6.2.1. Analysis of the aerodynamic characteristics.
1.
At
,
that corresponds to
or
(or simultaneously) we receive
.
That means, the result corresponds to the extreme low-aspect-ratio
wing in incompressible and subsonic gas flows. If the wing leading
edge becomes sound (
), then
,
i.e. we receive the characteristic of the airfoil of infinite aspect
ratio wing in supersonic gas flow.
2.
In case of sound and supersonic edges (
)
the non-linear additive to a lift coefficient is equal to zero, i.e.
.
3.
If
,
then
and at
a polar pull-off coefficient
,
i.e. wing. induced drag
;
so as for subsonic flow about high-aspect-ratio wing. It is visible
that the sucking force
reduces induced drag twice in comparison with that case, if it is not
taking into account.
4.
Ratios
and
are also functions of reduced aspect ratio
and airfoil plan form (for
).
6.3. Triangular wing with supersonic leading edges.
In
case, when leading edges of a triangular wing are supersonic -
.
The overflow is absent and the sucking force is not realized on the
leading edge. It is possible to mark out two characteristic flow
areas (Fig. 6.9). The wing areas
I
(shaded
sites) outside of Mach cone are streamlined as the isolated slipping
wing of infinite span, irrespective of other wing part. The pressure
in these areas is constant and pressure factor is determined:
,
(6.15)
where
- pressure factor on the airfoil;
or
- the leading edge characteristic.
Fig.
6.9. Triangular wing with supersonic edges
,
(6.16)
where
,
.
The integration of pressure distribution results in the following formulas for the aerodynamic characteristics:
,
,
,
, (6.17)
,
(6.18)
,
(6.19)
. (6.20)
It
is noteworthy, that the value of
for triangular wing with supersonic leading edges coincides to the
airfoil characteristic
(difference is in pressure distribution). The pressure rising at tip
sites compensates pressure decreasing in central area of the
triangular wing. It can be shown, that the share of tip sites in
total lift comes to
,
that at
corresponds to
and at
-
.
Just
as for a wing with subsonic edges, the ratios
and
also are functions of
and airfoil shape for wave drag (Fig. 6.10, 6.11). It is necessary to
note, that the formulas for a triangular wing with subsonic and
supersonic edges are theoretically joint to a fracture at
or
.
Experimentally
this fracture is smoothed out. In point
the leading edge passes from a subsonic flow mode to supersonic flow.
The application of wings with subsonic edges is evident on a curve of
wave drag (in this case induced drag decreases too due to realization
of sucking force). The most adverse flow mode - in zone of
numbers corresponded to a sound leading edge.
Fig. 6.10. Dependence of on reduced aspect ratio |
Fig. 6.11. Dependence of on reduced aspect ratio |
Fig.
6.12.
.
In this case lift coefficient
and induced drag
will be the same, as on the initial triangular wing. It is a
particular case of
the general theorem of reversibility.
According to this theorem, the lift of a flat wing of any plan form
at the direct and inverted flow will be identical, if the angles of
attack and speeds of undisturbed flow will be identical. For induced
drag the equality will be obeyed at supersonic leading edges (in
direct and inverted flows) or at identical values of sucking forces.
Considering
the load distribution along wing surface it is possible to make the
conclusion that the cut-out of trailing edge (form such as “swallow's
tail”) (Fig. 6.13,1) should result to increasing of
,
and additive of the area to a trailing edge
(Fig. 6.13,3) - to
decreasing of
.
It is possible to write down
.
Fig. 6.13. Various versions of trailing edge shape
At
the wings aerodynamic characteristic are determined by the
characteristics of the initial triangular wing by multiplication to a
factor dependent on the ratio of sweep angles on forward and trailing
edges
,
where
.
It is necessary to take the sweep angle on the trailing edge with its
own sign. So derivative of a lift coefficient
,
wave drag and location of aerodynamic center are defined by the
formulae
;
;
.
(6.21)