Air::Gas

Earth air at standard temperature and pressure

Argon::Gas

Argon gas

Bismuth::Gas

Bismuth vapor

Electron::Species

Electron

Krypton::Gas

Krypton gas

Lookup for thermal conductivity coefficients, from Table 1 of S. I. Braginskii, in Reviews of Plasma Physics (1965), Vol. 1, p. 205.

Mercury::Gas

Mercury vapor

NA

Number of atoms in a kg-mol (6.02214076e26 / kmol)

R0

Universal gas constant (8314.46261815324 J / kmol K)

Xenon::Gas

Xenon gas

e

Electron charge (1.602176634e-19 Coulomb)

kB

Boltzmann constant (1.380649e-23 J/K)

me

Electron mass (9.10938356e-31 kilograms)

AnomalousTransportModel

The abstract supertype of all types of anomalous transport models. Subtype this to define your own model.

Bohm(c) <: AnomalousTransportModel

Model where the anomalous collision frequency scales with the electron cyclotron frequency ωce times some scaling factor c

Braginskii conductivity model for fully-ionized plasmasa
S. I. Braginskii, in Reviews of Plasma Physics (1965), Vol. 1, p. 205.

struct Config{A<:HallThruster.AnomalousTransportModel, TC<:HallThruster.ThermalConductivityModel, W<:HallThruster.WallLossModel, IZ<:HallThruster.IonizationModel, EX<:HallThruster.ExcitationModel, EN<:HallThruster.ElectronNeutralModel, HET<:HallThruster.Thruster, S_N, S_IC, S_IM, S_ϕ, S_E, IC<:HallThruster.InitialCondition, HS<:HallThruster.HyperbolicScheme}

Hall thruster configuration struct. Only four mandatory fields: discharge_voltage, thruster, anode_mass_flow_rate, and domain.

Fields

• discharge_voltage::Float64

• cathode_potential::Float64

• anode_Te::Float64

• cathode_Te::Float64

• wall_loss_model::HallThruster.WallLossModel

• neutral_velocity::Float64

• neutral_temperature::Float64

• implicit_energy::Float64

• propellant::HallThruster.Gas

• ncharge::Int64

• ion_temperature::Float64

• anom_model::HallThruster.AnomalousTransportModel

• conductivity_model::HallThruster.ThermalConductivityModel

• ionization_model::HallThruster.IonizationModel

• excitation_model::HallThruster.ExcitationModel

• electron_neutral_model::HallThruster.ElectronNeutralModel

• electron_ion_collisions::Bool

• min_number_density::Float64

• min_electron_temperature::Float64

• transition_length::Float64

• initial_condition::HallThruster.InitialCondition

• magnetic_field_scale::Float64

• source_neutrals::Any

• source_ion_continuity::Any

• source_ion_momentum::Any

• source_potential::Any

• source_energy::Any

• scheme::HallThruster.HyperbolicScheme

• thruster::HallThruster.Thruster

• domain::Tuple{Float64, Float64}

• LANDMARK::Bool

• anode_mass_flow_rate::Float64

• ion_wall_losses::Bool

• background_pressure::Float64

• background_neutral_temperature::Float64

• neutral_ingestion_multiplier::Float64

• anode_boundary_condition::Symbol

• anom_smoothing_iters::Int64

• solve_plume::Bool

• apply_thrust_divergence_correction::Bool

• electron_plume_loss_scale::Float64

• reaction_rate_directories::Vector{String}

ExcitationLookup(;[directories::Vector{String} = String[]])

Default excitation model for HallThruster.jl. Reads excitation rate coefficients from file. Looks (preferentially) in provided directories and in the reactions subfolder for rate coefficient files

Electron neutral collision model as defined by Eq. 3.6-13, from Fundamentals of Electric Propulsion, Goebel and Katz, 2008.

Gas

A chemical element in the gaseous state. Container for element properties used in fluid computations.

