Journal of Computational Physics 448 (2022) 110695
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Journal of Computational Physics
www.elsevier.com/locate/jcp
A resolved CFD-DEM coupling model for modeling two-phase
fluids interaction with irregularly shaped particles
Zhihao Shen a , Gang Wang a,∗ , Duruo Huang b , Feng Jin b
a
b
Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong SAR, China
Department of Hydraulic Engineering, Tsinghua University, Beijing, China
a r t i c l e
i n f o
Article history:
Available online 28 September 2021
Keywords:
CFD-DEM coupling
Resolved method
Two-phase fluids
Irregularly shaped particle
Fluid-solid interaction
a b s t r a c t
In this paper, we develop a resolved coupling model to simulate interaction between twophase fluids and irregularly shaped particles by using Computational Fluid Dynamics (CFD)
and Discrete Element Method (DEM). The Volume of Fluid (VoF) method is introduced
to solve two-phase fluids, and irregularly shaped particles are represented by multisphere clumps in DEM. The resolved CFD-DEM coupling approach calculates meso-scale
flow around particles, and an integration scheme is proposed to directly calculate the
fluid forces on multi-sphere particles without resorting to empirical drag force models.
A number of benchmark cases are conducted, including a single particle settling in a twophase fluid, settling of two particles, disk-shaped particles and irregularly shaped particles.
The numerical simulations compare well with experimental works and previous studies,
showing the accuracy of this model. Finally, a case study of dambreak wave impact on a
rock pile demonstrates great potential to apply this model in coastal engineering.
© 2021 Elsevier Inc. All rights reserved.
1. Introduction
In geotechnical engineering, a better understanding of dynamic interactions of multi-phase fluids and granular materials
is needed. This knowledge can be applied in the analysis of many important engineering problems. For example, scouring
around offshore mono-pile foundation involves dynamic interaction between water, air, and sediment [1,2]; Internal erosion
of granular materials [3]; Interaction between interstitial fluid and particles, is one of the major causes of embankment
dam failure [4]. While, coupling process of multiphase fluids and granular particles in realistic practice is complex and
a multi-scale problem. On the one hand, hydraulic load on granular media can cause large impact force on structure in
macroscale, and localized force in microscale. The actual mechanics of the multi-scale problem is still difficult to determine.
Modeling pore-scale fluid-particle interaction remains to be a challenge, especially, when multi-phase fluids, air and water
are involved. On the other hand, granular particles are usually irregularly shaped. The surface shape will influence the
granular particle interlocking in microscale and plays an important role for the stability of the system. Therefore, modeling
realistic particle shapes and grain-scale contact are important to capture the macroscopic behavior of granular packing. Yet,
these factors have seldom been considered.
To model grain-scale behavior of granular material, Discrete Element method (DEM) [5] is often used. The DEM simulates individual particle movement following the Newton’s law. The trajectory of particle is solved, so DEM uses Lagrangian
description. Fluid motion is controlled by Navier-Stocks equations. To solve the N-S equations, various Computational Fluid
*
Corresponding author.
E-mail address: gwang@ust.hk (G. Wang).
https://doi.org/10.1016/j.jcp.2021.110695
0021-9991/© 2021 Elsevier Inc. All rights reserved.
Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Dynamics (CFD) methods, including Finite Difference Method (FDM) [6], Finite Element Method (FEM) [7], Finite Volume
Method (FVM) [8], etc. can be used. In these CFD methods, Eulerian description is commonly used and the space is discretized into mesh cells. In this study, FVM is used as the CFD method in coupling.
According to the relative scale of fluid mesh and particle size, the CFD-DEM coupling methods can be categorized into
unresolved [9–13] and resolved [14,15] approaches. In the unresolved method, the particle diameter should be smaller than
1/3 the mesh length [16–18] and local volume-averaged Navier-Stocks equations are solved. Interaction force (e.g. drag
force) between particle and fluid is based on empirical equations, which were obtained from laboratory experiments, such
as single particle settling in the fluid [19–21]. For example, Di Felice (1994) drag force model is often used in unresolved
CFD-DEM coupling, such that,
Fd = −
1
2
ρ f C d A (U p − u f )|U p − u f |ε1−χ ,
(1)
where, ρ f is the fluid density, A is reference area of particle size, U p and u f are particle velocity and averaged velocity
of a fluid cell, ε 1−χ is a correction coefficient considering multiple particles, C d is the drag coefficient calculated from
experimental results. The drag coefficient can be influenced by many factors including Reynolds number, particle shape, etc.
Apparently, the empirical drag force model cannot be reliably used if the particle is non-spherical, or, fine-scale interstitial interaction is concerned [22]. For fine-scale fluid motion simulation, the resolved method shall be used to solve
fluid-particle interaction force directly. For resolved CFD-DEM simulation, the particle diameter should be more than 8 times
the CFD mesh length [23]. It is worth noticing that the Lattice Boltzmann Method (LBM) [24] is another popular choice for
fluid modeling, and coupling between LBM and DEM has been developed in recent decades [25–27]. LBM is a mesoscopic
approach where discrete Boltzmann equation is solved with collision model. Distribution function is the basic variable and
macroscopic variables such as density, velocity, and pressure are calculated indirectly. While, the LBM usually requires a
very fine lattice scale and the simulation is very time-consuming and may become unaffordable for large-scale engineering
applications.
Note that the required mesh scale in the resolved CFD-DEM is smaller than the unresolved CFD-DEM, but larger than the
LBM-DEM. The resolved CFD-DEM is particularly suitable for simulating wave-granular system interaction containing large
boulders. For example, wave erosion on coastal riprap structures. Hager et al. [15] presented a resolved CFD-DEM coupling
scheme based on fictitious domain/immersed boundary method (FD/IBM) [28–30] and follows the implementation approach
by Shirgaonkar et al. [31]. Note that the FD/IBM is different from the conventional immersed boundary method (IBM). In the
conventional IBM [32,33], the solid boundary is described by a series of Lagrangian points and interactive forces between
the Lagrangian points and surrounding fluid cells are calculated. In FD/IBM, the void fraction of the solid/fluid mixture
is mapped into background grids, the boundary between the solid and fluid is calculated by the gradient of the “void
fraction” field. Patel et al. [34] use the FD/IBM to simulate multiphase flows with contact line dynamics through complex
solid geometries. Yu & Shao [35] used the FD/IBM to study the particle motions in Newtonian and non-Newtonian fluids.
Cheng et al. [36] presented a hybrid CFD-DEM coupling method to combine resolved and unresolved approaches using
FD/IBM. However, only the single-phase fluid or spherical, spheroidal particles were considered in most studies. A resolved
CFD-DEM coupling scheme considering both multi-phase fluid and irregular shaped particles has not been developed.
In this study, we develop a novel resolved CFD-DEM coupling framework to handle the fine-scale interaction between
two-phase fluids and irregular particles. Several salient features of this study are: (1) The two-phase fluid is solved using
Volume of Fluid (VOF) method [37], and irregular shaped particles are modeled using multi-sphere clumped model in DEM.
