13 fixed wing fighter aircraft- flight performance - i
TRANSCRIPT
Fixed Wing Fighter AircraftFlight Performance
Part I
SOLO HERMELIN
Updated: 04.12.12 28.02.15
1
http://www.solohermelin.com
Table of Content
SOLO Fixed Wing Aircraft Flight Performance
2
Introduction to Fixed Wing Aircraft Performance
Earth Atmosphere
Aerodynamics
Mach Number
Shock & Expansion Waves
Reynolds Number and Boundary Layer
Knudsen Number
Flight Instruments
Aerodynamic Forces
Aerodynamic Drag
Lift and Drag Forces
Wing Parameters
Specific Stabilizer/Tail Configurations
Table of Content (continue – 1)
SOLO
3
Specific Energy
Aircraft Propulsion Systems
Aircraft Propellers
Aircraft Turbo Engines
Afterburner
Thrust Reversal Operation
Aircraft Propulsion Summary
Vertical Take off and Landing - VTOL
Engine Control System
Aircraft Flight Control
Aircraft Equations of Motion
Aerodynamic Forces (Vectorial)
Three Degrees of Freedom Model in Earth Atmosphere
Comparison of Fighter Aircraft Propulsion Systems
Fixed Wing Fighter Aircraft Flight Performance
Table of Content (continue – 2)
SOLO Fixed Wing Fighter Aircraft Flight Performance
4
Parameters defining Aircraft Performance
Takeoff (no VSTOL capabilities)
Landing (no VSTOL capabilities)
Climbing Aircraft Performance
Gliding Flight
Level Flight
Steady Climb (V, γ = constant)
Optimum Climbing Trajectories using Energy State Approximation (ESA)Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)Maximum Range during Glide using Energy State Approximation (ESA)
Aircraft Turn Performance
Maneuvering Envelope, V – n Diagram
Fixed Wing Part
II
Table of Content (continue – 3)
SOLO Fixed Wing Fighter Aircraft Flight Performance
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Air-to-Air Combat
Energy–Maneuverability Theory
Supermaneuverability
Constraint Analysis
Aircraft Combat Performance Comparison
References
Fixed Wing
Part
II
SOLO
This Presentation is about Fixed Wing Aircraft Flight Performance.
The Fixed Wing Aircraft are•Commercial/Transport Aircraft (Passenger and/or Cargo)•Fighter Aircraft
Fixed Wing Fighter Aircraft Flight Performance
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7
Percent composition of dry atmosphere, by volume
ppmv: parts per million by volume
Gas Volume
Nitrogen (N2) 78.084%
Oxygen (O2) 20.946%
Argon (Ar) 0.9340%
Carbon dioxide (CO2) 365 ppmv
Neon (Ne) 18.18 ppmv
Helium (He) 5.24 ppmv
Methane (CH4) 1.745 ppmv
Krypton (Kr) 1.14 ppmv
Hydrogen (H2) 0.55 ppmv
Not included in above dry atmosphere:
Water vapor (highly variable) typically 1%
Gas Volume
nitrous oxide 0.5 ppmv
xenon 0.09 ppmv
ozone 0.0 to 0.07 ppmv (0.0 to 0.02 ppmv in winter)
nitrogen dioxide 0.02 ppmv
iodine 0.01 ppmv
carbon monoxide trace
ammonia trace
•The mean molecular mass of air is 28.97 g/mol.
Minor components of air not listed above include:
Composition of Earth's atmosphere. The lower pie represents the trace gases which together compose 0.039% of the atmosphere. Values normalized for illustration. The numbers are from a variety of years (mainly 1987, with CO2 and methane from 2009) and do not represent any single source
Earth AtmosphereSOLO
8
Earth AtmosphereSOLO
The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p.
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9
The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3.
v
mv ∆
∆=→∆ 0
limρ
The Temperature, T, with units in degrees Kelvin ( K). Is a measure of the average kinetic energy of gas particles.
The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area.
It has units of N/m2. Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg)
S
fp n
S ∆∆=
→∆ 0lim
kPamNbar 100/101 25 ==
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 ===
The Atmospheric Pressure at Sea Level is:
Earth Atmosphere
10
Physical Foundations of Atmospheric Model
The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude.
Atmospheric Equilibrium (Barometric) Equation
In figure we see an atmospheric element under equilibrium under pressure and gravitational forces
( )[ ] 0=⋅+−+⋅⋅⋅− APdPPHdAg gρ
or ( ) gg HdHgPd ⋅⋅=− ρ
In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas
where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume VR* - Universal gas constant
TRNVP ⋅⋅=⋅ *
V
m
M
mN == ρ&
MTRP /* ⋅⋅= ρ
Earth AtmosphereSOLO
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 ===
Earth AtmosphereSOLO
We must make a distinction between:- Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0)
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12
TM
MTM ⋅= 0
To simplify the computation let introduce:- Geopotential Altitude H- Geometric Altitude Hg
Newton Gravitational Law implies: ( )2
0
+
⋅=gE
Eg HR
RgHg
The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ
The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ
This means thatg
gE
Eg Hd
HR
RHd
g
gHd ⋅
+
=⋅=2
0
Integrating we obtaing
gE
E HHR
RH ⋅
+
=
Earth Atmosphere
13
Atmospheric Constants
Definition Symbol Value Units
Sea-level pressure P0 1.013250 x 105 N/m2
Sea-level temperature T0 288.15 K
Sea-level density ρ0 1.225 kg/m3
Avogadro’s Number Na 6.0220978 x 1023 /kg-mole
Universal Gas Constant R* 8.31432 x 103 J/kg-mole - K
Gas constant (air) Ra=R*/M0 287.0 J/kg--K
Adiabatic polytropic constant γ 1.405
Sea-level molecular weight M0 28.96643
Sea-level gravity acceleration g0 9.80665 m/s2
Radius of Earth (Equator) Re 6.3781 x 106 m
Thermal Constant β 1.458 x 10-6 Kg/(m-s- K1/2)
Sutherland’s Constant S 110.4 K
Collision diameter σ 3.65 x 10-10 m
Earth AtmosphereSOLO
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Physical Foundations of Atmospheric Model
Atmospheric Equilibrium Equation
HdgPd ⋅⋅=− 0ρAt altitude bellow 100 km we assume t6he Equation of an Ideal Gas
TRMTRP a
MRR
a
aa
⋅⋅=⋅⋅==
ρρ/
**
/
HdTR
g
P
Pd
a
⋅=− 0
Combining those two equations we obtain
Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant.
Earth AtmosphereSOLO
15
Layer Index
GeopotentialAltitude Z,
km
GeometricAltitude Z;
km
MolecularTemperature T,
K
0 0.0 0.0 288.150
1 11.0 11.0102 216.650
2 20.0 20.0631 216.650
3 32.0 32.1619 228.650
4 47.0 47.3501 270.650
5 51.0 51.4125 270.650
6 71.0 71.8020 214.650
7 84.8420 86.0 186.946
1976 Standard Atmosphere : Seven-Layer Atmosphere
Lapse RateLh;
K/km
-6.3
0.0
+1.0
+2.8
0.0
-2.8
-2.0
Earth AtmosphereSOLO
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Physical Foundations of Atmospheric Model
• Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km.
( ) HdHLTR
gHd
TR
g
P
Pd
aa
⋅⋅+
=⋅=−0
00
kmKLHLTT /3.60−=⋅+=
Integrating this equation we obtain
( )∫∫ ⋅⋅+
=−H
a
P
P
HdHLTR
g
P
PdS
S 0 0
0 1
0
( )0
00 lnln0
T
HLT
RL
g
P
P
aS
S ⋅+⋅⋅
−=
HenceaRL
g
SS HT
LPP
⋅−
⋅+⋅=
0
0
0
1
and
−
⋅=
⋅
10
0
0g
RL
S
S
a
P
P
L
TH
Earth AtmosphereSOLO
Stratosphere
Troposphere
17
Physical Foundations of Atmospheric Model
HdTR
g
P
Pd
Ta
⋅=− *0
Integrating this equation we obtain
( )TTaS
S HHTR
g
P
P
T
−⋅⋅
−= *0ln
Hence( )T
Ta
T
HHTR
g
SS ePP−⋅
⋅−
⋅=*
0
andS
STTaT P
P
g
TRHH ln
0
*
⋅⋅+=
∫∫ =−H
HTa
P
P T
S
TS
HdTR
g
P
Pd*
0
• Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 K = TT* is constant (isothermal layer), PST=22632 Pa
Earth AtmosphereSOLO
Stratosphere
Troposphere
18
Physical Foundations of Atmospheric Model
( )[ ] HdHHLTR
gHd
TR
g
P
Pd
SSTaa
⋅−⋅+⋅
=⋅=− *00
( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1* ===−⋅−=
Integrating this equation we obtain
( )[ ]∫∫ ⋅−⋅+
=−H
H SSTa
P
P S
S
SS
HdHHLTR
g
P
Pd*
0 1
( )[ ]*
*0 lnln
T
ST
aSSS
S
T
HHLT
RL
g
P
P −⋅+⋅⋅
=
Hence ( ) aRL
g
S
T
SSSS HH
T
LPP
⋅−
−⋅+⋅=
0
*1
and
−
⋅+=
⋅
10
* g
RL
SS
S
S
TS
aS
P
P
L
THH
Stratosphere Region (HS=20.0 km to 32.0 km).
Stratosphere
Troposphere
Earth AtmosphereSOLO
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1962 Standard Atmosphere from 86 km to 700 km
Layer Index GeometricAltitude
km
MolecularTemperature
,K
KineticTemperature
K
MolecularWeight
LapseRateK/km
7 86.0 186.946 186.946 28.9644 +1.6481
8 100.0 210.65 210.02 28.88 +5.0
9 110.0 260.65 257.00 28.56 +10.0
10 120.0 360.65 349.49 28.08 +20.0
11 150.0 960.65 892.79 26.92 +15.0
12 160.0 1110.65 1022.20 26.66 +10.0
13 170.0 1210.65 1103.40 26.49 +7.0
14 190.0 1350.65 1205.40 25.85 +5.0
15 230.0 1550.65 132230 24.70 +4.0
16 300.0 1830.65 1432.10 22.65 +3.3
17 400.0 2160.65 1487.40 19.94 +2.6
18 500.0 2420.65 1506.10 16.84 +1.7
19 600.0 2590.65 1506.10 16.84 +1.1
20 700.0 2700.65 1507.60 16.70
Earth AtmosphereSOLO
20
1976 Standard Atmosphere from 86 km to 1000 km
Geometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8)
78
/0.0
TT
kmKZd
Td
=
=
Geometric Altitude Range: from 91.0 km to 110.0 km (index 8 – 9)
2/12
8
2
8
2/12
8
1
1
−
−−
−⋅−=
−−⋅+=
a
ZZ
a
ZZ
a
A
Zd
Td
a
ZZATT C
kma
KA
KTC
9429.19
3232.76
1902.263
−=−=
=
Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10)( )
kmKZd
Td
ZZLTT Z
/0.12
99
+=
−⋅+=
Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11)
( ) ( )
( )
( )
+
+⋅−=
+
+⋅−⋅=
⋅−⋅−−=
∞
∞∞
ZR
ZRZZ
kmKZR
ZRTT
Zd
Td
TTTT
E
E
E
E
1010
1010
10
/
exp
ξ
λ
ξλ
KT
kmR
km
E
1000
10356766.6
/01875.03
=
×=
=
∞
λ
Earth AtmosphereSOLO
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Sea Level Values
Pressure p0 = 101,325 N/m2
Density ρ0 = 1.225 kg/m3
Temperature = 288.15 K (15 C)Acceleration of gravity g0 = 9.807 m/sec2
Speed of Sound a0 = 340.294 m/sec
Earth AtmosphereSOLO
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SOLO
Atmosphere
Continuum FlowLow-density and
Free-molecular Flow
Viscous Flow Inviscid Flow
Incompressible Flow
Compressible Flow
Subsonic Flow
Transonic Flow
Supersonic Flow
Hypersonic Flow
AERODYNAMICS
Fixed Wing Aircraft Flight Performance
AERODYNAMICS
23
SOLO
Dimensionless Equations
Dimensionless Field Equations
(C.M.): ( ) 0~~~~
=⋅∇+ ut
ρ∂
ρ∂
( ) ( )uR
uR
pGF
uut
u
eer
~~~~1
3
4~~~~1~~~~1~~~~
~~
2
⋅∇∇+×∇×∇−∇−=
∇⋅+ µµρ
∂∂ρ(C.L.M.):
( ) ( )TkPRt
QuG
Fu
t
pHu
t
H
rer
∇⋅∇−+⋅+⋅⋅∇+=
∇⋅+
∂∂ 11
~
~~~~1~~~
~~~~~
~
~~
2 ∂∂ρτ
∂∂ρ
(C.E.):
Reynolds:0
000
µρ lU
Re = Prandtl:0
0
k
CP pr
µ= Froude:
0
0
gl
UFr =
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ µµµ =
0/~
kkk =
Dimensionless Variables are:
Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)
0/~ λλλ =
Knudsenl
Kn0
0:λ=
AERODYNAMICS
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24
SOLO
Mach Number
Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.
