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Solo Hermelin
Solo Hermelin

Fighter Aircraft Performance, Part I of two, describes the parameters that affect aircraft performance. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

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Fixed Wing Fighter Aircraft Flight 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 F i x e d W i n g P a r t I I
Table of Content (continue – 3) SOLO Fixed Wing Fighter Aircraft Flight Performance 5 Air-to-Air Combat Energy–Maneuverability Theory Supermaneuverability Constraint Analysis Aircraft Combat Performance Comparison References F i x e d W i n g P a r t I I
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 Earth AtmosphereSOLO
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 m v ∆ ∆ = →∆ 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 f p n S ∆ ∆ = →∆ 0 lim 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 V R* - Universal gas constant TRNVP ⋅⋅=⋅ * V m M m N == 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) SOLO 12 T M M TM ⋅= 0 To simplify the computation let introduce: - Geopotential Altitude H - Geometric Altitude Hg Newton Gravitational Law implies: ( ) 2 0         + ⋅= gE E g HR R gHg The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ This means that g gE E g Hd HR R Hd g g Hd ⋅         + =⋅= 2 0 Integrating we obtain g gE E H HR R H ⋅         + = 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
14 Physical Foundations of Atmospheric Model Atmospheric Equilibrium Equation HdgPd ⋅⋅=− 0ρ At altitude bellow 100 km we assume t6he Equation of an Ideal Gas TRMTRP a MRR a aa ⋅⋅=⋅⋅= = ρρ / * * / Hd TR 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 Geopotential Altitude Z, km Geometric Altitude Z; km Molecular Temperature 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 Rate Lh; ͦ K/km -6.3 0.0 +1.0 +2.8 0.0 -2.8 -2.0 Earth AtmosphereSOLO
16 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. ( ) Hd HLTR g Hd TR g P Pd aa ⋅ ⋅+ =⋅=− 0 00 kmKLHLTT /3.60  −=⋅+= Integrating this equation we obtain ( )∫∫ ⋅ ⋅+ =− H a P P Hd HLTR g P PdS S 0 0 0 1 0 ( ) 0 00 lnln 0 T HLT RL g P P aS S ⋅+ ⋅ ⋅ −= Hence aRL g SS H T L PP ⋅ −       ⋅+⋅= 0 0 0 1 and           −         ⋅= ⋅ 1 0 0 0 g RL S S a P P L T H Earth AtmosphereSOLO Stratosphere Troposphere
17 Physical Foundations of Atmospheric Model Hd TR g P Pd Ta ⋅=− * 0 Integrating this equation we obtain ( )T TaS S HH TR g P P T −⋅ ⋅ −= * 0 ln Hence ( )T Ta T HH TR g SS ePP −⋅ ⋅ − ⋅= * 0 and S STTa T P P g TR HH ln 0 * ⋅ ⋅ += ∫∫ =− H HTa P P T S TS Hd TR 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 ( )[ ] Hd HHLTR g Hd 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 Hd HHLTR g P Pd * 0 1 ( )[ ] * * 0 lnln T ST aSSS S T HHLT RL g P P −⋅+ ⋅ ⋅ = Hence ( ) aRL g S T S SSS HH T L PP ⋅ −         −⋅+⋅= 0 * 1 and           −      ⋅+= ⋅ 1 0 * g RL SS S S T S aS P P L T HH Stratosphere Region (HS=20.0 km to 32.0 km). Stratosphere Troposphere Earth AtmosphereSOLO
19 1962 Standard Atmosphere from 86 km to 700 km Layer Index Geometric Altitude km Molecular Temperature , K Kinetic Temperature K Molecular Weight Lapse Rate K/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 kmK Zd 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 ZZ ATT C kma KA KTC 9429.19 3232.76 1902.263 −= −= =   Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10) ( ) kmK Zd Td ZZLTT Z /0.12 99  += −⋅+= Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11) ( ) ( ) ( ) ( )       + + ⋅−=       + + ⋅−⋅= ⋅−⋅−−= ∞ ∞∞ ZR ZR ZZ kmK ZR ZR TT Zd Td TTTT E E E E 10 10 10 10 10 / exp ξ λ ξλ  KT kmR km E  1000 10356766.6 /01875.0 3 = ×= = ∞ λ Earth AtmosphereSOLO
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 Earth AtmosphereSOLO Return to Table of Content
SOLO Atmosphere Continuum Flow Low-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 ~~~~ =⋅∇+ u t  ρ ∂ ρ∂ ( ) ( )u R u R pG F uu t u eer ~~~~1 3 4~~~~1~~~~1~~~ ~ ~ ~ 2   ⋅∇∇+×∇×∇−∇−=        ∇⋅+ µµρ ∂ ∂ ρ(C.L.M.): ( ) ( )Tk PRt Q uG F u t p Hu t H rer ∇⋅∇−+⋅+⋅⋅∇+=        ∇⋅+ ∂ ∂ 11 ~ ~ ~~~1~~~ ~ ~~~~ ~ ~ ~ 2 ∂ ∂ ρτ ∂ ∂ ρ  (C.E.): Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = Froude: 0 0 gl U Fr = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/ ~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/ ~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ 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 = Knudsen l Kn 0 0 : λ = AERODYNAMICS Return to Table of Content
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 U M = 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 TR a 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 <270 Most 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- 915 980-1,470 270-410 Transonic aircraft nearly always have swept wings, delaying drag- divergence, and often feature design adhering to the principles of the Whitcomb Area rule. Supersonic 1.2–5.0 915- 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 5.0–10.0 3,840– 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 10.0–25.0 7,680– 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 >16,25 0 >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 Return to Table of Content
27 SOLO - when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves. a V M M =      = − & 1 sin 1 µ 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 WAVES AERODYNAMICS
28 SOLO SHOCK & EXPANSION WAVES M < 1 M = 1 M > 1 Mach Waves AERODYNAMICS
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. Return to Table of Content AERODYNAMICS
30 SHOCK WAVES SOLO A shock wave occurs when a supersonic flow decelerates in response to a sharp increase in pressure (supersonic compression) or when a supersonic flow encounters a 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 is an 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 <<<<<<< MMMMMMMM SOLO AERODYNAMICS Return to Table of Content
32 where ρ0 = air density U0 = true speed l 0= characteristic length μ0 = absolute (dynamic) viscosity υ0 = kinematic viscosity NumberReynolds:Re 0 00 0 000 0 0 0 υµ ρ ρ µ υ lUlU = == 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 1884AERODYNAMICS
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 between Laminar and Turbulent Flow. SOLO AERODYNAMICS Boundary-Layer
36 Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity) AERODYNAMICS SOLO Return to Table of Content
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. Knudsen l Kn 0 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 TkB 20 2 σπ λ = • 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 ≈ 8x10-8 m. 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 ]. 0 2 1 λρµ c= m Tk c B π 8 = Tk m B2 0 π ρ µ λ = AERODYNAMICS
39 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:Re 0 000 µ ρ lU = Tk m lmTklallU aUM BB γρ µ γρ µ ρ µ µρ 00 0 00 0 000 0 0000 00 // / Re ==== Kn Tk m lTk m l BB == 22 00 0 00 0 π ρ µπγ γρ µ 2Re πγM Kn = 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 particles 2.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 πγM Kn = AERODYNAMICS
SOLO Knudsen Number (continue – 4) Inviscid Limit Free Molecular LimitKnudsen Number Boltzman Equation Collisionless Boltzman Equation Discrete Particle model Euler Equation Navier-Stokes Equation Continuum model Conservation Equation do not form a closed set Validity of conventional mathematical models as a function of local Knudsen 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. Knudsen l Kn 0 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) 0 0 T T MaTAS = qc is impact pressure P is static pressure         −      += 11 5 7 2 0 0 P q T T aTAS c Flight Instruments
SOLO Flight Instruments
SOLO 44 Flight Instruments Airspeed Indicators 2 2 1 vpp 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: 0ρ ρ VVE = V – True Airspeed ρ – Air Density ρ0 – Air Density at Sea Level
45 http://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 a Vehicle Surface SW. AERODYNAMICS
53 SOLO ( )       − − = L D F W A  VelocitytoNormalForceLiftL VelocitytooppositeForceDragD − − L D CSVL CSVD 2 2 2 1 2 1 ρ ρ = = ( ) ( ) tCoefficienLiftRMC tCoefficienDragRMC eL eD − − βα βα ,,, ,,, anglesideslipandattackofangle viscositydynamic lengthsticcharacteril soundofspeedHa numberReynoldslVR BodytoRelativeVelocityFlowV numberMachaVM e − − − − −= − −= βα µ µρ , )( / / AERODYNAMICS ( ) V W A nLVDF 11 −−=  Aerodynamic Forces Lift and Drag Forces
54 SOLO ( ) ( )∫∫ ∫∫ ⋅+⋅−= ⋅+⋅−= W W S VfVpL S fpD dsntCnnC S C dsVtCVnC S C 1ˆ1ˆ 1 1ˆ1ˆ 1 Wf Wp Ssurfacetheontcoefficienfriction V f C Ssurfacetheontcoefficienpressure V pp C −= − − = ∞ 2/ 2/ 2 2 ρ ρ ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − Aerodynamic Forces CD – Drag Coefficient CL – Lift Coefficient AERODYNAMICS
( ) [ ] [ ]( )∫ ∫ ∞∞ = −−−= −=′ Edge Trailing Edge Leading sideuppersidelower cos Edge Trailing Edge Leading sideuppersidelower pp cospcosp dxpp dsL sdxd USLS θ θθ Divide left and right sides of the first equation by cV 2 2 1 ∞ρ ∫             − − − = ′ ∞ ∞ ∞ ∞ ∞ Edge Trailing Edge Leading upperlower c x d V pp V pp cV L 222 2 1 2 1 2 1 ρρρ We get: Relationship between Lift and Pressure on Airfoil Lower Surface Upper Surface ( )∫ −=− Edge Trailing Edge Leading sideuppersidelower sinpsinp dsD USLS θθ Lift – Aerodynamic component normal to V Drag – Aerodynamic component opposite to V SOLO AERODYNAMICS Aerodynamic Forces
From the previous slide, ∫             − − − = ′ ∞ ∞ ∞ ∞ ∞ Edge Trailing Edge Leading upperlower c x d V pp V pp cV L 222 2 1 2 1 2 1 ρρρ The left side was previously defined as the sectional lift coefficient Cl. The pressure coefficient is defined as: 2 2 1 ∞ ∞− = V pp Cp ρ Thus, ( )∫ −= edge Trailing edge Leading upperplowerpl c x dCCC ,, Lower Surface Upper Surface 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 Upper xd yd ∞p∞p α   0,, 2 0 2 0 D TurbulentforMore LaminarforLess dragFriction fD TurbulentforLess LaminarforMore dragPressure pDD stall a L CCCC aC =+= <== = αααπα π Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR → ∞)  ( )       −=−= ARe C C L i a L π απααπ 22 0 ARe C be SC V w L SbAR Li i ππ α / 2 2 = === α π α π ARe a a ARe CL 0 0 1 2 1 2 + = + = ARe C C L Di π 2 =    AR C CCCCC L D drag induced D drag friction fD drag pressure 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) AERODYNAMICS Aerodynamic Drag
60 AERODYNAMICS Drag Variation with Mach Number SOLO Aerodynamic Drag
61 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 SOLO AERODYNAMICS Aerodynamic 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 AERODYNAMICS Aerodynamic Drag
63N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993 Parabolic Drag Polar of a typical High Subsonic Aircraft at different Mach Numbers SOLO AERODYNAMICS Aerodynamic Drag
