1. SECTION 1. AERODYNAMICS OF LIFTING SURFACES
TOPIC 5. THE AERODYNAMIC CHARACTERISTICS OF
WINGS IN GAS SUBSONIC FLOW
As it was spoken earlier, aerodynamics of subsonic speeds is limited by Mach
numbers 0 .4 ≤ M ∞ ≤ M* .
Potential of speeds of disturb flow satisfies the linearized equation of gas
dynamics in a linearized subsonic flow
(1 − M )ϕ2
∞ xx + ϕ yy + ϕ zz = 0 . (5.1)
Let's pass to new variables, as to receive the Laplas equation for speed potential
ϕ in incompressible fluid. Then, assuming flow parameters and aerodynamic
characteristics in incompressible fluid as known, we shall connect them to the required
characteristics in a subsonic gas flow.
The passage to incompressible flow is performed by replacement of coordinates:
⎧x = x 1 − M 2
⎪ n ∞
⎪
⎨ y = yn (5.2)
⎪z = z
⎪
⎩
n
where variables with an index n correspond to coordinates in incompressible gas flow.
At that ϕ ( x , y , z ) → ϕ ( x n , y n , z n ) and ϕ x n x n + ϕ y n y n + ϕ z n z n = 0 .
2 ∂ϕ
For the pressure factor in incompressible flow we have С pn = − and
V∞ ∂ x n
2 ∂ϕ 2 ∂ ϕ d xn 1
Сp = − =− = С pn . Finally we receive:
V∞ ∂ x V∞ ∂ x n d x 2
1 − M∞
С p incompr
Сp = . (5.3)
2
1 − M∞
Therefore, the aerodynamic characteristics connected to pressure forces are
determined analogously:
48

2. С ya n mz n С xi n
С ya = , mz = , С xi = . (5.4)
2 2 2
1− M∞ 1− M∞ 1− M∞
It is necessary to emphasize that the characteristics С ya n , m z n and С xi n
correspond to the transformed wing (Fig. 5.1) at passage to incompressible flow, i.e. to
a wing with geometry λ n , χ n and η n . Let's define these parameters:
n b0 n bw .t .
l n = l , b0 = , bw .t . = ,
2 2
1− M∞ 1− M∞
S n x0
Sn = , x0 = , ηn =η,
2 2
1− M∞ 1− M∞
2
λ n = λ 1 − M ∞ , - reduced aspect ratio,
n tgχ 0 n
tgχ 0 = , λtgχ 0 = λ n tgχ 0
2
1 − M∞
Now it is possible to obtain the
aerodynamic characteristics of the specified
wing in subsonic flow, knowing the
aerodynamic characteristics of the
transformed wing in a flow of incompressible
fluid. Let's notice, that the we consider linear
Fig. 5.1. a) - initial wing; components of C yа and mz . Therefore it is
possible to write C α and mα instead of C yа
b) - transformed wing.
yа z
and mz in the formulae (5.4).
5.1. Wing lift coefficient
In general case C yа = C yа лин + ΔC yа = C α (α − α0 ) + ΔC yа .
yа
At small angles of attack α and M ∞ < M∗ the lift coefficient of a wing is
determined by equality
49