Fields

name::String Full name of gas (i.e. Xenon)

short_name::String Short name/symbol (i.e. Xe for Xenon)

γ::Float64 Specific heat ratio / adiabatic index

M::Float64 Molar mass (grams/mol) or atomic mass units

m::Float64 Mass of atom in kg

cp::Float64 Specific heat at constant pressure in J / kg / K

cv::Float64 Specific heat at constant volume in J / kg / K

R::Float64 Gas constant in J / kg / K

Gas(name::String, short_name::String; γ::Float64, M::Float64)

Instantiate a new Gas, providing a name, short name, the adiabatic index, and the molar mass. Other gas properties, including gas constant, specific heats at constant pressure/volume, and mass of atom/molecule in kg will are then computed.

julia> Gas("Xenon", "Xe", γ = 5/3, M = 83.798)
Xenon

Grid1D(geometry, z_edge)

Given 1-D edge coordinates and thruster geometry, compute cell centers and cell volumes for grid

IonizationLookup(;[directories::Vector{String} = String[]])

Default ionization model for HallThruster.jl. Reads ionization rate coefficients from file. Looks (preferentially) in provided directories and in the reactions subfolder for rate coefficient files

LANDMARK, uses 10/9 μnϵ

LandmarkExcitationLookup()

Excitation model for the LANDMARK benchmark.

Reads excitation rate coefficients from the landmark/landmark_rates.csv file in the HallThruster.jl main directory. Supports only singly-charged Xenon.

LandmarkIonizationLookup()

Ionization model for the LANDMARK benchmark.

Reads ionization rate coefficients from the landmark/landmark_rates.csv file in the HallThruster.jl main directory. Supports only singly-charged Xenon.

Mitchner-Kruger conductivity model for partially-ionized plasmas
M. Mitchner and C. H. Kruger, Jr., Partially Ionized Gases (1973). Pg. 94

MultiLogBohm(zs, cs) <: AnomalousTransportModel

Model similar to that employed in Hall2De, where the mobility is Bohm-like (i.e. νan(z) = c(z) * ωce(z)) and z is in meters.

The function c(z) is defined by a sequence of nodes (z, c) provided by the user. At z = z[1], c(z) = c[1], and so forth.

At z[i] < z < z[i+1], log(c) is defined by linearly interpolating between log(c[i]) and log(c[i+1]).

For z < z[1], c = c[1] and for z > z[end], c(z) = c[end].

The user may also provide a single array of [z[1], z[2], ..., z[end], c[1], c[2], ..., c[end]]. The number of z values must be equal to the number of c values.

NoAnom <: AnomalousTransportModel

No anomalous collision frequency included in simulation

NoExcitation <: ExcitationModel

Model for neglecting excitation energy losses

ShiftedGaussianBohm(trough_location, trough_width, trough_depth, z0, dz, alpha, pstar) <: AnomalousTransportModel

Model in which the anomalous collision frequency is Bohm-like (νan ~ ωce), except in a Gaussian-shaped region defined by the parameters troughlocation, troughwidth, troughmax, and troughmin where the collision frequency is lower. The location of the trough is based on the background pressure and the user-provided coefficients.

Arguments

• trough_min: the minimum Hall parameter
• trough_max: the maximum Hall parameter
• trough_location: the axial position (in meters) of the mean of the Gaussian trough
• trough_width: the standard deviation (in meters) of the Gaussian trough
• z0: the furthest upstream displacement permitted at high back-pressures, relative to trough_location
• dz: the maximum allowable amount of axial displacement
• pstar: the background pressure at which the shift upstream halts/plateaus
• alpha: the slope of the pressure response curve, with a higher value corresponding to a steeper pressure response

ShiftedMultiBohm(zs, cs, z0, dz, alpha, pstar) <: AnomalousTransportModel

A version of MultiLogBohm where the coefficients are shifted depending on the background pressure.