64N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993 Variation of CD0 (M) for a supersonic aircraft Variation of aerodynamic characteristic for a typical subsonic transport aircraft Variation of aerodynamic characteristic for a typical supersonic fighter aircraft SOLO AERODYNAMICS Aerodynamic 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 1 1 2 1 1 − ∞ ∞             − + − + = γγ γ γ A A M M p p p∞, M∞ - Pressure and Mach Number upstream the Airfoil pA, MA- Pressure and Mach Number at a point A on the Airfoil Critical Mach Number The Pressure Coefficient Cp is computed using ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C 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 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C 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 or Laiton’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 of the 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 Number Drag Divergence Mach Number The Drag at small Mach number, due to Profile Drag with Induced Drag =0 (αi = 0) is constant (points a, b, and c) until M∞ = 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 of supersonic 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 separate From 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 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 Airfoil Df – Skin Friction Drag Dp – Pressure Drag due to Flow Separation Dw – 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 AERODYNAMICS Aerodynamic Drag
70 SOLO http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtChap5.pdf Categorization of Drag AERODYNAMICS Aerodynamic Drag
71 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag CD0 due to Flow Separation SOLO Aerodynamic Drag
72 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag due to Viscosity: 1.Skin Friction 2.Flow Separation (Drop in pressure behind body) ∫∫ ∫∫         ⋅+⋅ − −=         ⋅+⋅−= ∧∧ ∞ ∧∧ W W S S fpD ds w t V f w n V pp S ds w tC w nC S C xx xx 11 11 ˆ 2/ ˆ 2/ 1 ˆˆ 1 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 Return to Table of Content
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 and the average chord is the wing area (b x c = S). 4. Aspect Ratio, AR, is defined as: ( )∫− = 2/ 2/ b b dyycS ( ) b S dyyc b c b b == ∫− 2/ 2/ 1 S b AR 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 perpendicular to root chord. 8. Mean aerodynamic chord, rc Λ r t c c =λ tc λ ( )[ ]∫− = 2/ 2/ 21~ b b dyyc S 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-back Wing Delta Wing Elliptic Wing SOLO Return to Table of Content
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 Return to Table of Content
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 Return to Table of Content
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 Compressor Shaft 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 spool B. High pressure spool C. Stationary components 1. Nacelle 2. Fan 3. Low pressure compressor 4. High pressure compressor 5. Combustion chamber 6. High pressure turbine 7. Low pressure turbine 8. Core nozzle 9. 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 Return to Table of Content 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 2 11 2 22 2 1 2 1 VpVp ρρ +=+ 2 44 2 33 2 1 2 1 VpVp ρρ +=+ Bernoulli’s equations on each side of the Disk: Far from the Disk we have the same ambient pressure, hence: 41 pp = Therefore ( )2 1 2 423 2 1 VVpp −=− 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 ( )2 1 2 423 2 1 VVpp −=− ρ The Thrust provided by the Propeller Disk is given by: ( ) ( )143140 VVAVVVmT p −=−= ρ where - Fluid mass flow [kg/sec] through DiskpAVm 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 2 1 2 423 2 1 −=−= ρ From the two expressions of Thrust we obtain ( ) ( )2 1 2 4143 2 1 VVVVV −=− ( )413 2 1 VVV += 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 ⋅+= =−= ∞ρ ρ  We found Let compute vs as function of other parameters 0 2 vv 2 =−+ ∞ p pp A T V ρ 0 222 v 2 >+      +−= ∞∞ p p A TVV ρ This 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 TVV T DiskatVelocityFlowThrustVT Vm VmVm InFlowEnergyOutFlowEnergyP ρ222 ___v vv2 2 1 v2 2 1 2 0 2 0 2 0  
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. ( ) ( )aaVDVAT Va ppp p +=⋅+= ∞ = ∞ ∞ 1 2 vv2 22 /v: ρ π ρ ( ) aVV V VT VT PowerOutput PowerInput Va ppp P p + = + = + = +⋅ ⋅ == ∞= ∞∞ ∞ ∞ ∞ 1 1 /v1 1 vv /v: η ( ) p P P C JDV P aa 323 2 3 122 1 1 πρπη η ==+= − ∞ ( ) ( ) aaVDVTP p 232 1 2 v +=+= ∞∞ ρ π ( ) T P P C JDV T aa 2222 122 1 1 πρπη η ==+= − ∞ where Actuator Disk (One-Dimensional Momentum) Theory ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J T p n RD ρ ρ ππ = = Ω == ∞ Ω= = ∞
SOLO Propeller Aerodynamics The Efficiency of an Ideal Propeller ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J T p n RD ρ ρ ππ = = Ω == ∞ Ω= = ∞ E.Torenbeek, H.Wittenberg, “Flight Physics – Essentials of Aeronauical Disciplines and Technology, with Historical Notes”, Springer, 2009 Typical Propeller Diagram Actuator Disk (One-Dimensional Momentum) Theory T P P C J 22 121 πη η = − p P P C J 33 121 πη η = − 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, with Historical 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 J C C p T P =η ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J 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 V J 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 angleϕ [rad] V – air velocity [m/sec], normal to rotation plane V = V∞ + v Ω – 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 1 2 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 Element CL – 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 φφ φφ 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 summation of 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 cossin 2 1 cossin 2 1 sincos 2 1 φαφαρ φαφαρ φαφαρ ( ) r V r 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 Aerodynamics Blade Element Theory. 42 22 242 53 32 253 2 2 4 : 4 : : D T Dn T C R P Dn P C RatioAdvance R V Dn V J n RD T n RD p n RD Ω == Ω == Ω == Ω = = Ω = = ∞ Ω = = ∞ ρ π ρ ρ π ρ π π π π  ( ) ( ) ( ) ( )( )∫ = = ∞       −      + Ω = Ω = Rr r DLT R r drCrC R r R V R cB R T C 0 2 2 2 22 22 42 2 sincos 84 φαφαπ π π π ρ π σ  ( ) ( ) ( ) ( )( )∫ = = ∞       +      + Ω = Ω = Rr r DLP R r drCrC R r R rV R Bc R P C 0 2 2 2 2 22 53 3 cossin 84 φαφαππ π π π ρ π σ We have or Let use the definitions: ( ) Solidity R cB R RcB DiskSurface ElementsBladeSurface == == π π σ 2 : ( ) ( ) ( ) ( ) ( )( )∫ = = = −+= 1 0 222 / sincos 8 x x DL Rrx T xdxCxCxJC φαφαπσ π Thrust Coefficient ( ) ( ) ( ) ( ) ( )( )∫ = = = ++= 1 0 222 / cossin 8 x x DL Rrx P xdxCxCxxJC φαφαππσ π Power Coefficient
99 SOLO Propeller Aerodynamics Blade Element Theory. 42 22 242 53 32 253 2 2 4 : 4 : : D T Dn T C R P Dn P C RatioAdvance R V Dn V J n RD T n RD p n RD Ω == Ω == Ω == Ω = = Ω = = ∞ Ω = = ∞ ρ π ρ ρ π ρ π π π π Characteristic Curves of a Propeller Propeller Efficiency. J C C Dn V C C CDn VCDn P VT P T P T P T ==== ∞∞∞ 53 42 ρ ρ η
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 etropospher x P P x 1 75.0 00 ρ ρ
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 cossin000 2 00 2 00000 & mAUmmAU feee  =+= ρρUsing C.M. ( ) ( ) 00000 2 00 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 , ( ) 0 00000 00 00 /:111 mmf A A p p U U f Ap Um Ap f eee   =      −+      −+= T 2 0 2 0 00 002 0 0 2 00 0 2 00 00 2 000 00 00 MM TR TR M p a p U Ap UA Ap Um γ ρ γρρρρ =====  ( ) 0 000 2 0 00 /:111 mmf A A p p U U fM Ap f eee =      −+      −+= γ T 000 2 00 0 00 000000 MApaM TR Ap aUAam γρ === ( ) 0 0000 0 00000 /:1 1 11 1 mmf A A p p MU U fM ApMam f eee   =      −      +      −+=      = γγ TT 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 ForceThrust I ff sp TTT       ==== γ   Specific Impulse 0000 11 ApMfa gIsp T       = γ Specific Fuel Consumption (SFC) spIg f ThrustofPound HourperBurnedFuelofPound S 1 : ==== 0 f mT/T m   Air Breathing Jet Engines
107 Air Breathing Jet Engines PRESSURE Compressor Pressure Rise Turbine Pressure Drop (Turbojet) Heat Added in Combustion Chambers by 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 Compressor Pressure Rise Turbine Pressure Drop (Turbojet) Heat Added in Combustion Chambers by burning mfuel mass TOTAL TEMPERATURE Pressure corresponding to mfuel and ThrustA B C D1 F E Additional Turbine Pressure Drop in 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. ρ ρ = T T Sensitivity of Thrust and Specific Fuel Consumption with Velocity 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 with Velocity 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 with Velocity and Altitude for a Jet Engine Supersonic Conditions 1 2 2 1 1 −       − += γ γ γ M p p static total Ptotal is the pressure entering the Compressor from the Diffuser, that further increases the pressure and therefore the exit Velocity Ue and the Thrust. From the Figure we obtain that for the specific aircraft the Supersonic Thrust is given by ( )118.11 0 1 −+= = M MT T ..LSS.L. ρ ρ = 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 variation 1.Subsonic: T is constantwithV’ 2. Supersonic: T/Tm=1=1+1.18 (M’-1) Velocity variation 1. Subsonic: TSFC= 1.0+k M’ 2. Supersonic: TSFCis constant’ Altitude variation SFCis constantwith Altitude Specificfuel Consumption Power PA =T V’ Turbojet Engine Aircraft Propulsion Summary
117 Altitude variation T/T0=(ρ/ρ0 )m Velocity variation 1High bypassratio:T/TV=0=AM’ -n 2.Lowbypassratio:TfirstincreaseswithM’ then decreasesathigh supersonicM’ Velocity variation 1.High Bypassct =B(1.0+kM’ ) 2.LowBypass:ct graduatelyincreaseswithvelocity Altitude variation ct isconstantwith Altitude Specificfuel Consumption Power PA =TV’ Turbofan Engine J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186 Aircraft Propulsion Summary Aircraft Propulsion SystemSOLO
118 Velocityvariation PA isconstantwithM’ Altitude variation PA/PA,0 =(ρ/ρ0)m Velocityvariation CA isconstantwithV’ Altitude variation CA isconstantwithAltitude Specificfuel Consumption Power PA=(TP+Tj)V’ PA=hpr PS+Tj V’ PA=hpr Pes Turboprop Engine 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 variation 1. P/P0 = ρ/ρ0 2. (slightly more accurate) P/P0 =1.132 ρ/ρ0-0.132 Velocity variation Shaft Power P constant with V’ Velocity variation SFC is constant with V’ Altitude variation SFC is constant with Altitude Altitude variation T/T0 = ρ/ρ0 Velocity variation 1.Subsonic: T is constant with V’ 2. Supersonic: T/Tm=1=1+1.18 (M’-1) Velocity variation 1. Subsonic: TSFC = 1.0+k M’ 2. Supersonic: TSFC is constant’ Altitude variation SFC is constant with Altitude Altitude variation T/T0 =( ρ/ρ0 )m Velocity variation 1High bypass ratio: T/TV=0=A M’ -n 2. Low bypass ratio: T first increases with M’ then decreases at high supersonic M’ Velocity variation 1. High Bypass ct = B (1.0+k M’ ) 2.Low Bypass: ct graduately increases with velocity Altitude variation c t is constant with Altitude Velocity variation PA is constant with M’ Altitude variation PA/PA,0 = (ρ/ρ0)m Velocity variation CA is constant with V’ Altitude variation CA is constant with Altitude Specific fuel Consumption Specific fuel Consumption Specific fuel Consumption Specific fuel Consumption Power PA =T V’ Power PA =T V’ Power PA = hpr P hpr = f (J) J = V’/(N D) Power PA =(TP+Tj) V’ PA = hpr PS+Tj V’ PA = hpr Pes Reciprocating Engine/ Propeller Combination Turbojet Engine Turbofan Engine Turboprop Engine 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 0m T Aircraft Propulsion Summary SOLO
