Simulations of Corona and Glow Discharges with Adaptive Cartesian
Mesh
Vladimir I. Kolobov and Robert R. Arslanbekov
CFD Research Corporation, 215 Wynn Dr, Huntsville, AL 35805
Abstract: Corona discharges exist in a variety of forms (streamer and diffuse,
steady and pulsed) and have multi-scale nature (in both space and time). High
spatial resolution is required to analyze ionization fronts, streamers and Trichel
pulses observed in corona discharges. We present multi-dimensional simulations
of corona and glow discharges with adaptive Cartesian mesh using fluid models.
Effects of physical model on spatial structure of discharges are discussed.
Keywords: Adaptive Mesh Refinement, Octree Cartesian Mesh, Corona,
Streamer, Glow Discharges
1. Introduction
Fluid models have been widely used for simulations
of weakly ionized plasmas. These models utilize two
(density and velocity) or three (density, velocity, and
temperature) equations for electrons, ions and
neutral species coupled to electromagnetic solvers.
Adaptive Mesh Refinement (AMR) is crucial for
plasma problems characterized by sharp gradients of
parameters in localized areas of computational
domain. Many AMR codes have been developed for
large-scale computations in astrophysics.1 First
attempts to apply AMR to gas discharge simulations
were reported in 2 for a low-pressure ICP reactor.
Recently, AMR has been applied to simulations of
atmospheric pressure discharges with streamers,
filaments, sparks, and other dynamically evolving
spatial structures. The first paper utilizing
dynamically adaptive mesh for 2D axisymmetric
simulations of streamers 3 used an unstructured
triangular mesh and a finite element method with
flux corrected transport to analyze streamer
development from an initial Gaussian perturbation.
A similar problem was studied later using Cartesian
mesh. 4 Recently, 3D simulations of streamers with
adaptive Cartesian mesh have been reported.5,6 2D
discharge simulations using asynchronous AMR
were presented for plasma flow control. 7
2. Minimal Plasma Model
A minimal plasma model contains an equation for
the electron number density, Poisson equation for
the electric field, and equation(s) for ion number
density assuming immobile ions. Such a model has
been widely used for simulation of streamer
development assuming that the ionization rate is a
local
function
of
the
electric
field,
 i ~ exp(Bp / E) . We have implemented a plasma
model in GFS framework.8 The equations of the
minimal plasma model were solved using the finite
volume method with explicit scheme for time
marching. Electrode boundaries were represented
using the volume-of-fluid approach of GFS. The
large dynamic range of grid refinement/coarsening
(up to 10-12 levels) provided high resolution of the
streamer fronts. Good mesh adaptation was found to
be important for simulations of streamer
development in corona discharges. Our refinement
criterion was found by experimentation in the form:


ne
ne


 max(ne ) max(ne ) 
  20 
(1)
3  log10 (ne )  log10 (ni )  log10 ( i )  log10 ( E ) 
When the gradient of  exceeded a fixed value
  1 in a cell, this particular cell was refined.
When   1/ 4 in a cell, this cell was coarsened.
The procedure of mesh adaptation was performed
every 10 time steps.
In laboratories, streamers usually develop in corona
discharges in the regions of highly non-uniform
electric fields near electrodes.9,10 Figure 1 illustrates
streamer development near a rod-shaped anode in a
corona discharge. An off-axis toroidal structure was
formed in our simulations, as in 11. This
phenomenon was observed in our simulations with
more complicated physical models (ion transport,
electron thermal conductivity) and different
electrode shapes.
Figure 2. 3D simulations of a streamer formation: adapted grid
(top) and the electron densities at x-slices located at different
distance from the electrode tip at 20 ns
In all these simulations, the calculated fields near the
streamer tips were of the order of Bp, indicating that
non-local ionization should play an important role
neat the streamer fronts.
3. Ion Transport and Electron Energy
Transport
Figure 1. The computational grid (top), electrostatic potential
(center), and the electron density (bottom) for a streamer
developing near a rod shaped anode
3D simulations revealed that the 2D axi-symmetric
channels shown on Figure 1 break into separate
streamers (see Figure 2). Streamers form near the
anode surface with an almost periodic pattern and
propagate in different directions with different
speeds. Only two main streamers survive at a long
distance from the anode. The code generates about
1.5 million cells during the mesh adaptation process.
By adding ion transport to the minimal plasma
model, we simulated low-pressure discharges, which
are controlled by ion drift to the walls. To account
for electron thermal conductivity, the electron
energy transport equation was added to the model.
This model could reproduce qualitatively the basic
structure of DC glow discharges observed in
experiments.
Figure 3 illustrates the spatial distribution of plasma
parameters in a classical DC glow discharge in
Helium at a pressure of 3 Torr, tube radius R = 1 cm,
and electrode spacing L = 4 cm. An RC external
circuit is attached to the left electrode (anode) and
the right electrode (cathode) is grounded. The top
boundary is assumed to be a dielectric. The resistor
in the external circuit, Rext, was varied to change the
discharge current. Figure 4 (top) compares the axial
distributions of charged species using two models
for electrons. The first model assumes the ionization
rate a local function of the electric field. The second
model accounts for electron thermal conductivity
and calculates the ionization rate as a function of the
electron temperature.
Figure 3. Spatial distribution of electron density
kinetic theory of the cathode region 12 shows
deficiencies of the fluid model. Fluid models operate
with “an average electron” and cannot describe the
complicated structure of the electron energy
distribution function observed in the cathode region.
Furthermore, experimental observations and the
kinetic theory show that the electron temperature can
drop in the cathode region down to the gas
temperature, much lower than our fluid model
predicts.
Furthermore, we performed simulations of negative
corona in a pin-to-plane geometry for different gas
pressures. At the cathode surface the thermal flux
conditions were assumed with a secondary electron
emission coefficient  p =0.2. The discharge current
was changed by adjusting the resistor in the RC
external circuit. The results for 15 Torr, 32 µA are
shown in Figure 5 and Figure 6. The electron
temperature has a maximum of about 4 eV around
the tip of the cathode, which is about half of that
obtained in the 3 Torr case. The electric field is
monotonic at 15 Torr and changes sign for the 3 Torr
case, as in the case with the flat cathode described
above.
Figure 4. Axial distributions of electron and ion densities (top),
electrostatic potential and electron temperature (bottom) with
account of electron thermal conductivity.
This second model reproduces qualitatively the
complicated structure of the cathode region observed
in experiments. In particular, the nonmonotonic
distribution of the electrostatic potential is formed,
which corresponds to a double layer with two field
reversals in the plasma region (shown by an arrow in
Figure 4). A sharp maximum of plasma density is
observed in the vicinity of the first field reversal
near the cathode sheath. The electron temperature
reaches a minimum in the Faraday Dark Space, near
the second field reversal. Comparison with the
Figure 5. Computational grid and the electrostatic potential
(top), and electron density contours (bottom) for a glow
discharge between a rod-shaped cathode and a planar anode.
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Figure 6. The axial profiles of the electron and ion density (top),
the electron temperature and the x-component of the electric
field (zoom on the plasma region).
4. Conclusions
We have shown that fluid plasma models previously
developed for the traditional mesh techniques can be
extended for the adaptive Cartesian mesh. We have
illustrated the benefits of AMR for simulations of
corona and streamer discharges over a wide range of
gas pressures. We have shown that fluid models can
qualitatively reproduce many features of lowpressure DC glow discharges, but fail in quantitative
description of details and parts of the discharges,
which require kinetic treatment. Future work should
incorporate kinetic models for electrons in selected
areas of computational domains.
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