Free cooling granular system (2D)

Model/Simulation:

In this tutorial, the homogeneous and inhomogeneous regimes of a free cooling granular system (2D) are simulated [1]. This model is common in the study of the kinetic theory of granular gases [2-3]. For a 3D version of this code, see Free cooling granular system (3D).

Description:

Granular gasses are particulate systems in which the mean free path of the particles is greater than the typical particle size. Free cooling is the study of how a granular gas behaves when no external forces are applied. Such a free cooling model is characterized by a decay in kinetic energy. This decay of kinetic energy is due to effects such as dissipation during particle collisions, the formation of clusters, anomalous diffusion and characteristic shock waves, to name but a few [3].

Here, a short ParaView animation of the free cooling demo is shown.

Simulation set up:

To simulate the free cooling of a granular gas in 2D, particles are inserted into a square box, with periodic boundary conditions in both dimensions. First, a well-defined initial state with random particle positions and velocities is prepared in the following way:

1. The particles first sit on a regular lattice and get a random velocity with a total momentum of zero.
2. Then the simulation is started without dissipation and runs for a reasonable number of collisions per particle so that the system becomes homogeneous, "and the velocity distribution approaches a Maxwellian."
3. The above state is then used as the initial configuration for the dissipative regimes.

About the code

First, we need to include the following headers:

#include<iostream>
#include <Species/LinearViscoelasticSpecies.h>
#include <Boundaries/PeriodicBoundary.h>
#include "Mercury2D.h"

Then, we define a new class Free cooling granular system (2D). Because we want to run 2D simulations, this class inherits from the Mercury2D class.

class FreeCoolingDemoProblem : public Mercury2D 

Data members of the class:

The FreeCoolingDemo class has, but is not limited to, two (public) data members, namely a pointer to the species and number of particles

    int N = 1;
    LinearViscoelasticSpecies* FC2D_Species;
 

The key components of the class are explained in-turn, in the following:

Step 1: Define Walls/Periodic Boundaries
For experimental purposes, particles can be contained by a two dimensional box made up of four infinite walls. If this is required, then see the Free cooling granular system (2D) in walls. Otherwise, the simulation volume is replaced by periodic boundaries.

        PeriodicBoundary pb;
        pb.set(Vec3D(1,0,0), getXMin(), getXMax());
        boundaryHandler.copyAndAddObject(pb);
        pb.set(Vec3D(0,1,0), getYMin(), getYMax());
        boundaryHandler.copyAndAddObject(pb);

Step 2: Create Particles
Next, the particle species is defined. The particles in this problem use the linear visco-elastic (normal) contact model (LinearViscoelasticSpecies). The dissipation and stiffness defining the contact model can be set in different ways. In this example these contact model parameters are defined.

        auto species = speciesHandler.copyAndAddObject(LinearViscoelasticSpecies());
        FC2D_Species = species;
        species->setDensity(10000);
        species->setDissipation(0.0);
        species->setStiffness(1e3);

The particle properties are set subsequently. The particleHandler is cleared just to be sure it is empty, then the particle to be copied into the container is created and the set species is assigned to it.

        particleHandler.clear();
        SphericalParticle p0;
        p0.setSpecies(speciesHandler.getObject(0));

Step 3: Place Particles
After specifying the particle properties, the container is filled with copies of the particle. In this example, particles are placed in a lattice grid pattern, on evenly spaced positions.

        int N1=static_cast<int>(sqrt(N))+1;
        for (int i=0;i<N;i++) {
            int ix = i % N1;
            int iy = i / N1;
            // set particle position
            double x = (getXMax() - getXMin()) * (ix + 1) / (N1 + 1);
            double y = (getYMax() - getYMin()) * (iy + 1) / (N1 + 1);
            p0.setPosition(Vec3D(x, y, 0.0));
            // set random velocities for the particle
            p0.setVelocity(Vec3D(random.getRandomNumber(-0.001,0.001),  random.getRandomNumber(-0.001,0.001), 0.0));
            p0.setRadius(0.0002);
            p0.setSpecies(species);
            particleHandler.copyAndAddObject(p0);
        }

Step 4: Centre of mass velocity
Next, the center of mass velocity is subtracted to ensure a reduced random velocity. This results into a center of mass velocity nearly equal to zero.

        // Compute the center of mass velocity
        double particle_mass =  p0.getMass();
        double M_b = N*particle_mass; // mass of the bulk system
        Vec3D V_com = {0,0,0};
        for (int k = 0; k < particleHandler.getNumberOfObjects() ; k++){
            BaseParticle* p = particleHandler.getObject(k);
            V_com +=  (particle_mass*p->getVelocity())/M_b;
        }
        // Compute the reduced velocity for each particle
        for (int k = 0; k < particleHandler.getNumberOfObjects() ; k++){
            BaseParticle* p = particleHandler.getObject(k);
            p->setVelocity(p->getVelocity() - V_com);
        }

Actions After TimeStep:
The actionsAfterTimeStep() method specifies all actions that need to be performed in between time steps, i.e. Since, the simulation started with zero dissipation, after a reasonable number of collisions per particle, the system will be homogeneous, and the actionsAfterTimeStep() is used to set the initial configuration for the "next" dissipative regime.

    void actionsAfterTimeStep() override{
        // Time to switch on dissipation
        if (getTime() > 4e5*getTimeStep())
        {
            FC2D_Species->setDissipation(0.25);
        }
    }

Main Function

In the main program, the FreeCoolingDemo object is created, after which some of its basic properties are set: like, the number of particles, box dimensions, time step and saving parameters. Lastly, the problem is actually solved by calling its solve() method.

int main()
{
    // Problem setup
    FreeCoolingDemoProblem problem;
    problem.setName("FreeCoolingDemo");
    problem.N=2000;
    problem.setGravity(Vec3D(0.0,0.0,0.0));
    problem.setTimeStep(5e-5);
    problem.setSaveCount(4000);
    problem.setTimeMax(100.0);
    problem.setMax({0.032,0.032,0.032});
    problem.setHGridMaxLevels(1);
    problem.setHGridCellOverSizeRatio(1.2);
    problem.setHGridUpdateEachTimeStep(false);
    problem.setFileType(FileType::ONE_FILE);
    problem.setParticlesWriteVTK(true);
    problem.solve();
}

References:

1. Luding, S. (2005). Structure and cluster formation in granular media. Pramana, 64(6), 893-902.
2. Brilliantov, N. V., & Pöschel, T. (2010). Kinetic theory of granular gases. Oxford University Press.
3. Pöschel, T., & Luding, S. (Eds.). (2001). Granular gases (Vol. 564). Springer Science & Business Media

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