂux is calculated by solving the consequent Riemann problem;

• AVERAGE: The solution is ﬁnally updated through a ﬁnite-volume scheme (i.e. volume averaged conservative quantities).

Convert: States:

U V V V-,R;V +,L

Riemann: Update: 5 (V+L,V+R) F+ n+1 n U =U -(F+-F-) t/ x (V-L,V-R) F-

FIGURE 5.3: Flow diagram of PLUTO’s RSA strategy [12]

A solver is used to calculate the flux out of the Riemann problem. Which solver is specifically used, depends on the physics module implemented [11,12]. For this computational analysis more diffusive solvers have been eventually preferred since they are more suitable for relativistic flows [12]: as pointed out in Section 2.4.3 relativistic Alfvén speeds have been attained in some cells of the domain. Moreover, when an unsplit scheme is implemented, the solution is calculated by combining at the same time flux contributes from all the directions: this scheme being more accurate for multidimensional problems [11], has indeed been applied to attain the results hereby reported. After all the Riemann problems are solved, and continuity is achieved, the solution is evolved in time. The time steps that control the evolution are handled by the code itself. Specifically, the length of each new time step is selected by the code in accordance with [12]:

Ã d ! ∆lmin ∆t Ca min = d λd | max | d Thus, for each coordinate direction d, the ratio of the smallest cell length ∆lmin by the largest signal velocity λd is estimated. The minimum calculated ratio is then | max | multiplied by the Courant number Ca and the result is used as the next time step. The Alfvén speed is in fact related to signals in plasma [14], hence, its value directly affects the step length. In particular, when the Alfvén speed gets too big, the computation may become infeasible: an increasingly smaller ∆t is applied, until the computation crashes. The next chapter points out how this analysis have been particularly sensitive to this issue. Now, it is however useful to give some more details about how initial conditions are assigned in PLUTO. The next section fulfils that task, giving particular evidence to the magnetic field definition. 5.2.PLUTO 69

5.2.2. DEFINING INITIAL CONDITIONS As already mentioned in the previous section, primitive variables are defined in the whole computational domain. Hence, analytical functions are used to identify different parts of the domain (i.e. cells), in which specific conditions are to be reproduced. In addition, analytical functions are also used to define variable conditions for the same primitive quantity. For example, a uniform density distribution has to be set in the whole 3-dimensional domain, except for a limited sphere where the density has to progressively increase to a finite value towards the sphere centre. In that case, a uniform density is first assigned, without limitation, in the whole domain. Then, the mathematical function of the sphere is used to identify the desired part of the domain, and another to update the value of the density as a function of the sphere radius. Since the code has been programmed in C, this part, as well as all the others concerning the problem set-up, have to be written in the same language. 5 Particular attention has to be given to the definition of magnetic field components. In fact, the solenoidal condition has to be verified, and hence, included as initial condition.

SOLENOIDAL CONDITION When a magnetic field is analytically defined, its fundamental solenoidal condition is imposed by the expression of the function itself. If that is done correctly, the solenoidal condition is maintained throughout the temporal evolution of the field. That is due to the definition of magnetic field temporal evolution, which, according to the 3rd Maxwell’s Equation (i.e. Eq. 2.8), is:

∂B~ E~ ∂t = −∇ × Thus, since the divergence of the curl of any vector field is identically equal to 0 [34], the B~ 0 condition is not dynamical, and, if initially defined, is analytically and ∇ · = indefinitely maintained [55]. Nevertheless, due to possible discretization errors, the condition is not automatically preserved if numerical integration is employed to calculate the temporal evolution of the system [12, 55]. That is in fact a well known issue common to all computational codes developed for MHD applications. As a result, special routines have to be implemented in the code to force the preservation of the solenoidal condition at each new time step. Besides, as Chapter3 clearly explained, the solenoidal property of magnetic fields is indeed fundamental for the projected operation of the reaction chamber. Hence, the effectiveness of the routine, that controls the divergence-free condition, has been pointed out as particularly important for the quality of the results sought for this project. For the specific purpose, PLUTO implements 3 strategies [11]:

• Eight-Wave Formulation;

• Hyperbolic Divergence Cleaning;

• Constrained Transport. 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 70 COMPUTATIONAL CODE SELECTION

The first two are less sophisticated and do not preserve the solenoidal condition at machine accuracy [11]. Thus, for the specific application, following the advice of the main author of the code (i.e. Prof. A. Mignone), Constrained Transport has been finally selected. Furthermore, if the magnetic field is defined through its vector potential, the divergence-free condition is preserved at machine accuracy [11]. In fact, the code allows to define the magnetic field components either directly, or by expressing the related vector potential (see Sec. 3.1.1). In the latter case, the code derives the magnetic field components as B~ ~A in each cell, hence, the solenoidal = ∇× condition is assured everywhere [11]. Therefore, that has been the selected method to model the seed magnetic field of the reaction chamber, and its consequent temporal evolution. The next chapter describes the final configuration that has been used to attain 5 the results of this project. That has been derived from practical experience with the code, on one side, and the information extracted from previous similar analyses, on the other. In fact, according to what found in the literature, what has been completed for this thesis project is the first extensive computational analysis of the latest multi- coil parabolic chamber (i.e. Thio’s Concept [10]). Nonetheless, some partly usable results have still been extracted from similar computational analysis. Hence, the next section summarises such findings.

5.3. PREVIOUS NUMERICAL ANALYSIS The first attempt of estimating the performance of a magnetic flux compression reaction chamber was carried out by Hyde in 1983 [45]. In his study he proposed the modifications to the Daedalus’ reaction chamber that eventually brought to the most recent multi-coil parabolic design [45] (see. Sec. 3.2.3). In his study, he combined a routine to simulate the magnetic pressure as a function of the projected shape of the field with a 2D hydrodynamics code [45]. However, beside not including real Magneto-Hydrodynamics effects, its calculations were relative to a single-coil configuration [45]. Hence, they have been assumed not exhaustive for the reaction chamber design considered in this report. In addition, numerical simulations have been recently completed to estimate the thrust efficiency of the VISTA reaction chamber design [56–58]. However, those calculations did also not implement a computational MHD code: they used a Smoothed Particle Hydrodynamics (SPH) code, instead. SPH, and the more recently investigated Smoothed Particle Magneto-Hydrodynamics (SPMHD), uses a different approach for estimating the physical properties of the system described in the problem. However, the results attained with this method have been discovered in a late phase of the project: i.e. when the application of computational MHD had been already selected as the final alternative. Thus, due to the different nature of the results, the scarce documentation concerning available computational codes, and the limited time remained to complete the project, it has been decided to give priority to the use of a MHD model. In fact, in respect to computational MHD applied to plasma dynamics in space propulsion systems, several studies have been completed to simulate the operation 5.3.P REVIOUS NUMERICAL ANALYSIS 71 of steady-state magnetic nozzles (e.g. VASIMR’s nozzle) [1, 49, 51, 52, 59, 60]. In fact, steady-state magnetic nozzles being in a more advanced development phase, already, are more plausible of seeing real-life applications in the coming years [1, 61]. However, despite not being directly usable as a reference, all those analysis were indeed responsible for the conception of this very project. On the other hand, computational analysis of magnetic flux compression reaction chambers are still quite scarce. In fact, according to the literature, what presented by Stanic et al. [13] has so far been the first published attempt of implementing the use of a numerical code to solve the ideal-MHD of a reaction chamber. Their study is part of the Project Icarus, which has been inspired by Project Daedalus, and is still managed by the BIS[13]. In particular, Project Icarus is to use a nuclear pulse propulsion system, as well, hence, it implements a magnetic flux compression reaction chamber. The work of Stanic et al. has indeed been the only directly usable 5 reference that could support this thesis project. Thus, how they set up the computational problem is reported in the next.

5.3.1. COMPUTATIONAL PROBLEM SET-UP For their simulations, they decided to employ the Nautilus code. Nautilus is a code that is able to model MHD, as well as Hydrodynamics (HD). The former model is also implemented in various forms (e.g. ideal-MHD and resistive-MHD[13,62,63]). That allows the code to be very flexible and hence usable for many different kind of applications which can go from astrophysics, to engineering problems. The code was developed at Tech-X5, and has nowadays been replaced by a more recent solution (i.e. USim6). However, that is not freely distributed, and a copy was not attainable within the time scheduled for this thesis. Nevertheless, the results of the analysis have still been investigated for the sake of this project. In particular, it has been interesting to understand the approach they used to design and model the magnetic field, and the electric coils. As Stanic et al. mention in their report, due to the very little documentation on the matter, one very complex issue was to correctly design the geometry of the magnetic field that is generated by multiple current coils [13]. For that reason, they decided to employ the available data from the Daedalus Project [13], for which several simulations had been completed to reproduce the geometry of the field generated by the 4 coils. Figure 5.4 represents the graphical result of those simulations. Hence, Stanic et al. derived many of the data from Project Daedalus. Specifically, they reproduced the exact same initial conditions of the 1978 project. That includes the same initial plasma thermal energy (E 13.46GJ) and mass (m 0.287g) [13]. 0 = p = In addition, they implemented the same 4 electric coils and related current intensity, combined with a conductive wall. Still, despite the geometry and arrangement of the coils is analogous to what of Project Daedalus, Stanic et al. eventually implemented a parabolic chamber wall, instead than a hemispherical one. As a result, the modelled reaction chamber is actually an update of Project Daedalus’.

5https://www.txcorp.com 6https://www.txcorp.com/usim-for-basic-simulations 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 72 COMPUTATIONAL CODE SELECTION

FIGURE 5.4: Geometry of the seed magnetic ﬁeld in the reaction chamber of Project Daedalus [8]

Nonetheless, what is more interesting than the physical quantities used, is how Stanic et al. have reproduced that wall. As Chapter3 has described, the magnetic field compression is indeed the real fundamental aspect that grants the projected operation of a reaction chamber, and that is a direct consequence of the “plasma- magnetic field-conductive wall” interaction. However, despite the ideal-MHD module is sufficient to prevent the magnetic field lines from penetrating the volume of plasma (provided that proper initial conditions are set), the code may not include the possibility of explicitly including solid conductive structures in the domain. As a result, the field does not find any constraint to its deformation (i.e. no wall that confines it), and the compression is not attained. Hence, alternative methods have to be sought. In particular, Stanic et al. attained that by defining particular boundary conditions. Boundary conditions are in fact statements that tell the code how to update the computational variables when they cross the boundary of the domain. Thus, Stanic’s ploy was to extend the domain to include the interior of the chamber, only (i.e. the domain is extended until the surface of the conductive wall). As a result, an appropriate boundary condition could be applied to emulate the effect of the conductive wall on the magnetic field evolution. Thus, they first selected to exploit the symmetry of the system to limit the computational time. Hence, they opted for a 5.3.P REVIOUS NUMERICAL ANALYSIS 73

2-dimensional problem. Then, they selected a parabolic section as the lateral limit of the domain, which Figure 5.5 shows a picture of. That is the half-section of the chamber on the ρ-z plane (i.e. in cylindrical coordinates): the left side is the axis of the chamber itself (i.e. z-axis), the top its aperture, while the right its wall. The vertex of the chamber has been cut for integration stability reasons [13].

