Chapter 6
description
Transcript of Chapter 6
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Chapter 6
Elements of Airplane Performance
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
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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
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• 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
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• 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
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• 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
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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.
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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
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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
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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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Un-accelerated 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
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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
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• ■ Analytical Approach
• It is required to obtain an equation for TR as a function of V∞
• TR = D
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Required equation
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• Parasite and induced drag
•
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TR/D
V∞
CDo=CDi
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• 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
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• 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∞
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• 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
Aerospace Dept. Cairo University43
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• 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
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• 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
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Thrust Available TA
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49S
onic
spe
ed
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Thrust Available TA and Maximum Velocity Vmax
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• Power Required PR
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• Variation of PR with V∞
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PR
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• Power Available PA
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