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Phys. Fluids 34, 113301 (2022); https://doi.org/10.1063/5.0101365 34, 113301
© 2022 Author(s).
Effect of particle arrangement and density
on aerodynamic interference between twin
particles interacting with a plane shock wave
Cite as: Phys. Fluids 34, 113301 (2022); https://doi.org/10.1063/5.0101365
Submitted: 31 May 2022 • Accepted: 23 September 2022 • Accepted Manuscript Online: 25 September
2022 • Published Online: 01 November 2022
Shun Takahashi (高橋俊), Takayuki Nagata (永田貴之), Yusuke Mizuno (水野裕介), et al.
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Effect of particle arrangement and density
on aerodynamic interference between twin
particles interacting with a plane shock wave
Cite as: Phys. Fluids 34, 113301 (2022); doi: 10.1063/5.0101365
Submitted: 31 May 2022 .Accepted: 23 September 2022 .
Published Online: 1 November 2022
Shun Takahashi (高橋俊),
1,a)
Takayuki Nagata (永田貴之),
2,a)
Yusuke Mizuno (水野裕介),
1
Taku Nonomura (野々村拓),
2
and Shigeru Obayashi (大林茂)
3
AFFILIATIONS
1
Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan
2
Tohoku University, 6-6-01 Aramaki, Aoba-ku, Sendai, Miyagi 980-8579, Japan
3
Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan
a)
Authors to whom correspondence should be addressed: takahasi@tokai-u.jp and nagata@tohoku.ac.jp
ABSTRACT
Unsteady drag, unsteady lift, and movement of one or two moving particles caused by the passage of a planar shock wave are investigated
using particle-resolved simulations of viscous flows. The particle motion analysis is carried out based on particle-resolved simulations for
one or two particles under a shock Mach number of 1.22 and a particle Reynolds number of 49, and the particle migration and fluid forces
are investigated. The unsteady drag, unsteady lift, and particle behavior are investigated for different densities and particle configurations.
The time evolution of the unsteady drag and lift is changed by interference by the planar shock wave, Mach stem convergence, and the shock
wave reflected from the other particle. These two particles become closer after the shock wave passes than in the initial state under most con-
ditions. Two particles placed in an in-line arrangement approach each other very closely due to the passage of a shock wave. On the other
hand, two particles placed in a side-by-side arrangement are only slightly closer to each other after the shock wave passes between them. The
pressure waves resulting from Mach stem convergence of the upstream particle and the reflected shock waves from the downstream particle
are the main factors responsible for the force in the direction that pushes the particles apart. The wide distance between the two particles
attenuates these pressure waves, and the particles reduce their motion away from each other.
V
C2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://
creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/5.0101365
I. INTRODUCTION
Compressible particle-laden flows are found in many situations,
including rocket exhaust,
1
hybrid rocket motor combustion,
2
dust
explosions,
3
explosive dispersion of particles,
4–7
lithotripsy
treatment,
8–10
and shot peening.
11
Prediction of particle motion and
the effects of particles on the flow field are critical issues, but particle
behavior is different from that in incompressible particle-laden flows
because the flow over particles is governed by compressible low-
Reynolds-number conditions.
The characteristics of compressible flow over a particle have been
investigated experimentally and numerically. The drag coefficient was
measured through free-flight experiments for a wide range of
Reynolds and Mach numbers.
12–18
These drag data were assembled,
and particle drag models for numerical simulations of particle-laden
flows were constructed.
19–23
Further detailed investigations of the
Mach number effects on flow properties, such as flow stability,
24–26
vortex structures, drag coefficients,
27–29
the heat transfer characteris-
tics,
30
and the lift force due to particle rotation,
31
have been conducted.
Recently, numerical results obtained by Nagata et al.
29
were incorpo-
rated, and the drag model was improved.
23
Particles in a high-speed flow experience interaction with shock
waves. Tanno et al.
32
and Sun et al.
33
investigated unsteady drag both
experimentally and numerically. They found that the unsteady drag
coefficient is several times higher than the steady drag coefficient. In
addition, the convergence of the Mach stem at the rear stagnation
point of the particle was shown to provide instantaneous drag reduc-
tion. This phenomenon becomes more apparent as the Reynolds num-
ber increases, and the drag may become instantaneously negative.
Jourdan et al.
34
visualized the interaction of suspended particles by a
spider web with a planar shock, and the drag coefficient was estimated
Phys. Fluids 34, 113301 (2022); doi: 10.1063/5.0101365 34, 113301-1
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CAuthor(s) 2022
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by analyzing the trajectories of the particles. This kind of experiment
cannot capture instantaneous drag but can obtain the short-time-
averaged drag coefficient. Similar experiments were conducted with
particles placed on a shock tube floor,
35,36
with free-falling particles.
37,38
Although particle-laden flow is modeled based on knowledge of an iso-
lated particle, the effect of complex flow and particle motion generated
by aerodynamic interference between particles becomes important as
the number density of particles increases.
The change in the flow field due to particles has been investigated
using particle-resolved simulations. For compressible flow, the interac-
tion between a planar shock and a particle cloud has been studied, and
the change in the pressure profile, shock formation in front of the par-
ticle cloud, and the velocity fluctuation introduced by the particles
have been investigated.
39–44
Hosseinzadeh-Nik et al.
41
investigated the
unsteady characteristics of a flow produced by the impact of a shock
wave on many fixed particles using two-dimensional computational
fluid dynamics (CFD) and immersed boundary methods and the
velocity fluctuations, vorticity and kinetic energy were discussed.
Osnes et al.