ShiftedTwoZoneBohm(coeffs, z0, dz, alpha, pstar) <: AnomalousTransportModel

Model where the anomalous collision frequency has two values: c1 * ωce before some transition location and c2 * ωce after. Takes two arguments: c1 and c2. The transition between these values can be smoothed by the user-provided transition function. The location of the transition is based on the background pressure and the user-provided coefficients.

Species

Represents a gas with a specific charge state. In a plasma, different ionization states of the same gas may coexist, so we need to be able to differentiate between these.

julia> Species(Xenon, 0)
Xe

julia> Species(Xenon, 1)
Xe+

julia> Species(Xenon, 3)
Xe3+

Thermal_Conductivity

The abstract supertype of all types of thermal conductivity models. Subtype this to define your own model.

Thruster

Struct containing information about a Hall thruster. This includes a name, geometry (a Geometry1D object), magnetic_field (radial magnetic field along centerline, a function which takes z in meters and outputs B in Tesla), and a shielded (a flag indicating whether the thruster is magnetically-shielded).

Fields

• name::String

• geometry::HallThruster.Geometry1D

• magnetic_field::Any

• shielded::Bool

TwoZoneBohm(c1, c2) <: AnomalousTransportModel

Model where the anomalous collision frequency has two values: c1 * ωce inside the channel and c2 * ωce outside of the channel. Takes two arguments: c1 and c2. The transition between these values can be smoothed by the user-provided transition function.

abstract type WallLossModel

Abstract type for wall loss models in the electron energy equation. Types included with HallThruster.jl are:

• NoWallLosses No electron energy losses to the wall.
• ConstantSheathPotential(sheath_potential, inner_loss_coeff, outer_loss_coeff) Power to the walls scales with α * exp(-sheath_potential)), where α = inner_loss_coeff inside the channel and α = outer_loss_coeff outside.
• WallSheath(material::WallMaterial) Power to the walls is computed self-consistently using a sheath model, dependent on the secondary electron emission yield of the provided material. See WallMaterial for material options.

Users implementing their own WallLossModel will need to implement at least three methods 1) freq_electron_wall(model, params, i): Compute the electron-wall momentum transfer collision frequency in cell i 2) wall_power_loss!(Q, model, params): Compute the electron power lost to the walls in array Q

A third method, wall_electron_current(model, params, i), will compute the electron current to the walls in cell i. If left unimplemented, it defaults to Ie,w = e ne νew Vcell where Vcell is the cell volume.

A fourth method, wall_ion_current(model, params, i, Z), for computing the current of ions of charge Z to the walls in cell i, may also be implemented. If left unimplemented, it will default to computing the current assuming Ie,w = Ii,w.

EvenGrid(n)

Construct an evenly-spaced grid with n cells.

UnevenGrid(n, density = HallThruster.default_density)

Construct an unevenly-spaced grid according to provided density function. Defaults to twice as many grids inside of channel than outside. Provided density functions must have a signature of (z, z0, z1, Lch) where z is the location, (z0, z1) are the extents of the domain and Lch is the channel length

allocate_anom_variables(::AnomalousTransportModel, ncells)

Allocate arrays for anomalous transport state variables. ncells is the length of the arrays to be allocated. These anomalous transport variables are then stored in params.cache.anom_variables

backward_diff_coeffs(x0, x1, x2)

Generate finite difference coefficients for a backward first derivative approximation at the point x2 on a three-point stencil at points x0, x1, and x2

julia> backward_diff_coeffs(-2//1, -1//1, 0//1)
(1//2, -2//1, 3//2)
julia> backward_diff_coeffs(-3//2, -1//1, 0//1)
(4//3, -3//1, 5//3)

backward_difference(f0, f1, f2, x0, x1, x2)

Given three points x0, x1, and x2, and the function values at those points, f0, f1, f2, compute the second-order approximation of the derivative at x2

f(x) = x^4
x0, x1, x2 = 1.9999998, 1.9999999, 2
bd = backward_difference(f(x0), f(x1), f(x2), x0, x1, x2)
bd ≈ 32

