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for compressible media which are laws of conservation of mass, momentum
and energy in viscous gas:
∂
t
U
+
∇ ∙
~F
c
− ∇ ∙
~F
v
=
S
under relevant initial and boundary conditions. In these equations vector
U
= (
ρ, ρv
1
, ρv
2
, ρv
3
, E
)
T
is a conservative vector, where
ρ
is density,
E
is total energy and
~v
= (
v
1
, v
2
, v
3
)
∈
R
3
is velocity vector in the
cartesian coordinate system;
~F
c
(
U
)
are convective currents,
~F
v
(
U
)
are
viscous currents and
S
(
U
)
is the generic source component:
~F
c
i
=
ρv
i
ρv
i
v
1
+
pδ
i
1
ρv
i
v
2
+
pδ
i
2
ρv
i
v
3
+
pδ
i
3
v
i
(
E
+
p
)
, ~F
v
i
=
0
τ
i
1
τ
i
2
τ
i
3
v
j
τ
ij
+
k∂
i
T
, i
= 1
,
2
,
3
,
where
ρ
is static pressure,
T
is temperature,
δ
ij
is Kronecker symbol and
the viscous stress tensor is recorded as
τ
ij
=
μ
(
∂
j
v
i
+
∂
i
v
j
−
2
/
3
δ
ij
∇ ∙
~v
)
.
It should be noted that
i
,
j
indexes denote 3D cartesian coordinates, while
repeated indexes denote summation. Calorically perfect gas is used as the
operating medium, and the heat capacities ratio
γ
is assumed to equal 1.4.
The molecular viscosity
μ
is calculated from Sutherland’s formula, for the
thermal conductivity
k
the assumption of Prandtl number equalling 0.72 is
used.
The numerical method used is a variation of finite volume approach and
can be regarded as Godunov’s method variation. If the parameters within
cells (finite volumes) are assumed to be distributed constantly, the method
has only first-order accuracy in space. To achieve second-order accuracy
the piecewise linear reconstruction is used [13]. For example, variables
vectors leftside and right side the cell dividing the adjacent cells
i
and
j
can be defined as follows:
q
L
=
q
i
+
∇
q
i
∙
~r
L
,
q
R
=
q
j
+
∇
q
j
∙
~r
R
,
where
q
is a scalar variable;
∇
q
is this variable gradient;
~r
is the vector
passing from the cell centre into the face centre.
Non-viscous flows can be calculated with different variants of the exact
or approximate solution of Riemann problem. In the software used most of
the popular solvers are realized. This research used AUSM (advective
upstream splitting method) [14]. This approach to non-viscous flow
calculation is quite economical and suitable for viscous flows calculation.
12 ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2015. No. 1