3. C yа = C α (α − α 0 ) .
yа (5.5)
In compliance with the linear theory the lift coefficient C yа in compressed gas is
С ya n
determined by the formula С ya =
2
yа (
, where C yа n = C α n α − α 0 n is the lift )
1 − M∞
coefficient of the deformed wing in incompressible gas. It follows from the last ratios,
that
α Cα n
yа
C yа = , α0 = α0 n . (5.6)
2
1 − M∞
5.1.1. High-aspect-ratio wings
π λn
For incompressible fluid we have C α n =
yа ,
(1 − τ n ) + (1 − τ n ) 2
+ (π mn )
2
λn 2
where mn = , (refer to section 4.1.1). As λ n = λ 1 − M ∞ and
C α ∞ cos χ 0 .5
yа
n
1 λ 1 − M ∞ + tg 2 χ 0 .5
2
n
cos χ 0 .5 = , then mn = . Now C α should be
yа
1 + tg 2 χ 0 .5
n Cα n∞
yа
πλ
determined as Cα =
yа , where
(1 − τ n ) + (1 − τ n ) 2
+ (π mn )
2
λ 1 − M ∞ + tg 2 χ 0 .5
2
mn = .
Cα n∞
yа
If one assumes that χ 0 .5 = 0 and λ → ∞ , then we get a result of a thin airfoil
α Cα
, where C α n = C α ∞ = 2π − 1,69 4 c .
yаn
theory C yа = yа yа
2
1− M∞
( )
Parameter τ n = f λ n , χ n ,η n = f (λ , χ ,η ,M ∞ ) .
50

4. mn
Approximately τ n = τ 1 n ( mn ) ⋅ τ 2 n (η ) , where τ 1n ( mn ) = 0 .17 and
π
1
τ 2 n (η ) = η 2 + .
(5η + 1) 3
As non-linear lift component is absent for a wing of high aspect ratio ΔC yа = 0 ,
then C yа = C α (α − α0 ) .
yа
5.1.2. Low aspect ratio wings
It is possible to use the formula for determination of a derivative of a lift
coefficient by angle of attack C α :
yа
α C α ∞ λn
ya C α ∞ λ 1 − M∞
ya
2
C yа n = = , (5.7)
pn λ n + 2 pn λ 1 − 2
M∞ +2
where pn is the parameter determined for a transformed wing.
Particularly, we have for a tapered wing:
2
pn = 0 ,5 ⎛ 1 + tg 2 χ lne . + 1 + tg 2 χ tn.e . ⎞ +
⎜ ⎟ =
⎝ ⎠ λ n (η n + 1)
.
⎧
⎪ ⎜ 2 ⎫
⎪ 1
= ⎨0 ,5 ⎛ 1 − M ∞ + tg 2 χ l .e . + 1 − M ∞ + tg 2 χ t .e . ⎞ +
2 2 ⎟ ⎬ .
⎪ ⎝ ⎠ λ (η + 1) ⎪ 2
⎩ ⎭ 1 − M∞
α Cα ∞ λ
yа
Finally we shall write down the expression C yа = ~ ,
pλ + 2
where ~ = pn 1 − M ∞ ,
p 2
~ = 0 ,5
p ( 1 − M ∞ + tg 2 χ l .e . + 1 − M ∞ + tg 2 χ t .e . +
2 2
) 2
λ ( η + 1)
.
The non-linear additive is calculated under the formula:
8
1 − M ∞ cos 2 χ l .e . (α − α 0 ) .
2 2
ΔC yа = (5.8)
pλ + 2
51

5. It is necessary to note, that the conversion linear theory can not be used for
connection between compressed and incompressible flows while calculating the non-
linear additive ΔC yа .
Thus: C yа = C α (α − α0 ) + ΔC yа .
yа
5.1.3. Extreme small-aspect-ratio wings ( λ < 1 ).
α πλ n πλ 2 α Cα n
yа πλ
In such case we have C yа n = = 1− M∞ and C yа = = ,
2 2 1− 2
M∞ 2
πλ
therefore derivative C α =
yа does not depend on Mach numbers M ∞ .
2
The non-linear additive ΔC yа is determined by the above mentioned formula
(5.8).
Note: it is possible to notice in the above mentioned formulae for C α , that the
yа
ratio C α λ is a function of parameters λ 1 − M∞ , λ tgχ 0 .5 and η .
yа
2
2
Parameter λ 1 − M∞ is the reduced aspect ratio.
These parameters can be considered as parameters of similarity and used for
creation of the diagrams. So C α λ = f ⎛ λ 1 − M ∞ , λ tgχ 0 .5 , η ⎞ .
⎜ 2 ⎟
ya ⎝ ⎠
The analysis of wing lift in a subsonic gas flow:
1. The angle of zero lift α 0 does not depend on numbers M ∞ (Fig. 5.2).
2. The derivative C α grows with increasing of numbers M ∞ (Fig. 5.2).
ya
3. The effect of a compressibility (influence of Mach numbers M ∞ onto the
derivative C α ) decreases with reduction of λ (it is the reason of spatial flow of low
ya
aspect ratio wings) (Fig. 5.3).
4. The non-linear component ΔC ya decreases with increasing of numbers M ∞ .
52

