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 V∞ aircraft speed at flight altitude h characterized by pressure p and density ρ . Hydrostatics ∞ ∞

Fundamental principles Dynamic equilibrium Incompressible inviscid flow

Effects of viscosity L = WT = D (1)

Effects of compressibility Aircraft L lift ⊥V∞ lift-drag polar D drag k V∞ W aircraft weighta T thrust

a G is the center of gravity The aerodynamic force coefficients

Aerodynamics

R. Tognaccini 1 2 2 ρ∞V∞S 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 ρ∞V∞S 2 ρ∞V∞S 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 ρ∞V∞ 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 ρ∞ 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 = ∞ 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 → ∞ ideal flow Critical Mach numbers

Aerodynamics

R. Tognaccini 0 M∞,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 M∞,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 M∞ < M∞,cr subsonic regime 0 00 M∞,cr < M∞ < M∞,cr transonic regime 00 M∞ > M∞,cr supersonic regime Practical applications

Aerodynamics

R. Tognaccini Problem n. 3 Compute freestream Mach number of a given aircraft: Introduction

Hydrostatics V∞ Fundamental M∞ = (10) principles a∞ Incompressible √ inviscid flow a∞ = γRT∞, γ = 1.4, R = 287J/(KgK), T∞ = 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:

V∞L Re∞ = (11) ν∞ 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 V∞ eπ compressibility

Aircraft lift-drag polar m˙ mass flow rate of air interacting with aircraft 2 m˙ = eρ∞V∞πb /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˙ V∞ + ∆V − V∞ = 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 V∞: principles

Incompressible Di V∞ = ∆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 , Re∞, M∞,...) (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 ρ∞V∞c Hydrostatics 1 2 Drag d = Cd 2 ρ∞V∞c Fundamental 1 2 2 principles Pitching moment mle = Cmle 2 ρ∞V∞c 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 (M∞  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 (M∞  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 M∞ − 1 Incompressible inviscid flow

Effects of viscosity Wave drag:

Effects of compressibility 4α2 Aircraft Cdw ≈ (20) lift-drag polar p 2 M∞ − 1

Dramatic differences between subsonic and supersonic flow! Wing performance in iposonic flow

Aerodynamics R. Tognaccini Case of A  1:

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

Trapezoidal wing, A = 10, M∞  1, Re∞  1. Some remarks on wing performance in iposonic flow

Aerodynamics

R. Tognaccini At small α:

Introduction Hydrostatics CL = CLα(α − αzL) (21) Fundamental principles The lift curve is still a straight line.

Incompressible inviscid flow A Clα  1: CLα ≈ C Effects of 1+ lα viscosity πA A π A Effects of < 1: CLα ≈ 2 compressibility Lift grows slower against α on the wing respect to the Aircraft lift-drag polar airfoil. Completely different Aerodynamics of combat aircraft wings (small A) against transport aircraft wings (large A). Hydrostatics

Aerodynamics We assume the fluid occupies an infinite space where V = 0

R. Tognaccini everywhere. ∆S elementary fluid surface. Introduction

Hydrostatics n unit vector ⊥ ∆S identifying ∆S. Fundamental ∆F module of the force acting on ∆S, due to the molecular principles momentum exchange across ∆S. Incompressible inviscid flow Definition of hydrostatic pressure Effects of viscosity Effects of ∆F compressibility p = lim p > 0 (22) ∆S→0 ∆S Aircraft lift-drag polar Pascal’s principle

dF = −pndS (23) In a fluid at rest ∆F is orthogonal to ∆S. Stevino’s law

Aerodynamics We consider an infinitesimal volume dxdydz of fluid at rest.

R. Tognaccini z is a vertical axis bottom-top oriented and specifies altitude.

Introduction z-component of total pressure force: dp Hydrostatics p dxdy − (p + dp) dxdy = − dxdydz (24) Fundamental dz principles Equilibrium between gravity and pressure forces (g = 9.81m/s2 is Incompressible inviscid flow gravitational acceleration): Effects of dp viscosity p = p(z) , − dxdydz − ρgdxdydz = 0 (25) Effects of dz compressibility

Aircraft Stevino’s law lift-drag polar dp = −ρgdz (26) For a liquid (ρ = const):

p(z2) − p(z1) = −ρg(z2 − z1) (27) Some applications of Stevino’s law

Aerodynamics A straightforward consequence is: R. Tognaccini Archimedes’ principle Introduction Hydrostatics The buoyant force that is exerted on a body immersed in a Fundamental fluid is equal to the weight of the fluid that the body displaces principles and acts in the upward direction at the center of mass of the Incompressible inviscid flow displaced fluid. Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Simple device for measuring pressure: Torricelli’s barometer International Standard Atmosphere (ISA)

Aerodynamics Assumptions R. Tognaccini The air is dry. Introduction The air is a perfect gas: Hydrostatics

Fundamental principles p = ρRT (28)

Incompressible inviscid flow The air is at rest and Effects of Stevino’s law is valid: viscosity dp = −ρgdz Effects of compressibility 3 TSL = 288K, ρSL = 1.23Kg/m , pSL = 101000Pa Aircraft lift-drag polar 0 < z ≤ 11Km: troposphere, Tz = −6.5K/Km

11 < z ≤ 25Km: stratosphere, Tz = 0 25 < z ≤ 47Km: mesosphere, Tz = 3K/Km

Tz : temperature gradient. Application: compute ρ(z) and p(z) in the troposphere

Aerodynamics

R. Tognaccini Dividing Stevino’s law and perfect gas equation:

p z Introduction dp g dz Z dp¯ g Z dz¯ Hydrostatics = − ⇒ = − (29) p R T (z) pSL p¯ R 0 T (¯z) Fundamental principles In the troposphere: T = T − T z and the integral can be Incompressible SL z inviscid flow solved: Effects of g viscosity p  T  RTz Effects of = (30) compressibility pSL TSL Aircraft lift-drag polar By perfect gas equation:

g −1 ρ  T  RTz = (31) ρSL TSL Fluid particle kinematics Some definitions

Aerodynamics

R. Tognaccini

Introduction Hydrostatics Trajectory (or particle path), the curve traced out by a particle Fundamental as time progresses. principles

Incompressible Streamline for a fixed time is a curve in space tangent to inviscid flow particle velocities in each point. Effects of viscosity Strakeline the locus of all fluid particles which, at some time Effects of have past through a particular point. compressibility Aircraft In steady flow, trajectories, streamlines and strakelines are lift-drag polar coincident. Streamlines and strakelines

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Streamline visualization around a NACA airfoil,

M∞ ≈ 0, Re∞ ≈ 6000. Streakline visualization around a

wedge. Motion of a fluid particle

Aerodynamics R. Tognaccini A fluid particle translates (with velocity V) rotates (with

Introduction angular velocity Ω) and deforms (volume and shape

Hydrostatics change). Fundamental Rotation and deformation can be computed by performing principles

Incompressible spatial partial derivatives of the velocity field inviscid flow V(x, y, z) = (Vx , Vy , Vz ). Effects of viscosity Vorticity ω = 2Ω, fluid property linked to angular velocity of Effects of 2 compressibility fluid particle . Aircraft Dilatation Θ, percentage variation of the volume of a fluid lift-drag polar particle in the unit time 3. In incompressible flow (ρ = const) Θ = 0.

2ω = ( ∂Vy − ∂Vy , − ∂Vz + ∂Vx , ∂Vz − ∂Vx ) ∂x ∂Vx ∂x ∂z ∂x ∂y 3 ∂Vx ∂Vy ∂Vz Θ = ∂x + ∂y + ∂z Conservation of mass (continuity equation)

Aerodynamics Mass can be neither created nor destroyed. R. Tognaccini Continuity equation: Introduction

Hydrostatics

Fundamental principles ρ1V1A1 = ρ2V2A2 (32) Incompressible inviscid flow

Effects of viscosity dm Mass contained in the volume mass flow˙ m ≡ dt = ρVA Effects of A1V1dt: compressibility dm1 = ρ1A1V1dt. Aircraft lift-drag polar At time t2: dm2 = ρ2A2V2dt. The principle of mass conservation ensures that: Between two streamlines dm2 = dm1. m˙ = const Conservation of mass in incompressible flow (ρ = const)

Aerodynamics Continuity equation:

R. Tognaccini Ifm ˙ and A(x) are known: V1A1 = V2A2 (33) Introduction m˙ Hydrostatics V = (34) Fundamental ρA principles

Incompressible velocity is known in each inviscid flow section! Effects of viscosity Velocity increases along a Effects of convergent channel. compressibility Aircraft Velocity decreases along a lift-drag polar Quasi-1d flow divergent channel. Channel with small variations of Warning: the behavior is section area A(x). opposite in supersonic flow! Unknowns are average quantities in each section. Momentum equation (1/2)

Aerodynamics Momentum equation is second law of Dynamics F = ma R. Tognaccini specialized for a fluid particle P. Introduction Forces acting on P: Hydrostatics particle weight dm g; Fundamental principles pressure acting orthogonal to Incompressible each particle face; inviscid flow Effects of friction force acting tangent viscosity to each particle face. Effects of Hypotheses: compressibility Aircraft negligible friction force (µ = 0); lift-drag polar negligible effects of gravity Total force acting on the fluid particle P in streamwise direction:  dp  dp F = pdydz − p + dx dx dydz = − dx dxdydz Momentum equation (2/2)

Aerodynamics

R. Tognaccini

Introduction Fluid particle mass and acceleration: Hydrostatics dV dV dx dV dm = ρdxdydz; a = dt = dx dt = dx V . Fundamental principles Momentum equation Incompressible inviscid flow Effects of dp dV viscosity + ρV = 0 (35) Effects of dx dx compressibility

Aircraft lift-drag polar Conservation + momentum equations in inviscid flow are called Euler equations. Bernoulli’s theorem in inviscid and incompressible flow

Aerodynamics If ρ = const eq. (35) can be easily integrated along a streamline: R. Tognaccini Z p2 Z V2 Introduction dp + ρ V dV = 0 (36) p V Hydrostatics 1 1 Fundamental obtaining: principles Incompressible Bernoulli’s equation inviscid flow

Effects of viscosity 1 p + ρV 2 = const (37) Effects of 2 compressibility

Aircraft lift-drag polar In the case of a body immersed in an uniform stream, the constant is the same everywhere: 1 1 p + ρV 2 = p + ρV 2 (38) 2 ∞ 2 ∞ Some remarks on Bernoulli’s theorem

Aerodynamics R. Tognaccini It holds along a streamline for a steady, inviscid Introduction (frictionless) and incompressible flow. Hydrostatics q = ρV 2/2 is named dynamic pressure. Fundamental principles Standard pressure p is often named static pressure. Incompressible 2 inviscid flow p0 = p + ρV /2 is named total or stagnation pressure.

