Direct Numerical Simulation of Gas–Particle Flows with Particle–Wall

applied sciences Article

Direct Numerical Simulation of Gas–Particle Flows with Particle–Wall Collisions Using the Immersed Boundary Method Yusuke Mizuno 1, *, Shun Takahashi 2 , Kota Fukuda 3 and Shigeru Obayashi 4 1 2 3 4

*

Course of Science and Technology, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan Department of Prime Mover Engineering, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan; [email protected] Department of Aeronautics and Astronautics, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan; [email protected] Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8579, Japan; [email protected] Correspondence: [email protected]; Tel.: +81-0463-58-1211

Received: 13 October 2018; Accepted: 21 November 2018; Published: 26 November 2018







Abstract: We investigated particulate flows by coupling simulations of the three-dimensional incompressible Navier–Stokes equation with the immersed boundary method (IBM). The results obtained from the two-way coupled simulation were compared with those of the one-way simulation, which is generally applied for clarifying the particle kinematics in industry. In the present flow simulation, the IBM was solved using a ghost–cell approach and the particles and walls were defined by a level set function. Using proposed algorithms, particle–particle and particle–wall collisions were implemented simply; the subsequent coupling simulations were conducted stably. Additionally, the wake structures of the moving, colliding and rebounding particles were comprehensively compared with previous numerical and experimental results. In simulations of 50, 100, 200 and 500 particles, particle–wall collisions were more frequent in the one–way scheme than in the two-way scheme. This difference was linked to differences in losses in energy and momentum. Keywords: particulate flow; immersed boundary method; collision

1. Introduction The dynamics of the particle–wall and particle–particle collisions for Reynolds numbers below 1000 have been studied by various experiments and numerical simulations. Eames et al. [1] and Vanella et al. [2] explored the vortex structures generated by particle–wall collisions by the experiment and numerical simulation, and Griffith et al. [3] analyzed the dynamics associated with particle–particle collisions. Kajishima [4] and Deen et al. [5] conducted numerical simulations of the flows around multiple particles to investigate the clustering of particles and turbulence effects. Di Sarli et al. [6], using the numerical simulation based on the Eulerian approach for the fluid phase and the Lagrangian approach for the solid phase, studied the effect of the diameter of dust particles injected in a spherical vessel on the resulting turbulent flow field and dust distribution. However, the flow structures and the particle behavior of a system with multiple particle–wall collisions are still unclear. The shot peening process, which involves impacting a metal surface with a large number of shots (particles), is widely used in practical mechanical engineering processes. When the particles collide with the metal surface, a compressible residual stress is generated over its surface [7]. The important Appl. Sci. 2018, 8, 2387; doi:10.3390/app8122387

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parameter of this high-quality process is coverage level, which is defined as the percentage ratio of the total indentation area to the total treated surface area [8]. Kopp et al. [9] investigated the coverage level by noting the number of impressions. Garipy et al. [10] observed peen forming in experiments and numerical simulations. However, these studies focused on the plate, without adequately considering the interactions of the rebounding particles. Kato et al. [11] investigated the behavior of particles in experiments and multi-body dynamics simulations, however, they largely ignored the influence of the flow structure. Nguyen et al. [7] investigated the location of particle–surface impacts in the multiphase flow simulation. They employed the Euler–Lagrange approach and Reynolds-averaged Navier–Stokes equations, which cannot sufficiently model the particle–flow interactions and the detailed flow structures. Among the Cartesian mesh methods, the immersed boundary method (IBM) developed by Mittal et al. [12], has been widely used for its simplicity and applicability to multiple moving objects. We have developed an IBM–based two- and three-dimensional Euler–Euler compressible flow solver that analyzes the interaction between the particles and turbulence in a gas-particle flow [13,14]. In the present study, we developed IBM–based three-dimensional incompressible flow solver [15] for exploring the structures and behaviors of particles undergoing multiple particle–wall collisions. This present flow solver directly performs a numerical simulation using the Euler–Euler approach. This simulation can capture the particle–flow interactions and the detailed flow structures. The aim of this paper is to investigate the accuracy of the present IBM flow solver and to examine the flows around multiple particle–wall collisions to elucidate the interaction between the flow and particles using the Euler–Euler approach. To investigate the accuracy of the IBM flow solver, we computed the flow structure and the drag coefficient around a fixed sphere using different schemes, and compared the results of the solver with those of existing methods. We analyzed the flow when the moving particles impinge on a wall. Finally, for examining the interaction between the flow and a lot of moving particles, we compared the two-way coupled simulation with the one-way simulation that is generally applied in clarifying the particle kinematics in industry and in investigating the flow around particle–wall collisions. 2. Numerical Methods 2.1. Governing Equations The governing equations of the present flow simulations are the three-dimensional incompressible Navier–Stokes equations and the equation of continuity:  2  ∂ u ∂u ∂u 1 ∂p ∂2 u ∂2 u + u ∂u + v + w = − + ν + + , ρ ∂x ∂x ∂y ∂z ∂x2 ∂y2 ∂z2  2  ∂v ∂v ∂v 1 ∂p ∂ v ∂2 v ∂2 v ∂v ∂t + u ∂x + v ∂y + w ∂z = − ρ ∂y + ν ∂x2 + ∂y2 + ∂z2 ,  2  ∂w ∂w ∂w ∂w 1 ∂p ∂ w ∂2 w ∂2 w + u + v + w = − + ν + + , ρ ∂z ∂t ∂x ∂y ∂z ∂x2 ∂y2 ∂z2

(1)

∂u ∂v ∂w + + = 0, ∂x ∂y ∂z

(2)

∂u ∂t

where u, v and w are the components of fluid velocity, and p, ρ and ν are the pressure, density and kinematic viscosity, respectively. The fractional step method was applied for time marching. The mesh employed is an equally spaced three-dimensional Cartesian grid. The convection term was evaluated by the second-order skew-symmetric scheme [16], and the diffusion term was discretized using the second-order central-difference scheme. The pressure and diffusion terms were calculated by the second-order finite-difference method. The Poisson equation of the pressure was calculated using the successive over-relaxation (SOR) method.

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Our flow solver defined objects using the level set and ghost-cell methods [13,14]. The level set function φ was defined as the signed minimum distance between whole cells and the object 2.2. Immersed Boundary boundary. In theMethod present study, flows around moving multiple objects were solved using multiple level set functions based on the simple minimum–distance approach [17]. The level set function of Our flow solver defined objectsbyusing the level setparticle’s and ghost-cell methods [13,14]. The level set particle and wall was defined the position of the center and of the wall’s surface, Each classified as a fluid distance cell (FC), a between ghost cell (GC) or an object cellthe (OC) using boundary. function ϕrespectively. was defined ascell thewas signed minimum whole cells and object the level set function: In the present study, flows around moving multiple objects were solved using multiple level set

0  ,

FC functions based on the simple minimum–distance approach [17]. The level set function of particle and − 2.25 x   GC  0, (3) Each cell wall was defined by the position of the particle’s center and of the wall’s surface, respectively.  OC  −2.25 x. was classified as a fluid cell (FC), a ghost cell (GC) or an object cell (OC) using the level set function:

The ghost cells, which behave as guard cells between the fluid and object regions, were assigned in one or two layers (Figure 1). The flow quantities in a ghost cell are determined by the flow variables 0 < φFCby , the fluid cells. The image point is addressed at the of the image point within the region occupied edge of a probe extended from the ghost cell through the object (3) ≤ 0, boundary in the direction normal to −2.25∆x ≤ φGC the surface. To avoid recursive references, the probe length was set to 1.75 times the mesh size Δx. φOC < were −2.25∆x. The primitive variables on the image point interpolated from their values at points in the surrounding fluid. Finally, the value of a ghost cell was defined by the value of the image point. To The ghost cells, as guard fluid and were object regions, were determine thewhich velocitybehave components of the cells ghost between cells, linearthe extrapolations implemented with assigned the non-slip boundary1). conditions on the object surface. one or two layers (Figure The flow quantities in a ghost cell are determined by the flow variables The pressure for the ghost cell was defined by the value of the image point as follows:

in of the image point within the region occupied by the fluid cells. The image point is addressed d IP +  GC at the edge of a probe extended from (uIPthrough uGC = uthe − u IB ), the object boundary in the direction IP − ghost cell d IP normal to the surface. To avoid recursive references, the probe length was set to 1.75 times the mesh d IP +  GC ), vGCthe = vimage vIP − vIBinterpolated size ∆x. The primitive variables on point(were from their values at points in IP − d IP (4) the surrounding fluid. Finally, the value of a ghost cell was defined by the value of the image point. d IP +  GC (wIPlinear wGC =ofwIP − ghost cells, − wIB ), extrapolations were implemented with To determine the velocity components the d IP the non-slip boundary conditions on surface. PGC the = PIPobject .

