section 1. AERODYNAMICS OF LIFTING SURFACES
Topic 4. The aerodynamic characteristics of Wings in a flow of incompressible fluid
The
main aerodynamic characteristics of an aircraft moving with Mach
numbers
are considered in this lecture.
4.1. Wing lift coefficient
4.1.1. High-aspect-ratio wings
The
dependence
is linear on the segment of attached flow (Fig. 4.1).
In general
.
(4.1)
The
angle of zero lift
is determined by the airfoil shape and wing twist. For a flat wing
,
degree. (4.2)
Fig.
4.1. Dependence of the lift coefficient on the angle of attack
causes changing of
by the value
, which can be approximately estimated for the unswept wing by the
formula
.
If
the wing is swept then the absolute value of
should be reduced by size
.
The
derivative
depends on aspect ratio
and sweep angle
,
influence of taper
on derivative size is weak. The derivative
does not depend on wing twist. It is possible to offer the following
approximate formula for calculation of value of
:
,
(4.3)
Here
the parameter
takes into account the wing plan form and depends on
,
,
.
It is possible to assume in the first approximation
(in general
),
.
(4.4)
Parameter
is a derivative for the airfoil (wing with
)
and is calculated by the formula
.
(4.5)
It
follows from the formula, that at
and if in addition
,
then
.
It is also possible to use the following formula for calculation :
.
(4.6)
Here
- ratio of half-perimeter of the wing outline in the plan to span
(Fig. 4.2). The significance of the last formula - its universality
and capability to apply to any plan forms and aspect ratios (it is
especially useful for wings with curvilinear edges or edges with a
fractures).
Fig.
4.2.
,
or in case of tapered wing - by the formula
.
Fig.
4.3.
Fig.
4.4.
.
1.
With increasing of wing aspect ratio
the derivative of a lift coefficient on an angle of attack
grows (at conditions of
and
the value of a derivative tend to the airfoil characteristic
(Fig. 4.3).
2.
With increasing of sweep angle at half-line chord (
chord line
) the derivative value
decreases (Fig. 4.4). (It occurs due to effect of slipping, at
condition of
,
the sweep angles on the leading and trailing edges are identical
,
).
3.
The sweep influence on derivative
value decreases with decreasing of aspect ratio
(Fig. 4.5) (sweep practically does not influence on value of lift
coefficient derivative on an angle of attack
at small values of aspect ratio
).
Fig.
4.5.
influences a little bit onto the value of a derivative
(refer to formula (4.4),
parameter
).
4.1.2. Wings of small aspect ratio
It
is necessary to take into account the non-linear effects which occur
at flow about wings of small aspect ratio in dependence of a lift
coefficient on an angle of attack
(Fig. 4.6)
Fig.
4.6.
where
.
It
is also possible to define values of
and
by the formulae for large aspect ratio wings at
The value of derivative can be determined by the formula
for a wing of extremely small aspect ratio
,
and the angle of zero lift for the wing with unswept trailing edge is
equal to an angle of the trailing edge deflection
,
where
- equation of a surface of a wing,
,
- trailing edge coordinates.
The non-linear additive can be calculated by the formula which is fair at any Mach numbers (the linear theory of subsonic flow concerns only to a linear part of the dependence).
(4.7)
At
a supersonic leading edge (
)
the non-linear additive disappears and
.
With
decreasing of
the derivative
decreases, and the non-linear additive
grows
(Fig. 4.7, 4.8).
Fig. 4.7. Character of changing of the non-linear additive |
Fig. 4.8. Character of changing of the non-linear additive |
4.2. Maximum lift coefficient.
The
maximum lift coefficient is connected to occurrence and development
of flow stalling from the upper wing surface near the trailing edge
and depends on many factors, first of all, from the characteristics
of an airfoil (
,
nose section shape), wing plan form (
,
),
Reynolds number
.
The
value practically does not influence onto
for wings of large aspect ratio (Fig.
4.9),
at that with
decreasing
increases.
The
influence of
has
an effect as follows for wings of small aspect ratio:
grows
with
growing
from 0 up to
;
and then decreases - with increasing of
.
Small values of
(
)and
large values of
(
)
(fig. 4.10) are characteristic for wings with
.
A flow stalling delay in the latter case caused by influence of
vortex structures formed on the upper wing surface.
Fig.
4.9. Dependence
|
Fig.
4.10. Dependence
for wings of small aspect ratio
|
for wings of large aspect ratio
.
The
properties behaviour of curve
in area
depends on the nose section shape. The presence of
low values is characteristic for a wing with the pointed airfoil nose
section which do not depend on Reynolds numbers (Fig. 4.11). The
increasing of
(up to certain values) is characteristic for a wing with a rounded
airfoil nose section nose at increase of Reynolds numbers (Fig.
4.12).
Approximately,
value
of large aspect ratio wing (
)
can be determined by the formula
,
where
- the maximum value of an airfoil lift coefficient;
for
symmetrical airfoil
at Reynolds numbers
and
at
.
Fig. 4.11. Dependence for wings with a sharp leading edge |
Fig. 4.12. Dependence for wings with a classical airfoil |
Number - kinematic factor of similarity describing the ratio of inertial and viscous forces:
,
(4.8)
Where
- characteristic speed;
- characteristic length;
- kinematic factor of viscosity.