Effects of The stagnation pressure (p0) is the pressure of a fluid viscosity particle when decelerate to V = 0 in an adiabatic and Effects of compressibility frictionless process. Aircraft lift-drag polar In an incompressible, steady, inviscid quasi-1d flow (channel) V and p can be computed by continuity and Bernoulli’s equations provided the flow is known in one point of the channel. Energy equation (1/3)

Aerodynamics

R. Tognaccini Energy can be neither created nor destroyed, Introduction it can only change form. Hydrostatics

Fundamental principles It is nothing more than the first law of Thermodynamics for a

Incompressible Thermodynamic system: inviscid flow Effects of δq + δw = de (39) viscosity

Effects of compressibility δq heat flux entering the system (per unit mass), Aircraft lift-drag polar δw work done by the system (per unit mass), e internal energy per unit mass. 4

4 For a perfect gas e = Cv T , where Cv is the specific heat at constant volume. Energy equation (2/3)

Aerodynamics

R. Tognaccini R Introduction Mδw = −ds A pdA = −pdV 1 Hydrostatics δw = −pdv; v = ρ : specific volume Fundamental principles

Incompressible inviscid flow First law of Thermodynamics eq. (39) gives: Effects of viscosity

Effects of δq = de + pdv (40) compressibility Aircraft Introducing the specific enthalpy h = e + pv, differentiating it lift-drag polar and replacing in eq. (40):

δq = dh − vdp (41) Energy equation (3/3)

Aerodynamics

R. Tognaccini Assume the flow is adiabatic: δq = 0. Introduction Compute dp along a streamline from momentum equation eq. Hydrostatics 1 (35): dp = − v V dV . Fundamental principles Incompressible First law of Thermodynamics eq. (41) gives: inviscid flow Effects of dh + V dV = 0 (42) viscosity

Effects of compressibility Integrating along a streamline:

Aircraft lift-drag polar Energy equation (in a steady, adiabatic flow)

V 2 h + = const (43) 2 Some remarks on the energy equation

Aerodynamics

R. Tognaccini It holds along a streamline for a steady, inviscid and adiabatic flow. Introduction

Hydrostatics In the case of a body immersed in an uniform stream, the Fundamental constant is the same everywhere: principles

Incompressible 2 2 inviscid flow V V∞ h + = h∞ + (44) Effects of 2 2 viscosity Effects of For a perfect gas: h = CpT , compressibility Cp: specific heat at constant pressure. Aircraft lift-drag polar Energy equation allows to study the quasi-1d compressible flow. In incompressible, adiabatic and inviscid flow h = p/ρ: Bernoulli’s and energy equations are coincident. Isentropic flow

Aerodynamics

R. Tognaccini From the energy equation for a perfect gas in adiabatic flow Introduction (γ = Cp/Cv , R = Cp − Cv ): Hydrostatics T γ − 1 Fundamental 0 = 1 + M2 (45) principles T 2 Incompressible inviscid flow An adiabatic, inviscid and subsonic flow is also isentropic. For a Effects of viscosity perfect gas:

Effects of γ 1/(γ−1) γ/(γ−1) compressibility p ρ  ρ T  p T  0 = 0 ; 0 = 0 ; 0 = 0 Aircraft p ρ ρ T p T lift-drag polar (46)

T0, p0 and ρ0 are respectively named total or stagnation temperature, pressure and density in compressible flow. Practical applications

Aerodynamics

R. Tognaccini

Introduction Hydrostatics Problem n. 6 Fundamental principles For a given altitude and freestream Mach number compute the Incompressible inviscid flow maximum surface temperature on the aircraft

Effects of viscosity Tmax ≈ T0   Effects of γ−1 2 compressibility Tmax ≈ 1 + 2 M∞ T∞

Aircraft lift-drag polar The speed of sound (1/2)

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible Sound wave moving in a fluid at rest Reference system attached to the sound wave inviscid flow

Effects of Continuity equation: viscosity ρa = (ρ + dρ)(a + da) Effects of compressibility Neglecting smaller term dρ da: Aircraft lift-drag polar da a = −ρ (47) dρ From momentum eq. (35) with V = a: dp da = − ρa The speed of sound (2/2)

Aerodynamics

R. Tognaccini

Introduction Replacing in eq. (47): dp Hydrostatics a2 = (48) Fundamental dρ principles γ Incompressible The process is isentropic p = kρ , therefore inviscid flow

Effects of Speed of sound in a perfect gas: viscosity Effects of r p compressibility a = γ = pγRT (49) Aircraft ρ lift-drag polar since for a perfect gas p/ρ = RT . Application: airspeed measurement The Pitot tube

Aerodynamics The Pitot tube measures the freestream dynamic pressure. R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow Incompressible flow Effects of 1 The Pitot tube measures the pressure difference ∆p viscosity between the two pressure probes. Effects of compressibility 2 If properly placed, the static probe is at p = p∞, whereas Aircraft lift-drag polar the totale pressure probe is at p = p0: measured ∆p = p0 − p∞. 3 From the Bernoulli’s equation: q = ∆p. The Pitot tube measures the dynamic pressure of the freestream. Application: true and equivalent airspeeds in incompressible flow

Aerodynamics In incompressible flow, the freestream velocity (TAS, True Air R. Tognaccini Speed) can be obtained by the dynamic pressure q if the Introduction correct ρ∞ at flight level is known: Hydrostatics s Fundamental 2q principles TAS = V∞ = (50) Incompressible ρ∞ inviscid flow

Effects of viscosity The instrument display in the cockpit usually adopts standard

Effects of density at sea level ρSL to convert q in airspeed; therefore the compressibility displayed velocity is: Aircraft lift-drag polar s 2q EAS = Ve = (51) ρSL

EAS stands for Equivalent Air Speed, i.e. the true freestream velocity if ρ∞ = ρSL. Application: airspeed measurement in compressible subsonic flow

Aerodynamics As in incompressible flow, the pitot tube measures R. Tognaccini ∆p = p0 − p∞. Introduction From isentropic flow equations: Hydrostatics  γ/(γ−1) Fundamental p0 γ − 1 2 principles = 1 + M∞ (52) p∞ 2 Incompressible inviscid flow By some algebra: Effects of viscosity 2 " (γ−1)/γ # 2 2 2a∞ ∆p Effects of (TAS) = V∞ = 1 + − 1 (53) compressibility γ − 1 p∞ Aircraft lift-drag polar Replacing a∞ and p∞ with the ISA sea level values we obtain the Calibrated Air Speed (CAS):

2 " (γ−1)/γ # 2 2 2aSL ∆p (CAS) = V∞ = 1 + − 1 (54) γ − 1 pSL Application: remarks on airspeeds

Aerodynamics

R. Tognaccini

Introduction For an aircraft the real interest is in: Hydrostatics

Fundamental Dynamic pressure, responsible for the lift. principles

Incompressible The true freestream Mach number, which identifies the inviscid flow flow regime. Effects of viscosity Problem n. 7 Effects of compressibility For a given altitude and freestream velocity of an aircraft, Aircraft lift-drag polar compare TAS, EAS and CAS. Incompressible inviscid flow

Aerodynamics Hypotheses: R. Tognaccini The flow is steady.

Introduction The flow is incompressible (M = 0) or iposonic (M ≈ 0)5. Hydrostatics The flow is inviscid (Re → ∞) and adiabatic. Fundamental principles Solution strategy: Incompressible inviscid flow 1 Flow conditions V, p needs to be known in one point Effects of (freestream for instance). viscosity

Effects of 2 Velocity V is obtained solving continuity equation. compressibility 3 Aircraft Pressure p is obtained from Bernoulli’s equation. lift-drag polar An example: Quasi-1d (horizontal) channel flow A1(x) V (x) = A(x) 1 2 2
p(x) = p1 + 2 ρ V1 − V 5A rule of thumb is M < 0.3 everywhere Two-dimensional flow The simplest airfoil: the circular cylinder

Aerodynamics On a solid wall V = 0 because of viscosity. R. Tognaccini In inviscid flow V is tangent to the wall. Introduction Hydrostatics Pressure coefficient: Fundamental principles p − p Incompressible C = ∞ (55) inviscid flow p 1 2 ρ∞V∞ Effects of 2 viscosity

Effects of compressibility From Bernoulli’s equation:

Aircraft 2 lift-drag polar  V  Cp = 1 − (56) V∞ On the cylinder wall:

2 V = 2V∞ sin θ , Cp = 1 − 4 sin θ (57) Pressure and forces on the cylinder

Aerodynamics Two stagnation points: θ = 0◦ and R. Tognaccini θ = 180◦. Introduction Cpmax = 1 in stagnation points Hydrostatics (incompressible flow). Fundamental ◦ ◦ principles Cpmin = −3 at θ = 90 , 270 .

Incompressible Lift and drag (per unit length) can inviscid flow be obtained integrating pressure on Effects of viscosity the cylinder surface.

Effects of Pressure field is symmetric respect compressibility to x and y axes. Aircraft lift-drag polar Straightforward consequence: both lift and drag are zero. D’Alembert paradox: In inviscid, two-dimensional and subsonic flow the drag is zero. The relevance of vorticity

Aerodynamics Vorticity: ω = 2Ω, Ω: angular velocity of the fluid particle.

