SECTION 1. AERODYNAMICS OF LIFTING SURFACES

THEME 6. THE AERODYNAMIC CHARACTERISTICS OF

WINGS IN A SUPERSONIC GAS FLOW.

WING IN TRANSONIC RANGE OF SPEEDS

The wing aerodynamic characteristics in the supersonic gas flow

(1 20 1 25 4 5, ... , ...≤ ≤∞M ) depend on edges type: subsonic or supersonic. In the

beginning we shall consider the aerodynamic characteristics of wings of the individual

plan forms and on their example we shall reveal some common properties characteristic

for wings of derived plan forms. Then we shall consider features of the aerodynamic

characteristics of wings with various plan forms.

6.1. Rectangular wings.

The leading edge 1 and trailing

edge

2

4

−

3 − are supersonic, lateral edges

1 4− and 2 3− are subsonic on a wing (Fig.

6.1). The influence of lateral edges (pressure

equalization between the lower and upper

surface) has an effect only in areas 1 5

Fig. 6.1. Pressure distribution

On a surface of a wing

4− −

and 2 6 3− − , limited by Mach cones, in

contrast to the subsonic flow where the

pressure equalization (influence of lateral

edges) occurs all spanwise. Outside the area

of influence (surface of a wing 1 2 6 5− − − )

the flow is similar to the flow about flat plate

in two-dimensional flow. The pressure factor

in any point of wing surface outside the areas

59

of influence of lateral edges is equal

C C tgM

up loр . . р= = ± = ±−

∞∞

2 2

10 2

α μα . (6.1)

In the field of influence of lateral edges the factor of pressure is determined

C Cup loр . . р arcsin= ∞2π

tgtg ∞

υμ

. (6.2)

60

(Here the wing is considered as a flat plate). The last formula is

recorded for case when there is no mutual influence of the

right-hand and left-hand lateral edges (Fig. 6.2). This influence

appears at such Mach numbers, when the Mach cone of the left-

hand (or the right-hand) lateral edge crosses the right-hand

(left-hand) lateral edge.

The limit case of mutual influence absence is determined

by a condition λ M∞ − <2 1 1 . Fig. 6.3. shows probable cases of lateral edges mutual

influence.

Fig. 6.2.

λ M∞ − <2 1 1 λ M∞ − =2 1 1 λ M∞ − <2 1 2 λ M∞ − =2 1 2 λ M∞ − >2 1 2

Fig. 6.3.

The factor of pressure in the field of mutual influence of lateral edges is

determined by more complex formulae; they are shown in special literature.

Summarizing distributed pressure we shall define the aerodynamic characteristics.

The following formulas are received for the case when λ M∞ − ≥2 1 1 :

C Cya ya= ⋅α α ; (6.3)

CM M

yaα

λ=

−−

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∞ ∞

4

11 1

2 12 2; (6.4)

C C Cxa xw xi= + ; (6.5)

C Bc

M Mxw =

−−

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∞ ∞

2

2 211 1

2 1λ; (6.6)

; (6.7) C C ACxi ya ya= ⋅ =α 2

xM

MF =

− −

− −

∞

∞

0 51 2 3

1 1 2

2

2.λ

λ. (6.8)

At λ M∞ − <2 1 1 equations for the aerodynamic characteristics for rectangular

wings become considerably complicated.

Let's analyze the aerodynamic characteristics.

6.1.1. Lift.

1. The dependence ( )C fya = α is linear for any aspect ratio wing (λ ≥ 3 ).

2. With increasing of λ the value of a derivative Cyaα grows and at λ → ∞ tends

to the airfoil characteristic C CM

ya yaα α→ =

−∞

∞

4

12 (Fig. 6.4). This tendency takes

place faster than in subsonic flow, that is explained by the limited area of the lateral

edges influence at . M∞ > 1

3. The value of derivative Cyaα decreases and tends to Cya ∞

α with increasing of

Mach numbers , that is connected with narrowing of Mach cones at growth of

and reduction of lateral edges influence (Fig. 6.5). It is possible to assume, that

M∞ M∞

C Cya yaα α≈ ∞ already at λ M∞ − ≥2 1 7 8... (difference is less than 7% at

λ M∞ − =2 1 7 , at λ M∞ − =2 1 10 - less than ). 5%

61

Fig. 6.4. Dependence C fya = ( )α

at M cons∞ = t

Fig. 6.5. Dependence C fya = ( )α

at λ = const

4. The ratio Cyaα

λ is the universal dependence on the reduced aspect ratio

λ M∞ −2 1 .