(2) FD/IBM is used to solve solid-fluid interaction. New algorithms are developed to take account of the influence of solid
phase on two-phase fluids and calculate fluid force on clumped DEM particles of any shape. (3) The study developed a
new solver for three-phase coupling system based on the open-source C++ platform CFDEM [38]. Parallel computation
using MPI toolkit makes it efficient to solve large-scale problems. This model will greatly extend the capability of resolved
CFD-DEM simulation.
This paper is presented as follows. In section 2, formulations of the resolved coupling model containing two-phase fluid
and irregular shaped particle are introduced. In section 3, the implementation details of the algorithm are presented. In
section 4, validation cases are studied. Single sphere settling in two-phase fluids and clumped particle settling are compared
with experimental results. Two-particle non-contacting interaction in fluid is performed to validate the solver in situations
more than one particle. Disk particle settling with multi-sphere model is studied to show the capacity of fluid interaction
with non-spherical particles. Finally, a wave erosion case is simulated. Irregular shaped particles are considered in this case,
which shows the capability to analyze realistic two-phase fluid and particle interactions.
2. Formulation for resolved CFD-DEM coupling
In this section, we present the formulation of problems where two-phase fluids and non-spherical particles are considered. Fictitious domain method (FD/IBM) is applied to calculate the fluid forced on particles. Fig. 1(a) is a schematic
illustration of such problems. The materials in this system consist of two-phase fluids (air and water) and solid phase particles. In this system, f is the fluid domain; p is the particle domain, ∂ p is the particle surface which also means the
particle-fluid interface. The whole simulation domain is defined as = f ∪ p .
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Fig. 1. (a) Schematic of two-phase fluid interaction with irregularly shaped particles; (b) “Void fraction” to distinguish fluid and solid; (c) “Phase fraction”
to distinguish air and water.
Two field variables, α f and α1 are used to distinguish different materials. As shown in Fig. 1(b), the “void fraction” α f
[15] is used to divide the real fluid and the solid particle domain: α f = 0 in the solid domain, α f = 1 in the fluid domain.
The “Phase fraction” α1 in Fig. 1(c) is used to differentiate the water and the air. It is defined that, α1 = 0 in the air domain,
α1 = 1 in the water domain. The definition of phase fraction can be referred in VoF method [37,39].
With FD/IBM, the solid particle domain is considered as a fictitious fluid [15]. The N-S equations are solved in the whole
domain . After the calculation of N-S equations, rigid body motion constraint is enforced inside the particle domain p .
Then, forces on particles are integrated to update particle velocities and position. Based on above steps, the governing
equations are divided into the fluid part, the particle part and the coupling part.
2.1. Governing equations of fluid motion
For two-phase fluids, fluid phase fraction α1 is introduced to distinguish different phases. The value of α1 means the
proportion of water phase in a cell. To represent the proportion of air, α2 is used and it satisfies that, α2 = 1 − α1 . The
gravity and surface tension are considered in the control equations. Water and air are assumed to be incompressible fluids,
which is a reasonable assumption when the Mach Number is low. The control equations for fluid are solved over the whole
domain. The solid domain is considered as a fictitious fluid, which has the same viscosity as the surrounding fluid.
The momentum equation for the multi-phase fluid is,
∂(ρ f u f )
+ ∇ · (ρ f u f ⊗ u f ) = −∇ p + ∇ · (μ f ∇ ⊗ u f ) + fs + ρ f g in ,
∂t
in which, ρ f is the combined density of the multi-phase fluid, defined as,
ρ f = α1 ρ1 + (1 − α1 )ρ2 = α1 (ρ1 − ρ2 ) + ρ2 ,
(3)
ρ1 and ρ2 are the density of water and air; u f is the fluid velocity; p is the fluid pressure;
defined as,
μ f = α1 μ1 + (1 − α1 )μ2 = α1 (μ1 − μ2 ) + μ2 ,
where
(2)
μ f is the combined viscosity,
(4)
μ1 and μ2 are the viscosity of water and air; fs is the surface tension calculated as,
fs = σ κ ∇ α1 ,
(5)
where σ is the surface tension coefficient, κ is the surface curvature, α1 is the water fraction as mentioned before; g is the
gravity with magnitude 9.81 m/s2 . It is worth mentioning that, the pressure term and the gravity term are combined into
the “modified pressure” term p rgh [40] using the following equation,
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Fig. 2. (a) Spherical clump; (b) Disk clump; (c) Irregular shaped particle.
p rgh = p − ρ f g · h,
(6)
∇ prgh = ∇ p − ∇(ρ f g · h) = ∇ p − ρ f ∇(g · h) − g · h∇ ρ f = ∇ p − ρ f g − g · h∇ ρ f ,
(7)
in which, h the position vector. Substitution of Eqs. (5), (6) into the Eq. (2) results in the following form,
∂(ρ f u f )
+ ∇ · (ρ f u f ⊗ u f ) = −∇ prgh + ∇ · (μ f ∇ ⊗ u f ) − g · h∇ ρ f + σ κ ∇ α1 in .
∂t
(8)
The benefit of such processing is that, all the scaler terms on the right-hand side have a gradient form, which makes it
convenient to calculate numerically using a unified discrete form.
The continuity equation for an incompressible multi-phase fluid is,
Dρ f
Dt
=
∇ · uf = 0
in + u f · ∇ ρ f = 0 in ∂ρ f
∂t
(1)
,
(2)
(9)
Dρ
where, Eq. (9-1) is for fluid velocity and Eq. (9-2) is for the fluid density. The material derivative Dtf represents the
compressibility of fluid. Substitution of Eq. (3) into Eq. (9) leads to conservation of the water phase as the following phase
equation,
∂ α1
+ ∇ · (α1 u f ) = 0 in ,
∂t
(10)
and the density ρ f can be calculated using the solved α1 from Eq. (3). While in numerical modeling, the phase values
tend to diffuse around the phase interface, which can decrease the resolution of phase interface. Therefore, an artificial
compression term [41] is added in the above equation to ensure the sharpness of the phase interface,
∂ α1
+ ∇ · (α1 u f ) + ∇ · (α1 (1 − α1 )uc ) = 0 in ,
∂t
∇ α1
,
uc = c u f |∇ α1 |
(11)
(12)
where c is a coefficient which switches interface sharpening on (c = 1) or off (c = 0).
In the numerical implementation, different constitutive models and turbulence models are integrated into the viscosity
term ∇ · (μ f ∇ ⊗ u f ) in Eq. (8). Constitutive models include Newtonian and non-Newtonian fluids. Turbulence simulations
include Reynolds Averaged Navier-Stokes (RANS) simulation, Large Eddy Simulation (LES) and Direct Numerical Simulation
(DNS) [42]. In this study, Newtonian fluid model is used. To our knowledge, DNS requires that the mesh scale is smaller than
smallest turbulence vortex. Therefore, DNS has both high accuracy and high cost, which makes it difficult for engineering
calculation. Quasi-DNS procedure is developed and validated in some existing study [43], where the requirement of smallest
mesh scale is relaxed. In this study, quasi-DNS method is used.
