1. SECTION 2. AERODYNAMICS OF BODYS OF REVOLUTION
THEME 12. THE AERODYNAMIC CHARACTERISTICS OF
BODYS OF REVOLUTION, FUSELAGES AND THEIR ANALYSIS
12.1. Lifting force of a body of revolution.
According to the theory of an elongated body the factor of pressure on surface of
a body of revolution at flow about it at an angle of attack is determined by the formula
.
C p = −4α r( x ) cos ϕ , (12.1)
.
where r( x ) =
dr
.
dx
With taking that into account for a lift coefficient (11.9) we obtain 1
lf π lf π lf
8α 4πα
∫ ∫ ∫ ∫ ∫
2 . .
C ya = − r( x ) dx C p cos ϕ dϕ = r r dx cos 2 ϕ dϕ = r r dx .
Sm. f . Sm. f . Sm. f .
0 0 0 0 0
Let's consider distribution of a lift coefficient along the length of a body of
revolution
dC ya 4πα 2α dS ( x )
= r ( x ) r( x ) =
& , (12.2)
dx Sm. f . S m . f . dx
where S ( x ) = π r 2 ( x ) is the cross-sectional area of a body of revolution.
The last formula is more general and fair for any shapes of cross sections.
From the obtained expression it follows, that on a fuselage lift occurs only on
sites with the variable area of cross sections S ( x ) , at that, the sign of lift is determined
by the sign of derivative dS ( x ) dx . Therefore, on extending nose part positive lifting
π
1 cos 2 ϕ dϕ = π
∫ 2
Is used.
0
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2. force occurs, since here dS ( x ) dx > 0 , on the tapering rear part - negative lift, and on
the cylindrical part lift will be absent.
Experiments and more precise calculations show, that the above mentioned
qualitative analyses remain fair and for not thin fuselages. The quantitative results
according to this theory are satisfactory only for nose and cylindrical parts at M∞ ≤ 1 .
For the rear part the theory does not take into account the influence of a boundary layer
and flow stall, due to this influence the absolute value of lifting force decreases. At
supersonic speeds ( M∞ > 1 ) the theory does not take into account influence of nose
shape and numbers M ∞ , for cylindrical part - occurrence of lift due to “carry” from the
nose part.
The lift coefficient C ya of a body of revolution can be presented as a sum of the
factors of lifts of its parts. The lift coefficient is calculated separately for nose,
cylindrical and rear parts. For thin fuselages close to body of revolutions, the calculation
should be performed using the theory of an elongated body with consequent refinement
of influence of the various factors which are not taken into account by this theory.
So, generally it is possible to write down for the fuselage lift coefficient
C ya = C α (α − α 0 ) + ΔC ya ,
ya
where C α = C α nose + C α cil + C α rear .
ya ya ya ya (12.3)
The size of the derivative C α depends on the shape of the body of revolution and
ya
first of all on its nose part, angle of attack α , structure of a boundary layer, number
M ∞ and other factors.
12.1.1. Lift of a nose part.
In accordance with the theory the lifting force is distributed according to the law
dC ya 2α dS ( x )
= in subsonic range of speeds ( M ∞ < 1 ). So
dx S m . f . dx
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3. l nose Sm. f .
dS ( x )
∫ ∫ dS = 2(1 − ηnose ) ;
2 2
C α nose =
ya dx = 2
Sm. f . dx Sm. f .
0 S nose
12
⎛S ⎞
α
C ya nose = 2 ( 1 − ηnose
2
), η nose
d
= nose = ⎜ nose ⎟
d m. f . ⎜ Sm. f . ⎟
⎝ ⎠
. (12.4)
At absence of the air intake in the nose part ( η nose = 0 ) C α nose = 2 .
ya
It has to be noted, that at working engine, when air is sucked through the air
intake, an additional air intake lift occurs which should be taken into account in
C α nose .
ya
Approximately this force can be estimated by the formula
ya (
C α a .i . = 2ϕ ( 1 − S c .b . ) 1 − S c .b . ηnose ,
2
) (12.5)
where S c .b . = Sc .b . S nose is the relative area of the
body central part in input cross-section of the air intake
( S c .b . = d c .b . d nose ) (Fig. 12.1), ϕ
2 2
is the flow
coefficient of air flow rate (on computational
Fig. 12.1. operational mode of the air intake ϕ = 1 ).
