Chapter 6

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Chapter 6

Elements of Airplane Performance

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Simple Mission Profile for an Airplane

1 Switch on + Worming + Taxi


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Altitude

1 2

3 4

5 6

Takeoff

Climb

Un-accelerated level flight

(Cruising flight)Descent

Landing

Simple mission profile


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Airplane Performance

Equations of Motions

Static Performance(Zero acceleration

Dynamic Performance( Finite acceleration)

Thrust requiredThrust available Maximum

velocity

Power requiredPower available

Maximum velocity

Rate of climb

Gliding flight

Takeoff

Landing


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Time to climb

Maximum altitude

Service ceiling

Absolute ceiling

Range and endurance

Road map for Chapter 6

• 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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• 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:Prof. Galal Bahgat Salem

Aerospace Dept. Cairo University6

• 1- Lift force L

• 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


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Components of the resultant aerodynamic force R

• 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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■ Drag Types [ Kinds of Drag ]

Total Drag

Skin Friction Drag Pressure Drag

Form Drag )Drag Due to Flow separation( Induced Drag Wave Drag

Note : Profile Drag = Skin Friction Drag + Form Drag


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►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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►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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Skin-friction

Form drag


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►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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Formation of wing tip vortices


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Complete wing-vortex system


16 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

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The origin of downwash

The origin of induced drag


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►Wave Drag

This is the drag associated with the formation of shock waves in high-speed flight .


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■ 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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● 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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■ 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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Wing fillets


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● 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.

Prof. Galal Bahgat Salem

Aerospace Dept. Cairo University

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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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Schematic of the drag polar


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

Airplane Equations of Motion


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• Apply Newton’s 2nd low of motion:

In the direction of the flight path

Perpendicular to the flight path


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I-Steady Level Flight Performance


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Un-accelerated (steady) Level Flight Performance (Cruising Flight)


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• 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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● TR as a function of V∞ can be obtained by tow methods or approaches graphical/analytical■Graphical Approach

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∞Prof. Galal Bahgat Salem



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V∞CLCDCL/CDW/[CL/CD ]

6- Tabulate the results


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)TR(min occurs at )CL/CD(max

• ■ Analytical Approach

• It is required to obtain an equation for TR as a function of V∞

• TR = D


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Required equation

• Parasite and induced drag

•


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TR/D

V∞

CDo=CDi

• 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 [CL])TR(min = √CDo/K

• And [CDo])TR(min = 2CDoProf. Galal Bahgat Salem


• 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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L/D as function of angle of attack α L/D as function of velocity V∞

• L/D as function of V∞ :

• Since,

• But L=W

• Then

• or


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• 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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• Solving for q∞

• By replacing q∞ = ½ρ∞V2∞ we get

• Prof. Galal Bahgat Salem


• 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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●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

• 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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• 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∞


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V∞1 V∞2


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Thrust Available TA


50S

onic

spe

ed


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Thrust Available TA and Maximum Velocity Vmax

• For turbojet at subsonic speeds, )V∞)max can be obtained from:

• Just substitute (TA)max TR


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• Power Required PR


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• Variation of PR with V∞


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• Power Available PA


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• Power Available PA and Maximum Velocity Vmax

• The high speed

intersection

between PR and

)PA(max gives

Vmax

Vmax decreases

with altitudeProf. Galal Bahgat Salem



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• 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(maxProf. Galal Bahgat Salem


• Finally,

Vmin= Vstall = [2W/ ρ∞ S (CL)max ]½


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CL –α curve for an airplane

II-Steady Climb Performance


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• Steady Climb

• Assumptions:

1- dV∞/dt = 0

2- Climb along straight line, V2∞/ r = 0


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• 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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,for T = constant

[Turbojet]


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sin

Turbojet aircraft


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Turbojet aircraft

• 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+KC2

L)/CL3/2]

• For turbojet T = const

• For (R/C)max d(R/C)/dCL =0

• So, we getProf. Galal Bahgat Salem


• So, we get:

• And, V)R/C(max=[2W/ ρ∞ S CL(R/C)max ]½

• ( CD) (R/C)max = CDo +K C2

L(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


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CL(R/C)max = [ -(T/W) + √ (T/W)2 + 12 K CD0 ] / 2K

• For Propeller Aircraft

• For propeller aircraft )R/C(max occurs at

• )PR(min


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Propeller aircraft

• Analytical Solution for )R/C(max

• V)R/C(max= V)CL3/2/CD(max

• ( CD) (R/C)max = CDo +K C2

L(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 Prof. Galal Bahgat Salem



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• GLIDING (UNPOWERED) FLIGHT

• Assumptions

• 1- Steady gliding

• 2- Along straight line


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If PR ˃ PA the airplane will descend In the ultimate situation when T = 0, the airplane will be in gliding

• 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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• For Rmax ϴmin

• Where:


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CEILINGS


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max


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h(

R/C

)-1

• Minimum Time to Climb

• tmin =

• Assuming linear variation of )R/C(max with altitude h, then

• )R/C(max = a + b h

• a = )R/C(max at h = 0

• =1/b[ln)a+bh2(-lna]Prof. Galal Bahgat Salem


max

h

(R/C)max

b =slope

0

III-Range and Endurance


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W=Instantaneous weight


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Chapter 6

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