1. SECTION 1. AERODYNAMICS OF LIFTING SURFACES
THEME 3. FEATURES OF WINGS FLOW
The features of wings flow are connected with interaction of flows on the lower
and upper surfaces. The features of flow depend on Mach numbers M ∞ ( M ∞ < 1 or
M ∞ > 1 ), sweep angles, angles of attack and other parameters.
Let's consider an influence of these parameters onto process of wings flow.
3.1. Subsonic speeds of wings flow M ∞ < 1 .
3.1.1. Unswept high-aspect-ratio wings
The features of unswept high-aspect-ratio wings flow are determined by overflow
from the lower surface to the upper surface at the wing tips. The appearance of wing
spanwise flow is due to the fact that the pressure on the upper surface is less than
pressure on the lower surface.
So, we shall consider a finite-span wing
which is streamlined by a straight-line flow
having a constant velocity. If the wing creates
the lift force, then there is a zone of reduced
pressure above a wing, and under a wing is a
zone of increased pressure (Fig. 3.1). Under
influence of pressure difference there is an air
overflow passing through wing tip edges from
area of the increased pressure into area of
reduced pressure. A flow parallel to wing
Fig. 3.1. Flow lines curvature
span appears. The cross flow along wing span
caused by cross flow
is more intensive at the wing tips and
attenuates to central cross-section. This flow caused a curvature of flow lines: on the
upper surface towards the plane of symmetry, and on the lower - to the wing tips.
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2. Let's note, that the flow of a finite-span wing is not two dimensional parallel flow,
but three-dimensional, especially near its tips. At that wing tips effect the whole surface,
therefore aerodynamic characteristics of a finite-span wing differ from the aerodynamic
characteristics of an airfoil
While calculating the aerodynamic characteristics of high-aspect-ratio wings
( λ > 6 ) it is possible, in a whole, to neglect the mentioned above flow curvature and to
use a hypothesis of flat cross-sections. The hypothesis of flat cross-sections assumes,
that each cross-section of a wing is streamlined by its own two-dimensional parallel
flow.
In particular, the wing lifting force in this case is determined by summation of lift
of wing cross-sections, which is calculated under the Zhukovsky formula:
l
2
Ya = ρ∞V∞ ∫ Г ( z )dz ,
−l
(3.1)
2
Where Г ( z ) is the circulation of speed along the contour covering a wing in cross-
section of a chosen element.
The vortex sheet is formed as a result of interaction of the upper and lower flows
behind the wing. The vortex sheet consists of vortex threads, which occur due to various
direction of speeds on the trailing edge at the approach to the upper and lower surfaces,
however, values of these speeds are equal (postulate by Zhukovsky-Chaplygin) (Fig.
3.2,a).
The vorticity value is decreased with approaching to a plane of symmetry. In an
inviscid flow the vortex sheet behind a wing reaches infinity and at small angles of
attack (in the linear theory) is directed along speed of incoming flow.
In real conditions the vortex sheet is unstable and is turned off in two high-power
vortex cores (Fig. 3.2,b), which for transport airplanes stretch for tens kilometers. When
light airplanes happen to be in a bundle track behind a heavy airplane it can result in its
crash (failure).
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3. Fig. 3.2. Formation of a vortex sheet behind a wing.
The vortex sheet induces behind a wing a velocity field Vi (Fig. 3.3,a), which
deflects an undisturbed flow on an angle ε , which is called as downwash angle
Vi
ε ≈ tg ε = (3.2)
V∞
And the angle of attack of section (cross-section) receives the following true value
α real = α − ε . (3.3)
If it may be assumed , that in an inviscid flow the total aerodynamic force which
effecting wing cross-section is perpendicular to the true flow velocity
Vreal = V∞ + Vi2 , and the lift force is perpendicular to V∞ , then a force component
2
along the direction of incoming flow velocity appears (Fig. 3.3,b).
Fig. 3.3. Occurrence of a downwash behind a wing.