6. Fig. 5.2. Influence of number M ∞ Fig. 5.3. Influence of wing aspect ratio λ onto
onto dependence C ya = f ( α ) compressibility effect
5. The effect of a compressibility decreases with increasing of sweep angle χ
(Fig. 5.4). The reason of it is as follows: with rising up of sweep angle χ , the speeds
component normal to the leading edge from which depends the characteristic C ya
becomes less,.
6. The value of C ya max decreases with increasing of numbers M ∞ (the reason is
more earlier flow stalling) (Fig. 5.5). For example, for the airplane JaK − 40 : at
M ∞ = 0 .2 - C ya max = 1.44 , and at M ∞ = 0 .6 - C ya max = 1.1 .
Fig. 5.4. Influence of a sweep angle χ onto Fig. 5.5. Influence of number M ∞
effect of compressibility at λ = const onto dependence C ya = f ( α )
53

7. Using parameters of similarity the dependence
⎜ 2 ⎞
C α λ = f ⎛ λ 1 − M ∞ , λ tgχ 0 .5 , η ⎟ looks like it is shown in fig. 5.6.
ya ⎝ ⎠
Fig. 5.6. Dependence of a factor C α λ on parameters of similarity
ya
5.2. Induced drag.
The wing induced drag coefficient taking into account of compressibility is equal
C xi n
C xi = , (5.9)
2
1− M∞
2
Where C xi n = AnC ya n is the induced drag of the “deformed” wing in incompressible
2
gas flow; C ya n = C ya 1 − M ∞ ; the polar pull-off factor is equal to:
⎧ 1 + δn 1 + δn
⎪ πλ = 2
− for high - aspect - ratio wing;
⎪ n π λ 1 − M∞
⎪ 1 1
An = ⎨ = − for low - aspect - ratio wing.
⎪ Cα n Cα 1 − M 2
⎪ ya ya ∞
⎪.
⎩
2 2
After substitution C xi n = AnC ya n , C ya n = C ya 1 − M ∞ and An into (5.9) we
receive again
54

8. 2
C xi = AC ya , (5.10)
1 + δn 1
where А = - for high-aspect-ratio wings, А = α
- for low-aspect-ratio wings.
πλ C ya
Thus formula of induced drag in gas subsonic flow is kept in a prior form (at
angles of attack, where the linear dependence C ya = f ( α ) . Polar does not vary either
(at condition of C xp = const ).
5.3. Moment characteristics.
Location of center of pressure and aerodynamic center.
The factor of wing aerodynamic moment of pitch relatively to axis 0 z passing
through center of forces reductions is determined by the formula
mz = mz0 + ⎛ mz ya ⎞ C ya ,
C
⎜ ⎟ (5.11)
⎝ ⎠
at that factor of pitch moment for linear part of dependence is determined as
m z0 n
Where m z0 = is the factor of pitch aerodynamic moment at C ya = 0 ,
2
1− M∞
C dm z mα
xF = − m z ya =− =− z is the relative coordinate of aerodynamic center
dC ya Cα
ya
position.
The position of aerodynamic center x F = x F n and center of pressure
x c . p . = x c . p . n does not depend on Mach numbers M ∞ for a wing in subsonic and
incompressible gas flows!
Received result is approximate for low-aspect-ratio wings with taking into
account the non-linear effects. (it is exact at α → 0 ). For tapered high-aspect-ratio
wings it is possible to offer the following formula for definition of aerodynamic center
location relatively to wing top:
55