Figure Domain around object. Figure 1. 1. Domain aroundanan object.

The pressure for the ghost cell was defined by the value of the image point as follows: uGC = uIP −

dIP +|φGC | (uIP dIP

− uIB ),

vGC = vIP −

dIP +|φGC | (vIP dIP

− vIB ),

wGC = wIP −

dIP +|φGC | (wIP dIP

(4)

− wIB ),

PGC = PIP . 2.3. Force Evaluations The aerodynamic pressure and friction forces acting on the object surface were calculated on the cell face between the fluid and ghost cells. In the present method, the forces were estimated by a simple algorithm without surface polygons [18].

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2.3. Force Evaluations The pressure and friction forces acting on the object surface were calculated on the 2.4. Motion of aerodynamic Object cell face between the fluid and ghost cells. In the present method, the forces were estimated by a

The particles movement is described by [18]. the equations of motion for xyz-transportations and the simple algorithm without surface polygons angular momenta of a rigid body as follows: 2.4. Motion of Object


Z d mp up The particles movement is described by the for xyz-transportations and the = equations f dS + of mpmotion g, dt S angular momenta of a rigid body as follows: d (mpu p ) dXp = fdS + mp g, S = up , dt dt



dX pdωp = u p= , T, Idt p dt

(5)

(5)

(6)

(6)

(7)

dω p I p dθp = = Tω , , (7) (8) p dt dt p where mp and Ip denote the particles massdθ and moment, respectively. up = (up , vp , wp ) and = inertial ωp , (8) dt Xp = (xp , yp , zp ) represent the velocity and position of the particle’s center of mass, ωp = (ωpx , ωpy , ωpz ) mp and Ip denote the particles mass and inertial moment, respectively. up = (up, vp, wp) and Xp and θwhere p = (θ px , θ py , θ pz ) are the angular velocity and orientation of the particle. f and T express the = (xp, yp, zp) represent the velocity and position of the particle’s center of mass, ωp = (ωpx, ωpy, ωpz) and aerodynamics forces and torque acting on the particle by the fluid. S is the cell face between the fluid θp = (θpx, θpy, θpz) are the angular velocity and orientation of the particle. f and T express the and ghost cells, and g denotes the acceleration due to gravity. The distribution of the level set function aerodynamics forces and torque acting on the particle by the fluid. S is the cell face between the fluid changes by the position of particle’s center moving. The velocities of two particles (labeled 1 and 2) and ghost cells, and g denotes the acceleration due to gravity. The distribution of the level set function after changes their collision are respectively given by:moving. The velocities of two particles (labeled 1 and 2) by the position of particle’s center after their collision are respectivelymgiven by:

u0 p,1 =

p,2








(1 + e) up,2 − up,1 · c c + up,1 , mp,1m+p,m 2 p,2 u'p,1 = mp,1 (1 + e)(up, 2 − u p,1 )  cc +
u p,1 , mp,1 + mp, 2 (1 + e) up,1 − up,2 · c c + up,2 , u0 p,2 = m + m p,2 p,1

(9)

(9) mp,1 (1 + e)(u p,1 − u p, 2 )  cc + u p, 2 , mp,1 + mp, 2and c is a standardization vector [19,20]. In this solver, where e is the coefficient of restitution, u'p , 2 =

the collision to theofperfectly elastic e = 1. The collision detection of solver, particles where e is applied the coefficient restitution, and collision; c is a standardization vector [19,20]. In this the1 and is applied to an theinequality perfectly elastic collision; e = 1. Theformula collision as detection particles 1 and 2 wascollision determined using involving Pythagoras followsof(see Figure 2), 2 was determined using qan inequality involving Pythagoras formulaqas follows (see Figure 2), 2 2 2 2 ( x( x2 −− xx1))22 ++( y(y2−− )2 z1 ) (r≤+ r ()r21, + r2 ) , y y) 1 )+ (+ z (−z2z − 2

1

2

1

2

1

1

2

(10)

(10)

where r is particle the particle radius. where r is the radius.

Figure detection. Figure2.2. Collision Collision detection.

After a particle–wall collisions, the particle velocity becomes: 
 u0 p = (1 + e) −up · c c + up .

(11)

To detect a collision between a particle and the wall, the central position of the particle and the normal vector were obtained from the level set function. When the image point appeared in the ghost cell of the other object, the velocity and the pressure components of the ghost cell were not updated. Appl. Sci. 2018, 8, 2387 5 of 21 The velocity and the pressure of the object cell were constantly defined as the initial conditions. 3. Results

To detect a collision between a particle and the wall, the central position of the particle and the normal vector were obtained from the level set function. When the image point appeared in the ghost cell of the otheraobject, velocity and the pressure components of the ghost cell were not updated. 3.1. Validation around Fixedthe Particle The velocity and the pressure of the object cell were constantly defined as the initial conditions.

The accuracy of the flow solver was determined by comparing the calculated flow structure and Results the drag3.coefficient around a fixed sphere with those of previous studies and drag models [21–24]. The convection termaround was aevaluated using two schemes: U, the first-order upwind finite-difference 3.1. Validation Fixed Particle scheme and S, the second-order skew-symmetric scheme. The Reynolds numbers was set to Re = 300 The accuracy of the flow solver was determined by comparing the calculated flow structure and and 400 the depending on the freestream and particle diameter values. A and comparison was made using drag coefficient around a fixed sphere with those of previous studies drag models [21–24]. two mesh 0.05D (fine) and (coarse),U, where D is upwind the particle diameter. The The resolutions convection term was evaluated using0.1D two schemes: the first-order finite-difference scheme and S, the second-order skew-symmetric scheme. The computational Reynolds numbersdomain was set to(see Re =Figure 300 computational conditions are summarized in Table 1. The 3) is and 400 depending on the freestream and particle diameter values. A comparison was made using two 20D × 10D × 10D. The boundary values of the velocity and pressure were defined by Dirichlet mesh resolutions 0.05D (fine) and 0.1D (coarse), where D is the particle diameter. The computational boundary conditions at the inlet and outlet boundaries, and all variables were subject to Neumann conditions are summarized in Table 1. The computational domain (see Figure 3) is 20D × 10D × 10D. conditions the other boundaries. Alland variables normalized by the freestream values Theat boundary values of the velocity pressure were were defined by Dirichlet boundary conditions at of the density, the velocity and the boundaries, reference length taken D). to Neumann conditions at the other inlet and outlet and all (here variables wereassubject boundaries. All variables were normalized by the freestream values of the density, velocity and the reference length (here takenTable as D). 1. Test cases in the proposed model. Table 1. Test casesSize in the proposed model. Reynolds Number Mesh Scheme Case 0.10D Re300D010U Reynolds Number Mesh Size Scheme Case 300 0.05D upwind. Re300D005U 0.10D Re300D010U 300 0.05D upwind. Re300D005U 0.10D Re300D010S 0.10D Re300D010S 0.05D Re300D005S 0.05D Re300D005S 400 0.10D skew-sym. Re400D010S 400 0.10D skew-sym. Re400D010S 0.05D Re400D005S 0.05D Re400D005S

Figure 3. Computational domain. The red sphere is the object.