2.2. Governing equations of particle motion
In this study, particles are simulated in a Discrete Element framework implemented in an open source code LIGGGHTS
[44]. The particle motion follows Newton’s second law as described by Cundall & Strack [5]. All particles in the computational domain are updated by tracking their trajectories. In this study, multiple-sphere (MS) model is used to represent
irregular particles [45]. These objects built with MS model are called clumps. Fig. 2 shows three examples of clumps. In a
specific clump, component particles are fixed together and the clump is treated as a rigid body in the simulation process.
These component particles are denoted as “sub-particles”. These sub-particles in the same clump are “bonded” on the clump
and will move and rotate together in the simulation. There is no contact force between sub-particles in the same clump.
For an irregularly shaped clump, the moment of inertia Ii is a second order symmetric tensor.
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Fig. 3. A sketch of the Monte Carlo resampling procedure.
Fig. 4. (a) A clumped particle; (b) Mapped global “void fraction” field
α f (a section view).
The control equations solve the displacement and rotation motion of each clump based on force and moment:
mi
Ii
dUi
dt
dω i
dt
= mi g +
Fi , j + Fi , f ,
(13)
j
+ ωi × (Ii · ωi ) =
Mi , j + Mi , f ,
(14)
j
in which, mi is the mass of clump i, Ui is the clump velocity, Fi , j is the inter-particle contact force between sub-particles
in clump i and clump j, Fi , f is the fluid force on clump i. Ii is the inertia tensor, ω i is the rotation velocity, Mi , j is the
moment due to particle-particle contact, Mi , f is the moment caused by the fluid force. Only sub-particles from different
clumps can interact with each other, and sub-particles in the same clumps have no interaction. Therefore, the terms Fi , j
and Mi , j are only counted between sub-particles from different clumps. The contact force is calculated using Hertz-Mindlin
contact model in this study. The terms Fi , f and Mi , f are related to the CFD-DEM coupling procedure, which will be detailed
in the next section. Finally, the positions of sub-particles in a clump are then calculated based on the clump position and
rotation angle.
For the clump gravity, it cannot be the simple summation of the sub-particles, since there are overlapping parts between
two particles. In LIGGGHTS, a Monte Carlo resampling procedure is used to assess properties that are needed for the rigid
motion update [44]. Fig. 3 shows a sketch of Monte Carlo procedure using a clump with two sub-particles. Firstly, the
bounding box enveloping the clump is determined. Then, random points with a uniform distribution are generated in this
bounding box. The particles located inside the clump (black points) are marked as the clump constituents. Finally, the
marked points are used to calculate the clump volume, mass, center of mass, the inertia tensor and its eigenvalues.
2.3. Fluid force and moment on clumped particles
In this coupling process, fluids and particles interact with each other. The hydraulic force will be exerted on the particle;
in return, the particle will influence the fluid velocity and pressure. To determine the hydraulic force on particle, firstly the
particle is mapped into a “void fraction” field, α f as mentioned before. The mapping algorithm for a single sphere can refer
to Hager [23]. While for a clumped particle, we develop a new mapping scheme that considers locations of the cells with
respect to overlapping of sub-particles.
Here, we use a spherical cluster to illustrate the mapping results. Fig. 4(a) shows a clumped particle constructed using
54 spheres. In Fig. 4(b), section view of the mapped void fraction is represented. α f means the proportion of fluid in a cell.
After the mapping of “void fraction”, the next step is to calculate the forces on particles based on the α f field. The fluid
force on the particle surface is,
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Fig. 5. (a) All forces acting on the clumped particle; (b) Use a clump of overlapped spheres to approximately represent a non-spherical particle (c) The fluid
force Fi , f and fluid moment Mi , f on the clumped particle i, which are calculated but aggregating fluid forces on different sub-parts.
t = σ · n on ∂ p ,
where
(15)
σ is the fluid stress, expressed as the following for a Newtonian fluid,
σ = − pI + τ = − pI + μ f (∇ ⊗ u f + u f ⊗ ∇),
(16)
where n is the outward normal vector of the particle surface. The total fluid force and moment on a particle can be calculated by integrating the surface force around the whole surface, which can be further transformed into volume integration
using the divergence theorem as,
Fi , f =
tdS =
∂ p
σ · ndS =
∂ p
∇ · σ dV =
p
Mi , f =
r × tdS =
∂ p
(−∇ p + ∇ · τ )dV ,
r × (σ · n)dS =
∂ p
(17)
p
r × ∇ · σ dV =
p
r × (−∇ p + ∇ · τ )dV .
(18)
p
In a discretized form, the force and moment acting on a particle clump are calculated by summation of the stress
divergence terms over the cells occupied by that clump, using the fraction of solid as a weight,
(−∇ p + ∇ · τ ) V c (1 − α f ) =
f I B V c (1 − α f ),
=
r × (−∇ p + ∇ · τ ) V c (1 − α f ) =
r × fIB V c (1 − α f ),
Fi , f =
(19)
Mi , f
(20)
where (1 − α f ) is the solid fraction, fIB = (−∇ p + μ f u f ) is the fluid force vector for a Newtonian fluid. To compute the
pressure gradient ∇ p and laplacian of velocity u f , the Gauss linear interpolation can be used in the OpenFOAM. V c is the
volume of a cell.
Finally, the calculated fluid force and moment are loaded on the clumped particle centroid to update its motion. Fig. 5(a)
illustrates forces acting on a clumped particle i, including the fluid force and moment (Fi , f and Mi , f ) on the centroid
using Eqs. (19), (20), the contact forces from surrounding sub-particles (From Fi, j1 to Fi, j5 ), and the gravity G. Note that for
the clumped particle, summations in Eqs. (19), (20) are performed over cells occupied by individual spheres. As shown in
Fig. 5(b) (c), a cell at the overlapped regions should be assigned to only one sub-particle, such that it will not be counted
repeatedly. From particle P1 to particle P5, when a cell is counted in a former particle, it will not be counted to the next
particle.
2.4. Rigid body constraint
The fluid force on particles is calculated in section 2.3, and now we consider the influence of particle motion on the
fluid field. For a rigid body, the velocity field with the particle domain is constrained as,
u=U+ω×r
in p ,
(21)
where U is velocity of the clump centroid, r is the positive vector relative to the clump centroid. In numerical implementation, the velocity of solid particle i is mapped into a velocity field, us ,
us = c v (Ui + ωi × ric ) in p ,
(22)
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
Fig. 6. Flow chart of the numerical modules in CFD-DEM solver.
where, c v is a coefficient to control the influence of particle motion to the fluid. It equals to 1 if a no-slip boundary on
the particle surface is strictly enforced. When the particle is passing through the fluid interface, the particle surface fluid
changes from one phase to another phase. This value can be slightly relaxed to be smaller than one.
Then the mesh cells occupied by solid can be enforced with the solid velocity field us as the known velocity. The fluid
velocity field is corrected as,
u f = û f α f + us (1 − α f )
in (23)
where û f is the fluid velocity before correction. This correction means rigid body constraint is enforced in the particle
domain.
3. Numerical algorithm and implementation
The resolved CFD-DEM model is developed on the open-source C++ platform CFDEM [38]. CFDEM provides a coupling
interface between two open-source codes, OpenFOAM for CFD simulation and LIGGGHTS for DEM simulation, respectively.
