Environmental Engineering Reference
In-Depth Information
Fig. 2
h
region where the
kernel is applied
b
b
b
r
b
b
2
h
a
b
b
b
where the sum is over all particles (
b
) within the
h
region in the kernel function (see
Fig.
2
). The mass and density are denoted by m
b
and
ρ
b
, respectively, such that the
m
b
ρ
b
volume of a particle is defined by
V
b
=
, and for the position vector
r
=
r
a
,
W
ab
=
(
is the kernel between particles
a
and
b
.
The momentum equation is written in the form
r
a
−
r
b
,
h
)
dv
dt
=−
1
ρ
∇
P
+
g
+
,
(5)
8m/s
2
is the gravity acceleration. In the
DualSPHysics code it is possible to solve Eq. (
5
) by two options: artificial viscosity
(Monaghan
1992
) and laminar viscosity with sub-particle scale (Lo and Shao
2002
;
Gotoh et al.
2001
). Artificial viscositywas used in thiswork. Themomentumequation
in SPH notation is
is a dissipative term, and
g
=−
.
where
9
P
b
ρ
b
m
b
dv
a
dt
=−
P
a
ρ
b
+
a
+
ab
∇
a
W
ab
+
g,
(6)
2
where
ab
is a viscosity term (Gomez-Gestéira et al.
2012
).
The changes in the density are calculate by
b
m
b
v
ab
∇
a
W
ab
.
ρ
a
dt
=
d
(7)
The equation of state is based on Monaghan (
1994
), where the fluid is considered
as weakly compressible and Eq. (
8
) determines the pressure of the fluid which is
written in the form
B
ρ
ρ
0
γ
1
P
=
−
,
(8)
c
0
ρ
0
γ
1,000km/m
3
the reference density, and c
0
the
speed of sound at the reference density.
where
γ
=
7,
B
=
, being
ρ
0
=
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