42
carried out a numerical analysis of fixed particle curtains
using large eddy simulation (LES) with a three-dimensional unstruc-
tured mesh, varying the Mach number of the shock wave and the par-
ticle volume fraction, and the applicability of the sight-length index
and the generation mechanism for Reynolds stress was discussed.
Their results showed that the Reynolds stress was mainly caused by
reflection of the shock wave and the separation of the particle wake.
They also identified a tendency for attenuation of planar shock waves
passing through a stationary particle cloud for different particle
Reynolds numbers.
45
The interaction between particles may affect the aerodynamic force
coefficients for the particles. Sridharan et al.
46
identified a difference in
unsteady drag between isolated particles and multiple particles through
inviscid particle-resolved simulations. The maximum unsteady drag
coefficient for particles with an in-line arrangement interacting with a
shock wave decreases with an increasing Mach number, and the drag
coefficient for downstream particles was smaller than that for upstream
particles. Das et al.
47
studied the flow induced by a Mach-3.5 shock
wave passing through approximately 160 fixed particles with a particle
Reynolds number of 500 using the immersed boundary method through
two-dimensional particle-resolved simulations. They examined the aver-
age drag coefficient around many fixed particles in inviscid and viscous
flows and found that the viscous effect is small. The motion of particles
driven by the normal shock was also studied with moving particles.
These studies investigated the thermal and velocity relaxation of the
flow field.
48–51
Furthermore, machine learning-based mathematical
modeling has been studied for unsteady drag forces of multiple par-
ticles.
52,53
More recently, drag models have been reported for a wide
range of Mach numbers, Reynolds numbers, and volume fractions, con-
sidering interparticle interference in a steady flow.
54
The aerodynamic interference of multiple particles in a flow field
can produce a lift force, that is, not generated for isolated particles and
may affect the particle distribution. Thus, the lift coefficient is also
important for understanding aerodynamic interference between par-
ticles. Zarei et al.
55
investigated drag and lift forces by performing an
inviscid particle-resolved simulation of two oblique particles in a uni-
form flow at Mach 3.5. Three static zones of aerodynamic influence in
the wake of the lead particle were identified, and were denoted as the
entrainment, lateral attraction, and ejection zones. Laurence et al.
56
conducted a combined numerical and experimental study on unsteady
separation for two spheres in a Mach-4 flow. Mehta et al.
57
investi-
gated the lift coefficients for cylinders in a side-by-side arrangement
and in a particle transverse array arrangement using inviscid particle-
resolved simulations, and the influence of the Mach number and inter-
particle distance on the lift coefficient was clarified. Xiao et al.
58
studied
the interaction of planar shock waves with 300 randomly clustered
particles and showed that significant lateral forces can be generated
for particle clusters with high volume fractions. Although the flow over
an isolated particle and multiple particles has been studied in detail,
analyses of aerodynamic particle–particle interactions are necessary
to understand a complex particle-laden compressible flow better. In
particular, there are influences from the shock wave, the recompression
wave, and the expansion wave. Particle–particle interactions caused by
thesewavesareconsideredtobesignificant,eventhoughtheinter-
particle distance is large. In addition, most studies have been based on
particle-resolved simulations for systems in which particles are fixed. In
the case of moving particles, two-dimensional or inviscid simulations
must be used.
46,59
In the present study, the flow over unfixed twin particles interact-
ing with a planar shock was investigated by particle-resolved simula-
tions, and aerodynamic interference between the particles in the
compressible flow was examined for a simplified situation. The flow
simulation was performed by solving the equation of motion consider-
ing the migration of particles and the immersed boundary method
considering the moving boundary. The influence of the aerodynamic
interaction between particles was investigated by analyzing the effect
of particle density, interparticle distance, and particle arrangement on
flow properties such as flow structure, aerodynamic force coefficients,
and particle behavior.
The remainder of the present paper is organized as follows. Section
II describes the governing equations for the fluid and particles, as well as
the numerical methods. Section III provides the numerical results for a
fixed isolated particle interacting with a planar shock, and the numerical
code used in the present study is validated by comparing with the previ-
ous numerical results. In Sec. IV,theinfluenceofparticledensity,inter-
particle distance, and particle arrangement on the flow structure around
the two particles and the drag and lift coefficients are discussed. In Sec.
V, the influence of the particle arrangement is analyzed and discussed in
detail. Finally, conclusions are presented in Sec. VI.
II. NUMERICAL METHOD
A. Governing equations
The governing equations are the non-dimensionalized three-
dimensional compressible Navier–Stokes equations. The following
equations are the mass, momentum, and energy conservation equa-
tions, respectively:
@q
@tþ@
@xj
quj
ðÞ
¼0;(1)
@
@tqui
ðÞ
þ@
@xj
quiujþpdij
¼M
Re
@sij
@xj
;(2)
@
@tqE
ðÞ
þ@
@xj
qEþp
ðÞ
ujþqj
¼M
Re
@
@xj
uisij
ðÞ
:(3)
Here, the variables position x
i
,velocityu
i
,timet,totalenergyE,den-
sity q, pressure p, and viscosity lare non-dimensionalized by the
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Phys. Fluids 34, 113301 (2022); doi: 10.1063/5.0101365 34, 113301-2
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particle diameter d
p
, the reference sound speed c
0
, the dimensionless
time dp=c0, the square of the reference sound speed c2
0, the reference
density q
0
, the reference dynamic pressure q0c2
0, and the reference vis-
cosity l
0
, respectively. The particle Reynolds number and the shock
Mach number are Re ¼q0u0dp=l0and M¼u0=c0,respectively,
where u
0
is the velocity behind the shock wave. The non-
dimensionalized equation of state for an ideal gas [Eq. (4)], the viscous
stress [Eq. (5)], the heat flux [Eq. (6)], and Sutherland’slaw[Eq.(7)]
for obtaining various quantities in the Navier–Stokes equations are
expressed as follows:
p¼c1
ðÞ
qE1
2quiui
;(4)
sij ¼l@ui
@xj
þ@uj
@xi
2
3
@ui
@xi
dij
!