FD/IBM is used as the coupling strategy, and VoF is used to describe the two-phase fluid. Therefore, our new solver is
referred as “cfdemInterIB”. In the following section, the details of the solver implementation are introduced.
3.1. Algorithm of cfdemInterIB solver
The solver “cfdemInterIB” is consisted of three modules, namely, the DEM module, the CFD module and the coupling
module, as shown in Fig. 6. The main functions of the DEM and CFD modules are readily provided by open-source codes
LIGGGHTS and OpenFOAM, respectively. The coupling module is developed by the authors and implemented in the CFDEM
platform. The solver mainly contains the following steps:
(1). Setup variables for CFD simulation, including velocity, pressure, phase fields, etc. Create DEM particles and initiate
their properties.
(2). The DEM module is executed to solve the motion of the clumped particles by considering inter-particle and fluid
forces via Eqs. (13), (14). The multi-sphere model treats a clump as a rigid body. The fluid force from the last time step is
integrated to update the particles position and velocity.
(3). The particle positions and velocities are sent to the coupling module. The particles position data is used to update
the “void fraction” field.
(4). The phase equation (11) is solved based on the fluid velocity field from the last time step in the VoF module.
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
(5). In the CFD module, PISO (Pressure Implicit with Splitting of Operator) algorithm is carried out to solve the momentum equation and continuity equation of the multi-phase fluid. Eqs. (8), (9-1) are solved in this step. The solved velocity is
an intermediate velocity and need to be corrected in step (6).
(6). The particle velocity data is used to modify the fluid velocity field. Eqs. (22), (23) are used in this step. The intermediate fluid velocity û f is corrected into the final fluid velocity u f .
(7). The fluid field force fIB is calculated and integrated over the cells occupied by the clump. The total force and torque
from the fluid onto the particles are calculated via Eqs. (19), (20).
(8). Go to step (2) and iterate for the next time step.
Fig. 6 shows all the modules in the simulation procedure. The DEM module for particle motion and the CFD module for
fluid field are solved separately. The time steps for the DEM module ( t D E M ) and the CFD module ( t C F D ) can be different.
For the particle phase, Li et al. [46] suggested a critical time step based on the time of Rayleigh wave propagating along the
averaged particle radius R̄,
π R̄
2ρs (1 + ν )
tDEM =
,
0.163ν + 0.8766
E
(24)
where v is Poisson’s ratio, E is the Young’s Modules, ρs is the particle density. For the fluid phase, to achieve enough
accuracy and numerical stability, the Courant number is required smaller than 1. The Courant number Co is defined as,
Co = 0.5
ϕ
V cell
t C F D < 1,
(25)
where ϕ is the fluid flux through all faces of a cell, V cell is the volume of a mesh cell. In general, the critical time step in
DEM is much smaller than that in CFD. In resolved method, the value of t C F D is usually set to be 10 times of the t D E M
value. The implementation detail of several key steps will be discussed in section 3.2.
3.2. Discretized governing equations
For DEM calculation, the particle motion is updated by using Verlet integration scheme in LIGGGHTS framework [47].
When the multi-sphere model is used, forces from sub-particles are assembled and applied on the clump. The clump is
treated as a rigid body, and then the sub-particles motion is updated according to this rigid-body motion.
After the DEM calculation, the radius, position, and velocity of sub-particles are sent to the coupling module in OpenFOAM framework. Using the information, the “void fraction” field α f is set over the whole domain .
The phase Eq. (11) is solved using an explicit scheme MULES (Multidimensional Universal Limiter with Explicit Solution)
[48]. Assuming the current time step is n, superscript (n-1) denotes quantities at last step tn−1 (known step), and n denotes
these at current step tn (to be solved). The discretized form is shown as,
α1n − α1n−1
t
+ ∇ · (α1n−1 unf −1 ) + ∇ · (α1n−1 (1 − α1n−1 )unc −1 ) = 0 in .
(26)
PISO algorithm [49] is used to solve the velocity and pressure field in N-S Eqs. (8), (9-1). In the simulation process,
the modified pressure p rgh is first solved and the pressure field is calculated based on Eq. (6). Since the solid domain is
considered as a fictitious fluid, the N-S equation (2) and continuity equation (9-1) are discretized into a semi-implicit form
over the whole domain ,
ρ nf −1 (ûnf − unf −1 )
t
∇ · ûnf = 0 in ,
+ ∇ · (ρ nf −1 unf −1 ⊗ ûnf ) = ∇ · (μnf −1 ∇ ⊗ ûnf ) − ∇( pnrgh ) − g · h∇(ρ nf −1 ) + σ κ ∇(α1n−1 ) in , (27)
(28)
where, the temporal term uses a Eulerian implicit scheme, the convection term uses a semi-implicit scheme, diffusion term
uses an implicit scheme. The density, viscosity, and phase fraction values are explicit values obtained from the last time
step. By using the fluid fields unf −1 from the last time step and PISO algorithm to solve discretized Eqs. (27), (28), we get
the new fluid fields ûnf and pnrgh . Here, we denote ûnf with a hat to emphasize it is an intermediate result.
To account for influence (i.e. the blocking effect) of the solid phase, the fluid velocity field is corrected into a new field
unf as follows,
unf = ûnf αnf + uns (1 − αnf )
in ,
(29)
where uns is the solid velocity mapped to the fluid cell using Eq. (22), and
in section 2. Apparently, in the fluid domain (α = 1),
n
f
unf
=
ûnf ,
αnf is the void fraction of the cell mentioned
so there is no correction to the fluid velocity. In the solid
domain (αnf = 0), unf = uns , so the solid velocity replaces the fluid solution at the end of time step. In the solid-fluid interface,
fluid velocity is partially corrected since 0 < αnf < 1. The difference between ûnf and unf is,
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Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
D (unf ) = unf − ûnf = (uns − ûnf )(1 − αnf ) in .
(30)
The above are all the key steps to solve tn . Now we explain the influence of the solid phase on the fluid solution through
correction of the velocity field. Since the uncorrected fluid field is divergent free, ∇ · ûnf = 0, the divergence of the corrected
fluid velocity becomes,
∇ · unf = ∇ · ( D (unf ) + ûnf ) = ∇ · D (unf ).
Apparently, ∇
· unf
(31)
= 0 in the solid/fluid interface, which accounts for the convection of solid particle through the cell.
Denote the correction of fluid velocity at tn−1 as D (unf −1 ) = unf −1 − ûnf −1 = (uns −1 − ûnf −1 )(1 − αnf −1 ), Eq. (27) can be
rewritten as:
ρ nf −1 (ûnf − [ûnf −1 + D (unf −1 )])
t
n −1
∇
f
= ∇ · (μ
+ ∇ · (ρ nf −1 [ûnf −1 + D (unf −1 )] ⊗ ûnf )
⊗ ûnf ) − ∇( pnrgh ) − g · h∇( nf −1 ) +
ρ
n −1
)
1
σ κ ∇(α
(32)
in which can be reformed as,
ρ nf −1 (ûnf − ûnf −1 )
t
n −1
∇
f
∇ · (μ
+ ∇ · (ρ nf −1 ûnf −1 ⊗ ûnf ) + ∇ · (ρ nf −1 D (unf −1 ) ⊗ ûnf ) =
⊗ ûnf ) − ∇( pnrgh ) − g · h∇(ρ
n −1
)+
f
n −1
)+
1
σ κ ∇(α
ρ nf −1 D (unf −1 )
t
.