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity Stokes’ theorem: Effects of compressibility Aircraft Z I lift-drag polar n · ωdS = V · dl = Γ (58) S C

Γ : Circulation on the circuit C. dl : infinitesimal displacement tanget to C. Vortex lines, vortex tubes and vortices

Aerodynamics

R. Tognaccini Vortex line a curve in space in each point tangent to particle Introduction vorticity. Hydrostatics

Fundamental Vortex tube the set of vortex lines crossing a circuit. principles Γ = H V · dl intensity of the vortex tube. Incompressible C inviscid flow Vortex a vortex tube of infinitesimal section and finite Effects of viscosity intensity.

Effects of compressibility A region in which ω = 0 is named irrotational.

Aircraft lift-drag polar Crocco’s theorem: An inviscid, steady region characterized by an uniform upstream flow is irrotational. Helmholtz’s theorems

Aerodynamics

R. Tognaccini 1st Helmholtz’s theorem: Introduction The intensity of a vortex tube is constant along its axis. Hydrostatics

Fundamental principles Corollary: Incompressible A vortex tube is closed or starts and ends at the boundary of inviscid flow the flow domain. Effects of viscosity Isolated vortex tube a vortex tube immersed in an irrotational Effects of compressibility region. Aircraft lift-drag polar 2nd Helmholtz’s theorem: The circulations of all circuits surrounding an isolated vortex tube is the same and is equal to the vortex tube intensity. The infinite straight vortex in inviscid (ideal) flow

Aerodynamics

R. Tognaccini The flow is 2d in planes Introduction orthogonal to the vortex. Hydrostatics Streamlines are concentric Fundamental principles circumferences. Incompressible Γ inviscid flow V = 2πr , Γ: vortex intensity. Effects of From Bernoulli’s theorem viscosity pressure is constant on a Effects of compressibility streamline and infinitely low in Aircraft lift-drag polar the vortex core. In a real flow the vortex core has a finite thickness in which V ≈ Ωr. Rotating circular cylinder The Magnus effect

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility Due to viscosity on the body surface V = ΩR and a 2 Aircraft circulation Γ = 2πΩR is introduced. lift-drag polar The effect of rotation is obtained by assuming that in the cylinder centre there is a vortex of intensity Γ. the flow is no more symmetric and lift is generated. Kutta-Jukowskij theorem

Aerodynamics Hypotheses R. Tognaccini 2d body of arbitrary shape. Introduction Steady, inviscid, subsonic flow. Hydrostatics Fundamental Theorem: principles

Incompressible inviscid flow l = ρ∞V∞Γ (59) Effects of viscosity

Effects of compressibility The theorem highlights the presence of circulation if there Aircraft is lift and vice versa. lift-drag polar Stokes’ theorem highlights the necessity of vorticity to have circulation...... but the flow is irrotational! (see Crocco’s theorem). Again, another inconstistency of the inviscid flow theory. The Kutta condition (1/2)

Aerodynamics

R. Tognaccini In the case of a 2d body of smooth shape in inviscid,

Introduction subsonic flightΓ=0 and, according to Kutta-Jukowskij

Hydrostatics theorem, lift l = 0. Fundamental It is necessary to introduce a particular shape to obtain principles circulation Γ and therefore lift: the airfoil. Incompressible inviscid flow The airfoil is a 2d body characterized by a geometric Effects of viscosity discontinuity at trailing edge.

Effects of compressibility

Aircraft lift-drag polar

sharp trailing edge cusp trailing edge The Kutta condition (2/2)

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility Aircraft Solution A is unphysical, because it requires an infinite lift-drag polar acceleration at trailing edge. Kutta condition: V = 0 at a sharp trailing edge; V is continuous at a cusp trailing edge. Due to Kutta condition around the airfoil circulation is generated and therefore lift! The lift of an airfoil

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow pl : pressure on lower side; pu: pressure on upper side. Effects of viscosity Z TE Z TE Effects of l = pl cos θds − pu cos θds compressibility LE LE Aircraft Z c Z c lift-drag polar = (pl − p∞)dx − (pu − p∞)dx (60) 0 0

Z 1 x  Cl = (Cpl − Cpu )d (61) 0 c The pressure coefficient diagram

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

NACA 0012 airfoil. M∞ = 0, Re∞ = ∞, α = 9deg.

R 1 x
Airfoil load: ∆Cp = Cpl − Cpu Cl = 0 ∆Cpd c

The area within the red curve is equal to airfoil Cl ! Thin airfoil theory in inviscid incompressible flow

Aerodynamics Hypotheses: R. Tognaccini steady 2d, inviscid, incompressible flow; Introduction thin airfoil of small camber at small angle of attack. Hydrostatics Fundamental Lift coefficient: principles

Incompressible inviscid flow Cl = Clα(α − αzl ) (62) Effects of viscosity

Effects of Moment coefficient: compressibility

Aircraft lift-drag polar C C = C − l (63) mle m0 4

Cm1/4 = Cm0 (64)

Clα = 2π, αzl ∝ camber, Cm0 ∝ camber Main results of thin airfoil theory

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics Lift curve is a straight line with Clα = 2π. Fundamental principles αzl < 0 for positive camber and αzl ∝ camber. Incompressible C linearly decreases with α and C . inviscid flow mle l

Effects of The aerodynamic force is applied in the pressure centre viscosity C placed at xcp = − mle . Effects of c Cl compressibility The moment respect to the aerodynamic centre which is Aircraft lift-drag polar placed at x = c/4 is independent of the angle of attack. NACA airfoils

Aerodynamics

R. Tognaccini Modern subsonic airfoils are not thin! During WWI wind tunnel experiments in Germany showed Introduction relatively thick airoils have larger Cl . Hydrostatics max Fundamental Increased manouverability and reduced take-off lengths. principles

Incompressible Increased structure robustness and reduced weight. inviscid flow

Effects of Bracing not required: much lower drag. viscosity During ’30s extensive wind tunnel campaigns at NACA. Effects of compressibility Following WWII NACA data became public: NACA airfoils are Aircraft lift-drag polar now standard in subsonic flight.

Thick airfoil performances in inviscid incompressible flow:

Clα = 2π(1 + 0.77t), where t is the thickness ratio of the airfoil.

αzl well predicted by thin airfoil theory. NACA airfoils Geometry definition

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow Effects of x ∈ (0, 1), y = y (x): mean line equation, viscosity c y = yt (x): half-thickness equation, tan θ = dyc /dx. Effects of compressibility Aircraft Airfoil point coordinates: lift-drag polar

xU = x − yt sin θ yU = yc + yt cos θ

xL = x + yt sin θ yL = yc − yt cos θ 4 digit NACA airfoils

Aerodynamics Half-thickness distribution: R. Tognaccini t √ 2 yt = ± 0.29690 x − 0.12600x − 0.35160x Introduction 0.20 3 4
Hydrostatics + 0.28430x − 0.10150x . (65)

Fundamental principles t airfoil percentage thickness; maximum thickness at x = 0.3 Incompressible 2 inviscid flow rt = 1.1019t , curvature radius at LE Effects of viscosity Mean line: Effects of compressibility m 2 ∀x ≤ p : yc = 2 (2px − x ); (66) Aircraft p lift-drag polar m ∀x > p : y = (1 − 2p + 2px − x 2); (67) c (1 − p)2

px position of maximum camber m maximum camber (y value). 4 digit NACA airfoils Numbering system

Aerodynamics A 4 digit NACA airfoil is identified by 4 digits: D1D2D3D4 R. Tognaccini D1/100= m maximum camber Introduction D2/10= p position of maximum camber Hydrostatics

Fundamental D3D4 = t maximum thickness principles Example Incompressible inviscid flow NACA 2412 airfoil: 12% thickness, 2% camber, maximum camber Effects of viscosity placed at x/c = 0.4.

Effects of compressibility

Aircraft lift-drag polar

NACA 2412 airfoil 5 digit NACA airfoils

Aerodynamics

Designed for improved Clmax performance and reduced pitching R. Tognaccini moment.

Introduction Half-thickness: Hydrostatics same as 4 digit airfoils Fundamental principles Mean line:

Incompressible k1 3 2 2 inviscid flow ∀x ≤ m : yc = [x − 3mx + m (3 − m)x]; (68) Effects of 6 3 viscosity k1m ∀x > m : yc = (1 − x) . (69) Effects of 6 compressibility

Aircraft linea media m k1 lift-drag polar 210 0.0580 361.4 220 0.1260 51.64 230 0.2025 15.957 240 0.2900 6.643 250 0.3910 3.230 5 digit NACA airfoils Numbering system

Aerodynamics

R. Tognaccini A 5 digit NACA airfoil is identified by 5 digits: D1D2D3D4D5

Introduction D1D2D3 mean line

Hydrostatics D4D5 = t maximum thickness Fundamental principles Example Incompressible NACA 23012 airfoil: 12% thickness, 230 mean line. inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

NACA 23012 airfoil 4 digit vs 5 digit NACA airfoil performance

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar The wing of finite A in inviscid, incompressible flow

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

Is there the chance that the flow is 2d in planes parallel to (x, z)? The answer is yes if A  1 and if the sweep angle Λ ≈ 0. Genesis of the trailing vortices and lift induced drag 1/3

Aerodynamics We have the lift induced drag in a 3d wing but not for an airfoil in 2d

R. Tognaccini flow; why?

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Genesis of the trailing vortices and lift induced drag 2/3

Aerodynamics Lower-upper pressure difference causes a rotation of the flow R. Tognaccini around wing tips. Introduction Two strong counter-rotating vortices are generated at tips; they Hydrostatics are named free vortices. Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

water tunnel visualization of the trailing vortices around a finite wing. Genesis of the trailing vortices and lift induced drag 3/3

Aerodynamics R. Tognaccini The flow is 2d in planes parallel to

Introduction (x, z) but. . . Hydrostatics . . . free vortices induce a downwash Fundamental w for −b/2 < y < b/2 and an principles upwash for y < b/2 and y > b/2. Incompressible inviscid flow Each wing section works in a 2d Effects of viscosity flow but, due to downwash, at a