FIGURE 5.5: Integration domain of the computational problem [13]

Finally, to simulate the effect of the conductive wall, they set a reflective boundary condition on the right side of the domain (i.e. chamber wall). In fact, a reflective boundary condition has on the variables it is imposed on, the same effect a wall does on physical matter: i.e. that very variable cannot pass through the boundary limit, but can flow over it. That is indeed an interesting solution, for the magnetic field is stopped against the conductive armature, as the wind against a wall. However, if the same condition was set for all the variables, the mass flow would stop against the chamber wall, as well. Thus, ρ being 0 everywhere (see Sec. 2.4.3), 6= matter would pack against the chamber wall increasing the hydrostatic pressure in the same location. This pressure would hence add its contribute to the plasma shell reflection. Of course, given the small density used to reproduce vacuum conditions, this contribute might be negligible. Nevertheless, it is not clear whether Stanic et al. has tested it or not. Also, it is nowhere mentioned if the reflective condition had been extended to all the variables [13]. Regardless, this solution did not seem suitable for a multi-coil parabolic chamber as the one analysed for this thesis (i.e. with thrust rings replacing the conductive wall). Hence, a different and more appropriate approach has been selected. 5.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 74 COMPUTATIONAL CODE SELECTION

In addition, beside the non-implementability of perfect vacuum conditions, Stanic et al. pointed out another limitation of their analysis: i.e. Ideal-MHD does not model plasma detachment [13]. Plasma detachment is in fact the equivalent of flow separation in hydrodynamics. Specifically, due to its resistivity (i.e. not modelled in ideal- MHD), plasma particles may entangle to magnetic field lines [14,27, 64]. As a result, when expanding in a magnetic nozzle, or a reaction chamber, the plasma flow may be deviated from the axial direction, and hence, decrease the performance of the rocket [64]. Nevertheless, despite several investigations have again been completed in respect to plasma detachment in steady state magnetic nozzles [1, 65, 66], the same type of research applied to reaction chambers is still limited. Furthermore, since the same limitation has indeed been recognised in the computational analysis completed for this study, no further progress has been done on that 5 point. Nonetheless, the results of this project, which are presented in the next chapter, are still believed to be valuable. 6 MAGNETIC FLUX COMPRESSION REACTION CHAMBER: COMPUTATIONAL ANALYSIS

Therefore, the computational analysis of a multi-coil parabolic reaction chamber, has been executed with the aid of PLUTO. After some preliminary unsuccessful results, the chamber operation has eventually been reproduced: the plasma reflection, and the consequent thrust generation, has in fact been achieved. This chapter is thus to describe the final results of the analysis. According to what has been investigated for this thesis project, what hereby included is in fact the first computational ideal- MHD analysis of a multi-coil parabolic reaction chamber of the kind described by Thio et al. [10] and Adams et al. [4]. Section 6.1 thoroughly describes the final set-up that has been used. That includes the definition of the initial conditions, as well as all the ploys that have been implemented to better reproduce the reaction chamber constraints and characteristics. Then, the same set-up has been validated against the results of the study con- ducted by Stanic et al. [13]: the same initial conditions have been used to check the effectiveness of PLUTO’s model. That is reported in Section 6.2. Finally, Section 6.3 summarises the outcome of the computational analysis applied to the same multi- coil parabolic chamber that has been described in Chapter4. Specifically, the results of the simplified analytical model have been verified.

6.1. FINAL SET-UP This section describes the final configuration that has been employed to complete the numerical analysis. All the input files have been included in AppendixB of this report. To avoid computations involving quantities of too different orders of magnitudes, PLUTO is programmed to work with adimensionalised units [11]. Hence, the user has to select proper adimensionalisation units, so that the quantities involved in the problem are of similar order of magnitude: values around the unit are always prefer- able. However, the adimensionalisation units have to be set in Gaussian-cgs units

75 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 76 COMPUTATIONAL ANALYSIS

(i.e. cm-g-s). Thus, all the quantities expressed in this Chapter are given in cgs units, as well.

6.1.1. COMPUTATIONAL DOMAIN Given the axial symmetry of the system, a 2-dimensional rectangular domain expressed in cylindrical coordinates (i.e. ρ,z) has been chosen to reproduce one half only of the chamber. Although a 2-dimensional problem is more efficient in terms of computational time, it may lack of accuracy: as Section 3.1.2 already mentioned, perfect symmetry shall never be achieved in a real reaction chamber. In fact, im- plying a perfect axial symmetry of the system is the only assumption that has been kept from what of the simplified analytical model of Chapter4. However, the set- up hereby proposed can easily be adapted for a 3-dimensional problem, which may thus be completed in future. Hence, the length of the domain in the ρ-direction has been set to fully include the maximum radius of the chamber in accordance with Adams et al. design: it ex- 6 tends from 0 (to include the axis of the chamber) to 810 cm. On the other hand, the upper side has been prolonged to include a reasonable part of the plasma expansion. Thus, the domain goes from 0 to 1575 cm in the z-direction. Concerning the problem resolution (i.e. number of cells in each direction), that has been increased until the code could still complete the analysis in a reasonable amount of time (i.e. 10hour s). In fact, despite improving the definition of the resulting plot (i.e. im- ∼ age definition), a higher resolution increases the computational demand. Moreover, squared cells (i.e. having a finite side length) give better numerical stability and efficiency. As a result, Table 6.1 shows the final settings of the computational domain.

ρ z × size 810 1575 [cm cm] × × resolution 288 560 [NX 1 NX 2] × × TABLE 6.1: Computational domain characteristics

6.1.2. PLASMA PELLETAND SEED MAGNETIC FIELD DEFINITION Since the aim of this analysis was to verify the results of NASA’s simpliﬁed analytical model combined with the data discussed by Thio et al. (see Chap.4), the initial conditions have been derived from Table 4.4, and are hereby reported in Table 6.2. The initial conditions are implemented in PLUTO through the input ﬁle init.c included in Appendix B.3 of this report. Hence, the plasma has been set of a spherical shape, centred in the focus of the chamber parabola, and having a mass equal to m m . p0 = p The radius of the sphere has been selected to have an accurate representation (i.e. accurate shape, and mass) in combination with the chosen domain resolution. Also, it has been assumed that all the energy of the plasma was thermal energy, initially. Hence, plasma velocity has been set equal to 0, while its hydrostatic pressure derived from the ideal gas equation of state: 6.1.F INAL SET-UP 77

E jet p (γ 1) (6.1) 0 = 4 − 3 πR0 4 where, the initial speciﬁc energy (i.e. E jet divided by the initial plasma volume 3 πR0) includes the losses estimated by Thio et al. in their study [10]. Also, γ has been as- 5 sumed equal to 3 as from Section 2.3.4.

Plasma initial conditions

mp 2.26g R0 (initial plasma pellet radius) 30.0cm E (from Eq. 4.1) 905.0 1013 er g jet · v0 0.0cm/s

TABLE 6.2: Computational analysis initial conditions 6 Furthermore, for what it concerns the design of the reaction chamber, the data have been mainly determined from NASA’s study [5]. That includes the focal length of the parabolic chamber (i.e. l 200.0cm), the chamber aperture angle (i.e. θ f = = 5 π 75°), as well as the electric coils speciﬁcations of Table 6.3. There, a indi- 12 = n cates the big radius of each coil, zn the axial location of their centre (the vertex of the parabola coincides with the origin of the coordinate system), and the current is given in statampere (1A 3 109 statA) [40]. = · 1 2 3 4 I [statA] 7.56 1016 1.24 1016 5.82 1015 3.51 1015 · · · · an [cm] 27.8 84.4 144.0 211.0

zn [cm] 0.96 8.9 26.1 55.6 5 6 7 8 I [statA] 2.34 1015 1.63 1015 1.14 1015 7.89 1014 · · · · an [cm] 288.0 382.0 505.0 681.0

zn [cm] 104.0 182.0 319.0 579.0

TABLE 6.3: Electric coil speciﬁcations [5]

On the other hand, implementing the seed magnetic field did require to first discuss, and then implement a specific function. In fact, PLUTO does not allow to directly include any electric coil, and then derive the generated magnetic field: the magnetic field components are indeed among the primitive variables that have to be initially defined. Besides, the investigated numerical analyses did not give any useful hint or suggestion concerning the matter. Numerical codes such as Nautilus or USim do have specific routines to derive the magnetic field from assigned coil specifications, however, those are not accessible to the user. Nevertheless, an effective solution has 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 78 COMPUTATIONAL ANALYSIS

been eventually discovered, and the seed ﬁeld has been assigned by deﬁning its vector potential in every computational cell; in fact, no similar approach, applied to such kind of analysis, has been found in the literature.

INITIAL SEED MAGNETIC FIELD As already discussed in Section 3.1.1, the vector potential relative to the combination of 8 coils has been derived from the vectorial summation of each single coil contribute. To achieve that, Equation 3.4 has been applied. However, its deﬁnition has ﬁrst been expressed in cgs units as in [40]:

· 2 ¸ 4I an (2 k )K (k) 2E(k) ~Aφ(ρ,z) − − (6.2) = q k2 c a2 ρ2 z2 2aρ n + + − where the speed of light (c) is also given in cgs units (i.e. cm/s). Hence, the result of the square bracket of Equation 6.2 has first been tabulated for a series of 400 values of k equally spaced from 0 to 1.0. The table, exported in ASCII, 6 has been completed with the aid of MATLAB and its ellipke() function, which gives the value of both first and second complete elliptic integral. Then, PLUTO has been instructed to store the values of the table, and apply an iterative process to finally evaluate the magnetic vector potential of the combination of 8 coils. Specifically, for each coil, and in each computational cell:

• The exact value of k is estimated according to Equation 3.3 and the data from Table 6.3;

• The square bracket of Equation 6.2 is attained through a linear interpolation of the values stored in the ASCII table;

• The contribute of each coil is sequentially added to the estimated vector potential.

The result of this process is shown in Figure 6.1, where the coordinates are still in adimensionalised units, the coloured profile expresses the magnetic field magnitude in Gauss, and the black streamlines reproduce the field lines. Hence, despite not actually included in the domain, the position each coil would occupy in the real chamber can be guessed from the field magnitude profile. A series of circular and concentric patterns can in fact be recognised in the plotted profile; each series is centred in one of the coils. Given the size of Figure 6.1, that can probably be recognised for the coil that is closest to the chamber exit, only (i.e. z 125, r 160). ∼ ∼ Furthermore, to aid the localisation of the pictures, their geometry is expressed in adimensionalised units in Table 6.4.