(33)
in Compared with the momentum equation for the multiphase fluid in Eq. (8), the above equation includes additional terms
(underlined) caused by D (unf −1 ). One can further refer to [50,51] for more discussions of the fictitious domain method
[50,51].
4. Validations and case studies
In this section, several validation cases are presented to demonstrate the capability of the coupled CFD-DEM solver. In
section 4.1, a series of single sphere settling cases are simulated. In these cases, the particles settle through the interface
between two fluids. Different particle Reynold number (Re p ) are set, and the simulated drag coefficient (C d ) values are
compared with the experimental results. In section 4.2, a spherical clump is constructed using the multi-sphere model,
and the settling process is compared with the single sphere settling case. In section 4.3, two spherical clumps settling
case is simulated and the drafting-kissing-tumbling process is simulated. The result is compared with other literatures.
Furthermore, the solver is used to study the interaction between fluid and non-spherical particles. In section 4.4, a round
disk is constructed with the clump model, and the settling process is simulated to show the particle shape effect. Finally, in
section 4.5, a dambreak impact case is simulated with a particular emphasis on wave impact on an assemblage of irregular
particles.
4.1. Sphere settling in multi-phase fluid
Single particle settling is a classic example for validating fluid-particle interaction. When a spherical particle settles in
a viscosity fluid, the driving force is the gravity G, and the resistant force is the buoyancy force Fb and fluid drag force Fd .
Based on theoretical consideration,
π
ρ p d3p g,
π
Fb = − ρ f d3p g,
G=
(34)
6
(35)
6
where ρ p and ρ f are the density of the particle and fluid, respectively, d p is the particle diameter. The drag force can be
defined as a function of the particle settling velocity and particle shapes etc., for example, in the following form:
Fd = −C d
π d2p ρ f |U p |
4
2
(36)
Up ,
where C d is the drag coefficient, U p is settling velocity of the particle. When the particle settling reaches a steady state,
force balance equation G + Fb + Fd = 0 implies that the drag coefficient at the steady state becomes:
C d,num =
4 ρp − ρ f
3
ρf
gd p
.
U p ,steady 2
(37)
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Fig. 7. Geometry settings for the single sphere settling cases.
Table 1
Comparison between experimental and simulation results of single sphere settling.
Case
1
2
3
4
5
6
Box size
LxWxH (m)
ρf
0.1*0.1*0.1
0.1*0.1*0.15
0.4*0.4*0.4
0.01*0.01*0.02
0.01*0.01*0.02
0.02*0.02*0.04
1260
1260
1260
1000
1000
1000
(kg/m3 )
μf
ρp
(Pa·s)
1.5
1.5
1.5
0.001
0.001
0.001
dp
(m)
Numerical results
(kg/m3 )
U p ,steady (m/s)
Re p
C d,num
Experiment
C d,exp
2682
4251
1958.8
1289.7
1820.5
1582.4
0.01
0.01
0.04
0.001
0.001
0.002
0.0560
0.114
0.309
0.0499
0.103
0.147
0.47
0.96
10.4
49.9
103
294
55.60
28.71
4.02
1.52
1.06
0.68
55.64
28.65
3.58
1.25
0.87
0.60
It is worth mentioning that in our resolved CDF-DEM modeling, the drag force is directly calculated based on the fluid field.
The above equation is used to calculate the drag coefficient numerically, and compare it with experimental results.
In addition, the particle Reynolds Number is defined by:
Re p =
|U p |d p
ν
(38)
,
in which, U p is the magnitude of velocity, ν is the kinematic viscosity of the fluid. For a spherical particle, C d can be
experimentally obtained using the following relationship with Re p based on experimental tests [52],
C d,exp =
24
Re p
(1 + 0.15Re0p.681 ) +
0.407
1+
8710
Re p
(39)
.
In Table 1, we summarize six cases used to validate the numerical drag force. In these numerical tests, a wide range of
particle Reynolds numbers are achieved by assuming different material properties ρ f , ν , ρ p , and d p . In case 1, 2, 3, the Re p
values are small, glycerol is used. In case 4, 5, 6, water is used, and the Re p values are large. A rectangular box is used as
the simulation domain as shown in Fig. 7. The box cross section is a square, the width of which is always set to be 10 times
of the particle diameter and the height of the box changes accordingly. The diameter of the particle is 8 times of the mesh
cell length to get accurate results. Therefore, the mesh on the cross section is [80 × 80]. The particle is placed above the
fluid surface initially and released to settle into the fluid.
As shown in Fig. 8, the particle settling process can be divided into four stages in each case. In the first stage, the
particle settles in the air and there is almost no drag force on the particle. The particle accelerates under gravity in this
stage. In the second stage, the particle passes through the fluid surface and starts to decelerate because the drag force in
water and glycerol becomes larger than the effective gravity (gravity force subtracts buoyancy force). With the decreasing
of the particle velocity, the drag force on the particle decreases. When the particle is balanced under the fluid drag force
and effective gravity, it goes to the third stage and settles with a steady velocity. The steady settling velocities are shown in
Table 1. The simulation results of Re p and Cd value are compared with the experimental results in Fig. 9. Finally, in stage 4
when the particle is close to the bottom of the simulation box, the particle velocity gradually decreases to zero in glycerol
as compared to a sudden stop of the particle in the water when it reaches the bottom. This clearly shows the effects of
high viscosity of glycerol. In Table 1, we also compare the drag coefficients of numerical and experimental results. When
Reynolds number is small, the numerical result is close to experimental results. With the increase of Reynolds number, the
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Fig. 8. Settling velocity curves of these 6 cases (a) settling in glycerol (b) settling in water.
Fig. 9. Drag coefficient and Reynolds number of simulation results at the steady state.
numerical C d value is over-estimated by 13%-21% as compared with the experimental result. This implies that the accuracy
of this model needs to be improved for high Reynolds numbers.
The settling velocity curves of different cases are compared below. The first stage and the third stage for all these cases
are similar. There is obvious difference for the second stage. In the settling cases 1, 2, 3 in glycerol with a low Reynolds
number, the particle first decelerates to a velocity smaller than the steady state, and then accelerates to the steady state.
This phenomenon can be explained with the Fig. 10. Fig. 10(a) shows the simulation results in this study. When the particle
touches the fluid surface in case 1, the velocity is relatively high and the glycerol viscosity is large. In the particle entry
process, the cavity closure time is relative larger than the particle passing time. Before the fully closure of the cavity, the
domain behind the particle is air but not glycerol. The pressure in front of the particle is larger. When the particle is fully
immersed in glycerol, the pressure behind the particle increases, so, the total pressure on the particle decreases. For case
6, the fluid entry velocity is relative low and the water viscosity is small. The cavity behind the particle closes faster than
that in case 1. Therefore, this phenomenon is not obvious for cases 4, 5, 6. Fig. 10(b) shows the experimental results by
Schwalbach et al. [53]. Such results are also observed in their experiments study. Fig. 11 show the comparison of pressure
distribution between a partially immersed particle and a fully immersed particle in this study.
