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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