Effects of smaller effective angle of attack compressibility αeff = α − αi Aircraft lift-drag polar According to K-J theorem the αi induced angle of attack or downwash angle. aerodynamic force is orthogonal to Veff therefore a streamwise force parallel to V∞ arises: the lift induced drag. The vortex system of the wing

Aerodynamics

R. Tognaccini At each wing section y the flow is 2d. Introduction

Hydrostatics According to K-J theorem local lift Fundamental is: l(y) = ρV∞Γ(y). principles

Incompressible A bound vortex of variable intensity inviscid flow Γ(y) runs along the wing. Effects of viscosity Since Γ(y) varies, according to 1st Effects of Helmholtz’s theorem at each y a compressibility free vortex compensates circulation Aircraft lift-drag polar variation dΓ. The free vortices are aligned with the freestream and then roll-up. The free vortices induce the downwash w. The downwash on the wing

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible Downwash induced by an Downwash induced by an inviscid flow elementary infinite vortex: elementary semi-infinite vortex: Effects of viscosity dΓ dΓ Effects of dw = dw = compressibility 2π(y − y0) 4π(y − y0)

Aircraft lift-drag polar Downwash induced by all free vortices distributed along the wing:

Z +b/2 1 dΓ w = dy0 (70) −b/2 4π(y − y0) dy0 The downwash in the far wake

Aerodynamics

R. Tognaccini Free vortices are distributed along the whole wing. Wingtip vortices are more visible than the others because Introduction of their stronger intensity. Hydrostatics

Fundamental In the far wake the downwash is twice the one on the principles wing: w∞ = 2w. Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar The lifting line equation

Aerodynamics Downwash angle: R. Tognaccini Z +b/2 Introduction w 1 1 dΓ αi (y) ≈ = dy0 (71) Hydrostatics V∞ 4πV∞ −b/2 (y − y0) dy0 Fundamental principles 6 Effective AoA: αeff (y) = α(y) − αi (y). Incompressible inviscid flow Since dL = ρV∞Γdy and due to definition of Cl : Effects of 2Γ(y) viscosity = C (y)[α(y) − α (y)] (72) V c(y) lα i Effects of ∞ compressibility By equation (71): Aircraft lift-drag polar Lifting line equation:

2Γ(y) 1 Z +b/2 1 dΓ + dy0 = α(y) (73) V∞cClα 4πV∞ −b/2 (y − y0) dy0

6α measured respect to airfoil zero lift line. Lift and induced drag of the wing

Aerodynamics The load along the wing: γ = Γ = cCl .7 V∞b 2b R. Tognaccini R +b/2 R +b/2 L = −b/2 ldy and Di = −b/2 lαi dy, therefore Introduction Hydrostatics Lift coefficient: Fundamental principles Z +1 Incompressible CL = A γ(η)dη (74) inviscid flow −1 Effects of viscosity

Effects of Induced drag coefficient: compressibility Aircraft Z +1 lift-drag polar A CDi = γ(η)αi (η)dη (75) −1

y where η = b/2 .

7Last equality obtained thanks to the K-J theorem. The elliptic wing

Aerodynamics Only one explicit solution of the lifting line equation: the elliptic wing.

R. Tognaccini 1 elliptic distribution of the airfoil chords along the span: p 2 Introduction c = c0 1 − η ; Hydrostatics 2 same airfoil along the span; Fundamental principles 3 no twist. Incompressible inviscid flow

Effects of Elliptic wing performance: viscosity

Effects of compressibility 2CL p 2 CL γ(η) = 1 − η , αi = (76) Aircraft πA πA lift-drag polar Clα CL = CLα(α − αzL) , CLα = , αzL = αzl (77) Clα 1 + πA C 2 C = L (78) Di πA On the elliptic wing performance

Aerodynamics

R. Tognaccini Wing performance only depend on the load distribution. Introduction Hydrostatics The elliptic wing has an elliptic load p 2 Fundamental distribution: γ = γ0 1 − η . principles

Incompressible There are infinite way to obtain an inviscid flow elliptic load distribution. Effects of viscosity CLα < Clα. Effects of compressibility CL < Cl at the same AoA. Aircraft lift-drag polar Among the wings of simple shape the elliptic wing provides the

minimum CDi at the same CL. Lift performance of an arbitrary unswept wing are qualitatively similar to the ones of the elliptic wing. Practical applications

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental Problem n. 7 principles Given A and CL of an elliptic wing compute the corresponding Incompressible inviscid flow airfoil Cl . Effects of viscosity Problem n. 8 Effects of compressibility Given A and CL of a wing compute the average load γ. Aircraft lift-drag polar The load along the wing

Aerodynamics

R. Tognaccini

Introduction cCL Hydrostatics γ = 2b has important aerodynamic role but it is also Fundamental fundamental to design the wing structure. principles γ = 0 at wing tips. Incompressible inviscid flow An asymmetrical γ(η) respect to η = 0 allows for Effects of viscosity obtaining a roll moment necessary for aircraft veer.

Effects of compressibility Local load γ at a station η can be changed by changing

Aircraft local chord c or local lift coefficient Cl . Since wing AoA is lift-drag polar fixed, Cl can only be changed by wing twist. Basic and additional load

Aerodynamics Load decomposition: R. Tognaccini

Introduction γ(η) = γb(η) + γa(η) (79) Hydrostatics

Fundamental 8 principles γb basic load, the load distribution when CL = 0 . It depends on the Incompressible wing twist , γb(η) = 0 if  = 0. inviscid flow

Effects of γa additional load, the difference between the actual and the basic viscosity load. It depends on the wing planform. Effects of compressibility

Aircraft lift-drag polar

8Note that the area underlying γ(η) function is proportional to lift coefficient. Schrenk’s method for the computation of the wing load distribution

Aerodynamics Input data:

R. Tognaccini 1 Chord distribution c=c(y); Introduction 2 Airfoil performances Clα = Clα(y); Hydrostatics 9 Fundamental 3 Aerodynamic twist a = a(y). principles +1 +1 +1 Incompressible Z Z  Z inviscid flow CL = A γbdη + γadη = A γadη (80) Effects of −1 −1 −1 viscosity

Effects of CL only depends on γa. compressibility

Aircraft Since CL linearly varies: γa = CLγa1, where γa1 is the additional lift-drag polar load for CL = 1.

Therefore: γ = γb + CLγa1 (81)

9 a is referenced respect to zero lift lines. Schrenk’s method Computation of the additional load γa

Aerodynamics A → ∞: αi → 0 and γa = cCl /(2b), i.e. load is proportional R. Tognaccini to the chord. Introduction A → 0: the load is elliptic. Hydrostatics Schrenk’s hypothesis: for intermediate A, γa1 is the average Fundamental principles between the chord distribution and the one of an elliptic wing

Incompressible with same wing surface S: inviscid flow   1 c(η) cell Effects of γa1(η) = + (82) viscosity 2 2b 2b

Effects of compressibility p 2 4S c0 2 cell (η) = c0 1 − η = c0 sin θ; c0 = ; = ; Aircraft πb 2b πA lift-drag polar η = − cos θ; dη = sin θdθ.

Z +1 A Z +1 Z +1  4S p 2 A γa1dη = c(η)dη + 1 − η dη = 1 −1 4b −1 πb −1 (83) Schrenk’s method Computation of the basic load γb

Aerodynamics

R. Tognaccini 1 Compute αzL by approximate formula:

Introduction 2 Z b/2 Hydrostatics α = c (y) dy (84) zL S a Fundamental 0 principles

Incompressible 2 Compute a first guess of the basic AoA αb by formula: inviscid flow Effects of α¯ (y) = α −  (y) (85) viscosity b zL a

Effects of compressibility 3 Effective αb is obtained by averagingα ¯b and the basic Aircraft angle of the untwisted wing (for which αb0 = 0), lift-drag polar therefore αb =α ¯b/2 and cC cC α cC α¯ γ (η) = lb = lα b = lα b (86) b 2b 2b 4b Tapered and elliptic wings

Aerodynamics

R. Tognaccini

Introduction A tapered (trapezoidal) wing planform: Hydrostatics Fundamental c 1 principles = [(1 − η) + λη] A Incompressible 2b (1 + λ) inviscid flow Effects of where λ = ct /cr . viscosity Effects of A tapered wing with a proper twist distribution can compressibility provide an elliptic load, therefore the same performance of Aircraft lift-drag polar the elliptic wing. Wings of small aspect ratio

Aerodynamics

R. Tognaccini

Introduction Hydrostatics Flow in planes parallel to the wing symmetry plane is no Fundamental more 2d. principles

Incompressible CL = CLα(α − αzL) as for large A, but. . . inviscid flow ... C ≈ π A, much smaller! Effects of Lα 2 viscosity Stall angle αs much larger than for high aspect ratio wings Effects of compressibility (30deg and beyond). 2 Aircraft CL lift-drag polar Load distribution along y is elliptic and CDi ≈ πA The leading edge vortex

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics The flow is 2d in the

Fundamental crossflow plane! principles Lift is due to the low Incompressible inviscid flow pressure on the upper wing Effects of induced by LE vortices. viscosity

Effects of enhanced lift due to LE compressibility vortices. Aircraft lift-drag polar Stall due to LE vortex brekdown. The effects of viscosity Three unresolved questions of inviscid theory

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics 1 Why theoretical drag is zero in 2d subsonic flows against Fundamental the experience (D’Alembert paradox)? principles

Incompressible 2 How is the vorticity generated, necessary to obtain the inviscid flow circulation around a lifting airfoil? Effects of viscosity 3 In the inviscid model, velocity on the body wall is tangent Effects of but different than zero, but experience shows V = 0: why compressibility is possible a good prediction of pressure on the body by Aircraft lift-drag polar Bernoulli’s equation? The boundary layer theory