6.1.3. BOUNDARY CONDITIONS Boundary conditions are applied at the extremes of the computational domain (i.e. ghost zones), and are used to instruct the code on how to handle the variables when they reach the domain borders [11]. Thus, given the deﬁnition of the domain, an 6.1.F INAL SET-UP 79

1 2 3 4

an [cm] 6.2 18.8 32.0 46.9

zn [cm] 0.2 2.0 5.8 12.4 5 6 7 8

an [cm] 64.0 84.9 112.2 151.3

zn [cm] 23.1 40.4 70.9 128.7

TABLE 6.4: Electric coil geometry in adimensionalised units [5] axisymmetric condition has been set on the left-side, whereas an outflow condition on the remaining three sides (see App. B.1). The former is used to reproduce the axial symmetry of the system. Hence, all the scalar variables (q) are symmetrised in the ghost zones (i.e. q q), whereas the sign of all the vector fields components → (i.e. velocity, and magnetic field) is flipped (i.e. v v and B B ). 1;2;3 → − 1;2;3 1;2;3 → − 1;2;3 6 That has allowed to use a 2-dimensional geometry rather than a 3D one, and, hence, to compute the dynamics of one half-section of the chamber, only. As a result, the required total computational time has been greatly reduced. On the other hand, the outflow boundary condition is used to consider all the variables unchanged beyond the border (i.e. their spatial gradient in the normal direction to the domain border ~ (nˆ) is set equal to 0 ∂q ∂~v ∂B 0). ⇒ ∂nˆ = ∂nˆ = ∂nˆ = In addition, boundary conditions have been used for the purpose of assuring the compression of the magnetic field. At first, using Stanic et al. results as a reference, the implementation of a similar reflective condition has been examined. However, while imposing a reflective condition at the boundary of the domain is immediate, doing the same at computational cells within the domain (i.e. internal boundary conditions) is not as simple: a specific routine, not yet fully implemented in the code, would in fact be required. As a consequence, another approach has been discussed. In particular, the employed solution is believed to be even more reliable and accurate, than what Stanic et al. used: it has indeed effect on the magnetic field only, while all the other variables are not directly affected. Again, no similar examples have been found in the literature, hence, according to what has been investigated for this project, what hereby proposed is indeed a first. The final idea followed many other failed attempts to stop the magnetic field from crossing the location of the electric coils: thinking of a moving magnetic field, that is eventually reflected, was actually misleading, at first. The “motion” of the field is in fact related to its temporal and spatial variation. Therefore, if the field cannot pass through a specific location (i.e. it gets “reflected”), that is indeed because its magnitude cannot vary in that very spot. As a result, preventing any possible temporal variation of the magnetic field in some specific parts of the domain (i.e. freezing the magnetic field) has finally been the successful approach. In particular, for reasons of convenience, freezing the whole magnetic field located outside of the reaction chamber has been preferred. In fact, those very field lines have been discussed having a negligible influence on the system evolution (i.e. plasma expansion). Thus, a func- 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 80 COMPUTATIONAL ANALYSIS

FIGURE 6.1: Initial seed magnetic ﬁeld magnitude [Gauss] and streamlines (t 0) =

tion (f (ρ,z)) to express the chamber geometry (i.e. a parabola) has been defined. Then, the temporal evolution of the magnetic field has been identically set to 0 (i.e. ~ ∂B 0) in all the cells having the geometric centre located below the just defined ∂t = parabola (i.e. f (i, j) 0). < Furthermore, since there was no interest in evaluating the system evolution in those very cells, their update has been completely excluded from the computational analysis. That was in fact a suggestion of Prof. A. Mignone, and has greatly reduced the final computational time of the problem. Besides, some cells close to the origin of the magnetic field (i.e. electric coil locations) did show numerical integration problems, due to the too strong magnetic field combined with an extremely low density. That is in fact what has been mentioned in Section 2.4.3, and is related to relativistic and superluminal Alfvén speeds detected in the domain. As a consequence, the function defining the chamber wall has been shifted upwards along the z-axis (i.e. 6.1.F INAL SET-UP 81

ZOFFSET) to exclude those critical cells from the computational domain. Therefore, the ﬁnal equation that has been used to deﬁne the parabola as a function of the focal length of the chamber (i.e. l f ) is:

ρ2 f (ρ,z) z = − 4l f while the constraint has been imposed on all the cell centres (i, j), for which the following conditions are veriﬁed.

( f (i, j) ZOFFSET < j 603.0cm < The second condition has been added to limit the extension of the parabola and attain the final result pictured in Figure 6.2. There, the coloured pat- 6 tern is still related to the magnetic field magnitude as in Figure 6.1, while the red dashed line is the visual expression of the just defined geometric condition: the cells where the magnetic field is the highest (i.e. see red colours in the pattern) are in fact located below the parabola (i.e. excluded from the analysis). Therefore, to stop the temporal evolution of the magnetic field in those very cells, the asset of PLUTO that allows to have full control over the evolution of the variables within the computational domain has been exploited. Specifically, the routine that controls the evolution of the magnetic field, and aims to preserve its solenoidal condition, has been modified. In fact, that is not an option directly available to the user, but it rather requires to change part of PLUTO’s FIGURE 6.2: Coloured plot of the magnetic field magnitude [Gauss], and visualisation of source code. Nonetheless, the open the imposed geometric condition to attain the source nature of this code allowed to in- field compression (i.e. dashed line) (t 0) clude this modification with no more ef- = fort, than just seeking the specific routine’s function. Therefore, Constrained Transport being the selected strategy to preserve the solenoidal condition (see. Sec. 5.2.2), its related routine has been modified. That updates the magnetic field components at every time step, and in each computational cell, according to the curl of the EMF (i.e. applied electric potential due to a 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 82 COMPUTATIONAL ANALYSIS

moving circuit in respect to a magnetic field [40]). Moreover, the selected geometry of the problem being cylindrical, the azimuthal component of the electromotive force (i.e. Eφ) has been identified as the only one affecting the magnetic field evolution.

FIGURE 6.3: Compressed magnetic ﬁeld lines, and magnitude [Gauss] (t 2.85µs) '

Hence, in all the cells having the geometric centre (i, j) located below the dashed line of Figure 6.2, Eφ has been constrained to be constant (i.e. identically equal to 0) for all the computational time. As a result, the magnetic field swept by the expanding plasma cannot cross those cells, for its motion is stopped by the frozen magnetic field. Again, that is formally due to the imposed solenoidal constraint: magnetic field lines must remain closed lines. As a result, the sought field compression is attained as shown in Figure 6.3, where colours and streamlines are again related to the magnetic field and its magnitude. It can be clearly seen that the field has been swept by the expanding plasma, and compressed against the chamber structure: the semi- 6.1.F INAL SET-UP 83 circular light blue area, that is in contact with the left side of the domain, is indeed occupied by the expanding plasma, while the part of the domain below the dashed line is where the E 0 condition has been imposed. φ = Nevertheless, this solution may still not exactly reproduce the concept of the latest multi-coil parabolic chamber (i.e. from Adams et al. study [4]). In fact, the magnetic field is still stopped by a continuous number of adjacent cells. Thus, it is again as the effect of a conductive wall, and not of single coils, had been reproduced. As a consequence, further investigations are still required.

6.1.4. AMBIENT CONDITIONS According to the concept design, the plasma expands to vacuum, hence, the volume of the chamber that is not occupied by fusion plasma is “empty”. Nevertheless, as already discussed, implementing vacuum conditions can be a critical aspect for computational MHD. In particular, neither density, nor hydrostatic pressure can be set equal to 0, and still have a finite solution of the MHD system of equations [11,13]. As a consequence, particular attention has been given to the definition of ambient 6 conditions. Moreover, some investigations about real vacuum conditions have been completed for this project, and the results showed that vacuum in space is not empty, at all. Besides, in proximity of stars, the conditions do actually vary depending on their activity [67]. Pressure and particle density of deep space vacuum can respec- 12 3 23 3 tively be as low as 10− mbar and 10cm− (i.e. ρ 10− g/cm )[67]. ' Nevertheless, such extreme conditions have not been reproducible with this computational analysis: the calculation time would be excessively extended (i.e. due to the extremely small time steps), and the computation would eventually crash (see Sec. 5.2.1). The reason can be mainly attributed to the incredibly high Alfvén speeds that are that way attained: relativistic, and even superluminal speeds have been found. In particular, the cells of the domain where the magnetic field is particularly strong (see Fig. 6.1) were pretty sensitive to this issue. Therefore, the eventually implemented ambient conditions are reported in Table 6.5. Those have been derived from the available literature [13, 67], and finally selected to find a balance between accuracy and efficiency of the calculations. In fact, too high densities and pressures, although shortening the computational time, have been proven to partially obstruct the expansion of the plasma. In particular, the specific problem has been recognised more sensitive to the value of density than to the pressure’s: too high ambient densities actually limit the maximum speed reached by the plasma. Also, if proper conditions are not implemented, after an initial rapid acceleration, the expanding plasma starts to decelerate again. That is shown in Figure 6.4, where the same integration time (i.e. t 7.83µs) is pictured for two cases im- ' plementing different ambient densities. Due to a higher ambient density, the plot on the left shows plasma velocities one order of magnitude lower than what on the right. Also, the same picture, highlights that the already reflected plasma (i.e. the discontinuity in the coloured pattern between z 100 and z 150) is in fact at the same = = position in both the solutions. On the other hand, the portion of plasma that expands towards the chamber aperture is in a much more advanced location in the 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 84 COMPUTATIONAL ANALYSIS

Hydrostatic pressure p 30.0mbar 3.0 104 d yne/cm2 1 = · 5 2 2 p 1.0 10− mbar 1.0 10− d yne/cm 2 · = · Mass density 9 3 ρ 2.0 10− g/cm 1 · 11 3 ρ 2.0 10− g/cm 2 ·

TABLE 6.5: Different ambient conditions used for the computational analysis

plot on the right. Therefore, the more difﬁcult expansion of the plasma due to a higher ambient density clearly compromises the overall estimated performance. In- deed, ambient particles are put in motion by the expanding plasma, as well. Thus, 6 part of the plasma initial potential energy is actually spent to move the whole matter of the domain. As a consequence, higher the ambient density is (hence, the mass), greater the amount of energy spent to move it. The maximum speed reached by the expanding plasma is therefore affected.

FIGURE 6.4: Plasma velocity proﬁle at t 7.83µs and different ambient 9 3 ' 11 3 densities: ρ 2.0 10− g/cm (left), and ρ 2.0 10− g/cm (right) 1 = · 2 = ·

On the other hand, the value of ambient pressure has much less influence on the system. In fact, according to ideal-MHD, it is the thermo-magnetic pressure (i.e. the sum of ambient pressure and magnetic pressure) that affects the plasma dynamics. In particular, the completed solutions showed that magnetic pressure hereby plays the dominant role. That is proved in Figure 6.5, which shows the same integration 6.1.F INAL SET-UP 85 time of Figure 6.4 (i.e. t 7.83µs), and compares the thermo-magnetic pressure pro- ' file due to different ambient pressures. Hence, the very much similar plots, both in terms of general profile and pressure values, show that the initial ambient pressure has in fact a negligible influence on the overall system dynamics.

FIGURE 6.5: Thermo-magnetic pressure proﬁle p at t 7.83µs tm ' and different ambient pressures: p1 30.0mbar (left), and p2 1.0 5 = = · 10− mbar (right)

Furthermore, even though deep space vacuum conditions have not been reproduced, the more extreme values that have been implemented (i.e. p2,ρ2), are in fact close to the order of magnitude of what is achieved in the most advanced vacuum chambers. For example, NASA’s Space Simulation Vacuum Chamber (SSVC)1 can 6 12 3 reach pressures and air densities as low as 10− mbar, and 10− g/cm . Hence, ∼ ∼ despite not totally representative of a deep space application, the results of this computational analysis can still be a usable reference for future on-ground experiments and magnetic flux reaction chamber testing. However, those specific conditions have not been eventually implemented in any completed analysis. Indeed, they would worsen even more all the issues related to the consequently computed relativistic velocities, and cause extremely strong shock waves. Hence, despite PLUTO might be able to handle them, it still requires some fine tuning when dealing with such harsh situations. However, the scheduled time for this thesis was not sufficient to achieve that. In fact, the conditions reported in Table 6.5 do also cause similar problems; still, those being somehow less severe, have been finally mitigated as explained in the next section.