121 Air Breathing Jet Engines Aircraft Propulsion Summary SOLO
122 Air Breathing Jet Engines Aircraft 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 where ηp - propeller efficiency ηI - ideal propulsive efficiency ηnet-max ≈ 0.85 – 0.9 Efficiency decrease with airspeed Engine power decreases with altitude - Proportional with air density w/o supercharger V P V P T engine net engine 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
126 Stengel, MAE331, Lecture 6 Thrust of a Turbojet Engine ( )         −      +−      −      − = 11 11 2/1 00 0 c t c t t VmT τθ θ τ θ θ θ θ fuelair mmm  += ( ) heatsspecificofratio p p ambient stag =      = − γθ γγ , /1 0       = etemperaturambientfreestream etemperaturinletturbine 0θ       = etemperaturinletcompressor etemperaturoutletcompressor cτ • 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 SOLO Aircraft Propulsion System
128 Power and Thrust • Propeller • Turbojet • Throttle Effect airspeedoftindependenSVCVTPPower T ≈=•== 3 2 1 ρ airspeedoftindependenSVCTThrust T ≈== 2 2 1 ρ 10 2 1 2 max max ≤≤== TSVTCTTT T δρδδ Specific Fuel Consumption, SFC = cP or cT • Propeller aircraft • Jet aircraft [ ] [ ]       → → = −= −= lbf slb or kN skg c HP slb or kW skg c 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 Systems SOLO
130 Comparison of Fighter Aircraft Propulsion Systems SOLO
131 Comparison of Fighter Aircraft Propulsion Systems SOLO
132M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
133M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
134M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
135M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
136M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Return to Table of Content Comparison of Fighter Aircraft Propulsion Systems SOLO
137 SOLO Aircraft Propulsion System VTOL - Vertical Take off and Landing capability The advantages of vertical take off and landing VTOL are quite obvious. Conventional aircraft have to operate from a small number of airports with long runways. VTOL aircraft can take off and land vertically from much smaller areas. STOL - Short takeoff and landing These aircraft using thrust vectoring to decrease the distance needed for takeoff and landing but don’t have enough thrust vectoring capability to perform a vertical take off or landing. VSTOL - An aircraft that can perform either vertical or short takeoff and landings STOVL - Short takeoff and vertical land. An aircraft that has insufficient lift for vertical flight at takeoff weight but can 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 F35 Joint 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 System SOLO 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 Temperature NH – Speed of Rotation of Engine Shaft Tt - Total Temperature FCU – 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 Systems With Electrical Throttle Signaling : Engine Control System Return to Table of Content
145 Aircraft Flight ControlSOLO
146 center stickailerons 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 forwards rudder 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 Rudder Pedals Aircraft Flight ControlSOLO
148 The effect of left rudder pressure Four common types of flaps Leading edge high lift devices The 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 Return to Table of Content
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 rB BB Vx Vx VxVx Vx Vx xn α α sin 11cos 11 1111 11 11 11 − = × −• = × × ×= ( ) ( ) T rTB rB rrBB rB rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × •− = × × ×=     += −= 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 αρ αα    = = ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] 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 Attack Drag Polar Drag Polar Aircraft Equations of Motion
02/28/15 155 SOLO By changing αT from 0 to αMAX, and rotating around by 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 rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × ⋅− = × × ×= ( )σα,A  V  α MAXα ( )DL CC ,Σ σ MAXσ ( ) ( )αα 2 0 LDD CkCC += D σcosL σsinL σ MAXσ 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 ( ) ( )Rg m T V m MCq V r D    ++−= 10 and since for αT = 0 the aerodynamic forces will decrease the velocity. We can see that for αT ≠ 0, the deceleration due 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 rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × ⋅− = × × ×= ( ) ( ) ( ) ( ) ( ) 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 m V  ++= 1 ( ) ( ) ( ) VTrTT nMqLVMqDMqA 1,,1,,,, ααα +−=  By Integrating this Equation we obtain: ( ) ( )∫∫ +⋅=         ⋅− ⋅ =− t t t t dtTAV gm dt g Rg V g VV EE 00 000 0 1  ( ) ( ) ( )∫∫∫∫ ∫ +⋅=⋅ − − − =⋅− ⋅ =         ⋅− ⋅ =− t t R R dRR E R R t t V V dtTAV gm RdR Rgg VV Rd g Rg g VdV dt g Rg V g VV EE 0000 0 0 3 00 2 0 2 0000 0 11 2          µ Define Specific Energy Derivative: ( ) ( )TAV mg Rg V g VV E   +⋅=⋅− ⋅ = 1 : 00 2 0 0 : R g Eµ = ( )∫ +⋅=        −−      −=      −− − =− t t EEE dtTAV gmRgg V Rgg V RRgg VV EE 0 0000 2 0 00 2 000 2 0 2 0 1 22 11 2 µµµ Aircraft Equations of Motion
02/28/15 158 SOLO Specific Energy (continue – 1) ( ) ( )∫∫ +⋅=         ⋅− ⋅ =− t t t t dtTAV gm dt g Rg V g VV EE 00 000 0 1  0 2 0 2 00 20 0 g VV g VdV dt g VV t t V V − = ⋅ = ⋅ ∫ ∫    ( ) ( ) ( ) ( ) 0 3 0 3 2 0 02 0 2 2 2 0 3 2 0 00 0 0 0 0 00 3 2 0 0 00 3 2 21 hhhh R hhhd R h Rd R R RdR R R Rd g Rg dt g Rg V Rhh h hRR Rh R R dRRRdRR R R R Rg R g R R t t RddtV E E −≈−−−=        −≈ =⋅=⋅−=      ⋅− <<+= << =⋅ −= = = ∫ ∫∫∫∫       µ µ ( ) ( ) ( )[ ]∫∫ −⋅=+⋅ t t T t t dtMqDTTV gm V dtTAV gm 00 ,,11 1 00 α  Specific Kinetic Energy Specific Potential Energy ( ) ( )[ ]∫ −⋅=        +−      +=− t t T dtMqDTTV gm V h g V h g V EE 0 ,,11 22 0 0 0 2 0 0 2 0 α Specific Energy Gain due to Thrust and Loss due to Aerodynamic Drag ( )011 >⋅ TVif Aircraft Equations of Motion Return to Table of Content
SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       ≥==−= ≥+×Ω=−=++= === min00 min00 00 / 1 mmtmmtmcTm VVvRtVRpATTRgTA m V RtRRtRVR ffvacuum fwindaevacuum ff    Equations of Motion (State Equations): . ( ) ( ) fttttuxftx ≤≤= 0,,, π Controls: ( ) fttttu ≤≤0 VectorThrustT 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=W EC where σ – Roll Angle γ – Elevation Angle of the Trajectory χ – Azimuth Angle of the Trajectory Force Equation: amgmTFA  =++ where: • Aerodynamic Forces (Lift L and Drag D) ( )           − − = L D F W A 0  • Thrust T ( )           = α α sin 0 cos T T T W  • Gravitation acceleration ( ) ( )                     −           −           − == g cs sc cs sc cs scgCg EW E W 0 0 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ ( ) g cc cs s g W          − = γσ γσ 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 ( ) ( ) ( ) ( )WW W W VVa  ×+= → ω where: ( )           = 0 0 V V W  and ( )           = → 0 0 V V W   ( )                                         −+                     − +                     − =           = χ χχ χχ γ γγ γγσ σσ σσω     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 ( )           +− + − =           = γσχσγ γσχσγ γχσ ω ccs csc s r q p W W W W     therefore ( ) ( ) ( ) ( ) ( ) ( )           +− +−=           − =×+= → γσχσγ γσχσγω cscV ccsV V qV rV V VVa W W WW W W     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: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( )     ++−=+−=− =+−= −−= γσαγσχσγ γσγσχσγ γα ccgmLTcscVqV csgccsVrV sgmDTV W W /sin /)cos(    from which we obtain:       − + = = γσ α γσ coscos sin cossin V g Vm LT q V g r 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: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( ) σ σ σ σ γσαγσχσγ γσγσχσγ γα s c c s ccgmLTcscVqV csgccsVrV sgmDTV W W −− −      ++−=+−=− =+−= −−= /sin /)cos(    from which we obtain: ( ) ( )     += −+= −−= msLTcV cgmcLTV sgmDTV /sin /sin /)cos( σαγχ γσαγ γα    Define the Load Factor gm LT n + = 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 E W WE W E                          −           −          − =           0 0 0 0 001 0 010 0 100 0 0 V cs sc cs sc cs sc h y x σσ σσ γγ γγ χχ χχ         = = = γ χγ χγ sVh scVy ccVx    or • Energy per unit mass E g V hE 2 : 2 += Let differentiate this equation: ( ) W VDT W DT g g V V g VV hEps − =            − − +=+== α γ α γ cos sin cos sin:   Return to Table of Content
166 SOLO Flat 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 2 ρ= Aircraft Drag ( ) ( ) ARe k CkMCCMC iDC LDLD π 1 , 2 0 = +=  Parabolic Drag Polar gm LT n + = αsin ' Total Load Number ( ) 0/ 0 hh eh − = ρρ Air Density as Function of Height gm L n = 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 ( ) MAXM ha V ≤ Aerodynamic or heat limitation Three Degrees of Freedom Model in Earth Atmosphere
168 SOLO Constraints: • Maximum Load Factor ( ) MAXn W VhL n ≤= , • Maximum Roll Angle MAXMAX σσσ ≤≤− • Maximum Lift Coefficient or Maximum Angle of Attack ( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤ ( ) ( ) ( ) ( ) ( ) LSTALL LMAXL nVh W VhC VSh W VhC VShn ==≤ , , 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 δ If Thrust Vector Control (TVC) Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 170 ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) γ χ γ χγ χγσ γ α σ γ βα χ χγγ χγσ α σ βα γ χγγ γ βα χγ χγ γ cos sincossin cos sincoscostan2 tansincossin cos sin cos cos sincos cossinsincoscoscos coscos2coscos sin sin sincos sinsincoscossincos sin coscos sincos cos coscos cos sin * 2 * 2 2 * V a LatLat V R LatLat Lat R V Vm LT Vm CT V a LatLatLat V R Lat V g R V Vm LT Vm CT LatLatLatR ag m DT V R V R V td Latd LatR V LatR V td Longd V td 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) ( ) ( ) ( )Lat R V Vm LT Vm CT V g R V Vm LT Vm CT ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N tansincossin cos sin cos cos sincos coscos sin sin sincos sin coscos sincos cos coscos cos sin * χγσ γ α σ γ βα χ γσ α σ βα γ γ βα χγ χγ γ − + + + −=       −+ + + + = −− − = == == =    SOLO Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 172 σ γ α σ γ βα χ γσ α σ βα γ γ βα γ ξγ ξγ sin cos sin cos cos sincos coscos sin sin sincos sin coscos sin sincos coscos Vm LT Vm CT V g Vm LT Vm CT g m DT V Vz Vy Vx E E E + + + −= − + + + = − − = = = =       (c) Flat Earth SOLO Three Degrees of Freedom Model in Earth Atmosphere