• M is the Mach number,• U0 is the velocity of the source relative to the medium, and
• a0 is the speed of sound
Mach:0
0
a
UM =
The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret.
Ernst Mach (1838–1916)
Jakob Ackeret (1898–1981)
m
Tk
Mo
TRa Bγγ ==0
• R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1), [M1 L2 T−2 θ−1 'mol'−1]
• γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4.• T is the thermodynamic temperature [θ1]
• Mo is the molar mass, [M1 'mol'−1]
• m is the molecular mass, [M1]
AERODYNAMICS
25
SOLO
Mach Number – Flow Regimes Regime Mach mph km/h m/s General plane characteristics
Subsonic <0.8 <610 <980 <270Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges.
Transonic 0.8-1.2 610-915
980-1,470 270-410Transonic aircraft nearly always have swept wings, delaying drag-divergence, and often feature design adhering to the principles of the Whitcomb Area rule.
Supersonic 1.2–5.0915-3,840
1,470–6,150 410–1,710
Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behavior of flows above Mach 1. Sharp edges, thin airfoil-sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde.
Hypersonic 5.0–10.03,840–7,680
6,150–12,300
1,710–3,415
Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider
High-hypersonic
10.0–25.07,680–16,250
12,300–30,740
3,415–8,465
Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature.
Re-entry speeds >25.0
>16,250 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape
AERODYNAMICS
26
SOLO
Different Regimes of Flow
Mach Number – Flow Regimes AERODYNAMICS
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27
SOLO
- when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves.
a
VM
M=
= − &
1sin 1µ
Disturbances in a fluid propagate by molecular collision, at the sped of sound a,along a spherical surface centered at the disturbances source position.
The source of disturbances moves with the velocity V.
- when the source moves at supersonic velocity V > a, it will stay outside the family of spherical sound waves. These wave fronts form a disturbance
envelope given by two lines tangent to the family of spherical sound waves. Those lines are called Mach waves, and form an angle μ with the disturbance
source velocity:
SHOCK & EXPANSION WAVESAERODYNAMICS
28
SOLO
SHOCK & EXPANSION WAVES
M < 1
M = 1
M > 1
Mach WavesAERODYNAMICS
29
SOLO
When a supersonic flow encounters a boundary the following will happen:
When a flow encounters a boundary it must satisfy the boundary conditions,meaning that the flow must be parallel to the surface at the boundary.
- when the supersonic flow, in order to remain parallel to the boundary surface, must “turn into itself” a Oblique Shock will occur. After the shock wave the pressure, temperature and density will increase. The Mach number of the flow will decrease after the shock wave.
SHOCK & EXPANSION WAVES
- when the supersonic flow, in order to remain parallel to the boundary surface, must “turn away from itself” an Expansion wave will occur. In this case the pressure, temperature and density will decrease. The Mach number of the flow will increase after the expansion wave.
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AERODYNAMICS
30
SHOCK WAVES
SOLO
A shock wave occurs when a supersonic flow decelerates in response to a sharpincrease in pressure (supersonic compression) or when a supersonic flow encountersa sudden, compressive change in direction (the presence of an obstacle).
For the flow conditions where the gas is a continuum, the shock wave is a narrow region(on the order of several molecular mean free paths thick, ~ 6 x 10-6 cm) across which isan almost instantaneous change in the values of the flow parameters.
Shock Wave Definition (from John J. Bertin/ Michael L. Smith, “Aerodynamics for Engineers”, Prentice Hall, 1979, pp.254-255)
When the shock wave is normal to the streamlines it is called a Normal Shock Wave,
otherwise it is an Oblique Shock Wave.
The difference between a shock wave and a Mach wave is that:
- A Mach wave represents a surface across which some derivative of the flow variables (such as the thermodynamic properties of the fluid and the flow velocity) may be discontinuous while the variables themselves are continuous. For this reason we call it a weak shock.
- A shock wave represents a surface across which the thermodynamic properties and the flow velocity are essentially discontinuous. For this reason it is called a strong shock.
AERODYNAMICS
31
Movement of Shocks with Increasing Mach Number
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )87654321 ∞∞∞∞∞∞∞∞ <<<<<<< MMMMMMMM
SOLO AERODYNAMICS
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32
whereρ0 = air densityU0 = true speedl 0= characteristic lengthμ0 = absolute (dynamic) viscosityυ0 = kinematic viscosity
NumberReynolds:Re0
00
0
0000
00
υµρ ρ
µυlUlU
=
==
Osborne Reynolds (1842 –1912)
It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow.
Reynolds Number and Boundary Layer
SOLO 1884 AERODYNAMICS
33
Boundary Layer
SOLO 1904AERODYNAMICS
Ludwig Prandtl(1875 – 1953)
In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width•Dynamic friction coefficient μ•Friction Drag Coefficient CDf
34
The flow within the Boundary Layer can be of two types:•The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing.•The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time.There is usually a transition region between this two types of Boundary-Layer Flow
SOLO AERODYNAMICS
35
Normalized Velocity profiles within a Boundary-Layer, comparison betweenLaminar and Turbulent Flow.
SOLO AERODYNAMICSBoundary-Layer
36
Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity)
AERODYNAMICSSOLO
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37
SOLO
Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of the body in a fluid. The number is named after Danish physicist Martin Knudsen.
Knudsenl
Kn0
0:λ= Martin Knudsen
(1871–1949).
For a Boltzmann gas, the mean free path may be readily calculated as:
• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]p
TkB20
2 σπλ =
• T is the thermodynamic temperature [θ1]
λ0 = mean free path [L1]
Knudsen Number
l0 = representative physical length scale [L1].
• σ is the particle hard shell diameter, [L1]
• p is the total pressure, [M1 L−1 T−2].
See “Kinetic Theory of Gases” Presentation
For particle dynamics in the atmosphere and assuming standard atmosphere pressure i.e. 25 °C and 1 atm, we have λ0 ≈ 8x10-8m.
AERODYNAMICS
38
SOLO
Martin Knudsen (1871–1949).
Knudsen Number (continue – 1)
Relationship to Mach and Reynolds numbers
Dynamic viscosity,
Average molecule speed (from Maxwell–Boltzmann distribution),
thus the mean free path,
where
• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]
• T is the thermodynamic temperature [θ1]
• ĉ is the average molecular speed from the Maxwell–Boltzmann distribution, [L1 T−1]
• μ is the dynamic viscosity, [M1 L−1 T−1]
• m is the molecular mass, [M1]
• ρ is the density, [M1 L−3].
02
1 λρµ c=
m
Tkc B
π8=
Tk
m
B20
πρµλ =
AERODYNAMICS
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SOLO
Martin Knudsen (1871–1949).
Knudsen Number (continue – 2)
Relationship to Mach and Reynolds numbers (continue – 1)
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by
yields the Knudsen number.
The Mach, Reynolds and Knudsen numbers are therefore related by:
Reynolds:Re0
000
µρ lU=
Tk
m
lmTklallU
aUM
BB γρµ
γρµ
ρµ
µρ 00
0
00
0
000
0
0000
00
//
/
Re====
KnTk
m
lTk
m
l BB
==22 00
0
00
0 πρµπγ
γρµ
2Re
πγMKn =
AERODYNAMICS
40
SOLO
Knudsen Number (continue – 3)
Relationship to Mach and Reynolds numbers (continue –2)
According to the Knudsen Number the Gas Flow can be divided in three regions:1.Free Molecular Flow (Kn >> 1): M/Re > 3 molecule-interface interaction negligible between incident and reflected particles2.Transition (from molecular to continuum flow) regime: 3 > M/Re and M/(Re)1/2 > 0.01 (Re >> 1). Both intermolecular and molecule-surface collision are important.3.Continuum Flow (Kn << 1): 0.01 > M/(Re)1/2. Dominated by intermolecular collisions.
2Re
πγMKn =
AERODYNAMICS
SOLO
Knudsen Number (continue – 4)
InviscidLimit Free
MolecularLimitKnudsen Number
Boltzman EquationCollisionless
Boltzman Equation
DiscreteParticlemodel
Euler Equation
Navier-Stokes Equation
Continuummodel
Conservation Equationdo not form a closed set
Validity of conventional mathematical models as a function of localKnudsen Number
A higher Knudsen Number indicates larger mean free path λ, or the particular nature of the Fluid, meaning that Boltzmann Equations must be employed. Lower Knudsen Number means small free path, i.e. the flow acts as a continuum, and Navier-Stokes Equations must be used.
Knudsenl
Kn0
0:λ=
AERODYNAMICS
Return to Table of Content
42
The true airspeed (TAS; also KTAS, for knots true airspeed) of an aircraft is the speed of the aircraft relative to the air mass in which it is flying.
True Airspeed
TAS can be calculated as a function of Mach number and static air temperature: where a0 is the speed of sound at standard sea level (661.47 knots) M is Mach number, T is static air temperature in kelvin, T0 is the temperature at standard sea level (288.15ºK)
00 T
TMaTAS =
qc is impact pressureP is static pressure
−
+= 11
5 7
2
00 P
q
T
TaTAS c
Flight Instruments
SOLOFlight Instruments
SOLO
44
Flight Instruments
Airspeed Indicators
2
2
1vpp StatTotal ⋅+= ρ
The airspeed directly given by the differential pressure is called Indicated Airspeed (IAS). This indication is subject to positioning errors of the pitot and static probes, airplane altitude and instrument systematic defects. The airspeed corrected for those errors is called Calibrated Airspeed (CAS).Depending on altitude, the critic airspeeds for maneuver, flap operation etc. change because the aerodynamic forces are function of air density. An equivalent airspeed VE (EAS) is defined as follows:
0ρρ
VVE =V – True Airspeedρ – Air Densityρ0 – Air Density at Sea Level
45http://jim-quinn8.blogspot.co.il/2012_03_01_archive.html
Flight Instruments
46http://flysafe.raa.asn.au/groundschool/CAS_EAS.html
Calibrated Airspeed (CAS)
Flight Instruments
47
True Airspeed (TAS) and Calibrated Airspeed (CAS) Relationship with Varying Altitude and Temperature
Flight Instruments
48
TAS and CAS Relationship with Varying Altitude and Temperature (continue)
Flight Instruments
49
Mach Number vs TAS Variation with Altitude
Flight Instruments
50
Density Altitude Chart
Flight Instruments
51
http://digital.library.unt.edu/ark:/67531/metadc62400/m1/9/
Flight Instruments
Return to Table of Content
52
SOLO
Aerodynamic Forces
( )[ ]∫∫ +−= ∞WS
A dstfnppF
11
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ
−
−
( )
airflowingthebyweatedsurfaceVehicleS
SsurfacetheonmNstressforcefrictionf
Ssurfacetheondifferencepressurepp
W
W
W
−−
−−∞
)/( 2
Aerodynamic Forces acting on aVehicle Surface SW.