;(5)
qi¼lc
c1
M
RePr
@
@xi
p
q
;(6)
l¼T
Tref
3
2Tref þS
TþS;(7)
where all variables in the equation above, except for S,Tand Tref are
non-dimensional. The variables c,Pr,andSin the equations are the
specific heat ratio, the Prandtl number, and the individual gas con-
stants, respectively, which are assumed to be 1.4, 0.72 and 113 K,
respectively, for air. An inviscid flux is evaluated using a hybrid
scheme that combines a pseudo skew-symmetric central difference
scheme
60
with a monotonic upstream-centered scheme for conserva-
tion laws (MUSCL)-Roe scheme
61,62
proposed by the authors,
63
and
time integration is performed using a third-order accurate strong sta-
bility preserving Runge-Kutta scheme.
64
The accuracy of these devel-
oped numerical methods has been investigated in previous studies on
flow fields around particles in two and three dimensions.
63,65
B. Immersed boundary method
The numerical method used in the present study is an
Euler–Euler type particle-resolved simulation based on the immersed
boundary method. The computational mesh is a nonequally spaced
Cartesian mesh, and the boundary of the object is represented using
the immersed boundary method.
66–69
In the immersed boundary
method, the object boundary is represented by a level set function. The
level set function /0
iat position x
i
for grid index ifor a spherical parti-
cle with a center coordinate X
0
and radius r
p
used in the present study
is expressed as follows:
/0
i¼jxiX0jrp:(8)
All computational cells are classified as fluid cells (FCs), object cells
(OCs), or ghost cells (GCs) based on the value of this level set function
/i.Figure 1 shows a schematic diagram of the arrangement of classi-
fied cells, an image point (IP), and a point on the interface boundary
(IB). The flow variable for the IP is determined by trilinear interpola-
tion using the values of the eight FCs that surround the IP in three-
dimensional space. The values for the flow variables for the IP are
determined from the four FCs surrounded by the thick gray line in
Fig. 1.TheGCvelocityu
GC
is determined by the following equation
using u
IP
and U
IB
:
uGC ¼uIP DIP þj/GC j
DIP
uIP UIB
ðÞ;(9)
where /GC and D
IP
are the level set function for the ghost cell and the
distance between the image point and the interface boundary, respec-
tively. The value D
IP
is 1.75 times the grid cell width, and no recurrent
references are allowed. The density, the pressure, and the energy for
the ghost cell are determined by the following equations:
qGC ¼qIP;(10)
pGC ¼pIP;(11)
EGC ¼1
c1
pGC
qGC
þ1
2uGCuGC :(12)
The immersed boundary method based on these GCs represents a
nonslip and adiabatic wall boundary. The accuracy of these numerical
methods has been investigated in previous studies by the authors.
63,65
C. Equation of motion
The motion of the particles is tracked by the equations of motion
with fluid forces obtained from particle-resolved simulations. The
motion of the particles is solved by the equation of motion for a three-
degree-of-freedom translation as follows:
mp
dUj
dt ¼Fj;(13)
Fj¼ðC
pdij þl@ðuiUiÞ
@xj
þ@ðujUjÞ
@xi
!"#
njdS;(14)
where m
p
,U,F,C,n, and dSare the mass of the particles, the velocity
at which the particles move, the aerodynamic force, the cell boundaries
of the GCs and the FCs used in the immersed boundary method, the
unit normal vector for the cell boundaries, and the area of the bound-
ary cells, respectively. The mass of the particles is expressed as
mp¼4=3pðdp=2Þ3. The fluid forces acting on the particles are
FIG. 1. Schematic diagram of immersed boundary method. In the figure, GC repre-
sents a ghost cell, IB represents an interface boundary point, and IP represents an
image point.
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calculated using a simple surface stress calculation method proposed
in a previous study
70
that does not require polygonal geometry. The
gravitational force cannot be neglected for the practical motion of par-
ticles. However, the present study attempts to quantitatively clarify the
effects of aerodynamic interference between two particles. Therefore,
the present study considers only the fluid force for the motion of par-
ticles without taking gravity into account. Equation (13) is solved as
follows:
DU¼DtFn
j
mp
;(15)
Unþ1
j¼Un
jþDU;(16)
Xnþ1
j¼Xn
jþDtUn
jþ1
2DtDU;(17)
where Dtand X
j
are the time increment and the vector for particle
centroids, respectively. The level set function at each grid point is
updated using Eq. (8) after the particle center location is determined
using Eq. (17), and the GCs are rearranged for the flow simulation.
The accuracy of this numerical method has been validated in previous
studies.
71,72
We performed particle-resolved simulations considering
only translational motion, and not rotational motion, as in several pre-
vious studies.
39,46,47,73
On the other hand, the lift induced by particle
rotation depends on the volume fraction of the particles and also varies
with time in the case of a dusty shock flow containing a large number
of particles.
74,75
Therefore, six-degree-of-freedom motion must be
investigated in the future.
