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Study of shock interaction with a particle curtain using the
Multiphase Particle in Cell (MP-PIC) approach
Akhil V. Marayikkottu ∗and Deborah A. Levin †
The Department of Aerospace Engineering
University of Illinois at Urbana-Champaign
For multiphase flows with dense particulate distributions, the effect of momentum and
energy back coupling with the underlying gas flow and inter-particle interaction is significant. In
the present work, we present our preliminary work towards developing a two-way coupled solver
on the original framework of the FLASH[
1
] code. Particle force models that are compatible
with the underlying compressible gas flow are used in the solver. Inter-particle collisions are
implemented using the empirical model by Harris et al. [
2
]. The developed code is used to study
the canonical case of a planar shock interacting with a particle curtain. The temporal evolution
of the considered force terms is discussed in detail. The evolution of gas/particle phase features
is compared to the experimental results given in Ling et al.[3].
I. Introduction
The interaction of shockwaves with granular solid particulates is a common theme in various scenarios. These
interactions determine the evolution of ash dust clouds in the case of volcanic eruptions [
4
,
5
]. Such interactions are
also possible in high-altitude high-speed flights[
6
], and re-entry flights [
7
,
8
], dust explosions [
9
,
10
], and plume surface
interactions for inter-planetary explorations [
11
]. Theoretical and numerical studies in this regime are gaining interest in
recent times with respect to explosion safety, and mitigation efforts [
12
]. Therefore there is an increased requirement for
a modular compressible Eulerian-Lagrangian multiphase solver. The FLASH research code over the past three decades
has proved efficient in tackling compressible shock-dominated gas flow regimes (Eulerian). In this work, we report
the implementation of two-way coupled Lagrangian (particle) routines in the original Eulerian framework to study
high-speed gas-particle multiphase flows.
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Fig. 1 MP-PIC: fluid-particle solver coupling
∗Graduate student, Department of Aerospace Engineering, UIUC, Urbana, IL 61801, AIAA student Member.
†Professor, Department of Aerospace Engineering, UIUC, Urbana, IL 61801, AIAA Fellow.
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Copyright © 2023 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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II. Computational approach
The MP-PIC method proposed initially by Andrew et al. [
13
] solves the gas phase on the discretized grid, assuming
a continuum 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
𝜃
is mapped onto the Eulerian grid to facilitate phase coupling. Correspondingly, the gas
volume fraction for the same cell,
𝜀=1−𝜃
. The governing equation of the gas or fluid phase of the multiphase system
is given as follows:
𝜕(𝜀𝜌 𝑓)
𝜕𝑡 +∇.(𝜀 𝜌 𝑓u𝑓)=0(1)
𝜕(𝜀𝜌 𝑓u𝑓)
𝜕𝑡 +∇.𝜀 𝜌 𝑓u𝑓u𝑓+∇𝑃−𝜀𝜌 𝑓g=−𝐹𝑠(2)
𝜕(𝜀𝜌 𝑓𝐸)
𝜕𝑡 +∇.(𝜀 𝜌 𝑓𝐸+𝑃)u𝑓+𝑃𝜕𝜀
𝜕𝑡 −𝜀 𝜌 𝑓u𝑓.g=𝐸𝑠(3)
where
𝜌𝑓
,
u𝑓
,
𝑃
,
𝐸
and
g
represents the gas density, gas velocity, pressure, energy, and the acceleration due to gravity,
respectively. The momentum and energy source terms are represented by 𝐹𝑠and 𝐸𝑠respectively.
10 210 1100101102103104105
Re
100
101
102
103
C
D
Ma
0
Loth et al. Ma= 3.00
Loth et al. Ma= 1.50
Loth et al. Ma= 0.20
Loth et al. Ma= 0.02
Fig. 2 Drag coefficient
𝐶𝐷
variation with Reynolds number
𝑅𝑒
given by Loth et al.[
14
]. The model captures the
effect of gas compressibility and rarefaction.
The evolution of the particle distribution 𝜙is given as :
𝜕𝜙
𝜕𝑡 +∇.(𝜙u𝑝)+∇𝑢𝑝.(𝜙A)=0(4)
where
𝜙(x,u𝑝, 𝜌 𝑝, 𝑡)
is the probability distribution of particles and
u𝑝
and
A
are the particle velocity and the inter-phase
acceleration terms respectively. The acceleration term is represented by several surrogate or sub-grid force models as
follows:
𝑚A=FD+FSaff +FMag +Fpg +Fgran +Fthermo +Fgravity +Fam (5)
where
𝑚
is the mass of the particulate.
