4.3. Induced drag.
It
is a result of occurrences of a vortex sheet behind a wing, which
keeping demands energy equivalent to work of induced drag force. The
computational formulae for definition of
are described below.
4.3.1. Wings of large aspect ratio
The induced drag coefficient of a large aspect ratio wing is determined
,
(4.9)
Where
- induced drag of a flat wing;
- additive to induced drag caused by geometrical twist.
We have
,
(4.10)
Dependence of induced drag at small angles of attack is approximated by parabola (Fig. 4.13). The induced drag decreases at increasing of wing aspect ratio .
The
coefficient
is determined by the wing plan form and shows to what extent the
distribution of aerodynamic loading differs from the elliptical law,
for which
.
Generally
.
The expression for
is represented as
lately, where
- Osvald number,
.
Value
of
in contrast to
can not be equal to zero in general at
(because for non-flat wing various wing cross-sections may have lift
coefficients not equal to zero
at total
).
General
expression for
looks like:
,
where,
,
,
- twist angle of tip cross-section; the values of coefficients
and
are also undertaken from the diagrams depending on wing plan geometry
(
,
,
),
and twist law.
Fig. 4.14. shows the comparison between induced drag of a flat wing and wing having twist.
Fig. 4.13. Induced drag of a flat wing |
Fig. 4.14. Induced drag of a flat wing and wing with twist |
4.3.2. Wings of small aspect ratio
For a geometrically flat wing it is possible to assume:
.
(4.11)
This expression can be received if it assumes that the sucking force is not realized on the wing leading edge. The formula can be transformed to the following form if it neglects the non-linear additive ( ) we have:
,
(4.12)
In such form the formula is used in practice (it coincides to the formula for a flat wing of large aspect ratio)
(
For a wing of small aspect ratio
,
and
;
comparing with a wing of large aspect ratio
we shall notice that
).
4.4. Wing polar
Polar
of the first type is
called the dependence between coefficients of aerodynamic lift and
drag force
.
As well as induced drag coefficient the polar depends on twist
presence on a wing. Therefore we shall separately consider polar for
a flat wing and polar for a wing having twist, and we shall conduct
the qualitative analysis.
4.4.1. Flat wing.
Summing
induced drag
with profile drag
,
we shall receive expression for polar:
.
(4.13)
Fig.
4.15. Dependence of profile drag on lifting force
.
In this case we shall write down
,
where
- the wing drag coefficient at
(in this case it coincides to
).
It is necessary to note that in general the profile drag
rather changes by
especially at large values of
or
,
it is approximately proportional to
(Fig. 4.15); in some cases this change is taken into account in
parameter
:
,
(4.14)
Where
;
for high-aspect-ratio wings
and
,
).
Fig.
4.16. Flat wing polar
.
Wing profile drag
do not depends on
and polars for wings of various aspect ratio
are look like it is shown on a fig. 4.16. As it was spoken earlier,
the induced drag decreases with increasing of aspect ratio
,
and consequently the drag
for a wing of infinite aspect ratio
(airfoil) will be equal to profile drag
or
.
4.4.2. Non-planar wings (wings with geometrical twist).
The
general form of polar equation for a non-planar wing
,
where profile drag
and part of induced drag caused by wing geometrical twist enter into
parameter
,
i.e.
.
Parameter
is also determined by wing twist
.
The polar pull-off coefficient does not depend on wing twist.
Polar equation for a wing with geometrical twist is a square parabola with displaced peak (Fig. 4.17). It is possible to write down
,
(4.15)
Where
;
.
From comparison of polars for flat and non-planar wings (the Fig. 4.18) it is possible to reveal the advantages of using of a non-planar wing:
1.
At
twisted wing drag is less than
2.
The maximum lift-to-drag ratio of a non-planar wing is higher
.
(The wing is staying non-planar at mechanization deflection but
).
Fig. 4.17. Polar for a wing having twist (for a non-planar wing) |
Fig. 4.18. Polars for a flat and non-planar wings |
In
addition, due to geometrical twist it is possible to provide wing
balancing without increase of induced drag (non-planar wing is
self-balanced). It is easy to see, that the advantages of a
non-planar wing are shown at condition of
.
In this case polar peak displaces upwards.