Figure 4 plots the mean drag coefficient versus Reynolds number. The drag models of Figure 3.al. Computational domain. The red sphere is the object. Clift et al. [21] and Wen et [22] are respectively: 



24 0.0175Re 0.687 Figure 4 plots the mean Cdrag coefficient versus The drag models of Clift et al. 1 + 0.15Re + Reynolds number. , (12) D = Re 1 + 4.25 × 104 Re−1.16 [21] and Wen et al. [22] are respectively:

CD =

24  0.0175 Re  0.687 + , 1 + 0.15 Re 4 −1.16  Re  1 + 4.25  10 Re 

(12)

convection term by the first-order upwind finite-difference scheme, which cannot properly resolve the unsteady phenomena because of the large numerical dissipation. The same trend was observed at Re = 400 (see Figure 8e,f). In the present flow solver, the drag coefficient and the unsteady phenomena were resolved by Appl. Sci. 2018, 8, 2387 6 of 21 the second-order skew-symmetric scheme for the convection term.

Drag coefficients versus Reynoldsnumber number computed using the proposed model (red and(red and Figure 4.Figure Drag4.coefficients versus Reynolds computed using the proposed model blue symbols), drag model (black lines) and previous models (orange and green symbols). blue symbols), drag model (black lines) and previous models (orange and green symbols).

CD =

 24  1 + 0.15Re0.687 . Re

(13)

To show the adaptability of the scheme, the drag coefficients in schemes U and S were compared at Re = 300. The difference between these two schemes became small at high grid resolutions. Moreover, the difference between scheme S and the previous models was smaller than that of scheme U. The present results captured the trends produced by previous drag models and the drag coefficients obtained agreed well with those of previous numerical results. Figures 5 and 6 summarize the nondimensional time histories of the drag and the lift coefficients at Re = 300, respectively. The coefficients oscillated in all cases except Re300D010A. The amplitudes of the oscillations were higher for a fine-mesh than for a coarse-mesh and were almost twice as high in scheme S than in scheme U. From the distributions of the time-averaged pressure coefficients on the center plane (Figure 7); the time-average was processed in five cycles of the steady wake vortex. We observed the same non-axisymmetric flows as in a previous study [23]. We saw a weak vortex in the wake of scheme U instead of scheme S because of numerical dissipation. Furthermore, a smooth contour distribution was established as the grid resolution increased. The instantaneous wake structure was visualized using the second invariant of the velocity tensors Q (Q-criterion) in Figure 8. At Re = 300, Mittal et al. [23] observed a hairpin vortex in the wake of the particle. In the present study, all cases except Re300D010A showed a hairpin vortex in the wake structure. In scheme U on fine mesh (e.g., see Re300D005A in Figure 8b), the hairpin vortex was distorted. In scheme S, however, the vortex was highly resolved (see Figure 8c,d). The wake structure was missing in scheme U because of the convection term by the first-order upwind finite-difference scheme, which cannot properly resolve the unsteady phenomena because of the large numerical dissipation. The same trend was observed at Re = 400 (see Figure 8e,f).

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Figure 5. Variation of drag coefficient at Re = 300. Figure 5. Variationof of drag drag coefficient at Re 300. Figure 5. Variation coefficient at =Re = 300.

Figure 6. Variationof of lift lift coefficient at Re 300. Figure 6. Variation coefficient at =Re = 300. Figure 6. Variation of lift coefficient at Re = 300.

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Figure 6. Variation of lift coefficient at Re = 300.

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Figure 7. Time-averaged pressure coefficient distributions: (a) Re300D010U, (b) R300D005U, (c) Figure 7. Time-averaged pressure coefficientdistributions: distributions: (a) (b) (b) R300D005U, (c) Figure 7. Time-averaged pressure coefficient (a)Re300D010U, Re300D010U, R300D005U, R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S. R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S. (c) R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S.

Figure 8. Instantaneous distributions of second invariant of velocity tensor. (Q = 1.0 × 10−3): (a) Re300D010U, (b) R300D005U, (c) R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S. −3 ): Figure of of second invariant of of velocity tensor. (Q (Q = 1.0 × 10 Figure8.8.Instantaneous Instantaneousdistributions distributions second invariant velocity tensor. = 1.0 ×−3):10(a) 3.2. Collision of a Moving Particle with a Flat Wall Re300D010U, (b) R300D005U, (c) R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S. (a) Re300D010U, (b) R300D005U, (c) R300D010S, (d) R300D005S, (e) R400D010S, (f) R400D005S.

To validate the present flow solver, the flow structure obtained from the collision of a moving

In the present flow solver, the drag coefficient and thethe unsteady phenomena were resolved bybythe 3.2. Collision of a with Moving Particle with a Flat Wall particle a flat wall was compared with that from experiment and numerical simulation second-order skew-symmetric scheme for the convection term. Eames et al. [1] and Vanella et al. [2]. The flow and particle conditions were the same as the numerical To validate the present flow solver, the flow structure obtained from the collision of a moving simulations of Vanella et al. [2]. The simulation was normalized by the initial velocity of the particle particle with aofflat wall was compared withWall that from the experiment and numerical simulation by 3.2. Collision adiameter. Moving Particle with a Flat and its The computational mesh was fixed at 0.05D (fine) and 0.1D (coarse). The Eames etcomputational al. [1] and Vanella et (Figure al. [2]. The flow particle werewas thefixed sameatas domain 9) was setand to 10D × 10D conditions × 10D. The wall z =the 1D.numerical Dirichlet To validate the present flow solver, the flow structure obtained from the collision of moving simulations of Vanella et al. [2]. The simulation was normalized by the initial velocity of theaparticle boundary conditions were imposed on the velocity and pressure for the upper and side boundaries, particle with a flat wall was compared with that from the experiment and numerical simulation by and its diameter. The computational meshwere wasimposed fixed at (fine) and for 0.1D The whereas Neumann boundary conditions on 0.05D the upper boundary the(coarse). pressure and Eames etside al. boundaries [1]domain and Vanella et al. [2]. The flow and particle conditions same the numerical for velocity. computational (Figure 9) was set to 10D × 10D × 10D. The wallwere was the fixed at zas = 1D. Dirichlet simulations of Vanella et al. [2]. The simulation was normalized by the initial velocity of the particle and boundary conditions were imposed on the velocity and pressure for the upper and side boundaries, its diameter. The computational mesh was fixed at 0.05D and 0.1D (coarse). whereas Neumann boundary conditions were imposed on(fine) the upper boundary forThe the computational pressure and side boundaries for velocity.

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domain (Figure 9) was set to 10D × 10D × 10D. The wall was fixed at z = 1D. Dirichlet boundary 9 of 21 conditions were imposed on the velocity and pressure for the upper and side boundaries, whereas Neumann boundary conditions were imposed on the upper boundary for the pressure and side boundaries Appl. Sci. 2018,for 8, xvelocity. FOR PEER REVIEW 9 of 21

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Figure 9. Computational domain. The white object and gray plate represent particle and flat wall, Figure Computationaldomain. domain.The The white object represent particle andwall, flat Figure 9. 9. Computational white object andand graygray plateplate represent particle and flat respectively. wall, respectively. respectively.

