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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

+

i

1

ρv

i

v

2

+

i

2

ρv

i

v

3

+

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