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Chapter 6
Elements of Airplane
Performance
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Simple Mission Profile for an Airplane
Un-accelerated level flight
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(Cruising flight)
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Descent
Altitude
Climb
Landing
Takeoff
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1
2
1 Switch on + Worming + Taxi
Simple mission profile
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
6
Airplane Performance
Equations of Motions
Static Performance
(Zero acceleration
Dynamic Performance
(Finite acceleration)
Thrust required
Thrust available
Maximum
velocity
Takeoff
Power required
Power available
Landing
Maximum velocity
Rate of climb
Gliding flight
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Time to climb
Maximum altitude
Service ceiling
Absolute ceiling
Range and endurance
Road map for Chapter 6
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Study the airplane performance requires
the derivation of the airplane equations of
motion
• As we know the airplane is a rigid body
has six degrees of freedom
• But in case of airplane performance we
are deal with the calculation of velocities (
e.g.Vmax,Vmin..etc),distances (e.g. range,
takeoff distance, landing distance, ceilings
…etc), times (e.g. endurance, time to
climb,…etc), angles (e.g.climb angle…etc)
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• So, the rotation of the airplane about its
axes during flight in case of performance
study is not necessary.
• Therefore, we can assume that the
airplane is a point mass concentrated at its
c.g.
• Also, the derivation of the airplane’s
equations of motion requires the
knowledge of the forces acting on the
airplane
• The forces acting on an airplane are:
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
•
•
•
•
•
•
•
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Components of the resultant
1- Lift force L
aerodynamic force R
2- Drag force D
3- Thrust force T
Propulsive force
4- Weight W
Gravity force
Thrust T and weight W will be given
But what about L and D?
We are in the position that we can’t
calculate L and D with our limited
knowledge of the airplane aerodynamics
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• So, the relation between L and D will be
given in the form of the so called drag
polar
• But before write down the equation of the
airplane drag polar it is necessary to know
the airplane drag types
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
■ Drag Types [ Kinds of Drag ]
Total Drag
Skin Friction Drag
Pressure Drag
Form Drag (Drag Due to Flow separation)
Induced Drag
Note : Profile Drag = Skin Friction Drag + Form Drag
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Wave Drag
►Skin friction drag
This is the drag due to shear stress at the surface.
►Pressure drag
This is the drag that is generated by the resolved
components of the forces due to pressure acting normal
to the surface at all points and consists of [ form drag +
induced drag + wave drag ].
►Form drag
This can be defined as the difference between profile
drag and the skin-friction drag or the drag due to flow
separation.
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
►Profile Drag
● Profile drag is the sum of skin-friction and form drags.
● It is called profile drag because both skin-friction and
form drag [ or drag due to flow separation ] are
ramifications of the shape and size of the body, the
“profile” of the body.
● It is the total drag on an aerodynamic shape due to
viscous effects
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Skin-friction
Form drag
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
►Induced drag ( or vortex drag )
This is the drag generated due to the wing tip vortices ,
depends on lift, does not depend on viscous effects , and
can be estimated by assuming inviscid flow.
Finite wing flow tendencies
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Formation of wing tip vortices
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Complete wing-vortex system
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
The origin of downwash
The origin of induced drag
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
►Wave Drag
This is the drag associated with the formation of shock
waves in high-speed flight .
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
■ Total Drag of Airplane
● An airplane is composed of many components and each
will contribute to the total drag of its own.
● Possible airplane components drag include :
1. Drag of wing, wing flaps = Dw
2. Drag of fuselage = Df
3. Drag of tail surfaces = Dt
4. Drag of nacelles = Dn
5. Drag of engines = De
6. Drag of landing gear = Dlg
7. Drag of wing fuel tanks and external stores = Dwt
8. Drag of miscellaneous parts = Dms
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
● Total drag of an airplane is not simply the sum of the
drag of the components.
● This is because when the components are combined
into a complete airplane, one component can affect the
flow field, and hence, the drag of another.
● these effects are called interference effects, and the
change in the sum of the component drags is called
interference drag.