Aerodynamics

R. Tognaccini It must exist a region near the body of thickness δ in which

Introduction we have a velocity variation ∆V from zero to V ≈ V∞. Hydrostatics Experience shows that the larger is V∞, the smaller is δ. Fundamental ρ∞V∞L δ principles If Re =  1 :  1. Incompressible µ∞ L inviscid flow This region of thickness δ is named boundary layer. Effects of ∂V viscosity In the boundary layer, friction stress τ = µ ∂y , where Effects of ∂V V∞ ∂V compressibility ∂y ≈ δ . for δ → 0 ∂y → ∞: friction stress is Aircraft significant even if µ is very small. lift-drag polar Boundary layer theory results

Aerodynamics   Assuming τ ≈ ρV 2 we can obtain that δ ≈ O √1 in the L Re R. Tognaccini L boundary layer: confirmed that, as ReL → ∞ δ/L → 0. Introduction Inviscid state (ReL = ∞) is never obtained in practice, because Hydrostatics even if ReL  1 it cannot be really infinite. . . Fundamental ∂V principles . . . but outside boundary layer ∂y is no more very large and the Incompressible flow is effectively inviscid in practice. inviscid flow  ∂V  Effects of On the wall viscous stress τw = µ ∂y is not negligible and w viscosity is responsible for drag in 2d subsonic flows: D’Alembert paradox Effects of compressibility resolved. Aircraft Inside the boundary layer ω = ∂v : the boundary layer generates lift-drag polar ∂y vorticity necessary to obtain lift (as predicted by K-J theorem). ∂p Inside the boundary layer: ∂y = 0 ⇒ p = p(x): pressure on the wall is equal to pressure at the same x but at the border of the boundary layer where the flow is frictionless and Bernoulli’s theorem valid with all the results of inviscid theory. The laminar boundary layer on the flat plate at α = 0deg (p(x) = p∞)

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental ρ∞V∞x principles Local Reynolds number: Rex = . Incompressible µ∞ inviscid flow 5 Effects of Boundary layer thickness: δ(x) = √ x. viscosity Rex Effects of Z L compressibility Friction drag (per unit length): df = τw dx. Aircraft 0 lift-drag polar τ 0.664 Skin friction: C (x) = w = √ . f 1 2 2 ρ∞V∞ Rex Z 1 x  1.328 Friction drag coefficient: Cdf = Cf d = √ . 0 L ReL Practical applications

Aerodynamics Problem n. 9 R. Tognaccini Compute the drag per unit length of a flat plate at α = 0deg Introduction long 10cm in a stream at V∞ = 10Km/h in standard ISA Hydrostatics conditions. Fundamental principles

Incompressible Problem n. 10 inviscid flow Compute the boundary layer thickness at the end of the plate Effects of viscosity for the previous problem. Effects of compressibility

Aircraft Sutherland’s law for viscosity of air: lift-drag polar µ  T 3/2 T + 110K = 0 (87) µ0 T0 T + 110K

−5 T0 = 288K, µ0 = 1.79 × 10 Kg/(ms) The displacement thickness of a flat plate at α = 0deg

Aerodynamics Z ∞  V  R. Tognaccini δ∗(x) = 1 − dy (88) 0 V∞ Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar In order to obtain in inviscid flow the same mass flux rate above a body in a real viscous flow it is necessary to thicken the body ∗ ∗ of δ : shaded area = V∞δ . The (small) effect of the boundary on the outer inviscid flow can be obtained by thickening the body of δ∗ and repeat the inviscid analysis. The boundary layer over airfoils

Aerodynamics dp R. Tognaccini Since δ is small and dy is negligible in the boundary layer, a first good estimation of pressure on the airfoil can be Introduction obtained by inviscid analysis assuming δ∗ = 0. Hydrostatics

Fundamental Along the airfoil (x) we have pressure variations and the principles velocity profile along y inside the boundary layer can have Incompressible inviscid flow different characteristics near the wall.

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

∂p ∂p ∂x < 0 (favorable) ∂x > 0 (adverse) The separation point (1/2)

Aerodynamics R. Tognaccini ∂V  xs : = 0 Introduction ∂y y=0 Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Boundary layers with adverse pressure gradient could have a separation point. Following the separation point pressure is no more the one obtained by inviscid solution. The separation point (2/2)

Aerodynamics Separated boundary layers over airfoils have two main drawbacks of: R. Tognaccini 1 Large separated regions reduce the pressure peak at leading Introduction edge and lead to stall. Hydrostatics

Fundamental 2 Pressure recovery in the aft part of the airfoil is largely reduced principles therefore a new type of drag appears: the form or wake drag. Incompressible inviscid flow The term wake drag is used because bodies with large separated Effects of regions are characterized by a wake with a velocity defect. viscosity Effects of In cruise conditions there should not be separation on the airfoil. compressibility

Aircraft lift-drag polar The turbulence Reynolds’ experiment

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow Reynolds’ channel flow experiment evidenced the existence Effects of of two flow regimes radically different: viscosity

Effects of Transition from a regime to the other one depends on a compressibility critical Reynolds number Recr (In the case of the channel Aircraft flow Recr ≈ 2200): lift-drag polar 1 Re < Recr , laminar regime, the flow is stable and regular; 2 Re > Recr , turbulent regime, the flow is unstable and strongly irregular.

Recr is not universal, but depends on the particular flow. The turbulent flow over a flat plate 5 Flow visualization at V∞ = 3.3m/s, Recr ≈ 2 × 10

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

(a) upper view; (b) lateral view. The turbulent jet Laser induced fluorescence, ReD ≈ 2300

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Main characteristics of turbulent flows

Aerodynamics

R. Tognaccini Fluctuations of velocity and pressure. Velocity fluctuations in all 3 directions (even for a flow that should be 2d); Introduction fluctuations are around an average value. Hydrostatics

Fundamental Eddies of different dimensions (from 40mm to 0.05mm in principles the experiment of previous slide). Incompressible inviscid flow Random variations of fluid properties; not possible a Effects of viscosity deterministic analysis, the process is aleatory. Effects of Self-sustained motion. New vortices are created to compressibility substitute the ones dissipated due to viscosity. Aircraft lift-drag polar Flow mixing much stronger than in the laminar case. Turbulent eddies move in all 3 directions and cause a strong diffusion of mass, momentum and energy. Boundary layer thickness larger than in laminar case. Averages and fuctuations

Aerodynamics

R. Tognaccini V = (u, v, w): instantaneous velocity field. ¯ Introduction V = (¯u, v¯, w¯): average velocity field. Hydrostatics u0, v 0, w 0: velocity fluctuations. Fundamental principles

Incompressible inviscid flow Z t0+T 0 1 Effects of u =u ¯ + u u¯ = udt viscosity T t0 Effects of 1 Z t0+T compressibility v =v ¯ + v 0 v¯ = vdt (89) Aircraft T t0 lift-drag polar 1 Z t0+T w =w ¯ + w 0 w¯ = wdt T t0 p p p u¯0, v¯0, w¯0 = 0: fluctuations measured by u¯02, v¯02, w¯02 Average velocity profile in turbulent flows

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility p p p ¯02 ¯02 ¯02 Aircraft u , v , w → 0 at the wall: viscous sublayer. lift-drag polar Due to the increased mixing, turbulent velocity profile more “potbellied” than in the laminar case. . .  ∂V¯  . . . therefore µ ∂y much larger than in laminar flow. w In turbulent flows friction is much larger than in the laminar case. The turbulent boundary layer on a flat plate

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

δ −1/5 −1/5 −1/5 x ≈ 0.37Rex ; Cf ≈ 0.0592Rex ; Cd ≈ 0.074ReL . Practical applications

Aerodynamics

R. Tognaccini Problem n. 11 Introduction Compute the drag per unit length of a flat plate at α = 0deg Hydrostatics long 1m in a stream at V∞ = 10m/s in standard ISA Fundamental principles conditions. Incompressible inviscid flow Problem n. 12 Effects of viscosity Compute the boundary layer thickness at the end of the plate Effects of compressibility for the previous problem.

Aircraft lift-drag polar Problem n. 13 Recompute drag and boundary layer thickness of the previous problem assuming laminar flow. Laminar-turbulent transition

Aerodynamics The transition to turbulence depends on several factors. R. Tognaccini 1 Reynolds number, as already evidenced, is the first Introduction parameter. Hydrostatics 2 Pressure gradient. A laminar boundary layer is unstable if Fundamental ∂p principles there is an inflection in the velocity profile ( ∂x > 0). In Incompressible this case the boundary layer very quickly becomes inviscid flow

Effects of turbulent. viscosity 3 Freestream turbulence. Freestream flow has always a Effects of compressibility certain amount of turbulence: larger freestream turbulence Aircraft facilitates transition. lift-drag polar 4 Surface roughness. Roughness of the surface facilitates transition. 5 Contamination. Dust or bugs on the surface act as roughness. Airfoil boundary layer at very large Reynolds numbers

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental 1 Two boundary layers develop on the upper and lower sides, principles starting from front stagnation point. Initial boundary layer Incompressible inviscid flow thickness is not zero. Effects of 2 Initially the boundary layer is laminar. viscosity Effects of 3 (Possible) laminar separation. When separated the flow quickly compressibility becomes turbulent, acquires energy and reattaches (laminar Aircraft lift-drag polar separation bubble).

4 Transition to turbulence.

5 (Possible) turbulent separation.

6 Airfoil wake. Airfoil drag in subsonic flow

Aerodynamics

R. Tognaccini

Introduction In subsonic flow airfoil drag (2d flow) is completely of

Hydrostatics viscous origin and is named profile drag( dp). Fundamental Profile drag is built-up of two contributions: principles 1 friction drag (df ), due to the direct action of friction Incompressible inviscid flow stresses on the airfoil surface in both laminar and turbulent Effects of region. viscosity 2 form or wake drag (dwake ), due to the not complete Effects of compressibility recover of pressure in the aft part of the airfoil.

Aircraft lift-drag polar Profile drag decomposition

dp = df + dwake The wake drag

Aerodynamics

R. Tognaccini Due to the difference between the actual pressure distribution on the airfoil and the one in theoretical Introduction inviscid condition. Hydrostatics

Fundamental In real viscous flow, stagnation pressure is not recovered at principles the trailing edge implying loss of the thrust force present Incompressible inviscid flow in the back of the airfoil. Effects of viscosity It is always present but very significant when separation

Effects of occurs. compressibility For aerodynamic bodies friction drag is dominant. Aircraft lift-drag polar For blunt bodies wake drag is dominant.

