The integration of the Equations (27-28) is performed using backward
difference formulation (BDF). BDFs as implicit schemes have less
stability issues [74,
for adaptive time stepping schemes as described in
[71].
2.3. Tagging System
\label{tagging-system}
Due to the very high number of particles and necessity to use the binary
collision, it is very important to manage the data efficiently for the
sake of computational cost. In the following, a system of data
communication is proposed to study collision of flows laden with
numerous particles. The benefit of this system is that for each single
particle, it does not require to check the entire list of particles to
find the potential collision parameters. The following algorithm is
based on the assumption that no more than two particles’ overlapping
functions occupy each grid point of the domain:
- Each particle is tagged with an ID number. The walls also are tagged
with the value of zero.
- The reduced mass variable \(m_{\text{ij}}\) and a new field variable \(P_{\text{id}}\) is declared for each grid point of the domain. \(P_{\text{id}}\) or particle ID, is a two-dimensional array whose the
first argument represents the related cell and the second one with a
size of two stands for ID number of the particle. For the particle \(i\), \(m_{\text{ij}}\) and \(P_{\text{id}}\) are stored to the grid
points occupied by the particle when \(\psi_{i}>\varepsilon\). Here,
for simplicity we do not include the first dimension related to the
cells.
- At first \(m_{\text{ij}}=\infty\) and \(P_{\text{id}}=-1\) are
set in the entire domain. Then, for the grid points nearby the walls \(P_{\text{id}}\left(1\right)=0\).
- For each particle, the values of \(P_{\text{id}}\) in the cells with \(\psi_{i}>\varepsilon\) is set to the particle ID number and also \(m_{\text{ij}}=m_{i}\). If \(P_{\text{id}}\neq-1\) for the first
index, \(P_{\text{id}}\) of the second index is changed from \(-1\) to the particle ID number. Moreover, for those cells, \(m_{\text{ij}}=\left(\frac{1}{m_{i}}+\frac{1}{m_{j}}\right)^{-1}\)is
used to update the reduced mass.
- Particles with \(P_{\text{id}}\left(2\right)\neq-1\) are in the
process of collision and they are tagged to find the collision forces
and moments. For these particles \(m_{\text{ij}}\) is obtained by
using the data from grid points.