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field area occupied by the medium. The individual points have neither size
nor mass. All parameters of the medium are defined at the points, such as
radial and axial components of the velocity vector, density, and components
of the stress tensor. Radial and axial coordinates characterize each point.
Each Euler mesh cell is individualized by a pair of integer numbers
(
i, j
), here
i
is a cell number in the radial direction,
j
is a cell number in
the axial direction (see Fig. 1,
a
).
The same pair of numbers is used as an index for all parameters of the
medium in the individual (Lagrangian) point which is located inside this
Euler cell (
i, j
) at this moment. For calculation of the parameters evolution
in a Lagrangian point, which is located inside the Euler mesh cell (
i, j
),
the parameters of Lagrangian points from four adjacent Euler cells are
used. They are a lower cell (
i
−
1
, j
), an upper cell (
i
+ 1
, j
), a right
cell (
i, j
+ 1
), and a left cell (
i, j
−
1)
(Fig. 1,
b
). If some of these Euler
cells are empty (not occupied by the medium), they are considered to be
occupied by dummy points having the same velocity vector components as
the individual point at zero pressure. Introduction of the dummy Lagrangian
points allows calculating the evolution of all individual points’ parameters
similarly regardless of the number of their adjacent points.
For instance, a new density value
ρ
∗
(
i, j
)
(at the moment
t
+Δ
t
) at the
Lagrangian point located at the moment t inside the Euler mesh cell (
i, j
)
is calculated by a differential analog of the continuity equation
ρ
∗
(
i,j
)
=
ρ
(
i,j
)
1
−
Δ
t
v
r
(
i
+1
,j
)
−
v
r
(
i
−
1
,j
)
r
(
i
+1
,j
)
−
r
(
i
−
1
,j
)
+
+
v
r
(
i
+1
,j
)
+
v
r
(
i
−
1
,j
)
r
(
i
+1
,j
)
+
r
(
i
−
1
,j
)
+
v
z
(
i,j
+1)
−
v
z
(
i,j
−
1)
z
(
i,j
+1)
−
z
(
i,j
−
1)
!
.
Similarly, all the other medium motions and state parameters at
another time can be calculated by differential analogs of the corresponding
equations.
Then, new radial and axial coordinates of the individual points are
calculated
r
∗
(
i,j
)
=
r
(
i,j
)
+ Δ
t v
r
(
i,j
)
;
z
∗
(
i,j
)
=
z
(
i,j
)
+ Δ
t v
z
(
i,j
)
.
These coordinates are used for redistribution of the individual points among
the Euler mesh cells. For each individual point new indices,
i
and
j
, of
the Euler cell are defined while the point is located at new time. Besides,
if some Euler mesh cells contain several individual points, all these points
are replaced by one point which has parameters averaged by parameters of
ISSN 0236-3941. HERALD of the BMSTU Series “Mechanical Engineering”. 2015. No. 1 69