Flat plate at 0deg and 90deg AoA: Cd90 ≈ 100Cd0 . At 90deg flow separates at plate tips. Cp ≈ 1 on the front plate whereas, Cp ≈ 0 on the back. The drag on a circular cylinder The drag “crisis” of cylinders and spheres

3 Aerodynamics Up to ReD ≈ 10 , drag is essentially friction: it reduces as ReD grows. R. Tognaccini 3 For ReD > 10 , drag is mostly wake Introduction drag, therefore independent of ReD . Hydrostatics 5 Fundamental At ReD ≈ 10 separation moves from principles laminar to turbulent. Incompressible inviscid flow In the case of laminar flow, separation Effects of at θ = 100deg. viscosity

Effects of In the case of turbulent flow, separation compressibility at θ = 80deg. Aircraft lift-drag polar When separation at θ = 80deg, improved pressure recovery in the back: very significant reduction of wake drag. Roughness enhances turbulence and anticipates drag “crisis” (the golf ball). The laminar airfoils

Aerodynamics

R. Tognaccini Airfoil drag can be effectively reduced by increasing the Introduction extent of the laminar region. Hydrostatics Laminar airfoils are characterized by a very large extension Fundamental principles of the laminar region starting from stagnation point near Incompressible the leading edge. inviscid flow

Effects of This objective is obtained moving back the point where viscosity the pressure gradient becomes adverse (dp/dx > 0). Effects of compressibility This result can be obtained, for instance, in a limited Aircraft range of angles of attack, moving back the position of lift-drag polar maximum thickness. Laminar airfoils were independently introduced during WWII, by USA and Japan. NACA 6th-series laminar airfoils

Aerodynamics

R. Tognaccini

Introduction Numbering system example:

Hydrostatics NACA 65-215 a=0.6

Fundamental principles 6: series number;

Incompressible inviscid flow 5: laminar region up to x/c ≈ 0.5; Effects of viscosity 2: ideal lift coefficient Effects of compressibility Cli = 0.2; Aircraft lift-drag polar 15: thickness = 0.15% a=0.6: type of mean line. Laminar airfoil performance (1/2) in iposonic flow

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Laminar airfoil performance (2/2) in iposonic flow

Aerodynamics

R. Tognaccini Lift-drag polar curve is characterized by the typical laminar “bag”around the ideal lift coefficient (C ), where Introduction li

Hydrostatics the drag coefficient is significantly lower. Fundamental Up to 30% reduction of profile drag in cruise conditions. principles

Incompressible Far from Cli , due to the appearance of the pressure peak inviscid flow and consequent strong adverse pressure gradient, the Effects of viscosity laminar bag disappears and drag is again comparable to

Effects of standard airfoils. compressibility High lift performance worse than standard airfoils. Aircraft lift-drag polar Due to contamination and surface roughness, laminar flow conditions are very difficult to obtain in flight. Laminar airfoils were a success but not for obtaining laminar flow. . . 3D boundary layers

Aerodynamics

R. Tognaccini On a swept wing the inviscid Introduction streamline on the wing is curved. Hydrostatics Centrifugal force is balanced by Fundamental principles pressure gradient.

Incompressible inviscid flow Pressure gradient remains constant

Effects of inside boundary layer, but viscosity centrifugal force diminishes Effects of (V → 0), implying the rise of a compressibility crossflow velocity (w). Aircraft lift-drag polar w profile has always an inflection: Velocity profile in a 3d boundary layer boundary layers on swept wings are unstable: impossible to obtain a natural extended laminar region. Airfoil stall in iposonic flow.

Aerodynamics The airfoil stall is an essentially viscous phenomenon. R. Tognaccini

Clmax and stall angle αs increases as R∞ increases. Introduction Stall types: Hydrostatics

Fundamental 1 Turbulent stall for trailing edge separation, typical of thick principles airfoils (t > 12%). A separation point xs appears at Incompressible trailing edge on the upper side. It moves forward as α inviscid flow increases. Stall occurs when xs /c ≈ 0.5. It is a smooth Effects of viscosity stall.

Effects of 2 Burst of laminar separation bubble (LSB). Typical of compressibility medium thickness airfoils and/or lower Reynolds number Aircraft 6 lift-drag polar flows (Re∞ < 10 ). The LSB appears on the upper side; this stall is an abrupt and dangerous phenomenon. 3 Long bubble stall, typical of thin airfoils. A LSB appears, it increases in size with α; stall occurs when the LSB covers most of the upper airfoil. 4 Combined stall. Contemporary 1 and 2 phenomena. Wing stall

Aerodynamics

R. Tognaccini

Introduction Wing stall is a complex unsteady and strongly 3D Hydrostatics phenomenon. Fundamental principles Usually it is not symmetrical: it appears on one of the

Incompressible wings first. inviscid flow Therefore it is mandatory to have aircraft control on the Effects of viscosity roll axis: central part of the wing should stall first (ailerons Effects of must be still effective). compressibility Aircraft An approximate stall path can be determined by the lift-drag polar analysis of the curve Clmax (η) − Clb (η) assuming (false) the airfoils behave in 2d up to stall and beyond. Summary of airfoil performance Effects of the different parameters (iposonic flow)

Aerodynamics

R. Tognaccini

Introduction C ≈ C (α − α ). Hydrostatics l lα zl

Fundamental Clα ≈ 2π and (weakly) increases with thickness. principles

Incompressible |αzl | linearly increases with curvature. inviscid flow Cdmin at α = αi (ideal AoA), Cdmin ≈ 90dc, (thick airfoils), Effects of 6 viscosity Re∞ > 10 , fully turbulent flow. Effects of compressibility Clmax and αs increases with Re∞. Clmax ≈ 1.6 and 6 Aircraft αs ≈ 15deg (thick airfoils), Re∞ > 10 . lift-drag polar Stall type mainly influenced by Re∞ and thickness. High-lift devices

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics The task of a high-lift device is to increase CLmax and reduce Fundamental stall speed. principles

Incompressible 1 Mechanical systems. inviscid flow 2 Boundary layer control systems. Effects of viscosity 3 Jet-flaps. Effects of compressibility Here we just briefly discuss the most adopted solution: Aircraft lift-drag polar mechanical systems. Flaps

Aerodynamics Simple flap R. Tognaccini Flap rotation allows for a curvature

Introduction variation. δflap > 0 ⇒ |αzl | increase.

Hydrostatics δflap > 0 ⇒ increase of Cl at fixed AoA. Fundamental principles Stall angle decreases for δflap > 0, but Incompressible usually Clmax increases. inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Slotted flap and Fowler flap

Aerodynamics Slotted flap R. Tognaccini High pressure, high energy air Introduction on lower side airfoil moves Hydrostatics through the slot on the upper Fundamental side. principles Separation considerably Incompressible Double slotted flap inviscid flow delayed.

Effects of Stall angle αs increases, viscosity consequent increase of CLmax . Effects of compressibility Fowler flap Fowler flap Aircraft Main result: increase of wing lift-drag polar surface. Device type ∆Clmax Simple flap ≈ 0.9 Secondary action: increase of Slotted flap ≈ 1.5 curvature. Double slotted flap ≈ 1.9 It can also be slotted. Fowler flap ≈ 1.5 Slat

Aerodynamics

R. Tognaccini

Introduction Hydrostatics The slat is a small wing positioned in Fundamental front of the main body. principles The slot energizes the upper Incompressible inviscid flow side. Effects of Pressure peak on the slat: viscosity weaker adverse pressure Effects of compressibility gradient on the main body.

Aircraft ∆Clmax ≈ 0.5. lift-drag polar B737 high-lift system. Boundary layer around a flapped airfoil

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Sound wave propagation in a compressible inviscid flow

Aerodynamics

R. Tognaccini

Introduction dp ∆p a2 = = lim . Hydrostatics dρ ∆ρ→0 ∆ρ Fundamental 2 principles Incompressible flow: ∆ρ → 0 ⇒ a → ∞ Incompressible inviscid flow In an incompressible flow pressure perturbations propagate

Effects of at infinite speed: the presence of the body in an uniform viscosity stream is simultaneous felt in the complete infinite domain. Effects of compressibility In a compressible flow pressure perturbations propagate at Aircraft a finite speed: the space travelled by a pressure lift-drag polar perturbation during time t is finite and is s = at. Sound wave propagation in supersonic flow

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles a 1 Incompressible µ = arcsin = arcsin (90) inviscid flow V M Effects of viscosity µ: Mach angle. Effects of In supersonic flow pressure perturbations in the motion of the compressibility point from A to D are only felt in the region between the two Aircraft lift-drag polar Mach waves starting from point D. Flow perturbations travel on Mach waves. Along a Mach wave, fluid properties are constant. Outside the Mach cone flow perturbations (also sound) are not felt. Supersonic flow around an infinitesimal wedge

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics V cos µ = (V +dV ) cos(µ−dθ) (91)

Fundamental principles Since dθ  1 and neglecting 2nd

Incompressible order term sin µdV dθ: inviscid flow V sin µdθ + cos µdV = 0 (92) Effects of Just one perturbation in the viscosity flow at the wedge, where a dV dθ Effects of = − √ (93) compressibility Mach wave starts. V M2 − 1 Aircraft Downstream of the Mach wave From momentum equation: lift-drag polar the flow deviates of an angle dp dθ dθ. = √ (94) ρV 2 M2 − 1 p = const along the Mach wave: from momentum equation (35) V cos µ = const. Supersonic flow around an infinitesimal expansion corner

Aerodynamics

R. Tognaccini

Introduction Hydrostatics As for the infinitesimal wedge: Fundamental principles V cos µ = (V +dV ) cos(µ+dθ) (95)

Incompressible inviscid flow −V sin µdθ + cos µdV = 0 (96)

