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Up: 8.1 Compressible Flow without
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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 particlemesh method of
6.5.2; see Fig. 6.9.)
 Update the positions

(8.32) 
to complete the time step.
Particleincell 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," KluwerPlenum 2001