Figure 10 shows the instantaneous vorticity distributions at the timing of the collision, with Figure 10 shows shows the theinstantaneous instantaneousvorticity vorticity distributions at the timing of collision, the collision, Figure 10 distributions at the timing of the with Figure with 10a,bFigure pertaining pertaining to results on the coarse and fineand meshes, respectively. The distributions in the to results the coarse fine meshes, respectively. The distributions Figure 10a,b10a,b pertaining to results on theoncoarse and fine meshes, respectively. The distributions in the wake ofin the particle and between the particle and the wall qualitatively agreed with previous results the wake the particle and between the particle thequalitatively wall qualitatively with previous wake of the of particle and between the particle and theand wall agreedagreed with previous results [1,2]. Visually, the adhesion between the particle and the wall had no effect on the flow phenomena. results [1,2]. Visually, the adhesion theand particle andhad theno wall had on the flow [1,2]. Visually, the adhesion between between the particle the wall effect onno theeffect flow phenomena. This is phenomena. because the This visualization surface just the of the levelset set function. is visualization because of thethe visualization of theuses object surface uses just the of function. the level This is because the ofobject the object surface uses just theisosurface isosurface ofisosurface the level set function. Figure 11 shows the instantaneous vorticity distributions of the impacted (Figure Figure Figure 11 shows the instantaneous vorticity distributions (Figure 11a) 11a)11a) and 11 shows the instantaneous vorticity distributionsofofthe the impacted impacted (Figure and and rebounding (Figure particle at the same nondimensional t*, Figure 11a,b pertaining tothe rebounding 11b)11b) particle at same the same nondimensional timetime Figure 11a,b pertaining totothe rebounding (Figure(Figure 11b) particle at the nondimensional time t*,t*,Figure 11a,b pertaining the coarse and meshes, respectively. Ininstances, all instances, a vortex wake vortex appears the rebound coarse and finefine meshes, respectively. Ininstances, all a wake vortex appears fromfrom the after coarse and fine meshes, respectively. In all a wake appears from the rebound rebound after after the collision. The present distribution in the wake of the rebounding particle was consistent with the collision. The present distribution in the wake of the rebounding particle was consistent with the collision. The present distribution in the wake of the rebounding particle was consistent with previous results [1,2]. the had rebounded, the generation of flow vorticity previous [1,2]. After After the particle particle rebounded, generationof ofaaaflow flow of of opposite opposite previous resultsresults [1,2]. After the particle hadhad rebounded, thethe generation of oppositevorticity vorticity was clearly observed in the fine mesh. In the present flow solver, the accurate and stable simulation for was clearly observed in the fine mesh. In the present flow solver, the accurate and stable simulation was clearly observed in the fine mesh. In the present flow solver, the accurate and stable simulation the between moving particle and and the flat was was conducted. for collision the collision between moving particle the wall flat wall conducted. for the collision between moving particle and the flat wall was conducted.

Figure 10. Vorticity distributions at the particle–flat wall collision: (a) coarse mesh, (b) fine mesh. Figure 10. Vorticity distributions at particle–flat the particle–flat wall collision:(a) (a)coarse coarse mesh, mesh, (b) Figure 10. Vorticity distributions at the wall collision: (b)fine finemesh. mesh.

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Figure 11. Vorticity distributions, at the collision of the particle with the wall (a) and at the rebound of the particle (b): (a) coarse mesh, (b) fine mesh. Figure 11. Vorticity distributions, thecollision collision ofofthe particle withwith the wall and (a) at the rebound Figure 11. Vorticity distributions, at atthe the particle the(a)wall and at theofrebound of the particle (b): (a) coarse mesh, (b) fine mesh.

3.3. Collisionthe ofparticle a Moving Particle with(b) a Curved (b): (a) coarse mesh, fine mesh.Wall

Additionally, flow Particle distribution when a moving particle collides with a curved wall was 3.3. Collision ofthe a Moving with a Curved Wall evaluated. present Particle algorithm applied 3.3. CollisionThe of a Moving withwas a Curved Wallin the calculation of the flow for the collision of a Additionally, the flow distribution when a moving particle collides with a curved wall was evaluated. movingThe particle a curvedapplied wall as thecalculation collision ofdetection is the performed using theparticle normal vector presentwith algorithm in the the flow for collision of a moving Additionally, the flowwas distribution when a moving particle collides with a curved wall was with a curved wall as the collision detection is performed using the normal vector obtained from the 12), and obtained from the level set function. The initial particle position was fixed at z = 6D (Figure evaluated. The present algorithm was applied in the calculation of the flow for the collision of a level set function. The size initialwas particle position was fixed z = 6D (Figure 12), and computational the computational mesh 0.05D. Given the at freestream value andtheparticle diameter, the moving mesh particle a curved as the collision is performed using the was normal size with was 0.05D. Given wall the freestream value and detection particle diameter, the Reynolds number set vector Reynolds number was set to 400. Dirichlet boundary conditions were imposed on the velocity and obtainedtofrom the levelboundary set function. Thewere initial particle position at for z =the 6Dupper (Figure 400. Dirichlet conditions imposed on the velocitywas and fixed pressure and 12), and pressureside forboundaries, the upper and the side boundaries, whereas the Neumannonboundary conditions were whereas Neumann conditions were imposed the particle upper boundary the computational mesh size was 0.05D.boundary Given the freestream value and diameter, the imposedforon the upperand boundary for the pressure and side boundaries for velocity. the pressure side boundaries for velocity. Reynolds number was set to 400. Dirichlet boundary conditions were imposed on the velocity and pressure for the upper and side boundaries, whereas the Neumann boundary conditions were imposed on the upper boundary for the pressure and side boundaries for velocity.

12. Computational domainof ofaa particle particle (small sphere) colliding with awith curved Figure Figure 12. Computational domain (small sphere) colliding a wall. curved wall.

Figure 13 shows the instantaneous vorticity distributions at the instant of collision (nondimensional

Figure shows the after instantaneous vorticity distributions at the therebound instant time t*13 = 0.0, Figure 13a), the collision (t* = 0.1, Figure 13b) and during (t* =of 1.0, collision Figure 12. Computational domain of a particle (small sphere) colliding with a curved wall. (nondimensional time t* =13a, 0.0, Figure 13a), after collision = 0.1,wall Figure 13b) and during the Figure 13c). In Figure vorticity form around thethe particle and the(t* curved from the freestream. In Figure 13b, the vortex forming around the particle is reversed. With a flat wall, a similar flowfield rebound (t* = 1.0, Figure 13c). In Figure 13a, vorticity form around the particle and the curved wall Figure instantaneous vorticity distributions instant can be13 seenshows in Figurethe 13c. The present flow solver can simulate the flow withat the the collision betweenofthecollision from the freestream. In Figure 13b, the vortex forming around the particle is reversed. With a flat (nondimensional timeand t* =the0.0, Figure moving particle curved wall.13a), after the collision (t* = 0.1, Figure 13b) and during the wall, a similar flowfield can be seen in Figure 13c. The present flow solver can simulate the flow with rebound (t* = 1.0, Figure 13c). In Figure 13a, vorticity form around the particle and the curved wall the collision between the moving particle and the curved wall. from the freestream. In Figure 13b, the vortex forming around the particle is reversed. With a flat wall, a similar flowfield can be seen in Figure 13c. The present flow solver can simulate the flow with the collision between the moving particle and the curved wall.

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Figure 13. Vorticity distributions when a particle collides with a curved wall. (a) t* = 0.0, (b) t* = 0.1, (c) t* = 1.0. Figure 13. Vorticity distributions when a particle collides with curvedwall. wall. (a) (a) t* == 0.0, Figure 13. Vorticity distributions when a particle collides with a acurved 0.0,(b) (b)t*t*==0.1, 0.1, (c) t* = 1.0. (c) t* = 1.0.

3.4. Collision of Two Moving Particles

3.4. Collision of Two Moving Particles

3.4. The Collision Two Moving flowof around twoParticles moving colliding particles was compared with those of a previous study The flow around two moving colliding particles was compared with those of a previous study [3].

[3]. InThe the computational (Figure 14), the computational mesh was fixed at and 0.05D (fine) and around two domain moving(Figure colliding was compared with those of a (fine) previous study In flow the computational domain 14), particles the computational mesh was fixed at 0.05D 0.1D 0.1D (coarse). The Reynolds number was set to the value a previous study by setting the [3]. In the computational domain (Figure 14), thevalue computational meshinwas fixed at 0.05D (fine) and (coarse). The Reynolds number was set to the given in a given previous study by setting the diameter and initial velocity of thenumber particle. 0.1D (coarse). The Reynolds was set to the value given in a previous study by setting the diameter and initial velocity of the particle. diameter and initial velocity of the particle.