D. Data analysis
In the present study, the time tis evaluated using t¼t=s,nor-
malized by the characteristic time s¼dp=u0. The drag and lift coeffi-
cients are calculated using the following equations:
CD¼Fx
1
2q2u2
2pr2
p
;(18)
CL¼Fz
1
2q2u2
2pr2
p
;(19)
where r
p
is the radius of the particle, and q
2
and u
2
are the density and
velocity, respectively, of the flow behind the shock wave. The fluid
forces F
x
and F
z
in the xand zaxis directions are obtained by integrat-
ing the pressure and shear stress over the surface of the particle using
Eq. (14). These numerical results were evaluated based on procedures
used in previous studies.
33,46,76
The particle density q
p
is set to 5q
2
,
50q
2
, and 500q
2
for the analysis of moving particles discussed in Secs.
IV and V, and the effect of the particle weight is investigated. The dis-
tance between the centers of the two particles in the initial state is set
at 1.5d
p
,2.0d
p
,and2.5d
p
in the case of the analyses of two moving par-
ticles, and interference between the particles is investigated.
III. VALIDATION OF NUMERICAL METHOD
FOR STATIONARY ISOLATED PARTICLE
The accuracy of the numerical method was validated by analyz-
ing the shock wave impinging on a fixed particle under the same con-
ditions as in the previous study.
33
The shock Mach number and the
particle Reynolds number were Ms¼1:22 and Re
p
¼49, respectively.
Convergence was checked using three different grid resolutions with
particle diameters divided into 20 (CPD20), 40 (CPD40), and 80
(CPD80) grid cells. Table I,Table II,andFig. 2 show the calculation
conditions and the computational domain. The present calculation
was conducted using a distributed-memory parallel computer system
(2.7 PFLOPS theoretical performance) on a supercomputer at the
Advanced Fluid Information Research Center in the Institute of Fluid
Science at Tohoku University. The required computation time for the
CPD40 analysis was approximately 5 h, based on eight regions divided
for the Message Passing Interface (MPI) parallel computation and 160
threads of parallel computation within each region.
Figure 3 shows the time history of the drag coefficient associated
with shock wave propagation over the particle. The time of impact of
theshockwaveontheparticleisshiftedtozero.Theresultsfor
CPD40 and CPD80 are in good agreement with previous results
obtained by Sun et al.
33
The maximum values of the unsteady drag are
slightly lower than those previously obtained for all grid resolutions.
This underestimation of the peak C
D
is considered to be due to the
adaptive mesh refinement used in the numerical simulations of Sun
et al. The computational grid near the wall in their previous study is
much finer than that in the present study. Convergence was confirmed
and the entire history of unsteady drag in the shock–particle interac-
tion, which is the target of the present study, can be reproduced using
the present grid resolution. Hence, all of the subsequent numerical
analyses are performed with a grid size that divides the particle diame-
ters into 40 cells (CPD40).
IV. SIMULATION FOR MOVING PARTICLES
INTERACTING WITH PLANAR SHOCK
A. Isolated particle
A flow analysis around a single moving particle subjected to unsteady
drag was performed under the initial conditions shown in Table I.
TABLE I. Initial conditions.
Flow parameter Pre-shock Post-shock
Density q1¼1:4q2¼1:9269
Pressure p1¼1p2¼1:5698
Velocity u1¼0u2¼0:3361
Sound speed c1¼1c2¼1:0680
Viscosity l1¼0:013 217 l2¼0:013 217
FIG. 2. Computational grid used for analysis of isolated particles with diameters
divided by 20 cells. Every second grid line is drawn.
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The migration of particles was also investigated by varying the particle
density q
p
as 5q
2
,50q
2
and 500q
2
.
Figures 4–6show the drag coefficient, the velocity, and
the distance traveled by the particles as they move, respectively. The
time of impact of the shock wave on the particle is shifted to zero.
Figure 4 shows that the peak drag coefficient is clearly reduced only
for the lightest particles. The drag coefficients for the heavier particles
are almost the same as those for the fixed particle. Figure 5 shows the
change in the migration velocity of the particles over time. It illustrates
that the increase in velocity is reduced after t¼2. Mach stem conver-
gence at the stagnation point in the tail of the particle produces this
slowdown. This Mach stem convergence under high-Reynolds-num-
ber conditions can also generate an instantaneous negative drag.
33
The
time variation of the migration velocity, however, has little effect on
the migration distance of the particles. The reason is the very short
duration of the unsteady drag. Furthermore, lighter particles migrate
overs longer distances in inverse proportion to the density of the par-
ticles. The time histories of particle velocity and distance traveled
obtained in the present study exhibit a similar trend to the results of
Parmar et al. who predicted particle movement using a drag model,
76
FIG. 3. Time history of drag coefficient for a stationary isolated particle.
TABLE II. Computational grids.
Case
a
Grid width Number of grid cells
CPD20 0.05d
p
1.3 M ð160 90 90Þ
CPD40 0.025d
p
5.1 M ð260 140 140Þ
CPD80 0.0125d
p
26M ð460 240 240Þ
a
CPD: cells per diameter.
FIG. 4. Time history of drag coefficient for moving particle.
FIG. 5. Time history of translational velocity for moving isolated particle.
FIG. 6. Time history of travel distance for moving isolated particle.
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and Behrendt et al., who studied particle movement using an inviscid
CFD analysis.
