Environmental Engineering Reference
In-Depth Information
In a series of works (Galperin et al., 1994, 1995a, b, 1996a, b; Galperin, 1999,
2000) it was shown by Michael Galperin that the advantages of Lagrangian and
Eulerian approaches can be combined successfully into a non-diffusive positively
defined and reasonably monotonic advection scheme, which versions were pre-
sented by Galperin (1999, 2000), Galperin et al. (1996a), and Sofiev (2000). The
procedure mainly follows the methodology suggested in Galperin (2000), which
can be qualified as a pseudo-Lagrangian finite volume approach.
When deriving the Galperin's scheme, one should re-write the advection term
in Eq. 1 in the divergent form using the continuity equation with boundary
conditions defined separately for in- and outflow boundaries. Simulation grid is
defined as a set of
N
grid cells (
i= ). Centre of the
i-th
grid cell has a co-ordinate
equal to
x
i
, its left- and right-hand borders have co-ordinates
x
i-0.5
and
x
i+0.5
,
respectively. The 1-D cell volume is then
V
i
= x
i+0.5
−
x
i-0.5
. The advected field
is described by the total mass M
i
and position of the centre of mass X
i
,
]
1
N
X
−
∈
in each grid cell
i
. Initially, a piecewise constant mass
concentration ϕ
I
= M
i
/V
i
is defined and X
i
is set equal to the center-point of each
grid cell:
[
x
,
x
i
i
0
i
+
0
X
=
. At time step
k
, the mass distribution inside
i-th
cell is defined
as a rectangular pulse equal to 0 outside
x
i
i
ω and equal to
0
M
/
ω inside
ω,
i
i
(
)
X
ω
is the distance from the centre of
mass position
X
i
to the nearest border of the cell. Advection of the pulse does not
change its shape within the time step δ
t
assuming locally constant velocity field
“a”:
=
min
−
x
,
X
−
x
where
i
i
i
−
0
5
i
i
+
0
5
i
+
1
i
+
1
ϕ
(
x
)
=
ϕ
(
x
−
a
(
X
,
t
)
δ
t
)
. The global concentration distribution at
i
k
k
+
1
∑
ϕ
k
i
+
1
k+1
step is considered as a sum of pulses
ϕ
(
x
,
t
)
=
(
x
)
,
x
∈
[
−∞
,
∞
]
.
i
Finally, for each grid cell new mass and its position are computed following the
mass and momentum conservation equations
x
x
i
+
0
i
+
0
∫
∫
M
=
ϕ
(
x
)
dx
,
X
=
x
ϕ
(
x
)
dx
i
i
x
x
i
−
0
i
−
0
The Galperin's scheme is mass conservative, can operate at practically any
wind speed, in particular with Courant number exceeding 1, and its generalization
to 2D and 3D versions is trivial. It should be stressed that the algorithm is free
from any filtration, which is a pre-requisite for zero numerical viscosity (not
necessarily sufficient in general).
