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An Introduction to Aerodynamics

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Gestione dei sistemi aerospaziali per la difesa Universit`adi Napoli Federico II - Accademia Aeronautica di Pozzuoli Aerodynamics R. Tognaccini Introduction An introduction to Aerodynamics Hydrostatics Fundamental principles Renato Tognaccini1 Incompressible inviscid flow Effects of Dipartimento di Ingegneria Industriale, viscosity Universit`adi Napoli Federico II Effects of compressibility Aircraft lift-drag polar [email protected] Some definitions Aerodynamics R. Tognaccini Fluid a substance without its own shape; Introduction characterized by its own volume (liquid) or by Hydrostatics the volume of the container (gas). Fundamental principles Continuum each part of the fluid, whatever small, contains Incompressible a very large (infinite) number of molecules. inviscid flow Effects of Fluid particle an infinitely small volume in the (macroscopic) viscosity scale of our interest, but sufficiently large in the Effects of compressibility (microscopic) length scale of molecules in order Aircraft to contain an infinite number of molecules. lift-drag polar Aerodynamics branch of Fluid Mechanics concentrating on the interaction between a moving body and the fluid in which it is immersed. The aerodynamic forces Aerodynamics R. Tognaccini O(x; y; z) inertial reference system fixed to the aircraft. Introduction V1 aircraft speed at flight altitude h characterized by pressure p and density ρ . Hydrostatics 1 1 Fundamental principles Dynamic equilibrium Incompressible inviscid flow Effects of viscosity L = WT = D (1) Effects of compressibility Aircraft L lift ?V1 lift-drag polar D drag k V1 W aircraft weighta T thrust a G is the center of gravity The aerodynamic force coefficients Aerodynamics R. Tognaccini 1 2 2 ρ1V1S reference force Introduction S reference surface (usually wing surface Sw ) Hydrostatics Fundamental principles Lift and drag coefficients Incompressible inviscid flow L D Effects of CL = CD = (2) viscosity 1 2 1 2 2 ρ1V1S 2 ρ1V1S Effects of compressibility Aircraft lift-drag polar Aerodynamic efficiency L C E = = L (3) D CD Practical applications Aerodynamics R. Tognaccini Problem n. 1 Introduction Compute lift coefficient of an aircraft in uniform horizontal Hydrostatics flight: Fundamental 1 W principles C = (4) L 1 2 S Incompressible 2 ρ1V1 inviscid flow Effects of viscosity Problem n. 2 Effects of compressibility Compute stall speed of an aircraft in uniform horizontal flight: Aircraft lift-drag polar s r s 1 W 2 Vs = (5) CLmax S ρ1 Fundamental flow parameters Mach number Aerodynamics R. Tognaccini Mach number definition Introduction Hydrostatics V M = (6) Fundamental a principles Incompressible V : fluid particle velocity, a: local sound speed inviscid flow Effects of viscosity A flow with constant density everywhere is called Effects of incompressible. compressibility Aircraft Liquids are incompressible. lift-drag polar In an incompressible flow M = 0 everywhere. In some circumstances compressible fluids (gas) behave as incompressible (liquid): M ! 0. Fundamental flow parameters Reynolds number 1/2 Aerodynamics R. Tognaccini Reynolds number definition Introduction ρVL Hydrostatics Re = (7) Fundamental µ principles Kg Incompressible L: reference length, µ: dynamic viscosity ( ms ) inviscid flow Effects of viscosity Newtonian fluid Effects of compressibility @V Aircraft dF = µ dA (8) lift-drag polar @z Friction proportional to A fluid flowing on a solid plate. velocity gradient in the flow. Fundamental flow parameters Reynolds number 2/2 Aerodynamics The Reynolds number compares dynamic forces R. Tognaccini (associated with momentum of fluid particles) against Introduction friction forces (associated with momentum of molecules). Hydrostatics A flow in which µ = 0 is named inviscid or not dissipative. Fundamental principles In an inviscid flow Re = 1 and friction can be neglected. Incompressible inviscid flow kinematic viscosity: Effects of viscosity Effects of µ m2 compressibility ν = (9) Aircraft ρ s lift-drag polar −5 m2 For air in standard conditions: ν ≈ 10 s . In Aeronautics usually Re 1: in many aspects (but not all) the flow behaves as inviscid. Flow regimes Aerodynamics R. Tognaccini Based on Mach number: Introduction M = 0 (everywhere) incompressible flow Hydrostatics M 1 (everywhere) iposonic flow Fundamental principles M < 1 (everywhere) subsonic flow Incompressible M < 1 and M > 1 transonic flow inviscid flow Effects of M > 1 (everywhere) supersonic flow viscosity M 1 hypersonic flow Effects of compressibility Aircraft lift-drag polar Based on Reynolds number: Re ! 0 Stokes (or creeping) flow Re ! 