Twodimensional Modeling of the Hall Thruster Discharge
with Nonuniform Propellant Supply in Azimuth
Abstract
A twodimensional simulation is conducted to investigate the effect of crossfield electron transport enhancement in the Hall thruster discharge caused by the nonuniform propellant supply in azimuth. The Hall thruster operation with azimuthally nonuniform propellant supply is expected to be a good test case to understand the influences of azimuthal plasma property distributions on the axial electron transport. A particlefluid hybrid model is developed with a twodimensional magnetized electron fluid model incorporated with an empirical anomalous electron mobility model. The calculation results indicate that the azimuthal electric field is generated in the cases of nonuniform propellant supply. The azimuthal phase shift is observed between the ion number density and space potential at the downstream plume region. Owing to this azimuthal outofphase and the azimuthal electric field, the crossfield electron transport in the axial direction can be enhanced.
keywords:
plasma simulation, hybrid model, Hall thruster, EB device, electron transport1 Introduction
Hall thruster is one of the efficient electric propulsion devices used for spacecrafts. Hall thruster has an annular discharge channel with a crossedfield configuration where a radial magnetic field and an axial electric field are applied, and the Hall current is induced in the azimuthal (EB) direction. Owing to the magnetic confinement of electrons, an efficient ionization with short discharge channel is enabled and an electric field is maintained even with the plasma density of m. Research and development of Hall thruster systems have been competitively conducted in various countries including US, Europe, and Japan HamadaJSASS2017 (), and recently highly efficient thrusters with thrust efficiencies exceeding 60% have been developed. The plasma flow simulation technologies for the Hall thrusters have also been advanced, and the approach to developing Hall thrusters is shifting to the phase of computeraidedengineering. A deep understanding on the physics of the magnetized plasma in Hall thrusters remains important for accurate numerical simulations and efficient thruster developments.
The crossfield electron transport is one of the issues that is not fully understood in the physics of Hall thruster discharge. It is well known that the classical electron diffusion theory is not sufficient to account for the plasma property distributions and discharge currents obtained by experiments. To cope with this issue, most of the numerical works of Hall thrusters adopt empirical electron transport models for the agreement of simulation results with experimental data. The issue of the anomalous electron transport has been found in various crossfield devices, and the mechanism responsible for the enhanced electron transport has been investigated for decades. One theoretical mechanism proposed for the anomalous electron transport is related to the lowfrequency azimuthal plasma oscillations in 10 kHz–100 kHz, such as rotating spokes Keidar:2006aa (). The azimuthal oscillation induces an azimuthal electric field that causes the drift in electron motion in the axial direction. The existence of alone does not enhance the net electron current in the axial direction if one considers the integral of the full azimuth. However, if the electron density is also nonuniform and there is an outofphase between the electron density and space potential, the net electron current in the axial direction is changed. In other words, the axial electron transport can be enhanced if the following two conditions are satisfied: 1) the presence of , and 2) the outofphase between electron density and space potential.
Another research that is related to the azimuthal physics and enhanced electron transport in Hall thruster is the addition of artificial nonuniformities in azimuth. The azimuthally nonuniform propellant supply was tested with a 600 Wlevel thruster as shown in Fig. 1 FukushimaIEPC2009 (). In this research, the hollow anode for the xenon gas supply was azimuthally divided into four sections, and the mass flow rates for the gas inlet port in these sections were differentiated. The primary purpose of this nonuniform propellant supply was the mitigation of discharge current oscillation FukushimaIEPC2009 (). An additional important finding was that the discharge current was increased in the case of azimuthally nonuniform propellant supply whereas the ion beam current remained unchanged, compared with the case of uniform propellant supply BakIAPS2018 (). This means that the axial electron transport has been enhanced in the case of nonuniform propellant supply. This research exemplifies the electron transport enhancement due to the azimuthally nonuniform plasma properties.