FD
represents the drag force generated on an isolated particulate due to non-zero
relative velocity of the particle with the gas flow in its vicinity. Since the particulates in the system are exposed to
compressible and rarefied gas regimes, the model drag coefficient by Loth et al.[
14
] is used. The model efficiently
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captures these effects across wide ranges of gas compressibility and rarefaction. Figure 2 shows the variation of
drag coefficient
𝐶𝐷
across Reynolds numbers
𝑅𝑒
for different Mach numbers
𝑀𝑎
. The Saffman lift force (Saffman
et al.[
15
]) represented by
FSaff
is the lateral lift force generated due to the shearing flow in the particle exposed gas.
Pressure gradient force and thermophoretic forces are represented by
Fpg
and
Fthermo
respectively and are generated on
particulates due to pressure and temperature gradients in the flow. The stress force developed in the particle distribution
due to compaction is represented by the granular stress force represented by
Fgran
. Finally, the gravitational force on
particulates though negligible is given by Fgravity. The force models used in the solver are provided in Appendix A.
Figure 1 shows the coupling of the Lagrangian particle solver and the Eulerian fluid/gas solver. The fluid phase
volume fraction 𝜀is computed on the particle field by spatially integrating the particle distribution as,
𝜀=1−𝜃=1−∫ ∫ 𝜙 𝑑u𝑝𝑑𝜌 𝑝(6)
The evolution of the particle distribution through Newton’s equation along with particle mapping evolves the particle
and the fluid volume fraction fields (
𝜀=1−𝜃
) in space and time. The phase volume fraction and the momentum
and energy source terms
(𝐹𝑠and 𝐸𝑠)
from the particle solver is communicated to the Eulerian fluid solver through
cloud-in-cell (CIC) mapping. The particle acceleration (A) a function of the fluid macro parameters and gradients
residing on the Eulerian solver is communicated to the particle solver through isotropic shape functions provided in the
original FLASH framework.
The momentum source term is the net momentum transferred to the fluid due to a single particle acceleration in the
fluid, given as,
𝐹𝑠=∫𝑡𝑓
𝑡𝑖
𝑚A𝑑𝑡 (7)
where
𝑚
is the mass of the particle and Ais the acceleration of the particles. Note that the acceleration term does
not include the granular stress term. This is consistent with previous Eulerian-Lagrangian modeling for compressible
gas-particle multiphase systems by Dahal et al.[16].
10 210 1100101102103104105
Re
10 1
100
101
102
Nu
Ma
0
Fox et al. Ma= 3.00
Fox et al. Ma= 1.50
Fox et al. Ma= 0.20
Fox et al. Ma= 0.02
Fig. 3 Nusselt number
𝑁𝑢
variation across Reynolds numbers
𝑅𝑒
for different Mach numbers
𝑀𝑎
given by Fox
et al.[17]
The energy source term in this implementation is due to the (reactive) work done by the particle force term. Note
that the heat addition due to sensible heating or cooling of the particulates are not coupled to the Eulerian solver since
their magnitudes are found to be negligible for the application s under consideration. The term is also modeled as an
implicit source term in the FLASH Riemann solver as :
𝐸𝑠=∫𝑡𝑓
𝑡𝑖
𝑚A.u𝑝𝑑𝑡 (8)
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Since the gas temperature is higher than the initialized particles in blast wave systems, the particles are exposed to
high convective flux of heat. The quasi steady state heat convection
𝑄qs
between the gas and spherical particles is given
as:
𝑄qs =𝜋𝑑 𝑝𝑘𝑔(𝑇𝑔−𝑇𝑝)𝑁𝑢 (9)
where
𝑘𝑔
is the thermal conductivity of air in the vicinity of the particulate with temperature
𝑇𝑝
and diameter
𝑑𝑝
. The
Nusselt number 𝑁𝑢 quantifying the heat transfer is given by Fox et al.[17] as:
𝑁𝑢 =2 exp (−𝑀𝑎)
1+17𝑀 𝑎𝑅𝑒 +0.459𝑃𝑟 0.33 𝑅𝑒0.55 1+0.5 exp (−17𝑀 𝑎𝑅𝑒)
1.5(10)
where
𝑃𝑟
is the Prandtl number. The model captures the effect of gas rarefaction on heat transfer on spherical particles
through Knudsen number corrections as shown in figure 3.