AERODYNAMICS
53
SOLO
( )
−−
=L
DF W
A
VelocitytoNormalForceLiftL
VelocitytooppositeForceDragD
−−
L
D
CSVL
CSVD
2
2
2
12
1
ρ
ρ
=
= ( )( ) tCoefficienLiftRMC
tCoefficienDragRMC
eL
eD
−−
βαβα
,,,
,,,
anglesideslipandattackofangle
viscositydynamic
lengthsticcharacteril
soundofspeedHa
numberReynoldslVR
BodytoRelativeVelocityFlowV
numberMachaVM
e
−−−−
−=−−=
βαµ
µρ
,
)(
/
/
AERODYNAMICS
( )V
WA nLVDF 11 −−=
Aerodynamic Forces Lift and Drag Forces
54
SOLO
( )( )∫∫
∫∫
⋅+⋅−=
⋅+⋅−=
W
W
S
VfVpL
S
fpD
dsntCnnCS
C
dsVtCVnCS
C
1ˆ1ˆ1
1ˆ1ˆ1
Wf
Wp
SsurfacetheontcoefficienfrictionV
fC
SsurfacetheontcoefficienpressureV
ppC
−=
−−= ∞
2/
2/
2
2
ρ
ρ
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ
−
−
Aerodynamic Forces
CD – Drag Coefficient CL – Lift Coefficient
AERODYNAMICS
( )
[ ] [ ]( )∫
∫
∞∞
=
−−−=
−=′
EdgeTrailing
EdgeLeading
sideupper sidelower
cos
EdgeTrailing
EdgeLeading
sideupper sidelower
pp
cospcosp
dxpp
dsL
sdxd
USLS
θ
θθ
Divide left and right sides of the first equation by cV 2
2
1∞ρ
∫
−
−−=′
∞
∞
∞
∞
∞
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21 ρρρ
We get:
Relationship between Lift and Pressure on Airfoil
LowerSurface
UpperSurface
( )∫ −=−EdgeTrailing
EdgeLeading
sideupper sidelower sinpsinp dsD USLS θθ
Lift – Aerodynamic component normal to VDrag – Aerodynamic component opposite to V
SOLO AERODYNAMICS
Aerodynamic Forces
From the previous slide,
∫
−
−−=′
∞
∞
∞
∞
∞
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21 ρρρ
The left side was previously defined as the sectional lift coefficient C l.
The pressure coefficient is defined as: 2
21
∞
∞−=V
ppC p
ρThus, ( )∫ −=
edgeTrailing
edgeLeading
upperplowerpl c
xdCCC ,,
LowerSurface
UpperSurface
Relationship between Lift and Pressure on Airfoil (continue – 1)
SOLO AERODYNAMICS
Aerodynamic Forces
57
SOLO
Velocity Field
Sum of the elementary Forces on the Body
Lift as the Sum of the elementary Forces on the Body
AERODYNAMICS
Aerodynamic Forces
58
SOLO
Lift and Drag Coefficients
AERODYNAMICS
Subsonic Speeds
np
α−Upper
xd
yd
∞U
Upperxd
yd
∞p∞p α
0,,
2
0 20
D
TurbulentforMoreLaminarforLessdragFriction
fD
TurbulentforLessLaminarforMore
dragPressure
pDD
stall
a
L
CCCC
aC
=+=<==
=
αααπαπ
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR → ∞)
( )
−=−=
ARe
CC L
i
a
L παπααπ 22
0
ARe
C
be
SC
V
w LSbAR
Lii ππ
α/
2
2====
α
π
απ
ARe
aa
ARe
CL0
0
12
1
2
+=
+=
ARe
CC L
Di π
2
=
AR
CCCCCC L
D
draginduced
D
dragfriction
fD
dragpressure
pDD i π
2
0,, +=++=
e – span efficiency factor
Aerodynamic Forces
Return to Table of Content
59
SOLO
http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtChap5.pdf
Drag Breakdown Possibilities (internal flow neglected)
AERODYNAMICSAerodynamic Drag
60
AERODYNAMICS
Drag Variation with Mach Number
SOLO
Aerodynamic Drag
61Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
SOLO AERODYNAMICSAerodynamic Drag
62N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993
α =0 – corresponds to CL=0.α0 – minimize CD.α1 – minimize the ratio CD/CL
1/2.α2 – minimize the ratio CD/CL
2/3.α* – minimize the ratio CD/CL.α3 – minimize the ratio CD/CL
3/2.αmax – maximum CL.
A Realistic Drag Polar
SOLO AERODYNAMICSAerodynamic Drag
63N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993
Parabolic Drag Polar of a typical High Subsonic Aircraftat different Mach Numbers
SOLO AERODYNAMICSAerodynamic Drag
64N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993
Variation of CD0 (M) for a supersonicaircraft
Variation of aerodynamic characteristic for a typical subsonic transport aircraft
Variation of aerodynamic characteristic for a typical supersonic fighter aircraft
SOLO AERODYNAMICSAerodynamic Drag
65
Movement of Shocks with Increasing Mach Number
The Mach Number at witch M=1 appears on the Airfoil Upper Surface is called the Critical Mach Number for this Airfoil. The Critical Mach Number can be calculated as follows. Assuming an isentropic flow through the flow-field we have
( )1/
2
2
2
11
2
11
−
∞
∞
−+
−+=
γγ
γ
γ
A
A
M
M
p
p
p∞, M∞ - Pressure and Mach Number upstream the AirfoilpA, MA- Pressure and Mach Number at a point A on the Airfoil
Critical Mach Number
The Pressure Coefficient Cp is computed using
( )
−
−+
−+=
−=
−
∞
∞∞∞
1
2
11
21
121
2
1/
2
2γγ
γ
γ
γγA
ApA
M
M
Mp
p
MC
Definition of Critical Mach Number.Point A is the location of minimum pressure on the top surface of the Airfoil.
SOLO AERODYNAMICS
66
Movement of Shocks with Increasing Mach Number
Critical Mach Number
This relation gives a unique relation between the upstream values of p∞, M∞ and the respective values pA, MA at a point A on the Airfoil. Assume that point A is the point of minimum pressure, therefore maximum velocity, on the Airfoil and that this maximum velocity corresponds to MA = 1. Then by definition M∞ = Mcr .
( )
−
−+
−+=
−=
−
∞
∞∞∞
1
2
11
21
121
2
1/
2
2γγ
γ
γ
γγA
ApA
M
M
Mp
p
MC
( )
−
−+
−+=
−
1
2
11
21
12
1/2
γγ
γ
γ
γcr
crp
M
MC
cr
2
0
1 ∞−=
M
CC p
p
( )
−
−+
−+=
−
1
21
1
2
112
1/2
γγ
γ
γ
γcr
crp
M
MC
cr
2
0
1 ∞−=
M
CC p
p
To find the Mcr we need on other equation describing Cp at subsonic speeds. We can use the Prandtl-Glauert Correction
or the Karman-Tsien Rule orLaiton’s Rule
SOLO AERODYNAMICS
67
Movement of Shocks with Increasing Mach Number
Critical Mach Number
AirfoilThickAirfoilMediumAirfoilThin
AirfoilThickAirfoilMediumAirfoilThin
crcrcr
ppp
MMM
CCC
>>
<< 000
The point of minimum pressure, therefore maximum velocity, does not correspond to the point of maximum thickness of the Airfoil. This is because the point of minimum pressure is defined by the specific shape of the Airfoil and not by a local property.
The Critical Mach Number is a function ofthe thickness of the Airfoil. For the thin Airfoil the Cp0 is smaller in magnitude and because the disturbance in the Flow is smaller. Because of this the Critical Mach Number of the thin Airfoil is greater
SOLO AERODYNAMICS
68
Movement of Shocks with Increasing Mach NumberDrag Divergence Mach Number
The Drag at small Mach number, due toProfile Drag with Induced Drag =0 (αi = 0)is constant (points a, b, and c) untilM∞ = Mcr (point c). As the velocity increase above Mcr (point d), a finite region of supersonic flow (Weak Shock boundary)appears on the Airfoil. The Mach Number in this bubble ofsupersonic flow is slightly above Mach 1,typically 1.02 to 1.05. If M∞ increases more,We encounter a point, e, at which is a sudden increase in Drag. The Value of M∞ at which the sudden increase in Drag starts is defined as the Drag-divergence Mach Number, Mdrag-divergence < 1. At this point Shock Waves appear on the Airfoil. The Shock Waves are dissipative phenomena extracting energy (Drag) from the kinetic energy of the Airfoil. In addition the sharp increase of the pressure across the Shock Wave create a strong adverse pressure gradient, causing the Flow to separateFrom the Airfoil Surface creating Drag increase. Beyond the Drag-divergence Mach Number, the Drag Coefficient becomes very large, increasing by a factor of 10 or more. As M∞ approaches unity (point f) the Flow on both the top and the bottom surface is supersonic, both terminating with Strong Wave Shocks.
SOLO AERODYNAMICS
69
Summary of Airfoil Drag
The Drag of an Airfoil can be described as the sum of three contributions:
iwpf DDDDD +++=
where
D – Total Drag of the AirfoilDf – Skin Friction Drag Dp – Pressure Drag due to Flow SeparationDw – Wave Drag (present only at Transonic and Supersonic Speeds; zero for Subsonic Speeds below the Drag-divergence Mach Number)Di – Induced Drag
In terms of the Drag Coefficients, we can write:
iDwDpDfDD CCCCC ,,,, +++=
The Sum:
pDfD CC ,, + Profile Drag Coefficient
SOLO AERODYNAMICSAerodynamic Drag
70
SOLO
http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtChap5.pdf
Categorization of Drag
AERODYNAMICSAerodynamic Drag
71
Relative Drag Force as a Function of Reynolds Number (Viscosity)
AERODYNAMICS
Drag CD0 due toFlow Separation
SOLO
Aerodynamic Drag
72
Relative Drag Force as a Function of Reynolds Number (Viscosity)
AERODYNAMICS
Drag due to Viscosity:1.Skin Friction2.Flow Separation (Drop in pressure behind body)
∫∫
∫∫
⋅+⋅−−=
⋅+⋅−=
∧∧∞
∧∧
W
W
S
S
fpD
dswtV
fwn
V
pp
S
dswtCwnCS
C
xx
xx
11
11
ˆ2/
ˆ2/
1
ˆˆ1
22 ρρ
SOLO
Aerodynamic Drag
73
Effect of Mach Number on the Drag Coefficient for a given Angle of Attack (AOA) and on the Lift Coefficient
AERODYNAMICS
Summary of Mach Effect on Drag and Lift
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74
Wing Parameters
Airfoil: The cross-sectional shape obtained by the intersection of the wing with the perpendicular plane
1. Wing Area, S, is the plan surface of the wing.
2. Wing Span, b, is measured tip to tip.
3. Wing average chord, c, is the geometric average. The product of the span andthe average chord is the wing area (b x c = S).