02/28/15 173 (a) Spherical, Rotating Earth (Ω ≠ 0) (b) Spherical, Non-Rotating Earth (Ω = 0) (c) Flat Earth 0→Ω ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) χ γ χγ χγσ γ α σ γ βα χ χγγ χγσ α σ βα γ χγγ γ βα χγ χγ γ sincossin cos sincoscostan2 tansincossin cos sin cos cos sincos cossinsincoscoscos coscos2coscos sin sin sincos sinsincoscossincos sin coscos sincos cos coscos cos sin 2 2 2 * LatLat V R LatLat Lat R V Vm LT Vm CT LatLatLat V R Lat V g R V Vm LT Vm CT LatLatLatR ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N Ω + −Ω− − + + + −= + Ω + Ω+      −+ + + + = −Ω+ −− − = == == =    ( ) ( ) ( )Lat R V Vm LT Vm CT V g R V Vm LT Vm CT ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N tansincossin cos sin cos cos sincos coscos sin sin sincos sin coscos sincos cos coscos cos sin * χγσ γ α σ γ βα χ γσ α σ βα γ γ βα χγ χγ γ − + + + −=       −+ + + + = −− − = == == =    σ γ α σ γ βα χ γσ α σ βα γ γ βα γ ξγ ξγ sin cos sin cos cos sincos coscos sin sin sincos sin coscos sin sincos coscos Vm LT Vm CT V g Vm LT Vm CT g m DT V 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
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 Performance References (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 Performance References (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 Return to Table of Content
179 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – Stanford University 1983 – 1986 PhD AA
180
181
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” Return to Table of Content
SOLO 184 Aircraft Avionics
185 Ray Whitford, “Design for Air Combat” R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”, AIAA Publication, 2000
SOLO
SOLO
188 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
189 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
SOLO 193 Aircraft Flight Performance Drag
SOLO 194 Aircraft Flight Performance Drag
SOLO 195 Aircraft Flight Performance Drag
196 http://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=
203

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13 fixed wing fighter aircraft- flight performance - i

  • 1. Fixed Wing Fighter Aircraft Flight Performance Part I SOLO HERMELIN Updated: 04.12.12 28.02.15 1 http://www.solohermelin.com
  • 2. 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
  • 3. 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
  • 4. 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 F i x e d W i n g P a r t I I
  • 5. Table of Content (continue – 3) SOLO Fixed Wing Fighter Aircraft Flight Performance 5 Air-to-Air Combat Energy–Maneuverability Theory Supermaneuverability Constraint Analysis Aircraft Combat Performance Comparison References F i x e d W i n g P a r t I I
  • 6. 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. 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
  • 9. 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 m v ∆ ∆ = →∆ 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 f p n S ∆ ∆ = →∆ 0 lim kPamNbar 100/101 25 == ( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === The Atmospheric Pressure at Sea Level is: Earth Atmosphere
  • 10. 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 V R* - Universal gas constant TRNVP ⋅⋅=⋅ * V m M m N == ρ& MTRP /* ⋅⋅= ρ Earth AtmosphereSOLO
  • 11. ( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === Earth AtmosphereSOLO
  • 12. 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 T M M TM ⋅= 0 To simplify the computation let introduce: - Geopotential Altitude H - Geometric Altitude Hg Newton Gravitational Law implies: ( ) 2 0         + ⋅= gE E g HR R gHg The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ This means that g gE E g Hd HR R Hd g g Hd ⋅         + =⋅= 2 0 Integrating we obtain g gE E H HR R H ⋅         + = Earth Atmosphere
  • 13. 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
  • 14. 14 Physical Foundations of Atmospheric Model Atmospheric Equilibrium Equation HdgPd ⋅⋅=− 0ρ At altitude bellow 100 km we assume t6he Equation of an Ideal Gas TRMTRP a MRR a aa ⋅⋅=⋅⋅= = ρρ / * * / Hd TR 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. 15 Layer Index Geopotential Altitude Z, km Geometric Altitude Z; km Molecular Temperature 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 Rate Lh; ͦ K/km -6.3 0.0 +1.0 +2.8 0.0 -2.8 -2.0 Earth AtmosphereSOLO
  • 16. 16 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. ( ) Hd HLTR g Hd TR g P Pd aa ⋅ ⋅+ =⋅=− 0 00 kmKLHLTT /3.60  −=⋅+= Integrating this equation we obtain ( )∫∫ ⋅ ⋅+ =− H a P P Hd HLTR g P PdS S 0 0 0 1 0 ( ) 0 00 lnln 0 T HLT RL g P P aS S ⋅+ ⋅ ⋅ −= Hence aRL g SS H T L PP ⋅ −       ⋅+⋅= 0 0 0 1 and           −         ⋅= ⋅ 1 0 0 0 g RL S S a P P L T H Earth AtmosphereSOLO Stratosphere Troposphere
  • 17. 17 Physical Foundations of Atmospheric Model Hd TR g P Pd Ta ⋅=− * 0 Integrating this equation we obtain ( )T TaS S HH TR g P P T −⋅ ⋅ −= * 0 ln Hence ( )T Ta T HH TR g SS ePP −⋅ ⋅ − ⋅= * 0 and S STTa T P P g TR HH ln 0 * ⋅ ⋅ += ∫∫ =− H HTa P P T S TS Hd TR 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. 18 Physical Foundations of Atmospheric Model ( )[ ] Hd HHLTR g Hd 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 Hd HHLTR g P Pd * 0 1 ( )[ ] * * 0 lnln T ST aSSS S T HHLT RL g P P −⋅+ ⋅ ⋅ = Hence ( ) aRL g S T S SSS HH T L PP ⋅ −         −⋅+⋅= 0 * 1 and           −      ⋅+= ⋅ 1 0 * g RL SS S S T S aS P P L T HH Stratosphere Region (HS=20.0 km to 32.0 km). Stratosphere Troposphere Earth AtmosphereSOLO
  • 19. 19 1962 Standard Atmosphere from 86 km to 700 km Layer Index Geometric Altitude km Molecular Temperature , K Kinetic Temperature K Molecular Weight Lapse Rate K/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. 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 kmK Zd 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 ZZ ATT C kma KA KTC 9429.19 3232.76 1902.263 −= −= =   Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10) ( ) kmK Zd Td ZZLTT Z /0.12 99  += −⋅+= Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11) ( ) ( ) ( ) ( )       + + ⋅−=       + + ⋅−⋅= ⋅−⋅−−= ∞ ∞∞ ZR ZR ZZ kmK ZR ZR TT Zd Td TTTT E E E E 10 10 10 10 10 / exp ξ λ ξλ  KT kmR km E  1000 10356766.6 /01875.0 3 = ×= = ∞ λ Earth AtmosphereSOLO
  • 21. 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 Earth AtmosphereSOLO Return to Table of Content
  • 22. SOLO Atmosphere Continuum Flow Low-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. 23 SOLO Dimensionless Equations Dimensionless Field Equations (C.M.): ( ) 0 ~~~~ =⋅∇+ u t  ρ ∂ ρ∂ ( ) ( )u R u R pG F uu t u eer ~~~~1 3 4~~~~1~~~~1~~~ ~ ~ ~ 2   ⋅∇∇+×∇×∇−∇−=        ∇⋅+ µµρ ∂ ∂ ρ(C.L.M.): ( ) ( )Tk PRt Q uG F u t p Hu t H rer ∇⋅∇−+⋅+⋅⋅∇+=        ∇⋅+ ∂ ∂ 11 ~ ~ ~~~1~~~ ~ ~~~~ ~ ~ ~ 2 ∂ ∂ ρτ ∂ ∂ ρ  (C.E.): Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = Froude: 0 0 gl U Fr = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/ ~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/ ~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ 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 = Knudsen l Kn 0 0 : λ = AERODYNAMICS Return to Table of Content
  • 24. 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 U M = 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 TR a 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. 25 SOLO Mach Number – Flow Regimes Regime Mach mph km/h m/s General plane characteristics Subsonic <0.8 <610 <980 <270 Most 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- 915 980-1,470 270-410 Transonic aircraft nearly always have swept wings, delaying drag- divergence, and often feature design adhering to the principles of the Whitcomb Area rule. Supersonic 1.2–5.0 915- 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 5.0–10.0 3,840– 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 10.0–25.0 7,680– 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 >16,25 0 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape AERODYNAMICS
  • 26. 26 SOLO Different Regimes of Flow Mach Number – Flow Regimes AERODYNAMICS Return to Table of Content
  • 27. 27 SOLO - when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves. a V M M =      = − & 1 sin 1 µ 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 WAVES AERODYNAMICS
  • 28. 28 SOLO SHOCK & EXPANSION WAVES M < 1 M = 1 M > 1 Mach Waves AERODYNAMICS
  • 29. 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. Return to Table of Content AERODYNAMICS
  • 30. 30 SHOCK WAVES SOLO A shock wave occurs when a supersonic flow decelerates in response to a sharp increase in pressure (supersonic compression) or when a supersonic flow encounters a 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 is an 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. 31 Movement of Shocks with Increasing Mach Number <<<<<<< MMMMMMMM SOLO AERODYNAMICS Return to Table of Content
  • 32. 32 where ρ0 = air density U0 = true speed l 0= characteristic length μ0 = absolute (dynamic) viscosity υ0 = kinematic viscosity NumberReynolds:Re 0 00 0 000 0 0 0 υµ ρ ρ µ υ lUlU = == 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 1884AERODYNAMICS
  • 33. 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. 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. 35 Normalized Velocity profiles within a Boundary-Layer, comparison between Laminar and Turbulent Flow. SOLO AERODYNAMICS Boundary-Layer
  • 36. 36 Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity) AERODYNAMICS SOLO Return to Table of Content
  • 37. 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. Knudsen l Kn 0 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 TkB 20 2 σπ λ = • 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 ≈ 8x10-8 m. AERODYNAMICS
  • 38. 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 ]. 0 2 1 λρµ c= m Tk c B π 8 = Tk m B2 0 π ρ µ λ = AERODYNAMICS
  • 39. 39 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:Re 0 000 µ ρ lU = Tk m lmTklallU aUM BB γρ µ γρ µ ρ µ µρ 00 0 00 0 000 0 0000 00 // / Re ==== Kn Tk m lTk m l BB == 22 00 0 00 0 π ρ µπγ γρ µ 2Re πγM Kn = AERODYNAMICS
  • 40. 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 particles 2.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 πγM Kn = AERODYNAMICS
  • 41. SOLO Knudsen Number (continue – 4) Inviscid Limit Free Molecular LimitKnudsen Number Boltzman Equation Collisionless Boltzman Equation Discrete Particle model Euler Equation Navier-Stokes Equation Continuum model Conservation Equation do not form a closed set Validity of conventional mathematical models as a function of local Knudsen 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. Knudsen l Kn 0 0 : λ = AERODYNAMICS Return to Table of Content
  • 42. 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) 0 0 T T MaTAS = qc is impact pressure P is static pressure         −      += 11 5 7 2 0 0 P q T T aTAS c Flight Instruments