77
Figures 7 and 8show the instantaneous pressure distributions
at t¼1:26 and 3.92, respectively. The lower left, upper left, upper
right, and lower right panels in both figures show the pressure dis-
tributions around a fixed particle and a moving particle with rela-
tive densities of 5q2;50q2, and 500q2, respectively. The particle
moves only slightly at t¼1:26, just after the drag coefficient
reaches its peak, and the flow field around the particle is no differ-
ent from that around the fixed particle. The lightest particle moves
approximately 0.2d
p
at t¼3:92, and the high-pressure region
around the particle becomes smaller. The local pressure distribu-
tion in the vicinity of the particles changes for different particle
densities. The lightest particles are immediately driven by the shock
wave. Therefore, the overpressure on the upstream side of the parti-
cle is clearly relaxed compared to the other results.
B. Particle movement when shock wave impinges
on two particles placed in side-by-side arrangement
Here, we consider two spherical particles placed laterally with
respect to the direction of shock wave propagation (side-by-side
arrangement) to the flow, as shown in Fig. 9, with a distance between
their centers of 1:5dp;2:0dp,and2:5dp. The relative densities of the
particles are 5q2,50q2,and500q2based on the density q
2
behind the
shock wave. Figure 10 shows the time history of the drag coefficient
for the lower particle. The red, green, and blue lines represent the
results for particle densities of 500q
2
,50q
2
, and 5q
2
, respectively. In
addition, the dotted, dashed, and solid lines represent the results for
initial center-to-center distances of 2.5d
p
,2.0d
p
, and 1.5d
p
, respectively.
The time of impact of the shock wave on the particles is shifted to
zero. Figure 10 shows that the unsteady drag coefficient for two side-
by-side particles is almost the same as the value for a single isolated
particle. Figure 11 shows the time history of the lift coefficient of the
lower particles. Independent of the relative density and the interparti-
cle distance, the lower particle is first subjected to a negative lift, which
is subsequently reversed to a positive lift. The unsteady lift acts initially
in the direction of increasing distance between the two particles and
then in the direction of decreasing distance. The smaller the initial dis-
tance between the two particles, the larger the peak unsteady lift coeffi-
cient. Furthermore, the smaller the initial distance between the two
particles, the earlier the peak of the unsteady lift appears and con-
verges. The particle density has almost no effect on the unsteady lift
coefficient and only slightly mitigates the peak in the case of the light-
est particles.
Figure 12 shows that the high- and low-pressure regions act on
the interaction side and outside of the two particles, respectively, and,
FIG. 7. Instantaneous pressure distribution on x–zcross section through particle
center at t¼1:26. The dashed line represents the initial position of the particle
(lower left: fixed, upper left: 5q2, upper right: 50q2, lower right: 500q2).
FIG. 8. Instantaneous pressure distribution on x–zcross section through particle
center at t¼3:92. The dashed line represents the initial position of the particle
(lower left: fixed, upper left: 5q2, upper right: 50q2, lower right: 500q2).
FIG. 9. Computational grid for side-by-
side arrangement, coarsened in CPD20
for visualization. Every second grid line is
drawn.
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consequently, that the forces act in the direction of increasing particle
separation when t¼1.9. Figure 13 illustrates that the low- and high-
pressure regions around the particle act on the interaction side and the
outside, respectively, and the forces act in the direction of increasing
separation between the two particles, for the case which t¼4.0.
Naiman and Knight performed a numerical analysis of the attenuation
of a planar shock wave for an array of cylinders in a viscous flow.
6
They focused not on the fluid force but rather on the flow structure
and the peak pressure created by the shock wave hitting the array. The
recirculation region around the particles in the present study is more
diffuse than that reported by Naiman et al. because the Reynolds num-
ber in the present study is lower. Mehta et al. analyzed a flow field in
which a planar shock wave interacts with particles under slightly dif-
ferent configurations and flow conditions than those in the present
study. They showed that a narrow spacing between particles can lead
to large shock wave interference and changes in drag and lift.
57
The
interference of the reflected shock wave in front of the particle cap-
tured in the present study is similar to the result that they reported.
C. Particle movement when shock wave impinges
on two particles with in-line arrangement
The aerodynamic coefficients and flow fields for two particles
placed longitudinally with respect to the direction of shock wave prop-
agation (in-line arrangement) were next investigated. Figures 14 and
15 show the time history of the drag coefficient for the upstream and
downstream particles, respectively. The time of impact of the shock
wave on the upstream particle is shifted to zero. The unsteady drag
coefficient for the upstream particle before the arrival of the reflected
shock wave from the downstream particle at approximately t¼3, as
shown in Fig. 14, is the same as that for a single isolated particle. The
drag coefficient for the closest and lightest upstream particle shown in
Fig. 14 temporarily becomes negative because the reflected shock wave
FIG. 10. Time history of drag coefficient for particle located in negative zaxis direc-
tion for side-by-side arrangement.
FIG. 11. Time history of lift coefficient for particle located in negative zaxis direc-
tion for side-by-side arrangement.
FIG. 12. Instantaneous pressure distribution on x–zcross section through particle
center at t¼1.9 for side-by-side arrangement. The dashed line represents the ini-
tial position of the particle. (left: 1.5d
p
_5q
2
, right: 2.5d
p
_5q
2
.)
FIG. 13. Instantaneous pressure distribution on x–zcross section through particle
center at t¼4.0 for side-by-side arrangement. The dashed line represents the ini-
tial position of the particle. (left: 1.5d
p
_5q
2
, right: 2.5d
p
_5q
2
.)