The thruster operation with azimuthal nonuniform propellant supply has mainly two advantages for the investigation of crossfield electron transport. One is that the azimuthal plasma property distributions are steady so that the direct measurement of plasma property distributions is relatively easy. In the case of azimuthal plasma oscillations, the direct measurement is challenging since a highspeed measurement system is required to resolve the oscillations in 10 kHz–100 kHz. The other advantage is that the level of azimuthal nonuniformity is controllable by the differentiated mass flow rates. In the operation of azimuthally nonuniform propellant supply, the electron current changes as a function of the parameter representing the level of azimuthal nonuniformity BakIAPS2018 (). This characteristic will enable a detailed parametric study on the relationship between azimuthal plasma property distributions and axial electron transport. Therefore, the operation of a Hall thruster with azimuthally nonuniform propellant supply is expected to be a good test case to investigate the effects of azimuthal plasma properties on the crossfield electron transport.
The objective of this research is the numerical simulation of a Hall thruster operation with azimuthally nonuniform propellant supply. A numerical simulation based on a particlefluid hybrid model is performed in the two dimensions corresponding to the Efield and EB directions. We have checked the abovementioned two conditions for the electron transport enhancement in the case of azimuthally nonuniform propellant supply: 1) the presence of , and 2) the outofphase between electron density and space potential. In this paper, these points are qualitatively discussed.
2 Twodimensional Hybrid Model
2.1 Physical model and numerical methods
The physical model and computational methods for the twodimensional hybrid model are essentially consistent with the ones proposed in Ref. Kawashima:2018ab (). Ion and neutral particle flows are calculated by the twodimensionalthreevelocity (2D3V) particleincell (PIC) method. The collision frequencies are calculated by using the empirical reaction rate coefficients GoebelKatz2008 (). To simplify the calculation, only firstionization of xenon is treated as ions, and interparticle collisions between heavy particles are neglected.
The fundamental equations for electron fluid are the twodimensional electron mass and momentum conservation equations in quasineutral plasmas, and onedimensional (1D) energy conservation equation. In the mass conservation equation, the electron number density is treated as a timeconstant distribution by assuming the quasineutrality. The momentum conservation is expressed by the driftdiffusion equation obtained by neglecting the electron inertia. For a stable computation of magnetized electron fluids, the hyperbolic system approach using the pseudotime advancement technique is employed. The processes for deriving the hyperbolic system were presented elsewhere Kawashima201559 (). The space discretization is implemented by the secondorder upwind method, which has been verified for the magnetized electron fluid calculation in the coordinate of Efield and EB directions Kawashima:2018aa ().
A number of models have been proposed for accurate expression of the electron mobility in the orthogonal direction of magnetic lines of force. The most standard model for the crossfield electron mobility may be to include both the classical and anomalous electron mobilities as follows:
(1) 
where is the magnetic flux density and is the electron mobility of nonmagnetized electrons. The Bohmtype diffusion model is assumed for the anomalous electron mobility where the is an empirical Bohm diffusion coefficient. Figure 2 shows the distribution of assumed in this study, as a function of position normalized by the channel length . To select the value of , the threeregion model Hofer:2008aa () is referred in which is assumed to be 0.14, 0.02, and 1.0 at the anode, channel exit, and plume regions, respectively. To obtain a smooth distribution of , the Gaussian functions are used in the present model. The equations for these functions can be found in Ref. Kawashima:2018ab ().
2.2 Calculation target: SPT100
An annular Hall thruster similar to the SPT100 thruster is assumed as the calculation target. The SPT100 is a benchmark thruster in the Hall thruster community and several experimental and numerical researches have been conducted for this thruster. The thruster configuration and operation condition of the simulated thruster are presented in Table 1 Hofer:2007aa (); ReidIEPC2015 (). In the present 1D and 2D simulations that resolve Efield and EB directions, the radial effects including the radial gradients of plasma property and channel wall are neglected. The magnetic field is assumed to be only in the radial direction, with a distribution in the Efield direction Mitrofanova ().


