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7/28/2019 10 Lecture 41
1/15
AERODYNAMICS AND
FLIGHT MECHANICS
Thrust and Power Requirements
2
EXAMPLE: BEECHCRAFT QUEEN AIR The results we have developed so far for lift and drag for a finite wing may also be applied to a
complete airplane. In such relations:
CD is drag coefficient for complete airplane
CD,0 is parasitic drag coefficient, which contains not only profile drag of wing (cd) but also
friction and pressure drag of tail surfaces, fuselage, engine nacelles, landing gear and any
other components of airplane exposed to air flow
CL is total lift coefficient, including small contributions from horizontal tail and fuselage
Span efficiency for finite wing replaced with Oswald efficiency factor for entire airplane
Example: To see how this works, assume the aerodynamicists have provided all the information
needed about the complete airplane shown below
Beechcraft Queen Air Aircraft Data
W = 38,220 N
S = 27.3 m2
AR = 7.5
e (complete airplane) = 0.9
CD,0 (complete airplane) = 0.03
What thrust and power levels are required of
engines to cruise at 220 MPH at sea-level?How would these results change at 15,000 ft
3
OVERALL AIRPLANE DRAG
No longer concerned with aerodynamic details
Drag for complete airplane, not just wing
eAR
CCC LDD
2
0, eAR
CCC LdD
2
Wing or airfoil Entire Airplane
Landing Gear
Engine NacellesTail Surfaces
4
DRAG POLAR
CD,0 is parasite drag coefficient at zero lift (L=0) CD,i drag coefficient due to lift (induced drag)
Oswald efficiency factor, e, includes all effects from airplane
CD,0 and e are known aerodynamics quantities of airplane
iDDL
DD CCeAR
CCC ,0,
2
0,
Example ofDrag Polarfor complete airplane
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5
4 FORCES ACTING ON AIRPLANE Model airplane as rigid body with four natural forces acting on it
1. Lift, L
Acts perpendicular to flight path (always perpendicular to relative wind)
2. Drag, D
Acts parallel to flight path direction (parallel to incoming relative wind)
3. Propulsive Thrust, T
For most airplanes propulsive thrust acts in flight path direction
May be inclined with respect to flight path angle, T, usually small angle4. Weight, W
Always acts vertically toward center of earth
Inclined at angle, , with respect to lift direction
Apply Newtons Second Law (F=ma) to curvilinear flight path
Force balance in direction parallel to flight path
Force balance in direction perpendicular to flight path
6
GENERAL EQUATIONS OF MOTION (6.2)
cTlarperpendicu
Tparallel
r
V
mWTLF
dt
dVmWDTF
2
cossin
sincos
Apply Newtons 2nd Parallel to flight path:
Apply Newtons 2nd Perpendicular to flight path:
Free Body Diagram
7
LEVEL, UNACCELERATED FLIGHT
JSF is flying at constant speed (no accelerations)
Sum of forces = 0 in two perpendicular directions Entire weight of airplane is perfectly balanced by lift (L = W)
Engines produce just enough thrust to balance total drag at this airspeed (T = D)
For most conventional airplanesT is small enough such that cos(T) ~ 1
T D
L
W
8
LEVEL, UNACCELERATED FLIGHT
L
D
L
D
C
C
W
T
SCVWL
SCVDT
WL
DT
2
2
2
1
2
1
DL
W
CC
WT
D
L
R
TR is thrust required to fly at a given velocity in
level, unaccelerated flight
Notice that minimum TRis when airplane is at
maximum L/D
L/D is an important aero-perfo rmance quantity
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9
THRUST REQUIREMENT (6.3) TRfor airplane at given altitude varies with velocity
Thrust required curve: TRvs. V
10
PROCEDURE: THRUST REQUIREMENT1. Select a flight speed, V
2. Calculate CL
SV
WCL
2
2
1
eAR
CCC LDD
2
0,
D
L
R
C
C
WT
3. Calculate CD
4. Calculate CL/CD
5. Calculate TR
This is how much thrust engine
mustproduce to fly at selected V
Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil
Minimum TRwhen airplane
flying at (L/D)max
11
THRUST REQUIREMENT (6.3) Different points on TRcurve correspond to different angles of attack
eAR
CCSqSCqD
SCqSCVWL
LDD
LL
2
0,
2
2
1
At a:
Large q
Small CL and D large
At b:
Small q
Large CL (or CL2) and to support W
D large
12
THRUST REQUIRED VS. FLIGHT VELOCITY
eAR
CSqSCqT
CCSqSCqDT
LDR
iDDDR
2
0,
,0,
Zero-Lift TR(Parasitic Drag)
Lift-Induced TR(Induced Drag)
Zero-Lift TR~ V2
(Parasitic Drag)
Lift-Induced TR~ 1/V2
(Induced Drag)
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THRUST REQUIRED VS. FLIGHT VELOCITY
iDL
D
DR
RR
DR
CeAR
CC
eARSq
WSC
dq
dT
dq
dV
dV
dT
dq
dT
eARSq
WSCqT
,
2
0,
2
2
0,
2
0,
0
At point of minimum TR, dTR/dV=0
(or dTR/dq=0)
CD,0 = CD,i at minimum TR and maximum L/D
Zero-Lift Drag = Induced Drag at minimum TRand maximum L/D14
HOW FAST CAN YOU FLY? Maximum flight speed occurs when thrust available, TA=TR
Reduced throttle settings, TR< TA
Cannot physically achieve more thrust than TA which engine can provide
Intersection of TRcurve and maximum
TA defined maximum
flight speed of airplane
15
FURTHER IMPLICATIONS FOR DESIGN: VMAX Maximum velocity at a given altitude is important specification for new airplane
To design airplane for given Vmax, what are most important design parameters?
