# Particle Dynamics Modeling Using PII Scheme

#### Dynamics of impinging jets

Particle interaction mechanism is realized using a **probabilistic implicit
interaction** scheme (PII) [1]. The advantage of the
scheme is that it provides a linear dependence of the execution time on the
number of particles simulated.

PII is realized using a concept of *virtual* particles, which are the
carriers of the interaction force, similar to the description of quantum field
theory. Any real particle can convert a part of its energy, momentum, mass,
etc. into a virtual particle, which in tern should return these quantities to a
real particle. If there is no other real particle but the one that generated
the virtual particle, the quantities *borrowed* by the virtual particle
are returned back to the original particle. This would correspond to a free
propagation without interactions. If on the other hand, there are other
particles around, which could be the recepients of the quantities, an exchange
of quantities may occur, which constitutes an interaction. In the end of every
time-step all virtual particles should dissapear by dissipating all their
quantitis to the real particles, thereby satisfying the conservation laws.

Introducing virtual particles makes the execution time of the code linear in
the number of particles. This is due to the fact that instead of using a nested
double-loop over the particles to analyse the interaction distances and forces,
only two subsequent loops over real and virtual particles are needed to compute
real-virtual and virtual-real interactions.

The interaction events between the real and virtual particles are governed by
the interaction probabilities. In the current implementation the probabilities
are represented by a one-parametric family of functions. A simple change in
the single parameter may result in very different interaction mechanisms, which
are shown below. In the simulation three particle jets were injected at right
angles to each other from three faces of a cube, so that all will collide at a
sinble impingement point.

More on the PII scheme.

#### Hard-balls

[Hard-ball collision: three impinging jets]

**Animation**

#### Deflection

[Deflection: three impinging jets]

**Animation**

#### Soft-balls

[Soft-ball collision: three impinging jets]

**Animation**

####
Simulation of coalescing bubbles rising in a water duct

(small-frame animation, mpg, 25MB)

(big-frame animation, mpg, 80MB)

#### References

[1] Smirnov, A. and Celik, I. "A Lagrangian Particle Dynamics Model with an Implicit Four-Way Coupling Scheme, The 2000 ASME International Mechanical Engineering Congress and Exposition. Fluids Engineering Division, vol.FED-253, Orlando, FL, Nov.5-11, 2000, pp.93-100, 2000.