Resumen de Axisymmetric simulation codes for hall effect thrusters and plasma plumes
The development of reliable and versatile plasma discharges simulation codes is becoming of central importance, given the rapidly evolving electric propulsion landscape. These tools are essential for facilitating and complementing the design of new prototypes, significantly reducing development time and costs. Moreover, they can provide a deeper insight on already proven technologies, revealing optimization opportunities so as to improve the thruster performance and lifetime, and predicting the thruster operational parameters at different regimes of interest.
This Thesis is devoted to the numerical study of different plasma discharges and, in particular, the Hall effect thruster (HET) discharge. With special focus on particle-based modeling, two simulation codes have been developed.
The first one is named HYPHEN, which stands for HYbrid Plasma thruster Holistic simulation Environment. It is a new two-dimensional axisymmetric hybrid, particle-in-cell (PIC)/fluid multi-thruster simulation platform. HYPHEN overall architecture and structure are based on modularity, aiming to be a more flexible and capable simulation tool. Modularization strategy also facilitates code development and debugging, and the integration of new capabilities. Thus, this versatile plasma discharge simulator is potentially extensible to the simulation of different types of plasma thrusters, including the Helicon plasma thruster (HPT), the electron-cyclotron-resonance accelerator (ECRA) or the high efficiency multi-stage plasma thruster (HEMPT), as well as HET. Moreover, in order to maximize code sharing and standardization, HYPHEN has been designed with the same overall architecture, data structure and interfaces as those of the hybrid 3D plasma plume code EP2PLUS, developed by F. Cichocki. Both simulation tools make use of common baseline modules and dedicated subroutines whenever possible. As a result of the synergy between the two codes above (especially regarding the PIC module), HYPHEN is also able to simulate 2D axisymmetric plasma thruster plumes, yielding valuable estimates of the slow ion backflow responsible for the spacecraft (S/C) sensitive surfaces contamination and sputtering.
The two central modules of the hybrid simulator are the PIC module and the electron-fluid module. The former follows a Lagrangian-Eulerian approach which discretizes the heavy species (i.e. neutrals and ions) distribution function in terms of macroparticles, whose trajectories are integrated consistently with the existing electromagnetic fields. Therefore, it takes as inputs the prescribed magnetic field, the electric potential and the electron temperature, performs the different collisional processes involving the heavy species, such as ionization or charge-exchange (CEX) collisions, and propagates all the ion and neutral macroparticles one simulation timestep forward, obtaining the corresponding heavy species macroscopic properties, such as the particle densities and fluxes, through a particle weighting process. On the other hand, the electron-fluid module takes those heavy species macroscopic magnitudes from the PIC module and, applying quasineutrality, solves a given fluid model for the electron population computing the electric potential and the electron population related variables (e.g. the electron temperature), thus closing the loop. However, in general, each of those central modules operates on a different mesh of the simulation domain: a structured mesh for the PIC module (referred to as the PIC mesh hereafter), and an unstructured Magnetic Field Aligned Mesh (MFAM) for the electron-fluid module. Therefore, both modules communicate each other through a dedicated bidirectional interpolation module.
One of the main contributions of this Thesis corresponds to the development and testing of the versatile HYPHEN PIC module, which is applicable to all simulation scenarios mentioned above, and includes up-to-date optimized algorithms for the heavy species treatment. The main modeling novelties on the PIC module, oriented toward the improvement of the heavy species PIC-related statistics and the extension of the code capabilities, include: (i) the subdivision of the different ion and neutral species into different populations (or particle lists) based on their atomic mass, charge or energy content, which allows for an independent treatment and monitoring of heavy species featuring very different dynamics (such as those coming from a CEX collision), thus facilitating its population control and reducing the numerical noise in the PIC related statistics; (ii) the development of a new versatile PIC mesh generator providing ad hoc optimized structured non-uniform meshes with exponential node distribution and reduced cell deformation of typical simulation domains of interest, including HETs, HPTs and ECRAs, and enabling the simulation of the symmetry axis and the inclusion of active inner surfaces in the domain; (iii) the use of the 3D Cartesian leap-frog algorithm implemented in EP2PLUS for integrating the macroparticles trajectory, which avoids the singularity problem at the symmetry axis of a 2D cylindrical particle mover; (iv) the development of an ad hoc particle crossing check algorithm which enables the simulation of active inner surfaces in the simulation domain, which can inject a propellant mass flow with given properties into the domain, recombine collected ions into neutrals reinjected into the simulation domain, or reflect neutrals impinging on them; (v) the extension of the neutral-wall interaction algorithm for supporting mixed specular-diffuse neutral wall reflections through a constant specified probability of neutral-wall specular reflection; (vi) the incorporation of a new algorithm for the heavy species CEX collisions, which presents important improvements with respect to that of previous legacy codes such as HPHall-2 and extends the code capabilities enabling HYPHEN to simulate plasma plumes, in which CEX collisions are essential for determining the backscattering ion flux impinging sensitive S/C surfaces; and (vii) the development of a dedicated population control algorithm which, acting through all those processes generating new macroparticles in the domain (i.e. injection, collisions or wall recombination), maintains the number of macroparticles per cell of each simulated particle population within a specified range while limiting both the macroparticle weights dispersion and the computational time (when compared to particle resampling or merging/splitting approaches).
