Introduction
\label{introduction}
Particle laden flows are common phenomena appearing in natural and
industrial systems: fog drops, blood, sediments, spray dryers, and fuel
injectors are just a few of their extensive applications. While
numerical study of these flows is challenging due to fluid-structure
interaction of several particles with the carrier fluid, the interaction
of particles with each other and with the enclosing surfaces adds to the
complexity of the simulations. The current collision models mostly
include definition of case dependent parameters that can affect the
results. Moreover, the modeling becomes significantly more complicated
when the particles have non-spherical shapes. This includes particles
with arbitrary shapes such as sediments and dust storms or particles
with non-spherical standard shapes such as biconcave shape of red blood
cells, or the cylindrical shape of fibers. Thus, this study presents a
Simplified Spring Collision Model (SSCM) for the contact treatment of
spherical and non-spherical particles. It shows an inexpensive and
robust model with parameters easily obtainable from the flow and
particle motion.
The literatures studying various collision models span very diverse
approaches [1-5]. The direct numerical
simulation of collision of finite size particles as immersed objects
mostly utilizes the discrete element method (DEM)
[6] in conjunction with an immersed
boundary method [7]. DEM considers the
flow in an Eulerian framework and tracks the particles individually in
the Lagrangian framework based on the Newton’s second law of motion. DEM
soft sphere model (SSM) [8] is used
for the cases with limited values of particle Stokes number in which the
effect of flow on two nearby particles are not negligible i.e. when wet
collisions occurs. SSM uses the differential format of particle dynamic
motion and conservation laws update the particles’ velocities and
positions in time [9,
important parameters in the wet collision based on SSM.
1-1- Collision Forces
\label{collision-forces}
The simplest implementation of SSM obtains the collision forces by
defining a repulsive potential for each particle. Glowinski et al
[11] and also Wan & Turek
[12] suggested that the collision
force for spherical particles is proportional to the square of the
amount of overlapping. The overlapping is a parameter representing how
close the two particles are to each other. A similar approach uses a
Lenard-Jones type potential to apply a repulsion force
[13]. Even though the method is
simple to implement, there exists several problems with it. Firstly,
lack of physical reasoning for implementing the model can lead to
unrealistic results [14]. While the
method can prevent particles’ overlapping, it can enforce extra
repulsion leading to unphysical results. Therefore, the accuracy of the
results highly hinges on choosing case dependent parameters
[15-17]. Secondly, the method ignores
energy dissipation due to material damping
[14]. Thirdly, definition of
potential for non-spherical shape particles can be difficult and limited
to classical shapes such as ellipsoids
[18].
A common treatment in the particulate systems relies on modeling forces
using some form of spring and potentially a type of damper. Several
methods are suggested to find the spring system characteristics. One of
the common methods defines the coefficients based on the particle’s
solid structure characteristics. For the spherical particles Hertz
theory defines the spring force as \(k_{\text{Hz}}\delta^{\frac{3}{2}}\) where \(\delta\) is the displacement calculated from the amount of
overlapping [8, \(k_{\text{Hz}}\) is the Hertz spring constant obtained from the two
particles material and geometrical properties. The Hertz theory as an
elastic collision model agrees well with the experiments in dry
collisions [8,
but in a wet collision it neglects the viscous dissipation of energy. A
dashpot can enforce the dissipation of energy during the collision to
coincide the physical behavior. The damping coefficient of the dashpot
is suggested to be proportional to some exponent of displacement
[23,
includes the solid object characteristics in the calculations, the
implementation of the method to model wet collision can be inefficient
even for spherical particles. This is because the time scales related to
the actual solid contact mechanics is usually several orders of
magnitude smaller than the fluid due to very high spring coefficient
values [25].
Decrease of spring coefficient avoids problem of extra diverse time
scales as it extends the collision time to a scale comparable to the
flow scale. Consequently, while the momentum exchange is preserved, the
problem changes from high impulse in a short time to low impulse in a
long time. This situation is very helpful as it does not change the
problem in the large time scale in addition to the fact that the problem
becomes practically solvable. However, the question becomes how to set
the spring and dashpot variables to reach the actual physical behavior.
Kempe and Fröhlich [14] proposed the
adaptive collision model (ACM) by stretching the collision time to ten
times of the flow solver’s time step size. The model inspired by a
Hertz-like force definition for the spring, represents a second order
ODE as the spring and dashpot system. Using each particle’s rebound
velocity from the solid dry coefficient of restitution, the ODE
coefficients were obtained from an iterative solver. When the gap
between two particles is small the viscous forces become significant.
Therefore, the lubrication model of Ladd
[26] empowered the model to account
for the non-negligible viscous dissipation. By proposing an explicit
formulation to find the ODE coefficient instead of an iterative
approach, Ray et al [4] simplified the
implementation of the ACM. Both approaches ignore the hydrodynamic
forces on the particles when the particle is very close to the wall. To
avoid the unphysical behaviors during the enduring contacts, Biegert et
al [27] limited the ignorance of
hydrodynamic forces by including a critical Stokes number below which
the hydrodynamic forces should not be ignored.
The idea of preservation of momentum exchange not only can help to
increase the minimum time scale, it can also justify the implementation
of a linear spring. When the velocity of the bounced particle is
important and not the way this velocity is achieved, a linear ODE such
as harmonic damped oscillator can also represent the collision. This
concept inspired Izard et al. [28]
and Costa et al. [29] and to use the
idea of adjusting the spring and dashpot based on the contact duration.
