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The gradients for linear reconstruction can be calculated either from

Green-Gauss theorem, either with the least squares method. Green-Gauss

theorem [13] can be used to obtain the precise value of the linear function

gradient only for tetraedrical cells, and thus is not applicable for non

structured nets with different shape cells. Consequently in this research the

weighted least squares method is used by default for reconstruction.

It is well-known that second- or higher order reconstruction require

limiters to suppress false oscillations of the solution in the large gradients

area. The software in question employs Barth and Jespersen [13], Venkata-

Krishnan’s [15], Michalak and Ollivier-Gooch [16] limiters.

Speed and temperature gradients at the cell faces for viscous flows

are calculated as the mean value of the gradients in the cells centres with

Green–Gauss theorem or with the least squares method described above:

q

ij

~n

=

1

2

(

q

i

+

q

j

)

~n.

However in [17] it was demonstrated that this approach can result in

the discoordination of the solution for quadrangular or hexagonal meshes.

The following modified formula[18] is applied to reduce the method

discoordination error:

q

ij

~n

=

q

j

q

i

k

~r

j

~r

i

k

α

ij

+

1

2

(

q

i

+

q

j

)

(

~n

α

ij

~s

)

,

where

~n

is the normal to the cell face;

~s

is the normalised vector connecting

cells centres;

k

~r

j

~r

i

k

is the distance between

i

and

j

cells centres;

α

ij

is

the scalar product

α

ij

=

~s

~n

. We should remind that

q

i

gradient in

i

cell is calculated either with Green-Gauss theorem, either with the smallest

squares approach.

For the time discretization explicit Runge–Kutta methods of the second-

or third-order accuracy [19] can be used. The time step can be calculated

with regard to non-viscous and viscous limits to the step size.

Numerical modelling results for the gas dynamic duct flow. Compa-

rison with the experiment.

Numerical investigation was performed

according to a previously designed approach for two aerodynamic models

which were experimantally investigated in a two-membrane aerodynamic

shock tube at Institute for Problems in Mechanics of RAS. During the

numerical simulation the flow in the test chamber was viewed separately

from the shockwave motion in the shock tube starting at a short distance

from the nozzle. It is assumed that the receiver walls do not affect the flow

near the model. In this research it is also assumed that the flow after the

nozzle section is uniform in the lateral direction (the uniformity issues of

ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2015. No. 1 13