1 ideal flow Critical Mach numbers Aerodynamics R. Tognaccini 0 M1;cr lower critical Mach number: freestream Mach Introduction number producing at least one point in which M = 1 Hydrostatics whereas elsewhere M < 1 Fundamental 00 principles M1;cr upper critical Mach number: freestream Mach Incompressible number producing at least one point in which M = 1 inviscid flow whereas elsewhere M > 1 Effects of viscosity Effects of compressibility Aircraft lift-drag polar 0 M1 < M1;cr subsonic regime 0 00 M1;cr < M1 < M1;cr transonic regime 00 M1 > M1;cr supersonic regime Practical applications Aerodynamics R. Tognaccini Problem n. 3 Compute freestream Mach number of a given aircraft: Introduction Hydrostatics V1 Fundamental M1 = (10) principles a1 Incompressible p inviscid flow a1 = γRT1; γ = 1:4; R = 287J=(KgK); T1 = 272K Effects of viscosity (at sea level) Effects of compressibility Problem n. 4 Aircraft lift-drag polar Compute freestream Reynolds number for a given aircraft: V1L Re1 = (11) ν1 Genesis of lift Aerodynamics Newton's second law F = ma R. Tognaccini F: force, m: mass, a: acceleration Introduction Newton's third law for every action, there is an equal and Hydrostatics opposite reaction Fundamental principles Newton's second law for an aircraft in horizontal flight: Incompressible inviscid flow Effects of ∆V 2CL viscosity L =m _ ∆V = (12) A Effects of V1 eπ compressibility Aircraft lift-drag polar m_ mass flow rate of air interacting with aircraft 2 m_ = eρ1V1πb =4 where b is the wing span and e ≈ 1 ∆V downwash, vertical component of air speed after interaction with aircraft 2 A wing aspect ratio A = b =Sw Genesis of (lift) induced drag Di Aerodynamics Kinetic energy variation of the air flow interacting with aircraft: R. Tognaccini 1 2 2 2 1 2 ∆E = m_ V1 + ∆V − V1 = m_ ∆V (13) Introduction 2 2 Hydrostatics Due to the law of energy conservation the aircraft is making work Fundamental on the fluid, the only possibility is the presence of a drag force k V1: principles Incompressible Di V1 = ∆E (14) inviscid flow Effects of viscosity Due to definition of CD and downwash formula: Effects of compressibility 2 CL Aircraft CDi = (15) lift-drag polar eπA e Oswald factor (e ≤ 1) e = 1 elliptic wing: elliptic chord distribution with fixed airfoil and no twist Drag of an aircraft Aerodynamics R. Tognaccini Introduction Hydrostatics D = Di + Dp + Dw (16) Fundamental principles Incompressible D profile drag: associated with the direct action of viscosity inviscid flow p Effects of Dw wave drag: it appears in transonic and supersonic regimes. viscosity Effects of compressibility Lift-drag polar Aircraft lift-drag polar CL = CL(CD ; Re1; M1;:::) (17) Practical applications Aerodynamics R. Tognaccini Introduction Hydrostatics Fundamental Problem n. 5 principles Compute and compare lift induced drag of an aircraft in cruise Incompressible inviscid flow and landing Effects of viscosity What is the value of e? Effects of compressibility Warning: drag coefficient is not equivalent to drag Aircraft lift-drag polar Wing geometry Aerodynamics R. Tognaccini Introduction Hydrostatics Fundamental principles Incompressible α defined at wing root inviscid flow Effects of viscosity Effects of compressibility y η = b=2 Aircraft λ = ct lift-drag polar cr c = cr [1 − η(1 − λ)], chord distribution 2 R b=2 2 c¯ = S 0 c (y)dy, mean aerodynamic chord (M.A.C.) g twist: angle between tip and root chords The airfoil Aerodynamics R. Tognaccini Introduction Hydrostatics Fundamental principles Incompressible inviscid flow Effects of viscosity Effects of compressibility τ = thickness / chord, airfoil percentage thickness Aircraft lift-drag polar Assuming a rectangular wing of infinite A, the flow is two-dimensional, i.e. flow variables do not change in planes parallel to the symmetry plane of the wing. We can just study a two-dimensional flow around the airfoil. Airfoil aerodynamic characteristics Aerodynamics R. Tognaccini 1 2 Introduction Lift l = Cl 2 ρ1V1c Hydrostatics 1 2 Drag d = Cd 2 ρ1V1c Fundamental 1 2 2 principles Pitching moment mle = Cmle 2 ρ1V1c Incompressible referenced to airfoil leading edge inviscid flow Effects of viscosity For airfoils, force and moments are intended per unit Effects of length. compressibility Aircraft mle ; Cmle > 0 in case of pull up. lift-drag polar Alternatively, pitching moment can be referenced to quarter chord point (m1=4; Cm1=4 ) Lift and polar curve of an airfoil in iposonic flow (M1 1) Aerodynamics R. Tognaccini Introduction Hydrostatics Fundamental principles Incompressible inviscid flow Effects of viscosity Effects of compressibility Aircraft lift-drag polar Some remarks on airfoil performance in iposonic flow (M1 1) Aerodynamics There is a (small) range of α in which: R. Tognaccini Iposonic flow: Introduction Hydrostatics Fundamental Cl = Clα(α − αzl ) (18) principles Incompressible The lift curve is a straight line. inviscid flow Effects of viscosity Clα ≈ 2π is the lift curve slope and αzl is the zero lift Effects of compressibility angle. Aircraft At larger α evidenced the stall phenomenon, characterized lift-drag polar by the maximum lift coefficient (Clmax ) and the corresponding stall angle αs . Airfoil efficiency is very large (> 100) at small α, but drag very significantly grows near stall and beyond. Some remarks on airfoil performance in supersonic flow (M > 1 everywhere) Aerodynamics In the case of a thin airfoil at small α: R. Tognaccini Lift: Introduction Hydrostatics 4α Fundamental Cl ≈ (19) p 2 principles M1 − 1 Incompressible inviscid flow Effects of viscosity Wave drag: Effects of compressibility 4α2 Aircraft Cdw ≈ (20) lift-drag polar p 2 M1 − 1 Dramatic differences between subsonic and supersonic flow! Wing performance in iposonic flow Aerodynamics R.
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