Parameter  Value 
Channel centerline diameter  85 mm 
Channel width  15 mm 
Channel length  25 mm 
Total mass flow rate  5.0 mg/s 
Discharge voltage  300 V 
Anode temperature  850 K 

3 1D Simulation in Efield direction
3.1 Calculation condition
A 1D simulation in the Efield (axial) direction is conducted to validate the hybrid model. The 1D simulation is performed by ignoring the distributions and gradients in the EB direction. The calculation condition is shown in Fig. 3. The calculation domain of 80 mm is taken in the axial direction of the Hall thruster. All the xenon gas is introduced into the calculation domain from the anode side as neutral particles. The velocity distribution of the neutral particles from the anode is determined by the Maxwellian distribution function of the anode temperature. Discharge voltage is applied between the anode and cathode boundaries. A Dirichlet condition of electron temperature is given at the cathode boundary whereas a Neumann condition is given at the anode side.
3.2 Results
A simulation is continued for several milliseconds until a quasisteady state is reached. The simulated discharge current and thrust are 4.5 A and 108 mN, respectively. The onboard operation of SPT100 showed the discharge current and thrust of 4.6 A and 82 mN, respectively Manzella2001iepc (). Therefore, the discharge current is well reproduced by the simulation whereas the thrust is overestimated by 32%. The overestimated thrust is attributed to the model assumption that the channel wall effects are ignored. In the present simulation, all the ions are smoothly exhausted without colliding with the channel walls, yielding a large thrust.
The plasma property distributions simulated by the 1D model are shown in Fig. 4. The distribution tendencies of space potential , electron temperature , and ion number density are similar to the results of a previous numerical simulation Hofer:2007aa (), and the ion velocity distribution is consistent with the results of optical measurements Dorval:2002aa (). A closer look at the results proves that the peak and are 5.110 m and 67 eV, respectively. These values are larger than the typical values observed in the previous simulation and experiment Hofer:2007aa (); Dorval:2002aa (). The reason of the overestimated and is again supposed to be the assumption of no channel walls. By neglecting the channel walls, the ion and electron energy fluxes lost to the walls are assumed to be zero, resulting in the large and . Reflecting the channel wall effects to the models will be required to perform practical Hall thruster simulations.
The simulated electric field distribution is compared with experimental results Dorval:2002aa () in Fig. 5. In both the numerical and experimental results, the main acceleration region exists at the vicinity of the channel exit. The maximum electric fields obtained from the simulation and experiment are 4.410 V/m and 4.610 V/m, respectively, and the error is 4.1%. Hence the simulation result of electric field is considered to agree well with the experimental data. Since the potential is derived from the electron fluid model, this agreement indicates the validity of the present electron fluid model. In summary, although the model assumption of the neglected channel walls remains an issue for realistic simulation of the thruster, the basic characteristics of the magnetized electron flow is well reflected in the simulation.
4 2D Simulation in Efield and EB directions
4.1 Calculation condition
The calculation domain and boundary conditions for the 2D simulation in the Efield and EB directions are shown in Fig. 6. The full cylinder on the channel centerline is taken as the calculation domain. The effects of finite channel curvature are neglected, and the Cartesian coordinate is used in this simulation. The axis corresponds to the azimuthal direction and the position is normalized by the radius of the cylinder . The boundary conditions for the anode and cathode sides are consistent with those used in the 1D simulation. The periodic boundary condition is assumed on the top and bottom boundaries for both the particle and fluid calculations. Referring the preceding axialazimuthal simulation using a hybrid model Lam6922570 (), a rectangular mesh with 4848 grid is used in the present simulation to obtain a good spatial resolution with a moderate computational cost. The magnetic field is assumed to be directed to direction and symmetric in direction. The distribution of magnetic flux density is taken from the SPT100 data Mitrofanova ().