Recall that the fluid force on the particle is obtained by summation of the stress divergence term fIB = (−∇ p + ∇ · τ )
in Eqs. (19) (20). For case 5, Fig. 12 (a) show the distribution of the modified pressure field p rgh , which indicate the
hydrodynamic pressure around the moving particle. Fig. 12 (b) shows the fluid velocity around the particle. The stress
divergence term fIB is illustrated in Fig. 12 (c), which is integrated to generate the fluid force on the particle.
The examples presented in this section demonstrated the accuracy of the above numerical scheme using stress divergence.
4.2. Clumped particle settling
In this section, we validate the settling of clumped particles in a fluid using the coupling solver. A clumped spherical
particle and an irregularly shaped particle are constructed using 54 and 14 sub-particles, respectively, as shown in Fig. 13.
Here, we chose the case 4 in Section 4.1 as a reference. The clumped spherical particle and the irregularly shape particle
have the same density, volume and mass as these of the single sphere in Section 4.1 case 4. The equivalent sphere of the
spherical clumped particle has a diameter of 1 mm, while the circumscribed sphere of the irregular shaped particle has a
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Fig. 10. The cavity behind particle in the process passing through the surface; (a) comparison between case 1 and 6; (b) experimental result by Schwalbach
et al. [53].
Fig. 11. Pressure distribution in case 1 for; (a) a partially immersed particle at 0.15 s; (b) a fully immersed particle at 0.50 s.
diameter of 1.23 mm. Along some directions, the dimension of the irregular particle can be smaller than 1 mm due to its
irregular shape.
In this simulation, the simulation box and mesh size are the same with case 4. Along the diameter of these two particles,
there are about 8 mesh cells. As shown in Fig. 14, the settling velocity curves of the single sphere particle and the spherical
clumped particle are almost identical, which means the clumped spherical particle made by 54 sub-particles can closely
approximate the single sphere with the selected mesh size. For the single sphere particle, the “theoretical” steady settling
velocity is approximately 0.05 m/s (the black dash line) based on the experimental results [52] and Eqs. (37)–(39). The
results indicate the coupling model can correctly handle particle clumps. Meanwhile, the settling curve of the irregularly
shaped particle is different from that of the sphere particle. At the decelerating stage 2, the irregular particle decelerates to
a smaller velocity, indicating a higher resistance force acting on it when it enters into the fluid. In the stage 3, the settling
velocity of the irregular particle slightly oscillates around an averaged value of 0.052 m/s, which is slightly larger than that
of the spherical particle with the same volume and mass. This result shows the drag force on this irregular particle is
slightly smaller than sphere particle in the stage 3.
Fig. 15 shows the settling configurations of the irregular particle at different time instances and the fluid velocity at the
time step 0.3 s. It is obvious the settling trajectory is unsymmetrical, and the irregular particle is rotating and oscillating
in the settling process. This case shows the fluid force on the particle is influenced by the particle shape. It is obvious that
an unresolved coupling method cannot capture such phenomenon with irregular particle shape. While with the resolved
method, the particle shape is considered in “void fraction” and this solver is capable to capture this influence of particle
shape. However, there is still a great need to experimentally validate the accuracy of the numerical method for simulating
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Fig. 12. Distribution of (a) modified pressure p rgh , (b) velocity field, and (c) the stress divergence term fIB around the settling particle in Case 5, at 0.12
second.
Fig. 13. A spherical particle and an irregularly shaped particle.
Fig. 14. The settling velocity comparison between single particle, spherical clumped particle and irregular shaped particle.
irregularly shaped particles. In section 4.4, we will further study the setting of disk-shaped particles, and compare the
numerical simulation with experimental tests.
4.3. Two spherical clumps settling
Settling of two spherical particles is a classic example to demonstrate non-contacting interaction in fluid between particles. In the settling process, two particles will undergo drafting-kissing-tumbling motion [54].
The settling particles change the fluid field around. When the trailing particle is located in the wake behind the heading
particle, drag force on it is relatively low. Therefore, the trailing particle begins to accelerate, which is called “drafting”, until
two particles contact in the vertical direction (called “kissing”). In collision, velocity of the heading particle increases while
the trailing particle velocity decreases. Two-particle contact is unstable and they “tumble” and separate at the later stage.
The trailing particle will settle faster than the heading particle after “tumbling”.
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Fig. 15. (a) The settling configurations of the irregular particle from 0 s to 0.3 s with an interval of 0.03 s between two snapshots; (b) Velocity field at time
step 0.3 s.
Fig. 16. Initial settings of the two-particle settling case.
Table 2
Parameters of the two particles and fluid.
simulation box
size (cm)
1*1*4
particle
density (kg/m3 )
diameter (cm)
initial height (cm)
1140
0.167
3.16, 3.5
fluid
density (kg/m3 )
viscosity (m2 /s)
1000
1e-6
Here, we create two spherical clumps and simulate the settling case in a Newtonian fluid to validate the coupling model.
The same material properties as in Glowinski et al. [50] are used in this simulation, see Table 2. The simulation box is a
rectangular channel of 1 cm × 1 cm × 4 cm, as shown in Fig. 16.
The settling velocity time histories of these two particles are compared with results from Glowinski et al. [50] and
Sharma & Patankar [55], as shown in Fig. 17. Before 0.14 s, the trailing particle is out of the leading particle’s wake zone,
so two particles settling velocity are almost identical. From 0.14 s to 0.35 s, the trailing particle settles into the leading
particle’s wake. The fluid velocity in the wake domain is downwards, which drags the trailing particle to a higher speed and
close the gap between two particles. At 0.35 s, these two particles contact with each other, and tumble after the collision.
After 0.35 s, the trailing particle settles faster than the leading particle. Fig. 18 shows some gestures of these two particle
and the velocity distribution in wake domain at different times, which can better represent the non-contact interaction
between these two particles.
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Fig. 17. Comparison of the two particle settling curves between this study and previous studies.
Fig. 18. Ball positions at 0 s, 0.24 s, 0.36 s, 0.48 s, 0.60 s with velocity field in the fluid.
4.4. Disk-shaped particle settling
In this section, interaction between a disk-shaped particle and fluid is studied to highlight the solver’s capability further
in handling non-spherical particles. A disk-shaped particle is a simple non-spherical particle widely studied in the literature.
Experimental studies [56] show that, the settling process of a disk-shaped particle is apparently different from that of a
sphere particle. Three dimensionless parameters are used to characterize the system, i.e., the aspect ratio (λ), the moment
of inertia (I ∗ ) and particle Reynolds number (Re p ),
t
λ= ,
d
I∗ =
(40)
πρ p t
,
(41)
,
(42)
64ρ f d
Re p =
U pd
ν
where t is the thickness, d the diameter of the disk, U p is the particle velocity, v is the kinematic viscosity of the fluid. The
drag coefficient (C d ) of the disk particle is different from that of a sphere, written as,
Cd = 2
ρ p − ρ f gt
.
