A gas dynamic perspective on particle lifting in electrostatic discharge-like devices using multiphase particle-in-cell approach

Abstract

Although particle–laden electrostatic discharges are widely used in laboratories as well as in industrial applications, the mechanism of particle lifting for particles initially at rest in such highly unsteady systems is not well understood. A multiphase gas–particle solver is developed using the multiphase particle-in-cell (MP-PIC) approach to emulate the interaction of a compressible shock-dominated gas phase with the dense particle phase. First, the two-way coupled solver is initially used to study the interaction of a planar traveling shock with a vertical curtain of particulates. The gas and particle phase evolution was found to be in good agreement with a similar experimental study in Ling et al. [Phys. Fluids 24, 113301 (2012)]. Second, the MP-PIC code is used to study the interaction of an expanding blast wave with a thick bed of particles. The simulation considered forces such as quasi-steady drag, pressure-gradient, added-mass, Saffman, and Magnus forces. We observe that the vertical liftoff particles close to the shock impingement point in this configuration are associated with the quasi-steady drag, pressure gradient, and added-mass forces. Also, the Saffman lift and Magnus forces contribute to lifting particles located radially farther away from the shock impingement point. In addition, the study finds a decrease in particle lifting efficiency with decreasing plasma kernel length and shock strength.

Full text

A gas dynamic perspective on particle lifting
in electrostatic discharge-like devices using
multiphase particle-in-cell approach
Cite as: Phys. Fluids 35, 073320 (2023); doi: 10.1063/5.0158158
Submitted: 15 May 2023 .Accepted: 3 July 2023 .
Published Online: 18 July 2023
Akhil Marayikkottu Vijayan
a)
and Deborah A. Levin
AFFILIATIONS
The Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright St., Champaign,
Illinois 61801, USA
a)
Author to whom correspondence should be addressed: akhilmv@icloud.com
ABSTRACT
Although particle–laden electrostatic discharges are widely used in laboratories as well as in industrial applications, the mechanism of
particle lifting for particles initially at rest in such highly unsteady systems is not well understood. A multiphase gas–particle solver is
developed using the multiphase particle-in-cell (MP-PIC) approach to emulate the interaction of a compressible shock-dominated gas phase
with the dense particle phase. First, the two-way coupled solver is initially used to study the interaction of a planar traveling shock with a ver-
tical curtain of particulates. The gas and particle phase evolution was found to be in good agreement with a similar experimental study in
Ling et al. [Phys. Fluids 24, 113301 (2012)]. Second, the MP-PIC code is used to study the interaction of an expanding blast wave with a thick
bed of particles. The simulation considered forces such as quasi-steady drag, pressure-gradient, added-mass, Saffman, and Magnus forces.
We observe that the vertical liftoff particles close to the shock impingement point in this configuration are associated with the quasi-steady
drag, pressure gradient, and added-mass forces. Also, the Saffman lift and Magnus forces contribute to lifting particles located radially farther
away from the shock impingement point. In addition, the study finds a decrease in particle lifting efficiency with decreasing plasma kernel
length and shock strength.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0158158
I. INTRODUCTION
Interacting shockwaves with granular solid particulates is a com-
mon themein various scenarios. These interactions determine the evo-
lution of ash dust clouds in the case of volcanic eruptions.
2,3
Such
interactions are also possible in high-altitude high-speed flights,
4–6
and
reentry flights,
4,7
dust explosions,
8,9
and plume surface interactions for
inter-planetary explorations.
10–12
Furthermore, theoretical and numer-
ical studies in this regime have recently gained interest with respect to
explosion safety and mitigation efforts.
13
Therefore, there is an
increased requirement for a clear understanding of compressible gas–
particle multiphase systems. One interesting problem in the scope of
shock–particulate interaction rarely studied is the lifting of particles
from a bed exposed to blast waves. Such multiphase systems are
observed at various length scales, from laser surface cleaning (LSC)
devices
14
to near-ground thermal or nuclear explosions generating
dust clouds.
15
Such systems are also found in surface cleaning in semi-
conductor industries,
16
removing uranium oxide particles from surfa-
ces,
17
and even in cleaning antiquities.
18
Although this multiphase
configuration is ubiquitous, a clear understanding of the mechanism
of particle entrainment in these systems needs to be better understood.
In this work, we numerically study a multi-length scale, compressible
multiphase system using the multiphase particle-in-cell (MP-PIC)
approach implemented on the original FLASH
19
code framework.
Most experimental and computational studies in the past looked
at dust lifting behind a moving planar shock wave. In this configura-
tion, the particle bed is perpendicular to the planar shock wave, and
the direction of shock wave propagation is parallel to the particle bed
interface. Experimental work in this regard by researchers such as
Gerrard,
20
Borisov et al.,
21
Fletcher,
22
Bracht and Merzkirch,
23
Kauffman et al.,
24
Chowdhury et al.,
25,26
and _
Zydak et al.
27
studied the
dust cloud evolution behind a moving shock wave in a shock tube and
hypothesized various possible mechanisms for particle entrainment.
These mechanisms were sensitive to the gas and particle parameters
such as shock Mach number, particle diameter, particle material
density, and bed height. Various researchers performed numerical
simulations to understand the underlying mechanism of lifting. Such
Phys. Fluids 35, 073320 (2023); doi: 10.1063/5.0158158 35, 073320-1
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numerical works can be classified as Eulerian–Eulerian (EE) and
Eulerian–Lagrangian (EL) simulations. The gas and particle phases in
EE are modeled as continuum fluids, while in EL, the gas phase is
modeled as a continuum and the particle phase as discrete entities.
The EE simulations by researchers such as Kuhl et al.,
28
Fedorov
et al.,
29,30
Lai et al.,
31,32
and Ugarte et al.
33
studied a planar shock parti-
cle bed system but failed to incorporate the particle distribution’s poly-
dispersity.
34
In this regard, EL studies are more suited for individual
particles being simulated, and hence, polydispersity can be modeled
more realistically. In early works,
35,36
only monolayers with a small
number of particles were considered for these studies to reduce the
computational cost associated with the EL method. Note that these sys-
tems are dilute in nature and hence the effect of momentum transfer
between the gas and the particulate phase is negligible. In recent times,
denser bed cases are simulated by coupling the particle phase source
terms into the gas phase. Some of these works by Doi et al.,
37
Ilea
et al.,
38,39
and Crowe et al.
40
use different approaches for inter-particle
collisions and gas-induced force terms. The most recent works by
Shimura and Matsuo
41
and Li et al.
42
used the discrete element method
and the MP-PIC method for the planar shock wave problem. The inter-
action of shocks or compressible flow with particulates has also been
looked at length scales comparable to the particle size using particle-
resolved simulations.
43–46
These simulations are more accurate than
either the EL or the EE approaches but are restricted to a few numbers
of particulates. Therefore, for practical problems such as the ones con-
sidered in this paper, the EE and the EL approaches are widely used.
The blast wave systems considered in this study are very different
from the traveling shock wave systems discussed in the previous para-
graph. The strength of the blast decreases drastically with its propaga-
tion as compared to the shockwaves. Therefore, the strength of the
blast wave that impinges a bed vertically below the blast source varies
in space and time. The gas pressure gradient behind the moving blast
wave decreases gradually as opposed to the sudden decrease in the
case of a planar shock wave. These differences in the gas dynamics of
the blast wave evolution from the planar shock case imply different
mechanisms of dust entrainment or lifting behind the impinging blast
wave. Therefore, the current work explores the configuration of a blast
wave generated by a detonation interacting with a bed of particles rep-
resenting the ground. The general mechanism of lifting or entrainment
of particulates in this numerical system should be able to provide an
understanding of dust lifting in small-length scale devices such as
ESDs as well as large-scale near-surface explosions.
In the present work, we develop and use a numerical approach
for modeling the evolution of particle-dense plasma–particulate sys-
tems in laboratory electrostatic discharges (ESD) such as studied by
Mukhopadhyay et al.
47
The high-temperature plasma kernel generates
unsteady blast waves interacting with a bed of particles placed verti-
cally below the channel, causing particle lifting. The EL method based
on the MP-PIC approach used in this work incorporates surrogate
force models compatible with the underlying compressible carrier gas.
Our work studies the evolution of inert, non-reactive spherical par-
ticles emulating dust grains placed as a bed on a substrate or wall
directly below the plasma channel created by the ESD. We also look at
the effect of ESD parameters, such as the energy of discharge and
length of the plasma channel, on the dust lifting characteristics. Note
that, in our previous work,
48
we studied dust lifting from a monolayer
exposed to an ESD discharge. The work identified the gas velocity
gradient induced Saffman lift force as the predominant lifting mecha-
nism for the multiphase system and was in good agreement with the
experimental results of Huang et al.
49
The outline of the rest of the paper is as follows: Section III dis-
cusses the multiphase particle-in-cell (MP-PIC) approach and its
implementation in the FLASH research code.