The value of a derivative C α nose of the lift
ya
coefficient of the air intake C α a .i . is added to a derivative of the nose part.
ya
At supersonic speeds of flight the size of the derivative C α nose depends on the
ya
shape of the nose part and aspect ratio (parameter x n = M ∞ − 1 λ nose ) (Fig. 12.2).
2
102

4. Fig. 12.2. Influence of the shape of the nose part onto the derivative C α nose
ya
Examples:
- conical nose part without the air intake (w/o a.i.)
C α nose ≡ C α nose
ya ya w / o a .i . (
= 8 1 − 0 .2 x n exp − xn ) λ2 nose
4 λ nose + 1
2
; (12.6)
- shape of the nose part with curvilinear generative line without the air intake (w/o a.i.)
C α nose ≡ C α nose w / o a .i . = 1.65 + 0 .35(1 + 2 x n ) exp − 2 xn ;
2
ya ya (12.7)
- at presence of the air intake
( ) + 2ϕ (1 − S )(1 − ηnose
) 1 + 0 .46 x 2 , (12.8)
2
α α
C ya nose = C ya nose w / o a .i . 1 − ηnose
2
c .b . S c .b .
n
where x n = M ∞ − 1 λ nose .
2
12.1.2. Lift of the cylindrical part.
In subsonic flow ( M ∞ < 1 ) the cylindrical part of a fuselage does not create lift at
small angles of attack. According to the theory, as on the cylindrical part dS = 0 , then
C α cil = 0 .
ya
In the supersonic flow ( M ∞ > 1 ) there is a lift on the cylindrical part. It happens
because of influence of the nose part. At presence of lift on the nose part pressure they
have various values on its upper and lower parts. These pressures are propagated to the
102

5. cylindrical part as disturbances after reflection from a head shock wave. As a result,
there is a reduced pressure on the upper surface in comparison with the lower surface of
the cylindrical part, that causes occurrence of lift on the cylindrical part C α cil
ya
(Fig. 12.3). (In the subsonic flow disturbances are spread in all directions, therefore the
upper surface of the nose part effects both the upper and the lower parts of the cylinder
surface. The influence of the lower surface of the nose part is similar. As a result of
mutual influence at M∞ < 1 C α cil = 0 ).
ya
In general, the size of the derivative C α cil depends on the Mach number, aspect
ya
ratio of the nose part and type of coupling of nose and cylindrical parts (Fig. 12.4, 12.5)
C α cil = f
ya ( M ∞ − 1 λnose , λcil λnose , type of coupling .
2
)
Intersecting coupling
Tangent coupling
Fig. 12.3. Distribution of lift along length Fig. 12.4. Types of coupling of nose
of the cylindrical part and cylindrical parts
Approximately it is possible to estimate size of C α cil by the formula
ya
ya
b
(
C α cil = ax n exp − cx n 1 − exp − d xc
), (12.9)
where xc = M ∞ − 1 λ cil .
2
102

6. The values of factors a , b , c and d can also be adopted as the following:
- for conical nose part a = 1.3 , b = 0 .5 , c = 0 .05 , d = 1.29 ;
- for the nose with curvilinear generative line and tangent coupling a = 4 .5 , b = 3 .0 ,
c = 1.5 , d = 0 .88 .
It has to be noted, that the values
C α cil are a little bit larger at presence of
ya
nose cone in comparison with other shapes of
noses (Fig. 12.5).
Fig. 12.5. 12.1.3. Lift of the rear part.
The derivative of the lift coefficient of the rear part of the body of revolution does
not depend on the shape of the rear part and is determined by the following ratios.