FigureFigure 14. Computational domain of the particle–particle 14. Computational domain of the particle–particlecollision collision simulation. simulation.

Figure 14. Computational domain of the particle–particle collision simulation.

show the the instantaneous instantaneous vorticity Figure Figure 15a and15a,b 15b show vorticitydistributions, distributions,ononthe thecoarse coarseand andfine finemeshes, meshes,

Figure 15a andDuring 15b show the instantaneous vorticity distributions, on the coarse and fine meshes, respectively. the impact, vortices were formed thewake wakeof of each each particle. respectively. During the impact, twintwin vortices were formed ininthe particle.The Thepresent present respectively. During the impact, twin vortices were formedsimulation in the wake ofFigure each 16 particle. The present well with of the previous numerical simulation [3]. results results agreedagreed well with thosethose of the previous numerical [3]. Figure 16 shows showsthe the instantaneous distributions at the instant of contact (nondimensional time[3]. t* = 0.0, Figure 16a), results agreedvorticity wellvorticity with those of atthe simulation shows the instantaneous distributions the previous instant of numerical contact (nondimensional timeFigure t* = 0.0,16 Figure immediately after the collision (t* = 0.1, Figure 16b) and during the rebound (t* = 1.0, Figure 16c). instantaneous vorticity distributions the instant contact time t* = 16c). 0.0, Figure 16a), immediately after the collision (t* =at 0.1, Figure 16b)ofand during(nondimensional the rebound (t* = 1.0, Figure After the impact, the wake vortex moved forward along the surface of the particle. From these 16a), after thevortex collision (t* =forward 0.1, Figure and during rebound = 1.0, Figure 16c). Afterimmediately the impact, the wake moved along16b) the surface of thethe particle. From(t*these simple simple tests, the present flow solver could calculate flows during the collision and rebound of two After impact, the wake alongthe thecollision surface and of the particle. From these simple tests,the the present flow solvervortex could moved calculateforward flows during rebound of two moving moving particles. particles. tests, the present flow solver could calculate flows during the collision and rebound of two moving particles.

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Figure 15.15. Vorticity distributions during a particle–particle collision: (a) coarse mesh, (b)(b) fine mesh. Figure Vorticity distributions during Figure 15. Vorticity distributions during aa particle–particle particle–particle collision: collision: (a) (a) coarse coarse mesh, mesh, (b) fine fine mesh. mesh.

Figure 16.16. Vorticity distributions during a particle–particle collision: t* t* = 0.0, (b)(b) t* t* = 0.1, (c)(c) t* =t*1.0. Figure Vorticity distributions during a particle–particle collision:(a)(a) = 0.0, = 0.1, = 1.0. Figure 16. Vorticity distributions during a particle–particle collision: (a) t* = 0.0, (b) t* = 0.1, (c) t* = 1.0.

3.5. Collision of Multiple Particles with a Flat Wall 3.5. Collision of Multiple Particles with a Flat Wall 3.5. Collision of Multiple Particles with a Flat Wall a practical application, present flow solver was applied flow around multiple AsAs a practical application, thethe present flow solver was applied to to thethe flow around multiple As acolliding practicalwith application, theGiven present flow solver was applied to the flow around multiple particles a flat wall. the particle diameter and the relative velocity between particles colliding with a flat wall. Given the particle diameter and the relative velocity between thethe particles colliding with a flat wall. Given the particle diameter and the relative velocity between the freestream andthe theparticle particleafter afterits itsimpact impact with with the the wall, toto 400. The grid freestream and wall, the theReynolds Reynoldsnumber numberwas wassetset 400. The freestream andatthe particle after its was impact with the wall, the Reynolds number was set diameter to 400. The size waswas fixed The particle to be made hard of steel andsteel have same grid size fixed0.05D. at 0.05D. The particleassumed was assumed to beofmade hard and have same D. grid size was fixed at 0.05D. The particle was assumed to be made of hard steel and have same Adiabatic conditionsconditions and non-slip conditions were imposed on the particle surfaces. diameter D. Adiabatic and boundary non-slip boundary conditions were imposed on the particle diameter D. Adiabatic conditions andinitial non-slip boundary conditions were imposed on theFrom particle The particles were assigned random outside the computational domain. surfaces. The particles were assigned randompositions initial positions outside the computational domain.the surfaces. The particles were assigned random initial positions outside the computational domain. startthe of the many particles entered into the computational domain and wereand defined From startsimulation, of the simulation, many particles entered into the computational domain wereby From the start of the simulation, many particles entered into the computational domain and were defined by the level set function. The velocity of each particle was constant outside the computational defined by the level set function. The velocity of each particle was constant outside the computational domain but varied under gravitational and aerodynamics forces in the domain. The initial angular domain but varied under gravitational and aerodynamics forces in the domain. The initial angular

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the level set function. The velocity of each particle was constant outside the computational domain Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 21 but varied under gravitational and aerodynamics forces in the domain. The initial angular velocity ofvelocity the particle zero. The number of particles Np was 200200 or or 500. of thewas particle was zero. The number of particles Np set wastoset50, to 100, 50, 100, 500.The Thesize sizeofof the computational domain was 25.6D × 25.6D × 25.6D (Figure 17), and the wall was fixed at the computational domain was 25.6D × 25.6D × 25.6D (Figure 17), and the wall was fixed at z = 1D. z Dirichlet = 1D. Dirichlet boundary conditions for the velocity and pressure were set on the upper and outer boundary conditions for the velocity and pressure were set on the upper and outer side side boundaries, whereas Neumann boundary conditions the boundary upper boundary for the boundaries, whereas Neumann boundary conditions are setare onset theon upper for the pressure pressure and side boundaries for the velocity. The initial flow field was a steady flow without particles. and side boundaries for the velocity. The initial flow field was a steady flow without particles.

Figure 17. Computational domain used for the multiple-particle collisions with a flat wall (gray slab). Figure 17. Computational domain used for the multiple-particle collisions with a flat wall (gray slab). Observation points are z = 5D, 10D and 15D. Observation points are z = 5D, 10D and 15D.

Two coupling schemes were compared in this simulation: a one-way (A) and a two-way Two coupling schemes were compared in this simulation: a one-way (A) and a two-way (B) (B) scheme. The computational conditions are summarized in Table 2. In scheme A, the flow field was scheme. The computational conditions are summarized in Table 2. In scheme A, the flow field was a a steady flow and the aerodynamic forces for particle were calculated based on this steady flow. steady flow and the aerodynamic forces for particle were calculated based on this steady flow. In In scheme B, one the other hand, the flow field was influenced by the moving particle and the scheme B, one the other hand, the flow field was influenced by the moving particle and the aerodynamic forces were calculated based on the flow conditions. In this simulation, the lubrication aerodynamic forces were calculated based on the flow conditions. In this simulation, the lubrication force [25] was not considered, because the difference between the one-way and two-way scheme was force [25] was not considered, because the difference between the one-way and two-way scheme was focused on. The CPU times are shown in Appendix A. focused on. The CPU times are shown in Appendix A. Table 2. Test cases in the multiple particle–wall collision simulation. Table 2. Test cases in the multiple particle–wall collision simulation. Scheme Number of Particles Case