1http://facilities.grc.nasa.gov/spf/ 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 86 COMPUTATIONAL ANALYSIS

6.1.5. ENTROPY SWITCH The ETROPY_SWITCH is a special function of PLUTO’s that is implemented to increase its robustness when dealing with strong shock waves [11]. In fact, in the case too strong shock waves are detected, wrong values of hydrostatic pressure (i.e. negative) might be computed in the specific cells. The errors can be searched in the log file of the analysis. Among its output files, PLUTO includes a log in which it writes information about the integration steps. Each line has the following structure:

step:2540; t = 2.3080e-02; dt = 9.2070e-06; 1%; [7.297882, 0]

where, the number of steps completed, the integration time (i.e. t), the step length (i.e. dt), and the percentage of problem completion are ﬁrst given. Then, the maximum computed Mach number, and the number of completed iterations, in order to solve the Riemann problem, are noted in the square brackets. The latter is hereby always 0, since a non-iterative solver has been implemented for the analysis. On = 6 the other hand, the maximum computed Mach number is in fact greatly important to estimate the quality of the solution: it is a direct indicator of the robustness of the problem [11]. Very high numbers (i.e. 103) may indeed point out some issues > in the integration process. Besides, when that happens, negative pressures may be computed, and the following error is thus reported in the log.

step:250; t = 1.9737e-01; dt = 1.7535e-03; 2%; [3963.656737, 0] ! ConsToPrim: negative p(E) (-4.93e-06), zone [x1(235) = 54.492188, x2(99) = 55.132812], proc 1

where the speciﬁc cell coordinates are also included (i.e. x1 and x2). Nonetheless, that does not usually prevent the code from completing the solution: the error is eventually damped. However, since running the ﬁnal set-up produced an excessive number of warnings, the use of the ETROPY_SWITCH has been considered and evaluated. Basically, activating the switch adds the equation of entropy to the system of conservative laws; i.e. [11]:

∂σc (σc~v) 0 ∂t + ∇ · = where σ is ρσ, and the entropy σ is p/ργ for ideal-MHD[11]. The switch can c = = either be set to NO (i.e. deactivated), SELECTIVE, or ALWAYS, and has an effect on the recover part of the RSA strategy implemented by PLUTO: the hydrostatic pressure is recovered either from the total energy (etot ) or from σc , depending on the value of F(σ).

( p p(e ) i f F(σ) 0 = tot = p p(σ ) i f F(σ) 1 = c = In the case the switch is set to NO, then F(σ) is equal to 0 everywhere in the domain, and the pressure is recovered from the total energy. On the other hand, when set to 6.1.F INAL SET-UP 87

ALWAYS, the pressure is recovered everywhere from the entropy equation. However, the latter solution is consistent with smooth ﬂows, only (i.e. no shock waves) [11]. In addition, the user can instruct the code to selectively vary the recovery strategy of the pressure. Indeed, when the switch is set to SELECTIVE the value of F(σ) is decided in each cell according to the relation [11]:

( ˜ p 0 i f ˜ ~v 0 and k∇ k εp F(σ) ∇ · < p > = 1 other wi se where ˜ is a three-point undivided difference operator, and ε is a parameter related ∇ p to the shock wave strength. Thus, the user can select the value of εp to decide which is the minimum shock wave strength that triggers the activation of the switch, making this function more or less sensitive to shock waves. A SELECTIVE ETROPY_SWITCH has shown the most robust behaviour in the completed computational analyses. However, using this strategy, the general conservation of neither the total energy nor of the entropy are assured at numerical level [11]. As a consequence, its reliability and accuracy have been doubted during this 6 project. Therefore, the ﬁnal set-up has been run implementing all the three different switches, and the results have been then compared. Hence, the next section describes the routine of PLUTO (i.e. Runtime Analysis) that has been exploited to extract useful data from the output of the calculations: that has in fact allowed to efﬁciently derive the estimated performance of the chamber.

6.1.6. RUNTIME ANALYSIS A runtime analysis can be programmed to execute additional computations at each new integration step. Consequently, a possibly extensive post-processing analysis can be avoided. This function of PLUTO has been discovered while searching for an efficient and effective way of estimating the reaction chamber performance. The solution hereby presented followed many other failed attempts, and has in fact been inspired by the plasma momentum balance applied by Thio et al. [10]. Therefore, the variation of total axial momentum of the expanding plasma (i.e. p m (t)v (t)) x3 = p x3 has been computed, and a specific asset of PLUTO has been again exploited to successfully achieve that. In fact, ambient density being 0, distinguishing the expand- 6= ing plasma from the other matter filling the rest of the domain may be not that trivial. Besides, plasma do actually mixes with ambient matter during its expansion. Nevertheless, PLUTO can actually mark the matter within some specific cells of the domain, so that it is then able to recognise the evolution of that very matter from that of anything else: that is formally equivalent to the tracing of airflows in wind tunnels, where the air can actually be “coloured”. Thus, a tracer (i.e. TRC) has been set among the initial conditions of the problem to “colour” the plasma pellet. That tracer has then been set as the multiplying factor of the quantities that are used to estimate the axial momentum in each cell. Hence, the following expression has been used:

p TRC(ρv dV ) (6.3) x3 = x3 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 88 COMPUTATIONAL ANALYSIS

where dV is the volume of the specific cell (i.e. 2πρdρdz). Then, the contribute of each cell is added to finally attain the plasma total axial momentum at every new time step. Hence, since the integration (i.e. summation of each cell’s contribute) is executed in the whole computational domain, the multiplying tracer allows to exclude the cells that do not contain the expanding plasma. In addition, the same tracer gives a weight to each cell that is included in the summation. In fact, during the expansion, some of the “coloured” plasma mixes with the matter of the remaining domain (i.e. due to an ambient density 0). Hence, in the 6= cells where that mixture happens, the total density is the sum of plasma’s and ambient particles’, and the value of the assigned tracer is consequently affected. The tracer is in fact equivalent to an index of the percentage of expanding plasma density out of the total contained in the specific cell. Thus, TRC is initially 1, in the cells where the starting plasma pellet is defined, = and 0, everywhere else. Then, its value is affected by the evolution of the problem, = and varies in each cell between 0 and 1 depending on the percentage of mixing. Fig- 6 ure 6.6 shows this effect. The initial condition is shown on the left side, while a more advanced moment of the dynamics on the right. The coloured profile pictures the value of TRC, the white dashed line defines the chamber limit, and the streamlines represent the magnetic field. Initially (i.e. left plot), TRC is equal to 1 in the area occupied by the plasma (red semicircle), and 0 everywhere else (blue part). However, during the plasma expansion the mixture with the ambient density does happen, and that is shown in the plot on the right where TRC assumes more values between 0 and 1 (see the legend). Furthermore, employing the tracer has highlighted another important phenomenon involved in the numerical analysis: the profile of TRC is clearly in contact with the dashed line (i.e. the chamber). In fact, a loss of plasma through the limit of the updated computational domain (i.e. dashed line) has been identified. That is believed to be due to numerical resistivity. The resistivity of plasma is directly related to its diffusion in the transverse direction through the magnetic field lines (i.e. field penetration) [14]. According to ideal-MHD, plasma resistivity is 0, and that is an accurate simplification for high temperature plasma such as fusion plasma: resistivity is inversely proportional to plasma temperature [14]. Nonetheless, numerical effects can also lead to a possible increase of plasma resistivity (i.e. numerical resistivity), and that can even be orders of magnitude higher than natural resistivity [13]. It has not been possible to quantify the effect of this phenomenon, however, Stanic et al. highlighted the same kind of issue, and mentioned that numerical resistivity, if present, would anyway lead to conservative results [13]. Furthermore, Figure 6.7 shows that the plasma that do penetrate the field lines has a density orders of magnitude lower that what of the reflected plasma. The density profile is there pictured in both the plots at the same time step of Figure 6.6 (i.e. t 2.85µs). On the right side, ' the magnetic field lines have been removed to clearly show the colour of the density profile in that location. Therefore, the identified plasma loss has been considered having a negligible effect on the outcome of the analysis. As a result, the estimated axial momentum has been used to calculate the specific impulse (Isp ) and the efficiency of the reaction chamber (η j ) according to [10]: 6.2.V ALIDATION 89

FIGURE 6.6: Evolution of the TRC (t 0 left, t 2.85µs right) = '

∆px3 Isp = mp g0 ∆px3 η j p = mp 2Ecp where ∆px3 is the change of plasma axial momentum that is due to the reﬂection on the chamber’s wall. In fact, were the plasma free to expand, its axial momentum would stay identically 0 due to the radial symmetry of its motion. However, the chamber affects the dynamics of the plasma by slowing down its approach, ﬁrst, and redirecting it, eventually. Therefore, before discussing the results of this analysis in detail, a comparison with the work completed by Stanic et al. is included in the next section. Despite some inaccuracies have been recognised in their study, such a correlation can still be considered relevant for the validation of the just discussed set-up. Besides, what of Stanic et al. is the only other similar study that has been found in the literature.

6.2. VALIDATION The set-up discussed in Section 6.1 has thus been modiﬁed to implement the different initial conditions and chamber geometry of the Icarus Project [13] (i.e. m p = 0.287g, R 60cm, E 13.46 1013 er g, l 569.48cm, θ 48°). No meaningful 0 = jet = · f = α = 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 90 COMPUTATIONAL ANALYSIS

FIGURE 6.7: Density proﬁle at t 2.85µs (left and right). Magnetic ' ﬁeld lines included (left)

plot has been found in the literature, however, the results of the comparison are reported in Table 6.8. Also, the computational domain characteristics are included in Table 6.6, and the electric coil speciﬁcations in Table 6.7.

ρ z × size 2700.0 45140.625 [cm cm] × × resolution 256 428 × TABLE 6.6: Computational domain of the validation problem

Some discrepancies can in fact be observed in the estimated performance. Nev- ertheless, the ploy that has been used to attain the magnetic field compression (see Sec. 6.1.3) is completely different. Hence, the final result may have been affected. In addition, the outcome of the Icarus Project analysis is not reported in a very clear and consistent way, and few typos have indeed been recognised. Hence, there might be some inaccuracies in the extrapolated data (e.g. geometry of the system). However, the reaction chamber operation (i.e. the expanding plasma reflection) has been successfully achieved. Therefore, the discussed set-up has been considered validated, and employed for the computational analysis that is the subject of this research. The final outcome is summarised in the next section. Indeed, only the very 6.3.R ESULTS 91

n 1 2 3 4 I [statA] 2.7 1016 2.85 1015 1.683 1016 1.886 1016 · · · − · an [cm] 560.0 1800.0 2550.0 2550.0

zn [cm] 100.0 900.0 2100.0 2600.0

TABLE 6.7: Icarus Project electric coil speciﬁcations [13]

Reference value Results total impulse [g cm/s] 2.92 107 2.08 107 · · speciﬁc impulse [s] 1.1 105 7.2 104 · · nozzle efﬁciency 0.071 0.072

TABLE 6.8: Results of the validation problem last calculations are hereby discussed: those have been eventually accomplished af- 6 ter a long series of attempts, and a meticulous ﬁne tuning of the set-up. Part of the credit has to be given to Prof. A. Mignone, who has in fact supported the ﬁnal part of this project.

