Environmental Engineering Reference
In-Depth Information
ʵ
=
.
ˁ
ij
=
(ˁ
i
+
ˁ
j
)/
where
2. Thus, each SPH particle moves with a velocity
that is close to the average velocity in its neighborhood.
Finally, we need an equation of state, relating
P
and
0
5 and
. The fluid in SPH is treated
as weakly compressible. The compressibility is adjusted to reduce the speed of sound
so that the time step takes reasonable values. In SPH a Courant condition based on the
speed of sound is used. The compressibility is limited by the fact that the sound speed
should be about ten times faster than the maximum fluid velocity in order to keep
variations of density to within less than one percent. Therefore, we use Monaghan
and Kos (
1999
) relationship which is given by
ˁ
B
ˁ
ˁ
0
ʳ
1
P
=
−
,
(5)
c
0
ˁ
0
/ʳ
1,000Kg/m
3
and
where
ʳ
=
7 and
B
=
, with the re
ference d
ensity
ˁ
0
=
(ˁ
0
)
√
(∂
the sound speed at this density
c
0
=
/∂ˁ)
|
ˁ
0
.
For more details on the SPH method and its numerical implementation we refer
the reader to the papers by Liu and Liu (
2003
) and Gesteira et al. (
2010
).
c
P
2.2 Dissipative Particle Dynamics
Normally, complex fluids consist of different solvents and macromolecules such
as polymers, surfactants, ions, and solid structures interacting among one another.
All these molecules have very different sizes and many of the interesting phenom-
ena occur at different time scales. By their nature, microscopic molecular dynamic
simulations demand a great amount of computational resources and one option to
alleviate this problem is to use
coarse graining
simulations such as
dissipative par-
ticle dynamics
simulations (DPD) (Hoogerbrugge and Koelman
1992
). The DPD
method as was originally introduced by Hoogerbrugge and Koelman (2010) con-
sists of grouping numerous molecules or fraction of molecules, in a representative
way into soft mesoscopic beads. In a similar way as in ordinary molecular dynam-
ics simulations, in DPD one has to integrate the equations of motion to get the
particles' velocities and positions, but in this case three contributions to the total
force are present:
conservative
,
dissipative
and
random
. Not all these forces are
independent because the random force compensates the energy dissipated to keep
the temperature
T
constant, and so they act as a regulating thermostat. This fact
leads to the
fluctuation-dissipation
theorem which gives:
2
2
k
B
T
where
k
B
is the
˃
ʳ
=
Boltzmann's constant.
Dissipative forces account for the local viscosity of the medium, and are of the
form
r
ij
)
ˆ
v
ij
ˆ
F
ij
D
=−
ʳˉ
(
e
ij
·
e
ij
(6)
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