Aerodynamics PDF

Aerodynamics Why aircraft fly Ae111 – Introduction to Aeronautics...

Aerodynamics Why aircraft fly Ae111 – Introduction to Aeronautics E.Boniol 1 Forces on the airplane Weight Aerodynamic forces (Lift, Drag) (= Aerodynamic resultant) Propulsive force (Thrust) ( may not exist (gliders)) E.Boniol 2 Slides named NOTE are bonus information, not to be learnt. 1  Air, a Fluid  Held by gravity  Obeys:  Thermodynamics  Laws of Fluid motion  Behaviour extremely difficult to predict accurately Navier-Stokes equations (incompressible) E.Boniol 3 Fluid mechanics Basic physical quantities  Pressure Pa (N/m2)  Temperature K  Density kg/m3  Flow velocity m/s  Viscosity Pl (Pa.s) E.Boniol 4 Slides named NOTE are bonus information, not to be learnt. 2  Reynolds number Unitless quantity Characterize the fluid flow (« what shape the flow has »)  l : characteristic length for the system considered Wind tunnel tests :  To have an airflow being representative of the full scale aircraft, need for the same Reynolds number (and Mach number) E.Boniol 5 NOTE (previous slide) Translates the proportion of ‘inertial forces’ ( ) VS ‘viscous forces’ ( ) l characterizes the size of the system : the flow around a model airplane shall be different than around a full-size airplane  For wind tunnel tests on model airplanes, same Reynolds number (and Mach number) must be put to have the same airflow as on the real full-size airplane. One can express the viscosity of the air with respect to Temperature (in Kelvin), thanks to the Sutherland’s law : E.Boniol 6 Slides named NOTE are bonus information, not to be learnt. 3  Flow around a Sphere at different Reynolds numbers E.Boniol 7 Laminar / Turbulent flow For a certain Reynolds number value Below Laminar flow Turbulent  Deterministic Laminar Above Turbulent flow  Chaotic  In aeronautics, airflow is mostly turbulent. E.Boniol 8 Slides named NOTE are bonus information, not to be learnt. 4  NOTE (previous slide) There is a value for the Reynolds number above which the flow pattern changes radically (sometimes refered as Critical Reynolds number)  This value changes depending on the situation (pipe, wing, etc.) Below Laminar flow : Particules follow the streamlines in a deterministic way (particles are linked one to another by viscosity) Above Turbulent flow : Particules randomly move around the streamlines (particles inertia overcomes viscosity) For a cigaret : Reynolds increases as we climb from the cigaret (l increases in the Reynolds number), the smoke goes from laminar to turbulent at one point. E.Boniol 9 Boundary Layer When flowing across a solid surface, the fluid sticks to the surface  Velocity in the immediate viscinity of the solid is zero (no-slip condition) The transition region from zero to free-stream velocity is called the Boundary Layer (BL) The BL profile shape and thickness depends on the Reynolds number. Laminar BL : Parabolic shape Turbulent BL : transition from zero to freestream is much thicker(the flow has more kinetic energy) E.Boniol 10 Slides named NOTE are bonus information, not to be learnt. 5  Effects of Reynolds number Viscosity = Friction (Energy loss)  BL is the only region where viscosity must be considered (viscous region) Full-size airplane : BL thickness is negligible (few milimeters).  Thus , we shall consider inviscid flow for a full size airplane. E.Boniol 11 NOTE (previous slide) The higher the Reynolds, the smaller the viscous region is (in proportion to the size of the system) The flow will be radically different for a full size A320 and for its equivalent model airplane (Reynolds is much smaller (l is much smaller)  viscous region much more important for the model airplane). For a full-size airplane, the BL thickness is negligible compared to the wing size (few milimeters).  For this reason, we shall consider inviscid flow for a full size airplane. E.Boniol 12 Slides named NOTE are bonus information, not to be learnt. 6  Aerodynamic studies  Analytical methods :  Ex : Potential flow theory : considered with no energy loss Inviscid Suplemmental assumptions can be added, such as : Stationnary (Steady) Incompressible  CFD (Computational Fluid Dynamics)  Numerical (Computer based) methods to simulate the Navier-Stokes equations with some assumptions E.Boniol 13 Lift First thoughts We consider Incompressible (very subsonic) and Inviscid flow (no friction), through a converging pipe. Conservation of matter (mass flow rate between the inlet and the oulet) Incompressibility implies So : E.Boniol 14 Slides named NOTE are bonus information, not to be learnt. 