6.3. RESULTS By implementing an ideal-MHD model, the first aim of the completed computational analysis was to reproduce the projected operation of a magnetic flux compression reaction chamber. In particular, plasma reflection was the critical aspect to verify, and that has in fact been achieved. Hence, the next figures show the evolution of the calculated plasma expansion. There, the implemented ambient conditions 5 11 3 are p 1.0 10− mbar and ρ 2.0 10− g/cm , and the ENTROPY SWITCH is set to 2 = · 2 = · SELECTIVE. Nevertheless, as it is better discussed in the next, the other configurations did show a similar general behaviour, as well. Thus, as Figure 6.8 shows, at time t 1.42µs (plot on top) part of the plasma has ' already been stopped by the magnetic field. One can notice that, once stopped, the plasma does assume a parabolic shape, as supposed by Adams et al. [4]. Moreover, the plot at the bottom of Figure 6.9 shows a more advanced situation when most of the plasma density is indeed concentrated where the field stopped it. Besides, the reflection has been already achieved in both the plots there included; however, that is more clear in the plot on top, where the plasma is already moving away from the chamber wall. Then, the plot at the bottom of Figure 6.10 shows the moment when the farthest coil (i.e. n 8) has been reached by the plasma. On the other hand, the = plot on top pictures the final time step of the computation. Therefore, as the plots demonstrate, plasma reflection is indeed achieved. 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 92 COMPUTATIONAL ANALYSIS

FIGURE 6.8: Density proﬁle at t 0 (bottom), and t 1.42µs (top). = ' Selective entropy switch (bottom and top). Ambient conditions p2 5 11 3 = 1.0 10− mbar and ρ 2.0 10− g/cm (bottom and top) · 2 = · 6.3.R ESULTS 93

FIGURE 6.9: Density proﬁle at t 3.56µs (bottom), and t 6.41µs ' ' (top). Selective entropy switch (bottom and top). Ambient conditions 5 11 3 p 1.0 10− mbar and ρ 2.0 10− g/cm (bottom and top) 2 = · 2 = · 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 94 COMPUTATIONAL ANALYSIS

FIGURE 6.10: Density proﬁle at t 9.26µs (bottom), and t 14.24µs ' ' (top). Selective entropy switch (bottom and top). Ambient conditions 5 11 3 p 1.0 10− mbar and ρ 2.0 10− g/cm (bottom and top) 2 = · 2 = · 6.3.R ESULTS 95

Furthermore, as already mentioned, all the completed simulations have demonstrated the same general behaviour of the plasma in reaction to the magnetic field of the chamber. Nonetheless, the estimated performance are in fact affected by the specific set-up implemented. Thus, the runtime analysis of Section 6.1.6 has been employed to extract that information, and the results are hereby presented. Figure 6.11 shows the trend of plasma axial momentum versus the total integration time. The applied values of pressure and density are from Table 6.5, and the ETROPY_SWITCH is set to SELECTIVE in all the computations. The black line is related to the case in 9 3 which the ambient density is the highest (i.e. 2.0 10− g/cm ). Hence, the overall · lower axial momentum of that solution, proves the effect of ambient density on the dynamics of the system. Besides, the following decrease of momentum is due to the excessive plasma energy spent to accelerate the ambient matter, which has evenu- tally resulted in a deceleration of the plasma (see FIg. 6.4). On the other hand, the green and red plot, which differs in the initial ambient pressure, only, show very similar results. Again, that proves that magnetic pressure plays the dominant role in affecting plasma expansion. There, the decrease of 6 plasma momentum may be related to the energy spent to push the ambient matter, on one side, and to losses due to numerical resistivity, on the other. In fact, due to the mixing of plasma and ambient mass mentioned in Section 6.1.6, part of the plasma is lost trough the chamber wall (i.e. it crosses the dashed parabola of the figures). As a result, since the cells below the parabola are excluded from the computational analysis, all the mass that goes there is excluded from the solution. Nonetheless, it has not been possible to highlight which issue added the most affecting contribute to the total momentum decrease. In addition, Figure 6.12 compares how the results are affected by the state of the ETROPY_SWITCH. The same ambient conditions (i.e. p2,ρ2) are set for all the three analysis, while the legend expresses the respective state of the ETROPY_SWITCH. As the graph proves, the non-conservation of the total energy due to an always enabled entropy switch greatly affects the outcome of the results. However, as already mentioned, such a setting is consistent with smooth flows, only. Thus, since that is not the case for the specific problem, an always enabled entropy switch should not be used: every discontinuity in the plotted profile is in fact due to shock waves. On the contrary, the similar trend of the green and magenta line shows that a selective entropy switch, despite the total energy is not conserved at a numerical level [11], may have a limited impact on the outcome of the analysis. However, the solution of the magenta curve (i.e. NO ENTROPY_SWITCH), has been attained by completely ignoring the warnings for computed negative pressures. Thus, its result may not be totally reliable. Nonetheless, the same procedure (i.e. comparison between SELECTIVE and NO ENTROPY_SWITCH) has been repeated for the case of ambient conditions (p1,ρ2), and the results are shown in Figure 6.13. The outcome is thereby interesting for it shows very much similar trends to the case of Figure 6.12. However, the solution relative to the yellow curve, despite having the ENTROPY_SWITCH turned off, did not display any warnings concerning negative pressures. Therefore, that is believed to confirm the reliability of the completed calculations. 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 96 COMPUTATIONAL ANALYSIS

FIGURE 6.11: Variation of plasma axial momentum due to different 5 initial ambient conditions (p1 30.0mbar, p2 1.0 10− mbar and 9 =3 11= · 3 ρ 2.0 10− g/cm , ρ 2.0 10− g/cm ) 1 = · 2 = ·

As a result, despite enabling a selective entropy switch increases the robustness of the code when dealing with strong shock waves, the pointed out negligible effect of further decreasing the ambient hydrostatic pressure may suggest that implementing the switch is not necessary. In fact, strong shock waves have not been detected for higher values of pressure (e.g. p1). On the other hand, implementing real ambient conditions is fundamental if an estimation of the varying temperature of the system is sought. Indeed, the temperature (T ) of plasma is directly related to its pressure and particle density (n) by the relation [11]:

p nk T (6.4) = B That same relation has indeed been used to calculate an initial temperature of the plasma of 108 K (i.e. in accordance with the estimated temperature of fusion plasma ∼ [10]). However, estimating the temperature of the particles that get in touch with the chamber structure would be more interesting in the speciﬁc scenario. In fact, in low particle density conditions (i.e. such as vacuum is), the dominant form of heat transfer is conduction (i.e. through a direct contact) [4]. Hence, the temperature should be computed in the cells that are adjacent to the chamber wall. However, due to the non accurate vacuum conditions reproduced, as well as the mixture of plasma with ambient particles (see Sec. 6.1.6), completing an accurate analysis of the system temperature evolution has not been possible within the scheduled time for this 6.3.R ESULTS 97

FIGURE 6.12: Variation of plasma axial momentum due to different 5 ENTROPY_SWITCH states (ambient conditions p2 1.0 10− mbar, ρ2 11 3 = · = 2.0 10− g/cm ) · project. Nevertheless, the other aim of this research was to verify the results of the simpli- ﬁed analytical model, and that has in fact been achieved.

6.3.1. SIMPLIFIED ANALYTICAL MODEL VERIFICATION Therefore, as Section 6.1.6 already introduced, the change of plasma axial momentum, can be related to the speciﬁc impulse and thrust efﬁciency of the chamber:

∆px3 Isp = mp g0 ∆px3 η j p = mp 2Ecp

Where, since the initial plasma momentum is identically equal to 0, ∆px3 has been considered equal to the maximum axial momentum recorded (i.e. maximum values of Figure 6.12 and 6.13). In fact, as already pointed out in this section, the pointed out decrease of plasma momentum has been considered not representative of the real chamber operation. Nevertheless, before more accurate results can be achieved, that behaviour has first to be clearly justified. However, further calculations could not be completed within the scheduled time for this thesis. Hence, the final results are 6.M AGNETIC FLUX COMPRESSION REACTION CHAMBER: 98 COMPUTATIONAL ANALYSIS

FIGURE 6.13: Variation of plasma axial momentum due to different ENTROPY_SWITCH states (ambient conditions p1 30.0mbar, ρ2 2.0 11 3 = = · 10− g/cm )

reported in Table 6.9, where they are also compared to what from NASA’s simpliﬁed analytical model (i.e. ref.). There, the results attained with both a non-active (i.e. NO), and a selective (i.e. SEL) entropy switch have been included.

p1,ρ2 p2,ρ2 ref. NO SEL NO SEL ∆p [g cm/s] 17.02 107 15.62 107 15.19 107 15.63 107 15.22 107 x3 · · · · · I [s] 76.76 103 69.35 103 67.44 103 69.41 103 67.60 103 sp · · · · · η j 0.81 0.73 0.71 0.73 0.71

TABLE 6.9: Veriﬁcation of Thio et al.’s results

In conclusion, the estimated performance are in all the cases worse than what projected by the simplified analytical model. Of course there might still be some inaccuracies in the implemented numerical model, and the result might be not totally representative of the real case. Nonetheless, the simplified model is based on many strong simplifying assumptions (see Sec. 4.3.4), that might justify the higher estimated values. First of all, this numerical analysis proved that neglecting the whole first part of the plasma expansion (i.e. assuming the plasma of a parabolic shape, and all its thermal energy converted in kinetic energy) might have a too strong impact on 6.3.R ESULTS 99 the calculated dynamics. Nevertheless, the discrepancies in the results may still be due to the not yet properly modelled ambient conditions: exact vacuum could in fact justify some of the assumptions of the simplified model. Hence, implementing a Relativistic Magneto- Hydrodynamics (RMHD) model may be the next necessary step to attain a more accurate reproduction of real conditions: the computational code would indeed be more robust when dealing with extremely low densities and pressures [11].

7 CONCLUSIONS

A preliminary computational MHD analysis of the plasma dynamics in a magnetic flux compression reaction chamber has thus been completed for this thesis project. Specifically, the process of thrust generation has been modelled, and the projected operation confirmed. Hence, no clarifications and verifications concerning neither the fusion reaction ignition, nor the generation of the plasma pellet have been included in this report. Nonetheless, a magnetic flux compression reaction chamber, combined with a pulsed-fusion rocket, is believed to be one of the most promising options for making interplanetary space travel much more efficient and affordable in the quite near future. The possibility of reaching Mars within 90 days has in fact been proven in several completed mission analyses [3–5]. Besides, completing this project has given an answer to the following research questions.

• How is the thrust generated in a magnetic ﬂux compression reaction chamber?

– What is the theoretical background at support of the projected working principle of a reaction chamber? – What is the latest and most promising reaction chamber concept?

To understand the working principle of a reaction chamber, and hence how the thrust is generated, investigating some of the plasma physics principles has first been necessary. Thus, as Chapter2 has clearly discussed, plasma dynamics requires a specific physics model to be described. In fact, plasma has a double nature and behaves both as a fluid, and as a group of charged particles. As a result, the MHD model, which combines classical fluid dynamics with electromagnetism principles, has been introduced. In particular, according to this model, a magnetic field does have on plasma a similar effect that hydrostatic pressure has on a fluid: that is due to the so called magnetic pressure. Moreover, the same chapter has pointed out how ideal-MHD can be suitable for the description of fusion plasma dynamics. Hence, Chapter3 has extensively described how the thrust is produced in a reaction chamber. In fact, a fusion reaction is ignited within a bowl-shaped chamber,

101 102 7.C ONCLUSIONS

and the consequent expanding plasma is reflected by the chamber itself to attain a thrust in reaction. Also, the magnetic field, that is generated within the chamber, has been recognised responsible for the plasma reflection. In fact, due to electromagnetic reactions, any magnetic field that happens to be between two fast moving armatures, cannot penetrate through them; as a result, it gets compressed. In addition, due to the solenoidal nature of magnetic fields that compression is directly related to an increase of the field magnitude. Therefore, the magnetic pressure being proportional to the field magnitude, such a compression results in a raise of the former. In a reaction chamber, the role of conductive armatures is indeed played by the highly conductive expanding plasma, on one side, and by the conductive structure of the chamber, on the other. As a result, due to the plasma expansion, the magnetic pressure within the chamber increases. At some point, that pressure gets high enough to slow down, stop, and eventually reflect the plasma motion. Hence, due to the total momentum conservation of the system, a resultant thrust is applied to the chamber. Moreover, Chapter3 has been used to point out a multi-coil parabolic reaction chamber as the latest and most promising concept. That is in fact the concept included in the NASA’s HOPE mission design. Besides, it is the most thoroughly described in the literature, both in terms of design and estimated performance. 7 • How have the so far projected performance been derived?