ρ f U 2p
(43)
Different settling patterns have been observed experimentally [56]. When I ∗ < 0.04 and Re p < 100, the settling pattern
of the disk is steady falling. Under this pattern, the disk with any initial orientation quickly settles down to reach a steady
state, where the flat surface of the disk remains horizontal. When I ∗ < 0.04 and Re p > 150, the settling pattern changes
into periodical oscillation [57]. A transitional state from the steady state to periodical oscillation can be found when Re is
between 100 and 150. Tumbling patterns can be observed when I ∗ > 0.04. In this section, we set three cases to study the
steady settling pattern, periodical oscillation pattern and their transition.
A disk-shaped particle clump is constructed as shown in Fig. 19. From the front view of the clump, there are totally 69
identical sub-particles in each layer. Assuming the diameter of the disk is 1d, and diameter of the sub-particle is 0.1d. We
use 2 overlapping layers in the clump, so, the thickness of the disk is 0.3d and there are total 138 sub-particles in this
clump. The aspect ratio of this circular disk is 0.3.
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Fig. 19. (a) Front view of the disk particle; (b) Oblique view of the clump surface.
Table 3
The parameters for the disk settling cases.
Case
ρ p (kg/m3 )
d (mm)
box size (mm)
1
2
3
1300
1750
2700
1
2
2
6*6*15
12*12*30
12*12*48
Fig. 20. Particle settling positions (a) case 1, interval of 0.05 s; (b) case 2, interval of 0.02 s; (c) case 3, interval of 0.02 s.
For simplification, only one phase fluid is considered in the simulation. The simulation box is set to be a rectangular
channel with a square cross-section. The disk is placed with an initial slanting angle of 45◦ . The properties of fluid are that:
fluid density is 1000 kg/m3 , and viscosity is 0.001 Pa·s. Other material properties used in each case are show in Table 3.
Firstly, we plot the settling trajectories and disk orientation for these 3 cases in Fig. 20 (viewed in x-z plane). In case
1, the disk settles into a relative steady state after release, and there is no obvious particle oscillation. For case 2 and 3,
the particle is oscillating periodically. Fig. 21 shows the fluid velocity field, especially in the wake of the disk. A stable fluid
velocity field can be found in the wake zone for case 1, while for cases 2 and 3, the velocity is also periodically oscillating
around the vertical axis, and the tail of the wake oscillates to the left and right regularly.
The path velocity histories of the disks are plotted in Fig. 22. The path velocity is defined as the magnitude of centroid
velocity of the disk,
|U | =
(U2x + U2y + U2z ).
(44)
The path velocity for case 1 has no obvious fluctuation, which comes to a nearly constant value. The path velocity for
case 2 and 3 shows periodical oscillation. While, in case 2, the amplitude of velocity oscillation decreases with time. In case
3, the oscillation amplitude remains to be rather constant in the settling process.
We average the path velocity between 0.1-0.3 s as the mean path velocity, and list these values in Table 4. Based on
the mean path velocity, the Reynolds number and drag coefficient values are calculated via Eqs. (42), (43). Fig. 23 compares
relationships between the drag coefficients versus the Reynolds numbers, as obtained from numerical simulations in this
study against the experimental result [57]. Clearly, the numerical simulation and the experimental result share a very
consistent trend.
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Fig. 21. Velocity field in the wake domain when the disk is close to the bottom.
Fig. 22. Path velocity of the disks in three cases.
Fig. 23. The Rep and Cd value in this study and from experimental results by Willmarth et al. [57].
To better characterize the disk oscillation, a dimensionless parameter Strouhal number is introduced:
St =
fd
U z,mean
(45)
,
where f is the oscillation frequency. Experimental study from [58] presents the Strouhal number for disk with different
aspect ratios. In their study, mean vertical velocity, but not path velocity, is used to calculate Strouhal number. The mean
vertical velocity and Strouhal number comparison between this study and experimental study [58] are listed in Table 4.
4.5. Dambreak wave impact on a rock pile
Irregularly shaped granular materials, such as rock piles, are usually used for coastal wave erosion protection. In recent
years, the interaction between wave and coastal structures has been studied experimentally [59] and numerically [60]. In
this section, we present a wave impact case using our newly developed model. A small-scale example is presented to study
interaction of dambreak wave with irregular rock particles. The schematic figure of geometries in this case is shown as
Fig. 24.
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Table 4
Simulation velocities and dimensionless numbers of the disks.
Case
1
2
3
I∗
0.0191
0.0257
0.0398
U z,mean
(m/s)
Re p
Cd
f (s−1 )
St sim
St exp
(m/s)
0.021
0.0883
0.1335
0.021
0.0861
0.1087
21.0
176.6
267.0
4.004
1.132
1.123
6.41
8.70
0.149
0.160
0.142
0.155
|U|mean
Fig. 24. Schematic figure of the initial setup.
Table 5
Material properties.
Particle density (kg/m3 )
Volume (m3 )
Radius of circumscribed sphere (m)
Water density (kg/m3 )
viscosity (m2 /s)
Air density (kg/m3 )
viscosity (m2 /s)
2700
3.011e-5
0.0255
1000
1e-6
1.2
1.5e-5
The simulation box is a 150 cm long channel with a [50 cm × 20 cm] cross section. The top of the channel is set an
“inlet and outlet” boundary to allow for inflow and outflow of the air. The other boundaries are set as impermeable wall
boundaries. A water column of [60 cm × 20 cm × 40 cm] is located at the left corner of the simulation box. The properties
of fluids are listed in Table 5. The rock pile is located at the right corner, which is formed by collapse of a cubic assemblage
under gravity. The shape of the rock pile is nearly triangular with a bottom length of 57 cm and height of 30 cm. The slope
angle is approximately 27.8◦ . The simulation domain is meshed with [225 × 30 × 75] cells. The computational time step for
solving fluid motion is 2 × 10−4 s and the time step for particle motion is 2 × 10−5 s. It takes 45.8 hours on a 30-core High
Performance Computer equipped with Intel Xeon E5-2690 to simulate a 6 s dam break scenario.
There are a total of 299 clumps in the rock pile, and the porosity of the rock pile is 0.423. As shown in Fig. 24, each
clump consists of 14 overlapping sub-particles. The properties of the clumped particles used in this case are shown in
Table 5. The inertia tensor of a clumped particle is calculated by Monte Carlo procedure, and its eigenvalues are λ1 =
1.145 × 10−5 , λ2 = 1.442 × 10−5 , λ3 = 1.378 × 10−5 kg·m4 .