7  Lift First thoughts We consider Incompressible (very subsonic) and Inviscid flow (no friction), through a converging pipe. Section A1 is big compared to A2, so V2 is big compared to V1. An incompressible flow accelerates through a converging section ! E.Boniol 15 Airfoils Airfoil / Aerofoil / Wing profile : 2D slice of a wing  Plunged into an airflow with a relative wind (at speed V (or 𝑽 )), coming with a relative angle called the Angle of Attack (AoA) (incidence en français) E.Boniol 16 Slides named NOTE are bonus information, not to be learnt. 8  Airfoil in a wind tunnel  Smoke trails put into the flow to represent the streamlines  The two points are called Stagnation points (where the particles hit the surface, making them stop), where the Dynamic pressure is fully converted into Static pressure (see later) E.Boniol 17 Airfoil in a wind tunnel By looking at the flow lines :  The flow over the upper surface can be seen as a converging pipe  The flow over the lower surface can be seen as a diverging pipe  The flow over the upper surface is overall faster than the one on the lower surface ! E.Boniol 18 Slides named NOTE are bonus information, not to be learnt. 9  Bernoulli’s equation  Conservation of energy (« Pressure energy » + « Kinetic energy » conserved for a particule (along a streamline))  Valid for : Incompressible Inviscid Steady flow  Valid along a streamline1 : Static pressure Dynamic pressure Total pressure P q (conserved) E.Boniol 19 NOTE (previous slide) 1 (thus, explaining lift by blowing over a sheet of paper is not a valid explanation : air coming from your lungs is not at the same energetic level as the atmosphere, contrary to an airfoil within a homogeneous flow) As a proof of that the sheet of paper demo fails to explain lift, try to blow on one side of a sheet of paper held strictly vertical : No sunction on the side you blow ! E.Boniol 20 Slides named NOTE are bonus information, not to be learnt. 10  NOTE (previous slide) Bernoulli is only applicable to fluids that are considered :  Incompressible : meaning constant 𝜌  Air is not incompressible, but at low speed (by convention Mach < 0,3), the compression of the air around a body is negligible (thus Bernoulli is not to be used when Mach > 0,3)  Inviscid : meaning frictionless, enabling to use the concept of conservation of energy within the flow  Friction is to be considered in the BL only, thus inviscid may be a valid assumption outside the BL (+ Steady & Irrotational flow)  Bernoulli’s equation can be easily derived by applying Newton’s Second law on a cubic shaped particule (just like in Lecture 2, where we only considered forces equilibrium)… E.Boniol 21 NOTE (previous slide) « Gravitational Potential energy » The complete form of Bernoulli equation is : Static Pressure Conserved « Kinetic energy »  In aerodynamics, the last term « Potential energy » is often neglected as it is very small compared to the other ones. Nevertheless, remember Hydrostatic differential equation : Integrating it considering constant 𝝆 (incompressible fluid, which is not true to describe ISA at great scale as seen in Lecture 2), would give us :  Which is exactly Bernoulli’s equation when the fluid is static (V = 0) E.Boniol 22 Slides named NOTE are bonus information, not to be learnt. 11  Lift Qualitative explanation From the flow pattern, the streamlines are much less spaced on the upper surface than on the lower surface. So :  The air goes much faster on the upper surface than on the lower surface.  From the Bernoulli’s equation, (static) pressure is much weaker on the upper surface than on the lower surface.  Pressure difference means there is a force pulling (aspiring) the wing upward  This is Lift ! E.Boniol 23 NOTE (previous slide) This qualitative explanation, as relying on the Bernoulli’s equation, is only valid for incompressible flow (Mach < 0,3 by convention). Also, Bernoulli’s equation only holds for inviscid flow, so only outside the Boundary Layer, which is, as stated above, very thin for a large scale aircraft. For compressible flow and beyond (Mach > 0,3), the same equations are kept to express Lift and Drag forces (see later), compressibility shall only have effects on the Lift coefficient and the Drag coefficient. (see later) E.Boniol 24 Slides named NOTE are bonus information, not to be learnt. 12  Wrong explanation for lift, often presented to the public The « Equal transit time » theory Particules have no reason to met at the same time at the trailing edge ! (upper surface ones go much faster)  Furthermore, this give much weaker values for lift than reality ! E.Boniol 25 Another Lift explanation Newton’s third law Action-Reaction : Air pulled down by the wing  Wing pulled up by the Air  Give same result as Bernoulli E.Boniol 26 Slides named NOTE are bonus information, not to be learnt. 