An investigation revealed that the projected performance of a multi-coil parabolic reaction chamber have been so far estimated through simpliﬁed analytical models and energy balances. However, the same research pointed out the strong assumptions that are indeed included in the simpliﬁed models. Those have been reported in Chapter4. Among them, no electromagnetism principles are included, and a great part of the plasma expansion is neglected. As a consequence, the validity of the model has been doubted, and a more thorough way of modelling the plasma dynamics in a reaction chamber has been sought.

• Can the same performance be reproduced by a more extensive computational analysis?

– Can the computational analysis be performed on a commercial laptop?

As Chapter2 has clearly pointed out, ideal-MHD has been recognised as possibly being suitable for applications involving fusion plasma (i.e. a collisionless, highly-conductive, and strongly magnetised plasma). Hence, computational ideal- MHD has been highlighted as a valuable candidate to complete the verification of the simplified models of the reaction chamber. Thus, Chapter5 has investigated previous computational analyses published in the literature. The results has shown that codes officially programmed for space propulsion applications are in fact not freely available to the user. Hence, it has not been possible to attain a copy of them. Nonetheless, searching for less specialised codes increased the number of viable options. Eventually, PLUTO has been selected among the others as the final candidate for completing this research. In fact, PLUTO is a freely-distributed and modular code for computational astrophysics that, thanks to a user-friendly interface, the open source nature, and strong adapting capabilities, has eventually been able to replicate the reaction chamber dynamics. The same chapter has also been used to point out the scarceness of other completed reaction chamber computational analyses. Indeed, only one published result has been discovered. That, despite being related to a different reaction chamber concept, has been still used as a reference. As a result the capabilities of PLUTO have been extensively probed before the fi- 7 nal set-up reported in Chapter6 could be attained. Besides, it has been necessary to elaborate some specific ploy to accurately reproduce a reaction chamber and its dynamics. In particular, the initial magnetic field has been designed employing an approximate expression of the vector potential of a current coil: an exact analytical expression of such a magnetic field is in fact not available. Nevertheless, despite not an exact solution, this strategy assures the generation of a solenoidal field: that has indeed been recognised as more crucial for the correct operation of the chamber. Moreover, the source code of PLUTO has been modified to assure the compression of the magnetic field. In fact, the conductive coils of the chamber being not implementable in the computational domain, adding a specific ploy has been needed. The final solution, despite not being a perfect reproduction of the real system, is still believed to be more accurate and reliable than what found used in the literature. The final configuration that has been employed to set-up the computational problem is reported in Chapter6. The validity of the model has been successfully proven against the results of the only other similar computational analysis found in the literature. Therefore, the same analysis has been finally applied to the latest multi-coil parabolic chamber design as what discussed by Adams et al. [4] and Thio et al. [10] (i.e. a multi-coil parabolic chamber). Besides, according to what found in the literature, what has been completed for this thesis is in fact the first computational ideal-MHD analysis of that very reaction chamber concept. Thus, recalling that:

it can be fairly said that the purpose has been accomplished. Besides, the result of this analysis has been compared to what of the simplified analytical model: analogous initial conditions have been imposed. As a result, the similar estimated performance may prove that the simplified model is in fact sound. Therefore, this research has confirmed some hypothesis, raised some questions, and brought some contributes to the field of research of nuclear space propulsion. That is summarised in the next section.

7.1. RESEARCH CONTRIBUTIONS First of all, this research has completed a thorough review of the available material that concerns the concept of a magnetic flux compression reaction chamber. Several errors and inconsistencies have in fact been highlighted in the literature, and those have been hereby corrected. As a result, this report can be considered as a reviewed summary of what has been so far published about the operation of a reaction chamber. In particular, the simplified analytical model that has been developed at NASA has been extensively examined in order to verify and prove the validity of its assumptions. Besides, the first attempt (i.e. according to the found literature) of replicating the reaction chamber operation (i.e. thrust generation) by applying a computational 7 ideal-MHD analysis has been completed for this thesis. That has in fact proven that the analytical model might be over-simplified, and its performance estimations too optimistic. Specifically, assuming all the plasma thermal energy converted in kinetic energy long before its reflection is completed has been pointed out as affecting the most the results of the simplified model. Another interesting aspect which has not been found extensively treated in the investigated literature is the modelling of the ambient conditions in the chamber (i.e. ambient density and pressure). In fact, despite the issue of the impossibility of implementing a density 0 is everywhere stated, the consequences that a ρ 0 has = 6= on the plasma expansion were in fact nowhere mentioned. The value of the ambient density has indeed been recognised directly affecting the performance of the chamber: the mass that fills the rest of the domain has to be pushed away by the expanding plasma, hence, part of its energy is lost in the process. Nevertheless, ambient density could not be reduced enough to understand whether its effect was negligible or not: the last completed simulation (i.e. run with the lowest implemented density) still shows an anomaly in the computed plasma total momentum (i.e. after a preliminary rise due to the plasma reflection, its value drops). In fact, the low density combined with the strong magnetic field has resulted in relativistic Alfvén speeds that compromised the correct execution of the calculations. Therefore, implementing a RMHD model might be a possible option to overcome this problem. Indeed, Prof. A. Mignone, author of the code, pointed out that the relativistic model is more robust in handling low density profiles, and that it imposes more strict constraints on the computed velocities. Again, that would be a novelty in the computational analysis of a magnetic flux compression reaction chamber. On the other hand, ambient pressure has been proved having a negligible effect on the plasma expansion. Therefore, if no exact estimation of the system temper- 7.2.R ECOMMENDED FUTURE WORK 105 ature profile is sought, the constraints on the value of pressure are less strict. As a result, one can implement a value of pressure seeking the numerical stability of the problem, and hence limit the integration time. Moreover, it has been investigated that the implemented conditions (i.e. both density and pressure) can in fact be representative of the most advanced vacuum chambers capabilities. Hence, the results of this analysis could be useful for testing modelling purposes. In addition, the modelling of the magnetic field within the chamber has been extensively studied in order to attain the final result. Hence, the proposed solution, besides being quite innovative, is believed to be more reliable and accurate than what so far published. Furthermore, despite the reproduced system is still not an exact copy of the concept design (i.e. a conductive wall has been implemented rather than single coils), its adaptability may assure more accurate results in future. Indeed, what hereby presented can be as well considered as a validation of the chosen method: some simplifications (e.g. including a wall rather than the single coils) have been implemented for the sake of efficiently testing new approaches to the problem. As a consequence the recommended future work can now be included.

7.2. RECOMMENDED FUTURE WORK The first next step shall be the implementation of a computational RMHD analy- 7 sis aimed to definitely quantify the effect of the ambient density on the plasma expansion. In fact, the relativistic module is much more robust when dealing with extremely low density profiles. Nevertheless, the anomalous behaviour of the plasma momentum could also be due to numerical resistivity phenomena. In fact, this possible issue has not been analysed in details, yet. However, it might be interesting to clearly quantify the relation between that kind of resistivity (if actually present), and natural plasma resistivity. Indeed, plasma resistivity is directly related to plasma detachment (i.e. higher the resistivity is, more difficult the plasma detachment). Hence, it might also cause similar losses to what attained from this analysis (i.e. plasma momentum decrease). Therefore, were the decrease of plasma momentum related to numerical resistivity, it might in fact be exemplary of natural plasma resistivity effects on a real system. As a result, the correctness of an ideal-MHD model must yet be confirmed. In addition, the modelling of the magnetic field can be further improved by including the effect of separate coils. That could in fact be accomplished by modifying the implemented geometrical condition of the E 0 constraint (see Sec. 6.1.3). In- φ = deed, rather than using a parabola, 8 circumferences, each centred on one of the electric coil location, might be defined. Then, the same E 0 constraint might be φ = limited to the cells contained within each circumference. The radius of the circles could be derived from the completed study of the electric coil design included in [5]. Furthermore, the same calculations could be run implementing a 3-D domain (the completed set-up can easily be adapted). That would indeed be necessary to evaluate the assumption of a perfect axial symmetry of the system dynamics (i.e. the only assumption that has been borrowed from the simplified analytical model). Lastly, no exhaustive investigations have been completed concerning the evolu- 106 7.C ONCLUSIONS tion of the plasma temperature and the consequent heat transfer to the nozzle. In fact, the magnetic field is meant to completely insulate the chamber structure from any possible contact with the hot fusion plasma. However, some leakage of plasma through the modelled chamber wall has been identified; again, that might be due to numerical resistivity issues, which must hence be accurately clarified. A MATLAB SCRIPT

The sections of the MATLABTM script that have been used to reproduce the results of the simpliﬁed analytical model, and that are mentioned in the body of this report are hereby included.

A.1. ARRAYS DEFINITION

1 t(1,1) = t0;% initial time step(i.e. one-line horizontal array)

2 r(1,:) = r0;% initial radial position of the plasma shell

3 % sections

4 a(1,:) = a0;% initial focal length of the plasma shell

5 theta(1,:) = theta0;% initial colatitudinal coordinates of

6 % each plasma shell, and hence every

7 % thrust ring, as well 8 v0r(1:no) = sqrt(2*Ecpn/Mp);% initial radial velocity of each 9 % plasma section in spherical

10 % coordinates[m/s]

11 v0t(1:no) = 0;% initial colatitudinal velocity of each plasma

12 % section in spherical coordinates[m/s] 13 vrad(1,:) = vr(1,:).*sin(theta(1,:))+vt(1,:).*cos(theta(1,:)); 14 % initial radial velocity of each plasma section in

15 % polar coordinates[m/s]

16 B(1,:) = B0;% intiial magnetic field

17 pB(1,:) = pB0;% initial magnetic pressure 18 dvax(1,:) = -4*pi/Mp*pB0.*r(1,:).^2; 19 % initial axial acceleration in polar

20 % coordinates[m/s^2] 21 dvt(1,:) = -4*pi/Mp*pB0.*r(1,:).^2.*(cot(theta(1,:)/2)); 22 % initial colatitudinal acceleration in spherical

23 % coordinates[m/s^2] 24 dvr(1,:) = -4*pi/Mp*pB0.*r(1,:).^2; 25 % initial radial acceleration in spherical

26 % coordinates[m/s^2]

107 108 A. MATLAB SCRIPT A A.2. INTEGRATION PROBLEM

1 for n = 1:N-1% for loop used to increment the time step

2 for k = 1:no% for loop to calculate each plasma section

3 % separately

5 t(n+1,1) = t(n,1) + h;% time increment(h=Delta_t) 6 theta(n+1,k) = theta(n,k) + vt(n,k).*h./r(n,k); 7 % angular position increment of each

8 % plasma section(spherical) 9 r(n+1,k) = r(n,k) + vr(n,k).*h; 10 % radial position increment of each

11 % plasma section(spherical) 12 a(n+1,k) = (1-cos(theta(n,k))).*r(n+1,k)/2; 13 % focal length increment

14 % of the parabola related to each plasma section

16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

17 % the next if/else condition is used to exclude the contribute

18 % of the plasma acceleration after the rebound

19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

21 if (vrad(n,k) > 0)

22 23 B(n+1,k) = PHI*sin(theta(n+1,k)/2).^2./(2*pi . . . 24 *(al^2-a(n+1,k).^2).*cos(theta(n+1,k)/2)); 25 % magnetic field update