While the HYPHEN PIC formulation for the heavy species is standard for most of the aforementioned simulation scenarios, the electron-fluid module has been designed to include different models of the electron population depending on the particular physical processes driving its behavior on each case. Thus, in order to simulate near-collisionless and unmagnetized plasma plumes, HYPHEN includes the simple polytropic electron-fluid closure of EP2PLUS. On the other hand, regarding HET simulations, an electron-fluid model for the isotropic electron pressure case, named NOMADS (NOn-structured Magnetically Aligned plasma Discharge Simulator), was incorporated by Pérez-Grande. In this Thesis, an improved model formulation is presented with the main purposes of identifying the role of the electron turbulence in the different model equations and enabling a future treatment of the electron inertial effects. The contributions to the electron-fluid module for HET simulations implemented in the frame of this Thesis are: (i) a preliminary treatment of electron inertial effects through an electron drift velocity limiter; (ii) the incorporation to the electron internal energy equation of the equivalent Joule heating term corresponding to the anomalous electron transport, according to the collisional version of turbulent effects; and (iii) the improved treatment of the volumetric cathode source term in the electron conservation equations, which eliminates unphysical effects present in the original version of the code when changing the location of the MFAM cell corresponding to the volumetric cathode in the simulation domain.
With the purpose of validating and showing its capabilities, in this Thesis HYPHEN has been applied to the simulation of three different scenarios of interest. First, the simulation of a typical plasma plume expansion scenario based on an ion thruster has been considered to compare and benchmark HYPHEN against EP2PLUS. The non-trivial difficulties in the particle modeling inherent to the 2D cylindrical geometry have been satisfactorily overcome, and an excellent agreement has been found between the two codes, which are both capable of reproducing, with an acceptable noise level, the properties of heavy particle populations with densities differing by several orders of magnitude (i.e. the injected and CEX ions populations). It has been found that the optimal population control approach depends on the dynamics of the simulated macroparticle population, and it becomes essential to limit the noise level at the symmetry axis. As expected, the simulation of an axisymmetric plasma plume expansion into vacuum greatly benefits from a 2D formulation, which allows for a significant reduction of the computational time (a factor of 10) while keeping a similar PIC statistics noise level.
Second, the simulation of a simple scenario consisting of an unmagnetized plasma discharge in a surface-dominated cylindrical channel with isothermal electrons has permitted to further assess the performance of the HYPHEN hybrid code and to show the effect of the neutral-wall interaction on the plasma discharge considering the limit cases corresponding to pure diffuse and pure specular neutral-wall reflection. The main conclusions are: (i) the minimum propellant injection mass flow for which a steady and self-sustained plasma discharge is obtained greatly depends on the neutral-wall reflection process for the surface-dominated simulation domain considered; (ii) the diffuse neutral-wall reflection leads to a significant increase of the neutral residence time with respect to that of the specular reflection cases, thus enhancing the neutral ionization in the channel and explaining the existence of different neutral injection mass flow thresholds for the discharge ignition depending on the neutral-wall reflection type; and (iii) once the plasma discharge ignition takes place, it becomes dominated by the ion population, which is mostly generated through the ionization of the recombined neutrals, and the neutral-wall reflection type has been found to have a marginal effect on the discharge structure and performance.
Finally, a typical SPT-100 HET simulation scenario has allowed for a preliminary evaluation of the performance of an improved version of NOMADS for the isotropic electron pressure case, showing its capabilities and limitations. The simulations reproduce the typical HET breathing mode, although several numerical issues have been identified, which are related to (i) the large oscillations of the discharge current induced by the low frequency ionization instability, (ii) the determination of plasma properties such as the electric potential and the electron pressure (or temperature) at the simulation boundary through gradient reconstruction techniques, and (iii) the MFAM quality and the relative size of both the PIC and the MFAM cells, which may yield important interpolation errors. The neutral-wall specular reflection type has been found to increase up to a 7% the breathing mode frequency with respect to the diffuse neutral-wall reflection case. The reciprocal of the neutral gas average residence time in the thruster chamber turned out to closely approximate the resulting breathing mode frequency. Moreover, preliminary results for different electron turbulent parameter values featuring step-out profiles have shown that the breathing mode frequency slightly increases when considering both increasing turbulence in the near plume region and decreasing turbulence in the thruster chamber. The electric potential flattens in the first part of the thruster chamber for higher values of the turbulent parameter, while a larger potential fall is found for all the step-out profiles in the last part of the chamber. The peak in the axial component of the electric field increases accordingly and moves upstream into the thruster chamber in those cases. Larger thrust efficiency values, favored by the lower average discharge current values (closer to experimental ones), have been obtained for lower electron turbulent contribution inside the thruster chamber. On the other hand, a local perturbation of the plasma solution due to the concentrated electron injection performed through the volumetric cathode has been reported. The cathode magnetic line is far from being isothermal, specially near the cathode position. Similar plasma profiles inside the thruster chamber and thruster performances have been obtained when moving the cathode position along the same magnetic field streamline in the near plume region. In contrast, larger discrepancies have been found when the cathode magnetic streamline crosses the simulation domain further downstream.
