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Modification of the LS-STAG Immersed Boundary Method for Simulating Turbulent Flows
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
29
backward differentiation formula (AB/BDF 2) scheme. Predictor step leads to discrete
analogues of the Helmholtz equation for velocities prediction
,
x
U
y
U
at the time
1
= ( 1) :
n
t
n t
1
, ,
1 1
, , 1
,
,
1
, ,
1 1
, , 1
,
,
3 4
2
2
2
(
)
= 0;
3 4
2
2
2
(
)
n n
x
x
x
n n
ib c n n n
ib c n
x
x x
x
x x
x
T n n
n n
ib
x
xx
x xy
x x
x
n n
y
y
y
n n
ib c n n n
ib c n
y
y y
y
y y
y
T n n
n n
ib
y
yy
y xy
y y
y
U U U
M
C U S
C U S
t
D P T D T K U S
U U U
M
C U S
C U S
t
D P T D T K U S
= 0.
(14)
Here
t
is the constant time discretization step. Corrector step is the following:
1
1
3
(
) = 0;
2
n
T n
n
x
x
x
x
U U
M
D P P
t
1
1
3
(
) = 0;
2
n
y
y
T n
n
y
y
U U
M
D P P
t
,
1
1
1
= 0.
ib n
n
n
x x
y y
D U D U U
It leads to the following discrete analogue of Poisson equation for pressure
function
1
= 2 (
)/3:
n
n
t P P
, 1
,
ib n
x x
y y
A D U D U U
(15)
1
1
= ( ) ( )
( ) ( ) .
x x
x T y
y
y T
A D M D D M D
Then flow variables at the time point
1
n
t
are computed by the following formulae:
1
1
=
;
n
T
x
x
x x
U U M D
1
1
=
;
n
T
y
y y
U Uy M D
1
3=
.
2
n
n
P
P
t
(16)
After this, new values of Reynolds or subgrid stresses
1
,
n
xx
T
1
,
n
yy
T
1
n
xy
T
are
computed according to (8), (9) by solving the discrete analogues of the equations
from the used turbulence model:
1
, ,
[ ]
[ ,
]
[ ]
[ ,
,
]
= 0.
n
n
n
xy
xy
n ib c n
ib
xy n n
xy
n
ib
xy
n
xy
M
C U S
K G
t
S G
M PDA
(17)
Numerical experiments.
The flow past circular airfoil was simulated using the
developed modification of the LS-STAG method. The time averaged drag coefficient
D
C
and the Strouhal number
St
were computed. The coefficient
D
C
was obtained by
averaging over a large period of time the unsteady load
2
( )
( ) =
.
/ 2
xa
D
F t
C t
V