  • 44. SOLO 44 Flight Instruments Airspeed Indicators 2 2 1 vpp 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: 0ρ ρ VVE = V – True Airspeed ρ – Air Density ρ0 – Air Density at Sea Level
  • 47. 47 True Airspeed (TAS) and Calibrated Airspeed (CAS) Relationship with Varying Altitude and Temperature Flight Instruments
  • 48. 48 TAS and CAS Relationship with Varying Altitude and Temperature (continue) Flight Instruments
  • 49. 49 Mach Number vs TAS Variation with Altitude Flight Instruments
  • 52. 52 SOLO Aerodynamic Forces ( )[ ]∫∫ +−= ∞ WS A dstfnppF  11 ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − ( ) airflowingthebyweatedsurfaceVehicleS SsurfacetheonmNstressforcefrictionf Ssurfacetheondifferencepressurepp W W W − − −−∞ )/( 2 Aerodynamic Forces acting on a Vehicle Surface SW. AERODYNAMICS
  • 53. 53 SOLO ( )       − − = L D F W A  VelocitytoNormalForceLiftL VelocitytooppositeForceDragD − − L D CSVL CSVD 2 2 2 1 2 1 ρ ρ = = ( ) ( ) tCoefficienLiftRMC tCoefficienDragRMC eL eD − − βα βα ,,, ,,, anglesideslipandattackofangle viscositydynamic lengthsticcharacteril soundofspeedHa numberReynoldslVR BodytoRelativeVelocityFlowV numberMachaVM e − − − − −= − −= βα µ µρ , )( / / AERODYNAMICS ( ) V W A nLVDF 11 −−=  Aerodynamic Forces Lift and Drag Forces
  • 54. 54 SOLO ( ) ( )∫∫ ∫∫ ⋅+⋅−= ⋅+⋅−= W W S VfVpL S fpD dsntCnnC S C dsVtCVnC S C 1ˆ1ˆ 1 1ˆ1ˆ 1 Wf Wp Ssurfacetheontcoefficienfriction V f C Ssurfacetheontcoefficienpressure V pp C −= − − = ∞ 2/ 2/ 2 2 ρ ρ ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − Aerodynamic Forces CD – Drag Coefficient CL – Lift Coefficient AERODYNAMICS
  • 55. ( ) [ ] [ ]( )∫ ∫ ∞∞ = −−−= −=′ Edge Trailing Edge Leading sideuppersidelower cos Edge Trailing Edge Leading sideuppersidelower pp cospcosp dxpp dsL sdxd USLS θ θθ Divide left and right sides of the first equation by cV 2 2 1 ∞ρ ∫             − − − = ′ ∞ ∞ ∞ ∞ ∞ Edge Trailing Edge Leading upperlower c x d V pp V pp cV L 222 2 1 2 1 2 1 ρρρ We get: Relationship between Lift and Pressure on Airfoil Lower Surface Upper Surface ( )∫ −=− Edge Trailing Edge Leading sideuppersidelower sinpsinp dsD USLS θθ Lift – Aerodynamic component normal to V Drag – Aerodynamic component opposite to V SOLO AERODYNAMICS Aerodynamic Forces
  • 56. From the previous slide, ∫             − − − = ′ ∞ ∞ ∞ ∞ ∞ Edge Trailing Edge Leading upperlower c x d V pp V pp cV L 222 2 1 2 1 2 1 ρρρ The left side was previously defined as the sectional lift coefficient Cl. The pressure coefficient is defined as: 2 2 1 ∞ ∞− = V pp Cp ρ Thus, ( )∫ −= edge Trailing edge Leading upperplowerpl c x dCCC ,, Lower Surface Upper Surface Relationship between Lift and Pressure on Airfoil (continue – 1) SOLO AERODYNAMICS Aerodynamic Forces
  • 57. 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. 58 SOLO Lift and Drag Coefficients AERODYNAMICS Subsonic Speeds np α− Upper xd yd ∞U Upper xd yd ∞p∞p α   0,, 2 0 2 0 D TurbulentforMore LaminarforLess dragFriction fD TurbulentforLess LaminarforMore dragPressure pDD stall a L CCCC aC =+= <== = αααπα π Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR → ∞)  ( )       −=−= ARe C C L i a L π απααπ 22 0 ARe C be SC V w L SbAR Li i ππ α / 2 2 = === α π α π ARe a a ARe CL 0 0 1 2 1 2 + = + = ARe C C L Di π 2 =    AR C CCCCC L D drag induced D drag friction fD drag pressure pDD i π 2 0,, +=++= e – span efficiency factor Aerodynamic Forces Return to Table of Content
  • 60. 60 AERODYNAMICS Drag Variation with Mach Number SOLO Aerodynamic Drag
  • 61. 61 Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2 SOLO AERODYNAMICS Aerodynamic Drag
  • 62. 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 AERODYNAMICS Aerodynamic Drag
  • 63. 63N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993 Parabolic Drag Polar of a typical High Subsonic Aircraft at different Mach Numbers SOLO AERODYNAMICS Aerodynamic Drag
  • 64. 64N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993 Variation of CD0 (M) for a supersonic aircraft Variation of aerodynamic characteristic for a typical subsonic transport aircraft Variation of aerodynamic characteristic for a typical supersonic fighter aircraft SOLO AERODYNAMICS Aerodynamic Drag
  • 65. 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 1 1 2 1 1 − ∞ ∞             − + − + = γγ γ γ A A M M p p p∞, M∞ - Pressure and Mach Number upstream the Airfoil pA, MA- Pressure and Mach Number at a point A on the Airfoil Critical Mach Number The Pressure Coefficient Cp is computed using ( )             −             − + − + =      −= − ∞ ∞∞∞ 1 2 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C Definition of Critical Mach Number. Point A is the location of minimum pressure on the top surface of the Airfoil. SOLO AERODYNAMICS
  • 66. 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 1 1 2 1 1 2 1 2 1/ 2 2 γγ γ γ γγ A A pA M M Mp p M C ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C p p ( )             −             − + − + = − 1 2 1 1 2 1 1 2 1/ 2 γγ γ γ γ cr cr p M M C cr 2 0 1 ∞− = M C C 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 or Laiton’s Rule SOLO AERODYNAMICS
  • 67. 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 of the 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. 68 Movement of Shocks with Increasing Mach Number Drag Divergence Mach Number The Drag at small Mach number, due to Profile Drag with Induced Drag =0 (αi = 0) is constant (points a, b, and c) until M∞ = 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 of supersonic 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 separate From 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 SOLO AERODYNAMICS
  • 69. 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 Airfoil Df – Skin Friction Drag Dp – Pressure Drag due to Flow Separation Dw – 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 AERODYNAMICS Aerodynamic Drag
  • 71. 71 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag CD0 due to Flow Separation SOLO Aerodynamic Drag
  • 72. 72 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag due to Viscosity: 1.Skin Friction 2.Flow Separation (Drop in pressure behind body) ∫∫ ∫∫         ⋅+⋅ − −=         ⋅+⋅−= ∧∧ ∞ ∧∧ W W S S fpD ds w t V f w n V pp S ds w tC w nC S C xx xx 11 11 ˆ 2/ ˆ 2/ 1 ˆˆ 1 22 ρρ SOLO Aerodynamic Drag
  • 73. 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 Return to Table of Content
  • 74. 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 and the average chord is the wing area (b x c = S). 4. Aspect Ratio, AR, is defined as: ( )∫− = 2/ 2/ b b dyycS ( ) b S dyyc b c b b == ∫− 2/ 2/ 1 S b AR 2 = AERODYNAMICSSOLO
  • 75. 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 perpendicular to root chord. 8. Mean aerodynamic chord, rc Λ r t c c =λ tc λ ( )[ ]∫− = 2/ 2/ 21~ b b dyyc S c c~ AERODYNAMICSSOLO
  • 76. 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-back Wing Delta Wing Elliptic Wing SOLO Return to Table of Content
  • 77. 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 Return to Table of Content
  • 78. 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
  • 79. 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 Return to Table of Content
  • 80. 80 SOLO Aircraft Propulsion Systems Classification of Engine Concepts , mostly used in Aviation
  • 81. 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. 82 SOLO The Rotating Parts of Jet Engine Compressor Shaft Turbojet animation Turbine Air Breathing Jet Engines Run This
  • 83. 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. 84 SOLO Animation of a 2-spool, high-bypass turbofan. A. Low pressure spool B. High pressure spool C. Stationary components 1. Nacelle 2. Fan 3. Low pressure compressor 4. High pressure compressor 5. Combustion chamber 6. High pressure turbine 7. Low pressure turbine 8. Core nozzle 9. Fan nozzle Turbofan Air Breathing Jet Engines Run This
  • 85. 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
  • 86. 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 Return to Table of Content SOLO Air Breathing Jet Engines
  • 87. 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. 88 SOLO Propeller Aerodynamics Actuator Disk 2 11 2 22 2 1 2 1 VpVp ρρ +=+ 2 44 2 33 2 1 2 1 VpVp ρρ +=+ Bernoulli’s equations on each side of the Disk: Far from the Disk we have the same ambient pressure, hence: 41 pp = Therefore ( )2 1 2 423 2 1 VVpp −=− 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. 89 SOLO Propeller Aerodynamics Actuator Disk ( )2 1 2 423 2 1 VVpp −=− ρ The Thrust provided by the Propeller Disk is given by: ( ) ( )143140 VVAVVVmT p −=−= ρ where - Fluid mass flow [kg/sec] through DiskpAVm 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 2 1 2 423 2 1 −=−= ρ From the two expressions of Thrust we obtain ( ) ( )2 1 2 4143 2 1 VVVVV −=− ( )413 2 1 VVV += Conservation of Momentum Actuator Disk (One-Dimensional Momentum) Theory
  • 90. SOLO Propeller Aerodynamics Model of the Flow through Propeller according to the Actuator Disk Concept ( ) ( ) ppp pp VA mVVVAT v2v v20143 ⋅+= =−= ∞ρ ρ  We found Let compute vs as function of other parameters 0 2 vv 2 =−+ ∞ p pp A T V ρ 0 222 v 2 >+      +−= ∞∞ p p A TVV ρ This 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 TVV T DiskatVelocityFlowThrustVT Vm VmVm InFlowEnergyOutFlowEnergyP ρ222 ___v vv2 2 1 v2 2 1 2 0 2 0 2 0  
  • 91. 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. ( ) ( )aaVDVAT Va ppp p +=⋅+= ∞ = ∞ ∞ 1 2 vv2 22 /v: ρ π ρ ( ) aVV V VT VT PowerOutput PowerInput Va ppp P p + = + = + = +⋅ ⋅ == ∞= ∞∞ ∞ ∞ ∞ 1 1 /v1 1 vv /v: η ( ) p P P C JDV P aa 323 2 3 122 1 1 πρπη η ==+= − ∞ ( ) ( ) aaVDVTP p 232 1 2 v +=+= ∞∞ ρ π ( ) T P P C JDV T aa 2222 122 1 1 πρπη η ==+= − ∞ where Actuator Disk (One-Dimensional Momentum) Theory ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J T p n RD ρ ρ ππ = = Ω == ∞ Ω= = ∞
  • 92. SOLO Propeller Aerodynamics The Efficiency of an Ideal Propeller ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J T p n RD ρ ρ ππ = = Ω == ∞ Ω= = ∞ E.Torenbeek, H.Wittenberg, “Flight Physics – Essentials of Aeronauical Disciplines and Technology, with Historical Notes”, Springer, 2009 Typical Propeller Diagram Actuator Disk (One-Dimensional Momentum) Theory T P P C J 22 121 πη η = − p P P C J 33 121 πη η = − JV Dn P VT C C P p T η == ∞ ∞
  • 93. SOLO Propeller Aerodynamics The Efficiency of an Ideal Propeller E.Torenbeek, H.Wittenberg, “Flight Physics – Essentials of Aeronauical Disciplines and Technology, with Historical 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 J C C p T P =η ( ) .: .: : 42 53 2/ 2 CoeffThrust Dn T C CoeffPower Dn P C RatioAdvance R V Dn V J T p n RD ρ ρ ππ = = Ω == ∞ Ω= = ∞
  • 94. 94 AERODYNAMICS Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997 Actuator Disk (Momentum) Theory SOLO ( ) RatioAdvance R V Dn V J n RD Ω == ∞ Ω= = ∞ ππ2/ 2 : We can see that by varying the Propeller Pitch β we can operate at maximum efficiency ηmax.
