Environmental Engineering Reference
In-Depth Information
The compressibility reduces the speed of sound making the time step requirement
more reasonable. Another limitation on weak compressibility is to restrict the sound
speed to be at least ten times faster than the maximum fluid velocity, thereby keeping
density variations to within less than 1%.
Following Batchelor (
1974
) and Monaghan et al. (
1999
), the expression that
relates pressure and density is written as:
B
ˁ
ˁ
0
ʳ
1
P
=
−
,
(9)
where the parameter
B
is related to the compressibility of the fluid;
ˁ
0
=
1000
.
0
m
3
is a reference density, taken as the density of the fluid at the surface, and
kg
ʳ
is the polytropic constant that usually ranges from 1 to 7. The latter value has been
adopted in DualSPHysics.
/
2.6 Moving the Particles
Particles are moved using the XSPH variant (Monaghan
1989
):
b
d
r
a
dt
=
m
b
ˁ
ab
v
a
+
ʵ
v
ba
W
ab
,
(10)
1
where
ˁ
ab
=
2
(ˁ
a
+
ˁ
b
)
and
ʵ
is a constant, whose value ranges from zero to unity.
Here
5 is commonly used.
This method is a correction for the velocity of particle
a
. This velocity is recalcu-
lated taking into account the velocity of the particle and the average velocity of all
particles that interact with particle
a
. Only the closest neighbours will be included,
due to the compact support of the kernel. This correction allows particles to be more
organized and, for high fluid velocities, helps to avoid particle penetration through
the boundaries.
ʵ
=
0
.
2.7 Time Stepping
The physical magnitudes (velocity, density, position and density) change every time
step due to the forces calculated during particle interactions. The time integration
scheme to compute the new values of these quantities at the next time step in SPH
should be at least of second order to obtain accurate enough results.
Consider the equations of momentum (
6
), density (
8
), and position (
10
) written
in the following form:
d
v
a
dt
=
F
a
,
(11)
d
ρ
a
dt
=
D
a
(12)
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