21
0,
0,
2
maxmax
max
2
0,
2
2
0,22
2
0,
2
0,
4
0
D
DAA
D
DD
L
LDD
C
eAR
C
W
T
S
W
S
W
W
T
V
eARS
WTqSCq
eARSq
WSCq
eARSq
WCSqT
Sq
WC
eAR
CCSqSCqTD
Steady, level flight: T = D
Steady, level flight: L = W
Substitute into drag equation
Turn this equation into a quadratic
equation (by multiplying by q)
and rearranging
Solve quadratic equation and setthrust, T, to maximum available
thrust, TA,max
16
FURTHER IMPLICATIONS FOR DESIGN: VMAX
TA,max does not appear alone, but only in ratio (TA/W)max
S does not appear alone, but only in ratio (W/S)
Vmax does not depend on thrust alone or weight alone, but rather on ratios
(TA/W)max: maximum thrust-to-weight ratio
W/S: wing loading
Vmax also depends on density (altitude), CD,0, eAR
Increase Vmax
by
Increase maximum thrust-to-weight ratio , (TA/W)max
Increasing wing loading, (W/S)
Decreasing zero-lift drag coefficient, CD,0
21
0,
0,
2
maxmax
max
4
D
DAA
C
eAR
C
W
T
S
W
S
W
W
T
V
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AIRPLANE POWER PLANTS
Two types of engines common in
aviation today
1. Reciprocating piston engine with
propeller
Average light-weight, general
aviation aircraft
Rated in terms ofPOWER
2. Jet (Turbojet, turbofan) engine
Large commercial transports
and military aircraft
Rated in terms ofTHRUST
18
THRUST VS. POWER Jets Engines (turbojets, turbofans for military and commercial applications) are
usually rate in Thrust
Thrust is a Force with units (N = kg m/s2)
For example, the PW4000-112 is rated at 98,000 lb of thrust
Piston-Driven Engines are usually rated in terms ofPower
Power is a precise term and can be expressed as:
Energy / time with units (kg m2/s2) / s = kg m2/s3 = Watts
Note that Energy is exp ressed in Joules = kg m2/s2
Force * Velocity with units (kg m/s2) * (m/s) = kg m2/s3 = Watts
Usually rated in terms of horsepower (1 hp = 550 ft lb/s = 746 W)
Example:
Airplane is level, unaccelerated flight at a given altitude with speed V
Power Required, PR=TR*V
[W] = [N] * [m/s]
19
POWER AVAILABLE (6.6)
Jet EnginePropeller Drive Engine
20
POWER AVAILABLE (6.6)
Jet EnginePropeller Drive Engine
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POWER REQUIRED (6.5)
PRvs. V qualitatively
(Resembles TRvs. V)
22
POWER REQUIRED (6.5)
D
LL
DR
L
D
LR
L
L
D
L
RR
CCSC
CWP
SC
W
CC
WP
SC
WVSCVWL
V
CC
WVTP
233
23
2
12
2
2
2
1
PRvaries inversely as CL3/2/CD
Recall: TRvaries inversely as CL/CD
23
POWER REQUIRED (6.5)
eAR
CSVqVSCqP
VCCSqVSCqDVVTP
LDR
iDDDRR
2
0,
,0,
Zero-Lift PR Lift-Induced PR
Zero-Lift PR~ V3
Lift-Induced PR~ 1/V
24
POWER REQUIRED
03
1
2
3
2
12
1
,0,
2
2
0,
3
iDDR
DR
CCSVdV
dP
eARSV
WSCVP
iDD CC ,0,3
1
At point of minimum PR, dPR/dV=0
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POWER REQUIRED V
for minimum PRis less than V for minimum TR
iDD CC ,0,
iDD CC ,0,3
1
26
WHY DO WE CARE ABOUT THIS? We will show that for a piston-engine propeller combination
To fly longest distance (maximum range) we fly airplane at speed
corresponding to maximum L/D
To stay aloft longest (maximum endurance) we fly the airplane at minimum
PRor fly at a velocity where CL3/2/CD is a maximum
Power will also provide information on maximum rate of climb and altitude
27
POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)
Propeller Drive
Engine
1 hp = 550 ft lb/s = 746 W
PR
PA
28
POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)
Jet Engine
PR
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ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)
