Openfoam Course - Lagrangian Particle Interaction

OpenFOAM course - Lagrangian particle interaction
Erik Larsson
2nd February 2009
A brief introduction to Lagrangian particle tracking

There are four levels of particle tracking or coupling between the continous and the particle
phase in discrete element modelling. In one way coupling, the continous pahse is not aected
by the particles. The ow eld is solved before the particles is let into the ow and tracked
as they go. The particles does not know of any other particles in the ow.
For two way coupling, the uid ow is solved together with the movement for Lagrangian
particles. The particle inuence on the continous phase is taken into account. Both through
the momentum transfer between the phases and the volume fraction of the particles.
As more knowledge is needed a third ond fourth way coupling is considered. In four way
coupling, particle- particle interaction is also taken into account. In the less used three way
coupling, the particles ar interacting without collisions. To move futher, by e.g. resolve the
ow eld around eachindividual particle, would be going into DNS.
Governing equation
The governing equation for the particles is Newtons second law.
d2 xi X
mp = Fi (1)
dt2
The force vector is a matter of choice. The level of detail in the vector can be large by adding
many forces or simple by choosing the largest forces. The largest forces to account for are
depending on the specic case but the drag-, gravity-, and bouyancy forces are important
in many cases. Other forces may be the Basset force (accounting for particle history), the
Samann force (Velocity gradients in the main ow) and the Magnus lift force (particle rotat-
ing). The particles also give rise to an extra source term in the Navier- Stokes equation that
has to be included when solving the continous phase.
Particle collisions
Collisions between particles can be treated in dierent ways. One must choose between a hard
and a soft spere approach. The hard sphere approach is more suited for collision dominated
dilute ows and the soft spere approach is better for contact dominated dense ows.
The hard sphere approach is the more simple of the two, the collisions are instantaneous and
is simple to calculate through the conservation of momentum before and after collision except
1
TME 050 OpenFOAM Erik Larsson
for losses. The losses are calculated with the aid of two constants, e the coecient of normal
restitution and ζ the coecient of tangential restitution.
n · v012 = −en · v12 (2)
n × v012 = −ζn × v12 (3)
v012 is the relative speed between collision partners after collision, v12 is the relative speed
before the collision and n is the normal vector from the contact surface. The particles may
also stick to or slide against each other depending on the relation between the tangential and
the normal components of the collision, n×vn·v and the Coulomb friction, µ.
The soft sphere approach to collision modelling is less straight forward. The collision be-
tween particles must be allowed to last a number of timesteps. In order to model the collision,
the deformation of the particles and the contact between the particles must be taken into ac-
count. In this project, only the hard sphere model is implemented and the soft sphere model
will not be discussed further.
Numerical modelling of hard sphere collisions

In this section, the governing equations of the hard sphere collisoin model are presented.
First, the possibility of collision within the next time step has to be calculated. This can be
done by deterministic or by a stochastic method, here the deterministic method is presented.
The possibility can be expressed by the following equation.
|nt + k(nt+dt − nt |2 = (r1 + r2 )2 (4)
Collision occurs if 0 < k < 1. nt denotes the relative position at time t and nt+dt denotes the
relative position at time t + dt.
The particles are assumed to slide against each other, or in other words, µ < n×vn·v . The post
collision velocities are expressed as:
2 mj
vi 0 = vi − {(1 + e)(n · vij )n + |vct |t} (5)
7 mi + mj
5 mj
ωi 0 = ωi − |vct |(n × t) (6)
7ri mi + mj
vct is the relative tangential velocity at the point of contact. The equations are from Crowe,
Sommerfeld and Tsuji (1998).
The solidParticleCloud class in OpenFOAM

The solidParticleCloud class in OpenFOAM is a class that calculates the movement of parti-
cles.
Particle properties
The particles are assumed to be rigid and spherical and are only described by their constants
density, coecient of restitution, coecient of friction and diameter.
The solver only solves for the particle position and velocity. Above all, a particle rotation
would improve the physics of the ow.
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TME 050 OpenFOAM Erik Larsson
Particle forces
The forces acting on the particles are the drag, gravity and bouyancy force. The drag force is
given by the expression
24νc 3ρc
FD = (1 + 0.15Re0.687
p ) (7)
d 4dρp
The drag coecient is dependent on the Reynolds number and this correlation gives a good
correspondance for Re < 800.

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OpenFOAM course - Lagrangian particle interaction

A brief introduction to Lagrangian particle tracking

Numerical modelling of hard sphere collisions

The solidParticleCloud class in OpenFOAM

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