19
The surrogate force
models used in the study are given in detail in this section. Section III
studies the interaction of a planar shock wave with a thin curtain of
particles and comparesthe numerical results with previous experimen-
tal as well as other numerical simulations. The interaction of expand-
ing blast wave with a dense bed of particles is studied in Sec. IV.The
section discusses the general mechanism of particle lifting behind a
blast wave and also considers the effect of shock strength and source
geometry on particle lifting. The main results from the studies and the
major differences between the shock–curtain and blast–bed interac-
tions are discussed in Sec. V.
II. IMPLEMENTATION OF MULTIPHASE PARTICLE-IN-
CELL (MP-PIC) APPROACH IN FLASH
19
RESEARCH
CODE
The MP-PIC method proposed initially by Andrews and
O’Rourke
50
solves the gas phase on the discretized grid, assuming a con-
tinuum phase assumption. The particle phase is represented as discrete
entities embedded in the continuous Eulerian phase. For an Eulerian cell
containing particulates, the fraction of the total cell volume occupied by
the particle phase or the volume fraction his mapped onto the Eulerian
grid to facilitate two-phase coupling. Correspondingly, the gas volume
fraction for the same cell is given as e¼1"h.Thegoverningequation
of the gas, or fluid phase of the multiphase system, is given as follows:
@ðeqfÞ
@tþr&ðeqfufÞ¼0;(1)
@ðeqfufÞ
@tþr&eq fufufþrP"eqfg¼"Fs;(2)
@ðeqfEÞ
@tþr&ðeq fEþPÞufþP@e
@t"eq fuf&g¼Es;(3)
where q
f
,uf,P,E, and grepresent the gas density, gas velocity, pres-
sure, energy, and the acceleration due to gravity, respectively. The
momentum and energy source terms due to particle phase back cou-
pling are represented by Fsand E
s
, respectively.
The evolution of the particle distribution function /is given as
follows:
@/
@tþr&ð/upÞþr
up&ð/AÞ¼0;(4)
where /ðx;up;tÞis the probability distribution of particles with parti-
cle velocity upand the inter-phase acceleration terms A, and xand t
represent space and time, respectively. The particle distribution /
evolves in the system as individual particles in the domain are moved
based on the solution of Newton’s equation of motion. The accelera-
tion term is represented by the sum of several surrogate or sub-grid
force models as follows:
mA¼FDþFSaff þFMag þFpg þFgran þFthermo þFgravity þFam;
(5)
where mis the mass of the particulate, and FDrepresents the drag
force generated on a particulate due to non-zero relative velocity of the
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20 July 2023 13:17:53

particle with the gas flow in its vicinity. Since the systems under con-
sideration emulate dense as well as dilute particulate densities, the
drag formulation by Tenneti et al.
51
for dense regime and Loth
52
for
dilute regime are used with switching criteria of 5% volume fraction
marking the transition from dense to dilute. Detailed description of
the drag model is given in Appendix. Since the particulates in the sys-
tem are exposed to compressible and rarefied gas regimes, the model
drag coefficient by Loth,
52
which efficiently captures these effects
across wide ranges of gas compressibility and rarefaction, is used. The
Saffman lift force (Saffman
53
), represented by FSaff , is the lateral lift
force generated by shearing flow in the vicinity of the particle.
Pressure gradient and thermophoretic forces, represented by Fpg and
Fthermo, respectively, are generated on the particulate due to pressure
and temperature gradients in the flow. The inter-particle granular
force developed in the particle bed due to its compaction is repre-
sented by the granular stress force term Fgran. Finally, the gravitational
force on particulates, though negligible, is given by Fgravity. The force
models used in the Lagrangian solver are summarized in Table I, and
a detailed description is provided in Appendix for completeness. Note
that the individual computational particles in the system represent n
number of real particulates. This course-graining is required to reduce
the computational cost associated with the method.
A Cartesian AMR grid is used by the Lagrangian particle and the
Eulerian fluid/gas solvers. The fluid phase volume fraction eis com-
puted on the particle field by spatially integrating the particle distribu-
tion with respect to u
p
as follows:
eðx;tÞ¼1"hðx;tÞ¼1"ðu
/dup:(6)
Therefore, the temporal evolution of the particle distribution /
through Newton’s equation also evolves ein time. The phase volume
fraction and the momentum and energy source terms ðFsand EsÞ
from the particle solver are communicated to the Eulerian fluid solver
through cloud-in-cell (CIC) mapping provided in the original frame-
work of FLASH
19
code. The fluid or Eulerian macroparameters and
gradients required to compute the particle acceleration term Ais also
mapped to the particle or Lagrangian solver using the same shape
function or weights. The mapping scheme, also called the cloud-in-cell
(CIC), uses constant weights to map particle quantities to the Eulerian
grid in a radius of 1 cell with the particle containing cell as the center.
The weights for a one-dimensional mapping is W¼[wi"1¼1=4;
wi¼1=2;wiþ1¼1=4], where icorresponds to the cell containing the
particle. Higher dimensional weights are obtained by using outer prod-
ucts of this linear stencil, i.e., weight in three dimension
¼W'W'W.
The momentum source term representing the net momentum
transferred to the fluid due to the particle acceleration in the fluid is
given for N
p
representative particles residing in a fluid cell of volume
V
c
as follows:
Fs¼n
VcX
Np
i¼1
miA0
i;(7)
where m
i
is the mass of the particle i, and A0
iis the acceleration of the
particle iexcluding the granular stress given in Eq. (A20), and gravita-
tional term. Note that nis a constant for all the particles in the system.
Here, A0
iis given as follows:
A0
i¼ðFD;iþFSaff ;iþFMag;iþFpg;iþFthermo;iþFam;iÞ=mi;(8)
where the subscript iidentifies the ith particle. The granular stress
term models the momentum exchange within the granular or particle
phase and does not couple back to the fluid. Similarly,the gravitational
term is a body force term that does not couple back to the fluid. The
term is numerically integrated over a time step tto obtain the source
term for the FLASH
19
Riemann solver. This is consistent with the pre-
vious Eulerian–Lagrangian modeling for compressible gas–particle
multiphase systems by Dahal and McFarland.
56
The energy source term in this implementation is due to the
work done by the particle force term on the fluid. Note that the heat
addition due to sensible heating or cooling of the particulates is not
coupled to the Eulerian solver since their magnitudes were found to be
negligible for the applications under consideration, as was also shown
by Ling et al.
57
for particle laden blast wave systems. This term is also
TABLE I. Mathematical expressions and causes of various forces acting on the particles.
Force model Causes Mathematical formulation
Drag force
52
Relative motion between particle and gas phase FD¼qf
2CDAðuf"upÞjuf"upj
Saffman force
53
Driven by gas shear FSaff ¼Cs
4d2
pðlfqfÞ1=2jxfj"1=2½ðuf"upÞ)xf*
Magnus force
54
Driven by gas rotation FMag ¼p
8d3
pqf½ðuf"upÞ)xf*
Pressure-gradient force Pressure gradient in the gas Fpg ¼ "r&PVp
Collisional force
55
Inter-particle collisions Fgran ¼1
qphprsp
Thermophoretic force Temperature gradients in gas Fthermo ¼"p
2lf!fdp
Kn rTf
Tf
"#Kn1:7
1:15 þKn1:7
Added-mass force
1
Relative acceleration between particle and gas Fam ¼VpCMðMa;hÞ "rP"dðqfupÞ
dt
$%
Gravity Gravitational field Fgravity ¼mpg
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modeled for a fluid cell of volume V
c
as a source term in the FLASH
19
Riemann solver by numerically integrating the work done by N
p
par-
ticles on the gas (E
s
) over the time step tas follows:
Es¼n
VcX
Np
i¼1
miA0
i&up
i;(9)
where up
iis the velocity of particle i. Since the gas temperature is high
in blast wave systems, the particles are exposed to non-negligible con-
vective heat flux. The quasi-steady-state heat convection Qqs between
the gas and spherical particles is given as follows:
Qqs ¼pdpkgðTg"TpÞNu;(10)
where k
g
is the thermal conductivity of air in the vicinity of the partic-
ulate with temperature T
p
and diameter d
p
. The Nusselt number Nu
quantifying the heat transfer is given by Fox et al.
58
as follows:
Nu ¼2 exp ð"MaÞ
1þ17Ma=Re þ0:459Pr0:33 Re0:55 1þ0:5 exp ð"17Ma=ReÞ
1:5;
(11)
where Pr is the Prandtl number. This model was selected since it cap-
tures the effect of gas rarefaction on heat transfer on spherical particles
through Knudsen number corrections.
III. INTERACTION OF A PLANAR MOVING SHOCK
WAVE WITH A CURTAIN OF PARTICLES
The newly developed MP-PIC solver is first used to study the
interaction of a planar shock wave with a thin curtain of particles.