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4. This force is called as force of induced drag
l
2
X i = ρ∞ ∫ Vi ( z ) Г ( z )dz .
−l
(3.4)
2
The occurrence of induced drag is the important point in the course of studying
the features of finite-aspect-ratio wings flow.
Naturally the question of obtaining minimum value of induced drag X i at
specified values of lift force Ya and wing span l emerges. This problem is of
variational type, which solution is got in case of a constancy of induced speed Vi
spanwise. Thus the distribution of speed circulation Г ( z ) should have the elliptical law
(for a wing in a boundless flow).
What should an optimum wing be? The answer to this question is an ambiguous:
• chords of a flat wing should change under the elliptical law (elliptical wing
plan form);
• twist spanwise should vary under the elliptical law for the rectangular wing
plan form;
• the minimum value of induced drag is also reached by twist application which
law depends on the wing plan form for a wing of any form.
3.1.2. Swept high-aspect-ratio wings
There is an effect of slipping on swept wing in addition to the considered above
features: an additional curvature of flow lines caused by spanwise flow from centre to
the wing tips (for sweepback) appears.
Let's consider a slipping wing. We shall write down undisturbed speed of
incoming flow as: V∞ = Vn + Vτ2 , where Vn is the velocity component normal to the
2
leading edge; Vτ is the tangent component (fig. 3.4).
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5. The wing disturbs a flow. Let's designate tangent Vτ∗ and normal Vn velocity
∗
∗
components of a perturbed flow at a wing surface and we shall assume that Vn
anywhere does not reach value of sound velocity.
Tangent component is constant:
Vτ∗ = Vτ = const and Vn in normal cross-
∗
section varies in a flow in the same way as the
flow above airfoil in this cross-section.
∗
Summing Vτ and Vn , we receive total speed
V and we find flow lines.
The effect of slipping on swept wing is
realized in less degree in the middle and tip
area because of Vτ ≠ const (smaller values in
central and tip areas). Finite span and the
existence of a plane of symmetry result in
redistribution of pressure on wing airfoils in
various cross-sections. At the wing tips the
peak of rarefaction on the upper surface is
Fig. 3.4. Flow lines on a slipping wing: increased and displaces forward as a result of
a) - the angle of attack is equal to zero; slipping effect and overflow influence
b) - positive angle of attack (fig. 3.5, b). Simultaneously positive pressure
gradient considerably increases behind peak of rarefaction (along flow). Pressure
difference decreases near a plane of symmetry at the leading edge (fig. 3.5, b). As a
result the peak of rarefaction displaces back. The positive pressure gradient behind peak
of rarefaction is reduced.
On a swept-forward wing (fig. 3.5, c) the peak of rarefaction grows and displaces
forward in a plane of symmetry. The positive pressure gradient behind it increases also
with the peak of rarefaction growth.
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6. Fig. 3.5. Sweep influence on load distribution chordwise:
a) - unswept wing; b) - sweepback wing;
c) - swept-forward wing.
Existence of the spanwise wing flow to the wing tip and the displacement of
rarefaction peak to the leading edge results in probability of flow stalling on the swept-
back wing tips. Aerodynamic fences or leading edges fractures are used to prevent flow
stall at the wing tips (Fig. 3.6).
Fig. 3.6. The design solutions for prevention of flow stall:
a) - aerodynamic fence; b) - aerodynamic “dog-tooth”; c) - “slit ((saw)kerf)”
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7. 3.1.3. Small-aspect-ratio wings
Additional effects are essential influence of cross-sectional flow spanwise onto
longitudinal flow (curvature of flow lines) and occurrence of vortex structures above a
wing descending from lateral and leading edges. The nonlinearity of the characteristics
C ya = f (α ) and m z = f (α ) is the result of effects mentioned above.
The narrowing of flow lines on the upper surface
(
causes occurrence of additional speeds + ΔVupper and )
pressure decreasing; the divergence of flow lines on the
lower surface reduces flow rates ( − ΔVlower ) and
increases pressure (Fig. 3.7). An additional lift appears.