  • 95. 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 angleϕ [rad] V – air velocity [m/sec], normal to rotation plane V = V∞ + v Ω – 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. 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 1 2 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 Element CL – 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 φφ φφ cossin sincos Blade Element Theory. The Aerodynamic Moment and Power of the Propeller Blade Element are QdFdrFdUPd FdrQd xxT x Ω=⋅Ω== ⋅=
  • 97. 97 SOLO Propeller Aerodynamics The net force acting on the blades are the summation of 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 cossin 2 1 cossin 2 1 sincos 2 1 φαφαρ φαφαρ φαφαρ ( ) r V r 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. 98 SOLO Propeller Aerodynamics Blade Element Theory. 42 22 242 53 32 253 2 2 4 : 4 : : D T Dn T C R P Dn P C RatioAdvance R V Dn V J n RD T n RD p n RD Ω == Ω == Ω == Ω = = Ω = = ∞ Ω = = ∞ ρ π ρ ρ π ρ π π π π  ( ) ( ) ( ) ( )( )∫ = = ∞       −      + Ω = Ω = Rr r DLT R r drCrC R r R V R cB R T C 0 2 2 2 22 22 42 2 sincos 84 φαφαπ π π π ρ π σ  ( ) ( ) ( ) ( )( )∫ = = ∞       +      + Ω = Ω = Rr r DLP R r drCrC R r R rV R Bc R P C 0 2 2 2 2 22 53 3 cossin 84 φαφαππ π π π ρ π σ We have or Let use the definitions: ( ) Solidity R cB R RcB DiskSurface ElementsBladeSurface == == π π σ 2 : ( ) ( ) ( ) ( ) ( )( )∫ = = = −+= 1 0 222 / sincos 8 x x DL Rrx T xdxCxCxJC φαφαπσ π Thrust Coefficient ( ) ( ) ( ) ( ) ( )( )∫ = = = ++= 1 0 222 / cossin 8 x x DL Rrx P xdxCxCxxJC φαφαππσ π Power Coefficient
  • 99. 99 SOLO Propeller Aerodynamics Blade Element Theory. 42 22 242 53 32 253 2 2 4 : 4 : : D T Dn T C R P Dn P C RatioAdvance R V Dn V J n RD T n RD p n RD Ω == Ω == Ω == Ω = = Ω = = ∞ Ω = = ∞ ρ π ρ ρ π ρ π π π π Characteristic Curves of a Propeller Propeller Efficiency. J C C Dn V C C CDn VCDn P VT P T P T P T ==== ∞∞∞ 53 42 ρ ρ η
  • 100. 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 etropospher x P P x 1 75.0 00 ρ ρ
  • 101. 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. 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. 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. 104 Turbojet SOLO Thrust Computation for Air Breathing Engines ( ) ( )         DRAGFRICTION A WA DRAGPRESURE A WA THRUST eeeeex WW AdAdppAppAUAUF ∫∫∫∫ −−−−+−= θτθρρ cossin000 2 00 2 00000 & mAUmmAU feee  =+= ρρUsing C.M. ( ) ( ) 00000 2 00 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. 105 Turbojet SOLO Thrust Computation for Air Breathing Engines (continue – 1) since and 00 ,Up 0A eA ee Up , ( ) 0 00000 00 00 /:111 mmf A A p p U U f Ap Um Ap f eee   =      −+      −+= T 2 0 2 0 00 002 0 0 2 00 0 2 00 00 2 000 00 00 MM TR TR M p a p U Ap UA Ap Um γ ρ γρρρρ =====  ( ) 0 000 2 0 00 /:111 mmf A A p p U U fM Ap f eee =      −+      −+= γ T 000 2 00 0 00 000000 MApaM TR Ap aUAam γρ === ( ) 0 0000 0 00000 /:1 1 11 1 mmf A A p p MU U fM ApMam f eee   =      −      +      −+=      = γγ TT Air Breathing Jet Engines
  • 106. 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 ForceThrust I ff sp TTT       ==== γ   Specific Impulse 0000 11 ApMfa gIsp T       = γ Specific Fuel Consumption (SFC) spIg f ThrustofPound HourperBurnedFuelofPound S 1 : ==== 0 f mT/T m   Air Breathing Jet Engines
  • 107. 107 Air Breathing Jet Engines PRESSURE Compressor Pressure Rise Turbine Pressure Drop (Turbojet) Heat Added in Combustion Chambers by 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. 108 Air Breathing Jet Engines PRESSURE Compressor Pressure Rise Turbine Pressure Drop (Turbojet) Heat Added in Combustion Chambers by burning mfuel mass TOTAL TEMPERATURE Pressure corresponding to mfuel and ThrustA B C D1 F E Additional Turbine Pressure Drop in Turboprop
  • 109. 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. ρ ρ = T T Sensitivity of Thrust and Specific Fuel Consumption with Velocity 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. 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 with Velocity and Altitude for a Jet Engine
  • 111. 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 with Velocity and Altitude for a Jet Engine Supersonic Conditions 1 2 2 1 1 −       − += γ γ γ M p p static total Ptotal is the pressure entering the Compressor from the Diffuser, that further increases the pressure and therefore the exit Velocity Ue and the Thrust. From the Figure we obtain that for the specific aircraft the Supersonic Thrust is given by ( )118.11 0 1 −+= = M MT T ..LSS.L. ρ ρ = 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. 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. 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. 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. 115 Thrust Reversal Operation (Used during Landing) Aircraft Propulsion SystemSOLO Return to Table of Content
  • 116. 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 variation 1.Subsonic: T is constantwithV’ 2. Supersonic: T/Tm=1=1+1.18 (M’-1) Velocity variation 1. Subsonic: TSFC= 1.0+k M’ 2. Supersonic: TSFCis constant’ Altitude variation SFCis constantwith Altitude Specificfuel Consumption Power PA =T V’ Turbojet Engine Aircraft Propulsion Summary
  • 117. 117 Altitude variation T/T0=(ρ/ρ0 )m Velocity variation 1High bypassratio:T/TV=0=AM’ -n 2.Lowbypassratio:TfirstincreaseswithM’ then decreasesathigh supersonicM’ Velocity variation 1.High Bypassct =B(1.0+kM’ ) 2.LowBypass:ct graduatelyincreaseswithvelocity Altitude variation ct isconstantwith Altitude Specificfuel Consumption Power PA =TV’ Turbofan Engine J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186 Aircraft Propulsion Summary Aircraft Propulsion SystemSOLO
  • 118. 118 Velocityvariation PA isconstantwithM’ Altitude variation PA/PA,0 =(ρ/ρ0)m Velocityvariation CA isconstantwithV’ Altitude variation CA isconstantwithAltitude Specificfuel Consumption Power PA=(TP+Tj)V’ PA=hpr PS+Tj V’ PA=hpr Pes Turboprop Engine 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. 119 Altitude variation 1. P/P0 = ρ/ρ0 2. (slightly more accurate) P/P0 =1.132 ρ/ρ0-0.132 Velocity variation Shaft Power P constant with V’ Velocity variation SFC is constant with V’ Altitude variation SFC is constant with Altitude Altitude variation T/T0 = ρ/ρ0 Velocity variation 1.Subsonic: T is constant with V’ 2. Supersonic: T/Tm=1=1+1.18 (M’-1) Velocity variation 1. Subsonic: TSFC = 1.0+k M’ 2. Supersonic: TSFC is constant’ Altitude variation SFC is constant with Altitude Altitude variation T/T0 =( ρ/ρ0 )m Velocity variation 1High bypass ratio: T/TV=0=A M’ -n 2. Low bypass ratio: T first increases with M’ then decreases at high supersonic M’ Velocity variation 1. High Bypass ct = B (1.0+k M’ ) 2.Low Bypass: ct graduately increases with velocity Altitude variation c t is constant with Altitude Velocity variation PA is constant with M’ Altitude variation PA/PA,0 = (ρ/ρ0)m Velocity variation CA is constant with V’ Altitude variation CA is constant with Altitude Specific fuel Consumption Specific fuel Consumption Specific fuel Consumption Specific fuel Consumption Power PA =T V’ Power PA =T V’ Power PA = hpr P hpr = f (J) J = V’/(N D) Power PA =(TP+Tj) V’ PA = hpr PS+Tj V’ PA = hpr Pes Reciprocating Engine/ Propeller Combination Turbojet Engine Turbofan Engine Turboprop Engine Propulsion Systems J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999, pg.186 Aircraft Propulsion Summary Block Diagram Aircraft Propulsion SystemSOLO
  • 120. 120 Air Breathing Jet Engines 0m T Aircraft Propulsion Summary SOLO
  • 121. 121 Air Breathing Jet Engines Aircraft Propulsion Summary SOLO
  • 122. 122 Air Breathing Jet Engines Aircraft Propulsion Summary SOLO
  • 123. 123 SOLO Propulsive Efficiency Characteristics of Turboprop, Turbofan and Turbojet Engines Air Breathing Jet Engines Return to Table of Content Propulsive Efficiency Summary
  • 124. 124Stengel, MAE331, Lecture 6 Thrust of a Propeller- Driven Aircraft • With constant r.p.m., variable-pitch propeller where ηp - propeller efficiency ηI - ideal propulsive efficiency ηnet-max ≈ 0.85 – 0.9 Efficiency decrease with airspeed Engine power decreases with altitude - Proportional with air density w/o supercharger V P V P T engine net engine Ip ηηη == Variation of Thrust and Power of a Propeller-Driven Aircraft with True Airspeed Aircraft Propulsion Summary SOLO Aircraft Propulsion System
  • 125. 125 Thrust as a function of airspeed for different Propulsion Systems Aircraft Propulsion Summary SOLO Aircraft Propulsion System
  • 126. 126 Stengel, MAE331, Lecture 6 Thrust of a Turbojet Engine ( )         −      +−      −      − = 11 11 2/1 00 0 c t c t t VmT τθ θ τ θ θ θ θ fuelair mmm  += ( ) heatsspecificofratio p p ambient stag =      = − γθ γγ , /1 0       = etemperaturambientfreestream etemperaturinletturbine 0θ       = etemperaturinletcompressor etemperaturoutletcompressor cτ • 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. 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 SOLO Aircraft Propulsion System
  • 128. 128 Power and Thrust • Propeller • Turbojet • Throttle Effect airspeedoftindependenSVCVTPPower T ≈=•== 3 2 1 ρ airspeedoftindependenSVCTThrust T ≈== 2 2 1 ρ 10 2 1 2 max max ≤≤== TSVTCTTT T δρδδ Specific Fuel Consumption, SFC = cP or cT • Propeller aircraft • Jet aircraft [ ] [ ]       → → = −= −= lbf slb or kN skg c HP slb or kW skg c weightfuelw where thrusttoalproportionTcw powertoalproportionPcw T P f Tf Pf // // SOLO Aircraft Propulsion System Return to Table of Content