Scheme

Number of Particles Case 50 A050 50 A050 100 A100 one–way A100 200100 A200 one–way A200 500200 A500 500 A500 50 B050 B050 10050 B100 two–way 200100 B200 B100 two–way 500 B500 200 B200 500 B500 The instantaneous wake structure of the particle in scheme B was visualized by the second The of instantaneous structure of the particle in scheme was200 visualized the second invariant the velocity wake tensors (Q-criterion). The results for 50, B100, and 500 by particles are invariant in ofFigures the velocity (Q-criterion). The results fortime 50, t* 100, and 500 particles are displayed 18–21,tensors respectively. The nondimensional was200 defined by the particle displayed infreestream Figures 18,values. 19, 20 and respectively. The nondimensional t* was by the diameter and At t*21, = 40, multiple particles collided with time the flat wall defined and rebound particle(a)diameter and freestream values. At rebounded t* = 40, multiple particleswith collided the flat wall (b) and (panels in Figures 18–21). At t* = 50, they and collided otherwith particles (panels in collision Figures 18–21). Atwall, t* = 50, they structure rebounded andthe collided with particles inrebound Figures (panels 18–21). (a) After with the a wake from particle wasother observed as in Figures 18–21). After with wall,velocity. a wake structure the of particle was a (panels hairpin (b) vortex and developed with collision the change in the relative When thefrom number particles observed as athe hairpin vortex and developed withwith the change in relative velocity. When the number became large, vortices frequently interfered the particles and other vortices. Moreover, of particles became large, the vortices frequently interfered with the particles and other vortices. Moreover, many particles moved not only in the direction of the falling particles, but also in all other directions, being scattered by the freestream and in collisions with particle–particle.

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many particles moved not only in the direction of the falling particles, but also in all other directions, being scattered by the freestream and in collisions with particle–particle. Figure 22 shows the instantaneous distributions of the velocity magnitudes at the center of the computational domain. As for scheme A in Figure 22a, the steady flow field was maintained because the one-way scheme has been often used in the Lagrangian method as the momentum of the fluid acts on the particle surface unidirectionally. The flow decelerated as approaching the wall. The flow was Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 21 fastest in the center of the freestream and slowed down by the influence of the wall as it was going to the outside domain. In scheme B in Figure 22b, the flow field was complicated by the interference Appl. Sci. 2018, 8, x FOR PEER REVIEW    14 of 21  of the particles; nevertheless, the flow was determined. Although the initial particle positions were the same in8,both schemes, the particle distributions were clearly different. The simulations using Appl. Sci. 2018, x FOR PEER REVIEW 14 ofscheme 21 A were unable to predict the movement of particles due to the lack of the interaction.

Figure 18. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B050:   (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. Figure 18. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B050:  (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.  Figure 18. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) scheme B050:B050: Figure 18. Instantaneous distribution of second invariant of velocity tensors (Q =for 0.1) for scheme (a) t* (a) = 40, (b)40, t* (b) = 50. of theof isosurface represents the magnitude of theofvelocity. t* = t* The = 50.color The color the isosurface represents the magnitude the velocity.

Figure 19. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B100:   (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. Figure 19. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B100:  Figure 19. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B100: (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.  (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. Figure 19. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B100: (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.

Figure 20. Instantaneous distribution of second invariant of velocity tensors (Q =for 0.1) for scheme Figure 20. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) scheme B200:B200:   (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. Figure 20. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B200:  (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.  Figure 20. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B200: (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.

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Figure 21. Instantaneous distribution of second invariant of velocity tensors (Q = 0.1) for scheme B500: (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity.

Figure 22 shows the instantaneous distributions of the velocity magnitudes at the center of the computational domain. As for scheme A in Figure 22a, the steady flow field was maintained because the one-way scheme has been often used in the Lagrangian method as the momentum of the fluid acts on the particle surface unidirectionally. The flow decelerated as approaching the wall. The flow was fastest in the center of the freestream and slowed down by the influence of the wall as it was going to the outside domain. In scheme B in Figure 22b, the flow field was complicated by the interference of the particles; nevertheless, the flow was determined. Although the initial particle Figure 21. Instantaneous of second invariant of velocity tensors (Q = 0.1) scheme B500:B500: Figure 21.the Instantaneous distribution of second invariant of velocity tensors (Q clearly =for 0.1) fordifferent. scheme positions were same distribution in both schemes, the particle distributions were The (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. (a) t* = 40, (b) t* = 50. The color of the isosurface represents the magnitude of the velocity. simulations using scheme A were unable to predict the movement of particles due to the lack of the interaction. Figure 22 shows the instantaneous distributions of the velocity magnitudes at the center of the computational domain. As for scheme A in Figure 22a, the steady flow field was maintained because the one-way scheme has been often used in the Lagrangian method as the momentum of the fluid acts on the particle surface unidirectionally. The flow decelerated as approaching the wall. The flow was fastest in the center of the freestream and slowed down by the influence of the wall as it was going to the outside domain. In scheme B in Figure 22b, the flow field was complicated by the interference of the particles; nevertheless, the flow was determined. Although the initial particle positions were the same in both schemes, the particle distributions were clearly different. The simulations using scheme A were unable to predict the movement of particles due to the lack of the interaction.

Figure 22.22. Flow schemesA050 A050(a)(a) and B050 (b). color The color map indicates the Figure Flowdistributions distributionsfor for schemes and B050 (b). The map indicates the magnitude magnitude of the velocity. of the velocity.

Figure 23 shows the number of rebounding particles = 5D, Figure 23 shows the number of rebounding particles Np2 atNobservation points points z = 5D,z10D and10D p2 at observation 15D in the computational domain 17), respectively. In the all instances, number of 15Dand in the computational domain (Figure 17),(Figure respectively. In all instances, number of the rebounding rebounding particles was smaller had fallen, because particles sometimes particles was smaller than those thatthan had those fallen,that because particles sometimes moved outsidemoved the outside the computational domain. Moreover, the number of rebounding particles of the higher computational domain. Moreover, the number of rebounding particles of the higher observation observation point became smaller compared those hadobservation fallen at a lower point. point became smaller compared to those that hadto fallen at that a lower point. observation At z = 5D–10D, z = 5D–10D, the number of rebounding particles was halfofthan theparticles. number of particles. theAt number of rebounding particles was half than the number falling Atfalling z = 15D, the At z =of15D, the number of rebounding particles indomain the computational domain less than half the number rebounding particles in the computational was less than half thewas number of falling Figure The 22. Flow forinitial schemes A050position (a) and B050of (b). The map indicates thethat number of falling particles. The particle and initial flowcolor field of Scheme A was the same particles. initialdistributions particle position and initial flow field Scheme A was the same as of magnitude of the velocity. as that of Scheme B; however, Scheme A was unable to predict the movement particles. Scheme B; however, Scheme A was unable to predict the movement particles. Figure 24 shows the kinetic energy along with the normal direction to the flat wall of a rebounding Figure 23normalized shows the by number rebounding particles Nsubscript p2 at observation points z = 5D, 10D and particles that ofofthe falling (pre-collision; 1) particles at the observation point 15D zin=the (Figure 17),B,respectively. In the all instances, number 5D,computational 10D and 15D domain in schemes A and where, along z-axis, thethe energy wasoferebounding n = ∑0.5ρp(wp2 ). particles was smaller than those that had fallen, because particles sometimes moved outside the the After the collision between the particles and the wall, the energy of particles decreased along computational domain. Moreover, the number of rebounding particles of the higher observation z–axis because the particles moved in many directions. The energy of the higher observation point was pointsmaller becamethan smaller compared toobservation those that had fallen at a lower observation point. At z = 5D–10D, that at the lower point because the particles moved horizontally because of the number of rebounding particles was half than the number of falling particles. At z =more 15D,complex the the freestream. Scheme A shows a different trend than scheme B because the flow was number of rebounding particles in the computational domain was less than half the number of falling in the former. When the number of particles was large, the particle energy was reduced by the frequent particles. The initial collisions. particle position and initial flow field of Scheme A was the same as that of particle–particle Scheme B; however, Scheme A was unable to predict the movement particles.

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Figure 23. Number of passes of the multiple particles.