III. Evolution of gas and particle phases in a multiphase shocktube
Fig. 4 Simulation domain for the three-dimensional multiphase shock tube problem
The newly developed MP-PIC solver is used to study the interaction of a planar shock wave with a thin curtain of
particles. Researchers such as Ling et al.[
3
], Sugiyama et al.[
18
]and Theofanous et al.[
19
] have studied the system
before using various numerical approaches. Figure 4 shows the three-dimensional domain for the multiphase shock tube
problem. The simulation domain has a lateral length of 0.88 m (𝑏1+𝑏2+𝑏3+𝐿) and square edge length 𝑎=0.08 m.
The high pressure driver gas (air) is contained in the purple colored portion of the tube with
𝑏1
= 0.1 m. The particle
curtain is initialized as a thin sheet of particles of width
𝐿
= 2 mm at a distance of
𝑏2
= 0.65 m from the shock tube
diaphragm. The particle curtain is generated using spherical particles of diameter 115
𝜇
m and material density of 2520
kg/m
3
. Each representative parcel in the simulation represents 26 real particles. The particle volume fraction of the
curtain is maintained at 21 % as previously done by Ling et al.[
3
] for one-dimensional simulations of the same system.
The driven section of the tube is maintained at a pressure of 0.0827 MPa, while the temperature of both the sections are
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IS
TS
TS
PW
PW
RS
RS
RS
CD
CD PW
PC
PC
PC
PC
(a) t = 0.90 ms
(b) t = 1.05 ms
(c) t = 1.20 ms
(d) t = 1.43 ms
Fig. 5 Evolution of the multiphase shocktube in time close to the particle curtain. The expansion of the particle
curtain is evident. The time series shown in the series are in order, 0.9, 1.05, 1.20, and 1.425 milliseconds after the
shock initialization.
maintained at 296.4 K. The driver section is initialized at a pressure of 0.98132 MPa to generate a Mach 1.666 flow
similar to the case given in Ling et al. [
3
]. The shock tube length is smaller than the original one-dimensional study of
[
3
] to reduced the computational requirements of a full three-dimensional rendering of a one-dimensional problem. The
computational domain is discretized using the PARAMESH library to a maximum AMR level of three and block size of
20
×
20
×
20. The minimum size of the fluid cell is
Δ𝑥∼1×10−3
m. A fifth order HLLC Riemann solver is used to
solve the fluid governing equation. CFL criteria for the gas solver is 0.7 and the particle motion is restricted such that
u𝑝Δ𝑡Δ𝑥<0.05
to maintain stability while the two way coupling terms are activated. The time window of interest is
such that the particle curtain as well as the reflected shock in the period of interest does not get contaminated by the
contact discontinuity or the expansion waves following it. For this study, the effect of gravity, Saffman force, Magnus
force and the thermophoretic forces are not considered to be consistent with the previous study by Ling et al.[3].
For a shock Mach number
𝑀𝑠=1.66
, the shock speed is given as
𝑢𝑠=𝑀𝑠.𝑎 =573.31
m/s, where
𝑎=345.37
m/s is
the speed of sound in the medium (air). For the current configuration of the three-dimensional multiphase shock tube,
the shock takes
𝑏2𝑢𝑠∼9.767 ×10−4
s to reach the particle curtain. The contact discontinuity behind the shock has a
velocity of
∼304.5
m/s. Thus this configuration gives us a time window of
∼0.014
s to analyze the interaction of the
shock with the particle curtain.
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RS RS RSTS TS TS
0.64 0.66 0.68 0.70
x(m)
0.64 0.66 0.68 0.70
x(m)
0.64 0.66 0.68 0.70
x(m)
dp/dx du/dx ωz
PC
PW
(a) (b) (c)
+ve
-ve
Fig. 6 Particle wake (PW) behind the particle curtain (PC) indicated using the gas vorticity in z-direction
at time
𝑡=
1.05 ms after shock initialization. The variation of
𝑑𝑝𝑑𝑥
and
𝑑𝑢𝑑𝑥
in the vicinity of the particle
distribution at 𝑡= 1.05 ms after shock initialization.