In the subsonic flow ( M ∞ < 1 ) distribution of lift along body length according to
dC ya 2α dS
the theory of an elongated body = , so
dx S m . f . dx
S rear
C α rear =
ya
2
Sm. f . ∫
dS
dx
dx = −2(1 − S base ) = −2(1 − ηrear ) .
2
(12.10)
Sm. f .
In real flow (Fig. 12.6) boundary layer δ ∗ rising happens in the rear part due to
∗
influence of viscosity, that results in the body thickening S base and decreasing of angle
of declination of generative line.
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7. Fig. 12.6. Thickening of the rear part due to the boundary layer
As a result, the size of parameter C α rear should decrease on an absolute value.
ya
The account of viscosity influence results in the following computational formula
ya (
C α rear = −0 .4 1 − η rear .
2
) (12.11)
In the supersonic flow ( M∞ > 1 ) the Mach numbers M ∞ effect the amount of the
derivative C α rear and determination of C α rear is performed by the formula
ya ya
α 1 − η rear
2 M∞ − 1
2
C ya rear = −0 .4 , xr = . (12.12)
1 + 0 .4 x r η rear
2 2 λ rear
With increasing of numbers M ∞ the amount of C α rear decreases in an absolute
ya
value.
It is necessary to note one more effect, which is not taken into account in the
theory of the elongated body. This is an occurrence of the non-linear component on the
fuselage due to formation of vortical structures on the upper surface (it is similar to the
wing).
The values ΔC ya are essential in general size C ya for thin body of revolutions at
large angles of attack. For fuselages of airplanes the occurrence of the non-linear
component, as a rule, is not considered. Also it is necessary to remember, that in the
system of an airplane the non-linear components from a wing and fuselage decreases.
The size of zero lift angle of the fuselage α 0 is determined by chamber of its axis
which is caused by nose deflection and splayed rear part. The value of α 0 is calculated
by the formula
[ ( )
α 0 = 1.25 β nose λ nose λ f + 0 .1β rear λ rear λ f ( )] , (12.13)
where β nose is the angle of nose deflection; β rear is the angle of taper of the rear part.
The angles β nose and β rear also are taken with positive sign, if the nose is
deflected downwards, and the rear part is tapered upwards.
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8. 12.2. Aerodynamic moment of a body of revolution.
Coordinate of aerodynamic center.
According to the theory of a thin (elongated) body the longitudinal moment is
determined under the formula
l π
∫ x r dx ∫ C p cos ϕ dϕ
2
mz = (12.14)
SL
0 0
As the lifting (normal) force was determined for separate parts of a body of
revolution (for nose, cylindrical and rear parts), and moment characteristics should be
also calculated for parts of body of revolution.
12.2.1. Aerodynamic moment of a nose and coordinate of an
aerodynamic center.
Let's use the results of the theory of an elongated body, according to which the
factor of pressure on surface of the body of revolution at streamlining under the angle of
.
attack is determined by the formula (12.1) C p = −4α r cos ϕ .
We have
l nose π l nose π
8α
∫ ∫ ∫ ∫
2 .
m z nose = x r dx C p cos ϕ dϕ = − x r r dx cos 2 ϕ dϕ ,
S m . f .l nose S m . f .l nose
0 0 0 0
l nose
4πα
∫
.
m z nose =− x r r dx . (12.15)
S m . f .l nose
0
l nose
∫
.
Let's consider the integral function x r r dx :
0
102

9. l nose
l nose
l nose 2
∫ ∫ ∫ r dx =
. 1 1
x r r dx = x r dr = xr 2 − 2
2 0
2
0 0
1 1 1 ⎛ Wnose ⎞
= l nose rm . f . −
2
Wnose = l nose rm . f . ⎜ 1 −
2
⎜ ⎟ .
2 2π 2 ⎝ l nose S m . f . ⎟
⎠
Having accounted it an aerodynamic moment of the nose part
m z nose = −2α (1 − W nose ) , (12.16)
where W nose = Wnose S m . f .l nose - relative volume of the nose part.