Effects of viscosity dV dθ = √ (97) Effects of V M2 − 1 compressibility Just one perturbation in the dp dθ Aircraft flow at the corner, where a = − √ (98) lift-drag polar Mach wave starts. ρV 2 M2 − 1 Downstream of the Mach wave the flow deviates of dθ. Supersonic flow around a flat plate at α  1 Ackeret linearized theory

Aerodynamics

R. Tognaccini p − p α l ∞ = √ (99) 2 2 ρV∞ M∞ − 1 Introduction pu − p∞ α Hydrostatics = − √ (100) 2 2 ρV∞ M∞ − 1 Fundamental principles

Incompressible Lift coefficient: inviscid flow

Effects of 4α Cl = √ (101) viscosity 2 M∞ − 1 Effects of Solid lines: compression wave: compressibility dV < 0, dp > 0. Aircraft Wave drag coefficient: lift-drag polar Dashed lines: expansion wave: dV > 0, dp < 0. 4α2 Cd = √ (102) Lift: l = (pl − pu)c w 2 M∞ − 1 Drag: d = (pl − pu)cα. Performance of supersonic flat plate airfoil Ackeret linearized theory

Aerodynamics ·10−2 0.1 1 R. Tognaccini )

−2 2 ) 8 · 10 0.8 Introduction deg deg Hydrostatics −2 / / 6 · 10 0.6 (1 Fundamental (1 −2 principles 4 · 10 2 0.4 /α /α

Incompressible l w C inviscid flow 2 · 10−2 d 0.2 C Effects of viscosity 0 0 0 1 2 3 0 1 2 3 Effects of compressibility M∞ M∞ Aircraft lift-drag polar Pressure is constant along the upper and lower plate (not so in subsonic flow).

Pressure centre: xcp/c = 0.5 moves backward with respect to subsonic flow (xcp/c = 0.25).

Cmle = −Cl /2. Supersonic, thin and cambered airfoil Ackeret linearized theory

Aerodynamics Airfoil geometry: y = γC(x) ± τT (x) γ = y /c of mean line (curvature); R. Tognaccini max τ = airfoil maximum thickness (percentage). Introduction 4α 1 C = √ Hydrostatics l 2 M∞−1 Fundamental 4 h 2 γ2 R c 02 τ 2 R c 02 i principles 2 Cd = √ α + C (x)dx + T (x)dx w M2 −1 c 0 c 0 Incompressible ∞ inviscid flow C xcp 3 l Cmle = − 2 , c = 0.5 Effects of viscosity

Effects of Lift not influenced by both curvature and thickness. . . compressibility . . . but curvature and thickness add considerable wave Aircraft lift-drag polar drag. A supersonic airfoil is symmetric and thin with a sharp leading edge. Curvature variation cannot be used for lift control: ailerons and flaps are not effective for supersonic airfoils. Shock waves

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Small pressure perturbations travel at the speed of sound. Strong pressure perturbations can travel faster: the shock waves.

A shock wave is an extremely thin region (≈ 10−5cm) across which fluid properties change drastically. Normal shock waves

Aerodynamics Across a shock wave the flow variables are discontinuous. R. Tognaccini Governing equations Introduction 1 Continuity: ρ1u1 = ρ2u2 Hydrostatics 2 2 2 Momentum: p1 + ρ1u1 = p2 + ρ2u2 Fundamental principles 2 2 u1 u2 3 Energy: h1 + = h2 + Incompressible 2 2 inviscid flow Solution: Effects of 2 viscosity 2 1 + [(γ − 1)/2]M1 1 M2 = 2 Effects of γM1 − (γ − 1)/2 Large positive pressure jump compressibility 2 across a shock wave. ρ2 u1 (γ + 1)M1 Aircraft 2 = = 2 lift-drag polar ρ1 u2 2 + (γ − 1)M1 The flow downstream of a normal shock wave is subsonic. p2 2γ 2 3 = 1 + (M1 − 1) p1 γ + 1 Downstream flow only depends C on upstream Mach number M . γ = p : specific heat ratio 1 Cv (1.4 for air). Oblique shock waves and expansion fans

Aerodynamics

R. Tognaccini Mach waves only allow for very small deviations θ. Introduction

Hydrostatics What happens for larger θ?

Fundamental Case (b): Mach waves principles inclination µ reduces while flow Incompressible expands (velocity increases) inviscid flow through an expansion fan. Effects of viscosity The flow across an expansion Effects of fan is continuous. compressibility Case (a): on the contrary, there Aircraft lift-drag polar is a coalescence of compression waves into a single oblique shock wave. Across the oblique shock wave the flow is discontinuous. Oblique shock wave chart

Aerodynamics

R. Tognaccini For each M1 there is a θmax .

Introduction If θ > θmax it is necessary the presence of a normal shock (at least locally). Hydrostatics

Fundamental Given M1 and θ, two obliques shocks principles are possible: a strong shock and a Incompressible weak shock. inviscid flow Strong or weak depends on the Effects of viscosity boundary conditions. Effects of Usually there are weak shocks around compressibility isolated airfoils and wings. Aircraft lift-drag polar Downstream of a strong shock the flow is subsonic. Except for the small region between red and blue curves, downstream of a weak shock the flow is supersonic. Procedure for oblique shock wave calculation

Aerodynamics

R. Tognaccini 1 Assign upstream Mach number M1 and deflection angle θ. Introduction

Hydrostatics 2 Enter in oblique shock wave chart with M1 and θ and find

Fundamental shock wave angle β. principles 3 Compute upstream Mach number normal to shock wave: Incompressible inviscid flow Mn1 = M1 sin β. Effects of viscosity 4 Compute downstream Mach number normal to shock wave Effects of Mn2, ρ2/ρ1 and p2/p1 by normal shock wave relations or compressibility tables. T2/T1 can be obtained by perfect gas state Aircraft lift-drag polar equation. 5 Compute downstream Mach number: M2 = Mn2/ sin(β − θ). Prandtl-Meyer expansion waves

Aerodynamics

R. Tognaccini Computation of the expansion fan, given M1 and θ Introduction 1 Compute ν(M1) from Hydrostatics eq. (103). Fundamental principles 2 Compute ν(M2) from Incompressible eq. (104). inviscid flow 3 Compute ν(M2) from Effects of viscosity Prandtl-Meyer function eq. (103). Effects of 4 In the expansion fan, compressibility flow is isentropic; p, ρ r γ + 1 r γ − 1 Aircraft ν(M) = arctan (M2 − 1) and T can by computed lift-drag polar γ − 1 γ + 1 from eqs. (45) and p − arctan M2 − 1 (103) (46). Procedure simplified by table in the next slide.

θ = ν(M2) − ν(M1) (104) Prandtl-Meyer function and Mach angle Table for Mach range [1, 50]

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

Shock-expansion theory for the analysis of the supersonic airfoil

Aerodynamics Linearized Ackeret theory can be used R. Tognaccini to analyze a very thin double wedge airfoil. Introduction More accurate analysis by exact Hydrostatics computations of the oblique shocks Fundamental principles and expansion fans. Incompressible If α 6= 0 and α > ε at leading edge on inviscid flow the upper surface there is an expansion Effects of fan! viscosity

Effects of Flow in region 4 computed by imposing compressibility that pressures downstream of trailing Aircraft edge coming from lower and upper lift-drag polar flows are the same. Shock-expansion theory can also be Supersonic airfoil: an airfoil applied to a biconvex airfoil (i.e. designed with attached circular arc upper and lower surface) oblique shocks at leading subdividing the upper and lower edge. surface in small straight segments. Practical applications

Aerodynamics

R. Tognaccini

Introduction Problem n. 14 Hydrostatics Given M and wedge angle θ, compute the oblique shock. Fundamental 1 principles Incompressible Problem n. 15 inviscid flow Effects of Given M1 and corner angle θ, compute the expansion fan. viscosity

Effects of compressibility Problem n. 16 Aircraft lift-drag polar Given a double wedge airfoil, M∞ > 1 and α compute the pressure distribution around the airfoil. The wing of finite A The delta wing in supersonic flight

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible Supersonic leading edge Subsonic leading edge inviscid flow

Effects of viscosity Recall that Mach number orthogonal to a Mach wave

Effects of Mn = 1. compressibility Supersonic LE: the flow orthogonal to the LE is supersonic. Aircraft lift-drag polar Subsonic LE: the flow orthogonal to the LE is subsonic. Wave drag is considerably lower with a subsonic LE. In case of a subsonic LE a conventional blunt airfoil can be used implying better wing performance in subsonic flight. Wing-body performance in supersonic flight The area rule

Aerodynamics The wave drag depends on the R. Tognaccini distribution along the longitudinal

Introduction x-axis of the frontal area A(x).

Hydrostatics Wave drag is minimized reducing

Fundamental discontinuities of the function A(x). principles Result obtained, for instance, reducing Incompressible inviscid flow cross section of fuselage where wing starts. Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

Area rule applied to F-15A Sonic boom

Aerodynamics

R. Tognaccini

Introduction Sonic booom is given by the trace at

Hydrostatics sea level of the complex shock-expansion wave patterns Fundamental principles produced by the aircraft. Incompressible Due to the interaction of the waves it inviscid flow results on earth in the classical Effects of viscosity N-shape pressure graph. Effects of Sonic boom is currently limiting the compressibility supersonic flight on land. Aircraft lift-drag polar Future civil supersonic flight on land depends on the development of low-boom configurations. Quasi-1d flow and the subsonic-supersonic nozzle

Aerodynamics Continuity + momentum equations: R. Tognaccini d(ρVA) = 0 ; dp + ρV dV = 0 (105) Introduction With some calculus (dρ/dp = 1/a2): Hydrostatics Fundamental dρ dV dA dρ dV + + = 0 ; + M2 = 0 (106) principles ρ V A ρ V Incompressible inviscid flow Eliminating dρ/ρ in continuity equation Effects of viscosity Area variation against velocity variation: Effects of compressibility dA 2 dV Aircraft = −(1 − M ) (107) lift-drag polar A V

M < 1: dA > 0 ⇒ dV < 0 M > 1: dA > 0 ⇒ dV > 0 M = 1: dA = 0 Under-expanded and over-expanded supersonic nozzle

