1. SECTION 2. AERODYNAMICS OF BODY OF REVOLUTIONS
THEME 9. FUSELAGE GEOMETRICAL PARAMETERS
9.1. General
Fuselage is the main part of an airplane structure. It serves for joining of all its
parts in a whole, and also for arrangement of crew, passengers, equipment and freights.
The exterior shape of a fuselage is determined by the airplane assigning, range of
speeds of flight, arrangement of engines and other factors.
The airplane fuselages (Fig. 9.1) and engine nacelles have the shape of the body
of revolutions or close to it. For the airplanes having integral configurations the wing
passes smoothly into the fuselage and the shape of the fuselage cross-section can
essentially differ from circular. The fuselages of transport airplanes frequently have tail
unit deflected upwards. The noses of modern fighters, as a rule, are rejected downwards.
Fig. 9.1. The basic geometrical characteristics of a fuselage
The basic geometrical parameters of a fuselage are the following ones.
Length of a fuselage l f is the greatest size of a fuselage along its centerline.
The area of fuselage midsection S m . f . is the greatest area of fuselage cross-
section by a plane, perpendicular to its centerline.
The shape of a fuselage cross section essentially influences the interference
aerodynamic characteristics at the installation of a wing and tail unit. While calculating
the aerodynamic characteristics of an isolated fuselage it is approximately substituted by
a body of revolution with the equivalent area of cross sections.
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2. Maximum equivalent diameter of a fuselage d m . f . is the diameter of a circle,
which area is equal to the fuselage midsection area.
d f = d m. f . = 4 Sm. f . π . (9.1)
The distinction of a fuselage from the body of revolution is taken into account by
an addend in the aerodynamic characteristics depending on design features. Therefore
we shall consider the geometric and aerodynamic characteristics of body of revolutions,
connecting them with the specific fuselage below.
Fuselage aspect ratio λ f is the ratio of the fuselage length to its maximum
equivalent diameter,
λf = lf d f . (9.2)
In some cases, especially when the fuselage is the body of revolution, it is
possible to allocate nose (head), cylindrical (central) and rear parts (fig. 9.1) and to
introduce the appropriate geometric parameters for them. As the total fuselage length
l f = l nose + lcil + l rear , then its aspect ratio
λ f = λ nose + λ cil + λ rear , (9.3)
where λ nose , λ cil , λ rear is the aspect ratio of nose, cylindrical and rear parts,
λ nose = l nose d f , λ cil = lcil d f , λ rear = l rear d f .
Frequently nose parts of fuselages have bluntness.
In some cases, nose can have an inside channel - the engine air intake.
The rear part of a fuselage can have a blunt base.
Then, the following additional parameters for the description of a nose and rear
part are used:
Nose tapering η nose = d nose d f is the ratio of a fuselage nose diameter to its
maximum equivalent diameter.
Tapering of a rear part η rear = d base d f is the ratio of diameter of a base of a
fuselage to its maximum equivalent diameter.
The relative area of the blunt base - S base = d base d 2 .
2
f
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3. The following angles are used at presence of a cam tail part. An angle of mean
line deviation β rear , the deviation of the nose part is determined by the angle β nose
(angle β nose is considered as positive at nose part deflection downwards).
9.1. Shape of a nose part and its geometrical parameters
The conical shape of a fuselage nose:
1 1
λ nose = , tgβ 0 = ;
2 tgβ 0 2λ nose
1
Wnose = l nose S m . f . - volume of the nose part.
3
1 − η nose 1 − η nose
λ nose = , tgβ 0 = ;
2 tgβ 0 2λ nose
Wnose =
1
3
( 2
)
1 + η nose + η nose l nose S m . f . .
The parabolic shape of a fuselage nose:
x
r = 0 .5 d f x( 2 − x ) , x = , 0 ≤ x ≤ 1;
l nose
1
η nose = 0 , tgβ0 = .
λ nose
[
r = 0 .5 d f ηnose + (1 − ηnose ) x( 2 − x ) ; ]
Wnose =
1
15
( 2
)
8 + 4η nose + 3η nose l nose S m . f . .
The ogival shape formed by arcs of a circle is close to the parabolic one, which
β0
aspect ratio is equal to λ nose = 0 .5 ctg . Volume of the nose part at λ nose ≥ 2 .5 is
2
determined by expression Wnose =
2
3
( 2
)
1 + 0 .5η nose l nose S m . f . .