4. Aspect Ratio, AR, is defined as:
( )∫−
=2/
2/
b
b
dyycS
( )b
Sdyyc
bc
b
b
== ∫−
2/
2/
1
S
bAR
2
=
AERODYNAMICSSOLO
75
Wing Parameters (Continue)
5. The root chord, , is the chord at the wing centerline, and the tip chord, is the chord at the tip.
6. Taper ratio,
7. Sweep Angle, is the angle between the line of 25 percent chord and the perpendicularto root chord.
8. Mean aerodynamic chord,
rc
Λ
r
t
c
c=λ
tc
λ
( )[ ]∫−
=2/
2/
21~b
b
dyycS
c
c~
AERODYNAMICSSOLO
76
Wing Parameters (Continue)
AERODYNAMICS
Illustration of Wing Geometry
Planform, xy plane
Dihedral (V form), yz plane
Profile, twist xz plane
Geometric Designation of Wings of various planform
Swept-backWing
DeltaWing
EllipticWing
SOLO
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77
Wing Design Parameters
•Planform - Aspect Ratio - Sweep - Taper - Shape at Tip - Shape at Root•Chord Section - Airfoils - Twist•Movable Surfaces - Leading and Trailing-Edge Devices - Ailerons - Spoilers•Interfaces - Fuselage - Powerplants - Dihedral Angle
AERODYNAMICSSOLO
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SOLO
78
Aircraft Flight Control
Specific Stabilizer/Tail Configurations
Tailplane
Fuselage mounted Cruciform T-tail Flying tailplane
The tailplane comprises the tail-mounted fixed horizontal stabilizer and movable elevator. Besides its planform, it is characterized by:
• Number of tail planes - from 0 (tailless or canard) to 3 (Roe triplane)• Location of tailplane - mounted high, mid or low on the fuselage, fin or tail
booms.• Fixed stabilizer and movable elevator surfaces, or a single combined stabilator or
(all) flying tail.[1] (General Dynamics F-111)
Some locations have been given special names:• Cruciform: mid-mounted on the fin (Hawker Sea Hawk, Sud Aviation Caravelle)• T-tail: high-mounted on the fin (Gloster Javelin, Boeing 727)
Sud Aviation Caravelle
Gloster Javelin
SOLO
79
Aircraft Flight Control
Specific Stabilizer/Tail Configurations
Tailplane
Some locations have been given special names:
• V-tail: (sometimes called a Butterfly tail) • Twin tail: specific type of vertical stabilizer arrangement found on the empennage of
some aircraft. • Twin-boom tail: has two longitudinal booms fixed to the main wing on either side of
the center line.
The V-tail of a Belgian Air Force Fouga Magister
de Havilland Vampire T11, Twin-Boom Tail
A twin-tailed B-25 Mitchell
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80
SOLO Aircraft Propulsion Systems
Classification of Engine Concepts , mostly used in Aviation
81
Run This
http://lyle.smu.edu/propulsion/Pages/propeller.htm
In small aircraft, the propeller is normally powered by a piston engine as shown above. In larger vessels like nuclear submarines, the propeller may be powered by a nuclear power plant. The basic operation of a propeller propulsion system is described in the interactive animation below. Use the arrows to step through descriptions of the different components.
SOLO Propeller Propulsion
82
SOLO
The Rotating Parts of Jet Engine
CompressorShaft
Turbojet animation
Turbine
Air Breathing Jet Engines
Run This
83
http://lyle.smu.edu/propulsion/Pages/variations.htm
Run This
A turbofan still has all the main components of a turbojet, but a fan and surrounding duct are added to the front as shown in the animation below. A fan is basically a propeller with a lot of blades specially designed to spin very quickly. Its function is essentially identical to a propeller, namely, the blades accelerate the oncoming air flow to create thrust. In a turbofan, however, the fan is driven by turbines in the attached turbojet engine, rather than by an internal combustion engine. Use the arrows in the interactive animation below to step through descriptions of the different components and obtain more detailed information about their operation.
Turbofan
84
SOLO
Animation of a 2-spool, high-bypass turbofan.A. Low pressure spoolB. High pressure spoolC. Stationary components1. Nacelle2. Fan3. Low pressure compressor4. High pressure compressor5. Combustion chamber6. High pressure turbine7. Low pressure turbine8. Core nozzle9. Fan nozzle
Turbofan
Air Breathing Jet Engines
Run This
85
SOLO
Turboprop
A turboprop engine is a type of turbine engine which drives an aircraft propeller using a reduction gear.The gas turbine is designed specifically for this application, with almost all of its output being used to drive the propeller. The engine's exhaust gases contain little energy compared to a jet engine and play only a minor role in the propulsion of the aircraft.The propeller is coupled to the turbine through a reduction gear that converts the high RPM, low torque output to low RPM, high torque. The propeller itself is normally a constant speed (variable pitch) type similar to that used with larger reciprocating aircraft engines.Turboprop engines are generally used on small subsonic aircraft, but some aircraft outfitted with turboprops have cruising speeds in excess of 500 kt (926 km/h, 575 mph). Large military and civil aircraft, such as the Lockheed L-188 Electra and the Tupolev Tu-95, have also used turboprop power. The Airbus A400M is powered by four Europrop TP400 engines, which are the third most powerful turboprop engines ever produced, after the Kuznetsov NK-12 and Progress D-27.
Air Breathing Jet Engines
Run This
86http://lyle.smu.edu/propulsion/Pages/variations.htm
Turboprop Engines: A turboprop engine is basically a propeller driven by a turbojet. Alternatively, it can be viewed as a very large bypass ratio turbofan. It is not exactly a turbofan because there is no shroud or "duct" surrounding the propeller and the propeller does not spin as fast as a fan. The basic components of a turboprop are illustrated in the interactive animation below. Use the arrows to step through descriptions of the different components.
A turboprop engine enjoys the high efficiency of a propeller, owing to the large bypass ratio it provides. In fact, nearly all of the thrust generated by a turboprop is from the propeller. A turboprop also enjoys the high power-to-weight ratio of turbojet engines, resulting in a powerful compact propulsion system.
Run This
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SOLO Air Breathing Jet Engines
87
SOLO Aircraft Propulsion System
Aircraft propellers or airscrews[1] convert rotary motion from piston engines, turboprops or electric motors to provide propulsive force. They may be fixed or variable pitch.
Aircraft Propellers
Diesel Engine developed in the GAP program. Credit: NASA
The simplest theory describing the operation of the propeller, assumes that the rotating propeller can be approximated by a thin Actuator Disk producing a uniform change in the velocity of the air stream passing across it.
Actuator Disk (One-Dimensional Momentum) Theory
88
SOLO Propeller Aerodynamics
Actuator Disk
211
222 2
1
2
1VpVp ρρ +=+
244
233 2
1
2
1VpVp ρρ +=+
Bernoulli’s equations on each side of the Disk:
Far from the Disk we have the same ambient pressure, hence: 41 pp =
Therefore ( )21
2423 2
1VVpp −=− ρ
Conservation of Mass through the Propeller Disk
pp AVAVm 320 ρρ ==32 VV =
Conservation of Energy on both sides of the Propeller Disk
Actuator Disk (One-Dimensional Momentum) Theory
89
SOLO Propeller Aerodynamics
Actuator Disk
( )21
2423 2
1VVpp −=− ρ
The Thrust provided by the Propeller Disk is given by:
( ) ( )143140 VVAVVVmT p −=−= ρ
where
- Fluid mass flow [kg/sec] through Disk pAVm 30 ρ=
ρ – Flow density [kg/m3]
Ap – Disk area [m2]
The Thrust also equals the Force on the Disk Surface due to Pressure jump:
( ) ( ) pp AVVAppT 21
2423 2
1 −=−= ρ
From the two expressions of Thrust we obtain
( ) ( )21
24143 2
1VVVVV −=− ( )413 2
1VVV +=
Conservation of Momentum
Actuator Disk (One-Dimensional Momentum) Theory
SOLO Propeller Aerodynamics
Model of the Flow through Propeller according to the Actuator Disk Concept
( )( ) ppp
pp
VA
mVVVAT
v2v
v20143
⋅+=
=−=
∞ρρ
We found
Let compute vs as function of other parameters
02
vv 2 =−+ ∞p
pp A
TV
ρ
0222
v2
>+
+−= ∞∞
pp A
TVV
ρThis solution corresponds to a Propeller, where Energy is added to the Flow.
Actuator Disk (One-Dimensional Momentum) Theory
Ideal Power Consumed by the Rotor
( )( )
( )
+
+=
⋅=+=
+=
−+=
−=
∞∞
∞
∞
∞∞
p
p
pp
p
A
TVVT
DiskatVelocityFlowThrustVT
Vm
VmVm
InFlowEnergyOutFlowEnergyP
ρ222
___v
vv22
1v2
2
1
2
0
20
20
SOLO Propeller Aerodynamics
The Efficiency of an Ideal Propeller
This is called the idea1 efficiency of a propeller, which represents the upper limit of the efficiency that cannot be exceeded whatever the shape of the propeller.
( ) ( ) aaVDVATVa
ppp
p
+=⋅+= ∞
=
∞
∞
12
vv2 22/v:
ρπρ
( ) aVV
V
VT
VT
PowerOutput
PowerInput Va
pppP
p
+=
+=
+=
+⋅⋅==
∞=
∞∞
∞
∞
∞
1
1
/v1
1
vv
/v:
η
( ) pP
P CJDV
Paa
323
2
3
1221
1
πρπηη ==+=−
∞
( ) ( ) aaVDVTP p232 1
2v +=+= ∞∞ ρπ
( ) TP
P CJDV
Taa
2222
1221
1
πρπηη ==+=−
∞
where
Actuator Disk (One-Dimensional Momentum) Theory
( )
.:
.:
:
42
53
2/
2
CoeffThrustDn
TC
CoeffPowerDn
PC
RatioAdvanceR
V
Dn
VJ
T
p
n
RD
ρ
ρ
ππ
=
=
Ω== ∞
Ω=
=
∞
SOLO Propeller Aerodynamics
The Efficiency of an Ideal Propeller ( )
.:
.:
:
42
53
2/
2
CoeffThrustDn
TC
CoeffPowerDn
PC
RatioAdvanceR
V
Dn
VJ
T
p
n
RD
ρ
ρ
ππ
=
=
Ω== ∞
Ω=
=
∞
E.Torenbeek, H.Wittenberg, “Flight Physics – Essentials of Aeronauical Disciplines and Technology, withHistorical Notes”, Springer, 2009
Typical Propeller Diagram
Actuator Disk (One-Dimensional Momentum) Theory
TP
P CJ 22
121
πηη =−
pP
P CJ 33
121
πηη =−
JV
Dn
P
VT
C
C P
p
T η==∞
∞
SOLO Propeller Aerodynamics
The Efficiency of an Ideal Propeller
E.Torenbeek, H.Wittenberg, “Flight Physics – Essentials of Aeronauical Disciplines and Technology, withHistorical Notes”, Springer, 2009
Propeller Efficiency and Advance Ratio for various flight speeds.The Blade Pitch β is given. The change in Efficiency is due to the change in Angle-of-Attack (due to change in Velocity V∞ or Ω),
Actuator Disk (Momentum) Theory
JC
C
p
TP =η
( )
.:
.:
:
42
53
2/
2
CoeffThrustDn
TC
CoeffPowerDn
PC
RatioAdvanceR
V
Dn
VJ
T
p
n
RD
ρ
ρ
ππ
=
=
Ω== ∞
Ω=
=
∞
94
AERODYNAMICS
Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
Actuator Disk (Momentum) Theory
SOLO
( )RatioAdvance
R
V
Dn
VJ
n
RD Ω== ∞
Ω=
=
∞ ππ2/
2:
We can see that by varying the Propeller Pitch β we can operate at maximum efficiency ηmax.
95
SOLO Propeller Aerodynamics
E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”, pg. 236
Propeller Blade Geometry
Variation of Angles and Velocities along a Propeller Blade
Propeller Blade have a variation of•Twist β•Chord c•Thickness t
r
V
Ω=φtan
From the Propeller Blade Geometry
– advance angle ϕ [rad]V – air velocity [m/sec], normal to rotation plane V = V∞ + vΩ – rotation rate [rad/sec]r – rotation radii [m] of blade section element
φβα −=α – angle of attack [rad] of the section element (between section chord and resultant velocity)β – angle [rad] between section chord and rotation plane
Blade Element Theory.
96
SOLO Propeller Aerodynamics
( ) ( )2222 v++Ω=+= ∞VrUUV pTres
Given a Propeller Blade Element at a distance r from the Hub, the Resultant Velocity is given by
We have
( ) ( ) ( )
( ) ( ) ( )αραρ
αραρ
DDres
LLres
CcrVCcVDd
CcrVCcVLd
22222
22222
2
1
2
12
1
2
1
Ω+==
Ω+==
∞
∞ Section Lift, normal to Vres
Section Drag, opposite to Vres
Simplified view of the forces on a Propeller Blade Element
c – chord of Propeller Blade ElementCL – Lift Coefficient of Propeller Blade Element CD – Drag Coefficient of Propeller Blade Element
The resultant forces Normal (d T) and in the Disk Plane (d Fx) are
−=
Dd
Ld
Fd
Td
x φφφφ
cossin
sincos
Blade Element Theory.
