12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

FINITE-TIME BLOWUP RELATED TO PARTICLE COLLISION IN DIRECTEULERIAN SIMULATION

Seulgi Lee
Department of Prosthodontics
Yonsei University College of Dentistry50-1 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Republic of Korea
Department of Mechanical Engineering
Yonsei University
50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Republic of Korea
sg.lee@yonsei.ac.kr

Changhoon Lee
Department of Mechanical Engineering & School of Mathematics and Computing
Yonsei University
50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Republic of Korea
clee@yonsei.ac.kr

ABSTRACTnation and understanding of the basic mechanism of turbulent
particle interaction. This is because the underlying mechanismof turbulence that causes particle movement is not fully understood due to its nonlinearity and multiscale properties. Moreover, the movement of particles, which has nonlinearity different from fluids, contributes more to the complexity. Choi et al.
(2016) attempted to investigate directly the collision mechanism tracking colliding finite-sized particle in particle-laden
isotropic turbulence. They showed that colliding pattern varies
depending on Stokes number. However, an investigation of acollision event in the Lagrangian simulation of particles has
limitations since it requires expansive computational cost, andluck to have a collision.
An inspiring characterization of particle collision wasproposed by Falkovich et al. (2002), who derived a formulafor the collision rate of small inertial particle in turbulence using a form that traces random trajectories (Shraiman & Siggia, 2000; Falkovich et al., 2001). They introduced a particlevelocity field ~v(~x,t) instead of a particle motion equation established. The governing equation of particle velocity field isdetermined by relation with particle velocity gradient and particle time scale. When particle collision happens the particle
velocity gradient goes to infinity, that is finite-time blowup.
This singularity of particle velocity gradient implies that theparticle velocity develops blowup and becomes multi-values;
in other word, a particle can have two different velocities atthe same position. Therefore, the eventual goal of this study isto gain a better understanding of collision process of particles
using directly rigorous Eulerian approach.

This study inspired by Falkovich et al. (2002) aims toinvestigate the motion of particles in turbulence using Eulerian approach under the assumption that particle velocity isa smooth function in space and is uniquely determined by
the particle position until collisions between particles occur.
When the first collision happens, a discontinuity in the particle velocity gradient appears, resulting in finite-time blowup.
Using this approach, the singularities in particle velocity gradient, particle number density, and particle vorticity for various Stokes numbers and gravity factors are investigated numerically in a simple and intuitive Taylor-Green vortex flow,
two-dimensional turbulence, and three-dimensional isotropicturbulence.

INSTRUCTION
Particle-laden turbulence has been actively studied for theinvestigation of natural phenomena and utilization in various
industrial fields. Often, collisions between particles play a rolein the determination of particle behavior. For example, the precipitation process in clouds is triggered by the collision rate
between settling small water droplets affected by backgroundturbulence, thus much research has been carried out to predictthe rainfall.
To identify collision between particles, the preferential
concentration or clustering of inertial particles has been alternatively investigated using direct numerical simulation (DNS)
for decades. The primary mechanism of the preferential concentration is the centrifuge effect of the inertial particles
around rotating structures of turbulence (Reade & Collins,2000). Moreover, a new type of preferential accumulation incolumnar structures was observed (Bec et al., 2014; Park &Lee, 2014). Such a clustering naturally leads to more collision
between particles. Still, there is a lack of mathematical expla-METHODOLOGYWith specific parameters, it is possible to find a phenomenon in which the behavior of particles moves like a fluid
called flow of particles. (Park & Lee, 2014; Fouxon et al.,1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1. Plot of representing collision of particle (St = 1,W = 0): Colored contours represent the vorticity level of the backgroundflow (left) and particle number density (right). The solid lines for the negative divergence level of particles are denoted.

RESULTS AND DISCUSSION

2015). Assuming that the behavior of these particles is smoothin space, the particle velocity equation and particle numberdensity (n(~x,t)) can be derived in the Eulerian frame as follows:∂~v~u −~v+ (~v · ∇)~v =+~g∂tτp∂n+ (~v · ∇) n = −∇ · (n~v)∂tTaylor-Green vortex has an exact solution in twodimensional space, and it is easy to intuitively understand thebehavior of particles. It was confirmed in Figure 1 that when
the particle number density has the maximum value, the negative value of the particle velocity gradient increases at the sametime where the collision happens in the x- and y-axis direction. Moreover, since there is a high probability that the collision direction is not in the x-axis and y-axis directions in twodimensional and three-dimensional turbulence, the minimum
value of eigenvalue on symmetric part of particle velocity gradient (λD : D is dimension of simulation) was investigated, andit was confirmed the results for the Taylor-Green vortex. Finally, such statistics were examined for three-dimensional turbulence.
To understand the mutual mechanism of particles andfluids, the particle equation of motion at the Eulerian lattice
points obtained by the new approach can be used to investigate the finite-time blowup (Beale et al., 1984; Majda, 1991;
Bustamante & Kerr, 2008). Therefore, the solution of Eq. (1)can be differentiated to obtain a solution that diverges within afinite time, and is as follows (Falkovich et al., 2002):