The elliptical (ellipsoidal) shape of the nose part.
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4. x
r = 0 .5 d f 1 − ( x − 1) = 0 .5 d f x( 2 − x ) , x =
2
;
l nose
2
Wnose = l nose S m . f .
3
The particular case of an elliptical nose part is the hemisphere.
The shapes of rear parts are designed the same as nose ones.
THEME 10. FEATURES OF FLOW ABOUT BODYS OF
REVOLUTION
Let's consider a body of revolution, which is streamlined by undisturbed flow at
angle of attack α . The flow can be represented as a result of superposition of two flows
- longitudinal with speed V∞ cos α and transversal with speed V∞ sinα . At large angles
of attack the flow is determined by transversal flow, and at small angles of attack - by
longitudinal. The transversal flow is always subsonic at small angles of attack.
Conditionally we shall point out three flow modes about body of revolution.
1. Attached flow at small angles of attack ( α = 0 ...5 o ). At small angles of attack
the flow about cross-sections differs only by thickness and status of the boundary layer.
The laminar boundary layer of small thickness is in the nose. Further thickness of a
boundary layer gradually arises along the length of the body of revolution. Its character
varies, the boundary layer becomes turbulent.
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5. 2. At moderate angles of attack
( α ≤ 20 o ) the axis-symmetrical character of
flow is upset. The flow about fuselage takes
place with separation of the boundary layer at
lateral areas. The separated boundary layer is
turned into two vortex bundles (Fig. 10.1, a).
The location of the point of boundary layer
separation depends on the shape and aspect
ratio of the nose part, Mach numbers and
some other factors.
3. At large angles of attack ( α > 30 o )
the disturbance of a symmetry of a vortex
system takes place. Vortex bundles separate
from the surface of the body of revolution,
not having reached its rear part. At
Fig. 10.1. The flow scheme disturbance of flow rotational symmetry the
whole system of vortexes on the upper
surface of a body of revolution (Fig. 10.1, a)
is formed. It results into formation of sizable
transversal forces and moments of yaw.
At supersonic speed M ∞ > 1 and large
angles of attack the internal (hanging) shock
waves appear because of a large positive
Fig. 10.2. Flow near a cone: a 1-cone;
gradient of pressure on leeward fuselage side
2-head shock wave;
(Fig. 10.2).
3-vortexes; 4-hanging internal shock
At that, the flow structure is similar to
waves.
the structure of the track behind the cylinder
with circular cross section. The hanging shock waves prevent from the loss of symmetry
of the fuselage vortical system. Therefore, at large supersonic speeds transversal forces
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6. and the moments of a yaw conditioned by non-symmetry of the fuselage vortical are
absent.
THEME 11. DEFINITION OF THE AERODYNAMIC
CHARACTERISTICS OF A BODY OF REVOLUTION
USING PRESSURE DISTRIBUTION
It is possible to define aerodynamic forces effecting the body of revolution
knowing the law of pressure distribution along its surface. The pressure in the given
point of the surface is determined by two factors: parameters of incoming flow and
geometrical features of the streamlined body. The angle of attack α and number M ∞
exert the essential influence onto pressure distribution along outline of the body of
revolution. If we compare pressure distribution along wing airfoil with pressure
distribution along body of revolution, then it is possible to note the following: the
rarefaction on the body of revolution is much less than rarefaction on the wing. It takes
place due to spatial flow about the body of revolution. The flow spatiality enfeebles the
influence of flow compressibility onto character of flow about the body of revolution. In
subsonic flow for the body of revolution the factor of pressure is equal
C p incomp
Cp = , (11.1)
4 2
1 − M∞
C p incomp
while at flow about the wing C p = , i.e. the factor of pressure C p of the body
2
1− M∞
of revolution depends on number M ∞ less than C p for the wing.
Let's separately consider the pressure on a lateral surface of the body of
revolution and the pressure on the blunt base. Let's define the aerodynamic
characteristics of the body of revolution in body axes, and then by the transition
formulae we shall receive the aerodynamic characteristics in wind axes. Let's assume
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7. that an overpressure p − p∞ (Fig. 11.1) acts on an elementary site of the lateral surface
dS = r dl dϕ .
Fig. 11.1. A nose part of a body of revolution.
