Next: 8.1.3 Smoothed Particle Hydrodynamics
Up: 8.1 Compressible Flow without
Previous: 8.1.1 Explicit Eulerian Methods
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:
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:
with
|
(8.25) |
- Now rewrite 8.19 as
|
(8.26) |
and first treat only the part
:
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
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001