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This results in the numerical method of solving the quasi-one-dimensi-
onal equations of gas-dynamics (physically specified well), underlaid by
the predictor-corrector method and a variant of the non-linear quasi-
monotonous compact difference scheme of the higher order of accuracy.
In this case, for the “predictor” stage, the system of Euler’s quasi-one-
dimensional equations is used in a general non-divergent characteristic
form, in which the unknown values are written relative to the Riemann
quasiinvariants. At the “corrector” stage, the divergent form of Euler’s
quasi-one-dimensional equations is used.
Time increment
Δ
t
, necessary for integrating the difference scheme
given above, is selected from the conditions of satisfying the Courant-
Friedrichs-Lewy stability criterion.
To approximate the second derivatives that are included into the
“viscous” part of the equations set of the dynamic and thermal boundary
layer, we used the finite-difference presentation of
(
μf
)
0
z
the variables on
the mesh introduced earlier by the following formulae [7]:
(
μf
)
0
z
≈
h
μ
i
−
1
f
0
i
−
1
+ 8
μ
i
+1
/
2
f
0
i
+1
/
2
−
μ
i
−
1
/
2
f
0
i
−
1
/
2
−
μ
i
+1
f
0
i
+1
i
6Δ
;
μ
i
±
1
/
2
=
[
μ
i
±
2
+ 9 (
μ
i
+
μ
i
±
1
)
−
μ
i
∓
1
]
16
;
f
0
i
±
1
/
2
≈
[27 (
∓
f
i
±
f
i
±
1
)
±
(
f
i
∓
1
∓
f
i
±
2
)]
12Δ
;
f
0
i
±
1
/
2
≈
[
∓
f
i
∓
2
±
6
f
i
∓
1
∓
18
f
i
±
10
f
i
±
1
±
3
f
i
±
2
]
12Δ
.
Mixed derivatives
∂
∂x
μ
∂f
∂y
i,j
have been approximated by the known
central differencies of the fourth order (or with the formulas given):
(
μf
)
0
z
≈
[
f
i
−
2
+ 8 (
f
i
+1
−
f
i
−
1
)
−
f
i
+2
]
12Δ
. The approximation effectiveness
of using “viscous” members of the boundary layer of high order accuracy
operators is particularly noticeable in the tasks relating to the computation
of the aerodynamic shock tubes. Let us use the displacement operator
T
m
u
j
=
u
j
+
m
=
u
(
z
j
+
m
Δ)
and the symbol of unit operator
E
=
T
0
to
determine the derivatives
∂v
∂r
r
=
R
,
∂ζ
∂r
r
=
R
−
δ
at the boundary of the design
area:
u
0
|
r
=
R
=
1
60Δ
[
−
12
T
−
5
+ 75
T
−
4
−
−
200
T
−
3
+ 300
T
−
2
−
300
T
−
1
+ 137
E
] +
Δ
5
6
u
VI
;
ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1 15