6.1.2. Drag.

There is only pressure drag which determines wave drag and induced drag

in the inviscid supersonic flow. As the wing leading edge is

supersonic, then there is no sucking force and C C

C C Cxa xw xi= +

xi ya= ⋅α (for flat wing). As

C Cya ya= ⋅α α and α α=1C

Cya

ya , then C C ; Axi ya= ⋅2 ACya

=1α . So influence of λ onto

at is weaker than in subsonic flow (for example at Cxi M∞ > 1 λ ≥ 4

Cxi ya=+1 2Cδπλ

). The wave drag C is defined by airfoil drag xw C Bc

Mxw∞

∞

=−

2

2 1 and

multiplier 1 1

2 12−

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∞λ M which is taking into account span finite. Let's notice, that

62

in case of unswept wing, the same multiplier is included in the formula for Cyaα (6.4).

As well as C , the value of parameter C with increasing of and ya Cxw xw→ ∞ M∞ λ

(more precisely λ M∞ −2 1 ). It is explained by narrowing of Mach cone and reduction

of lateral edges influence. It is possible to consider that at λ M∞ − ≥2 1 7 8...

(error ). The ratio C Cxw xw≈ ∞ ≈ 7% C

cxw

λ 2 is dependent only on reduced aspect ratio

and factor of the airfoil plan form B i.e. ( )C

cf Mxw

λλ2

2 1= −∞ , airfoil planform .

6.1.3. Location of aerodynamic center.

"Loss" of lift in the wing areas falling inside

the Mach cones will cause displacement of pressure

center and aerodynamic center forward, to the leading

edge, in comparison with location of the airfoil

aerodynamic center xF∞ = 0 5. (Fig. 6.6). The

location of aerodynamic center is a function of aspect

ratio x f MF = −⎛⎝⎜

⎞⎠⎟∞λ 2 1 .

Fig. 6.6

At λ M∞ − → ∞2 1 x xF F→ ∞ . Approximately x xF F≈ ∞ at λ M∞ − ≥2 1 7

with an error less than . At 3% λ M∞ − =2 1 5 xF = ⋅ =0 96 0 5 0 48. . . difference from

xF∞ is . 4%

63

6.2. Triangular wings with subsonic leading edges.

64

90o − < ∞χ μl e. .

Let's consider a triangular wing with subsonic

leading edges. With the help of fig. 6.7. we get an

internal sweep angle and

nctgtg

M ctg Ml el e= = − = − <

Fig. 6.7. A triangular wing

with subsonic edges

Fig. 6.8. Pressure distribution

in wing cross-section

χ

∞∞ ∞μ

χ λ. .. .

2 21 14

1 1 .

Thus triangular wing with subsonic edges has

reduced aspect ratio less than (4 λ M∞ − <2 1 4 ). In

this case there is an overflow from the lower surface to

the upper surface. The sucking force is realized on the

leading edge and reduces induced drag. There is also

non-linear additive to the lift coefficient ΔCya .

Pressure distribution along wing surface (fig. 6.8)

submits to the law within the linear theory (α << 1 )

Ctg

E tup l

l eр . .

. .= ±−

2

1 2

α γ, .9)

where t tgtg l e

=ϑγ . .

and (refer to fig.