Researchers such as Ling et al.,
1
Sugiyama et al.,
59
and Theofanous
and Chang
60
have studied this system before using various numerical
approaches. Comparison of our simulation results with the existing lit-
erature for this canonical problem is essential in establishing the gen-
eral features of a shock–particle interaction study.
A. Computational setup
Figure 1 shows the three-dimensional domain for the multiphase
shock tube problem. The simulation domain has a total length of
0.88 m ¼b1þb2þb3þLand a square edge length a¼0.08m. The
high-pressure driver gas (air) is contained in the purple-colored por-
tion of the tube with b
1
¼0.1 m. An outflow boundary condition is
used at the left (x¼0 m) and right (x¼0.88 m) boundaries, while all
other boundaries are reflective. The particle curtain is initialized as a
thin sheet of width L¼2 mm at a distance of b
2
¼0.65 m from the
shock tube diaphragm at x¼b
1
filled with a particle curtain containing
spherical particles of diameter 115lm with a material density of
2520 kg/m
3
. The particle diameter (d) corresponds to an approximate
particle Knudsen number (Kn ¼k=d)of5)10"4for an air mean
free path (k) of 60 nm. Each representative computational particulate
(or parcel) in the simulation represents 26 real particles (n¼26). The
particle volume fraction of the curtain is initialized to be 21% as previ-
ously calculated by Ling et al.
1
for their one-dimensional simulations
of the same system. The driven section of the tube is maintained at a
pressure of 0.0827 MPa, while the gas temperature of both the sections
is kept constant at 296.4K. The driver section is initialized at a pres-
sure of 0.98132 MPa to generate a Mach 1.66 flow similar to the case
given in Ling et al.
1
The shock tube length is smaller than the original
one-dimensional study of Ling et al.
1
to reduce the computational
requirements of our full three-dimensional rendering of the one-
dimensional problem. The computational domain is discretized using
the PARAMESH
61
library to a maximum AMR level of three and
block (The PARAMESH library discretizes the domain into three-
dimensional blocks. Each block is further divided into cells. These
blocks are the fundamental computational units in the FLASH code.)
size of 20 )20 )20. The minimum size of a fluid cell is
Dx+1)10"3m. A fifth-order Harten-Lax-van Leer Contact
(HLLC)
62
Riemann solver is used to solve the fluid governing equa-
tion. The CFL criteria for the gas solver are 0.7 and the particle motion
is restricted such that jupjDt=Dx<0:05 to maintain stability, while
the two-way coupling terms are activated. For this study, the effect of
gravity, Saffman force, Magnus force, and the thermophoretic forces
are not considered in order to be consistent with the previous compu-
tation of Ling et al.
1
B. Evolution of the system of shock waves and particle
curtain
For a shock Mach number Ms¼1:66, the shock speed is given
as us¼Ms&a¼573:31 m/s, where a¼345.37 m/s is the speed of
sound in the medium (air) at 296.4 K. For the current configuration of
the three-dimensional multiphase shock tube, the shock takes
b2=us+9:767 )10"4s to reach the particle curtain. The contact dis-
continuity behind the shock has a velocity of +304:5m/s, which is
also the post-shock gas velocity u
ps
. This configuration gives us a time
window of +0:014 s to analyze the interaction of the shock with the
particle curtain. In this time window, the right-moving contact discon-
tinuity (CD) does not interact with the left-moving reflected shock
(RS) and the expanding particle curtain (PC).
The temporal evolution of the multiphase shock tube is shown as
numerical schlieren in Fig. 2 at different time instances by generating
density gradient maps. The blue-colored contour levels indicate the
evolution of the particle cloud as it spreads and moves slowly to the
right. We define a normalized timescale t,¼ðt"t0Þ=ðL=usÞwith
origin at time t0¼0:9767 ms corresponding to the initial interaction
FIG. 1. Simulation domain for the three-dimensional multiphase shock tube
problem.
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of the incident shock (IS) with the particle curtain (the physical time t
of the simulation can be obtained from the normalized time t,as
t¼t,L=usþ0:9767ms). The recentered time is scaled with
L=us+3:49 )10"6s, where L¼2 mm is the thickness of the particle
curtain and u
s
is the speed of the incident shock that is a constant. The
incident shock (IS) wave interacts with the particle curtain (PC) at a
normalized time of ðt"t0Þ=ðL=usÞ¼0 and splits into a reflected
shock (RS) that travels in the negative xdirection and a transmitted
shock wave (TS) traveling in the positive xdirection. The reflected
shock is much weaker than the transmitted shock and travels
upstream with a velocity approximately 50% of the transmitted shock.
The shock refraction that occurs on the upstream and downstream
sides of the particle curtain creates a negative pressure gradient [as
shown in Fig. 3(a)] and a positive velocity gradient [as shown in Fig.
3(b)] along the xaxis, which causes the particle curtain to expand as
time progresses. A particle wake (PW) is generated behind the particle
cloud as the gas interleaves the particle curtain as shown in Fig. 3(c).
The phase plot shown in Fig. 4 compares the simulation results
with experimental results reported in Ling et al.
1
The x–t diagram is
centered at the particle curtain location, i.e., x¼0.66 m, normalized
with the curtain thickness L¼2mm. Sincethe experimentallyobtained
trajectory for the transmitted shock (TS) is limited to ðt"toÞ=
ðL=usÞ<30 due to a limitation in the experimental viewing window,
the simulation results for TS are compared with an extrapolation of the
FIG. 2. Evolution of the multiphase shock tube at physical time t¼0.9, 1.05, 1.20,
and 1.425 ms after the shock initialization. Subfigures (b)–(d) correspond to normal-
ized times of t,¼ðt"t0Þ=ðL=usÞ¼21;64, and 130, respectively. The user-
defined coloration for the gas phase identifies the contact discontinuity (CD) as a
bold black band. The reflected shock (RS) is marked by a think black band, while
the transmitted shock (TS) is identified using a white band. The particle curtain
(PC) is identified as a blue band.
FIG. 3. The variation of (a) dp/dx, (b) du/
dx, and (c) gas vorticity in z direction x
z
in the vicinity of the particle curtain at
t¼1.05 ms after shock initialization. Here,
the PC is at 0:660 <x<0:662 m.
FIG. 4. Temporal evolution of gas and particle features in the three-dimensional
multiphase shock tube simulation. The numerical evolution of reflected shock (RS)
and transmitted shock (TS) is marked by square and cross, respectively. The upper
particle front (UPF) is marked with diamond and circle for with and without added-
mass force term, respectively. The lower particle front (LPF) is marked with dia-
mond and circle for with and without added-mass force term, respectively. The
corresponding dashed lines are experimental results given in Ling et al.
1
Here,
ðt"t0Þ=ðL=usÞ¼0(y
1
axis) corresponds to the instance of shock (IS) particle
curtain interaction. The y
2
or secondary axis shows the physical time in milliseconds.
The x=L¼0 location corresponds to x¼b1þb2¼0:75 m of Fig. 1. The pink line
shows the extrapolation of the experimental trajectory for TS.
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experimental trajectory for ðt"t0Þ=ðL=usÞ>30 (pink line). Our sim-
ulation results are generally in good agreement with the measurements
given in Ling et al.
1
The discrepancies in the particle phase evolution are
possibly due to the differences in the surrogate force models being used
in the two studies as well as the difference in the modeling of the dimen-
sionality of the problem. The three-dimensionality of the current simu-
lations tends to generate drafts in the lateral (y and z) directions that
reduce the x-directional forcing on the particle curtain. The inter-
particle aerodynamic interaction in the system under consideration is
modeled using the volume fraction-based first-order drag correction
formulation given in Eq. (A11).However,inreality,theseaerodynamic
interactions generate a distribution of drag as well as lateral lift forces on
the particle system and the standard deviation of the lateral lift forces
can be of the same order of magnitude as the average drag force itself.
63
These second-order effects that are not currently modeled in the system
when considered can accelerate a fraction of the particles in the curtain
to larger lateral displacements as compared to the particle curtain dis-
placement predicted in the current work. The importance of the added-
mass term in simulating the particle curtain evolution, as shown by Ling
et al.,
1
is also seen in our study by comparing the particle curtain evolu-
tion with and without the added-mass term in Fig. 4.
Finally, the temporal variation of the active force terms (F
D
,F
pg
,
F
gran
,andF
am
) on the entire particle system is shown in Fig. 5.Ascan
be seen from the figure, the dominant force in the system is the drag
force. The magnitude of the average drag force peaks during the shock
curtain interaction and decays with time as the particles approach gas
velocities. The effect of pressure gradient force and the added mass
force are also important in the current configuration. Both these forces
decrease continuously as the shock moves away from the curtain. The
collisional force Fgran given by the empirical model of Harris and
Crighton
55
does not contribute significantly to the evolution of the
particle phase as was also noted in the numerical work of Ling et al.