  • 129. 129Dr. Carlo Kopp, Air Power Australia, Sukhoi Su-34 Fullback, Russia's New Heavy Strike Fighter Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 130. 130 Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 131. 131 Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 132. 132M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 133. 133M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 134. 134M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 135. 135M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 136. 136M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Return to Table of Content Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 137. 137 SOLO Aircraft Propulsion System VTOL - Vertical Take off and Landing capability The advantages of vertical take off and landing VTOL are quite obvious. Conventional aircraft have to operate from a small number of airports with long runways. VTOL aircraft can take off and land vertically from much smaller areas. STOL - Short takeoff and landing These aircraft using thrust vectoring to decrease the distance needed for takeoff and landing but don’t have enough thrust vectoring capability to perform a vertical take off or landing. VSTOL - An aircraft that can perform either vertical or short takeoff and landings STOVL - Short takeoff and vertical land. An aircraft that has insufficient lift for vertical flight at takeoff weight but can land vertically at landing weight. TVC - Thrust Vector Control Vertical Take off and Landing - VTOL
  • 138. 138 SOLO Vertical Take off and Landing - VTOL
  • 139. 139M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
  • 140. 140 Lockheed_Martin_F-35_Lightning_II STOVL The Unique F-35 Fighter Plane, Movie USP 3” part F35 Joint Strike Fighter ENG, Movie SOLO Aircraft Propulsion System Thrust vectoring nozzle of the F135-PW-600 STOVL variant Return to Table of Content
  • 141. 141 Aircraft Propulsion System SOLO 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. 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. 143 Aircraft Propulsion SystemSOLO A Simple Engine Control Systems : Pilot in the Loop A Simple Limited Authority Engine Control Systems TGT – Turbine Gas Temperature NH – Speed of Rotation of Engine Shaft Tt - Total Temperature FCU – Fuel Control Unit Engine Control System
  • 144. 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 Systems With Electrical Throttle Signaling : Engine Control System Return to Table of Content
  • 146. 146 center stickailerons 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 forwards rudder 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
  • 148. 148 The effect of left rudder pressure Four common types of flaps Leading edge high lift devices The stabilator is a one-piece horizontal tail surface that pivots up and down about a central hinge point. Aircraft Flight ControlSOLO
  • 150. SOLO 150 Aerodynamics of Flight Aircraft Flight Control Return to Table of Content
  • 151. 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
  • 152. 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 rB BB Vx Vx VxVx Vx Vx xn α α sin 11cos 11 1111 11 11 11 − = × −• = × × ×= ( ) ( ) T rTB rB rrBB rB rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × •− = × × ×=     += −= TBTBV TBTBr nxn nxV αα αα cos1sin11 sin1cos11     +−= += TVTrB TVTrB nVn nVx αα αα cos1sin11 sin1cos11 Aerodynamic Forces (Vectorial) Aircraft Equations of Motion
  • 153. 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 αρ αα    = = ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] 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. 154 SOLO Drag ,Lift Coefficients as functions of Angle of Attack Drag Polar Drag Polar Aircraft Equations of Motion
  • 155. 02/28/15 155 SOLO By changing αT from 0 to αMAX, and rotating around by 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 rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × ⋅− = × × ×= ( )σα,A  V  α MAXα ( )DL CC ,Σ σ MAXσ ( ) ( )αα 2 0 LDD CkCC += D σcosL σsinL σ MAXσ L L Aerodynamic Forces (Vectorial) Aircraft Equations of Motion
  • 156. 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 ( ) ( )Rg m T V m MCq V r D    ++−= 10 and since for αT = 0 the aerodynamic forces will decrease the velocity. We can see that for αT ≠ 0, the deceleration due 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 rB rV Vx Vx VVxx Vx Vx Vn α α sin 1cos1 11 1111 11 11 11 − = × ⋅− = × × ×= ( ) ( ) ( ) ( ) ( ) VTLrTD VTrTT nMCqVMCq nMqLVMqDMqA 1,1, 1,,1,,,, αα ααα +−= +−=  Aerodynamic Forces (Vectorial) Aircraft Equations of Motion Return to Table of Content
  • 157. 02/28/15 157 SOLO Specific Energy ( ) ( )RgTA m V  ++= 1 ( ) ( ) ( ) VTrTT nMqLVMqDMqA 1,,1,,,, ααα +−=  By Integrating this Equation we obtain: ( ) ( )∫∫ +⋅=         ⋅− ⋅ =− t t t t dtTAV gm dt g Rg V g VV EE 00 000 0 1  ( ) ( ) ( )∫∫∫∫ ∫ +⋅=⋅ − − − =⋅− ⋅ =         ⋅− ⋅ =− t t R R dRR E R R t t V V dtTAV gm RdR Rgg VV Rd g Rg g VdV dt g Rg V g VV EE 0000 0 0 3 00 2 0 2 0000 0 11 2          µ Define Specific Energy Derivative: ( ) ( )TAV mg Rg V g VV E   +⋅=⋅− ⋅ = 1 : 00 2 0 0 : R g Eµ = ( )∫ +⋅=        −−      −=      −− − =− t t EEE dtTAV gmRgg V Rgg V RRgg VV EE 0 0000 2 0 00 2 000 2 0 2 0 1 22 11 2 µµµ Aircraft Equations of Motion
  • 158. 02/28/15 158 SOLO Specific Energy (continue – 1) ( ) ( )∫∫ +⋅=         ⋅− ⋅ =− t t t t dtTAV gm dt g Rg V g VV EE 00 000 0 1  0 2 0 2 00 20 0 g VV g VdV dt g VV t t V V − = ⋅ = ⋅ ∫ ∫    ( ) ( ) ( ) ( ) 0 3 0 3 2 0 02 0 2 2 2 0 3 2 0 00 0 0 0 0 00 3 2 0 0 00 3 2 21 hhhh R hhhd R h Rd R R RdR R R Rd g Rg dt g Rg V Rhh h hRR Rh R R dRRRdRR R R R Rg R g R R t t RddtV E E −≈−−−=        −≈ =⋅=⋅−=      ⋅− <<+= << =⋅ −= = = ∫ ∫∫∫∫       µ µ ( ) ( ) ( )[ ]∫∫ −⋅=+⋅ t t T t t dtMqDTTV gm V dtTAV gm 00 ,,11 1 00 α  Specific Kinetic Energy Specific Potential Energy ( ) ( )[ ]∫ −⋅=        +−      +=− t t T dtMqDTTV gm V h g V h g V EE 0 ,,11 22 0 0 0 2 0 0 2 0 α Specific Energy Gain due to Thrust and Loss due to Aerodynamic Drag ( )011 >⋅ TVif Aircraft Equations of Motion Return to Table of Content
  • 159. SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       ≥==−= ≥+×Ω=−=++= === min00 min00 00 / 1 mmtmmtmcTm VVvRtVRpATTRgTA m V RtRRtRVR ffvacuum fwindaevacuum ff    Equations of Motion (State Equations): . ( ) ( ) fttttuxftx ≤≤= 0,,, π Controls: ( ) fttttu ≤≤0 VectorThrustT ForcescAerodynamiA − −   Three Degrees of Freedom Model in Earth Atmosphere
  • 161. 161 SOLO • Rotation Matrix from Earth to Wind Coordinates [ ] [ ] [ ]321 χγσ=W EC where σ – Roll Angle γ – Elevation Angle of the Trajectory χ – Azimuth Angle of the Trajectory Force Equation: amgmTFA  =++ where: • Aerodynamic Forces (Lift L and Drag D) ( )           − − = L D F W A 0  • Thrust T ( )           = α α sin 0 cos T T T W  • Gravitation acceleration ( ) ( )                     −           −           − == g cs sc cs sc cs scgCg EW E W 0 0 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ ( ) g cc cs s g W          − = γσ γσ γ  α T V L D Bx Wx Bz Wz Wy By Flat Earth Three Degrees of Freedom Aircraft Equations
  • 162. 162 SOLO α T V L D Bx Wx Bz Wz Wy By • Aircraft Acceleration ( ) ( ) ( ) ( )WW W W VVa  ×+= → ω where: ( )           = 0 0 V V W  and ( )           = → 0 0 V V W   ( )                                         −+                     − +                     − =           = χ χχ χχ γ γγ γγσ σσ σσω     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 ( )           +− + − =           = γσχσγ γσχσγ γχσ ω ccs csc s r q p W W W W     therefore ( ) ( ) ( ) ( ) ( ) ( )           +− +−=           − =×+= → γσχσγ γσχσγω cscV ccsV V qV rV V VVa W W WW W W     Flat Earth Three Degrees of Freedom Aircraft Equations
  • 163. 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: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( )     ++−=+−=− =+−= −−= γσαγσχσγ γσγσχσγ γα ccgmLTcscVqV csgccsVrV sgmDTV W W /sin /)cos(    from which we obtain:       − + = = γσ α γσ coscos sin cossin V g Vm LT q V g r W W
  • 164. 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: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( ) σ σ σ σ γσαγσχσγ γσγσχσγ γα s c c s ccgmLTcscVqV csgccsVrV sgmDTV W W −− −      ++−=+−=− =+−= −−= /sin /)cos(    from which we obtain: ( ) ( )     += −+= −−= msLTcV cgmcLTV sgmDTV /sin /sin /)cos( σαγχ γσαγ γα    Define the Load Factor gm LT n + = αsin :
  • 165. 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 E W WE W E                          −           −          − =           0 0 0 0 001 0 010 0 100 0 0 V cs sc cs sc cs sc h y x σσ σσ γγ γγ χχ χχ         = = = γ χγ χγ sVh scVy ccVx    or • Energy per unit mass E g V hE 2 : 2 += Let differentiate this equation: ( ) W VDT W DT g g V V g VV hEps − =            − − +=+== α γ α γ cos sin cos sin:   Return to Table of Content
  • 166. 166 SOLO Flat 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 2 ρ= Aircraft Drag ( ) ( ) ARe k CkMCCMC iDC LDLD π 1 , 2 0 = +=  Parabolic Drag Polar gm LT n + = αsin ' Total Load Number ( ) 0/ 0 hh eh − = ρρ Air Density as Function of Height gm L n = Load Factor
  • 167. 167 SOLO Constraints: State Constraints • Minimum Altitude Limit minhh ≥ • Maximum dynamic pressure limit ( ) ( )hVVorqVhq MAXMAX ≤≤= 2 2 1 ρ • Maximum Mach Number limit ( ) MAXM ha V ≤ Aerodynamic or heat limitation Three Degrees of Freedom Model in Earth Atmosphere
  • 168. 168 SOLO Constraints: • Maximum Load Factor ( ) MAXn W VhL n ≤= , • Maximum Roll Angle MAXMAX σσσ ≤≤− • Maximum Lift Coefficient or Maximum Angle of Attack ( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤ ( ) ( ) ( ) ( ) ( ) LSTALL LMAXL nVh W VhC VSh W VhC VShn ==≤ , , 2 1, 2 1 2_2 αρρ α Control Constraints (continue): ( ) fttttuU ≤≤≤ 00, Three Degrees of Freedom Model in Earth Atmosphere
  • 169. 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 δ If Thrust Vector Control (TVC) Three Degrees of Freedom Model in Earth Atmosphere
  • 170. 02/28/15 170 ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) γ χ γ χγ χγσ γ α σ γ βα χ χγγ χγσ α σ βα γ χγγ γ βα χγ χγ γ cos sincossin cos sincoscostan2 tansincossin cos sin cos cos sincos cossinsincoscoscos coscos2coscos sin sin sincos sinsincoscossincos sin coscos sincos cos coscos cos sin * 2 * 2 2 * V a LatLat V R LatLat Lat R V Vm LT Vm CT V a LatLatLat V R Lat V g R V Vm LT Vm CT LatLatLatR ag m DT V R V R V td Latd LatR V LatR V td Longd V td 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