6.7); it is possible to define approximately

γ l e l e. . . .= −90o χ

E - the

elliptical integral of II type dependent on

parametern M= ∞14

12λ −

n

, by the formula

. ( )E n≈ + −1 0 0 6 2, ,

It follows from the formula (6.9) for that on each ray Cp ϑ = const value of

, that is the feature of conical flows. At C consp = t ϑ γ→ l e. . i.e. at approach to the

leading edge C (the similar situation takes place in a subsonic flow). For a

sharp edge the site in nose section is equal to zero. Multiplying C onto the nose

p→ ±∞

p

section area and expanding the uncertainty ∞ ⋅0 , we receive final force value which

projection onto incoming flow direction creates force F reducing the induced drag.

This is the sucking force. The aerodynamic characteristics of a triangular wing with

subsonic edges are determined by the following formulae:

C C Cya ya ya= ⋅ +α α Δ , C C Cxa xw xi= ; (6.10) +

CM

nE Eya

α π π λ=

−⋅

⋅=

∞

4

1 2 22; (6.11)

ΔCC

Myaya

l e= − ∞4

1 2 2α

π λχ αcos . . ⋅ ; (6.12)

( )C Bc

Mn nxв =

−+

∞

2

2 11 1 0 2. . ; (6.13)

; C C C ACxi ya F ya= ⋅ − ≈α 2 AC

n

ya= −

−1 1 2

α ξπ λ

, (6.14)

where ξ is a factor of realization of sucking force (ξtheor = 1), it is possible to adopt for

sharp leading edges, that ξ ≈ 0 ; for rounded edges - ξ ≈ −0 8 1 2. . C ya at

, further 0 0≤ ≤Cya .66 ξ ≈ 0 .

For a triangular wing x xb

F F= =0

23

. The last result follows from consideration of

conical flow with constant pressure C consp t= on rays outgoing from the wing top.

6.2.1. Analysis of the aerodynamic characteristics.

1. At , that corresponds to n→ 0 λ → 0 or (or simultaneously) we

receive

M∞ → 1

Cyaα πλ

=2

. That means, the result corresponds to the extremely low-aspect-ratio

wing in incompressible and subsonic gas flows. If the wing leading edge becomes sound

65

( ), then n = 1 CM

yaα =

−∞

4

12, 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 (M l e∞ ≥cos . .χ 1 ) the non-linear

additive to a lift coefficient is equal to zero, i.e. ΔCya = 0 .

3. If , then n→ 0 Cyaα πλ→

2 and at ξtheor = 1 a polar pull-off coefficient

A = 1πλ

, i.e. wing. induced drag Cxi ya=1 2Cπλ

; so as for subsonic flow high-aspect-

ratio wing. It is clear that the sucking force C CF ya2 reduces the induced drag twice in

comparison with that case, if it is not taken into account.

4. Ratios Cyaα

λ and C

cxw

λ 2 are also functions of reduced aspect ratio λ M∞ −2 1

and airfoil plan form (for C

cxw

λ 2 ).

6.3. Triangular wing with supersonic leading edges.

If the leading edges of a triangular wing are supersonic - n M= −∞14

1 12λ > .

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 the 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:

CC

pIp=−

∞

1 2σ, (6.15)

66

where CM

p∞∞

= ±−

2

12α is the pressure factor on the airfoil; σ

λ= =

−∞

1 4

12n M or

σ μγ

=tgtg l e

0

. . is the leading edge characteristic.

There is an influence of the angular point (wing

top) in the wing central part falling into Mach cone.

Conical flow takes place in this area II, for which the

constancy of pressure on each ray outgoing from wing

top is characteristic, but pressure on different rays is

various. The pressure factor for such case is determined

as

67

CC

pIIp=−

−⎛

⎝⎜⎜

∞

11 2

2σ πarcsin t

t−−

⎞

⎠⎟⎟1

2 2

2σ , (6.16)

where t tg tg l e= ϑ γ . . , 0 ≤ ≤t σ .