1
The magnitude of the drag force F
D
dominates followed by the added-
mass force F
am
and the pressure-gradient force F
pg
. The added-mass
force F
am
and the pressure-gradient force F
pg
decrease more gradually
compared to F
D
.
IV. DUST LIFTING BY AN EXPANDING BLAST WAVE
INTERACTING WITH A BED OF PARTICLES
The interaction of an expanding blast wave with a thick bed of
particulates is studied in this section. A high-energy plasma channel
generates the blast wave, and the particle bed is used as a surrogate for
dust layers in the case of ground or contaminated surfaces. The general
characteristics of gas and particle phase evolution seen in the simula-
tion results will be discussed in this section.
A. Computational setup
Figure 6 shows the three-dimensional domain used in this study.
The base of the computational domain has a square footprint with an
edge length of a¼10 mm with a particle bed of height h
p
¼0.6 mm.
The cylindrical high-temperature gas charge of surface X, radius
R
c
¼0.5 mm, and length L
c
is positioned at a height of h
c
. For all the
studies considered in this paper, the simulation domain height h¼5
and hc¼2:5 mm. Note that the ends of the charge are closed with a
hemispherical cap to avoid sharp discontinuities in the initialization.
The pressure, density, and temperature of the gas charge are
FIG. 5. The variation of (a) drag, (b) pressure gradient, (c) collisional force, and (d) added-mass force with time. The curves show the evolution of the average force with time.
Standard deviations of the force within the distribution of particles are shown using error bars.
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represented by P
c
,q
c
,andT
c
, respectively, and the ambient pressure,
density, and temperature are indicated by P
0
,q
0
,andT
0
, respectively.
The ambient pressure and temperature for all the cases considered
here are maintained at 1 atm and 300 K, respectively.
The initialization of the bed of particles in the simulation setup is
an essential step in maintaining the stability of the MP-PIC implemen-
tation. A uniform initialization of the particle distribution is required
to maintain a smooth or continuous mapping of the particle proper-
ties, such as e;Fs,andE
s,
to the Eulerian grid, which in turn is required
in the Eulerian solver to avoid spurious numerical oscillations of
shocks in the gas field solution. Therefore, particles are placed at the
fluid cell centers to maintain a continuous bed in the slab shown for
0-x-a;0-z-a,and0-y-hpin Fig. 6. The typical number
of particles used in a 40% particle bed is +4)10
6
. The Eulerian com-
putational domain is discretized using the PARAMESH
61
library to a
maximum AMR (adaptive mesh refinement) level of four and block
size of 20 )20 )20. The adaptive mesh refinement strategy typically
discretizes the computational domain into +165 000 cells. A fifth-
order HLLC
62
Riemann solver provided with the original framework
of FLASH
19
code is used to time integrate the fluid equations. The
CFL criteria for the gas solver are 0.7, and the particle motion is
restricted so that jupjDt=Dx<0:05. The solver uses the minimum
value from the time step restriction imposed by the Eulerian/fluid and
the Lagrangian solver for the time accuracy of the predicted solution
for a total time of 16 ls after spark initialization for all the cases con-
sidered in this paper.
The second challenge is achieving a high particle bed volume
fraction in the simulation. High volume fraction requires increasing
the nof the represented particles in the system. For light particles, nis
much higher than for larger particles corresponding to the same bed
volume fraction. Since the light particles readily equilibrate with the
gas by transferring momentum into the gas, the source terms [Fsand
E
s
in Eqs. (2) and (3)] generated by light particles with high nare very
large. Such large source terms (energy and momentum) can cause
divergence in the base Riemann solver. Therefore, a high-volume frac-
tion bed cannot be created solely by using light particles. We use a bi-
disperse bed with large and small particles to circumvent this problem.
The large particles dictate the volume fraction required for the bed
while selecting a global nnecessary to maintain the high particle bed
volume fraction. Since the response time of these large particles is
much higher than the flow evolution time, the instantaneous source
terms from the larger particles do not impact the stability of the base
solver. The particle bed is generated using a bi-disperse distribution of
1 and 23 lm particles for the cases considered in this study. The
approximate particle Knudsen number corresponding to 1 and 23 lm
particles is 0.06 and 0.003, respectively, based on a gas mean free path
of 60nm for atmospheric air. For the cases discussed in this paper,
n¼2 to maintain a bed volume fraction of +40%. The particles are
assumed to have a specific heat of 840J/kgK and are initialized with a
temperature of 300 K corresponding to the ambient temperature T
0
.
The geometric parameters, such as the length L
c
, and physical
parameters, such as charge pressure and temperature P
c
and T
c
of the
cylindrical charge, are varied while keeping the simulation domain
dimensions and particle bed composition the same as the base case to
achieve different shock strengths and shock configurations. The values
of these specific parameters are summarized in the Table II.
B. General evolution of the ESD multiphase system
As mentioned earlier, the interaction of a blast wave generated by
the plasma kernel with a bed of particles is a different scenario com-
pared to the planar shock–particle curtain interaction discussed in the
Sec. III. The expansion of the blast wave is associated with a continu-
ous decrease in its strength as it propagates. In the case of planar
shocks, the shock strength is constant, and therefore, the particle
FIG. 6. Simulation domain for the particle bed case.
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curtain was exposed to a constant post-shock gas velocity
ups +304:5 m/s behind the planar shock wave. Due to the spatial and
temporal variation of shock strength for the blast wave, the particles in
the bed are exposed to spatially and temporally varying gas velocities.
The mechanism of particle lifting due to these differences in the gas
phase evolution for the ESD system is not well understood, and the
timescales of the evolution of the gas and particle phases in this multi-
phase system are very different due to the disparity in their masses.
The evolution of the ESD-like system exposed to a bed of par-
ticles can be broadly classified based on the chronology of the events
as: (1) plasma kernel expansion and the emanation of blast wave, (2)
shock interaction with the particle bed and particle lifting, and (3)
reflected blast interaction with the plasma kernel, as will be discussed
in more detail below. To study the evolution of an ESD-generated blast
with a bed of particles, we choose a plasma kernel of length L
c
¼2 mm
and pressure ratio Pc=P0¼Tc=T0¼16:67 as our base case, as indi-
cated in Table II, since this case is geometrically similar to the experi-
mental setup provided in Mukhopadhyay et al.
47
Figure 7 identifies the gas phase features of the ESD system at an
intermediate time after the main shock (MS) generated by the expand-
ing plasma kernel (PK) has interacted with the particle bed interface
II0. The reflected shock at the interface and the portion of the shock
transmitted into the particle bed are identified as RS and TS, respec-
tively. The reflected shock interacts with the expanding PK generating
a diffracted shock DS as shown in the figure. The figure also indicates
the initialized plasma kernel surface as Xand the shock impingement
point S.
Furthermore, Table III summarizes the time scales associated
with the gas and the particle phases in the ESD multiphase system.
From this table, it is evident that there are multiple length scales asso-
ciated with the overall evolution of this system.
1. Plasma kernel expansion and emanation of blast
wave
In the early stages of system evolution, sg¼act=Rc<1, a shock
wave (indicated using dashed lines in Fig. 8) and an expansion fan
(identified using star symbol in Fig. 8) are generated at the surface X
of the initialized plasma kernel. Here, a
c
and R
c
represent the speed of
sound in the high-temperature kernel and the kernel radius, respec-
tively. s
g
represents the timescale at which the expansion fans gener-
ated at the interface Xof the plasma kernel reach the center of the
kernel. For the base case, sg¼0:56 <1. This expansion fan cools the
plasma kernel as it propagates into the kernel. The main shock moves
radially outwards solely by the shock–area relation approximated by
Friedman et al.
64
Figures 8(a) and 10(a) show this stage of gas phase
evolution. A line extraction of the gas pressure and temperature at
time t¼0:2ls shown in Fig. 9 corresponds to the early stages of sys-
tem evolution and identifies the expansion fan propagating inwards
FIG. 7. Diagrammatic representation of gas phase features in the ESD multiphase
system at a time instance after shock particle bed interaction.
TABLE II. Computational cases considered in this study. For all the cases consid-
ered in this study, R
c
¼0.5 mm, P
0
¼1 atm, and T
0
¼300 K. The particle bed for all
the cases considered here has a nominal volume fraction of 40%, and each particle
has a density of 2500 kg/m
3
.
Parameters Base case Case 1 Case 2 Case 3 Case 4
L
c
2 mm 4 mm 1 mm 2 mm 2 mm
Pc=P016.67 16.67 16.67 33.33 6.67
Tc=T016.67 16.67 16.67 33.33 6.67
M
i
2.59 2.59 2.59 3.08 1.96
TABLE III. Table summarizing the different time scales associated with the base case ESD system.