13  NOTE (previous slide) By the action-reaction law (the wing deflects air down (downwash), so the air pulls the wing up in return), lift is found to exist.  This is an explanation equivalent to the Bernoulli’s one (There is no debate to make between Bernoulli and Newton explanations !)  Provides exactly the same results as Bernoulli E.Boniol 27 Another Lift explanation Coandă effect Near the Leading edge, on the upper surface :  Also Action-Reaction : Air follows the surface Submitted to Centripetal force Wing submitted to a Centrifugal force by reaction (= Lift)  Give same result as Bernoulli Lift (resultant) Centrifugal force on the wing (Reaction) Centripetal force on the air (Coandă effect) E.Boniol 28 Slides named NOTE are bonus information, not to be learnt. 14  NOTE (previous slide)  Again, it can be proven mathematically that Bernoulli, Newton and Coandă give the same results.  They describe the same phenomenon with different approaches, each of them being correct.  There is no debate to make between Bernoulli, Newton and Coandă !  The only remaining question is : Why does air flow around an airfoil that way ? Because, from now, we have only assumed the flow pattern experimentally, but we will briefly address this question (see Kutta-Joukowski) E.Boniol 29 Quantify lift Let’s consider the average velocity over the upper and the lower surfaces (respectively and )   E.Boniol 30 Slides named NOTE are bonus information, not to be learnt. 15  Quantifying lift E.Boniol 31 From pressure to lift force What surface to use ? Wetted surface  Surface in contact with the air (both sides of the wing) Wing reference area S E.Boniol 32 Slides named NOTE are bonus information, not to be learnt. 16  NOTE (previous slide) Wing area (Surface alaire en français) Reference surface in order to properly define CL (and CD, see later), several conventions exist : European definition of Wing area S :  Surface of the wing planform, including a ‘fictional’ rectangular section inside the fuselage American definition of Wing area S :  Surface of the wing planform, with leading and trailing edges extended inside the fuselage Obviously, Lift (and Drag) forces must have the same numerical value, whatever the convention used, so that CL (and CD) must be rigourously defined with respect to the Wing area definition used ! When dealing with a whole airplane, the Lift and Drag formulas include the Lift and the Drag produced by the whole airplane : Additional lift (positive or negative) is produced by the empenage, and the fuselage produces a little bit of lift Drag is obviously greater when adding a fuselage, empenage, landing gear, etc. E.Boniol 33 Lift formula Unit : N (Force) Direction :  Always perpendicular to the relative wind Air Density Wing area Airspeed Lift coefficient E.Boniol 34 Slides named NOTE are bonus information, not to be learnt. 17  Lift coefficient  At the heart of aerodynamics  Dependancies : E.Boniol 35 Lift coefficient and AoA From the Thin Profiles Theory (from Potential flow), 𝐶 has been shown to be linear with the Angle of Attack. 𝐶 is called the Lift slope. for the ideal Thin Profiles case for a symmetrical airfoil E.Boniol 36 Slides named NOTE are bonus information, not to be learnt. 18  Lift coefficient and AoA Reality Stall : Major flow separation over a certain AoA Flow separation Reality Often around 12° to 15° E.Boniol 37 NOTE (previous slide) The lift coefficient is indeed linear with AoA, but only until a certain limit, from where the lift coefficient value drops. This limit is called the Stall Angle of attack (around 15°), and the associated lift coefficient is called 𝐶 (around 1.2 to 1.6). This phenomenon is due to large flow separation (separation of the boundary layer, see later) Drag sharply increases past the stall AoA. The 𝐶 value does not become zero after the Stall AoA, it’s just not worth displaying such data as a stalled airplane flies under sharply increased drag and may experience loss of control. E.Boniol 38 Slides named NOTE are bonus information, not to be learnt. 19  Deep stall Stalled wing wake hitting the horizontal stabilizer (then becoming ineffective). This situation is not recoverable. E.Boniol 41 NOTE (previous slide) Deep stall : Situation where the wake of the stalled main wing hits the horizontal stabilizer. The horizontal stabilizer then becomes ineffective. This situation is not recoverable. This phenomenon (that must be avoided at all cost), must be taken into account in the design of the tail. E.Boniol 42 Slides named NOTE are bonus information, not to be learnt. 