Recall PR= f()Subscript 0 denotes seal-level conditions
21
00,,
21
00
RALTR
ALT
PP
VV
30
ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)Propeller-Driven Airplane
Vmax,ALT < Vmax,sea-level
31
RATE OF CLIMB (6.8)
Boeing 777: Lift-Off Speed ~ 180 MPH
How fast can it climb to a cruising altitude of 30,000 ft?
32
RATE OF CLIMB (6.8)
cos
sin
WLWDT
Governing Equations:
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RATE OF CLIMB (6.8)
sin/
sin
sin
sin
VCR
VW
DVTV
WVDVTV
WDT
Vertical velocity
Rate of Climb:
TV
is power available
DV
is level-flight power required (for small neglect W)TV
- DV
is excess power
34
RATE OF CLIMB (6.8)
Jet EnginePropeller Drive Engine
Maximum R/C Occurs when Maximum Excess Power
35
EXAMPLE: F-15 K Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-
seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact
at high speed (kinetic energy weapon)
F-15 can bring ALMV under the ground track of its target, as opposed to a ground-
based system, which must wait for a target satellite to overfly i ts launch site.
36
GLIDING FLIGHT (6.9)
DL
L
D
1tancos
sin
cos
sin
0
WL
WD
T
To maximize range, smallest
occurs at (L/D)max
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EXAMPLE: HIGH ASPECT RATIO GLIDER
To maximize range, smallest occurs at (L/D)maxA modern sailplane may have a glide ratio as high as 60:1
So = tan-1(1/60) ~ 1
RANGE AND ENDURANCE
How far can we fly?
How long can we stay aloft?
How do answers vary for propeller-driven vs. jet-engine?
39
RANGE AND ENDURANCE
Range: Total distance (measured with respect to the ground) traversed by airplane
on a single tank of fuel
Endurance: Total time that airplane stays in air on a single tank of fuel
1. Parameters to maximize range are different from those that maximize endurance
2. Parameters are different forpropeller-poweredandjet-powered aircraft
Fuel Consumption Definitions
Propeller-Powered:
Specific Fuel Consumption (SFC)
Definition: Weight of fuel consumed per unit powerper unit time
Jet-Powered:
Thrust Specific Fuel Consumption (TSFC)
Definition: Weight of fuel consumed per unit thrustper unit time
40
PROPELLER-DRIVEN: RANGE AND ENDURANCE
SFC: Weight of fuel consumed per unit power per unit time
hourHPfueloflb
SFC
ENDURANCE: To stay in air for longest amount of time, use minimum
number of pounds of fuel per hour
HPSFC
hour
fueloflb
Minimum lb of fuel per hour obtained with minimum HP
Maximum endurance for a propeller-driven airplane occurs whenairplane is flying at minimum power required
Maximum endurance for a propeller-driven airplane occurs when
airplane is flying at a velocity such that CL3/2/CD is a maximized
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PROPELLER-DRIVEN: RANGE AND ENDURANCE SFC: Weight of fuel consumed per unit power per unit time
hourHPfueloflb
SFC
RANGE: To cover longest distance use minimum pounds of fuel per mile
V
HPSFC
mile
fueloflb
Minimum lb of fuel per hour obtained with minimum HP/V
Maximum range for a propeller-driven airplane occurs when airplane
is flying at a velocity such that CL/CD is a maximum
42
PROPELLER-DRIVEN: RANGE BREGUET FORMULA
To maximize range:
Largest propeller effici ency,
Lowest possible SFC
Highest ratio of Winitial to Wfinal, which is obtained with the largest fuel weight
Fly at maximum L/D
final
initial
D
L
W
W
C
C
SFCR ln
43
PROPELLER-DRIVEN: RANGE BREGUET FORMULA
final
initial
D
L
W
W
C
C
SFCR ln