As suggested by Costa et al. [29]
obtaining the contact during from the flow time scale softens the
collision and saves the computational cost significantly. A modified
lubrication force can enforce the dissipation
[29]. Applying the dissipation
through the lubrication forces as the way used in
[14, 28,
Firstly, the implementation of lubrication forces requires resolved grid
at the places near the contact place which reduces the efficiency of the
simulation. Secondly, as shown in
[27] the dissipation can depend on
the lubrication gap thickness. Biegert et al.
[27] could decrease the dependency on
the lubrication gap by using higher order sub-stepping for the particle
motion. To avoid the complications related to implementation of a
lubrication model, this paper offers a collision model with simplified
definition of collision parameters to make the study of collision
physics easier. With no implementation of lubrication forces, the SSCM
model leads to having less expensive simulations and accurate results.
This model overcomes the difficulties and shortcomings of the available
collision models e.g. iterative schemes, necessity of using case
sensitive parameters, and disparities between the time scales.
1-2- Collision of Non-Spherical
Particles
\label{collision-of-non-spherical-particles}
Study of particle collision becomes much more challenging when the
particles’ shapes are not spherical. Difficulties with studying these
particles collision stems from the fact that finding the related
collision parameters i.e. contact point and the collision forces is a
tedious task. While for the spherical particles the contact point is
located on the intersection of the particles’ interface and the center
to center connecting line, for non-spherical objects the contact point
location depends also on the the more complex geometry. Besides, for
non-spherical objects the contact normal vector mostly does not pass
through the gravity center; therefore, a moment is induced to the
particle. Moreover, the collision forces depend on the amount of
overlapping and the radii of curvature which are also more difficult to
find when the particles do not have a spherical shape.
When the particles have a classical shape e.g. ellipsoids and polygons,
the geometrical features help finding the contact/collision point. There
are several ways to find the collision point of the non-spherical
particles which mostly require an iterative approach to find the contact
point [30-37]. Unfortunately, there
is no common way to define the collision point for non-spherical
particles. The collision point can be defined as the geometric center of
the intersection volume [38]. Lin and
Ng [30,
line connecting the surface of the two particles as the collision point.
The connection line can connect the deepest part of each particle into
the other one’s similar point [39,
common normals of the two particles are located
[30]. The collision point of the
ellipsoids also could be obtained from the closest distance between two
particles as described by [32,
approach sets up two balls inside each particle and tangent to the
surface and updates the positions of the balls by sliding the balls to
the intersection of the ellipsoids and the line connecting the ball
centers. For the polygonal particles, Nezami et al.
[33] used the common plane
[34] between the particles to detect
the collision point. Other contact point detection methods include
polyhedrons [35,
[37,
[43] as well as “potential
particles” [44,
particles by defining a potential field which can morph from a sphere to
cube. The resulted shape can have a shape of sphere, cube or a
transition of sphere to cube. The method was extended to other classical
shapes such as tetrahedral-shaped, prism and rounded tetrahedral-shapes.
For arbitrary particles, no special feature of the geometry can help to
find the collision point. However, it is common to compose the particle
from small geometric elements to form the particle. Thus, the original
particle with arbitrary shape is a composite of the elements that are
glued. The geometric elements can be spheres
[46-53], cylinders
[54] , ellipsoids
[55] or a combination of several
shapes [56]. The problem with
combining different shapes is that the accuracy depends on the number of
used elements. While fewer number of elements are easier and faster to
form the final clumped geometry, the resulted surface will be rough and
inaccurate and multiple collision points might be captured. To avoid
this problem, recently use of “spherosimplices” for complex objects
has gained popularity. Spherosimplices constructs the shapes by moving a
cylinder or sphere over a frame named skeleton
[57-61]. The method is shown to be
faster than the composed particle method. However, the generation of the
shape becomes complicated for the complex objects especially in 3D as it
depends on how accurate the shape of skeleton is set up. Moreover, the
sharp edges will be smoothed because of using a cylinder or sphere to
generate a shape.
The other way to find the contact point of the arbitrary shape particles
is to classify the simulation grid points into fluid and each individual
particle. Wachs [62] implemented this
concept by using a spatial algorithm to define the grid points in the
context of fictitious domain method
[63]. We believe classification of
grid point is the most general method in definition of arbitrary shape
particles as no analytical representation will restrict the definition
of the particles’ shapes. Moreover, finding the contact point which is a
challenging task for non-spherical particles becomes easy. The method
presented by Wachs [62] suggests a
range for the spring/dashpot stiffness based on the problem and an
initial estimation is required for the for their values. Moreover, the
direction of normal forces needs to be defined clearly.
To overcome the shortcoming of the current collision models, this paper
proposes a soft collision model for the simulation of arbitrary shape
particles by using a field variable to find the overlapping places. This
helps us to avoid many difficulties related to finding the collision
points, collision normals, local particle interface curvature and the
amount of overlapping. Moreover, as a non-iterative scheme, the method
saves time in finding the collision parameters. We also propose a novel
spring method for the collision which facilitates implementation of the
model significantly. A particle tagging system is also presented to
conduct the model efficiently for high particle numbers. The method does
not include any case dependent parameters and can handle any particle
shape. The results show that the model is able to simulate
particle-particle and particle-wall collision with proper accuracy.