The integration of pressure distribution results in

the following formulas for the aerodynamic characteristics:

Fig. 6.9. Triangular wing

with supersonic edges

C Cya ya= ⋅α α , C C Cxa xw xi= + , C C , ACxi ya ya= ⋅ =α 2

CM

yaα =

−∞

4

12, (6.17)

C Bc

Mn

nxw =

−+⎛

⎝⎜⎞⎠⎟

∞

2

2 211 0 3. , (6.18)

AC

M

ya= =

−∞1 14

2

α , (6.19)

x xb

F F= =0

23

. (6.20)

It is noteworthy, that the value of Cyaα for triangular wing with supersonic leading

edges coincides with the airfoil characteristic Cya ∞α (the 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 nn−+11

, that at n = 1 5. corresponds to ≈ 45% and at n = 2 - . ≈ 57%

Just as for a wing with subsonic edges, the ratios Cyaα

λand C

cxw

λ 2 are also functions

of λ M∞ −2 1 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 n = 1 or λ M∞ − =⎛⎝⎜

⎞⎠⎟

2 1 4 .

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 the induced drag decreases too

due to realization of sucking force). The most adverse flow mode is in zone of

numbers corresponding to a sound leading edge.

A

M∞

Fig. 6.10. Dependence of Cyaα

λ on reduced Fig. 6.11. Dependence of C

cxw

λ 2 on reduced

68

aspect ratio λ M∞ −2 1 aspect ratio λ M∞ −2 1

It is interesting to note, that if triangular wing is put into flow by the reverse side

(Fig. 6.12), then pressure distribution along the inverted wing will be the same as for a

wing of infinite aspect ratio, i.e. C CM

p p= = ±−

∞∞

2

12α . In this case lift coefficient

Cyaα and induced drag C 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 are 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.

xi

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 the increasing of Cyaα , and

additive of the area to a trailing edge

(Fig. 6.13,3) - to decreasing of Cyaα . It is possible to write down C C Cya ya ya1 2

Fig. 6.12.

3α α α> > .

69

Fig. 6.13. Various versions of trailing edge shape

At χ t e. . ≤ 20o the wings aerodynamic characteristics 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 ( )1

1 + ε, where ε χ

χ= −

ctgctg

l e

t e

. .

. ..

It is necessary to take the sweep angle on the trailing edge with its own sign. So

derivative of a lift coefficient Cyaα , wave drag and location of aerodynamic center are

defined by the formulae

CC

yayαα

ε=

+Δ

1; C

Cxw

xw=+

Δ

1 ε; x x

FF=+Δ

1 ε. (6.21)

6.4. Wings of any plan form.

The qualitative analysis of the aerodynamic characteristics.

The main feature of the aerodynamic characteristics of all wings: with increasing

of Mach numbers (more precisely - reduced aspect ratio M∞ λ M∞ −2 1 ) the

aerodynamic characteristics Cyaα , tend to the airfoil characteristics, i.e. Cxw

C CM

ya yaα α= =

−∞

∞

4

12; C C Bc

Mxw xw= =

−∞

∞

2

2 1. It can be explained, by the

fact that the Mach cone is narrowing with increasing of (at M∞ λ = const ) and

each cross-section of a wing will be also isolated streamlined. It also follows, that

the wing aerodynamic center (or center of pressure) displaces into the center of

mass of the plan form, i.e. x xF c g= . ..

Let's analyze the aerodynamic characteristics of wings.

70

6.4.1. Lift.

Generally for flat wing C C Cya ya ya= ⋅ +α α Δ , where is the non-linear

additive exists only at a subsonic leading edge. It can be estimated by the formula

ΔCya

ΔC C Mya ya l e= − ∞4 1 2 2π λ

χ αα cos . . ⋅2

At M l e∞ ≥cos . .χ 1 (supersonic edge) ΔCya = 0 . There are schedules

constructed in a generalized form for definition of the derivative Cyaα , as the

dependence Cyaα

λ on parameters of similarity:

Cf M tgya

α

λλ λ χ= −⎛⎝⎜

⎞⎠⎟∞

20 51, ,. η .