Time scale Description Approximate magnitude
sg¼act=RcNormalized timescale of kernel evolution. Defined as the time taken by the
expansion fan generated at the kernel surface Xto reach the kernel center. +0:56
sMS ¼Mia0t=ðhc"RcÞNormalized timescale of main shock (MS) propagation. Defined as the time
taken by the expanding main shock (MS) to reach the particle bed interface II0+0:86
sRS +sMS Normalized timescale of reflected shock (RS) propagation. Defined as the time
taken by the reflected shock (RS) to reach the kernel surface Xfrom the inter-
face II0
+0:86
tint ¼dp=ðMia0ÞTime scale at which the expanding main shock (MS) interacts with the particle
of diameter d
p
.+OðnsÞ
tp+sV¼qpd2
p=18lThe timescale of particle phase evolution. This timescale is of the same order of
magnitude as the momentum relaxation time s
V
of the particles. +OðlsÞ
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into the plasma kernel as well as the shock front propagating outward
into the ambient gas. The figure clearly shows the decrease in the core
kernel pressure and temperature due to its interaction with the expan-
sion fan.
Since the particle force terms such as the Saffman lift and
Magnus forces given in Table I are also dependent on the gas vorticity
in the vicinity of the particle, a good understanding of the gas vorticity
evolution in space and time is required to evaluate the evolution of the
gas particle multiphase system under consideration. Therefore, the
mechanism of gas vorticity generation in the vicinity or at the surface
of the kernel as well as the whole multiphase system is evaluated by
looking at the gas vorticity equation obtained by taking the curl of the
momentum equation [Eq. (2)]offluidasfollows:
Dxf
Dt ¼ðxf&rÞuf"xfðr&ufÞþrqf)rPf
ðqfÞ2þr)Fs;(12)
where x
f
and u
f
are the gas vorticity and velocities, respectively. In the
early stages of system evolution shown in Fig. 10, the gas vorticity gen-
eration is only due to the baroclinic vorticity source term through the
misaligned pressure and density gradients. At time t¼0.2 ls(sg<1),
shown in Fig. 10(a), the shock and the plasma kernel surface Xare
close. As the plasma kernel surface expands radially outward and then
decelerates, gas vorticity is deposited along the expanding plasma ker-
nel surface as shown in Figs. 10(b) and 10(c) for sg>1. At later stages
of kernel evolution, the gas vortices penetrate the kernel collapsing the
same and generating ejecta as observed in the sequence of Figs.
12(b)–12(d) of our previous work.
48
At later stages of system evolu-
tion, when the main shock interacts with the particle bed, an addi-
tional source of vorticity generation due to particle momentum source
term (fourth term of RHS) becomes important.
2. Shock interaction with the particle bed and particle
lifting
The radially expanding blast wave (main shock) moves toward
the particle bed at non-dimensional timescales of s
MS
. The time of
propagation of the main shock from the kernel surface Xis normal-
ized with ðhc"RcÞ=Miaoto obtain an approximate value of s
MS
,
where the numerator defines the distance between the boundary of the
plasma kernel and the interface II0and the denominator is the initial
speed of the shock at the boundary Xof the plasma kernel shown in
Fig. 6. Here, the initial Mach number of the blast front M
i
is approxi-
mated using the planar shock relationship at the kernel boundary X
and a0¼ffiffiffiffiffiffiffiffiffiffi
cRT0
pis the speed of sound in the ambient air (Rand care
the gas constant and ratio of specific heats for air). For the base case,
sMS +0:86 for the main shock to reach the interface II0,whichis
greater than the kernel evolution time defined in Sec. IV B 1
sg+0:56. Figure 11 shows the interaction of the expanding main
shock (MS) with the particle bed at the interface II0at time t¼4ls
(sMS ¼2:59) after plasma kernel initialization. The y-directional gas
velocity contour given in Fig. 11(a) suggests that the gas velocity is
negligible through the bulk of the bed, and only particles at the inter-
face II0experience significant relative velocity or slip velocity
(uf"up) capable of generating substantial drag force FD. We can also
see that a considerable positive gas velocity of Vy+250 m/s is
observed close to the shock impingement point Sbehind the reflected
shock at the interface. In the same region, the gas pressure gradient
dp/dy shows a slightly negative value in Fig. 11(b). The figure also
shows a high positive magnitude of y-directional pressure gradient dp/
dy close to the foot of the main shock interacting with the interface II0.
The interaction of the expanding blast wave with the interface also
develops a vortex sheet at the gas–particle bed interface II0
(5.4 mm <x<8 mm) as shown in the x
z
contours in Fig. 11(c).The
vorticity generation is through the third and the fourth terms of Eq.
(12). The misalignment of the pressure and the density at the interface
contributes to the second term, while the particle source term Fscou-
pling back to the fluid solver introduces a new source for vorticity gen-
eration. The strength of gas vorticity in the vicinity of the shock
impingement point Sis negligible as the pressure gradient rPand the
density gradient rqof the gas are approximately at an angle of 180.
at this point. The granular stress contour calculated using Eq. (A21)
shown in Fig. 11(d) indicates no significant gradients in s
p
. A slight
gradient in the stress term is generated close to the shock impingement
point as the initial shock impingement pushes the particle bed in the
negative y direction at the shock impingement point S. The high iner-
tia of the particles does not allow significant motion of them in the
FIG. 8. Gas temperature evolution of the
plasma kernel at (a) 0.2 (sg¼0:55),
(b) 1.0 (sg¼2:75), and (c) 1.8 (sg
¼4:95) ls after kernel initialization. The
fluctuation of core temperature due to
expansion wave and secondary shock
interaction is evident through these time
snapshots. Dashed line indicates the posi-
tion of the expanding blast/shock front.
Note that the line extractions shown in
Fig. 9 are taken along the direction ras
indicated in Fig. 6 with center at ðx;y;zÞ
¼ða=2;h=2;a=2Þ¼ð5;2:5;5Þmm
(base case).
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FIG. 9. Line extraction of gas pressure and temperature at time t¼0.2 ls
(sg¼0:55) after spark initialization along direction r(Fig. 6)withoriginatðx;y;zÞ
¼ða=2;h=2;a=2Þ¼ð5;2:5;5Þmm. The line extraction shows the initial condition in
blue dashed line and marks the expansion fan that propagates into the kernel as well
as the shock front that moves radially out from the plasma kernel (base case).
FIG. 11. Contours of (a) v
y
, (b) dp/dy, (c) x
z
, and (d) s
p
given in Eq. (A21) at time
t¼4ls after plasma kernel initialization. The subfigures (a)–(c) correspond to the
gas phase, while (d) shows the granular or particle phase. The subplots show the
solution in the region close to the interface II0of the particle bed (base case).
FIG. 10. Gas vorticity evolution in the
vicinity of the plasma kernel at (a) 0.2
(sg¼0:55), (b) 1.0 (sg¼2:75), and (c)
1.8 (sg¼4:95) ls after kernel initializa-
tion. The dashed lines in the figure repre-
sent the location of the shock front (base
case).
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negative y direction to generate substantial stress gradients in the sim-
ulation time under consideration. The significant spatial variation of
these gas and particle properties imposes complex force fields on the
particle system, eventually dispersing the particles. Also, note that the
gradients of the gas macroscopic properties are not significant inside
the bulk of the particle bed, implying that the top layer particles are
the most affected by the blast interaction compared to the rest of the
bed.
The interaction of the expanding shock with the particle bed gen-
erates forces on the particles. The timescale of this interaction between
the shock and particle, t
int
, can be approximated as the time taken
by the shock wave to move through a particle diameter +dp=Mia0.
Here, the numerator is the diameter of the particle, and the denomina-
tor is the initial speed of the blast wave or the main shock in the ambi-
ent gas. For the base case and a particle diameter, dp¼1lm,
tint +2 ns /s
MS
. The spatial variation of the y-directional drag FD;y,
pressure gradient Fpg;y, and added-mass Fam;yforces acting on a
quarter-section of the particle bed is shown in Fig. 12 for time t¼4ls
after plasma kernel initialization. At this instance, the gas main shock
has expanded over the bed covering a circular area of radius +2 mm
with respect to the shock impingement point S. Note that the y-
directional forces are symmetric across the other quadrants of the par-
ticle bed. The high negative y-directional gas velocity behind the main
shock at the point of interaction with II0generates a very high negative
drag value at the shock foot on the particles. The positive pressure gra-
dient (y-directional) in the same area generates a high negative pres-
sure gradient force since Fpg;yis proportional to "rP. Since the
added-mass force Fam;yis also strongly proportional to "rP, the
added-mass force also has a high negative y-directional value close to
the shock foot.
The scenario is very different for the particles close to the shock
impingement point S. Particles in the vicinity of Sexperience a positive
drag force due to the positive y-directional gas velocity V
y
around this
area, as shown in Fig. 11. The pressure gradient and the added-mass
force also have a positive value in the vicinity of Sdue to the negative
pressure gradient rPin gas in this region, as shown in Fig. 11.