The Aerodynamic Moment and Power of the Propeller Blade Element are
QdFdrFdUPd
FdrQd
xxT
x
Ω=⋅Ω==⋅=
97
SOLO Propeller Aerodynamics
The net force acting on the blades are the summationof the forces acting upon the individual elements.We must multiply by the number of blades B of the Rotor.
We have
Blade Element Theory.
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )( )∫∫
∫∫
∫∫
=
=∞
=
=
=
=∞
=
=
=
=∞
=
=
+Ω+Ω=⋅Ω=
+Ω+=⋅=
−Ω+==
Rr
r
DL
Rr
r
x
Rr
r
DL
Rr
r
x
Rr
r
DL
Rr
r
rdrCrCrrVBcFdrBP
rdrCrCrrVBcFdrBQ
rdrCrCrVBcTdBT
0
222
0
0
222
0
0
222
0
cossin2
1
cossin2
1
sincos2
1
φαφαρ
φαφαρ
φαφαρ
( )r
Vr
Ω+= ∞ v
tanφ ( ) ( ) ( )rrr φβα −=
The Thrust, Aerodynamic Moment and Power of the Propeller (B blades) are
The β (r) must be twisted to have the function α (r) optimal at each section r for given V∞ and Ω. If V∞ changes by rotating the Propeller around it’s axis (Pitch) we change β (r) to optimize again α (r).
98
SOLO Propeller AerodynamicsBlade Element Theory.
42
22
242
53
32
253
2
2
4:
4:
:
D
T
Dn
TC
R
P
Dn
PC
RatioAdvanceR
V
Dn
VJ
n
RDT
n
RDp
n
RD
Ω==
Ω==
Ω==
Ω=
=
Ω=
=
∞
Ω=
=
∞
ρπ
ρ
ρπ
ρ
π
π
π
π
( ) ( ) ( ) ( )( )∫=
=
∞
−
+
Ω=
Ω=
Rr
r
DLT R
rdrCrC
R
r
R
V
R
cB
R
TC
02
22
22
22
42
2
sincos84
φαφαπππ
πρπ
σ
( ) ( ) ( ) ( )( )∫=
=
∞
+
+
Ω=
Ω=
Rr
r
DLP R
rdrCrC
R
r
R
rV
R
Bc
R
PC
02
22
2
22
53
3
cossin84
φαφαππππ
πρπ
σ
We have
or
Let use the definitions:
( )
SolidityR
cB
R
RcB
DiskSurface
ElementsBladeSurface
==
==
π
πσ
2:
( ) ( ) ( ) ( ) ( )( )∫=
=
=−+=
1
0
222/
sincos8
x
x
DL
Rrx
T xdxCxCxJC φαφαπσπThrust Coefficient
( ) ( ) ( ) ( ) ( )( )∫=
=
=++=
1
0
222/
cossin8
x
x
DL
Rrx
P xdxCxCxxJC φαφαππσπ Power Coefficient
99
SOLO Propeller AerodynamicsBlade Element Theory.
42
22
242
53
32
253
2
2
4:
4:
:
D
T
Dn
TC
R
P
Dn
PC
RatioAdvanceR
V
Dn
VJ
n
RDT
n
RDp
n
RD
Ω==
Ω==
Ω==
Ω=
=
Ω=
=
∞
Ω=
=
∞
ρπ
ρ
ρπ
ρ
π
π
π
π
Characteristic Curves of a Propeller
Propeller Efficiency.
JC
C
Dn
V
C
C
CDn
VCDn
P
VT
P
T
P
T
P
T ==== ∞∞∞53
42
ρρη
100
SOLO Propeller Aerodynamics
Fuel Consumption
For
VTPP ppA ⋅⋅=⋅= ηηThe Available Power is
ηp – propulsive efficiency
For a given throttle setting, a regular piston engine, that aspire atmospheric air, produces power that is almost constant with velocity but decreases as the altitude increases (air density decreases).
VTP ⋅=
Propeller Propulsion
The fuel mass flow is proportional to engine power P
pApp PcPcWf η/==−=
cp – power specific fuel consumption
VPT /=The engine power is
=
=
restratosphe
etropospherx
P
Px
1
75.0
00 ρρ
101
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Asselin, M., “Introduction to Aircraft Aerodynamics”, AIAA Education Series, 1997
Return to Table of Content
102
Most jet engines are Turbofans and some are Turbojets which use gas turbines to give high pressure ratios and are able to get high efficiency, but a few use simple ram effect or pulse combustion to give compression.Most commercial aircraft possess turbofans, these have an enlarged air compressor which permit them to generate most of their thrust from air which bypasses the combustion chamber.
AIR BREATHING JET ENGINESSOLO
Operation of Aircraft Turbojet EngineAircraft Turbo Engines
The turboprop engine : Turboprop engine derives its propulsion by the conversion of the majority of gas stream energy into mechanical power to drive the compressor , accessories , and the propeller load. The shaft on which the turbine is mounted drives the propeller through the propeller reduction gear system . Approximately 90% of thrust comes from propeller and about only 10% comes from exhaust gas.
The turbofan engine : Turbofan engine has a duct enclosed fan mounted at the front of the engine and driven either mechanically at the same speed as the compressor , or by an independent turbine located to the rear of the compressor drive turbine . The fan air can exit separately from the primary engine air , or it can be ducted back to mix with the primary's air at the rear . Approximately more than 75% of thrust comes from fan and less than 25% comes from exhaust gas.
103
Propulsion Force = Thrust
SOLO
The net Thrust ( T ) of a Turbojet is given by
where: ṁ air = the mass rate of air flow through the engine
ṁ fuel = the mass rate of fuel flow entering the engine
Ue = the velocity of the jet (the exhaust plume)
U0 = the velocity of the air intake = the true airspeed of the aircraft
(ṁ air + ṁ fuel )Ue = the nozzle gross thrust (FG)
ṁ air U0 = the ram drag of the intake air
Aircraft Propulsion System
( )[ ] ( ) airfueleeeair mmfAppUUfmTHRUST /:1 00 =−+−+==T
Jet Engines Thrust Force Introduction to Air Breathing Jet Engines
00 ,Up
0A
eA
ee Up ,
104
Turbojet
SOLO
Thrust Computation for Air Breathing Engines
( ) ( )
DRAGFRICTION
A
WA
DRAGPRESURE
A
WA
THRUST
eeeeex
WW
AdAdppAppAUAUF ∫∫∫∫ −−−−+−= θτθρρ cossin000200
2
00000 & mAUmmAU feee =+= ρρUsing C.M.
( ) ( ) 00000200
2 UmUmmAppAUAUTHRUST efeeeee −+=−+−= ρρ
or
we obtain
( )[ ] ( ) 0000 /:1 mmfAppUUfmTHRUST feee =−+−+==T
and ( )
DRAGFRICTION
A
WA
DRAGPRESURE
A
WA
WW
AdAdppDRAGD ∫∫∫∫ +−== θτθ cossin0
00 ,Up
0A
eA
ee Up ,
Air Breathing Jet Engines
Pressure force
Friction force
Wetted Surface
Aerodynamic Forces on Wetted Surfaces
105
Turbojet
SOLO
Thrust Computation for Air Breathing Engines (continue – 1)
since
and
00 ,Up
0A
eA
ee Up ,
( ) 000000
00
00
/:111 mmfA
A
p
p
U
Uf
Ap
Um
Ap feee
=
−+
−+=T
20
20
00
0020
0
200
0
200
00
2000
00
00 MMTR
TRM
p
a
p
U
Ap
UA
Ap
Um γρ
γρρρρ =====
( ) 0000
20
00
/:111 mmfA
A
p
p
U
UfM
Ap feee =
−+
−+= γT
000200
0
00000000 MApaM
TR
ApaUAam γρ ===
( ) 00000
000000
/:11
111
mmfA
A
p
p
MU
UfM
ApMam feee
=
−
+
−+=
=
γγTT
Air Breathing Jet Engines
106
Turbojet
SOLO
Thrust Computation for Air Breathing Engines (continue – 2)
00 ,Up
0A
eA
ee Up ,
000
0
00
00 11:
ApMg
a
famg
a
m
m
gmWeightFuelBurned
ForceThrustI
ffsp
TTT
====
γ
Specific Impulse
0000
11
ApMfa
gI sp T
=
γ
Specific Fuel Consumption (SFC)
spIg
f
ThrustofPound
HourperBurnedFuelofPoundS
1: ====
0
f
mT/T
m
Air Breathing Jet Engines
107
Air Breathing Jet Engines
PRESSURE
CompressorPressureRise
TurbinePressureDrop(Turbojet)
Heat Added in Combustion Chambersby burning mfuel mass
TOTAL TEMPERATURE
mfuel_1
mfuel_2
mfuel_3
mfuel_1 >mfuel_2>mfuel_3
Pressure corresponding to mfuel_1 and Thrust1
Pressure corresponding to mfuel_2 and Thrust2
Pressure corresponding to mfuel_3 and Thrust3A
B1
B2
B3
C1
D1
C2
D2
C3
D3
Thrust1 >Thrust2>Thrust3
E
108
Air Breathing Jet Engines
PRESSURE
CompressorPressureRise
TurbinePressureDrop(Turbojet)
Heat Added in Combustion Chambersby burning mfuel mass
TOTAL TEMPERATURE
Pressure corresponding to mfuel and ThrustA
B C
D1
F
E
Additional TurbinePressure Dropin Turboprop
109
( )[ ] ( ) 0000 /:1 mmfAppUUfm feee =−+−+=T
00 ,Up
0A
eA
ee Up ,
Aircraft Propulsion SystemSOLO
0000 UAm ρ=
The change in altitude (air density) will affect the thrust as follows
As U0 increases Ue doesn’t change (at the first order), since the value of Ue depends more of the internal compression and combustion processes inside the engine than on the U0. Therefore Ue – U0
will decrease. Since increase in U0 increases ṁ0 , the Thrust T will remain, at first order, constant.
0UwithconstantelyapproximatisT
..LSS.L. ρρ=
T
T
Sensitivity of Thrust and Specific Fuel Consumption withVelocity and Altitude for a Jet Engine
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
The Specific Fuel Consumption increases with Mach at subsonic velocity (see Figure next slide)
11 00 <+= MMkTSFC
The Specific Fuel Consumption is constant with altitude at subsonic velocity (see Figure next slide)
altitudewithconstantisTSFC
110
Typical results for the variation of Thrust and Thrust Specific Fuel Consumption with Subsonic Mach number for a Turbojet
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
Aircraft Propulsion SystemSOLO
Sensitivity of Thrust and Specific Fuel Consumption withVelocity and Altitude for a Jet Engine
111
Typical results for the variation of Thrust and Thrust Specific Fuel Consumption with Supersonic Mach number for a Turbojet
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
Aircraft Propulsion SystemSOLO
Sensitivity of Thrust and Specific Fuel Consumption withVelocity and Altitude for a Jet Engine
Supersonic Conditions
12
2
11
−
−+=
γγ
γM
p
p
static
total
Ptotal is the pressure entering theCompressor from the Diffuser, that further increases the pressure and therefore the exitVelocity Ue and the Thrust.
From the Figure we obtain that for the specific aircraft the Supersonic Thrust is given by
( )118.11 01
−+==
MMT
T
..LSS.L. ρρ=
T
T
The Specific Fuel Consumption is constant with Mach at supersonic velocity (see Figure)
The Specific Fuel Consumption is constant with altitude at supersonic velocity (see Figure)
altitudewithconstantisTSFC
10 >MMachwithconstantisTSFC
112
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Turbojet Performance
Aircraft Propulsion SystemSOLO
Return to Table of Content
113
SOLO
Thrust Augmentation – Reheat in an Afterburner
Aircraft Propulsion System
To achieve Take-Off from a Short Runway a Fighter Aircraft needs additional Thrust. This is also necessary in Dogfight Combat to increase Aircraft Maneuverability. A very effective and widely used method to increase Thrust is by Reheat or Afterburning which enables Thrust to be increased by 50 percent. The technology of Reheat is possible because the hot gas after passing the Turbine, still contains enough oxygen to allow a Second Combustion given additional Fuel is Injected. (Only part of the air is discharged by the Compressor is used for Combustion, the greater part is used for Cooling).