Approximately it is possible to consider, that the taper η practically does not

influence the lift coefficient. For each wing the function

Cf M tgya

α

λλ λ χ= −⎛⎝⎜

⎞⎠⎟∞

20 51, ,. η is various. However, as it was mentioned above, at

λ M∞ − → ∞2 1 we receive for all wings C

Myaα

λ λ=

−∞

4

12. Practically this

dependence can be used at λ M∞ − ≥2 1 6 7... .

The general view of dependence Cyaα

λ

is shown in a Fig. 6.14. The presence of

fractures at changing of flow modes about

edges is characteristic for it:

Fig. 6.14.

In point - the trailing edge passes

from subsonic to supersonic flow mode;

A

In point B - the leading edge passes

from subsonic to supersonic flow mode.

In experiment these fractures are

smoothed out.

71

6.4.2. Wave drag.

Generally C

cf M tgxw

λλ λ χ η2

20 51= −⎛

⎝⎜⎞⎠⎟∞ , , ,. airfoil shape . At

λ M∞ − → ∞2 1 irrespectively of the wing plan form we shall have

C

с

B

Mxw

λ λ2 2 1=

−∞

as the characteristic of an airfoil. Practically this formula can be

used at λ M∞ − ≥2 1 6 7... . It is necessary to note weak influence of taper onto wave

drag. The presence of fractures on a curve (Fig. 6.15). is characteristic for general

dependence C

cxw

λ 2 on the reduced aspect ratio.

72

The fracture in point - wing trailing

edge passes from a subsonic flow mode to

supersonic; in point C - transition of the

maximum thickness line from subsonic to

supersonic flow; in a point

A

B - the leading

edge passes from subsonic to supersonic flow

mode. These fractures are not present in

experimental dependencies, they are

smoothed out.

73

The maximum of curves is observed in

the area of transition of the maximum

thickness line ( χc ) from a subsonic flow

mode to supersonic (point C ). For a sound

line of maximum thickness

λ χ λtg Mc = − 1∞2 . It is necessary to note

that at subsonic lines of maximum thickness

the wave drag of swept wings is less than

drag of unswept wing. Thus the longer wing

aspect ratio (at χc const= ), sweep (at

λ = const ) or parameter λ χtg c is, then the

bigger profit is received in drag. On the contrary, at a supersonic line of maximum

thickness the wave drag of swept wing is more than of unswept one (Fig. 6.16).

Fig. 6.15.

Fig. 6.16.

6.4.3. Induced drag.

If the leading edge is supersonic, then C Cxi ya= α or C , where ACxi ya= 2

A C ya= 1 α - for the flat wing. At the subsonic leading edge it is necessary to take into

account the sucking force. In this case C C Cxi ya F= −α , or approximately

; C ACxi ya= 2 AC

CCya

F

ya= −1

2α ; CC

M SS

C

CF

ya

l e

l e

ya

ya2

2 2 214

=− ⎛

⎝⎜⎜

⎞

⎠⎟⎟

∞ξχ

π χ

α

αcoscos

. .

. .

Δ Δ ; (6.22)

Where ξ is the factor of sucking force realization (refer to

item 6.2);

Fig. 6.17.

C ya Δα and are parameters of a triangular wing, which

leading edge coincides with the leading edge of the

wing under consideration (fig. 6.17).

SΔ

6.4.4. Location of aerodynamic center.

Generally, there is the dependence x f M tgF = −⎛⎝⎜

⎞⎠⎟∞λ λ χ2 1, , η , in which the

influence of taper is essential in contrast to the characteristics Cyaα and C . Location

of aerodynamic center tends to the position of the center of mass of a figure presenting

the wing plan form with increasing of reduced aspect ratio. In particular, for wings with

unswept edges of the tapered plan form we have

xb0

x xb

x tF F c g l e= = =+ ++

++⎛

⎝⎜⎜

⎞

⎠⎟⎟

0

213

11

14

. . . .ηη η

ηη λ χg . (6.23)

The reduced formula can be used already at λ M∞ − ≥2 1 5 6... :

(For a rectangular wing - η = 1 , χ l e. . = 0 ; xF = 0 5,

for a triangular wing - η = ∞ , λ χtg l e. . = 4 , xF = 2 3 ).