20  3D Stall  3D stall progression depends on planform shape Rectangular :  Root stalls first High taper or Swept wings :  Tip stalls first E.Boniol 43 NOTE (previous slide) Stall characteristics highly depends on the planform shape. Tip stall, like on high taper wings or swept wings, can induce loss of control on the roll axis (see later), the airplane may then roll upside down and dive to the ground. Stall progression is defined by local angle of attack, varying along the wingspan due to 3D finite wing aerodynamic effects, as described later for the induced drag. E.Boniol 44 Slides named NOTE are bonus information, not to be learnt. 21  Laten the flow separation Vortex generators Generating vortices to create highly turbulent flow The more turbulent  The later the stall Useful on : Wing upper surface, Engine nacelles, Fighter aircraft, etc. E.Boniol 45 NOTE (previous slide) Vortex Generators : Small surfaces raising from the wing or the engines nacelle Goal : Re-energize the airflow to prevent it from detaching from the wing surface (laten the stall, or minimizing the flow separation on the wing due to the engines nacelles wake)  Must be positionned upstream of where flow separation is likely to occur. As a highly turbulent flow increases friction drag, this is a compromise between : Without vortex generators : Local lift decrease and Pressure drag increase (local flow separation) With vortex generators : Friction drag increase E.Boniol 46 Slides named NOTE are bonus information, not to be learnt. 22  d’Alembert Paradox Potential flow theory (Inviscid flow) predict zero aerodynamic force. Potential flow theory :  Non Physical pattern  Zero Lift, Zero Drag Lift and Drag can’t be explained by pure potential flow theory Why ? The key is viscosity (to explain Lift, Drag, Stall, etc.) E.Boniol 47 NOTE (previous slide) We based our explanation of lift on wind tunnel observation. But, if we strictly apply the Potential flow theory to an airfoil geometry obstacle, we find zero aerodynamic force !  It does not predict Lift nor Drag  This is called the d’Alembert Paradox Furthermore, our explanation of lift did not predict the Stall phenomenon  From now, we are not able to describe what Drag is.  This is a strange situation where we want to neglect the Boundary Layer size (to use well known and simple Potential flow equations), but we have to take into account its effects (to explain Lift and Drag). Otherwise, without Boundary Layer (Inviscid), Lift and Drag are zero in theory ( d’Alembert Paradox) E.Boniol 48 Slides named NOTE are bonus information, not to be learnt. 23  Kutta-Joukowski theory Analytically quantity the lift Circulation : rotation of the flow around the airfoil to respect the Kutta condition Kutta condition : the second stagnation point must be at the trailing edge (non- physical otherwise). Circulation is then explaining why the upper surface flow is accelerated ! E.Boniol 49 NOTE (previous slide) Martin Wilhelm Kutta and Nikolaï Joukovski developped a very powerful mathematical method to predict the lift coefficient, depending on the airfoil geometry (with no computer). 𝛤 : Circulation  They still consider Incompressible, Steady and Inviscid flow, but introducing a mathematical term, called the Circulation, enabling to take into account the viscosity effects for the lift generation.  The goal of this method is to change the potential flow prediction so that the aft stagnation points is at the trailing edge, by introduction Circulation ; this is, in the real world, an effect of viscosity  Method largely used until World War II E.Boniol 50 Slides named NOTE are bonus information, not to be learnt. 24  Drag Represents the air resistance.  Force that need to be compensated by the engines (gliders cannot fly indefinetly)  Divided into several types : o Parasitic drag Pressure / Form drag Friction drag o Induced drag o Wave drag E.Boniol 51 Friction drag Primary effect of the Boundary Layer Boundary layer = Friction (Energy loss) Due to its shape : (Friction Drag)Turbulent BL > (Friction Drag)Laminar BL E.Boniol 52 Slides named NOTE are bonus information, not to be learnt. 25  NOTE (previous slide)  As Boundary Layer arise due to Friction, Kinetic energy is taken to the air.  It results in shear stress, whose resultant is a resistance force called Friction drag. A Turbulent Boundary Layer is steeper near the surface.  