Therefore, there is a spatial variation of the forces with respect to the
shock impingement point Sfor FD;y;Fpg;y,andFam;y, which pushes
the particle in the negative y direction in the vicinity of the shock foot
and pulls the particles in the positive y direction near the shock
impingement point. Note that the magnitudes of Fam;yand Fpg;yare
approximately 50% of the drag force FD;y.
Similarly, the spatial variation of the gas vorticity dependent y-
directional Saffman force FSaff;yand Magnus force FMag;yare shown in
Fig. 13 at time t¼4ls. Both the Saffman force FSaff;yand Magnus
force FMag;yare zero in the vicinity of the shock impingement point S
since the gas vorticity at this spatial location is zero, as seen in Fig. 11
from the x
z
contour. The particles behind the shock foot experience a
positive FSaff;yand FMag;ydue to the positive ðuf"upÞ)xfterm in
the formulations of the Saffman and the Magnus forces. The slip or
the relative velocity vector (uf"up) is directed radially away from the
shock impingement point Sas shown in the vorticity contour of Fig.
11. The gas vorticity pseudo-vector indicated by !indicated in Fig.
11(c) has a clockwise rotation (into the paper) due to the low gas
velocity at the interface increasing progressively with increasing ydis-
tance. This combination of vector directions generates a positive mag-
nitude for the y-directional component of the vector ðuf"upÞ)xf.
The magnitudes of these two forces were found to be similar but an
order of magnitude lower than the drag (FD;y), pressure-gradient
(Fpg;y), and added-mass (Fam;y) force. It should be noted that the for-
mulation of the Magnus and Saffman forces is based on incompress-
ible flow assumptions. The coefficients of these force models for
FIG. 12. Spatial y-directional of drag FD;y, pressure gradient Fpg;y, and added mass
Fam;yforce at time t¼4ls after shock initialization (base case).
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compressible regimes, such as our simulations, are still an area of
active research. Therefore, these force models, in general, only give a
qualitative estimate of the vorticity-induced lifting in our simulations.
More accurate quantitative estimates are possible only with better
models for these force terms that are compatible with the underlying
compressible gas phase. Finally, although the thermophoretic force
Ftherm term was considered in this study, it is +Oð10"13ÞNwithno
significant effect on the particle phase evolution. Its small magnitude is
due to the small thermal gradient at the shock front. Also, the gravita-
tional force on the particles is negligible compared to other fluid-
induced forces due to their low mass.
The timescale of evolution of the particle phase t
p
is longer than
the interaction time t
int
due to the high inertia of the particle. This
timescale t
p
is on the same order of magnitude as the momentum
relaxation time sV¼qpd2
p=18lf.Fora1lm particle, tp+sV+1ls
0tint +2 ns. The ratio sV=tint +500 indicates the true multi-
timescale aspect of the shock particle interaction in this system. The
particle evolution happens at timescales that are at least two orders of
magnitude longer than the shock–particle interaction timescale that
generates lifting forces on the particles. The spatial and temporal varia-
tion of the particle forces experienced by the bed causes the particle
phase in the system to evolve. Figure 14(a) shows the evolution of the
particle pathlines or Lagrangian pathlines exposed to the blast wave.
Only the top layer of the particle bed, which lifts from the surface, is
tracked in this pathline representation of the particle trajectories since
the bulk of the particle bed is not lifted in this high particle bed volume
fraction case. The evolution of the particle system is shown by plotting
the Lagrangian pathlines for the 10 054 1 lm particles occupying the
upper layer of the particle bed within a radius of 2.5mm with center at
S¼ðx;zÞ¼ð5;5Þmm. The particle lifting was observed to be sym-
metric for the other three-quarters of the simulation. These particles
were tagged in the particle bed during bed initialization so that only
their particle time-accurate data were written every time step; there-
fore, reducing the computational time required in the I/O routine of
the FLASH
19
code for particle Lagrangian pathline data output. The
pathlines are plotted until time t¼16 ls after plasma kernel initializa-
tion. We observe that the particles close to the shock impingement
point Slift higher compared to the other particles. This trend is
expected as the shock system imposes a high positive y-directional lift
force on the particles close to the shock impingement point Sthrough
the FD;y;Fpg;y,andFam ;yforce terms as seen in the earlier Fig. 12.As
the positive lifting effect of FD;y;Fpg;y,andFam;ydiminishes for par-
ticles located radially from the shock impingement point S, lifting is
dominated by the FMag;yand the FSaff;yforce terms. Since these terms
are much weaker than FD;y;Fpg;y, and Fam;y, particles radially away
from Sare lifted less readily than the central particles. Similar behavior
was observed in our previous work with a monolayer of particles.
48
Figures 14(b) and 14(c) show the instantaneous particle velocities in
the X and Y directions, respectively, for the 1-lm particles from the
top layer of the particle bed by coloring the particle pathlines or
Lagrangians with the instantaneous velocity experienced by the parti-
cle as it evolves in time and space. The y-velocity V
y
suggests that the
particles close to the shock impingement point Sat x ¼5 mm achieve
higher lift velocities +80 m/s compared to the particles radially away
from this point +45 m/s. The x-directional velocity V
x
is negligible for
particles close to Sbut achieves significant magnitudes +30 m/s for
the radially positioned particles. Because the direction of V
x
is toward
S, the particles positioned radially away from the shock impingement
point are directed back toward Sat latter times.
The literature suggests a disparate range of values for the Saffman
coefficient C
s
given in the formulation Eq. (A14), with typical values of
Cs¼6:44 (our baseline), 32, and 160. To quantify the effect of chang-
ing C
s
on the system evolution, we performed two extra simulations
with C
s
¼32 and 160, by keeping all other simulation parameters the
same as the base case. Figure 15 compares the particle trajectories for
the cases with C
s
¼32 and 160 in Figs. 15(b) and 15(c) with base case
in Fig. 15(a). We can see that the particle lifting height is not effected
significantly with an increase in C
s
. Although the top-layer of the parti-
cle bed experiences a higher Saffman force for the C
s
¼160 compared
to the C
s
¼32 case due to the systematic increase in the coefficient, the
drag, pressure-gradient, and added-mass forces dominate in particle
lifting compared to the gas-vorticity-based Saffman force.
3. Reflected blast interaction with the plasma kernel
The reflected shock (RS) generated at the interface II0moves in
the positive ydirection toward the expanding plasma kernel. The time-
scale of this process s
RS
is slightly less than s
MS
since the main shock
(MS) increases the temperature of the gas column between the
FIG. 13. Spatial variation of y-directional Saffman force FSaff;yand Magnus force
FMag;yat time t¼4ls after shock initialization (base case).
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interface II0and the plasma kernel, which in turn increases the speed
of the reflected shock in the gas. For practical considerations, we can
assume that sRS +sMS. The interaction of the reflected blast wave
with the expanding plasma kernel is shown in Fig. 16 using the x
z
gas
vorticity and gas temperature contours at time t¼4ls after plasma
kernel initialization. The pressure and the density gradients in the gas
are misaligned when the reflected blast initially interacts with the ker-
nel. The third term of Eq. (12) deposits high gas vorticity at the bound-
ary defining the contact discontinuity. The temperature contour
indicates the slight increase in the gas temperature behind the dif-
fracted shock wave (DS). At later times, the reflected shock advects the
plasma kernel in the positive ydirection, and the deposited vorticity
perturbs the edges of the contact discontinuity that defines the plasma
kernel. The shock moves faster through the hot plasma kernel. Since
the temperature of the gas through which the reflected shock interacts
is higher than the gas behind it, the pressure jump across the shock is
reduced significantly, quenching the shock to become a weak pressure
wave.
C. Effect of plasma kernel length and blast wave
strength on particle lifting
The length of the plasma kernel generated in ESD devices
depends on the distance between the pin electrodes used to create the
dielectric breakdown. This length L
c
of the kernel can have an effect
on the particle lifting characteristics. To evaluate the impact of plasma
kernel length or shape on the particle lifting features, we performed
two additional simulations with two different lengths L
c
for the kernel.
A long kernel has a length of L
c
¼4 mm, which is twice the base case
discussed in the Sec. IV B (case 1 in Table II). The short kernel is gen-
erated with an L
c
¼1 mm (case 2 in Table II). Note that the radius R
c
and the pressure ratio Pc=P0for both these cases are the same as the
base case and equal to 0.5mm and 16.67, respectively. Figure 17 shows
the Lagrangian pathlines and the corresponding instantaneous particle
velocities for these two cases with L
c
¼1 and 4mm. Clearly, we can
see that the fraction of particles lifted increases with increasing kernel
FIG. 14. (a) The Lagrangian pathlines for 10,054 1 lm particles occupying the upper layer of the particle bed within a radius of 2.5 mm with center at S¼ðx;zÞ¼ð5;5Þmm.