(1)
(2)

where ~v is the particle velocity, ~u is the fluid velocity, ~g is thegravity acceleration, and τ p is the time scale of particle.
Direct numerical simulation was performed to understand the mechanism of particles suspended in turbulence.
For Taylor-Green vortex, the exact solutions were used andfor the other background fluids as two-dimensional decaying turbulence and three-dimensional isotropic turbulence, thespatial discretization was calculated using the Fourier spectral method, and the time advancement was carried out using the 3rd-order Runge-Kutta method. Both fluid and particle
governing equations were calculated in the Eulerian framework. For the two-dimension and three-dimension computational regions, periodic boundary conditions were set for asquare domain [2π × 2π] and a regular hexahedron domain[2π × 2π × 2π], respectively.
The dimensionless numbers that determine the physicalproperties of suspended particles through dimensional analysis
are the Stokes number (St) and the gravity factor (W ), which
are defined as follows:
St =τp,τf

W=σ (t) ∼1t − tc(3)

where the blowup of σ = 5~v means the collision of two particles, and tc is the collision time.
For particle number density, the solution of Eq. (2) for the
initial uniform particle number density can be expressed as:vTvf

 Zt
n(t) = n0 exp − 5 ·~v(~x(t 0 ),t 0 )dt 0(4)0where τ f and v f depend on each background fluid. In the case√√of 2D flow, τ f = 1/ ωi ωi and v f = vi vi were used and inthe case of 3D flow, the Kolmogorov scale was used.
In this study, we investigated the case where the Stokes
number is less than or greater than 1 and the gravitational constant is 0-20 using Eulerian direct simulation. That is, the analysis was performed in the range of the case where the collision
did not occur and the case where the collision occurred depending on the presence or absence of gravity.

If 5 ·~v ∼ 1/(t −tc ) near blowup, the solution of Eq. (4) can bederived as:n(t)tc∼|t − tc |n0where n0 is the initial uniform particle number density.

2(5)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. Time evolution of particle velocity gradient, particle number density, and particle vorticity for various Stokes numbers withgravity factors in three background fluids until the collision between particles: Cλ , Cn , Cω are the compensation factors.
For particle vorticity ω p in finite-time singularity, the derived equation follows as:ω p (t) ∼1t − tc

though the collision happens before increasing gravity. Thismeans that flow of particles exists. This study has started with
an ideal assumption. From this, particle vorticity ω p is ideallyderived. Thus, the finite-time blowup of particle vorticity can
not be observed for 2D Taylor-Green vortex.

(6)

that is equivalent to the particle velocity gradient.
The derived Eqs. (3), (5), and (6) look simple, but theirsolutions are determined by very complex processes. In particular, when the motion of particles is driven by turbulence,the particle velocity gradient can easily go to infinity in a finite
time due to quadratic nonlinearity. A discontinuity in the particle velocity gradient means a collision between two particles
physically. The major results in all background fluids depicted
in Figure 2 show the process of finite-time blowup for particle velocity, particle number density, and particle vorticity. The
solutions of particle velocity gradient, particle number density,and particle vorticity follow well Eqs. (3), (5), and (6), respectively. Therefore the results are collected altogether in a linear
line. With increasing gravity, the particles do not collide even

CONCLUSION
In order to provide a complete mathematical descriptionof a particle collision in particle-laden turbulence, we studied
Eulerian approach to the motion of particles, which was derived under the assumption that particle velocity is a smoothfunction of space. Finite-time blowups of particle velocity gradient, particle number density, and particle vorticity were observed as a result of the mutual collision between particles. The
simulation results showed that it is consistent with the solution
obtained by Eulerian approach. Therefore, the characteristicsof the background fluid can be grasped in the vicinity of thecollision of the three-dimensional turbulence, and this leads toan understanding of the mechanism of collision.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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