26 pB(n+1,k) = B(n+1,k).^2/2/mu0;% magnetic pressure update

28 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

29 % accelerations update

30 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 31 dvax(n+1,k) = -4*pi/Mp*pB(n+1,k).*r(n+1,k).^2; 32 dvt(n+1,k) = -4*pi/Mp*pB(n+1,k).*r(n+1,k).^2 . . . 33 .*(cot(theta(n+1,k)/2)); 34 dvr(n+1,k) = -4*pi/Mp*pB(n+1,k).*r(n+1,k).^2; 35

36 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

37 % velocities update

38 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 39 vr(n+1,k) = vr(n,k) + dvr(n,k).*h; 40 vt(n+1,k) = vt(n,k) + dvt(n,k).*h; 41 vrad(n+1,k) = vr(n+1,k).*sin(theta(n+1,k))+vt(n+1,k) . . . 42 .*cos(theta(n+1,k)); 43

44 else

46 % magnetic field and magnetic pressure constant after the rebound

47 B(n+1,k) = B(n,k);

48 pB(n+1,k) = B(n+1,k).^2/2/mu0;

50 % accelerations equal to0 after the rebound

51 dvax(n+1,k) = 0; A.2.I NTEGRATION PROBLEM 109 A

52 dvt(n+1,k) = 0;

53 dvr(n+1,k) = 0;

55 % velocities kept constant after the integration 56 vr(n+1,k) = vr(n,k) + dvr(n,k).*h; 57 vt(n+1,k) = vt(n,k) + dvt(n,k).*h; 58 vrad(n+1,k) = vrad(n,k);

59 end

60 end

62 for i = 2:no-1% for each plasma section

64 % istantaneous thrust calculation 65 Fax(n,i) = Mp/2*dvax(n,i).*(cos((theta(n,i+1)+theta(n,i))/2) . . . 66 -cos((theta(n,i)+theta(n,i-1))/2)); 67 Fax(n,no) = Mp/2*dvax(n,no).*(cos((Dtheta*pi/180/2 . . . 68 +2*theta(n,no))/2)-cos((theta(n,no-1)+theta(n,no))/2)); 69 Fax(n,1) = Mp/2*dvax(n,1).*(cos((theta(n,2)+theta(n,1))/2) . . . 70 -cos((theta(n,1)+Dtheta*pi/180/2)/2)); 71

72 % average thrust 73 T = frep*sum(sum(Fax*h));% double sum cause Fax is an array 74

75 end

76 end

B PLUTO

The PLUTO input ﬁles that have been implemented in order to complete the computational analysis are hereby included.

B.1. PLUTO.INI

1 [Grid]

3 X1-grid 1 0.0 288 u 180.0

4 X2-grid 1 0.0 560 u 350.0

5 X3-grid 1 0.0 1 u 1.0

7 [Chombo Refinement]

9 Levels 4

10 Ref_ratio 2 2 2 2 2

11 Regrid_interval 2 2 2 2

12 Refine_thresh 0.3

13 Tag_buffer_size 3

14 Block_factor 8

15 Max_grid_size 64

16 Fill_ratio 0.75

18 [Time]

20 CFL 0.45

21 CFL_max_var 1.1

22 tstop 2.0

23 first_dt 1.e-7

25 [Solver]

27 Solver hll

29 [Boundary]

111 112 B.PLUTO

31 X1-beg axisymmetric

32 X1-end outflow

33 X2-beg outflow

B 34 X2-end outflow

35 X3-beg outflow

36 X3-end outflow

38 [Static Grid Output]

40 uservar 4 pm ptot beta vz

41 output_dir ./results

42 dbl -1.0 -1 single_file

43 flt -1.0 -1 single_file

44 vtk 0.1 -100 single_file cgs

45 dbl.h5 -3.0e-1 -1 single_file

46 flt.h5 -1.0 -1

47 tab -1.0 -1

48 ppm -1.0 -1

49 png -1.0 -1

50 log 100

51 analysis -4.e-3 1

53 [Chombo HDF5 output]

55 Checkpoint_interval -1.0 0

56 Plot_interval 1.0 0

58 [Parameters]

60 ZOFFSET 5.0

61 PVAC 1.38e-2

62 RHOVAC 2.e-11

B.2. DEFINITIONS.C

1 #define PHYSICS MHD

2 #define DIMENSIONS 2

3 #define COMPONENTS 2

4 #define GEOMETRY CYLINDRICAL

5 #define BODY_FORCE NO

6 #define COOLING NO

7 #define RECONSTRUCTION LINEAR

8 #define TIME_STEPPING RK2

9 #define DIMENSIONAL_SPLITTING NO

10 #define NTRACER 2

11 #define USER_DEF_PARAMETERS 3

12 13 /* -- physics dependent declarations -- */ 14

15 #define EOS IDEAL

16 #define ENTROPY_SWITCH SELECTIVE

17 #define DIVB_CONTROL CONSTRAINED_TRANSPORT B.3. INIT.C 113

18 #define BACKGROUND_FIELD NO

19 #define RESISTIVITY NO

20 #define THERMAL_CONDUCTION NO

21 #define VISCOSITY NO B

22 #define ROTATING_FRAME NO

23 24 /* -- user-defined parameters (labels) -- */ 25

26 #define ZOFFSET 0

27 #define PVAC 1

28 #define RHOVAC 2

29 30 /* [Beg] user-defined constants (do not change this line) */ 31

32 #define UNIT_DENSITY 2.e-9

33 #define UNIT_VELOCITY 6.32e5

34 #define UNIT_LENGTH 4.5

35 #define EPS_PSHOCK_FLATTEN 2.0

36 #define VTK_VECTOR_DUMP YES

37 #define VTK_TIME_INFO YES

38 39 /* [End] user-defined constants (do not change this line) */ 40 41 /* -- supplementary constants (user editable) -- */ 42

43 #define INITIAL_SMOOTHING NO

44 #define WARNING_MESSAGES NO

45 #define PRINT_TO_FILE YES

46 #define INTERNAL_BOUNDARY YES

47 #define SHOCK_FLATTENING MULTID

48 #define CHAR_LIMITING NO

49 #define LIMITER VANLEER_LIM

50 #define CT_EMF_AVERAGE UCT_CONTACT

51 #define CT_EN_CORRECTION YES

52 #define ASSIGN_VECTOR_POTENTIAL YES

53 #define UPDATE_VECTOR_POTENTIAL NO

B.3. INIT.C

1 /* /////////////////////////////////////////////////////////// */ 2 /*! 3 \file

4 \brief Contains basic functions for problem initialization.

6 The init.c file collects most of the user-supplied

7 functions useful for problem configuration.

8 It is automatically searched for by the makefile.

10 \author A. Mignone ([email protected])

11 \date Sepy 10, 2012 12 */ 13 /* /////////////////////////////////////////////////////////// */ 114 B.PLUTO

14 #include "pluto.h"

15 16 /* *********************************************************** */ B 17 void Init (double *v, double x1, double x2, double x3) 18 /*! 19 * The Init() function can be used to assign initial conditions as 20 * as a function of spatial position. 21 * 22 * \param [out] v a pointer to a vector of primitive variables 23 * \param [in] x1 coordinate point in the 1st dimension 24 * \param [in] x2 coordinate point in the 2nd dimension 25 * \param [in] x3 coordinate point in the 3rdt dimension 26 * 27 * The meaning of x1, x2 and x3 depends on the geometry: 28 * \f[ \begin{array}{cccl} 29 * x_1 & x_2 & x_3 & \mathrm{Geometry} \\ 30 * \hline 31 * x & y & z & \mathrm{Cartesian} \\ 32 * R & z & - & \mathrm{cylindrical} \\ 33 * R & \phi & z & \mathrm{polar} \\ 34 * r & \theta & \phi & \mathrm{spherical} 35 * \end{array} 36 * \f] 37 * 38 * Variable names are accessed by means of an index v[nv], where 39 * nv = RHO is density, nv = PRS is pressure, nv = (VX1, VX2, VX3) 40 * are the three components of velocity, and so forth. 41 * 42 ************************************************************* */ 43 {

45 // adimensionalisation units

47 double p0, rho0, v0, l0;

48 49 p0 = UNIT_DENSITY*UNIT_VELOCITY*UNIT_VELOCITY; 50 rho0 = UNIT_DENSITY;

51 v0 = UNIT_VELOCITY;

52 l0 = UNIT_LENGTH;

54 // preallocate all the initial conditions

56 v[RHO] = g_inputParam[RHOVAC]/rho0;

57 v[PRS] = g_inputParam[PVAC]/p0;

58 v[VX1] = v[VX2] = v[VX3] = 0.0;

59 v[BX3] = v[BX1] = v[BX2] = 0.0;

60 v[AX1] = v[AX2] = v[AX3] = 0.0;

61 v[TRC] = 1.0;

62 v[TRC+1] = 0.0;

64 // plasma shell paramters

66 double f, zp, r, r0;

67 double E0, mp, V0; B.3. INIT.C 115

68 double pi;

70 pi = CONST_PI;

71 B

72 f = 200.0/l0;

73 zp = x2-f;

74 r0 = 30.0; 75 r = sqrt(x1*x1+zp*zp); 76

77 E0 = 905.e13;

78 mp = 2.26; 79 V0 = 4./3.*pi*r0*r0*r0; 80

81 if (r r0/l0) { ≤ 82 v[PRS] = E0/V0*(g_gamma-1.)/p0; 83 v[RHO] = mp/V0/rho0;

84 v[TRC+1] = 1.0;

85 }

87 // initial magnetic field

89 static int ntab;

90 static int n; 91 static double *k_tab, *S_tab; 92 double kk, S, kmid, dk, Z, R;

93 double c; // speed of light

94 double a[8] = {2.78e1, 8.44e1, 1.44e2, 2.11e2,

95 2.88e2, 3.82e2, 5.05e2, 6.81e2};

96 double z[8] = {0.96, 8.9, 26.1, 55.6,

97 104.0, 182.0, 319.0, 579.0};

98 double I[8] = {7.56e16, 1.24e16, 5.82e15,

99 3.51e15, 2.34e15, 1.63e15,

100 1.14e15, 7.89e14};

101 int nlo, nhi, nmid, i;

102 103 FILE *fk; 104

105 if (k_tab == NULL){

106 print1 (" > Reading table from disk...\n");

107 fk = fopen("./elliptic_integrals/table.dat", "r");

108 if (fk == NULL) {

109 print1 ("! table.dat could not be found.\n");

110 QUIT_PLUTO(1);

111 }

112 k_tab = ARRAY_1D(400,double);

113 S_tab = ARRAY_1D(400,double);

114 ntab = 0;

115 while (fscanf(fk, "%lf%lf\n", k_tab+ ntab,

116 S_tab + ntab) != EOF) {

117 ntab++;

118 }

119 }

120

121 c = CONST_c; 116 B.PLUTO

122

123 for ( i = 0; i < 8; i++ ) { 124 Z = x2*l0 - z[i]; B 125 R = x1*l0; 126 kk = 4*a[i]*R/(a[i]*a[i]+R*R+Z*Z+2*a[i]*R); 127 nlo = 0;

128 nhi = ntab - 1;

129 while (nlo != (nhi - 1)) {

130 nmid = (nlo + nhi)/2;

131 kmid = k_tab[nmid];

132 if (kk kmid) { ≤ 133 nhi = nmid;

134 } else if (kk > kmid) {

135 nlo = nmid;