50 40 30 20 10 0 10 20 30
x/L
0
20
40
60
80
100
120
140
t/(L/u
s
)
RS
TS
LPFUPF
Fig. 7 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
◻
and
×
respectively.
The upper particle front (UPF) is marked with
◇
and
◯
for with and without added-mass force term respectively.
The lower particle front (LPF) is marked with
◇
and
◯
for with and without added-mass force term respectively.
The corresponding dashed lines are experimental results given in Ling et al. [3].
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The temporal evolution of the multiphase shocktube is shown in figure 5 at different time instances. The incident
shock (IS) wave interacts with the particle curtain (PC) at
𝑡(𝐿𝑢𝑠)=0
and splits into a reflected shock (RS) that travels
in the negative
𝑥
direction and a transmitted shock wave (TS) traveling in the positive
𝑥
direction. The reflected shock is
much weaker than the transmitted shock and travels at different speeds. The reflected shock moves 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 positive velocity gradient (as shown in figure 6(a)) and a negative pressure gradient
(as shown in figure 6(b)) along the
𝑥
axis, 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 figure 6(c).
The phase plot shown in figure 7 compares the simulation with experimental results reported in Ling et al.[
3
]. The x-t
diagram is centered at the particle curtain location, i.e.,
𝑥=0.66
m and normalized with the curtain thickness
𝐿=2
mm.
The time axis is normalized with
𝐿𝑢𝑠∼3.49 ×10−5
s. Our simulation results are generally in good agreement with
the experimental given in the article by Ling et al.[
3
]. The discrepancies in the particle phase evolution are possibly due
to the differences in the surrogate force models being used in the study as well as the difference in the dimensionalities
of the problems. Three-dimensionality of the current simulations tend to generate drafts in the lateral (y & z) directions
that can reduce the x-directional forcing on the particle curtain. Also to be noted is that the effect of aerodynamic
interaction between particles in high speed flows can generate drag and lift forces on the particulate systems with
standard deviations in the same order of magnitude as the average drag force itself [
20
]. These extra forces can accelerate
a fraction of the particles in the curtain as compared to the average drag law in this implementation. Modeling these
extra force terms can increase the accuracy of Eulerian-Lagrangian prediction of the multiphase shock tube system.
Finally, the temporal variation of the active force terms on the particle system are shown in figure8. 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
form the curtain. The collisional force given by the empirical model of Harris[
2
] does not contribute significantly to the
evolution of the particle phase was also noted in the numerical work of Ling et al.[3].
IV. Conclusions and future works
The current work discusses the implementation of a two-way coupled Eulerian-Lagrangian framework using the
original FLASH research code framework [
1
,
9
,
21
] to study compressible gas-particle multiphase flows. Inter-particle
collisions are implemented using the empirical model by Harris et al. [
2
]. The implementation of the Lagrangian
routines is modular to be used across different simulation configurations. The modular implementation also makes
adding new force terms to the particle routine straightforward. The newly developed Eulerian-Lagrangian code was
used to study a three-dimensional multiphase shock tube problem. The interaction of a planar shock with a dense
particle curtain 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 negligible. The gas and particle phase features were in reasonable agreement with the
experimental studies provided in Ling et al. [3].
Since the accuracy of Eulerian-Lagrangian codes relies on the force models that couple the phases, particle-resolved
simulations are underway to develop such models for complex particulate geometries and configurations in the
compressible gas regime [
20
,
22
]. These newly developed models will be introduced into the Lagrangian force routines
in the two-way coupled solver for future use.
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 Comet-GPU and EXPANSE supercomputing resource provided by the
Extreme Science and Engineering Discovery Environment (XSEDE) TACC.
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0 20 40 60 80 100 120 140
t/(L/u
s
)
0
200
400
600
800
F
D
( N)
(a)
0 20 40 60 80 100 120 140
t/(L/u
s
)
0
20
40
60
80
100
120
F
pg
( N)
(b)
0 20 40 60 80 100 120 140
t/(L/u
s
)
2
0
2
4
6
8
F
gran
( N)
(c)
0 20 40 60 80 100 120 140
t/(L/u
s
)
0
50
100
150
200
250
300
350
F
am
( N)
(d)
Fig. 8 The variation of (a) drag, (b) pressure gradient, (c) collisional force, and (d) added-mass force with
time. The lines show the evolution of the average force with time. Standard deviation of the force within the
distribution of particles are shown using error bars for each plot.