The function for elementary normal force dY from pressure p − p∞ effecting
onto an element of the lateral surface dS looks like this:
dY = − ( p − p∞ ) ds cos ϕ cos ϑ = − q∞ C p r dl dϕ cos ϕ cos ϑ , (11.1)
taking into account, that dl cos ϑ = dx , we shall receive
dY = − q∞C p cos ϕ r( x ) dx . (11.2)
Analogously, we shall find a function for elementary longitudinal force dX from
pressure p − p∞
dX = ( p − p∞ ) ds sinϑ = q∞ C p r dϕ dl sinϑ , (11.3)
dr
taking into account, that dl sinϑ = dr and dr = dx , we shall receive
dx
.
dX = q∞C p r r dx dϕ . (11.4)
Integrating expressions (11.2) and (11.4) by length of the body of revolution from
0 up to l f and by an arc of a circle from 0 up to 2π we shall receive the formulae for
normal and longitudinal force without the account of forces of friction
lf 2π
Y = − q∞ ∫ r( x) dx ∫ C p ( x , ϕ ) cos ϕ dϕ ; (11.5)
0 0
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8. lf 2π
.
X = q∞ ∫ r( x) r( x)dx ∫ C p ( x ,ϕ ) dϕ . (11.6)
0 0
Let's write down the expression for the elementary moment dM z from normal
and longitudinal force
dM z = − dY x + dX r cos ϕ = ( p − p∞ )ds cos ϕ ( x cos ϑ + r sin ϑ ) ;
⎛ .⎞ (11.7)
dM z = q ∞ C p cos ϕ ⎜ x + r r⎟ rdxdϕ
⎝ ⎠
also we shall define the function of the longitudinal moment from pressure as
lf 2π
⎛ . ⎞
M z = q∞
∫ ⎝ ⎠ ∫
r( x )⎜ x + r( x ) r( x )⎟ dx C p ( x ,ϕ ) cos ϕ dϕ . (11.8)
0 0
Let's pass in the formulae (11.5), (11.6) and (11.8) from forces to their factors. As
the characteristic area we shall accept the area of midsection of a body of revolution
S m . f . , and as characteristic length - length of a fuselage l f . We can write down
lf π
2
C ya ≈ C y = −
Sm. f . ∫ r( x) dx ∫ C p cos ϕ dϕ ; (11.9)
0 0
lf π
2 .
Cx =
Sm. f . ∫ r( x) r( x) dx ∫ C p dϕ , (11.10)
0 0
at α = 0 the factor of pressure C p in the last formula does not depend on the angle ϕ ,
and depends only on coordinate x and in this case
lf
2π .
C xa0 = C x0 =
Sm. f . ∫ C p r( x ) r( x )dx ; (11.11)
0
lf π
2 ⎛ .⎞
mz =
Sm. f . l f ∫ ⎝ ⎠ ∫
r⎜ x + r r⎟ dx C p cos ϕ dϕ , (11.12)
0 0
For thin bodies ( r << 1 ) it is possible not to take into account a moment from
&
longitudinal force:
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9. lf π
2
mz ≈
Sm. f . l f ∫ r x dx ∫ C p cos ϕ dϕ . (11.13)
0 0
Dimensionless coordinate of center of pressure location can be defined by
formula for an aerodynamic moment relatively to the fuselage nose M z = −Y xc . p . :
xc . p . mz m
x c . p. = =− ≈− z . (11.14)
lf Cy C yа
Let's define pressure forces which act onto the blunt base. If the blunt base is
located along the normal to an axis of the body of revolution, there is only longitudinal
force, which at small angles of attack practically does not vary on α . This force is
called the force of base drag. The force of base drag can be determined at α = 0 . In
this case, value of ( pbase − p∞ ) along the circle with radius r is a constant
( pbase − p∞ ) = const (Fig. 11.2).
Fig. 11.2. The blunt base of the fuselage.
The function for elementary longitudinal force dX from pressure ( pbase − p∞ )
on an element of the base surface dS = 2π r dr looks like this:
dX = dX base = − ( pbaase − p∞ )2π rdr = − q∞ C p base 2π rdr . (11.15)
Let's pass to the factor of force of base drag
rbase
2π
C x base = −
Sm. f . ∫ C p base r dr . (11.16)
0
Practically, pressure on the blunt base C p base ( r ) ≈ C p base = const and in this
case it is possible to consider, that
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10. C x base = − C p base S base . (11.17)
Thus, the aerodynamic characteristics of the body of revolution can be calculated,
if the pressure distribution along its surface and blunt base is known.
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