The Afterburner is a Tube-like structure attached to the Gas Generator immediately behind the Turbine. The forward part is designed as a Diffuser (increasing cross-section) which decrease flow velocity from Mach 0.5 to 0.2. It consists of the following four components: - Flame Tube - Fuel Injection System - Flame Holder Assembly (prevent Flame for being carried away) - Variable Geometry Exhaust Nozzle
Afterburner
114
SOLO
Ideal Turbojet Engine with Afterburner
Pressure-Volume Diagram Temperature-Entropy Diagram
Ideal Turbojet with Afterburner
eA
ee Up ,00 ,Up
0A
Air Breathing Jet Engines
Typical afterburning jet pipe equipment.
Afterburner
Return to Table of Content
115
Thrust Reversal Operation (Used during Landing)
Aircraft Propulsion SystemSOLO
Return to Table of Content
116
Typical results for the variation of Thrust and Thrust Specific Fuel Consumption with Subsonic Mach number for Turbojet
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
Aircraft Propulsion SystemSOLO
Altitude variation T/T0 = ρ/ρ0
Velocity variation1.Subsonic: T is constant with V
2. Supersonic: T/Tm=1=1+1.18 (M-1)
Velocity variation1. Subsonic: TSFC = 1.0+k M
2. Supersonic: TSFC is constant
Altitude variationSFC is cons tant with Alti tude
Specific fuelConsumption
PowerPA =T V
TurbojetEngine
Aircraft Propulsion Summary
117
Altitude variation T/T0 =( ρ/ρ0 )
m
Velocity variation1High bypass ratio: T/TV=0=A M
-n
2. Low bypass ratio: T first increases with M
then decreases at high supersonic M
Velocity variation1. High Bypass ct = B (1.0+k M )2.Low Bypass: ct graduately increases with velocity
Altitude variationc t is constant with Alti tude
Specific fuelConsumption
PowerPA =T V
TurbofanEngine
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186
Aircraft Propulsion Summary
Aircraft Propulsion SystemSOLO
118
Velocity variationPA is constant with M
Altitude variationPA/PA,0 = (ρ/ρ0)
m
Velocity variationCA is constant with V
Altitude variationCA is constant with Alti tude
Specific fuelConsumption
PowerPA =(TP+Tj) V
PA = hpr PS+Tj V
PA = hpr Pes
TurbopropEngine
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186
Aircraft Propulsion Summary
Block Diagram
Aircraft Propulsion SystemSOLO
Aircraft Propulsion Summary
119
Altitude variation1. P/P0 = ρ/ρ0
2. (slightly more accurate) P/P0 =1.132 ρ/ρ0-0.132
Velocity variationShaft Power P constant with V
Velocity variationSFC is cons tant with V
Altitude variationSFC is cons tant with Alti tude
Al titude variation T/T0 = ρ/ρ0
Velocity variation1.Subsonic: T is constant with V
2. Supersonic: T/Tm=1=1+1.18 (M-1)
Velocity variation1. Subsonic: TSFC = 1.0+k M2. Supersonic: TSFC is constant
Altitude variationSFC is cons tant with Alti tude
Al titude variation T/T0 =( ρ/ρ0 )
m
Velocity variation1High bypass ratio: T/TV=0=A M
-n
2. Low bypass ratio: T first increases with M
then decreases at high supersonic M
Velocity variation1. High Bypass ct = B (1.0+k M )2.Low Bypass: ct graduately increases with velocity
Altitude variationc t is constant with Alti tude
Velocity variationPA is constant with M
Al titude variationPA/PA,0 = (ρ/ρ0)
m
Velocity variationCA is constant with V
Al titude variationCA is constant with Alti tude
Specific fuelConsumption
Specific fuelConsumption
Specific fuelConsumption
Specific fuelConsumption
PowerPA =T V
PowerPA =T V
PowerPA = hpr Php r = f (J)J = V/(N D)
PowerPA =(TP+Tj) V
PA = hpr PS+Tj V
PA = hpr Pes
Reciprocating Engine/Propeller Combination
TurbojetEngine
TurbofanEngine
TurbopropEngine
Propulsion Systems
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186
Aircraft Propulsion Summary
Block Diagram
Aircraft Propulsion SystemSOLO
120
Air Breathing Jet Engines
0mT
Aircraft Propulsion Summary
SOLO
121
Air Breathing Jet EnginesAircraft Propulsion Summary
SOLO
122
Air Breathing Jet EnginesAircraft Propulsion Summary
SOLO
123
SOLO
Propulsive Efficiency Characteristics of Turboprop, Turbofan and Turbojet Engines
Air Breathing Jet Engines
Return to Table of Content
Propulsive Efficiency Summary
124Stengel, MAE331, Lecture 6
Thrust of a Propeller-Driven Aircraft
• With constant r.p.m., variable-pitch propeller
whereηp - propeller efficiencyηI - ideal propulsive efficiencyηnet-max ≈ 0.85 – 0.9
Efficiency decrease with airspeedEngine power decreases with altitude- Proportional with air density w/o supercharger
V
P
V
PT engine
netengine
Ip ηηη ==
Variation of Thrust and Power of a Propeller-Driven Aircraft with True Airspeed
Aircraft Propulsion Summary
SOLO Aircraft Propulsion System
125
Thrust as a function of airspeed for different Propulsion Systems
Aircraft Propulsion Summary
SOLO Aircraft Propulsion System
126Stengel, MAE331, Lecture 6
Thrust of aTurbojetEngine
( )
−
+−
−
−
= 1111
2/1
00
0
c
tc
t
tVmTτθ
θτθ
θθ
θ
fuelair mmm +=( )
heatsspecificofratiop
p
ambient
stag =
=
−
γθγγ
,/1
0
=
etemperaturambientfreestream
etemperaturinletturbine0θ
=
etemperaturinletcompressor
etemperaturoutletcompressorcτ
• Little change in thrust with airspeed below Mcrit
• Decrease with increasing altitude
where
Variation of Thrust and Power of a Turbojet Engine with True Airspeed
SOLO Aircraft Propulsion System
127
Stengel, MAE331, Lecture 6
John D. Anderson, Jr., “Introduction to Flight”, McGraw Hill, 1978, § 6.4, pg. 217
B. N. Pamadi, “Performance, Stability, Dynamics and Control of Aircraft”, AIAA Education Series, 1998, pp. 68-69
SOLO Aircraft Propulsion System
128
Power and Thrust• Propeller
• Turbojet
• Throttle Effect
airspeedoftindependenSVCVTPPower T ≈=•== 3
2
1 ρ
airspeedoftindependenSVCTThrust T ≈== 2
2
1 ρ
102
1 2max max
≤≤== TSVTCTTT T δρδδ
Specific Fuel Consumption, SFC = cP or cT
• Propeller aircraft
• Jet aircraft
[ ]
[ ]
→
→=
−=
−=
lbf
slbor
kN
skgc
HP
slbor
kW
skgc
weightfuelw
where
thrusttoalproportionTcw
powertoalproportionPcw
T
P
f
Tf
Pf
//
//
SOLO Aircraft Propulsion System
Return to Table of Content
129Dr. Carlo Kopp, Air Power Australia, Sukhoi Su-34 Fullback, Russia's New Heavy Strike Fighter
Comparison of Fighter Aircraft Propulsion SystemsSOLO
130
Comparison of Fighter Aircraft Propulsion SystemsSOLO
131
Comparison of Fighter Aircraft Propulsion SystemsSOLO
132M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Fighter Aircraft Propulsion SystemsSOLO
133M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Fighter Aircraft Propulsion SystemsSOLO
134M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Fighter Aircraft Propulsion SystemsSOLO
135M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Fighter Aircraft Propulsion SystemsSOLO
136M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”Return to Table of Content
Comparison of Fighter Aircraft Propulsion SystemsSOLO
137
SOLO Aircraft Propulsion System
VTOL - Vertical Take off and Landing capabilityThe advantages of vertical take off and landing VTOL are quite obvious.Conventional aircraft have to operate from a small number of airports withlong runways. VTOL aircraft can take off and land vertically from muchsmaller areas.STOL - Short takeoff and landingThese aircraft using thrust vectoring to decrease the distance needed fortakeoff and landing but don’t have enough thrust vectoring capability toperform a vertical take off or landing.VSTOL - An aircraft that can perform either vertical or short takeoff and landingsSTOVL - Short takeoff and vertical land.An aircraft that has insufficient lift for vertical flight at takeoff weight butcan land vertically at landing weight.TVC - Thrust Vector Control
Vertical Take off and Landing - VTOL
138
SOLO
Vertical Take off and Landing - VTOL
139M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
140
Lockheed_Martin_F-35_Lightning_II STOVL
The Unique F-35 Fighter Plane, Movie
USP 3” part F35Joint Strike Fighter ENG,
Movie
SOLO Aircraft Propulsion System
Thrust vectoring nozzle of the F135-PW-600 STOVL variant
Return to Table of Content
141
Aircraft Propulsion SystemSOLO
Engine Control System
Engine Control System Basic Inputs and Outputs
Engine Control System Input Signals:• Throttle Position (Pilot Control)• Air Data (from Air Data Computer) Airspeed and Altitude• Total Temperature (at the Engine Face)• Engine Rotation Speed• Engine Temperature• Nozzle Position• Fuel Flow• Internal Pressure Ratio at different Stages of the Engine
Output Signals• Fuel Flow Control• Air Flow Control
142
Aircraft Propulsion SystemSOLO
The Fighter Aircraft Propulsion Systems Consists of: - One or Two Jet Engines - The Fuel Tanks (Internal and External) and Pipes. - Engines Control Systems * Throttles * Engine Control Displays
Engine Control Systems – Basic Inputs and Outputs
143
Aircraft Propulsion SystemSOLO
A Simple Engine Control Systems : Pilot in the Loop
A Simple Limited Authority Engine Control Systems
TGT – Turbine Gas TemperatureNH – Speed of Rotation of Engine ShaftTt - Total TemperatureFCU – Fuel Control Unit
Engine Control System
144
Aircraft Propulsion SystemSOLO
A Simple Engine Control Systems : Pilot in the Loop
A Simple Limited Authority Engine Control Systems
Engine Control Systems : with NH and TGT exceedance warning
Full Authority Engine Control SystemsWith Electrical Throttle Signaling :
Engine Control System Return to Table of Content
145
Aircraft Flight ControlSOLO
146
center stick ailerons
elevators
rudder
Aircraft Flight Control
Generally, the primary cockpit flight controls are arranged as follows:a control yoke (also known as a control column), center stick or side-stick (the latter two also colloquially known as a control or B joystick), governs the aircraft's roll and pitch by moving the A ailerons (or activating wing warping on some very early aircraft designs) when turned or deflected left and right, and moves the C elevators when moved backwards or forwardsrudder pedals, or the earlier, pre-1919 "rudder bar", to control yaw, which move the D rudder; left foot forward will move the rudder left for instance.throttle controls to control engine speed or thrust for powered aircraft.
SOLO
147
Stick
Stick
RudderPedals
Aircraft Flight ControlSOLO
148
The effect of left rudder pressure Four common types of flaps
Leading edge high lift devicesThe stabilator is a one-piece horizontal tail surface that pivots up and down about a central hinge point.