A. Force models
1) Drag force on the particle is given as :
F𝐷=𝜌𝑓
2𝐶𝐷𝐴(u𝑓−u𝑝).(u𝑓−u𝑝)(11)
𝐶𝐷=24
𝑅𝑒 +4.4
√𝑅𝑒 +0.42 (12)
where
𝐴
is the projected area of the spherical particulate. The drag coefficient
𝐶𝐷
is a function of particle
Reynolds number
𝑅𝑒 =𝜌𝑓u𝑓−u𝑝𝑑𝜇𝑓
and Mach number
𝑀𝑎 =u𝑓−u𝑝𝛾𝑅𝑇𝑔
provided by the model of
Loth et al. [14] as:
𝐶𝐷=24
𝑅𝑒 1+0.15𝑅𝑒0.687𝐻𝑀+0.42𝐶𝑀
1+42,500𝐺𝑀𝑅𝑒−1.16 for 𝑅𝑒𝑝>45 (13)
𝐶𝐷=𝐶𝐷, 𝐾 𝑛, 𝑅𝑒
1+𝑀𝑎4+𝑀 𝑎4𝐶𝐷 , 𝑓 𝑚,𝑅 𝑒
1+𝑀𝑎4for 𝑅𝑒 ≤45 (14)
The coefficients for compression-dominated regime, 𝑅𝑒 >45 is given as:
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𝐶𝑀=1.65 +0.65 tanh(4𝑀𝑎 −3.4)for 𝑀𝑎 ≤1.5(15)
𝐶𝑀=2.18 −0.13 tanh(0.9𝑀𝑎 −2.7)for 𝑀𝑎 >1.5(16)
𝐺𝑀=166𝑀 𝑎3+3.29𝑀 𝑎2−10.9𝑀𝑎 +20 for 𝑀𝑎 ≤0.8(17)
𝐺𝑀=5+40
𝑀𝑎3for 𝑀𝑎 >0.8(18)
𝐻𝑀=0.0239𝑀 𝑎3+0.212𝑀 𝑎2−0.074𝑀 𝑎 +1for 𝑀𝑎 ≤1.0(19)
𝐻𝑀=0.93 +1
3.5+𝑀𝑎5for 𝑀𝑎 >1.0(20)
The coefficients of rarefaction dominated regime, 𝑅𝑒 <45 is given as:
𝐶𝐷, 𝑓 𝑚 ,𝑅𝑒 =𝐶𝐷, 𝑓 𝑚
1+𝐶′
𝐷, 𝑓 𝑚
1.63 −1𝑅𝑒
45
𝐶𝐷, 𝑓 𝑚 =(1+2𝑠2)exp(−𝑠2)
𝑠3√𝜋+(4𝑠4+4𝑆2−1)erf(𝑠)
2𝑠4+2
3𝑠𝜋𝑇𝑝
𝑇∞
𝐶′
𝐷, 𝑓 𝑚 =𝐶𝐷 , 𝑓 𝑚;𝑇𝑝=𝑇∞=(1+2𝑠2)exp(−𝑠2)
𝑠3√𝜋+(4𝑠4+4𝑆2−1)erf(𝑠)
2𝑠4
𝐶𝐷, 𝐾 𝑛, 𝑅𝑒 =24
𝑅𝑒 1+0.15𝑅𝑒0.687 𝑓𝐾 𝑛
𝑓𝐾 𝑛 =1+𝐾 𝑛(2.514 +0.8exp(−0.55𝐾 𝑛)−1
(21)
where
𝑅𝑒
,
𝑠
,
𝑀𝑎
, and
𝐾𝑛
represents the particle Reynolds number, speed ratio, Mach number and Knudsen
number respectively. Note that [
14
]’s model (Eq. 14) accounts for the gas rarefaction and compressibility in the
vicinity of the spherical particulate reduces to CG model for 𝐻 𝑀 , 𝐶 𝑀 , 𝐺 𝑀 =1.