The second code corresponds to a new version of the one-dimensional radial particle-in-cell model for the simulation of an annular Hall effect thruster discharge, originally developed by F. Taccogna. The model features secondary electron emission from the walls and a non-uniform radial magnetic field. With the main goals of (i) assessing the temperature anisotropy ratio of the VDF of both primary and secondary electrons combined with the asymmetries introduced by cylindrical geometry effects (which include the geometrical expansion, the centrifugal force, and the magnetic mirror), and (ii) analyzing the influence of this anisotropy and asymmetry in the macroscopic laws of interest in the steady-state discharge so as to obtain valuable information enabling a future improvement of the plasma-wall interaction models implemented in HYPHEN, the following improvements have been incorporated to the model, increasing its numerical consistency and extending its capabilities: (i) an ionization-controlled discharge (ICD) algorithm, which acting through the background neutral density, ensures a stationary discharge by compensating the wall losses through the ionization collisions in the HET acceleration region, thus maintaining the average plasma density in the domain without the need of an axial contribution (axial particle refreshing) of plasma; (ii) an extended volumetric weighting (EVW) algorithm, which, considering the information of the simulated particles during a given number of steps, allows to obtain much more reliable estimates of the relevant macroscopic magnitudes characterizing the low populated species, such as the secondary electrons emitted from the walls; and (iii) the elimination of the secular growth of ion axial energy and the subsequent refreshing of ion macroparticles by canceling the ion axial acceleration, so that new born ions are generated with a prescribed mean axial velocity instead.
Two different studies have been conducted using the improved radial PIC model. First, a reference simulation case has permitted to validate the model and analyze in detail the physics of the steady-state plasma discharge, with special focus on the radial dynamics and the VDF of the different electron populations. This study has permitted to draw the following major conclusions: (i) the weak collisionality regime typical of a HET discharge yields an important depletion of the high-radial-energy tails of the primary electrons VDF; (ii) the replenishment ratio of the high radial-energy tail of primary electrons is small, which leads to a reduced sheath potential fall; (iii) the parallel-to-perpendicular temperature anisotropy ratio is lower (greater) than one for the primary (secondary) electrons; (iv) the two secondary electron populations are partially recollected by the walls and partially converted into primary electrons (after a strong collision) in a proportion of about 60%-40%, respectively; (v) the resulting density of the secondary electrons is very low, thus being the radial electric potential profile almost exclusively shaped by the primary electrons; (vi) the radial profiles in both the plasma bulk and sheaths are asymmetric with respect to the mid-radius due to the magnetic mirror effect, which combines the temperature anisotropy and the geometric cylindrical expansion induced by the non-uniform (divergence-free) radial magnetic field, and the centrifugal force; (vii) the above cylindrical effects introduce significant differences between those magnitudes related to the plasma-wall interaction at the inner and outer walls, such as the collected electric currents, the mean impact energy (and thus the resulting SEE yields), and the wall and sheath electric potentials; (viii) the non-negligible contributions of the temperature anisotropy and non-uniformity, and the centrifugal force modify the electron radial momentum equilibrium, which deviates from the classical Boltzmann relation on electrons along the magnetic lines.
Finally, a parametric study for different field values, wall temperature and secondary electron emission, as well as the simulation of a plane case (i.e. at larger radius) have been carried out to further validate the 1D radial particle model and provide a deeper insight on the physics of the response. The main conclusions of this analysis are: (i) significantly larger SEE yields have been found in both walls when the dominant electron backscattering process is included in the model; (ii) the enhanced SEE reduces the sheath potential drops, which facilitates the electron wall collection, and could explain the decrease of the primary electron population temperature in the bulk plasma; (iii) the influence of the wall temperature for the SEE has a negligible effect on the structure of the plasma discharge; (iv) the cylindrical effects inducing asymmetries in the macroscopic profiles and in the different plasma magnitudes at the inner and outer walls turned out to vanish in the planar case analyzed, although the electron temperature anisotropy induced by the magnetic field and the wall losses in the low collisionality regime has been found to follow the same trend as in the cylindrical reference case; and (v) a smaller deviation from the Boltzmann relation has been reported for the primary electrons in the planar case, since the influence of the magnetic mirror effect and the centrifugal force tends to zero for increasing radius.