Only a quarter of the domain is shown in the figure. The pathlines are plotted for time t¼16 ls after shock initialization. Particle pathlines as viewed along the z direction for
the quarter domain of particles on the top layer of the particle bed. The pathlines are colored according to the magnitude of the particle velocity in the x direction V
x
(b) and y
direction V
y
(c). Note that figures (b) and (c) show all the 10 054 particle paths projected on to the X–Y plane.
FIG. 15. Lagrangian particle trajectories for case with Saffman coefficient (a) Cs
¼6:44 (base case), (b) 32, and (c) 160.
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length for the same pressure ratio. More particles are lifted in the
L
c
¼4 mm case compared to 2 and 1 mm. It is also interesting to note
that the maximum particle lifting height for the same pressure ratio
does not increase significantly with increasing kernel length as the
maximum height achieved by the particles in time t+16 lsfor
L
c
¼4 mm case after spark initialization is almost the same as the base
case +1:2 mm from the wall.
Different magnitudes of discharge energy are used in the devices
that generate the blast wave.
49,65
This variation in the breakdown
energy can change the shock strength that emanates from the kernel,
which in turn can affect the particle lifting mechanisms. To study the
effect of different discharge energies on particle lifting, we vary the
initial pressure ratio of the initialized plasma kernel. Note that in
these cases, the geometric measurements of the plasma kernel are the
same as the base case, and only the pressure ratio Pc=P0is changed.
In this section, we consider two cases Pc=P0¼33:3 (high-energy
kernel) and 6.67 (low-energy kernel), corresponding to cases 3 and 4
in Table II, respectively. The strength or the shock Mach number of
the blast wave impinging the particle bed interface for the Pc=P0
¼33:3 case will be much higher than the base case as the initial
shock strength M
i
generated at the plasma kernel surface is greater
than the base case according to the planar normal shock relation at
the plasma kernel interface as given in Table II.Figure 18 shows the
evolution of the particle pathlines and the particle velocities for the
two pressure ratio cases. The higher pressure ratio case shows an
increased value of lifted particle velocity near the shock impingement
point Sðx;y;zÞ¼ð5:0;0:6;5:0Þmm +100 m/s compared to +85 m/
s in the base case, which has a pressure ratio of 16.67.
D. Revisiting particle lifting from a monolayer
of particles
To quantify the difference in particle lifting for a monolayer of
particles vs a dense particle bed, we generate a uniform monolayer of
single particle diameter 1 lm(aty¼0 m). The gas parameters for this
simulation are the same as the base case given in Table II, and to emu-
late the dilute nature of the monolayer, the granular stress term is
switched off.
Figure 19 shows the spatial variation of the drag FD;y, pressure
gradient Fpg;y, and added-mass Fam;yforces on the monolayer com-
posed of 1 lm sized particles at time t¼4ls after spark initialization.
The shock expands over the monolayer radially to +2mm from the
shock impingement point Sat the center by 4 ls. As the 23 lmsized
particles in the particle bed experience higher drag magnitudes
(FD/d2
p), the magnitude of the drag experienced by particles in the
monolayer is three orders of magnitude lower than that of the bed as
given in Fig. 12. It should be noted that the particles in the vicinity of
the shock impingement point Sexperiences a high positive drag mag-
nitude in the case of the particle bed (Fig. 12), while for the monolayer,
the particles experience negligible drag in this region. This is due to
the penetration effect of the shock/blast wave in the former verses the
latter case where the reflected shock from the bed has a high gas veloc-
ity magnitude behind the reflected shock that establishes high drag
forces on these particles. At the shock foot, the particle bed as well as
the monolayer experiences a negative drag force pushing the particles
toward the wall. For the monolayer, as the blast wave reflects off of the
wall and leaves the particle distribution behind, there are no gas pres-
sure gradients in the vicinity of the particle distribution that would
have generated pressure gradient and added-mass forces in the vicinity
of S. At the shock foot, both these forces are active as the gas pressure
gradients at the foot of the expanding shock at the wall are large. The
gas vorticity-dependent forces, Saffman force FSaff;yand Magnus force
FMag;y,areshowninFig. 20. When we compare across the different
forces generated on the monolayer, the magnitude of the Saffman
force is at least an order of magnitude larger than all other force com-
ponents. This result is consistent with our previous publication
(Marayikkottu et al.
48
) and again suggests that the Saffman force is the
FIG. 16. The interaction of the reflected blast wave with the expanding plasma kernel at time t¼4ls after plasma kernel initialization shown using (a) x
z
and (b) tempera-
ture contours.
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predominant lifting mechanism in the case of monolayers of particu-
lates in contrast to the case of a dense bed where the drag, pressure-
gradient, and added-mass forces dominate.
Finally, Fig. 21 shows the Lagrangian pathlines for the particle
bed (base case) and the monolayer. We can clearly see that the spatial
variation in the particle lifting behavior is very different for the two
cases compared here. For the particle bed, particles in the zone close to
the shock impingement point S(5 <x<5:7 mm) experience the
highest particle lifting corresponding to the high magnitudes of drag,
pressure-gradient, and added-mass forces in the particle bed. On
the other hand, for the monolayer case, particles in the zone
(5:4<x<6 mm) experiences high lift compared to the particles in
the vicinity of the shock impingement point S.
V. SUMMARY AND CONCLUSIONS
The current work discusses the implementation of a two-way
coupled Eulerian–Lagrangian framework using the original FLASH
research code framework
8,19,48
to study compressible gas–particle mul-
tiphase flows. The Lagrangian solver considered forces such as the
quasi-steady drag, pressure-gradient, added-mass, thermophoretic,
gravitational, Saffman, and Magnus forces. Inter-particle collisions are
implemented using the empirical model by Harris and Crighton
55
The
implementation of the Lagrangian modules is modular, so it can be
used across different simulation configurations within the FLASH
19
code framework. First, the newly developed Eulerian–Lagrangian code
was used to study a three-dimensional multiphase shock tube prob-
lem. The interaction of a planar shock with a dense particle curtain
FIG. 17. Figures showing the effect of kernel length on particle evolution. The particle pathlines for L
c
¼4 and 1 mm kernel are shown in (c) and (f), respectively. Particle veloc-
ity evolution along the 16 ls particle pathline is shown by coloring particle pathlines with the instantaneous particle velocity magnitudes. (a) and (d) represent the x-directional
velocity, while (b) and (e) indicates the particle lifting velocity or the y-directional velocity for L
c
¼1 and 4 mm cases. The velocity contours are viewed in the z direction such
that the pathlines are projected onto the X–Y plane.
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generated reflected and transmitted shocks. The prominent forces in
the motion of the particle curtain were found to be the quasi-steady
drag, pressure gradient force, and the added-mass force. The effect of
inter-particle collisions on the system was found to be negligible. The
gas and particle phase features agreed with the experimental studies of
Ling et al.
1
Second, the MP-PIC FLASH
19
code was used to study particle
liftingwhen an expanding blast wave interacts with a dense bed of par-
ticles in a head-on orientation. The current work studied a millimeter
scale domain to understand the general mechanisms of gas and parti-
cle phase evolution in an ESD system. The emanation of the blast
wave at the plasma kernel interface and the plasma kernel cooling was
observed in the early stages of the system evolution. As the expanding
blast wave impinges the particle bed which is placed below the kernel,
complex interactions between the particles and the gas are observed.
This interaction splitsthe main shock intoa transmitted shock moving
into the bed and a reflected shock moving away from the bed. A vortex
sheet is also generated along the particle bed interface due to this inter-
action. Due to these spatially varying gas property gradients, fluid-
induced particle forces are generated on the particles close to the inter-
face, which results in particle lifting or dispersion. It was also observed
that only the top layer of the particle bed containing 1-lm particles
was lifted in these systems. The high volume fraction of the particle
bed reduces the percolation of the gas into the bulk of the bed and
hence does not expose particles within the bed to high fluid-induced
forces. Near the shock impingement point, the drag, pressure gradient,
FIG. 18. Figures showing the effect of shock strength or the discharge energy on particle evolution. Particle pathlines for Pc=P0¼33.3 and 6.67 kernel are shown in (c) and
(f), respectively. Particle velocity evolution along the 16 ls particle pathline is shown by coloring the particle pathlines with the instantaneous particle velocity magnitudes. (a)
and (d) represent the x-directional velocity, while (b) and (e) indicate the particle lifting velocity or the y-directional velocity for Pc=P0¼33.33 and 6.67 cases. The velocity con-
tours are viewed in the z direction such that the pathlines are projected onto the X–Y plane.
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and added mass forces are dominant and act as the major lifting
mechanism in these systems. The effect of gas vorticity-dependent
forces, Saffman, and Magnus forces was found to be relatively smaller
compared to the former in this multiphase system. Finally, decreasing
plasma kernel length and strength was found to decrease the overall
lifting efficiency of the ESD device.