Due to its shape, a Turbulent BL creates more Friction drag than a laminar one. E.Boniol 53 Pressure / Form Drag Flow separation In Adverse pressure gradient (near Trailing Edge) :  Flow is slowed down  In the Boundary Layer, it can cause flow reversal, leading to vortices (flow is separated)  Due to its shape, a laminar BL is more sensitive to this phenomenon E.Boniol 54 Slides named NOTE are bonus information, not to be learnt. 26  NOTE (previous slide) Airflow is slowed down when submitted to an adverse pressure gradient. In the Boundary Layer, it can cause flow reversal, leading to vortices (flow is separated)  Due to its shape, a laminar BL is more sensitive to this phenomenon Turbulent flow is not to be confused with Separated flow :  Turbulent flow is just a flow with a high Reynolds number  Separated flow is when boundary layer is detached (reversed) from the solid surface and creates vortices E.Boniol 55 Pressure drag Why flow separation at this location ? Adverse pressure Proverse pressure gradient Flow is slowed down gradient Flow is accelerated  Flow separation induces a decrease in Pressure at the trailing edge, so that the superior leading edge pressure creates the Pressure drag. E.Boniol 56 Slides named NOTE are bonus information, not to be learnt. 27  NOTE (previous slide) This precise kind of pressure drag (flow separation due to adverse pressure gradient) is called Form drag. For an airfoil, total drag is called Profile drag : Profile drag = pressure drag (through form drag) + friction drag. E.Boniol 57 Parasitic/Profile Drag Parasitic Drag = Pressure Drag + Friction Drag  Profile drag represents the total drag for a 2D solid section (Form Drag + Friction Drag)  For an complete airplane configuration, the term Parasitic Drag is used instead  Parasitic (or Profile ) drag is the addition of the uneven pressure forces (Pressure drag), and friction shear stresses (Friction drag) on the whole aircraft (or on the airfoil )  For a thin shape like an airfoil at low angles of attack (and low Mach), friction drag is greater than pressure drag E.Boniol 58 Slides named NOTE are bonus information, not to be learnt. 28  NOTE (previous slide)  Thin bodies : Friction drag is dominant (high wetted surface, but small flow separation)  Blunt bodies : Pressure drag is dominant (high flow separation)  For a relatively thin body, as its AoA increases, Pressure drag increases E.Boniol 59 NOTE (previous slide) For a complete aircraft configuration, instead of speaking in terms of profile drag, we use the term Parasitic drag : Parasitic drag = pressure drag + friction drag  Pressure drag is the sum of the form drags of individual components, plus additional pressure drag due to boudary layer interactions at the junctions of those different components (typically between the wing root and the fuselage).  This additional pressure drag is called Interference drag.  Friction drag is linked with the complete airplane surface in contact with the fluid (wetted surface), thus greatly increased for a complete airplane compared to an airfoil (extruded to the reference wing area being used).  For a complete airplane configuration, Parasitic Drag is not the Total Drag (cf. Induced Drag) E.Boniol 60 Slides named NOTE are bonus information, not to be learnt. 29  Airflow over an airfoil (subsonic) Synthesis E.Boniol 61 Drag formula As an aerodynamic force, the structure of the Drag formula can be proved to be the same as the Lift formula But, cannot be proved through Bernoulli’s equation (phenomena due to viscosity) Proved by the Buckingham Pi-theorem (cf. Dimensional Analysis) For a 2D airfoil, the Drag coefficient is : E.Boniol 62 Slides named NOTE are bonus information, not to be learnt. 30  NOTE (previous slide)  Again, Drag is caused by the whole airplane (Wing, Fuselage, Empenage, Landing Gear, etc…) (Friction Drag linked to the Wetted area).  Still, Wing area S is used as a reference (same as lift) (CD is defined with respect to the numerical value of S) E.Boniol 63 3D wing Vocabulary E.Boniol 64 Slides named NOTE are bonus information, not to be learnt. 31  Effects of a finite 3D wing Wing tip vortices Due to the difference in pressure between the lower and upper surfaces, the flow is drawn into vortices at the wing tip The higher the pressure difference at the tip, the stronger the vortex. These vortices induce a supplemental downwash across the wing E.Boniol 65 Wing tip vortices Induced drag Appart from being very dangerous for the airplanes behind,… They decrease the angle of attack, tilting the lift vector backward : Decreased ‘vertical’ component  Degraded Lift Added ‘horizontal’ component  Induced Drag E.Boniol 66 Slides named NOTE are bonus information, not to be learnt. 