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I.K. Marchevsky, V.V. Puzikova
28
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
,
,
,
,
,
,
,
,
,
,
,
,
( )
= 0;
( )
= 0;
u
u
u
u
i j
i j
i j
i j
t
t
xx
xy
u
u
i j
i j
v
v
v
v
i j
i j
i j
i j
t
t
yy
xy
v
v
i j
i j
d udV
udS p dS
u dS
dt
dS
dS
d vdV
vdS p dS
v dS
dt
dS
dS
x
x
y
y
y
x
v n
e n
n
e n
e n
v n
e n
n
e n
e n
(10)
,
,
,
,
( )
= (
)(
)
(Prod Dis Add) .
xy
xy
xy
xy
i j
i j
i j
i j
d dV
dS
dS
dV
dt
v n
n
(11)
In case of fixed immersed boundaries by analogy with the LS-STAG
discretization of Navier — Stokes equations [2] the general form of the LS-STAG
discretization for (10), (11) can be written as the following:
,
,
,
,
= 0;
(
)
(
)
= 0;
(
)
(
)
= 0;
ib
x
y
x
y
x
x
x
x
ib c
ib
x
x
xx
x xy
x x
x
y
y
y
y
ib c
ib
y
y
yy
y xy
y
y
y
D U D U U
d M U C U G P T D T K U S
S
dt
d M U C U G P T D T K U S
S
dt
(12)
, ,
[ ]
[ ,
]
[ ]
[ ,
,
]
= 0.
n
xy
xy
n ib c n
ib
xy n n
xy
n
ib
xy
n
xy
d M C U S
K G
dt
S G
M PDA
(13)
Here
P
is the discrete pressure,
x
U
and
y
U
are the discrete components of the
velocity vector,
xy
T
is the discrete shear Reynolds or subgrid stresses,
xx
T
and
yy
T
are
the discrete normal Reynolds or subgrid stresses,
is the discrete
,
is the
discrete
;
ib
U
denotes the mass flux arising in case of
;
ib
v 0
,
,
ib c
x
S
,
,
ib
x
S
,
,
ib c
y
S
,
,
ib
y
S
,
,
ib c
xy
S
xy
S
are source terms;
,
x
K
y
K
and
xy
K
represent the discretization of the
diffusive terms;
,
x
D
,
y
D
,
x
D
y
D
are the divergence discrete analogues on the
corresponding meshes;
,
x
C
y
C
and
xy
C
represent the discretization of the convective
terms;
=
x
T
x
G D
and
=
y
T
y
G D
are the gradient discrete analogues;
PDA
is the
discrete analog of
(Prod Dis Add);
,
=
[ ,
]
xy
ib g
ib
G G S
is the discrete
analogue of
/
x
and
/
y
which are computed in the middle of
,
i j
fluid
faces.
The time integration of the differential algebraic system (12) is performed with a
semi-implicit projection method based on the Adams — Bashforth/second-order