# output

true

central_diff_coeffs(x0, x1, x2)

Generate finite difference coefficients for a central first derivative approximation at the point x1 on a three-point stencil at points x0, x1, and x2

julia> central_diff_coeffs(-1//1, 0//1, 1//1)
(-1//2, 0//1, 1//2)
julia> central_diff_coeffs(-1//2, 0//1, 1//1)
(-4//3, 1//1, 1//3)

central_difference(f0, f1, f2, x0, x1, x2)

Given three points x0, x1, and x2, and the function values at those points, f0, f1, f2, compute the second-order approximation of the derivative at x1

f(x) = x^4
x0, x1, x2 = 1.9999999, 2, 2.0000001
cd = central_difference(f(x0), f(x1), f(x2), x0, x1, x2)
cd ≈ 32

# output

true

channel_area(outer_radius, inner_radius)

Compute the cross-sectional area of a Hall thruster channel from its dimensions

channel_perimeter(outer_radius, inner_radius)

Compute the perimeteter of the thruster channel, equal to the sum of the inner and outer circumferences

channel_width(outer_radius, inner_radius)

Compute the thruster channel width

compute_current(sol, location)

compute current at anode or cathode = outflow in 1D code.

coulomb_logarithm(ne, Tev, Z = 1)

calculate coulomb logarithm for electron-ion collisions as a function of ion charge state Z, electron number density in m^-3, and electron temperature in eV.

downwind_diff_coeffs(x0, x1, x2)

Generate finite difference coefficients for a downwind first derivative approximation at the point x2 on a three-point stencil at points x0, x1, and x2 (uses only points x1 and x2)

julia> downwind_diff_coeffs(-1//1, 0//1, 2//1)
(0//1, -1//2, 1//2)

electron_mobility(νe, B)

calculates electron transport according to the generalized Ohm's law as a function of sum of the classical and anomalous collision frequencies and the magnetic field.

forward_diff_coeffs(x0, x1, x2)

Generate finite difference coefficients for a forward first derivative approximation at the point x0 on a three-point stencil at points x0, x1, and x2

julia> forward_diff_coeffs(1.0, 2.0, 3.0)
(-1.5, 2.0, -0.5)
julia> forward_diff_coeffs(0//1, 1//2, 3//2)
(-8//3, 3//1, -1//3)

forward_difference(f0, f1, f2, x0, x1, x2)

Given three points x0, x1, and x2, and the function values at those points, f0, f1, f2, compute the second-order approximation of the derivative at x0

f(x) = x^4
x0, x1, x2 = 2.0, 2.000001, 2.000002
fd = forward_difference(f(x0), f(x1), f(x2), x0, x1, x2)
fd ≈ 32

# output

true

freq_electron_electron(ne, Tev)

Effective frequency at which electrons are scattered due to collisions with other electrons

freq_electron_ion(ne, Tev, Z)

Effective frequency at which electrons are scattered due to collisions with ions

freq_electron_neutral(model::ElectronNeutralModel, nn, Tev)

Effective frequency of electron scattering caused by collisions with neutrals

generate_grid(geometry, ncells)

Generate a one-dimensional uniform grid on the domain specified in the geomety. Returns number of cells, coordinates of cell centers (plus ghost cells face coordinates), interface/edges and volume of a cell for number density calculations.

interpolation_coeffs(x, x0, x1, y0, y1)

Compute the coefficients for interpolation between two points (x0, y0) and (x1, y1) such that y = c0 * y0 + c1 * y1 ```jldoctest;setup = :(using HallThruster: itpcoeffs, lerp) julia> c0, c1 = interpolationcoeffs(0.5, 0.0, 1.0, 0.0, 2.0) (0.5, 0.5) julia> c0 * 0.0 + c1 * 2.0 == lerp(0.5, 0.0, 1.0, 0.0, 2.0) true

lerp(x, x0, x1, y0, y1)