To add the nonuniformity in the mass flow density in the direction, the inflow neutral flux density from the anode is controlled by cosine functions as shown in Fig. 7. Here the strength of azimuthal nonuniformity is represented by the differential mass flow ratio defined as follows:
(2) 
where , , and are the maximum, minimum, and average neutral flux density from the anode, respectively. is used at the of and , and is used at the of 0, , and . In the case of = 0.0, the neutral particle flux density from the anode is uniform, whereas the case of = 1.0 gives the strongest azimuthal nonuniformity. In the present study, is varied as 0.0, 0.6, and 1.0. The velocity of inflow macroparticles is determined by the Maxwellian distribution function, whereas the velocity is given by the halfMaxwellian distribution with only positive values. In other words, the velocity direction is random in the hemisphere in the direction. Owing to this assumption, the inflow neutral particles diffuse in the direction until they reach the main ionization region.
4.2 Twodimensional plasma property distribution
The twodimensional distributions of ion number density and space potential obtained by the present simulations are shown in Fig. 8. In the case of uniform propellant supply ( = 0.0), the distributions are uniform in direction. The region of high ion number density is formed at around = 15 mm, as well as the result of the 1D simulation. On the other hand, in the cases of nonuniform propellant supply ( = 0.6 and = 1.0), plasma property distributions become nonuniform in direction. In the ion number density distributions, the nonuniformity becomes stronger as the differential mass flow ratio is increased. In the results of nonuniform propellant supply cases, the peaks of ion number density exist at around the of and , which correspond to the regions of high neutral particle flux densities from the anode. High neutral particle flux causes large ionization rates, resulting in the high ion number density.
The space potential distributions are also nonuniform in the direction in the cases of nonuniform propellant supply. At around = 15 mm, high potential regions are observed at of and , which correspond to the regions of high ion number density. In the case of = 0.0, the electric field has only the component, and hence the direction of EB is essentially in the direction. However, in the cases of nonuniform propellant supply, the azimuthal electric field exists and effect in the direction is generated. In the cylindrical coordinate of the annular Hall thruster, this effect corresponds to the effect in the axial direction. The electron motion is supposed to be affected by this effect in the thruster operation of azimuthal nonuniform propellant supply.
4.3 Outofphase between and
To enhance the axial electron transport in Hall thrusters, the outofphase between the electron number density and space potential distributions is required, in addition to the presence of . Owing to the quasineutral assumption, the electron number density equates the ion number density . The distributions in the direction of and space potential of the = 1.0 case are shown in Fig. 9. At the upstream region ( = 15 mm), the phases almost match between and . In this case, the same amount of electron fluxes are induced to both directions, and eventually they are cancelled out. However, at the downstream region ( = 40 mm), the phase shift is observed between and . The high regions exist in the azimuthal ranges of and , where is generated. On the other hand, at the low regions, is generated. As shown in Fig. 6, the magnetic field is in the direction. Therefore, the enhances the electron flux flowing to the upstream () direction whereas the causes an opposite effect. Eventually, the net electron flux toward the anode is increased within the full azimuth. One can consider that the azimuthally nonuniform propellant supply can enhance the axial crossfield electron transport.
The mechanism for generating the outofphase has not been elucidated. The key to understand this mechanism would be the electron transport throughout the discharge path. The similar outofphase between the and has also been reported in the rotating spokes, and its relationship with the axial electron transport has been concerned EllisonPoP2012 (). A detailed quantitative research on the effects of azimuthal nonuniformity on the crossfield electron transport is reserved for future study.
5 Conclusion
The effect of crossfield electron transport enhancement in the Hall thruster discharge with the azimuthally nonuniform propellant supply is qualitatively studied by a numerical simulation. A twodimensional hybrid model is used incorporated with an empirical anomalous electron mobility model. The model is validated by the comparison of the axial electric fields obtained by the present simulation and the experiment. Cosine functions of neutral flux density from the anode are assumed for the azimuthally differentiated propellant supply. The main findings are summarized as follows:

Nonuniform plasma properties and the generation of electric field in the EB direction is confirmed.

Outofphase is observed in the EB direction between the distributions of electron number density and space potential.

Owing to the presence of the electric field and outofphase, the crossfield electron transport can be enhanced in the Hall thruster operation with azimuthally nonuniform propellant supply.
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP17K14873.
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