Aircraft Flight ControlSOLO
SOLO
149
Flight Control
Aircraft Flight Control
SOLO
150
Aerodynamics of Flight
Aircraft Flight Control
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SOLO
- Aerodynamic Forces( ) ( ) ( )
( ) ( ) BTBT
VTrTT
nMqNxMqA
nMqLVMqDMqA
1,,1,,
1,,1,,,,
αα
ααα
+−=
+−=
( )MqD T ,,α - Drag Force
( )MqN T ,,α - Normal Force
Mq T ,,α - Dynamic Pressure, Total Angle of Attack, Mach Number
( ) ( )
( ) ( )MCSVhD
MCSVhL
TD
q
r
TL
q
r
,2
1
,2
1
2
2
αρ
αρ
=
=
Aerodynamic Forces (Vectorial)
( )MqA T ,,α - Axial Drag Force
( )MqL T ,,α - Lift Force
( ) ( )
( ) ( )MCSVhA
MCSVhN
TA
q
r
TN
q
r
,2
1
,2
1
2
2
αρ
αρ
=
=
+=
−=
TBTBV
TBTBr
nxn
nxV
αα
αα
cos1sin11
sin1cos11
−=
+=
TVTrB
TVTrB
nVn
nVx
αα
αα
cos1sin11
sin1cos11
Aircraft Equations of Motion
SOLO
- Aerodynamic Forces( ) ( ) ( )
( ) ( ) BTBT
VTrTT
nMqNxMqA
nMqLVMqDMqA
1,,1,,
1,,1,,,,
αα
ααα
+−=
+−=
are coplanar( )01,11,1 ≠TVBrB nnandVx α
( ) ( )T
rBT
rB
rBrB
rB
rBBB
Vx
Vx
VxVx
Vx
Vxxn
ααsin
11cos
11
1111
11
1111
−=×
−•=×
××=
( ) ( )T
rTB
rB
rrBB
rB
rBrV
Vx
Vx
VVxx
Vx
VxVn
αα
sin
1cos1
11
1111
11
1111
−=×
•−=×
××=
+=
−=
TBTBV
TBTBr
nxn
nxV
αα
αα
cos1sin11
sin1cos11
+−=
+=
TVTrB
TVTrB
nVn
nVx
αα
αα
cos1sin11
sin1cos11
Aerodynamic Forces (Vectorial)
Aircraft Equations of Motion
SOLO
- Aerodynamic Forces( ) ( ) ( )
( ) ( ) BTBT
VTrTT
nMqNxMqA
nMqLVMqDMqA
1,,1,,
1,,1,,,,
αα
ααα
+−=
+−=
( ) ( )
( ) ( )MCSVhD
MCSVhL
TD
q
r
TL
q
r
,2
1
,2
1
2
2
αρ
αρ
=
=
( ) ( )( )
( ) ( )MCSVhA
MCSVhN
TA
q
r
MC
TN
q
r
TN
,2
1
,2
1
2
2
αρ
αραα
=
=
( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]
( ) ( ) ( )[ ]
MAXT
VTTNTTAr
rTTNTTAr
VTLrTDT
nMCMCSVh
VMCMCSVh
nMCVMCSVhMqA
αα
ααααρ
ααααρ
ααρα
≤
++
+−=
+−=
1cos,sin,2
1
1sin,cos,2
1
1,1,2
1,,
2
2
2
Aerodynamic Forces (Vectorial)
Aircraft Equations of Motion
154
SOLO
Drag ,Lift Coefficients as functions of Angle of AttackDrag Polar
Drag Polar
Aircraft Equations of Motion
02/28/15 155
SOLO
By changing αT from 0 to αMAX, and rotating aroundby σ (from 0 to σMAX) we obtain a Surface of Revolution Σq (CA,CN) which defines the Achievable Aerodynamic Forces for the given dynamic pressure q.
rV1
( )( )
VzVyV
MAXT
soundr
r
windr
nnn
hVVM
VShq
vRVV
1sin1cos1
/2
1 2
σσ
αα
ρ
+=
≤=
=
−×Ω−=
( ) ( ) ( )( ) ( ) VTLrTD
VTrTT
nMCqVMCq
nMqLVMqDMqA
1,1,
1,,1,,,,
αα
ααα
+−=
+−=
( ) ( )T
rTB
rB
rrBB
rB
rBrV
Vx
Vx
VVxx
Vx
VxVn
αα
sin
1cos1
11
1111
11
1111
−=×
⋅−=×
××=
( )σα,A
V
αMAXα
( )DL CC ,Σ
σMAXσ
( ) ( )αα 20 LDD CkCC +=
D
σcosL
σsinL
σMAXσ
L
L
Aerodynamic Forces (Vectorial)
Aircraft Equations of Motion
02/28/15 156
SOLO
We can see that for αT = 0
( ) ( )( )
( )( )
MA
Ar
MD
DT
TT
MCqVMCqMqA,0
0
,0
0 1,0,==
−=−==αα
α
( ) ( )Rgm
TV
m
MCqV r
D
++−= 10
and since for αT = 0
the aerodynamic forces will decrease the velocity.
We can see that for αT ≠ 0, the decelerationdue to aerodynamics will only increase.
( ) ( ) ( )[ ] ( ) ( )MqDMCqMCMCqMqD TATTNTAT ,0,sincos,0, ==>+=≠ ααααα α
The most Energy Effective Trajectory is one with αT = 0.
( )( )
VzVyV
MAXT
soundr
r
windr
nnn
hVVM
VShq
vRVV
1sin1cos1
/2
1 2
σσ
αα
ρ
+=
≤=
=
−×Ω−=
( ) ( )T
rTB
rB
rrBB
rB
rBrV
Vx
Vx
VVxx
Vx
VxVn
αα
sin
1cos1
11
1111
11
1111
−=×
⋅−=×
××=
( ) ( ) ( )( ) ( ) VTLrTD
VTrTT
nMCqVMCq
nMqLVMqDMqA
1,1,
1,,1,,,,
αα
ααα
+−=
+−=
Aerodynamic Forces (Vectorial)
Aircraft Equations of Motion
Return to Table of Content
02/28/15 157
SOLO
Specific Energy( ) ( )RgTA
mV
++= 1
( ) ( ) ( ) VTrTT nMqLVMqDMqA 1,,1,,,, ααα +−=
By Integrating this Equation we obtain:
( ) ( )∫∫ +⋅=
⋅−⋅=−
t
t
t
t
dtTAVgm
dtg
RgV
g
VVEE
00 0000
1
( ) ( ) ( )∫∫∫∫ ∫ +⋅=⋅−−−=⋅−⋅=
⋅−⋅=−
t
t
R
R dRR
ER
R
t
t
V
V
dtTAVgm
RdRRgg
VVRd
g
Rg
g
VdVdt
g
RgV
g
VVEE
0000 0 03
00
20
2
00000
11
2
µ
Define Specific Energy Derivative:( ) ( )TAV
mg
RgV
g
VVE
+⋅=⋅−⋅= 1
:00
20
0 :R
g Eµ=
( )∫ +⋅=
−−
−=
−−−=−
t
t
EEE dtTAVgmRgg
V
Rgg
V
RRgg
VVEE
0 0000
20
00
2
000
20
2
0
1
22
11
2
µµµ
Aircraft Equations of Motion
02/28/15 158
SOLO
Specific Energy (continue – 1)( ) ( )∫∫ +⋅=
⋅−⋅=−
t
t
t
t
dtTAVgm
dtg
RgV
g
VVEE
00 0000
1
0
20
2
00 20 0
g
VV
g
VdVdt
g
VVt
t
V
V
−=⋅=⋅∫ ∫
( ) ( ) ( )
( ) 03
03
20
020
2
2
20
3
20
00
0
0
0
0
00
3
20
000
3
221 hhhh
Rhhhd
R
h
RdR
RRdR
R
RRd
g
Rgdt
g
RgV
Rhh
h
hRR
Rh
R
R
dRRRdRR
R
RR
Rg
Rg
R
R
t
t
RddtVE
E
−≈−−−=
−≈
=⋅=⋅−=
⋅−
<<+=
<<
=⋅−=
=
=
∫
∫∫∫∫
µ
µ
( ) ( ) ( )[ ]∫∫ −⋅=+⋅t
t
T
t
t
dtMqDTTVgm
VdtTAV
gm00
,,111
00
α
Specific Kinetic Energy
Specific Potential Energy
( ) ( )[ ]∫ −⋅=
+−
+=−
t
t
T dtMqDTTVgm
Vh
g
Vh
g
VEE
0
,,1122 0
00
20
0
2
0 α
Specific Energy Gain due to Thrustand Loss due to Aerodynamic Drag
( )011 >⋅ TVif
Aircraft Equations of Motion
Return to Table of Content
SOLO
( ) ( )( ) ( ) ( ) ( )
( ) ( )
≥==−=
≥+×Ω=−=++=
===
min00
min00
00
/
1
mmtmmtmcTm
VVvRtVRpATTRgTAm
V
RtRRtRVR
ffvacuum
fwindaevacuum
ff
Equations of Motion (State Equations): . ( ) ( ) fttttuxftx ≤≤= 0,,, π
Controls: ( ) fttttu ≤≤0VectorThrustT
ForcescAerodynamiA
−
−
Three Degrees of Freedom Model in Earth Atmosphere
160
SOLO Aircraft Equations of Motion
161
SOLO
• Rotation Matrix from Earth to Wind Coordinates
[ ] [ ] [ ]321 χγσ=WEC
whereσ – Roll Angleγ – Elevation Angle of the Trajectoryχ – Azimuth Angle of the Trajectory
Force Equation:
amgmTFA
=++
where:
• Aerodynamic Forces (Lift L and Drag D)( )
−
−=
L
D
F WA 0
• Thrust T ( )
=
α
α
sin
0
cos
T
T
T W
• Gravitation acceleration
( ) ( )
−
−
−==
g
cs
sc
cs
sc
cs
scgCg EWE
W 0
0
100
0
0
0
010
0
0
0
001
χχχχ
γγ
γγ
σσσσ
( ) g
cc
cs
s
g W
−=
γσγσ
γ
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
Flat Earth Three Degrees of Freedom Aircraft Equations
162
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Aircraft Acceleration
( )( )
( ) ( )WWW
W VVa ×+=
→
ω
where:
( )
=
0
0
V
V W
and( )
=
→
0
0
V
VW
( )
−+
−+
−=
=
χχχχχ
γγγ
γγσ
σσσσω
0
0
100
0
0
0
0
0
010
0
0
0
0
0
001
cs
sc
cs
sc
cs
sc
r
q
p
W
W
W
W
or ( )
+−+
−=
=
γσχσγγσχσγγχσ
ωccs
csc
s
r
q
p
W
W
W
W
therefore( )
( )( ) ( ) ( )
( )
+−+−=
−=×+=
→
γσχσγγσχσγω
cscV
ccsV
V
qV
rV
V
VVa
W
WWW
WW
Flat Earth Three Degrees of Freedom Aircraft Equations
163
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Aircraft Acceleration
Flat Earth Three Degrees of Freedom Aircraft Equations
From the Force equation we obtain:
( )( )
( ) ( ) ( ) ( )( ) ( )WWWA
WWW
W gTFm
VVa ++=×+=
→ 1ω
or
( )( ) ( )
++−=+−=−=+−=
−−=
γσαγσχσγγσγσχσγ
γα
ccgmLTcscVqV
csgccsVrV
sgmDTV
W
W
/sin
/)cos(
from which we obtain:
−+=
=
γσα
γσ
coscossin
cossin
V
g
Vm
LTq
V
gr
W
W
164
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Aircraft Acceleration
Flat Earth Three Degrees of Freedom Aircraft Equations
From the Force equation we obtain:
( )( )
( ) ( ) ( ) ( )( ) ( )WWWA
WWW
W gTFm
VVa ++=×+=
→ 1ω
or
( )( ) ( ) σ
σσσ
γσαγσχσγγσγσχσγ
γα
s
c
c
s
ccgmLTcscVqV
csgccsVrV
sgmDTV
W
W
−−−
++−=+−=−=+−=
−−=
/sin
/)cos(
from which we obtain:
( )( )
+=−+=
−−=
msLTcV
cgmcLTV
sgmDTV
/sin
/sin
/)cos(
σαγχγσαγ
γα
Define the Load Factor
gm
LTn
+= αsin:
165
SOLO
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
• Velocity Equation
Flat Earth Three Degrees of Freedom Aircraft Equations
( ) ( )
==
=
0
0
V
CVC
h
y
x
V EW
WEW
E
−
−
−=
0
0
0
0
001
0
010
0
100
0
0 V
cs
sc
cs
sc
cs
sc
h
y
x
σσσσ
γγ
γγχχχχ
=
==
γχγχγ
sVh
scVy
ccVx
or
• Energy per unit mass E
g
VhE
2:
2
+=
Let differentiate this equation:( )
W
VDT
W
DTg
g
VV
g
VVhEps
−=
−−+=+== αγαγ cos
sincos
sin:
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166
SOLOFlat Earth Three Degrees of Freedom Aircraft Equations
We have
Aircraft Thrust( ) 10, ≤≤= ηη VhTT MAX
( ) ( ) soundofspeedhaNumberMachMhaVM === &/
( ) ( )MSCVhL L ,2
1 2 αρ= Aircraft Lift
( ) ( )LD CMSCVhD ,2
1 2ρ= Aircraft Drag
( ) ( )
ARek
CkMCCMC
iDC
LDLD
π1
, 20
=
+= Parabolic Drag Polar
gm
LTn
+= αsin' Total Load Number
( ) 0/0
hheh −= ρρ Air Density as Function of Height
gm
Ln = Load Factor
167
SOLO
Constraints:
State Constraints
• Minimum Altitude Limit
minhh ≥
• Maximum dynamic pressure limit
( ) ( )hVVorqVhq MAXMAX ≤≤= 2
2
1 ρ
• Maximum Mach Number limit
( ) MAXMha
V ≤
Aerodynamic or heat limitation
Three Degrees of Freedom Model in Earth Atmosphere
168
SOLO
Constraints:
• Maximum Load Factor
( )MAXn
W
VhLn ≤= ,
• Maximum Roll Angle
MAXMAX σσσ ≤≤−
• Maximum Lift Coefficient or Maximum Angle of Attack
( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤
( ) ( ) ( ) ( ) ( ) LSTALLLMAXL nVh
W
VhCVSh
W
VhCVShn ==≤ ,
,
