Environmental Engineering Reference
In-Depth Information
Ta b l e 1
Different values of
ʱ
D
for 2D and 3D
Kernel
2D
3D
h
2
)
h
3
)
Cubic-spline
10/(7
ˀ
1/(
ˀ
7/(4
ˀ
h
2
)
21/(16
ˀ
h
3
)
Wendland
2.3 Momentum Equation
The equation of momentum conservation in a continuum field is
d
d
t
=−
1
ˁ
∇
P
+
g
+
ʓ
,
(5)
refers to the dissipative terms and
g
is the gravitational acceleration (0, 0,
where
ʓ
s
−
2
. There are several ways to solve the dissipative terms, but the artificial
viscosity proposed by Monaghan (
1992
) has been the most widely used due to its
simplicity.
In discrete notation and adding the artificial viscosity, the previous equation can
be rewritten as follows:
−
9.81) m
·
P
b
ˁ
b
m
b
d
v
a
d
t
P
a
ˁ
∇
a
W
ab
+
g
=−
b
+
a
+
ʠ
ab
(6)
2
where
P
k
represents the pressure of particle
k
(with
k
a
or
b
). The artificial
viscosity term depends on the relative position and motion of the computed particles
=
−
ʱ
c
ab
μ
ab
ˁ
ab
v
ab
·
r
ab
<
0
ʠ
ab
=
(7)
0
v
ab
·
r
ab
>
0
,
r
ab
+
ʷ
2
where
r
ab
=
r
a
−
v
ab
=
v
a
−
μ
ab
=
v
ab
·
r
ab
/(
)
,
c
ab
=
.
(
C
a
+
C
b
)
r
b
,
r
b
,
h
0
5
2
01
h
2
, and
ʷ
=
.
ʱ
is the mean value of the speed of sound,
is a free parameter that
should be tuned according to the configuration of the problem.
0
2.4 Continuity Equation
Themass of each particle is constant, so that changes in the fluid density are computed
by solving the conservation of mass or continuity equation in SPH form:
b
m
b
v
ab
·∇
a
W
ab
.
d
ˁ
a
d
t
=
(8)
2.5 Equation of State
Following the work of Monaghan (
1994
), the fluid is treated as weakly compressible
and an equation of state is used to determine the pressure as a function of density.
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