  • 171. 02/28/15 171 (b) Spherical, Non-Rotating Earth (Ω = 0) ( ) ( ) ( )Lat R V Vm LT Vm CT V g R V Vm LT Vm CT ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N tansincossin cos sin cos cos sincos coscos sin sin sincos sin coscos sincos cos coscos cos sin * χγσ γ α σ γ βα χ γσ α σ βα γ γ βα χγ χγ γ − + + + −=       −+ + + + = −− − = == == =    SOLO Three Degrees of Freedom Model in Earth Atmosphere
  • 173. 02/28/15 173 (a) Spherical, Rotating Earth (Ω ≠ 0) (b) Spherical, Non-Rotating Earth (Ω = 0) (c) Flat Earth 0→Ω ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) χ γ χγ χγσ γ α σ γ βα χ χγγ χγσ α σ βα γ χγγ γ βα χγ χγ γ sincossin cos sincoscostan2 tansincossin cos sin cos cos sincos cossinsincoscoscos coscos2coscos sin sin sincos sinsincoscossincos sin coscos sincos cos coscos cos sin 2 2 2 * LatLat V R LatLat Lat R V Vm LT Vm CT LatLatLat V R Lat V g R V Vm LT Vm CT LatLatLatR ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N Ω + −Ω− − + + + −= + Ω + Ω+      −+ + + + = −Ω+ −− − = == == =    ( ) ( ) ( )Lat R V Vm LT Vm CT V g R V Vm LT Vm CT ag m DT V R V R V td Latd LatR V LatR V td Longd V td Rd xWW E N tansincossin cos sin cos cos sincos coscos sin sin sincos sin coscos sincos cos coscos cos sin * χγσ γ α σ γ βα χ γσ α σ βα γ γ βα χγ χγ γ − + + + −=       −+ + + + = −− − = == == =    σ γ α σ γ βα χ γσ α σ βα γ γ βα γ ξγ ξγ sin cos sin cos cos sincos coscos sin sin sincos sin coscos sin sincos coscos Vm LT Vm CT V g Vm LT Vm CT g m DT V Vz Vy Vx E E E + + + −= − + + + = − − = = = =       SOLO ∞→R Three Degrees of Freedom Model in Earth Atmosphere Return to Table of Content
  • 174. 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. 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. 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 Performance References (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. 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 Performance References (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. 178 SOLO Aircraft Flight Performance References (continue – 4) Solo Hermelin Presentations http://www.solohermelin.com • Aerodynamics Folder • Propulsion Folder • Aircraft Systems Folder Return to Table of Content
  • 179. 179 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – Stanford University 1983 – 1986 PhD AA
  • 180. 180
  • 181. 181
  • 182. 182M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison Tables
  • 183. 183M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Return to Table of Content
  • 185. 185 Ray Whitford, “Design for Air Combat” R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”, AIAA Publication, 2000
  • 186. SOLO
  • 187. SOLO
  • 188. 188 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
  • 189. 189 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
  • 190. 190 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
  • 191. 191 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
  • 200. 200
  • 203. 203

Editor's Notes

  1. J.D. Anderson, Jr, “Fundamentals of Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  2. http://en.wikipedia.org/wiki/Earth%27s_atmosphere
  3. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  4. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  5. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  6. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  7. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  8. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  9. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  10. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  11. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  12. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  13. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  14. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  15. J.D. Anderson, Jr, “Fundamentals of Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  16. http://en.wikipedia.org/wiki/Mach_number
  17. http://en.wikipedia.org/wiki/Mach_number
  18. J.D. Anderson, Jr, “Fundamentals of Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  19. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  20. J.D. Anderson Jr., “Ludwig Prandtl ‘s Boundary Layer”, Physics Today, December 2005, http://www.aps.org/units/dfd/resources/upload/prandtl_vol58no12p42_48.pdf
  21. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  22. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  23. http://en.wikipedia.org/wiki/Knudsen_number Angelo Miele, “Flight Mechanics, Volume 1, Theory of Flight Paths”, Addison Wesley, 1962, pg. 71 F.J. Regan, S.M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993, Ch.11 “Flowfield Description” J.D. Anderson, Jr., “Hypersonic and High Temperature Gas Dynamics”, McGraw-Hill, 1989, pp. 473-476
  24. http://en.wikipedia.org/wiki/Knudsen_number Angelo Miele, “Flight Mechanics, Volume 1, Theory of Flight Paths”, Addison Wesley, 1962, pg. 71 F.J. Regan, S.M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993, Ch.11 “Flowfield Description”
  25. http://en.wikipedia.org/wiki/Knudsen_number Angelo Miele, “Flight Mechanics, Volume 1, Theory of Flight Paths”, Addison Wesley, 1962, pg. 71 F.J. Regan, S.M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993, Ch.11 “Flowfield Description”
  26. F.J. Regan, S.M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993, Ch.11 “Flowfield Description”
  27. J.D. Anderson, Jr., “Hypersonic and High Temperature Gas Dynamics”, McGraw-Hill, 1989, pg.22
  28. http://en.wikipedia.org/wiki/True_airspeed
  29. Chapter 7 Flight Instruments http://www.gov/library/manuals/aviation/pilot_handbook/media/PHAK%20-%20Chapter%2007.pdf
  30. “Aerodynamics for Naval Aviators “,00=80T-80 1-1-1965
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  32. http://www.rfcafe.com/references/electrical/ew-radar-handbook/mach-number.htm
  33. http://www.rfcafe.com/references/electrical/ew-radar-handbook/mach-number.htm
  34. http://www.free-online-private-pilot-ground-school.com/aircraft_performance.html
  35. http://www.diam.unige.it/~irro/
  36. http://www.diam.unige.it/~irro/
  37. http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtChap5.pdf
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  39. John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  40. John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  41. John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  42. John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001, pp.612-619
  43. John D._Anderson, “Fundamentals_of_Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001, pp.612-619
  44. http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtChap5.pdf
  45. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  46. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  47. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  48. H. Schlichting, E. Truckenbrodt, “Aerodynamics of the Airplane”, McGraw-Hill, 1979
  49. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
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  53. http://www.grc.nasa.gov/WWW/K-12/airplane/Animation/turbpar/enex.html http://www.nobullengineering.com/index_services.html
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  55. http://en.wikipedia.org/wiki/Turboprop
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  60. Richard Shiu Wing Cheung , “Numerical Prediction of Propeller Performance by Vertex Lattice Method ”, UTIAS Technical Note No. 265 CN ISSN 0082-5263, 1987 L. Sankar, “Helicopter Aerodynamics and Performance”, E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”, Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  61. Richard Shiu Wing Cheung , “Numerical Prediction of Propeller Performance by Vertex Lattice Method ”, UTIAS Technical Note No. 265 CN ISSN 0082-5263, 1987 L. Sankar, “Helicopter Aerodynamics and Performance”, E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”, Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  62. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  63. E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”,
  64. E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”,
  65. E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”,
  66. E. Torenbeek, H. Wittenberg, “Flight Physics, Essentials of Aeronautical Disciplines and Technology, with Historical Notes”, Springer, 2009, § 5.9, “Propeller Performance”,
  67. http://toffsworld.com/outdoor-activities/airplanes-aircraft/propeller-engines/ 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
  68. http://www.pilotfriend.com/aero_engines/aero_jet.htm Claire Soares, “Gas Turbines - A Handbook of Air, Land and Sea Applications”, Butterworth Heinemann
  69. http://en.wikipedia.org/wiki/Turbojet http://en.wikipedia.org/wiki/Components_of_jet_engines
  70. Philip G. Hill, Carl R. Peterson, “Mechanics and Thermodynamics of Propulsion”, 2nd Ed., Addison-Wesley,1992
  71. Brian J. Cantwell, “AA283 – Aircraft and Rocket Propulsion”, Stanford University, 2007 “The Aircraft Gas Turbine Engine and Its Operation”, Pratt &amp; Whitney Aircraft, August 1970 Philip G. Hill, Carl R. Peterson, “Mechanics and Thermodynamics of Propulsion”, Addison-Wesley
  72. Brian J. Cantwell, “AA283 – Aircraft and Rocket Propulsion”, Stanford University, 2007 Gordon C. Oates, “Aerothermodynamics of Gas Turbine and Rocket Propulsion”, AIAA Education Series, 1984 Philip G. Hill, Carl R. Peterson, “Mechanics and Thermodynamics of Propulsion”, Addison-Wesley
  73. http://www.aircraftenginedesign.com/custom.html4.html
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  75. “The Jet Engine”, Rolls Royce, 5th Ed., 1996 Claire Soares, “Gas Turbines - A Handbook of Air, Land and Sea Applications”,
  76. Klaus Hünecke, “Jet Engines – Fundamentals Of Theory, Design and Operation”, 1997
  77. Stengel, MAE331, Lecture 6, Cruising Flight Performance
  78. Asselin, M., “Introduction to Aircraft Aerodynamics”, AIAA Education Series, 1997, Fig. 9.12
  79. Stengel, MAE331, Lecture 6, Cruising Flight Performance
  80. Stengel, MAE331, Lecture 6, Cruising Flight Performance
  81. Stengel, MAE331, Lecture 6, Cruising Flight Performance
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