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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
27
1,
,
,
1,
1
1,
,
,
1,
1,
,
1,
1
1,
,
,
,
,
,
,
1,
,
,
,
if
0,
0;
(
) / 2
(
, )
=
, if
= 0,
0;
/ 2
( , )
=
,
if
0,
= 0.
/ 2
i
j
i j
v
v
i j
i
j
v
v
i
i
i
j
i j
ib w
ib
i
j
j
i
j
v
v
i j
i
j
v
i
i
j
i j
i j
ib e
ib
j
i j
i j
v
v
i j
i
j
v
i
i j
i j
v
v
x
x
v
v x y
v
v
x
x
x
v x y v
v
x
x
Components of hydrodynamic force acting on the immersed boundary can be
computed as the following:
,
1,
,
,
,
cut-cells ,
=
(
)
Quad
;
u
u
ib
xa
j
i j
i
j
i j
i j
ib
i j
i j
u
u
F
y p
x
y
y
e n
,
,
, 1
,
сut-cells
,
,
=
Quad
(
)
.
ib
v
v
ya
i
i j
i j
i j
i j
ib
i j
i j
v
v
F
x p
x
y
x
e n
Here
xa
F
is the drag force,
ya
F
is the lift force,
1
=
,
i
i
i
x x x
1
=
,
j
j
j
y y y
,
Quad
ib
i j
is the quadrature of the shear stresses.
,
Quad
ib
i j
has to be adapted to each type
of cut-cells. This quadrature is based on the location of point where the shear stresses
are sampled in Fig. 4 and the trapezoidal rule.
It is conveniently to sample the linear turbulence scale
turb
l
and the characteristic
filter size
for LES and DES at the same points as the
t
and
.
k
We recall that the
maximum mesh step at the given point of the computational domain is used as a filter
size
for DES approach. Since we deal with
xy
-mesh, the characteristic filter size is
defined as a following:
max
,
,
1,
,
1,
, 1
,
, 1
= = max{
,
,
,
,
,
},
xy
xy
xy
xy
xy
xy
i j
i j
i
j
i j
i
j
i j
i j
i j
y
y y
x
x x
where
1
,
,
, 1
1=
;
2
xy
u
u
j
j
i j
i j
i j
y
y
y
1
,
,
1,
1=
.
2
xy
v
v
i
i
i j
i j
i
j
x
x
x
Within LES approach the following filter can also be used on the LS-STAG mesh:
,
,
,
= = .
xy
vol
i j
i j
i j
M
According to the concept of the LS-STAG method, equations (1), (7) should be
written in integral form for cell of base mesh, cell of
x
-mesh, cell of
y
-mesh and cell of
xy
-mesh, respectively:
,
= 0;
i j
dS
v n
(10)