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow pexit > p∞ Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

pexit < p∞ Subsonic regime

Aerodynamics R. Tognaccini From eq. (107) and momentum equation:

Introduction Area variation against pressure variation: Hydrostatics dA dp Fundamental = (1 − M2) (108) principles A ρV 2 Incompressible inviscid flow Mach number amplifies pressure variation for a given area variation. Effects of viscosity Prandtl-Glauert compressibility rule: Effects of compressibility C (M = 0) Aircraft p ∞ Cp(M∞) ≈ √ (109) lift-drag polar 2 1 − M∞

Cl (M∞ = 0) Clα(M∞ = 0) Cl (M∞) ≈ √ ⇒ Clα(M∞) ≈ √ (110) 2 2 1 − M∞ 1 − M∞ Pressure coefficient in compressible regime

Aerodynamics

R. Tognaccini

Introduction Energy equation + isentropic relation + speed of sound formula: γ/(γ−1) Hydrostatics T0 γ−1 2 p0 T0
2 p T = 1 + 2 M ; p = T ; a = γ ρ Fundamental principles p − p 2  p  Incompressible C = ∞ = − 1 inviscid flow p 1 2 2 ρ∞V∞ γM∞ p∞ Effects of 2 viscosity  γ   γ−1 2  γ−1 Effects of 2 1 + 2 M∞ compressibility =  − 1 (111) 2    γ  Aircraft γM∞ γ−1 2 γ−1 lift-drag polar 1 + 2 M Pressure coefficients in stagnation and critical points

Aerodynamics

R. Tognaccini Stagnation point (M = 0):

Introduction " γ #   γ−1 Hydrostatics 2 γ − 1 2 Cp0 = 2 1 + M∞ − 1 (112) Fundamental γM∞ 2 principles Incompressible Critical point (M = 1): inviscid flow Effects of " γ γ #  − γ−1   γ−1 viscosity ∗ 2 γ + 1 γ − 1 2 Cp = 1 + M∞ − 1 (113) Effects of γM2 2 2 compressibility ∞

Aircraft lift-drag polar If M∞ 6= 0 then Cp0 > 1. ∗ If M∞ = 1 then Cp = 0. 0 On the lower critical Mach number M∞,cr of an airfoil at a given α

Aerodynamics Sonic conditions reached at first in the point of maximum speed, i.e.

R. Tognaccini where Cp = Cpmin .

Cpmin (M∞) given by the Prandtl-Glauert rule eq. (109). Introduction

Hydrostatics 10 10 8 8

Fundamental |

principles 6 ∗ | 6 min p p C Incompressible 4 C 4 | | inviscid flow 2 2 Effects of 0 0 viscosity 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Effects of M∞ M∞ compressibility 10 Aircraft Abscissa of intersection of lift-drag polar 8 |Cpmin | with |Cp ∗ | gives 0 |

p 6 M∞,cr . C | 4 0 M∞,cr = 0.48 in the picture 2 at right. 0 0 0.2 0.4 0.6 0.8 1 Increasing α implies a 0 M∞ reduction of M∞,cr . Practical applications

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible Problem n. 17 inviscid flow Assuming a given pressure peak of the airfoil, compute M0 . Effects of ∞,cr viscosity

Effects of compressibility

Aircraft lift-drag polar The transonic regime 0 00 M∞,cr < M∞ < M∞,cr

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Transonic flow around an airfoil

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar

Iso-Mach contours. Pressure coefficient on the airfoil.

6 RAE 2822 airfoil; α = 2.31deg, M∞ = 0.729, Re∞ = 6.5 × 10 . The sonic barrier

Aerodynamics

R. Tognaccini Cd is approximately constant up to M0 . Introduction ∞,cr 0 Hydrostatics For M∞ > M∞,cr steep increase of Fundamental Cd due to wave drag appearence. principles

Incompressible M∞DD , drag divergence Mach inviscid flow number: at M∞ = M∞DD we have Effects of dCd /dM∞ = 0.1. viscosity

Effects of Cd reaches a maximum value at compressibility M∞ ≈ 1 then it decreases with the Aircraft lift-drag polar law predicted by Ackeret theory.

Similarly Cl also reaches a maximum at M∞ ≈ 1 and then

NACA 0012-34; Cd = Cd (M∞). decreases with the law predicted by Ackeret theory: shock wave stall. Transonic regime peculiarities

Aerodynamics R. Tognaccini Strong normal shock on the upper surface. Introduction Important phenomenon of the shock -boundary layer interaction. Hydrostatics The strong adverse pressure gradient facilitates boundary layer Fundamental principles separation. Incompressible inviscid flow Boundary layer thickening, shock advances, boundary layer

Effects of reattaches, shock draws back, boundary layer separates again viscosity and so on: buffet phenomenon. Effects of compressibility Too strong stresses on wing in buffet: buffet is the operational Aircraft limit of commercial aircrafts. lift-drag polar Ailerons in the separated boundary layer are no more effective. Vortex generators on the wing are used to energize the turbulent boundary and so delay the shock induced separation. The swept wing

Aerodynamics 0 R. Tognaccini At a given AoA, the swept wing increases M∞,cr , therefore M∞DD and allows for an increased V∞ for a given thrust. Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility α Aircraft αeff ≈ ; (114) lift-drag polar cos Λ

V∞cosΛ Veff = ≈ V∞ cos Λ cos αeff Sketch of an infinite swept wing. (115) The infinite swept wing

Aerodynamics

R. Tognaccini

Introduction On an infinite swept wing the cross flow V∞ sin Λ is not

Hydrostatics effective. Fundamental The flow around the airfoil AC is 2d with a freestream Mach principles number M¯ = M cos Λ. Incompressible ∞ ∞ inviscid flow ¯ 0 Being M∞,cr the critical Mach number of airfoil AC, critical Effects of ¯ 0 viscosity conditions on the wing will be reached when M∞ cos Λ = M∞,cr . Effects of compressibility Therefore critical Mach number of the wing is 0 ¯ 0 Aircraft M∞,cr = M∞,cr / cos Λ lift-drag polar The critical Mach number has been increased of a factor 1/ cos Λ! The lift coefficient of the swept wing 1/2

Aerodynamics

R. Tognaccini Total lift can be computed by using both airfoil sections AB Introduction and AC: Hydrostatics 1 2 1 2 Fundamental Cl ρ∞V∞S = Cleff ρ∞V∞S (116) principles 2 2 Incompressible Therefore: inviscid flow Effects of 2 2 2 viscosity Cl V∞ = Cleff Veff ⇒ Cl = Cleff cos Λ ; (117) Effects of compressibility α Cleff = Clααeff = Clα ; (118) Aircraft cos Λ lift-drag polar α C = C cos2 Λ = C cos Λα . (119) l lα cos Λ lα The lift coefficient of the swept wing 2/2

Aerodynamics

R. Tognaccini The section lift coefficient of a swept wing is a factor Introduction cos Λ lower of the corresponding straight wing. Hydrostatics

Fundamental Low speed performance of a straight wing are better. principles The introduction of a positive Λ on a wing causes a wing Incompressible inviscid flow load displacement towards the tips. Effects of viscosity On the contrary Λ < 0 increases the load towards the root

Effects of section. compressibility Aeroelastic problems (flutter) currently limit the adoption Aircraft lift-drag polar of negative Λ. A positive swept wing usually requires twist and/or taper to obtain a proper wing load distribution. The airfoil for transonic wings

Aerodynamics 0 R. Tognaccini To obtain an increase of M∞,cr and therefore of M∞DD necessary to reduce pressure peak. Introduction

Hydrostatics Laminar airfoils better suited than standard airfoils, but with a cruise M < M0 . Fundamental ∞ ∞,cr principles 0 Supercritical airfoils are designed for a cruise M∞ > M∞,cr . Incompressible inviscid flow Blunt LE and flatter upper surface allows for a reduced Effects of viscosity maximum M: therefore weaker shock. Effects of Necessary lift recovery by rear camber. compressibility

Aircraft lift-drag polar Supercritical vs conventional airfoil

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Aircraft lift-drag polar

Aerodynamics

R. Tognaccini CL = CL(CD , Re∞, M∞, configuration, trim, engine) (120)

Introduction

Hydrostatics

Fundamental principles

Incompressible inviscid flow

Effects of viscosity

Effects of compressibility

Aircraft lift-drag polar Preliminary design polar in cruise:

C 2 C = C + L (121) D D0 πAe Remarks on the parabolic approximation of the aircraft polar

Aerodynamics

R. Tognaccini

Introduction

Hydrostatics CD0 takes into account for profile and wave drag. Fundamental The quadratic term essentially takes into account for the principles lift induced drag. Incompressible inviscid flow The parabolic approximation cannot clearly be used in Effects of viscosity high lift condition: it cannot reproduce stall! Effects of Minimum Drag is in general not obtained for C = 0. compressibility L Aircraft Profile and wave drag depend on CL: it can be taken into lift-drag polar account by a proper modification of the parameter e. Calculation of CD0

Aerodynamics Aircraft is subdivided in N components: wing, fuselage, nacelle,

R. Tognaccini horizontal tail, . . .

Introduction 1 N CD0 = Σk=1IFk CDk Sk (122) Hydrostatics S

Fundamental CDk : drag coefficient of component k. principles

Incompressible Sk : reference surface of component k. inviscid flow IFk : interference factor of the component k with the rest of the Effects of viscosity configuration. Effects of If component k is an aerodynamic body, the profile drag of the airfoil compressibility is essentially friction: Aircraft S lift-drag polar C = C¯ wet FF (123) Dk f S k

C¯f : flat plate average drag coefficient.

Swet : wetted area of the component k.

FFk : : form factor of the component k. Practical applications

Aerodynamics

R. Tognaccini

Introduction Hydrostatics Problem n. 18 Fundamental principles Compute the profile drag coefficient of a tapered wing. Incompressible inviscid flow 1 Z +b/2 Effects of viscosity CDp = Cdp cdy (124) S −b/2 Effects of compressibility Cdp : profile drag coefficient of the airfoil. Aircraft lift-drag polar