13 fixed wing fighter aircraft- flight performance - i

Fixed Wing Fighter AircraftFlight Performance

Part I

SOLO HERMELIN

Updated: 04.12.12 28.02.15

1

http://www.solohermelin.com

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


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


SOLO Fixed Wing Fighter Aircraft Flight Performance

5

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

Return to Table of Content

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


The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p.

SOLO

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


( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 ===


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)

SOLO

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


14


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.


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


16


• 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


Stratosphere

Troposphere

17


HdTR

g

P

Pd

Ta

⋅=− *0


( )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


Stratosphere

Troposphere

18


( )[ ] HdHHLTR

gHd

TR

g

P

Pd

SSTaa

⋅−⋅+⋅

=⋅=− *00

( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1* ===−⋅−=


( )[ ]∫∫ ⋅−⋅+

=−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


19

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


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

=

=


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

+=

−⋅+=


( ) ( )

( )

( )

+

+⋅−=

+

+⋅−⋅=

⋅−⋅−−=

∞

∞∞

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


21

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



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


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


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.


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


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


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

39

SOLO

Martin Knudsen (1871–1949).


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


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


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


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


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


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



Parabolic Drag Polar of a typical High Subsonic Aircraftat different Mach Numbers



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


65


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



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



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


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


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


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


SOLO

78



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



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


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


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.


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


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


89


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



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.


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


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


( )

.:

.:

:

42

53

2/

2

CoeffThrustDn

TC

CoeffPowerDn

PC

RatioAdvanceR

V

Dn

VJ

T

p

n

RD

ρ

ρ

ππ

=

=

Ω== ∞

Ω=

=

∞


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


TP

P CJ 22

121

πηη =−

pP

P CJ 33

121

πηη =−

JV

Dn

P

VT

C

C P

p

T η==∞

∞


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


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


( ) ( )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


The Aerodynamic Moment and Power of the Propeller Blade Element are

QdFdrFdUPd

FdrQd

xxT

x

Ω=⋅Ω==⋅=

97


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


( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )∫∫

∫∫

∫∫

=

=∞

=

=

=

=∞

=

=

=

=∞

=

=

+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


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


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 ,


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


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


107


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


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




111

Typical results for the variation of Thrust and Thrust Specific Fuel Consumption with Supersonic Mach number for a Turbojet




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


Turbojet Performance



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


Typical afterburning jet pipe equipment.

Afterburner


115

Thrust Reversal Operation (Used during Landing)



116

Typical results for the variation of Thrust and Thrust Specific Fuel Consumption with Subsonic Mach number for Turbojet



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


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


PowerPA =T V

TurbofanEngine

J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186



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


PowerPA =(TP+Tj) V

PA = hpr PS+Tj V

PA = hpr Pes

TurbopropEngine



Block Diagram



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


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


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





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



Block Diagram


120


0mT


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



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



125

Thrust as a function of airspeed for different Propulsion Systems



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


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


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

//

//



129Dr. Carlo Kopp, Air Power Australia, Sukhoi Su-34 Fullback, Russia's New Heavy Strike Fighter

Comparison of Fighter Aircraft Propulsion SystemsSOLO

130


131


132M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”


136M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”Return to Table of Content


137


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


138

SOLO


140

Lockheed_Martin_F-35_Lightning_II STOVL

The Unique F-35 Fighter Plane, Movie

USP 3” part F35Joint Strike Fighter ENG,

Movie


Thrust vectoring nozzle of the F135-PW-600 STOVL variant


141



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


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


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


144


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


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


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.


SOLO

149

Flight Control


SOLO

150

Aerodynamics of Flight



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

=

=


( )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


SOLO


( ) ( ) 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



SOLO


( ) ( ) 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



154

SOLO

Drag ,Lift Coefficients as functions of Angle of AttackDrag Polar

Drag Polar


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



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

+−=

+−=




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


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



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

−

−


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


163

SOLO

α

T

V

L

D

Bx

Wx

Bz

Wz

Wy

By



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



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


( ) ( )

==

=

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:


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


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,


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)


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

−+++−=

−++++=

−−−=

==

==

=


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


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



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


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

176

Courtland_D._Perkins,_Robert_E._Hage, “Airplane Performance Stability and Control”, John Wiley & Sons, 1949

SOLO

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

177

SOLO

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

178

SOLO Aircraft Flight Performance

References (continue – 4)

Solo Hermelin Presentations http://www.solohermelin.com

• Aerodynamics Folder

• Propulsion Folder

• Aircraft Systems Folder


179

SOLO

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


Comparison Tables

SOLO

184

Aircraft Avionics

185Ray Whitford, “Design for Air Combat”

R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”,AIAA Publication, 2000

188


189


190


191


192http://www.worldaffairsboard.com/military-aviation/62863-comparing-fighter-performance-same-generations-important-factor-war-2.html

SOLO

193


Drag

SOLO

194


Drag

SOLO

195


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/

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

13 fixed wing fighter aircraft- flight performance - i

Science

Transcript of 13 fixed wing fighter aircraft- flight performance - i