Note: While calculating the aerodynamic characteristics of the complex plan form

wings (curved edges or the edges with a fracture) approximate methods replacing

variables spanwise ( )χm z , ( )c zm , ... by their mean values are used together with

precisely numerical calculations of a particular wing.

74

6.5. Wing in transonic range of speeds.

Speeds corresponding to Mach numbers (M M* , ... ,≤ ≤∞ 1 20 1 25 ) are called

transonic. All range can be divided on to: area of subsonic speeds (M M* ≤ ≤∞ 1 ), area

of supersonic speeds ( ), flow mode with Mach numbers 1 1 20< ≤∞M , ... ,1 25 M∞ = 1 .

Features of the aerodynamic characteristics in subsonic part of transonic speeds

are determined by existence of mixed flow

including subsonic (outside of the wing) and

supersonic (on the wing and near to it) flow

areas. The forward border of supersonic flow

area represents so-called sound line, along

which the transition from subsonic to

supersonic flow takes place. The flow

remains subsonic outside the zones limited by

the sound line. Fig. 6.18 shows the

approximate borders of supersonic zones at

various Mach numbers . With increasing

of Mach number the shock waves are

originally formed on the upper surface and

move to the trailing edge. Then the supersonic

area is formed on the lower surface. The

development of supersonic area on the airfoil

lower surface proceeds more intensively, than on lower. The supersonic areas are

finished by shock waves, which with increasing of numbers displace back and

enlarge the extent in the vertical direction. At

M∞

M∞

M∞

M∞ = 1 a shock wave is theoretically

distributed into infinity, at that there is a head shock wave before the wing also in

infinity. Further increasing of causes movement of a head shock wave and shock

wave on the wing surface downwards the flow. The supersonic wing flow mode comes

M∞

Fig. 6.18.

75

76

M∞ = 1 20 1 25, ... ,at values , when the shock waves practically do not move any more,

and reduce their angle of inclination with increasing of . M∞

The appearance of transonic parameter of similarity λ с3 , as a result of the

non-linear theory, is characteristic for transonic area. Parameter λ с3 influences onto

changing of the aerodynamic characteristics so, that:

C

f M tg сyaα

λλ λ χ η= −⎛⎝⎜

⎞⎠⎟∞

2 31 , , , λ (Fig. 6.19);

Cс

f M tg сxw

λλ λ χ η λ2

2 31= −⎛⎝⎜

⎞⎠⎟∞ , , , , airfoil shape (Fig. 6.20);

x f M tg сF = −⎛⎝⎜

⎞⎠⎟∞λ λ χ η2 31 , , , λ (Fig. 6.21).

At : the dimensionless parameters M∞ = 1C yaα

λ, C

сxb

λ 2 , xF depend on λ χtg ,

η , shape of the airfoil and λ с3 . Let's remind, that the taper η continues to play a small

role in changing of Cyaα and , in some cases its influence can be neglected.

However, the parameter

Cxb

η plays an essential role for characteristic of aerodynamic

center location, because it effects onto aerodynamic loading distribution wing spanwise.

For wings of arbitrary shape the influence of λ с3 is investigated a little. More

detail research is carried out on rectangular wings with rhomboid airfoil

( χ = 0 ,η = 1 , or ). it was proved, that for such wings at and B = 4 Kп р = 1 M∞ → 1

λ с3 1≤ the value of Cyaα

λπ

→ ≈2

1 57, . It can be explained that at reduced

aspect ratio

M∞ → 1

λ M∞ − →2 1 0 and we pass to the very low-aspect-ratio wing, for which

Cyaα π λ

=2

. If λ с3 2≥ , then the theory of transonic flows for a rectangular wing gives

C

с3yaα

λ λ≈2 3, .