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from the downstream particle reaches the upstream particle and
changes its drag coefficient significantly. The lighter the particle, the
more quickly the shock wave reflected from the downstream particles
reaches the upstream particle. Furthermore, the larger the initial dis-
tance between the two particles, the smaller the effect of the reflected
shock wave. The time history of the unsteady drag coefficient for
downstream particles is shown in Fig. 15, and lighter particles produ-
ces more quickly attenuated peaks. Figure 15 also shows that the
greater the distance between particles in the initial state, the greater the
peak unsteady drag coefficient for the downstream particle. This
enhancement of the peak drag coefficient for the downstream particle
is caused by the convergence of the Mach stem that occurs directly
behind the upstream particles. A similar situation was studied by
Sridharan et al.,
46
but they focused on the effect of the Mach number
on particle arrays by means of inviscid flow simulations. However,
they also concluded that the unsteady drag coefficient for the down-
stream particles was greater than that for the upstream particles. The
current study investigates two moving particles and flow fields caused
by a shock wave using viscous simulations. Mehta et al. noted that the
dissipation of viscous effects is significant for a flow field under low-
shock-Mach-number conditions.
78
This viscous effect possibly sup-
presses the increase in unsteady drag in the present study.
Figure 16 shows the time history of the distance between the two
particleswhentheyareinitiallyplacedalongthepropagationdirection
of the shock wave. The variable DXisthetimeevolutionofthedis-
tance between the center position x
1
of particle 1 and the center posi-
tion x
2
of particle 2, expressed by the following equation:
DX¼XX0;(20)
where Xis the distance between the centers of the two particles obtained
from X¼jx1x2j.NegativevaluesofDXindicate that the two par-
ticlesarecloserthantheywereintheinitialstate.Thereductionininter-
particle distance caused by unsteady drag before t¼3 is independent
of the interparticle distance for all conditions, whereas that after t¼3
is dependent on the interparticle distance. The unsteady drag generated
by the shock wave impinging on the upstream particles causes a reduc-
tion in the interparticle distance before t¼3inFig. 16.Interferenceof
the shock wave with the upstream particle is independent of the distance
between the two particles, because this interference occurs before the
shock wave reaches the downstream particle. For two particles that are
initially closely spaced, the reduction in their separation is smaller after
t¼3inFig. 16. The reflected shock wave generated by the downstream
particles reaches the upstream particle immediately when the interparti-
cle distance is small and this attenuation occurs immediately. Two par-
ticles with an initially large interparticle distance eventually become
significantly closer, because the reflected shock wave is attenuated dur-
ing propagation in the viscous flow.
FIG. 14. Time history of drag coefficient for upstream particle with in-line
arrangement.
FIG. 15. Time history of drag coefficient for downstream particle with in-line
arrangement.
FIG. 16. Time history of distance between centers of two particles with in-line
arrangement.
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Figures 17 and 18 show instantaneous pressure distributions
at t¼3:8 and 5.8 around two particles with a relative density of
5q2for initial particle center-to-center distances of 1:5dpand 2:5dp,
respectively. Figure 17 shows reflected shock waves generated by
the downstream particle impinging on the upstream particle in the
case of 1:5dp_5q
2
, and the bottom panel of Fig. 18 illustrates a simi-
lar phenomenon. The smaller interparticle spacing results in a
strong reflected shock wave arriving from the downstream particle,
and the upstream particle suffers a significantly reduced/negative
drag.
Figure 18 shows that upstream particle migration is similar for
different interparticle distances, but downstream particle migration
is clearly greater for a small interparticle distance. The greater the
initial distance between the two particles, the greater the decrease in
the distance between the two particles after the shock wave impinges
on these particles. The reflected shock wave generated by the down-
stream particle propagates over a larger distance and becomes atten-
uated, and the force in the direction of increasing interparticle
distance diminishes.
D. Particle movement when shock wave impinges
on two diagonally placed particles
The behavior of two particles positioned diagonally at an angle of
45relative to the direction of propagation of the shock wave, and the
flow field around these particles were investigated. Figures 19 and 20
show the time histories of the drag coefficient for the upstream and
downstream particles, respectively. The time of impact of the shock
wave on the upstream particles is shifted to zero in the graph. The
upstream particle is initially aligned downward in the figure, i.e., in the
negative zaxis direction. Figures 19 and 20 show that the drag coeffi-
cients for the two diagonally placed particles exhibit a similar trend to
that for two particles placed along the direction of the shock wave, as
FIG. 17. Instantaneous pressure distribution on x–zcross section through particle
center at t¼3:8 for in-line arrangement. The dashed line represents the initial
position of the particle. (top: 1.5d
p
_5q
2
, bottom: 2.5d
p
_5q
2
.)
FIG. 18. Instantaneous pressure distribution on x–zcross section through particle
center at t¼5.8 for in-line arrangement. The dashed line represents the initial posi-
tion of the particle. (top: 1.5d
p
_5q
2
, bottom: 2.5d
p
_5q
2
.)
FIG. 19. Time history of drag coefficient for two diagonally placed particles, one of
which is placed upstream and in negative zaxis direction.
FIG. 20. Time history of drag coefficient for two diagonally placed particles, one of
which is placed downstream and in positive zaxis direction.
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shown in Figs. 14 and 15, respectively. The drag reduction for the
upstream particle and the peak drag coefficient for the downstream
particle are, however, smaller for the diagonal configuration.