Figure 24 shows the kinetic energy along with the normal direction to the flat wall of a rebounding particles normalized by that of the falling (pre-collision; subscript 1) particles at the observation point z = 5D, 10D and 15D in schemes A and B, where, along the z-axis, the energy was en = ∑0.5ρp(wp2). After the collision between the particles and the wall, the energy of particles decreased along the z–axis because the particles moved in many directions. The energy of the higher observation point was smaller than that at the lower observation point because the particles moved horizontally because of the freestream. Scheme A shows a different trend than scheme B because the 23.23. Number of passes of the multiple particles. Figure Number of passes of the particles. flow was more complex inFigure the former. When the number of multiple particles was large, the particle energy was reduced by the frequent particle–particle collisions. Figure 24 shows the kinetic energy along with the normal direction to the flat wall of a rebounding particles normalized by that of the falling (pre-collision; subscript 1) particles at the observation point z = 5D, 10D and 15D in schemes A and B, where, along the z-axis, the energy was en = ∑0.5ρp(wp2). After the collision between the particles and the wall, the energy of particles decreased along the z–axis because the particles moved in many directions. The energy of the higher observation point was smaller than that at the lower observation point because the particles moved horizontally because of the freestream. Scheme A shows a different trend than scheme B because the flow was more complex in the former. When the number of particles was large, the particle energy was reduced by the frequent particle–particle collisions.

Figure 24.24. Kinetic energy of rebounding particles, where thethe energy is eisn e=n0.5ρp(w p2).p2 ). Figure Kinetic energy of rebounding particles, where energy = 0.5ρp(w

Figure 25 shows kinetic energy ofrebounding the rebounding (post–collision; subscript 2) particles Figure 25 shows the the kinetic energy of the (post–collision; subscript 2) particles at the at the observation = 5D, 10D and for schemes A where and B, in where in the x-y the plane, the energy observation point zpoint = 5D,z10D and 15D for15D schemes A and B, the x-y plane, energy was et = ∑p20.5ρp(u + vp2 ). After the collision, ofincreased particles increased because themoved particles et =was ∑0.5ρp(u + vp2). p2 After the collision, the energythe of energy particles because the particles movedwith laterally with the freestream and the collisions. The the low observation was laterally the freestream and the collisions. The energy at energy the lowatobservation point waspoint larger larger the high observation point and tended to be . Whenof the number than that than at thethat highatobservation point and tended to be opposite to opposite en. When to theennumber particle of large, particle large,increased the energy increased because the flow became and particle–particle was thewas energy because the flow became complex andcomplex particle–particle collisions collisions occurred morethe frequently; the gap schemes A andScheme B is larger. Scheme Ato was occurred more frequently; gap between the between schemes the A and B is larger. A was unable Kinetic ofas particles, where the energy is eparticles n = 0.5ρp(w p2). unablethe toFigure produce the same result scheme Bthe because theofnumber of rebounding particles obtained produce same 24. result as energy scheme Brebounding because number rebounding obtained from from Scheme A was different one obtained from Scheme B (Figure 23). Scheme A was different from thefrom one the obtained from Scheme B (Figure 23). Figure 25 shows the kinetic energy of the rebounding (post–collision; subscript 2) particles at the of From schemes A and B, Figure 26 plots the number of particle–wall collisions as functions observation point z = 5D, 10D and 15D for schemes A and B, where in the x-y plane, the energy was particle number. The number of collisions was double the number of falling particles in each scheme. et =However, ∑0.5ρp(up2the + vresults p2). After the collision, the energy of particles increased because the particles moved of scheme A became progressively higher than those of scheme B as particle laterally with the freestream andparticles the collisions. Thewall energy at thelaterally low observation was larger number increased. Multiple near the moved in schemepoint A, because these than that at the high observation point and tended to be opposite to e n. When the number of particle particles interacted stronger with the freestream than in scheme B. As a result, the laterally moving was large, the energy and increased flow became complex and particle–particle particles increased fallingbecause particlesthe collided with these particles more frequently.collisions Therefore, occurred more frequently; the gap between the schemes A and B is larger. Scheme A was unable to produce the same result as scheme B because the number of rebounding particles obtained from Scheme A was different from the one obtained from Scheme B (Figure 23).

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with a large number of particles, the estimate in scheme B was difficult to obtain. The kinetic energy of the rebounding particle in each scheme (Figures 24 and 25) was almost the same; however, the number of particle–wall collisions yielded different results. In the particulate flow simulation with the wall, the interaction between the flow and the moving particles can be significant to capture more Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 21 realistic characteristics.

Figure 25. Kinetic energy of rebounding particles, where the energy is et = 0.5ρp(up2 + vp2).

From schemes A and B, Figure 26 plots the number of particle–wall collisions as functions of particle number. The number of collisions was double the number of falling particles in each scheme. However, the results of scheme A became progressively higher than those of scheme B as particle number increased. Multiple particles near the wall moved laterally in scheme A, because these particles interacted stronger with the freestream than in scheme B. As a result, the laterally moving particles increased and falling particles collided with these particles more frequently. Therefore, with a large number of particles, the estimate in scheme B was difficult to obtain. The kinetic energy of the rebounding particle in each scheme (Figures 24 and 25) was almost the same; however, the number of particle–wall collisions yielded different results. In the particulate flow simulation with the wall, the interaction between the flow energy and the moving particles canthe be significant to capture realistic 25. energy Kinetic of rebounding particles,where where is eis ). more t =et0.5ρp(u p2 + v p2+ Figure 25.Figure Kinetic of rebounding particles, theenergy energy = 0.5ρp(u p2 vp2). characteristics. From schemes A and B, Figure 26 plots the number of particle–wall collisions as functions of particle number. The number of collisions was double the number of falling particles in each scheme. However, the results of scheme A became progressively higher than those of scheme B as particle number increased. Multiple particles near the wall moved laterally in scheme A, because these particles interacted stronger with the freestream than in scheme B. As a result, the laterally moving particles increased and falling particles collided with these particles more frequently. Therefore, with a large number of particles, the estimate in scheme B was difficult to obtain. The kinetic energy of the rebounding particle in each scheme (Figures 24 and 25) was almost the same; however, the number of particle–wall collisions yielded different results. In the particulate flow simulation with the wall, the interaction between the flow and the moving particles can be significant to capture more realistic characteristics.

Figure 26. of Number of collisions between the wall manyparticles. particles. Red and blue plotsplots pertain Figure 26. Number collisions between the wall andand thethemany Red and blue pertain to scheme A and B, respectively. to scheme A and B, respectively.

Figure 27 visualizes the distribution of the collision points on the wall in the one cell painted.

FigureThe 27 irregular visualizes the distribution of the collision the wall in the Although one cell the painted. distribution was attributed to the initially points random on particle dispositions. initial particle positions were the same in both schemes, the distributions were clearly Although different. the The irregular distribution was attributed to the initially random particle dispositions. For schemed A200 andthe B200, the numbers of collisionsthe between particle and wallclearly were almost the For initial particle positions were same in both schemes, distributions were different. same in Figure 26, however, the locations of collision points were different. schemed A200 and B200, the numbers of collisions between particle and wall were almost the same in Figure 26, however, the locations of collision points were different. In the shot peening process, the evaluation of the flow field using scheme B was important because the location and number of collisions between particle and wall influenced the accuracy of predictions of the process. Figure 26. Number of collisions between the wall and the many particles. Red and blue plots pertain to scheme A and B, respectively.

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Figure 27. onon thethe wall in schemes A050 (a), B050 (b), A100 (c), B100 A200 27. Distribution Distributionofofcollisions collisions wall in schemes A050 (a), B050 (b), A100 (c), (d), B100 (d), (e), B200 A500 (g) and A200 (e), (f), B200 (f), A500 (g)B500 and (h). B500 (h).

