chapter 6
DESCRIPTION
Chapter 6. Elements of Airplane Performance. Un-accelerated level flight. Simple Mission Profile for an Airplane 1 Switch on + Worming + Taxi. (Cruising flight). 4. 3. Descent. Altitude. Climb. Landing. Takeoff. 5. 6. 1. 2. Simple mission profile. Airplane Performance. - PowerPoint PPT PresentationTRANSCRIPT
1
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
2
Altitude
1 2
3 4
5 6
Takeoff
Climb
Un-accelerated level flight
(Cruising flight)Descent
Landing
Simple mission profile
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
3
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
4
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(
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
5
• 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
7
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
8
9
■ 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
10
►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.
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
11
►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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
12
Skin-friction
Form drag
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
13
►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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
14
Formation of wing tip vortices
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
15
Complete wing-vortex system
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
16 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
17
The origin of downwash
The origin of induced drag
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
18
►Wave Drag
This is the drag associated with the formation of shock waves in high-speed flight .
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
19
■ 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
20
● 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
21
■ 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.
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
22
Wing fillets
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
23
● 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
24
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
25
Schematic of the drag polar
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
Airplane Equations of Motion
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
26
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
27
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
28
• Apply Newton’s 2nd low of motion:
In the direction of the flight path
Perpendicular to the flight path
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
29
I-Steady Level Flight Performance
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
30
Un-accelerated (steady) Level Flight Performance (Cruising Flight)
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
31
• Thrust Required for Level Un-accelerated Flight
)Drag(
Thrust required TR for a given airplane to fly at V∞ is given as : TR = D
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
32
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
33
● 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
Aerospace Dept. Cairo University34
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
35
V∞CLCDCL/CDW/[CL/CD ]
6- Tabulate the results
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
36
)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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
37
Required equation
• Parasite and induced drag
•
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
38
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
Aerospace Dept. Cairo University39
• 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
40
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
41
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
42
• 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∞
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
43
• Solving for q∞
• By replacing q∞ = ½ρ∞V2∞ we get
• Prof. Galal Bahgat Salem
Aerospace Dept. Cairo University44
• 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
45
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
46
●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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
47
• 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∞
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
48
V∞1 V∞2
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
49
Thrust Available TA
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
50S
onic
spe
ed
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
51
Thrust Available TA and Maximum Velocity Vmax
• For turbojet at subsonic speeds, )V∞)max can be obtained from:
• Just substitute (TA)max TR
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
52
• Power Required PR
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
53
• Variation of PR with V∞
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
54
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
55
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
56
• Power Available PA
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
57
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
58
• 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
Aerospace Dept. Cairo University59
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
60
• 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
Aerospace Dept. Cairo University61
• Finally,
Vmin= Vstall = [2W/ ρ∞ S (CL)max ]½
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
62
CL –α curve for an airplane
II-Steady Climb Performance
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
63
• Steady Climb
• Assumptions:
1- dV∞/dt = 0
2- Climb along straight line, V2∞/ r = 0
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
64
• 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
65
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
66
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
67
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
68
,for T = constant
[Turbojet]
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
69
sin
Turbojet aircraft
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
70
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
71
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
Aerospace Dept. Cairo University72
• 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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
73
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
74
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
Aerospace Dept. Cairo University75
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
76
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
77
• GLIDING (UNPOWERED) FLIGHT
• Assumptions
• 1- Steady gliding
• 2- Along straight line
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
78
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
79
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
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
80
• For Rmax ϴmin
• Where:
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
81
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
82
CEILINGS
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
83
max
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
84
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
Aerospace Dept. Cairo University85
max
h
(R/C)max
b =slope
0
III-Range and Endurance
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
86
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
87
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
88
W=Instantaneous weight
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
89
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
90
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
91
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
92
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
93