● Thus,
(Drag)1+2 = (Drag)1 + (Drag)2 + (Drag)interference
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
■ Buid-up Technique of Airplae Drag D
● Using the build-up technique, the airplane total drag D is
expressed as:
D = Dw + Df + Dt + Dn +De + Dlg + Dwt + Dms + Dinterference
► Interference Drag
● An additional pressure drag caused by the mutual
interaction of the flow fields around each component of
the airplane.
● Interference drag can be minimized by proper fairing and
filleting which induces smooth mixing of air past the
components.
● The Figure shows an airplane with large degree of wing
filleting.
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Wing fillets
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
● No theoretical method can predict interference drag,
thus, it is obtained from wind-tunnel or flight-test
measurements.
● For rough drag calculations a figure of 5% to 10% can
be attributed to interference drag on a total drag, i.e,
Dinterference ≈ [ 5% – 10% ] of components total drag
■ The Airplane Drag Polar
● For every airplane, there is a relation between CD and
CL that can be expressed as an equation or plotted on a
graph.
● The equation and the graph are called the drag polar.
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
For the complete airplane, the drag coefficient is written as
CD = CDo + K CL2
This equation is the drag polar for an airplane.
Where: CDo drag coefficient at zero lift ( or
parasite drag coefficient )
K CL2 = drag coefficient due to lift ( or
induced drag coefficient CDi )
K = 1/π e AR
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
e
Oswald efficiency factor = 0.75 – 0.9
(sometimes known as the airplane efficiency factor)
AR wing aspect ratio = b2/S ,
b wing span
and
S wing planform area
Schematic of the drag polar
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Airplane Equations of Motion
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Apply Newton’s 2nd low of motion:
In the direction of the flight path
Perpendicular to the flight path
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
I-Steady Level Flight
Performance
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Un-accelerated (steady) Level Flight
Performance (Cruising Flight)
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Thrust Required for Level Un-accelerated Flight
(Drag)
Thrust required TR for a given airplane to fly at
V∞ is given as :
TR = D
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
● TR as a function of V∞ can be obtained by tow methods
or approaches graphical/analytical
■Graphical Approach
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
1- Choose a value of V∞
2 - For the chosen V∞ calculate CL
L = W = ½ρ∞ V2∞S CL
CL = 2W/ ρ∞ V2∞S
3- Calculate CD from the drag polar
CD = CDo + K CL2
4- Calculate drag, hence TR, from
TR = D = ½ρ∞ V2∞S CD
5- Repeat for different values of V∞
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
6- Tabulate the results
V∞
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CL
CD
CL/CD
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
W/[CL/CD ]
(TR)min occurs at (CL/CD)max
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• ■ Analytical Approach
• It is required to obtain an equation for TR as a
function of V∞
•
TR = D
Required equation
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Parasite and induced drag
•
TR/D
CDo=CDi
V∞
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Note that TR is minimum at the point of
intersection of the parasite drag Do and
induced drag Di
• Thus
Do = Di at [TR]min
• or
CDo = CDi
•
= KCL2
• Then
• And
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[CL](TR)min = √CDo/K
[CDo](TR)min = 2CDo
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Finally, (L/D)max = (CL/CD)max
•
= √CDo/K /2CDo
•
•
(CL/CD)max = 1/√4KCDo
• Also,[V∞](TR)min =[V∞] (CL/CD)max is obtained
from:
W=L
•
= ½ρ∞[V]2(TR)minS [CL](TR)min
• Thus:
•
[V] (TR)min= {2(W/S)(√K/CDo)/ρ∞}½
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
L/D as function of angle of attack α
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L/D as function of velocity V∞
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• L/D as function of V∞ :
• Since,
• But
• Then
L=W
• or
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Flight Velocity for a Given TR
•
TR = D
• In terms of q∞ = ½ρ∞V2∞ we obtain
• Multiplying by q∞ and rearranging, we have
• This is quadratic equation in q∞
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Solving for q∞
• By replacing q∞ = ½ρ∞V2∞ we get
•
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Let
• Where (TR/W) is the thrust-to-weight-ratio
•
(W/S) is the wing loading
• The final expression for velocity is
• This equation has two roots as shown in
figure corresponding to point 1 an 2
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
●When the discriminant equals zero ,then only
one solution for V∞ is obtained
●This corresponds to point 3 in the figure,
namely at (TR)min
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Or,
(TR/W)min = √4CDoK
• Then the velocity V3 =V(TR)min is
• Substituting for (TR/W)min = √4CDoK we have
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Effect of Altitude on (TR)min