Coordinate of the nose aerodynamic center relatively to nose of the
body of revolution in shares of length of the nose part
x F nose = x F nose l nose = m z
C ya
(
= mα C α
z ya ) nose
:
- at absence of the air intake in the nose part x F nose = 1 − W nose .
1 − W nose
- at presence of the air intake in the nose part x F nose = .
1 − η nose
2
The obtained formulae can be used at any Mach numbers M ∞ (despite of the fact
that the theory of an elongated body was applied which is fair for calculation of the
derivative C α nose only at subsonic speeds M ∞ < 1 ).
ya
For conical nose part
1 2 + η nose
x F nose = .
3 1 + η nose
For chambered nose part
1 7 + 3η nose
x F nose = .
15 1 + η nose
l nose
∫ r 2dx =
2 Wnose Is used
π
0
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10. It is necessary to note, that for bodies with the parabolic nose part coordinate of
the aerodynamic center practically does not vary with the increase of Mach numbers
M∞ .
12.2.2. Coordinate of the aerodynamic center of the cylindrical part.
In the subsonic flow ( M ∞ < 1 ) the lift of the cylindrical part C ya cil = 0 ,
therefore the moment characteristics of the cylindrical part are not calculated.
In the supersonic flow ( M∞ > 1 ) coordinate of the aerodynamic center x F cil , as
well as the derivative C α cil , depends on Mach number, aspect ratio of the nose and
ya
type of coupling of nose and cylindrical parts (Fig. 12.7):
( )
x F cil
x F cil = = f M ∞ − 1 λnose , λcil λnose , type of coupling .
2
l nose
Let's express a coordinate of the
aerodynamic center of the cylindrical part
x F cil
x F cil = in shares of fuselage nose
l nose
length
−1
⎛ d λ cil ⎞
xn λcil ⎜ x n λ nose ⎟ M∞ − 1
2
x F cil = 1+ − exp − 1⎟ , xn = , (12.17)
d λnose ⎜
⎜ ⎟ λ nose
⎝ ⎠
where the factor d value can be accepted as the following ones:
- for conical nose part d = 1.29 ;
- for the nose with chambered generative line and tangent coupling d = 0 .88 .
102

11. M∞ − 1
2
Let's note, that at → ∞ the
λ nose
coordinate of the aerodynamic center depends
only on the attitude of aspect ratios of
cylindrical and nose parts of the body of
revolution x Fcil → 1 + 0 .5( λ cil λ nose ) .
At presence of smooth coupling of the
nose and cylindrical parts the aerodynamic
center xFcil is located a little bit distant, than
Fig. 12.7. Coordinate of the aero-
in the case of conical nose and intersecting
dynamic center of the cylindrical part
coupling.
12.2.3. Coordinate of the aerodynamic center of the rear part.
Irrespectively of the shape of the rear part, for any Mach numbers M ∞ coordinate
of the aerodynamic center of rear part can be calculated by the formula
xFrear = l f − 0 .5 l rear =
( )
(12.18)
= l f 1 − 0 .5 λrear λ f ,
i.e. we accept, that the rear part aerodynamic
center is located in its middle.
12.2.4. Coordinate of the aerodynamic center
of body of revolution in a whole.
Let's consider the configuration of a body of revolution (Fig. 12.8). In this case
the coordinate of the aerodynamic center relatively to the nose is determined as
xF = − mα C α :
z ya
102

12. Fig. 12.8.
C α nose xFnose + C α cil xFcil + C α rear xFrear
ya ya ya
xF = , (12.19)
Cα
ya
where C α = C α nose + C α cil + C α rear .
ya ya ya ya
It is necessary to note, that at subsonic speeds ( M ∞ < 1 ) and small angles of
attack, at which C α cil = 0 the aerodynamic center of the body of revolution can be
ya
placed ahead of a nose, i.e. xF < 0 .
102