9. xF ⎛ η − 1⎞ η + 1
xF = = x F∞ ⎜ 1 − 4 ⎟+ λ tg χ l .e .
b0 ⎝ 3π η ⎠ 3π η
(
x F∞ = 0 ,25 1 − 1,6 f
2
) - airfoil.
Fig. 5.7.
It is noteworthy, that if concepts of bА and x А
are used, then position of aerodynamic center relatively to the leading edge MAC in
shares of bА for wings of large aspect ratio is determined by the formula:
xFA
x FA = = 0 .25 ;
bА
xFA = xF − x A .
It is possible to consider this
ratio as fair for wings with curvilinear
edges or with fracture (Fig. 5.8). The
Fig. 5.8.
position of aerodynamic center
relatively to wing top is determined by the formula:
l 2
2
xF = x A + 0 .25 bA =
S ∫ [ xl .e .( z) + 0 .25 b( z)] b( z)dz .
0
There is no common formula for low-aspect-ratio wings. In particular cases:
λ 1
xF = is for rectangular wing, x F = is for triangular wing.
2 .2 + 3 .6 λ 1.52 + 0 .12λ
It is noteworthy, that the aerodynamic center displaces forward with decreasing
of λ for rectangular wing (at λ → 0 , x F → 0 and all aerodynamic load is concentrated
on the leading edge), and the aerodynamic center with λ reduction displaces back for
triangular wing (at λ → 0 , x F → 0 .66 , more precisely 2
3 ).
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10. 5.4. Wing critical Mach number M* .
The critical Mach number M* determines the upper border of subsonic flows and
the above mentioned formulae are fair at condition of M∞ ≤ M* . Generally
( )
M* = f η , λ , χ , c , C ya . Parameters χ , c and C ya have the greatest influence. The
value M* can be defined by
theoretical curve by
S.A. Christianovich (Fig. 5.9),
having the diagram of
distribution of pressure factors
Cp along wing surface in
incompressible flow. It is also
possible to use the following
formula for assessment M* of
wings with ordinary airfoils
Fig. 5.9. Christianovich dependence
(Fig. 5.10) at lift coefficient
value C y = 0 :
0 ,7 λ 2 c
M* = 1 − cos χ c , (5.12)
2
λ + 0 ,1
Where χc is the sweep angle at a line of maximum thickness.
Other formula
M* =
1 − mλ 2 c*
cos χ ; c =
c
(c + 17 f ) ,
*
2
c
(5.13)
λ 2 + 0 ,1 xc
Where m = 0 .35 is for classical airfoil, m = 0 .27 is for supercritical airfoil.
As it is visible from the above mentioned formulae, the value of M* depends on
relative thickness c and airfoil camber f , on the airfoil shape (first of all on maximum
thickness location x c ) and on the wing plan form λ , χ c .
57

11. It is possible to increase M* by application of supercritical airfoils (Fig. 5.11).
They are characterized by more uniform distribution of a pressure factor chord
lengthwise.
Fig. 5.10. Pressure distribution on the Fig. 5.11. Pressure distribution on the
upper surface for ordinary airfoil upper surface for supercritical airfoil
The account of C y influence can be done by the following formula
⎛ 3
2 cos 2 χ ⎞ λ2
M* = 1 − ⎜ 0 .7 c cos χ c + 3 .2 c C ya c⎟ 2 . (5.14)
⎝ ⎠ λ + 0 .1
It is possible to use dependence for supercritical airfoil and wings with such
airfoils:
(
M* = 1 − 0 .55 c cos χ c + 3
c C ya 2
cos χ c )λ 2
λ2
+ 0 ,1
. (5.15)
At C ya = 0 the last formula gives
0 .55 λ 2 c
M* = 1 − cos χ c for supercritical airfoils
2
λ + 0 .1
58