FIG. 19. Spatial variation of y-directional drag FD;y, pressure gradient Fpg;y, and
added-mass Fam;yforces on quarter portion of the monolayer at time t¼4ls after
spark initialization.
FIG. 20. Spatial variation of y-directional Saffman lift force FSaff ;yand Magnus force
FMag;yon a quarter portion of the monolayer at time t¼4ls after spark
initialization.
FIG. 21. Comparison of particle pathlines for the dense bed (left) and monolayer
(right). The pathlines are colored according to the time of particle phase evolution.
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Additionally, we found that the Saffman coefficient is not a sensi-
tive parameter for particle lifting from dense beds. However, for particle
beds represented as a monolayer, the Saffman force remains the pre-
dominant lifting mechanism as observed in our previous work
(Marayikkottu et al.
48
)sincetheeffectofunsteadyforcessuchasthe
added-mass and pressure gradient forces is negligible in these systems.
Through this study, we have quantified the time scales associated with
the gas and particle phase evolution, and particle lifting mechanisms.
Since the accuracy of Eulerian–Lagrangian codes relies on the force
models coupling the phases, particle-resolved simulations are under-
taken to develop force models for complex particulate geometries and
configurations in the compressible gas regime.
43,63,66
Of particular inter-
est may be the average drag model of Osnes et al.,
66
in capturing the
effect of gas compressibility in shock-dominated gas–particulate multi-
phase systems with dense particulate distributions. These newly devel-
oped models will be incorporated into the Lagrangian force routines in
the two-way coupled solver, increasing the solver’s predictive accuracy.
ACKNOWLEDGMENTS
This work is supported by the Defense Threat Reduction Agency
(DTRA) through Grant No. HDTRA1-20-2-0001. The computational
research is supported by the Stampede2 supercomputing resource
provided by the Extreme Science and Engineering Discovery
Environment (XSEDE) TACC. The software used in this work was
developed in part by the DOE NNSA and DOE Office of Science-
supported FLASH Center for Computational Science at the University
of Chicago and the University of Rochester.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
All authors contributed equally to this work.
Akhil Marayikkottu Vijayan: Conceptualization (equal); Data
curation (equal); Formal analysis (equal); Investigation (equal);
Methodology (equal); Software (equal); Validation (equal);
Visualization (equal); Writing – original draft (equal). Deborah Levin:
Conceptualization (equal); Funding acquisition (equal); Methodology
(equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
APPENDIX: SURROGATE FORCE MODELS USED
IN THE MP-PIC CODE
1. Drag force on the particle is given as follows:
FD¼qf
2CDAðuf"upÞjuf"upj;(A1)
where Ais the projected area of the spherical particulate. The
drag coefficient C
D
for a spherical particle in a dilute system
where the particle volume fraction /<5% is formulated as a
function of particle Reynolds number Re ¼qfjuf"upjd=lfand
Mach number Ma ¼juf"upj=ffiffiffiffiffiffiffiffiffiffi
cRTg
pby Loth
52
as follows:
CD¼24
Re 1þ0:15Re0:687
½*
HMþ0:42CM
1þ42;500GMRe"1:16
for Rep>45;(A2)
CD¼CD;Kn;Re
1þMa4þMa4CD;fm;Re
1þMa4for Re -45:(A3)
The coefficients for compression-dominated regime, Re >45, are
given as follows:
CM¼1:65 þ0:65 tanhð4Ma "3:4Þfor Ma -1:5;(A4)
CM¼2:18 "0:13 tanhð0:9Ma "2:7 for Ma >1:5;(A5)
GM¼166Ma3þ3:29Ma2"10:9Ma þ20 for Ma -0:8;(A6)
GM¼5þ40
Ma3for Ma >0:8;(A7)
HM¼0:0239Ma3þ0:212Ma2"0:074Ma þ1 for Ma -1:0;
(A8)
HM¼0:93 þ1
3:5þMa5for Ma >1:0:(A9)
The coefficients of rarefaction dominated regime, Re <45, are
given as follows:
CD;fm;Re ¼CD;fm
1þC0
D;fm
1:63 "1
"#
ffiffiffiffiffi
Re
45
r;
CD;fm ¼ð1þ2s2Þexpð"s2Þ
s3ffiffiffi
p
pþð4s4þ4S2"1ÞerfðsÞ
2s4
þ2
3sffiffiffiffiffiffiffiffiffiffi
pTp
T1
s;
C0
D;fm ¼CD;fm;Tp¼T1¼ð1þ2s2Þexpð"s2Þ
s3ffiffiffi
p
p
þð4s4þ4S2"1ÞerfðsÞ
2s4;
CD;Kn;Re ¼24
Re ð1þ0:15Re0:687ÞfKn ;
fKn ¼1þKnð2:514 þ0:8 expð"0:55KnÞ*"1;
'
(A10)
where Re,s,Ma, and Kn represent the particle Reynolds number,
speed ratio, Mach number, and Knudsen number, respectively.
Note that Loth’s
52
model [Eq. (A3)] accounts for the gas rarefac-
tion and compressibility in the vicinity of the spherical particu-
late reduces to CG model for HM;CM;and GM ¼1.
The particles in dense particle distributions of high particle con-
centration (volume fraction />5%) are modeled using the
model by Tenneti et al.,
51
FDðh;RemÞ¼FisolðRemÞ
ð1"hÞ3þFhðhÞþFh;Remðh;RemÞ;(A11)
where his the particle volume fraction and FisolðRemÞis the iso-
lated sphere drag law by Schiller,
67
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FisolðRemÞ¼qf
2
24
Remð1þ0:15Re0:687
mÞAðuf"upÞjuf"upj:(A12)
The other two terms in the formulation are given as follows:
FhðhÞ¼ 5:81h
ð1"hÞ3þ0:48 h1=3
ð1"hÞ4
"#
^
n;
Fh;Remðh;RemÞ¼ h3Rem0:95 þ0:61h3
ð1"hÞ2
!"#
^
n;
(A13)
where Rem¼qfjuf"upjdð1"hÞ=lfis the modified Reynolds
number for Tenneti’s
51
model and ^
nis a unit vector in the direc-
tion of Fisol.
2. Force on the particle due to gas velocity gradient is given by
Saffman
53
as follows:
FSaff ¼Cs
4d2
pðlfqfÞ1=2jxfj"1=2ðuf"upÞ)xf
'(
;(A14)
where Cs¼6:44 is the Saffman coefficient. ~
xf¼r)~
ufis the
gas vorticity.
3. The force on the particle due to gas temperature gradient rTfis
given by the thermophoretic force given by Loth
52
as follows:
Fthermo ¼FrT;Kn01F,
rT;
¼"p
2lf!fdp
Kn rTf
Tf
"#Kn1:7
1:15 þKn1:7:(A15)
4. Magnus force on an irrotational particle is given by Rubinow
and Keller
54
as follows:
FMag ¼p
8d3
pqfðuf"upÞ)xf
'(
:(A16)
5. Force on the particulate due to the spatial gas pressure gradient
in the vicinity of the particulate, pressure gradient force,
68
is
given as follows:
Fpg ¼þcS "P^
ndS;(A17)
Fpg ¼þcV "r&P dV:(A18)
By assuming the pressure gradient to be constant in the vicinity
of the particulate,
Fpg ¼ "r&PVp;(A19)
where the volume of the spherical particulate is given by V
p
.
6. The force on the particulate distribution due to the multiple
inter-particle interactions is given by Harris and Crighton
55
expressed as follows:
Fgran ¼1
qphprsp;(A20)
sp¼Pshb
max hcp "h;eð1"hÞ
'(
;(A21)
where h
cp
is the closed packing factor, and his the particle volume
fraction. Ps+Oð105Þ;1-b-5, and e+Oð10"7Þare constants
in the model. The scaling study by Ling et al.
1
shows that the para-
metric space for P
s
and bgiven above models the interaction of
shock waves with dense particle distributions efficiently.
7. Added mass force: when a body is accelerated through a fluid,
there is a corresponding acceleration of the fluid. The difference
in acceleration between the particle and the fluid induces an
additional force, which is similar to adding an additional mass or
virtual mass to the particle. This additional force is given as the
added mass force
Fam ¼VpCMðMa;hÞ "rP"dðqfupÞ
dt
$%
;(A22)
where V
p
is the particle volume, CMðMa;hÞ¼CM;stdg1
)ðMaÞg2ðhÞ;CM;std ¼1=2 is the standard coefficient for added
mass force. his the particle volume fraction. The compressibility
and volume fraction coefficients are given by Parmar et al.
69
and
Zuber
70
as follows:
g1ðMaÞ¼1þ1:8Ma2þ7:6Ma4;(A23)
g2ðhÞ¼1þ2h:(A24)
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20 July 2023 13:17:53