Interpolate between two points (x0, y0) and (x1, y1) ```jldoctest;setup = :(using HallThruster: lerp) julia> lerp(0.5, 0.0, 1.0, 0.0, 2.0) 1.0

Charge exchange fits from Hause, Prince and Bemish, 2013

load_reactions(model::ReactionModel, species)::Vector{IonizationReaction}

Load ionization reactions for the provided species and ionization model

maximum_charge_state(model::ReactionModel)::Int

Return the maximum supported charge state for a given reaction model. If 0 is returned, then no charge state restriction is applied.

nodes_from_density(density, x0, x1, N)

Given bounds x0, x1, a number of points N, and a density function density(x), generate N nodes betweeen x0 and x1 spaced according to the provided desity function using inverse CDF transformation.

num_anom_variables(::AnomalousTransportModel)::Int

The number of variable arrays that should be allocated for the provided anomalous transport model. These arrays are used to save state beyond the anomalous collision frequency, and are useful for defining more complex anomalous transport models. If not defined by the user, this defaults to zero.

By default, rate_coeff looks for a lookup table stored in the reaction struct

read_restart(path::AbstractString)

Load a JLD2 restart file from path.

run_from_setup(U, params; duration, nsave, verbose)

Given a state vector U and params struct generated by setup_simulation, run for provided duration, saving nsave snapshots.

run_simulation(config; duration, nsave, verbose, kwargs...)

Run a Hall thruster simulation using the provided Config object.

Arguments

• config: a Config containing simulation parameters.
• dt: The timestep, in seconds. Typical values are O(10 ns) (1e-8 seconds).
• duration: How long to run the simulation, in seconds (simulation time, not wall time). Typical runtimes are O(1 ms) (1e-3 seconds).
• ncells: How many cells to use. Typical values are 100 - 1000 cells.
• nsave: How many frames to save.

second_deriv_central_diff(f0, f1, f2, x0, x1, x2)

Given three points x0, x1, and x2, and the function values at those points, f0, f1, f2, compute the second-order approximation of the second derivative at x1

f(x) = x^4
x0, x1, x2 = 1.9999, 2.0, 2.0001
sd = second_deriv_central_diff(f(x0), f(x1), f(x2), x0, x1, x2)
sd ≈ 48

# output

true

second_deriv_coeffs(x0, x1, x2)

Generate finite difference coefficients for a central second derivative approximation at the point x1 on a three-point stencil at points x0, x1, and x2

julia> second_deriv_coeffs(-2//1, 0//1, 2//1)
(1//4, -1//2, 1//4)
julia> second_deriv_coeffs(-1//2, 0//1, 1//1)
(8//3, -4//1, 4//3)

sheath_potential(Tev, γ, mi))

compute wall sheath to be used for radiative losses and loss to wall. Goebel Katz equ. 7.3-29, 7.3-44. Assumed nₑuₑ/nᵢuᵢ ≈ 0.5 Sheath potentials are positive by convention in HallThruster.jl.

supported_gases(model::ReactionModel)::Vector{HallThruster.Gas}

Check which gases are supported by a given reaction model. If an empty vector is provided, then there are no restrictions on what gases can be used.

time_average(sol, tstampstart)

compute time-averaged solution, input Solution type and the frame at which averaging starts. Returns a Solution object with a single frame.

upwind_diff_coeffs(x0, x1, x2)

Generate finite difference coefficients for a upwind first derivative approximation at the point x2 on a three-point stencil at points x0, x1, and x2 (uses only points x0 and x1)

julia> upwind_diff_coeffs(-3//1, 0//1, 2//1)
(-1//3, 1//3, 0//1)

write_restart(path::AbstractString, sol)

Write a JLD2 restart file to path`.

This can be reloaded to resume a simulation.