2
1,
2
1 2_2 αρρ α
Control Constraints (continue): ( ) fttttuU ≤≤≤ 00,
Three Degrees of Freedom Model in Earth Atmosphere
02/28/15
169
SOLO
Control Constraints: ( ) fttttuU ≤≤≤ 00,
• Thrust Controls options are:
Thrust Direction
Thrust Magnitude
( ) throttableVhTT rMAX 10, ≤≤= ηη
Deflector Nozzle
Thrust Reversal Operation
F-35 Propulsion
If no Thrust Vector Control (No TVC)
BxT 11 =
1cos111 max ≤≤•≤− TBxT δIf Thrust Vector Control (TVC)
Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 170
( ) ( )
( ) ( ) ( )( )
( )
( ) ( ) ( )( )
( )
( ) ( )( ) ( ) ( )γ
χγ
χγ
χγσγ
ασγβαχ
χγγ
χγσασβαγ
χγγ
γβα
χγ
χγ
γ
cossincossin
cossincoscostan2
tansincossincos
sincos
cos
sincos
cossinsincoscoscos
coscos2coscossin
sinsincos
sinsincoscossincos
sincoscos
sincos
cos
coscos
cos
sin
*2
*2
2
*
V
aLatLat
V
RLatLat
LatR
V
Vm
LT
Vm
CTV
aLatLatLat
V
R
LatV
g
R
V
Vm
LT
Vm
CT
LatLatLatR
agm
DTV
R
V
R
V
td
Latd
LatR
V
LatR
V
td
Longd
Vtd
Rd
yWW
zWW
xWW
E
N
−Ω+−Ω−
−+++−=
−+Ω+
Ω+
−++++=
−Ω+
−−−=
==
==
=
(a) Spherical, Rotating Earth (Ω ≠ 0)
SOLO Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 171
(b) Spherical, Non-Rotating Earth (Ω = 0)
( ) ( )
( )LatR
V
Vm
LT
Vm
CT
V
g
R
V
Vm
LT
Vm
CT
agm
DTV
R
V
R
V
td
Latd
LatR
V
LatR
V
td
Longd
Vtd
Rd
xWW
E
N
tansincossincos
sincos
cos
sincos
coscossin
sinsincos
sincoscos
sincos
cos
coscos
cos
sin
*
χγσγ
ασγβαχ
γσασβαγ
γβα
χγ
χγ
γ
−+++−=
−++++=
−−−=
==
==
=
SOLO Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 172
σγ
ασγβαχ
γσασβαγ
γβαγ
ξγξγ
sincos
sincos
cos
sincos
coscossin
sinsincos
sincoscos
sin
sincos
coscos
Vm
LT
Vm
CT
V
g
Vm
LT
Vm
CT
gm
DTV
Vz
Vy
Vx
E
E
E
+++−=
−+++=
−−=
===
(c) Flat Earth
SOLO Three Degrees of Freedom Model in Earth Atmosphere
02/28/15173
(a) Spherical, Rotating Earth (Ω ≠ 0)
(b) Spherical, Non-Rotating Earth (Ω = 0)
(c) Flat Earth
0→Ω
( ) ( )
( ) ( ) ( )( )
( )
( ) ( ) ( )( )
( )
( ) ( )( )
( ) ( ) χγ
χγ
χγσγ
ασγβαχ
χγγ
χγσασβαγ
χγγ
γβα
χγ
χγ
γ
sincossincos
sincoscostan2
tansincossincos
sincos
cos
sincos
cossinsincoscoscos
coscos2coscossin
sinsincos
sinsincoscossincos
sincoscos
sincos
cos
coscos
cos
sin
2
2
2
*
LatLatV
R
LatLat
LatR
V
Vm
LT
Vm
CT
LatLatLatV
R
LatV
g
R
V
Vm
LT
Vm
CT
LatLatLatR
agm
DTV
R
V
R
V
td
Latd
LatR
V
LatR
V
td
Longd
Vtd
Rd
xWW
E
N
Ω+
−Ω−
−+++−=
+Ω+
Ω+
−++++=
−Ω+
−−−=
==
==
=
( ) ( )
( )LatR
V
Vm
LT
Vm
CT
V
g
R
V
Vm
LT
Vm
CT
agm
DTV
R
V
R
V
td
Latd
LatR
V
LatR
V
td
Longd
Vtd
Rd
xWW
E
N
tansincossincos
sincos
cos
sincos
coscossin
sinsincos
sincoscos
sincos
cos
coscos
cos
sin
*
χγσγ
ασγβαχ
γσασβαγ
γβα
χγ
χγ
γ
−+++−=
−++++=
−−−=
==
==
=
σγ
ασγβαχ
γσασβαγ
γβαγ
ξγξγ
sincos
sincos
cos
sincos
coscossin
sinsincos
sincoscos
sin
sincos
coscos
Vm
LT
Vm
CT
V
g
Vm
LT
Vm
CT
gm
DTV
Vz
Vy
Vx
E
E
E
+++−=
−+++=
−−=
===
SOLO
∞→R
Three Degrees of Freedom Model in Earth Atmosphere
Return to Table of Content
174
References
SOLO
Miele, A., “Flight Mechanics , Theory of Flight Paths, Vol I”, Addison Wesley, 1962
Aircraft Flight Performance
J.D. Anderson, Jr., “Introduction to Flight”, McGraw Hill, 1978, Ch. 6, “Elements of Airplane Performance”
A. Filippone, “Flight Performance of Fixed and Rotary Wing Aircraft”, Elsevier, 2006
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
Stengel, MAE 331, Aircraft Flight Dynamics, Princeton University
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993
F.O. Smetana, “Flight Vehicle Performance and Aerodynamic Control”, AIAA Education Series, 2001
L. George, J.F. Vernet, “La Mécanique du Vol, Performances des Avions et des Engines”, Librairie Polytechnique Ch. Béranger, 1960
L.J. Clancy, “Aerodynamics”, Pitman International Text, 1975
175
Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
SOLO Aircraft Flight Performance
J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
Naval Air Training Command, “Air Combat Maneuvering”, CNATRA P-1289 (Rev. 08-09)
Patrick Le Blaye, “Agility: Definitions, Basic Concepts, History”, ONERA
Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
References (continue – 1)
B. N. Pamadi, “Performance, Stability, Dynamics, and Control of Airplanes”, AIAA Educational Series, 1998, Ch. 2 , Aircraft Performance
L.E. Miller, P.G. Koch, “Aircraft Flight Performance”, July 1978, AD-A018 547,AFFDL-TR-75-89
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Courtland_D._Perkins,_Robert_E._Hage, “Airplane Performance Stability and Control”, John Wiley & Sons, 1949
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Asselin, M., “Introduction to Aircraft Aerodynamics”, AIAA Education Series, 1997
Aircraft Flight PerformanceReferences (continue – 2)
Donald R. Crawford, “A Practical Guide to Airplane Performance and Design”,Crawford Aviation, 1981
Francis J. Hale, “ Introduction to aircraft performance, Selection and Design”, John Wiley & Sons, 1984
J. Russell, ‘Performance and Stability of Aircraft“, Butterworth-Heinemann, 1996
Jan Roskam, C. T. Lan, “Airplane Aerodynamics and Performance”, DARcorporation, 1997
Nono Le Rouje, “Performances of light aircraft”, AIAA, 1999
Peter J. Swatton, “Aircraft performance theory for Pilots”, Blackwell Science, 2000
S. K. Ojha, “Flight Performance of Aircraft “, AIAA, 1995
W. Austyn Mair, David L._Birdsall, “Aircraft Performance”, Cambridge University Press, 1992
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E.S. Rutowski, “Energy Approach to the General Aircraft Performance Problem”, Journal of the Aeronautical Sciences, March 1954, pp. 187-195
Aircraft Flight PerformanceReferences (continue – 3)
A.E. Bryson, Jr., “Applications of Optimal Control Theory in Aerospace Engineering”, Journal of Spacecraft and Rockets, Vol. 4, No.5, May 1967, pp. 553
W.C. Hoffman, A.E. Bryson, Jr., “A Study of Techniques for Real-Time, On-Line Optimum Flight Path Control”, Aerospace System Inc., ASI-TR-73-21, January 1973, AD 758799
A.E. Bryson, Jr., “A Study of Techniques for Real-Time, On-Line Optimum Flight Path Control. Algorithms for Three-Dimensional Minimum-Time Flight Paths with Two State Variables”, AD-A008 985, December 1974
M.G. Parsons, A.E. Bryson, Jr., W.C. Hoffman, “Long-Range Energy-State Maneuvers for Minimum Time to Specified Terminal Conditions”, Journal of Optimization Theory and Applications, Vol.17, No. 5-6, Dec 1975, pp. 447-463
A.E. Bryson, Jr., M.N, Desai, W.C. Hoffman, “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 6, Nov-Dec 1969, pp. 481-488
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References (continue – 4)
Solo Hermelin Presentations http://www.solohermelin.com
• Aerodynamics Folder
• Propulsion Folder
• Aircraft Systems Folder
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TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –
Stanford University1983 – 1986 PhD AA
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182M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison Tables
183M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
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Aircraft Avionics
185Ray Whitford, “Design for Air Combat”
R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”,AIAA Publication, 2000
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188
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
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H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
190
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
191
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
192http://www.worldaffairsboard.com/military-aviation/62863-comparing-fighter-performance-same-generations-important-factor-war-2.html
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193
Aircraft Flight Performance
Drag
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194
Aircraft Flight Performance
Drag
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Aircraft Flight Performance
Drag
196http://indiandefence.com/threads/comparing-modern-western-fighters.41124/page-16
197
http://selair.selkirk.bc.ca/training/aerodynamics/range_jet.htm
198
http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
199
http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
200
201http://www.ausairpower.net/jsf.html
202http://www.ilbe.com/index.php?document_srl=2330174362&mid=military&page=406&sort_index=readed_count&order_type=desc
203