It is caused by influence of cross-sectional flow onto
longitudinal flow. Cross-sectional flow does not create
lift itself, its influence has an effect on friction drag.
The additional contribution to the aerodynamic
Fig. 3.7.
characteristics is introduced by vortex structures above
the wing (Fig. 3.8), in particular, playing a role of washers interfering pressure
compensation between the lower and upper surfaces. At angles of attack increasing
dependencies C ya = f (α ) and m z = f (α ) also become non-linear (Fig. 3.9). In a
general the lift coefficient value is possible to represent as the sum of two items:
C ya = C ya line + ΔC ya , (3.5)
Where C ya line the lift coefficient is determined without the account of effects of small-
aspect-ratio wing; ΔC ya is non-linear additive;
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8. Fig. 3.8. Fig. 3.9.
It is possible to achieve the increasing of non-linear additive caused by formation
of vortex structures above the wing by application of curvilinear edges, saws etc.
3.2. Supersonic velocities of wings flow M ∞ > 1 .
The features of wings flow are determined by basic property of supersonic flows -
existence of influence areas limited by Mach cones (Fig. 3.10).
Fig. 3.10. A Mach cone.
The areas of influence divide wing edges into subsonic and supersonic ones,
with various flow features.
Leading edges.
Subsonic edges. Velocity component (Mach number M ∞ ) perpendicular to the
leading edge is subsonic one ( M ∞n = M ∞ cos χ l .e . < 1 ). Suction force appears at edge
flow, so it is necessary to apply rounded edges (subsonic airfoils) to the greater
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9. realization of suction force and smooth flow or flap sharp edges providing shock-free
flow (Fig. 3.11). Let's define parameter n describing character of edge flow.
The leading edge will be subsonic if the following conditions are satisfied
χ l .e . > 90 °− μ ∞ , or tgχ l .e . > tg(90 °− μ ∞ ) and 2
tgχ l .e . > M ∞ − 1 . We have
2
n = M ∞ − 1 ctgχ l .e . < 1 .
For example, for a triangular wing the aspect ratio is equal to λ = 4ctgχ l .e . , so we
1 2
receive n = λ M∞ − 1 < 1 .
4
Supersonic edges. Velocity component perpendicular to the leading edge is
supersonic one and M ∞ cos χ l .e . > 1 . Edges should be sharp (supersonic airfoil) for
wave drag decreasing (Fig. 3.12).
2
We have for a supersonic leading edge n = M ∞ − 1 ⋅ ctgχ l .e . > 1 .
Fig. 3.11. A subsonic leading edge Fig. 3.12. A supersonic leading edge
The concepts of subsonic, sound and supersonic lateral and trailing edges of
finite-span wing are analogous ones.
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10. Lateral edges.
Subsonic edge. There is an overflow
and pressure compensation between the lower
and upper surfaces (analogously to subsonic
flow).
Supersonic edge. The upper and lower
surfaces are separately streamlined without
Fig. 3.13. A subsonic and supersonic mutual influence.
lateral edge.
Trailing edges.
Subsonic edge. The postulate Chaplygin-Zhukovsky about flow stall from the
trailing edge and relations Vupper = Vlower and ΔC р = C р lower − C р upper = 0 should
be executed that is conditioned by mutual influence of the upper and lower surfaces of
the trailing edge (Fig. 3.14).
Supersonic. The upper and lower surfaces are separately streamlined. The flow
departures from the trailing edge without the requirement of fulfillment of Zhukovsky-
Chaplygin postulate (Fig. 3.15).
Fig. 3.14. A subsonic trailing edge Fig. 3.15. A supersonic trailing edge
It is necessary to note, that the concepts of subsonic and supersonic edges are
connected both with wing plan form, and with Mach number M ∞ . The edge can be
subsonic or supersonic, depending on Mach number M ∞ at specified wing geometry.
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