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In the shot peening process, the evaluation of the flow field using scheme B was important because the location and number of collisions between particle and wall influenced the accuracy of predictions of the process. 4. Conclusions A three-dimensional flow solver based on IBM was developed and applied to the flows around multiple particles colliding with a wall. The performance of the present flow solver was compared with those of previous studies of the two types of collisions; a moving particle colliding with a flat wall, and collision between two moving particles. Differences between the two-way and one-way schemes were identified for multiple particles colliding with the wall. The main results are summarized below: (1)

(2)

(3) (4)

The drag coefficient of a fixed particle computed by the proposed solver agreed well with the drag coefficient of previous studies. The skew–symmetric scheme could capture the unsteady flow characteristics such as the wake structure. The wake structures of the moving particle colliding with the flat wall, the curved wall and another moving particle exhibited the same trends as the wake structures computed in previous studies. We demonstrated that the present flow solver using a simple collision algorithm appropriately solves the collision problem. When the number of particles is large, the flow structure becomes complex because particle–particle collisions caused frequent lateral movements of the particles. For many particles, the one-way scheme could not accurately predict the flows, because it overestimated the number of collisions and ignored the influence of the particles on the fluid, which altered the flow phenomena.

The interaction of a fluid and multiple colliding particles remains as an open problem, requiring more details to be able to compare the results with experimental results in a large region. Moreover, the difference of the flow characteristics between the particles–flat wall and particles–curved wall collisions will need to be investigated in the case of the large number of particles. Author Contributions: Conceptualization, Y.M, and S.T.; Methodology, S.T.; Software, Y.M. and S.T.; Formal analysis, Y.M.; Investigation, Y.M.; Resources, S.O.; Project Administration, S.T., K.F. and S.O. Funding: The research received no external funding. Acknowledgments: Part of the present simulations were implemented by the High Performance Computer Infrastructure (HPCI) hp150130, hp160150 and hp170111. This work was supported by JSPS KAKENHI Grant Number 16K18018, 17K06167 and 18K03937. Part of the work was carried out under the Collaborative Research Project J15052 “Study for accurate prediction of unsteady aerodynamic characteristics around moving objects” with the Institute of Fluid Science, Tohoku University. This study was partially supported by the Research Project of Tokai University “Development of measurement system for unsteady flows around moving objects by numerical simulations and experiments”. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A Table A1 shows the CPU times of test cases in Section 3.5. These simulations were carried out by SX-ACE of Cyberscience Center Tohoku University. The CPU time increased with the number of particles. However, there was no big difference between the computational time of the scheme A and B.

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Table A1. CPU times. Case

CPU Time (s)

A050 A100 A200 A500 B050 B100 B200 B500

24,966.707 24,538.322 26,742.404 26,214.485 21,369.633 23,787.872 25,996.342 26,384.007

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

14.

15.

16. 17. 18. 19.

Eames, I.; Dalziel, S.B. Dust resuspension by the flow around an impacting sphere. J. Fluid Mech. 2000, 403, 305–328. [CrossRef] Vanella, M.; Balaras, E. A moving–least–squares reconstruction for embedded–boundary formulations. J. Comput. 2009, 228, 6617–6628. [CrossRef] Griffith, D.M.; Schouveiler, L.; Thompson, C.M.; Hourigan, K. Dynamics of the flow around colliding spheres. J. Fluids Struct. 2011, 27, 1349–1356. [CrossRef] Kajishima, T. Influence of particle rotation on the interaction between particle clusters and particle-induced turbulence. Int. J. Heat Fluid Flow 2004, 25, 721–728. [CrossRef] Deen, N.G.; Peters, E.A.J.F.; Padding, J.T.; Kuipers, J.A.M. Review of direct numerical simulation of fluid–particle mass, momentum and heat transfer in dense gas–solid flows. Chem. Eng. Sci. 2014, 116, 710–724. [CrossRef] Di Sarli, V.; Russo, P.; Sanchirico, R.; Di Benedetto, A. CFD simulations of the effect of dust diameter on the dispersion in the 20 L bomb. Chem. Eng. Trans. 2013, 31, 727–732. [CrossRef] Nguyen, B.V.; Poh, J.H.; Zhang, W.Y. Predicting shot peening coverage using multiphase computational fluid dynamics simulations. Powder Technol. 2004, 256, 100–112. [CrossRef] Kirk, D.; Abyaneh, M.Y. Theoretical basis of shot peening coverage control. Shot Peener 1995, 9, 28–30. Kopp, R.; Wustefeld, F. Modern Simulation and Optimization of Peen Forming Processes. In Proceedings of the International Conferences on Shot Peening, Tokyo, Japan, 1990; pp. 561–572. Garipy, A.; Larose, S.; Perron, C.; Bocher, P.; Levesque, M. On the effect of the peening trajectory in shot peen forming. Finite Elem. Anal. Des. 2013, 69, 48–61. [CrossRef] Kato, Y.; Omiya, M.; Hoshino, H. Modeling of Particle Behavior in Shot Peening Process. J. Mech. Eng. Autom. 2014, 4, 83–91. Mittal, R.; Iaccarino, G. IMMERSED BOUNDARY METHODS. Fluid Mech. 2005, 37, 239–261. [CrossRef] Mizuno, Y.; Takahashi, S.; Nonomura, T.; Nagata, T.; Fukuda, K. A Simple Immersed Boundary Method for Compressible Flow Simulation around a Stationary and Moving Sphere. Math. Probl. Eng. 2015, 2015, 1–17. [CrossRef] Takahashi, S.; Nonomura, T.; Fukuda, K. A Numerical Scheme Based on an Immersed Boundary Method for Compressible Turbulent Flows with Shocks: Application to Two–Dimensional Flows around Cylinders. J. Appl. Math. 2014, 2014, 1–21. [CrossRef] Mizuno, Y.; Inoue, T.; Takahashi, S.; Fukuda, K. Investigation of A Gas–particle Flow with Particle–particle And Particle–wall Collisions by Immersed Boundary Method. Int. J. Comput. Meth. Exp. Meas. 2018, 6, 132–138. [CrossRef] Morinishi, Y. Skew–symmetric form of convective terms and fully conservative finite difference schemes for variable density low–Mach number flows. J. Comput. Phys. 2010, 229, 276–300. [CrossRef] Takahashi, S.; Monjugawa, I.; Nakahashi, K. Unsteady Flow Computations around Moving Airfoils by Overset Unstructured Grid Method. Trans. Jpn. Soc. Aeronaut. Space Sci. 2008, 51, 78–85. [CrossRef] Nonomura, T.; Onishi, J. A comparative Study on Evaluation Methods of Fluid Forces on Cartesian Grids. Math. Probl. Eng. 2017, 2017, 1–15. [CrossRef] Kosinski, P.; Hoffmann, A.C. Extension of the hard-sphere particle-wall collision model to account for particle deposition. Phys. Rev. 2009, 79, 1–11. [CrossRef] [PubMed]

Appl. Sci. 2018, 8, 2387

20. 21. 22. 23.

24. 25.

21 of 21

Kosinski, P.; Hoffmann, A.C. An extension of the hard–sphere particle–particle collision model to study agglomeration. Chem. Eng. Sci. 2010, 65, 3231–3239. [CrossRef] Clift, R.; Ganvio, H.W. Motion of entrained particles in gas streams. Can. J. Chem. Eng. 1971, 49, 439–448. [CrossRef] Wen, C.Y.; Yu, Y.H. A generalized method for predicting the minimum fluidization velocity. Chem. Eng. Prog. Symp. Ser. 1699, 6, 100. [CrossRef] Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F.M.; Vargas, A.; von Loebbecke, A. A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 2008, 227, 4825–4852. [CrossRef] [PubMed] Luo, K.; Wang, Z.; Fan, J. A modified immersed boundary method for simulations of fluid–particle interactions. Comput. Methods Appl. Mech. Eng. 2007, 197, 36–46. [CrossRef] Zhang, W.; Noda, R.; Horio, M. Evaluation of lubrication force on colliding particles for DEM simulation of fluidized beds. Powder Technol. 2005, 158, 92–101. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).