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tubes should be supported by the following: methodical calculations,
accuracy control, comparison of numerical results with analytical solutions
and published designed and experimental results.
One-dimensional numerical method of the medium parameters
computing in the working section of a shock tube.
Despite the one-
dimensional nature of the task relating to the multiple pass calculation,
reflection (from end faces of the shock tube) and interaction of previously
described system of waves, it steps up the demands for the numerical
method used in addressing it. First of all the design model must have the
improved dispersion and dissipation properties; it must be cost-effective,
algorithmically simple, and monotonous; it must approximate smooth
solutions with the highest order of accuracy.
These requirements can be satisfied by using a numerical solution
method for the quasi-one-dimensional synthermal one-fluid equations of
gas dynamics, which is based on the subincremental method. In this case
it consists of two steps [2]. The systems of equations mentioned above
can be solved using a version of the nonlinear quasimonotonous compact
difference scheme of the higher order of accuracy, developed by the authors.
Let us describe the ways to find the numerical solution of these fractional
steps.
The first fractional step involves gas-dynamic processes (the hyperbolic
part of the equations set in question agrees with these processes) that occur
in the shock tube after the rupture of the diaphragm, which separates the
driver gas from the test (driven) gas; the second fractional step involves
the quasi-one-dimensional geometry of the facility.
Mathematical formulation of the first fractional step and the solution to
the hyperbolic part of the equations set are based on the divergence form
and can be formulated by the following:
∂ ~U
∂t
+
∂F ~U
∂ξ
=
~F
2
, F
ρ
=
−
ρv
d
ln
F
dz
;
F
ρv
=
−
ρv
2
d
ln
F
dz
, F
E
=
−
(
ρEv
+
vP
)
d
ln
F
dz
;
or
∂ ~U
∂t
=
L
(
U
)
, L
=
−
∂F ~U
∂ξ
+
~F
2
;
here the parameter
ξ
can take one of the values from the value set
(
r, z
)
,
the solution vector is
~U
= (
ρ, ρu
ξ
, ρE
)
T
, the flow variable vector can
be written as
F ~U
=
ρu
ξ
, ρu
2
ξ
+
P, ρEu
ξ
+
Pu
ξ
T
, and the right part
vector is presented as
~F
2
= (
F
ρ
, F
ρu
, F
E
)
T
.
ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1 7