136 }

137 }

138 dk = k_tab[nhi] - k_tab[nlo]; 139 S = S_tab[nlo]*(k_tab[nhi] - kk)/dk + 140 S_tab[nhi]*(kk-k_tab[nlo])/dk; 141 v[AX3] += I[i]*4*a[i]*S/sqrt(a[i]*a[i]+R*R+Z*Z+2*a[i]*R)/c; 142 }

143 144 v[AX3] /= sqrt(4*pi*rho0)*v0*l0; 145

146 } 147 /* *********************************************************** */ 148 void Analysis (const Data *d, Grid *grid) 149 /*! 150 * Perform runtime data analysis. 151 * 152 * \param [in] d the PLUTO Data structure 153 * \param [in] grid pointer to array of Grid structures 154 * 155 ************************************************************* */ 156 {

157

158 int i, j ,k; 159 double *dr, *dphi, *dz; 160 double *r, *phi, *z; 161 double ***V1, ***V2, ***V3; 162 double ***rho, ***trc; 163 double m, A, V, Mp;

164 double dI, qz, qz_;

165 double dummy;

166 double p0, v0, l0, rho0, t0;

167 double t, dt;

168 double tr;

169 170 FILE *fp; 171

172 dr = grid[IDIR].dx;

173 dphi = grid[KDIR].dx;

174 dz = grid[JDIR].dx;

175 B.3. INIT.C 117

176 r = grid[IDIR].x;

177 phi = grid[KDIR].x;

178 z = grid[JDIR].x;

179 B

180 V1 = d->Vc[VX1];

181 V2 = d->Vc[VX2];

182 V3 = d->Vc[VX3];

183

184 rho = d->Vc[RHO];

185 trc = d->Vc[TRC+1];

186

187 v0 = UNIT_VELOCITY; rho0 = UNIT_DENSITY; l0 = UNIT_LENGTH; 188 p0 = rho0*v0*v0; 189 t0 = l0/v0; t = g_time*t0; dt = g_dt*t0; 190

191 Mp = qz = 0.0;

192

193 DOM_LOOP(k,j,i) {

194 tr = MAX(trc[k][j][i],0.0); 195 A = CONST_PI*((r[i]+dr[i]/2)*(r[i]+dr[i]/2)- 196 (r[i]-dr[i]/2)*(r[i]-dr[i]/2)); 197 V = A*dz[j]; 198 Mp += tr*rho[k][j][i]*V; 199 qz += tr*rho[k][j][i]*V*V2[k][j][i]; 200 }

201

202 #ifdef PARALLEL

203 MPI_Allreduce (&Mp, &dummy, 1, MPI_DOUBLE, MPI_SUM,

204 MPI_COMM_WORLD); 205 Mp = dummy*rho0*l0*l0*l0; 206

207 MPI_Allreduce (&qz, &dummy, 1, MPI_DOUBLE, MPI_SUM,

208 MPI_COMM_WORLD); 209 qz = dummy*v0*rho0*l0*l0*l0; 210

211 MPI_Barrier (MPI_COMM_WORLD);

212 #endif

213

214 if (prank == 0) {

215 char fname[512];

216 static double tpos = -1.0;

217 sprintf (fname, "%s/momentum.dat", RuntimeGet()->output_dir);

218 if (g_stepNumber == 0) {

219 fp = fopen(fname,"w");

220 fprintf (fp,"#%7s %12s %12s %12s\n",

221 "t", "dt", "qz", "Mp");

222 }

223 else{

224 if (tpos < 0.0) {

225 char sline[512];

226 fp = fopen(fname,"r");

227 while (fgets(sline, 512, fp)) {}

228 sscanf(sline, "%lf\n",&tpos);

229 fclose(fp); 118 B.PLUTO

230 }

231 fp = fopen(fname,"a");

232 }

B 233 if (g_time > tpos) {

234 fprintf (fp, "%12.6e %12.6e %12.6e %12.6e\n",

235 t, dt, qz, Mp);

236 }

237 fclose(fp);

238 }

239

240 }

241 #if PHYSICS == MHD 242 /* ************************************************************ */ 243 void UserDefBoundary (const Data *d, RBox *box, int side, 244 Grid *grid) 245 /*! 246 * Assign user-defined boundary conditions. 247 * 248 * \param [in,out] d pointer to the PLUTO data structure 249 * containing cell-centered primitive quantities 250 * (d->Vc) and staggered magnetic fields 251 * (d->Vs, when used) to be filled. 252 * \param [in] box pointer to a RBox structure containing the 253 * lower and upper indices of the ghost 254 * zone-centers/nodes or edges at which data 255 * values should be assigned. 256 * \param [in] side specifies the boundary side where ghost 257 * zones need to be filled. It can assume the 258 * following pre-definite values: X1_BEG, X1_END, 259 * X2_BEG, X2_END, 260 * X3_BEG, X3_END. 261 * The special value side == 0 is used to control 262 * a region inside the computational domain. 263 * \param [in] grid pointer to an array of Grid structures. 264 * 265 /* ************************************************************ */ 266 {

267

268 int i, j, k, nv; 269 double *x1, *x2, *x3; 270 double h, eq;

271 double P0, rho0;

272 273 P0 = UNIT_DENSITY*UNIT_VELOCITY*UNIT_VELOCITY; 274 rho0 = UNIT_DENSITY;

275

276 x1 = grid[IDIR].x;

277 x2 = grid[JDIR].x;

278 x3 = grid[KDIR].x;

279 280 h = 1.248e-3*UNIT_LENGTH; 281

282 if ( side == 0 ) {

283 TOT_LOOP(k,j,i) { B.4. USERDEFOUTPUT.C 119

284 eq = x2[j] - x1[i]*x1[i]*h; 285 if (eq < g_inputParam[ZOFFSET] && x2[j] < 603.0/UNIT_LENGTH){

286 d->Vc[TRC][k][j][i] = 0.0;

287 d->flag[k][j][i] |= FLAG_INTERNAL_BOUNDARY; B

288 } else{

289 d->Vc[TRC][k][j][i] = 1.0;

290 }

291 }

292 }

293

294 }

B.4. USERDEFOUTPUT.C

1 #include "pluto.h"

2 3 /* ************************************************************ */ 4 void ComputeUserVar (const Data *d, Grid *grid) 5 /* 6 * 7 * PURPOSE 8 * 9 * Define user-defined output variables 10 * 11 * 12 * 13 /* ************************************************************ */ 14 {

15 int i, j, k; 16 double ***pm, ***ptot; 17 double ***bx, ***by, ***bz, ***ps; 18 double ***beta, ***vz; 19 double B;

20 double pm0, p0;

22 pm = GetUserVar("pm");

23 ptot = GetUserVar("ptot");

24 beta = GetUserVar("beta");

25 vz = GetUserVar("vz");

27 bx = d->Vc[BX1];

28 by = d->Vc[BX2];

29 bz = d->Vc[BX3];

31 ps = d->Vc[PRS];

32 33 pm0 = 4*CONST_PI*UNIT_DENSITY*UNIT_VELOCITY*UNIT_VELOCITY; 34 p0 = UNIT_DENSITY*UNIT_VELOCITY*UNIT_VELOCITY; 35

36 DOM_LOOP (k,j,i) { 37 EXPAND ( B = bx[k][j][i]*bx[k][j][i]; , 38 B += by[k][j][i]*by[k][j][i]; , 120 B.PLUTO

39 B += bz[k][j][i]*bz[k][j][i]; ) 40 pm[k][j][i] = B/2.0*pm0; 41 ptot[k][j][i] = pm[k][j][i] + ps[k][j][i]*p0; B 42 43 beta[k][j][i] = ps[k][j][i]*p0/pm[k][j][i]; 44 vz[k][j][i] = d->Vc[VX2][k][j][i];

45 }

46 }

B.5. CT.C

1 /* /////////////////////////////////////////////////////////// */ 2 /*! 3 \file

4 \brief Update staggered magnetic field.

6 Update face-centered magnetic field in the constrained transport

7 formulation using a discrete version of Stoke's theorem.

8 The update consists of a single Euler step:

9 \f[

10 \mathtt{d->Vs} = \mathtt{Bs} + \Delta t R

11 \f]

12 where \c d->Vs is the main staggered array used by PLUTO,

13 \c Bs is the magnetic field to be updated and \c R is the

14 right hand side already computed during the unsplit integrator.

15 \c d->Vs and \c Bs may be the same array or may be different.

17 \b References

18 - "A staggered mesh algorithm using high-order Godunov fluxes to

19 ensure solenoidal magnetic field in MHD simulations"\n

20 Balsara \& Spicer, JCP (1999) 149, 270

22 \author A. Mignone ([email protected])

23 \date April 1, 2014 24 */ 25 /* /////////////////////////////////////////////////////////// */ 26 #include "pluto.h"

27 28 /* ************************************************************ */ 29 void CT_Update(const Data *d, Data_Arr Bs, double dt, 30 Grid *grid) 31 /*! 32 * Update staggered magnetic field using discrete version of 33 * Stoke's theorem. 34 * Only \c d->Vs is updated, while \c Bs is the original array:\n 35 * \c d->Vs = \c Bs + \c dt * \c R, where R = curl(E) is the 36 * electric field. 37 * 38 * \param [in,out] d pointer to PLUTO Data structure. 39 * d->Vs will be updated. 40 * \param [in] Bs the original array (not updated). 41 * \param [in] dt step size B.5. CT.C 121

42 * \param [in] grid pointer to an array of Grid structures 43 * 44 /* ************************************************************ */ 45 { B

46 int i, j, k, nv;

47 int ibeg, jbeg, kbeg;

48 int iend, jend, kend;

49 double rhs_x, rhs_y, rhs_z; 50 double *dx, *dy, *dz, *A1, *A2, *dV1, *dV2; 51 double *r, *rp, *th, r_2; 52 double ***ex, ***ey, ***ez; 53 double *x1, *x2, *x3; 54 double h; 55 EMF *emf; 56 57 /* ---- check div.B ---- */ 58

59 #if CHECK_DIVB_CONDITION == YES

60 if (g_intStage == 1) CT_CheckDivB (d->Vs, grid);

61 #endif

63 emf = CT_GetEMF (d, grid);

65 #if UPDATE_VECTOR_POTENTIAL == YES

66 VectorPotentialUpdate (d, emf, NULL, grid);

67 #endif

69 ex = emf->ex;

70 ey = emf->ey;

71 ez = emf->ez;

72 73 /* ------74 Correct EMF if the ShearingBox module is enabled.

75 For CTU, this step is carried only during the final stage. 76 ------*/ 77

78 #ifdef SHEARINGBOX

79 #ifdef CTU

80 if (g_intStage == 2) SB_CorrectEMF (emf, Bs, grid);

81 #else

82 SB_CorrectEMF (emf, d->Vs, grid);

83 #endif

84 #endif

86 dx = grid[IDIR].dx;

87 dy = grid[JDIR].dx;

88 dz = grid[KDIR].dx;

90 x1 = grid[IDIR].x;

91 x2 = grid[JDIR].x;

92 x3 = grid[KDIR].x;

94 r = grid[IDIR].x;

95 rp = grid[IDIR].xr; 122 B.PLUTO

96 A1 = grid[IDIR].A;

97 dV1 = grid[IDIR].dV;

99 th = grid[JDIR].x;

100 A2 = grid[JDIR].A;

101 dV2 = grid[JDIR].dV;

102 103 h = 1.248e-3*UNIT_LENGTH; 104

105 TOT_LOOP(k,j,i) { 106 if (x2[j]-x1[i]*x1[i]*h < g_inputParam[ZOFFSET] && x2[j] < 134.0) { 107 ez[k][j][i] = 0.0;

108 }

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