• We know that
•
(TR/W)min = √4CDoK
• This means that (TR)min is independent of altitude as
show in Figure
• (TR)min occurs at higher V∞
V∞1
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V∞2
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Thrust Available TA
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Sonic speed
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Thrust Available TA and Maximum Velocity Vmax
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• For turbojet at subsonic speeds, (V∞)max
can be obtained from:
• Just substitute (TA)max
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
TR
• Power Required PR
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Variation of PR with V∞
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
CD= 4C
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Power Available PA
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Power Available PA and Maximum
Velocity Vmax
• The high speed
intersection
between PR and
(PA)max gives
Vmax
 Vmax decreases
with altitude
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Minimum Velocity: Stall Velocity
• Airplane minimum velocity Vmin is usually
dictated by its stall velocity
• Stall velocity Vstall is the velocity
corresponds to the maximum lift coefficient
(CL)max of the airplane
• Hence,
Vmin = Vstall
• But,
L = W = ½ρ∞ V2∞S CL
V∞ = (2W/ ρ∞ S CL )½
• Substitute for CL
(CL)max
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Finally,
Vmin= Vstall = [2W/ ρ∞ S (CL)max ]½
CL –α curve for an airplane
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
II-Steady Climb
Performance
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Steady Climb
• Assumptions:
1- dV∞/dt = 0
2- Climb along straight line, V2∞/ r = 0
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• The equations of motion in this case
become:
•
T cos ε – D – W sin ϴ = 0
•
L + T sin ε – W cos ϴ = 0
• Assuming , ε = 0
• Then,
T – D – W sin ϴ = 0
•
L– W cos ϴ = 0
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
[Turbojet]
,for T = constant
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
sin
Turbojet aircraft
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
Turbojet aircraft
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
•
•
•
•
•
•
•
•
•
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Analytical Solution for (R/C)max
R/C = V∞ sin ϴ
= (2W/ ρ∞ S CL )½ [ T/W- D/L]
= (2W/ ρ∞ S CL )½ [T/W-CD/CL]
= (2W/ ρ∞ S CL )½ [T/W-CDo +KCL2/CL]
=(2W/ ρ∞ S )½ [CL-½(T/W)-(CDo+KC2L)/CL3/2]
For turbojet T = const
For (R/C)max
d(R/C)/dCL =0
So, we get
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• So, we get:
CL(R/C)max = [ -(T/W) + √ (T/W)2 + 12 K CD0 ] / 2K
•
•
•
•
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And, V(R/C)max=[2W/ ρ∞ S CL(R/C)max ]½
( CD) (R/C)max = CDo +K C2L(R/C)max
(Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max
(R/C)max = V(R/C)max (sin ϴ) (R/C)max
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• For Propeller Aircraft
• For propeller aircraft (R/C)max occurs at
• (PR)min
Propeller aircraft
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Analytical Solution for (R/C)max
• V(R/C)max= V(CL3/2/CD)max
• ( CD) (R/C)max = CDo +K C2L(R/C)max
•
= CDo +K [√3CDo/K ]2 = 4CDo
• (Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max
• (R/C)max = V(R/C)max (sin ϴ) (R/C)max
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
•
•
•
•
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GLIDING (UNPOWERED) FLIGHT
Assumptions
1- Steady gliding
2- Along straight line
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
 If PR ˃ PA the airplane will descend
 In the ultimate situation when T = 0, the
airplane will be in gliding
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Maximum Range
• For an airplane at a given altitude h, the
max. horizontal distance covered over the
ground is denoted max. range R
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• For Rmax
• Where:
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ϴmin
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
CEILINGS
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Aerospace Dept. Cairo University
max
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
(R/C)-1
h
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
• Minimum Time to Climb
• tmin =
max
• Assuming linear variation of (R/C)max with
altitude h, then
h
b =slope
•
(R/C)max = a + b h
• a = (R/C)max at h = 0
0
•
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=1/b[ln(a+bh2)-lna]
Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
(R/C)max
III-Range and Endurance
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Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
W=Instantaneous weight
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Aerospace Dept. Cairo University
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Aerospace Dept. Cairo University
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Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University
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Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University