   Next: 8.1.3 Smoothed Particle Hydrodynamics Up: 8.1 Compressible Flow without Previous: 8.1.1 Explicit Eulerian Methods

8.1.2 Particle-in-Cell Method (PIC)

Consider an ideal gas; assume adiabatic equation of state (fast flow or slow conduction of heat): constant in a flowing element Lagrangian time derivative . Therefore (8.17)

(continuity equation for ). Flow equations:   (8.18)   (8.19)   (8.20)

Strategy:
- Discretize the time: - Assume a 2D (or 3D) Euler lattice with - Represent the (variable) local density by a number of particles in each cell
- Each particle represents a fluid element (not a molecule!) and carries a vector of properties, (8.21)

The properties of the Eulerian (space fixed) cells are sums over the particles they contain:   (8.22)   (8.23)   (8.24)

with (8.25)

- Now rewrite 8.19 as (8.26)

and first treat only the part :   (8.27)   (8.28)

with - Update the particle properties and , thus: (8.29)

- Now treat the Lagrangian part of equation 8.26, by letting the fluid particles move with appropriate velocities: Defining (8.30)

compute the particle velocities as a weighted sum over the adjacent Eulerian cells: (8.31)

where the weights are the overlap areas of a square of side length centered around particle and the nearest Euler cells . (See the particle-mesh method of 6.5.2; see Fig. 6.9.)
- Update the positions (8.32)

to complete the time step. Particle-in-cell method. Note that pressure gradients are evaluated using an Eulerian grid, while the transport of mass, momentum and energy is treated in continuous space.   Next: 8.1.3 Smoothed Particle Hydrodynamics Up: 8.1 Compressible Flow without Previous: 8.1.1 Explicit Eulerian Methods
Franz J. Vesely Oct 2005