PropulsionAerodynamics
Structures and Materials
44
PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA
To maximize endurance:
Largest propeller effici ency, Lowest possible SFC
Largest fuel weight
Fly at maximum CL3/2/CD
Flight at sea level
2
12
12
12
3
2 initialfinalD
L WWSC
C
SFCE
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JET-POWERED: RANGE AND ENDURANCE TSFC: Weight of fuel consumed per thrust per unit time
hourthrustoflbfueloflb
TSFC
ENDURANCE: To stay in air for longest amount of time, use minimum
number of pounds of fuel per hour
ThrustTSFC
hour
fueloflb
Minimum lb of fuel per hour obtained with minimum thrust
Maximum endurance for a jet-powered airplane occurs when
airplane is flying at minimum thrust required
Maximum endurance for a jet-powered airplane occurs when
airplane is flying at a velocity such that CL/CD is a maximum
46
JET-POWERED: RANGE AND ENDURANCE TSFC: Weight of fuel consumed per unit power per unit time
hourthrustoflbfueloflb
TSFC
RANGE: To cover longest distance use minimum pounds of fuel per mile
V
ThrustSFC
mile
fueloflb
Minimum lb of fuel per hour obtained with minimum Thrust/V
Maximum range for a jet-powered airplane occurs when airplane isflying at a velocity such that CL1/2/CD is a maximum
D
L
D
L
R
CC
CSC
WS
V
T
21
12
2
1
47
JET-POWERED: RANGE BREGUET FORMULA
To maximize range:
Minimum TSFC
Maximum fuel weight
Flight at maximum CL1/2/CD
Fly at high altitudes
21
212
1
122 finalinitial
D
L WWC
C
TSFCSR
48
JET-POWERED: ENDURACE BREGUET FORMULA
To maximize endurance:
Minimum TSFC
Maximum fuel weight
Flight at maximum L/D
final
initial
D
L
W
W
C
C
TSFCE ln
1
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SUMMARY: ENDURANCE AND RANGE Maximum Endurance
Propeller-Driven
Maximum endurance for a propeller-driven airplane occurs when airplane is
flying at minimum power required
Maximum endurance for a propeller-driven airplane occurs when airplane is
flying at a velocity such that CL3/2/CD is a maximized
Jet Engine-Driven
Maximum endurance for a jet-powered airplane occurs when airplane is
flying at minimum thrust required
Maximum endurance for a jet-powered airplane occurs when airplane is
flying at a velocity such that CL/CD is a maximum
Maximum Range
Propeller-Driven
Maximum range for a propeller-driven airplane occurs when airplane is flying
at a velocity such that CL/CD is a maximum
Jet Engine-Driven
Maximum range for a jet-powered airplane occurs when airplane is flying at avelocity such that CL1/2/CD is a maximum
EXAMPLES OF DYNAMIC
PERFORMANCE
Take-Off Distance
Turning Flight
51
TAKE-OFF AND LANDING ANALYSES (6.15)
F
mVs
Vdtds
t
m
FV
dtm
FdV
dt
dVmmaF
2
2
dtdV
mLWDTRDTF r
Rolling resistance
r= 0.02
s: lift-off distance
52
NUMERICAL SOLUTION FOR TAKE-OFF
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USEFUL APPROXIMATION (T >> D, R)
Lift-off distance very sensitive to weight, varies as W2
Depends on ambient density
Lift-off distance may be decreased:
Increasing wing area, S
Increasing CL,max
Increasing thrust, T
TSCg
Ws
L
OL
max,
2
..
44.1
sL.O.: lift-off distance
54
EXAMPLES OF GROUND EFFECT
55
TURNING FLIGHT
V
ng
R
V
dt
d
ng
VR
R
VmF
nWF
W
Ln
WLF
WL
r
r
r
1
1
1
cos
2
2
2
2
2
22
Load Factor
R: Turn Radius
: Turn Rate56
EXAMPLE: PULL-UP MANEUVER
V
ng
ng
VR
R
VmF
nWWLF
r
r
1
1
1
2
2
R: Turn Radius
: Turn Rate
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V-n DIAGRAMS
SW
CVn
W
SCV
W
Ln
L
L
max,2
max
2
2
1
2
1
58
STRUCTURAL LIMITS