The analogous results are received for wave drag: C

cxw

λ 2 3 0≈ , at λ с3 1≤ ,

C

c сxw

λ λ2 33 65

≈, at λ с3 2≥ .

Fig. 6.19. Dependence Cyaα

λ on reduced

aspect ratio λ M∞ −2 1

Fig. 6.20. Dependence C

cxw

λ 2 on reduced

aspect ratio λ M∞ −2 1

77

It is necessary to note, that in the latter

case

78

C cxw ~5 3 , i.e. the wave drag grows

with increasing of c , though not so fast as in

the supersonic flow, in which Cxw ~2c . The

change of the aerodynamic center location

dependently on λ с3 is similar to changes of

Cyaα (Fig. 6.21). The more λ с3 is, then

aerodynamic center changing by Mach

numbers behaves more irregularly:

drastic displacement forward in subsonic

range is probable with the subsequent

displacement backward into position

corresponding to supersonic speeds

(displacement of aerodynamic center for a triangular wing at passage from

M∞

M∞ < 1 to

is determined as M∞ > 1 x bF 0 0 12≈ , λ ).

Fig. 6.21. Dependence of the

aerodynamic center location on

λ M∞ −2 1

The main measures providing reduction of wing wave drag, improvement of its

lifting properties and smooth change of aerodynamic center in supersonic range of

speeds by Mach numbers are: reduction of M∞ c and λ (decreasing of parameter

value λ с3 ) and increasing of χ .

6.6. Wing induced drag at M M∞ ≤ * with taking into account local

supersonic flows.

Let's consider the problem on the account of additional drag occurring at values

of C and , outgoing of critical values (the approximate method of the account is

offered by S. I. Kuznetsov). It is necessary to take into account this drag while

constructing the wing polar for specified

ya M∞

M con 1. s∞ t= , M∞ <

Let's assume that the dependence of critical Mach number on CM* Cya ya= * ,

i.e. or ( )M f Cya* *= ( )C f Mya * *= . Let's construct this dependence in a plane of

, (Fig. 6.22). Cya M∞

It is obvious, that all values of C

and , lying below the curve

Fig. 6.22

ya

M∞

( )C f Mya * *= fall into subsonic speeds

area. However if at specified

M con∞ st= will be C , then

the flow supersonic area closed by shock

waves is formed on the wing. In this case

there is an additional drag caused by lift

Cya ya> *

ΔCxi (at C Cya ya≤ * ΔCxi = 0 ). If one

assumes, that the growth of lift is not

accompanied by growth of sucking force

with increasing of angles of attack at

(that at presence of the broad

supersonic area on a wing is permissible),

then for a flat wing we shall have

C Cya ya> *

ΔC Cxi ya= α or . ( )ΔC C Cxi ya ya= − * α

In addition adopting, that on

transonic flow modes the proportion

C Cya ya= α α is executed, then finally we

receive

Fig. 6.23. Wing polar with the account of

ΔCxi

( )

ΔCC C C

Cxi

ya ya ya

ya=

− *α

.

This parameter is added to induced drag, and thus we have:

79

( )[ ]

C AC C C M const

C AC C C C C C C

xi ya ya ya

xi ya ya ya ya ya ya ya

= ≤ =

= + − >

⎧⎨⎪

⎩⎪

∞2

2

, если при ;

, если .*

* *α

(6.22)

Wing polar take the form as it is shown in fig. 6.23.

Values of lift coefficients C corresponding to the beginning of wave crisis at

; i.e. the dependence

ya *

M∞ ( )C f Mya * *= can be found from the formula (M M∞ ≡ * ):

( )Cк c

M m cyac

c

n

* cos. cos= − +

⎛⎝⎜

⎞⎠⎟−

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪∞

1 1 1 0 12 2χ λ

χ

where the factors , , for a wing with a classical airfoil are equal k m n k = 3 2. ,

m = 07. , n = 23 ; for a wing with a supercritical airfoil - k = 1 2. , m = 0 65. , n = 13 .

80