Figures 21 and 22 show the time histories of the lift coefficient
for the upstream and downstream particles, respectively. The lift coef-
ficient for the lower particle on the zaxis shown in Fig. 21 shows a
similar trend to that for the lower particle in the side-by-side case
shown in Fig. 11.AlthoughFig. 11 shows that the two side-by-side
particles experience unsteady lift forces of the same magnitude and
that these forces always act in the opposite direction, Figs. 21 and 22,
which illustrate the time histories of the unsteady lift forces on the two
diagonally placed particles, show that their directions have the same
time variation. The first negative peak in the lift coefficient for the
upstream particle shown in Fig. 21 is caused by reflection of the shock
wave from the downstream particle. Figure 21 shows that the peak
unsteady lift coefficient relaxes more quickly and significantly for ligh-
ter particles, which is similar to the time variation observed in Fig. 11.
Figure 22 indicates that the peak negative lift coefficient for the down-
stream particle appears almost simultaneously with the peak of the
negative lift coefficient for the upstream particles.
Figure 23 shows the time history of the distance between two par-
ticles initially placed diagonally at an angle of 45relative to the direc-
tion of propagation of the shock wave. The distance between the
particles decreases with time for all conditions, as shown in Fig. 23,
and the two particles gradually approach each other.
Figure 24 shows the change in distance in the xaxis direction
between two particles at t¼7:0. Negative values indicate that the xaxis
distance has decreased from its initial value. This illustrates that the sepa-
ration between two light widely spaced particles decreases by approxi-
mately 20% of their diameter as the shock wave passes through them.
Figure 25 shows the change in the zaxis separation between two
particles at t¼7:0. Positive values represent an increase in separa-
tion after the shock wave has passed. Only the two lightest particles,
which are initially the furthest apart, reduce their separation after the
shock wave passes through them. The reflected shock wave generated
by the downstream particle increases the zaxis separation when the
distance between the two particles is small. On the other hand, the sep-
aration decreases slightly for two lighter, widely spaced particles. The
reduction in separation along the zaxis is, however, two orders of
magnitude smaller than that along the xaxis.
Figures 26 and 27 show the instantaneous pressure and Mach
number distributions for the central cross section at t¼2:8 for initial
distances between particles of 1:5dpand 2:5dp, respectively. Both par-
ticles receive a large negative lift at t¼2:8. Figure 26 shows that the
reflected shock wave generated by the downstream particle impinges
on the upstream particle, and the upstream particle receive a signifi-
cant downward lift. A compression region, on the other hand, is
observed upstream of the downstream particle. The Mach stem
FIG. 21. Time history of lift coefficient for two diagonally placed particles, one of
which is placed upstream and in negative zaxis direction.
FIG. 22. Time history of lift coefficient for two diagonally placed particles, one of
which is placed downstream and in positive zaxis direction.
FIG. 23. Time history of distance between centers of two particles placed
diagonally.
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convergence of the upstream particles cancels the compression region
for the downstream particle, and a low-pressure region is generated in
the area surrounded by the upstream and downstream particles. This
low-pressure region interacts with the downstream particles, and a sig-
nificant downward lift is induced. A higher flow velocity in the region
between the particles, which corresponds to the low-pressure region
described above, is observed at a distance of 1:5dpbetween the two
particles in the Mach number distribution shown in Fig. 27.Theshock
wave impinges on two moving particles placed diagonally, and these
particles move in the same direction at approximately the same
moment. Whether the two particles are closer as a result will be dis-
cussed in detail in Sec. V.
V. CONFIGURATION OF PARTICLES, UNSTEADY FLUID
FORCES, AND DISTANCE BETWEEN PARTICLES
We analyzed the unsteady fluid forces and particle migration
characteristics by defining the angle from upstream to downstream,
using the position of the particle on one side as the center position, as
shown in Fig. 28.Notethattheparticlesofinterestherearenotthe
center particle, but particles surrounding it. The particles with hfrom
90to 90areupstreamfromthecenterparticle,asshowninFig. 28,
because of the propagation direction of the shock wave is from left
to right.
Figure 29 shows the maximum drag coefficient for particles at
each position. The maximum drag coefficient arises from the shock
wave passing through the particles, and the heavier the particle, the
larger the value. Lighter particles reduce the overpressure on the
upstream side by moving immediately after the passage of the shock
wave, and their maximum drag coefficient also decreases. The maxi-
mum drag coefficient for the upstream particle due to the planar shock
wave is independent of the downstream particle, and the coefficient is
constant up to hof 90. The maximum drag coefficient for particles
FIG. 24. Variation of distance between two particles in xaxis direction from initial
state when t¼7:0.
FIG. 25. Variation of distance between two particles in zaxis direction from initial
state when t¼7:0.
FIG. 26. Instantaneous pressure distribution on x–zcross section through particle
center at t¼2:8. The dashed line represents the initial position of the particle.
(Left: 1.5d
p
_5q
2
, right: 2.5d
p
_5q
2
.)
FIG. 27. Instantaneous Mach number distribution on x–zcross section through
particle center at t¼2:8. The dashed line represents the initial position of the par-
ticle. (Left: 1.5d
p
_5q
2
, right: 2.5d
p
_5q
2
.)
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located downstream of h¼135with a smaller interparticle distance,
on the other hand, is lower than that for particles located upstream of
h90. The maximum drag coefficient for narrowly spaced down-
stream particles is damped by the cancelation between the Mach stem
convergence created around the upstream particle and the compres-
sion region created around itself. The effect of Mach stem convergence
is significant for downstream particles at h¼180,andtheirmaxi-
mum drag coefficient increases.
Figure 30 shows the minimum values for the lift coefficients for
particles at each location. The particles of interest here are located in
the positive zaxis direction with respect to the central particle, and the
negative lift coefficient indicates the force in the direction of reduced
distance from the center particle. Negative minimum lift coefficients
toward the center particle are produced at angles other than h¼0
and 180. The minimum lift coefficient exhibits a peak at h¼135,
and particles placed diagonally downstream are subjected to a signifi-
cant lift in the negative zaxis direction.