The particles in dense particle distributions of high particle concentration is modeled using the model by Tenneti
et al.[23]:
F𝐷(𝜃, 𝑅𝑒)=24
𝑅𝑒
𝑓isol(𝑅𝑒)
(1−𝜃)3+𝑓𝜃(𝜃)+𝑓𝜃,𝑅𝑒 (𝜃, 𝑅𝑒)
(22)
where 𝜙is the particle volume fraction and 𝐹isol(𝑅𝑒)is the isolated sphere drag law by Schiller[24]:
𝑓isol(𝑅𝑒)=1+0.15𝑅𝑒0.687 (23)
The other two terms in the formulation are given as:
𝑓𝜃(𝜃)=5.81𝜃
(1−𝜃)3+0.48 𝜃1/3
(1−𝜃)4
𝑓𝜃,𝑅𝑒 (𝜃, 𝑅𝑒)=𝜃3𝑅𝑒0.95 +0.61𝜃3
(1−𝜃)2(24)
2) Force on the particle due to gas velocity gradient is given by Saffman[15] as:
FSaff =𝐶𝑠
4𝑑2
𝑝(𝜇𝑔𝜌𝑔)1/2𝜔𝑔−1/2(u𝑔−u𝑝)×𝜔𝑔(25)
where 𝐶𝑠=6.44 is the Saffman coefficient.
𝜔𝑔=∇×
𝑢𝑔is the gas vorticity.
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3)
The force on the particle due to gas temperature gradient
∇𝑇𝑔
is given by the thermophoretic force given by Loth
et al.[14] as follows:
Fthermo =𝐹∇𝑇, 𝐾 𝑛>>1𝐹∗
∇𝑇
=−𝜋
2𝜇𝑔𝜈𝑔
𝑑𝑝
𝐾 𝑛
∇𝑇𝑔
𝑇𝑔
𝐾𝑛1.7
1.15 +𝐾𝑛1.7
(26)
4) Magnus force on the particle is given by Rubinow and Keller[25] as:
FMag =𝜋
8𝑑3
𝑝𝜌𝑔1
2∇×u𝑔−𝜔𝑝×(u𝑔−u𝑝)(27)
5)
Force on the particulate due to the spatial gas pressure gradient in the vicinity of the particulate, pressure gradient
force [26], is given as:
F𝑝𝑔 =∮𝑐𝑆 −𝑃ˆ
n𝑑𝑆 (28)
F𝑝𝑔 =∮𝑐𝑉 −∇. 𝑃 𝑑𝑉 (29)
By assuming the pressure gradient to be constant in the vicinity of the particulate (30)
F𝑝𝑔 =−∇.𝑃𝑉𝑝(31)
where the volume of the spherical particulate is given by 𝑉𝑝.
6)
The force on the particulate distribution due to the multiple inter-particle interactions is given by Harris et al.[
2
]
expressed as:
F𝑔𝑟 𝑎𝑛 =1
𝜌𝑝𝜃𝑝∇𝜏𝑝(32)
𝜏𝑝=𝑃𝑠𝜃𝛽
max[𝜃𝑐 𝑝 −𝜃, 𝜖(1−𝜃)] (33)
where
𝜃𝑐 𝑝
is the closed packing factor,
𝜃
is the particle volume fraction.
𝑃𝑠∼O(105)
,
1≤𝛽≤5
and
𝜖∼O(10−7)
are constants in the model. The scaling study by Ling et al.[
3
] shows that the parametric space for
𝑃𝑠
and
𝛽
given 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
which is at the expense of work done by the body. This additional work relates to the virtual mass effect or added
mass effect given for compressible regimes as:
F𝑎𝑚 =𝑉𝑝𝐶𝑀(𝑀 𝑎, 𝜃)−∇𝑃−𝑑(𝜌𝑓u𝑝)
𝑑𝑡 (34)
where
𝑉𝑝
is the particle volume,
𝐶𝑀(𝑀𝑎, 𝜃)=𝐶𝑀, 𝑠𝑡 𝑑 𝜂1(𝑀 𝑎)𝜂2(𝜃)
;
𝐶𝑀, 𝑠𝑡 𝑑 =12
is the standard coefficient
for added mass force.
𝜃
is the particle volume fraction. The compressibility and volume fraction coefficients are
given by Parmer et al[27]. and Zuber et al.[28] as follows:
𝜂1(𝑀𝑎)=1+1.8𝑀 𝑎2+7.6𝑀 𝑎4(35)
𝜂2(𝜃)=1+2𝜃(36)
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