Fig. 25 shows the wave flow and impact on the rock pile at different time steps between 1.0-6.0 s in the simulation
channel. In these figures, the interface between water and air is represented in a shading color to represent the fluid
velocity on it. At the same time, the particles are colored to show the particle velocity. In this process, the wave fluctuates
in the simulation channel for three times. Meanwhile, the water infiltrates into the stone pile through the pore space
between rock clumps. Fig. 26 are section views to show the details of the water infiltration process close to the rock pile. In
these figures, red represents the water body and blue represents the air. The development of infiltration front (the red/blue
interface) inside the rock pile can be clearly observed. During the water infiltration, the air trapped inside the rock pile is
gradually displaced out through the top of the rock pile. Generally, the seepage process is non-uniform, and there are still
some air bubbles trapped inside the rock at the end of the simulation.
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Fig. 25. Wave flow and impact on the rock pile at 1.0 s, 2.0 s, 3.0 s, 4.0 s, 5.0 s, 6.0 s. (Color shading of particles: particle velocity; Color shading of air/water
interface: fluid velocity). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Because the right and the bottom boundaries of the simulation domain are set to be impermeable walls, both air and
water cannot pass through them. Fig. 27 shows the velocity field at 1.0 s in the air and water phases. When the wave surges
onto the rock pile, a vortex is formed in the free water body, and water infiltrates into the rock pile mainly in a downward
direction. Fig. 27 also highlights the air flow path inside the rock pile when it is expelled out during water infiltration
process. The air flows upward through the pore space and gathers on the top of the rock pile. Eventually, an air chimney is
formed to eject the air through the water body to the outside.
Finally, the fluid forces and inter-particle contact forces in the rock pile are analyzed. The averaged fluid force F f ,ave is
defined as,
N clump
F f ,ave = (
F j , f )/ N clump ,
(46)
j =1
where, F j , f is the fluid force on a clumped particle j as illustrated in Fig. 5, N clump is the number of clumped particles in
the rock pile. The averaged contact force F c,ave is defined as,
F c ,ave = (
N
contact
Fi ,c )/ N contact ,
(47)
i =1
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Fig. 26. Development of the seepage line from 0 s to 6.0 s. (Red: water; Blue: air).
Fig. 27. Velocity field around the rock pile at 1.0 s. (Red: water; Blue: air).
Fig. 28. Average fluid force on clumped particle and average contact force over all contact points.
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Fig. 29. Particle velocity in the rock pile at 0.2 s, 0.32 s, 0.48 s, 0.56 s.
where Fi ,c is the contact force at contact point i shown in Fig. 5(a), and N contact is the number of contact points in the rock
pile. F f , ave and F c, ave are indicators of wave impact load and inter-particle forces, respectively. Fig. 28 shows the variation
of F f , ave and F c, ave during the wave impact process. F f , ave , is initially zero before the water impact. Upon the first wave
impact at 0.5 s, F f , ave reaches to a peak value of 0.24 N. During the wave impact, some rock clumps at the toe the rock
pile are displaced and move up towards the slope. Fig. 29 shows velocities of moving particles from 0.20 s to 0.56 s with
a maximum particle velocity exceeding 1 m/s. As the wave is subdued at 6 s, the average fluid force approximately equals
to 0.18 N. At this moment, the fluid force on particle mainly consists of the buoyancy force, and the dynamic fluid force is
already not as significant as in the first impact. The average contact force, F c,ave , is initially equal to 1.18 N when the rock
pile is dry and in static equilibrium under gravity load. During the fluid impact process, F c,ave increases to a peak point at
the first wave surge and decreases subsequently. It is interesting to note that at 6 s, the contact force is even smaller than
the initial value, as some part of the rock pile is immersed in water and buoyancy reduces inter-particle contact forces.
The change of the average fluid and contact forces on clumped particles indicates that, internal force states of the rock
pile are influenced by the wave impacting significantly. While, the analysis in this section represents the wave influence
on the rock pile qualitatively. Quantitative description of the influence needs to be further studied and compared with
experimental results.
5. Conclusions
Numerical study of the interaction between porous structure and two-phase fluid in pore-scale is important in both
industry and academia. In this paper, we present a CFD-DEM coupling model for two-phase fluid and irregular shaped particles using a resolved scheme. This model is implemented based on the open source C++ libraries code CFDEM. Compared
with unresolved coupling method, this resolved method can help us to get the meso-scale perspective of fluid flow in porous
media, without using empirical drag-force models. In the validations and case studies, we validate many situations including
two-phase and single-phase fluids, single and clumped spherical particles. Interaction between fluid and disk shaped/irregularly shaped particles is also studied. The study of wave impact on rock pile shows the possibility of using this model to
study interaction between wave and riprap coastal structures. Based on the analysis from above studies, we conclude that:
(1) An extended resolved coupling solver is developed for two-phase fluid. Validation cases with a single sphere settling
show that the dynamic fluid force on the particle agrees well with experimental results when Reynolds number is between
0.5-300. Comparison of different water entry cases shows the model can effectively simulate the influence of air-water
interface on particle settling.
(2) Multi-sphere model is used for particles with complex shapes. From the validation cases of a single clumped particle
settling and drafting-kissing-tumbling simulation, it is demonstrated that the solver is capable of simulating interaction
between the fluid and clumped particles. Although a large number of overlapping spheres are used in constructing a single
clump with complex shape, the gravity and dynamic fluid force are correctly calculated.
(3) Disk settling cases also demonstrate that this solver is capable for modeling non-spherical particles. The simulated
disk settling phenomena compare very well with the observed disk settling patterns in experiments.
(4) The simulation of dambreak wave impact on a rock pile case demonstrates great potential of the model in analyzing
wave erosion problems in coastal engineering. Although the settings of this case are relatively simple, a variety of phenomena in the wave impact process on the rock pile is well simulated, including air-water exchange, seepage process within the
rock pile, as well as particle movement upon wave impact, have been well observed from the simulation. The detailed variation of fluid force and inter-particle contact forces have been analyzed in details, which can help us to better understand
stability of the granular system under wave impact.
Even though this resolved CFD-DEM solver has been successful to handle non-spherical particles, there are still problems
remaining to be solved in the future study. First is about computational efficiency. The resolved coupling requires very small
21
Z. Shen, G. Wang, D. Huang et al.
Journal of Computational Physics 448 (2022) 110695
computational cells, making it very time-consuming. Except for using parallel computing, dynamic mesh refinement could
be used to improve the efficiency. Another problem is about the turbulence modeling. In this study, the Reynolds number
of validation cases ranges from 0.1 to several hundreds. For further study, we will test the model’s capability in solving
fluid-solid coupling problems with high Reynolds numbers and validate these cases. Future work will be continued to apply
this model to more realistic coastal engineering applications, such as coastal embankment erosion and protection.
CRediT authorship contribution statement
Zhihao Shen: Formal analysis, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing.
Gang Wang: Funding acquisition, Methodology, Supervision, Writing – original draft, Writing – review & editing. Duruo
Huang: Conceptualization, Project administration, Writing – original draft, Writing – review & editing. Feng Jin: Funding
acquisition, Methodology, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgements
The study is supported by the Key Program of the National Natural Science Foundation of China (Grant No.
52039005), NSFC/RGC Joint Research Scheme (Grant No. 51861165102 from National Natural Science Foundation of China,
N_HKUST621/18 from Hong Kong Research Grants Council), and General Research Fund 16214220 from Hong Kong Research
Grants Council.
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