Figure 31 summarizes the maximum lift coefficient for particles
at each position. The particles of interest here are located in the posi-
tive zaxis direction with respect to the center particle, and a positive
lift corresponds to a force in the direction away from the central parti-
cle. The maximum value of this positive lift coefficient is found at an
upstream diagonal position of h¼45. The maximum lift coefficient
fortheupstreamparticlesiscausedbyreflectedshockwavesfrompar-
ticles located downstream (in the center).
This positive lift coefficient can increase the distance between the
particles and the central particle, but the distance between these par-
ticles will be discussed later, because the actual distance between them
is related to the motion of both particles. Figure 32 shows the change
in the distance between the centers of the two particles at t¼7:0
with respect to the angle h.ThedistanceDXat t¼7:0 is adopted,
because the time evolution of the distance between the two particles
FIG. 28. Angle between particles with respect to center particle.
FIG. 29. Maximum drag coefficient for particles at each position.
FIG. 30. Minimum lift coefficient for particles at each position.
FIG. 31. Maximum lift coefficient for particles at each position.
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becomes approximately stable. Negative values on the vertical axis
indicate a decrease from the original distance, i.e., the particles are
closer together. The distance between the two particles remains almost
unchanged at h¼90but it decreases significantly at 0and 180,as
shown in Fig. 32. Furthermore, Fig. 32 also shows that the distance
between the two particles has almost a linear dependence on h.
VI. CONCLUSIONS
The interaction between two particles was investigated using
particle-resolved simulations as a preliminary study of particle clouds
and particle curtains involving the interaction of multiple moving par-
ticles and shock waves. The particle-resolved simulations and the par-
ticle motion analyses were carried out for one or two particles with a
shock Mach number of 1.22, a particle Reynolds number of 49, and an
initial separation ranging from 1.5 to 2.5 times the particle diameter
under various configurations. The relative density of the two particles
varied from 5 to 500 times greater than that of gas. First, the grid
dependency of the numerical solution was examined by comparing
this solution with that obtained in a previous numerical study. Next,
the unsteady drag coefficient and behavior for a single isolated particle
were investigated by solving a flow field in which the particle was
moved by the shock wave. Finally, the unsteady drag, unsteady lift,
and behavior of two moving particles were investigated based on pas-
sage of a shock wave for several particle configurations. The following
findings were obtained from these numerical investigations.
•A planar shock wave impinges a free-moving particle, and the
particle is pushed with a large acceleration. The shock wave pro-
duces Mach stem convergence in the downstream region of the
particle immediately afterward, and the acceleration of the parti-
cle is reduced.
•Lighter particles move immediately in response to the passage of
the shock wave, and the pressure difference around the particles
relaxes immediately. The time history of the unsteady drag
coefficient almost converges when the density of the particles is
50 times larger than that of the gas.
•In most cases, the distance between two particles decreases after
the shock wave passes through them. The main reason is the
impulsion of the upstream particle by the shock wave.
Conversely, reflected shock waves generated by the downstream
particle affect the upstream particles and tend to increase the dis-
tance between the particles.
•The greater the initial distance between two particles, the greater
the reduction rate for the distance between the particles after the
passage of the shock wave. This is due to damping of the reflected
shock wave produced by the downstream particle over long prop-
agation distances.
•The unsteady lift coefficient is approximately two orders of mag-
nitude smaller than the unsteady drag coefficient under the flow
conditions in the present study. In other words, the effect of the
unsteady lift force between the two particles is negligibly small.
ACKNOWLEDGMENTS
The present study was supported in part by the Japan Society
for Promotion Sciences, KAKENHI, Grant Nos. JP18K03937 and
JP21K14071. T. Nagata was supported by the Japan Science and
Technology Agency through CREST Grant No. JPMJCR1763. Part
of the study was carried out under the Collaborative Research
Project of the Institute of Fluid Science, Tohoku University.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Shun Takahashi: Conceptualization (equal); Data curation (lead);
Formal analysis (equal); Funding acquisition (lead); Investigation
(lead); Methodology (lead); Project administration (lead);
Resources (lead); Software (lead); Supervision (lead); Validation
(lead); Visualization (lead); Writing –original draft (lead); Writing –
review & editing (lead). Takayuki Nagata: Conceptualization
(equal); Data curation (equal); Formal analysis (equal); Funding
acquisition (equal); Investigation (equal); Methodology (equal);
Project administration (equal); Resources (supporting); Software
(supporting); Supervision (equal); Validation (supporting);
Visualization (supporting); Writing –original draft (supporting);
Writing –review & editing (supporting). Yusuke Mizuno:
Conceptualization (equal); Data curation (equal); Formal analysis
(equal); Investigation (equal); Methodology (equal); Project
administration (supporting); Software (equal); Validation (sup-
porting); Visualization (supporting); Writing –original draft
(supporting); Writing –review & editing (supporting). Taku
Nonomura: Conceptualization (supporting); Data curation (sup-
porting); Formal analysis (supporting); Investigation (supporting);
Methodology (equal); Software (equal); Validation (supporting);
Visualization (equal); Writing –original draft (equal); Writing –
review & editing (equal). Shigeru Obayashi: Formal analysis (sup-
porting); Investigation (supporting); Resources (lead); Validation
FIG. 32. Variation of distance between two particles from initial state when
t¼7:0.
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CAuthor(s) 2022
(supporting); Writing –original draft (supporting); Writing –
review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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