32  NOTE (previous slide)  Along the span of the wing, wing tip vortices create a downwash (flow pushed downward)  As a result, local AoA is decreased : Because Lift is perpendicular to local airspeed, Lift is tilted backwards Lift has then two components with respect to global airspeed vector (aircraft dispacement in the air) : o Perpendicular to global airspeed (dZ on the scheme) : Lift is reduced by a cos( ) factor (and inherently by effective AoA decrease) o Parallel to global airspeed (dX on the scheme) : Resistive force is added by a sin( ) factor  Induced Drag  This downwash may not be uniform, so that every elemental airfoil along the wing span will be submitted to different AoAs and not stall at the same time. E.Boniol 67 Wing tip vortices Prandtl lifting line theory  Analytical method to estimate the induced drag Induced drag decreases with aspect ratio Induced drag coefficient Elliptic lift distribution : the most : Aspect Ratio efficient lift distribution to mitigate : Span efficiency factor induced drag (= 1 for elliptic distrib.) E.Boniol 68 Slides named NOTE are bonus information, not to be learnt. 33  NOTE (previous slide) Elliptic lift distribution minimizes Induced drag  Corresponds to a uniform downwash along the wingspan This is why an elliptical planform stalls uniformly (each elemental airfoil submitted to the same AoA decrease due to this uniform downwash) es is the Span efficiency factor, representing « how far » we are from the elliptical lift distrib. Indeed, elliptical lift distribution is often not suitable (sudden uniform stall, manufacturing complexity) Anyway, presence of the fuselage and other structural parts shall disturb this elliptical lift distribution. E.Boniol 69 NOTE (previous slide) In addition to creating induced drag, the tilt of the effective (2D) lift vector also reduces the lift force of the 3D finite wing compared to the 2D airfoil value. This reduction in lift is expressed through the lift coefficient , more specifically through the lift coefficient slope : : 3D finite wing lift slope : airfoil lift slope  For a same airfoil, when the aspect ratio decreases, the 3D finite wing lift curve slope decreases. E.Boniol 70 Slides named NOTE are bonus information, not to be learnt. 34  Spitfire  (Pseudo-) Elliptic wing planform  Excellent mitigation of induced drag But, Increased manufacturing cost Very bad stall characteristics (Sudden uniform stall along the wing) E.Boniol 71 Reduce the Induced Drag Target the elliptic distribution  Act on the Span efficiency factor E.Boniol 72 Slides named NOTE are bonus information, not to be learnt. 35  Reduce the induced drag Act on the Aspect Ratio Higher aspect ratio means less induced drag…But not always possible due to structural stress or wingspan Thus, winglets (sharklets) enable to artificially increase the Aspect ratio, thus migiating the wing tip vortices But : Increased Parasitic drag Added weight & stresses on the wing structure (torsional stresses) E.Boniol 73 Total Drag (Subsonic) Total drag = Parasitic Drag (2D) + Induced Drag (3D) Thus, in terms of coefficients : Friction Drag + Pressure Drag Induced Drag E.Boniol 74 Slides named NOTE are bonus information, not to be learnt. 36  Symmetrical Parabolic Drag polar Simplified cases :  Assuming parasitic drag does not depend on AoA : Span efficiency factor : Zero lift drag coef. E.Boniol 75 NOTE (previous slide) This is the simplest form of the drag polar, and is valid under certain constraining assumptions :  Parasitic Drag does not depend on AoA, thus CL (not true mainly due to flow separation increasing with AoA)  Minimum Drag achieved at zero CL, not well suited for an aircraft whose minimum drag configuration should be encountered for a certain non-zero lift E.Boniol 76 Slides named NOTE are bonus information, not to be learnt. 37  Symmetrical Parabolic Drag polar  Considering parasitic drag dependance on AoA (primarily through gradual flow seperation (even before stall), thus pressure drag increase) Low AoA, small flow separation Greater AoA, greater flow separation  Small pressure drag  Increased pressure drag : Oswald factor Zero-lift Drag coef. Lift-Dependant Drag coef. E.Boniol 77 NOTE (previous slide) This slide takes into account the parasitic drag dependance on AoA through pressure drag. As a first approximation, Pressure Drag may be expressed as : So that total drag coefficient is the sum (Friction + Pressure + Induced Drags coefs): CD0 : Zero-Lift Drag coefficient (Drag coef. at zero lift) k.CL2 : Lift-Induced / Dependant Drag, all the dependances of drag to lift (coef. for Induced Drag & part of the Pressure Drag, included through the Oswald factor) E.Boniol 78 Slides named NOTE are bonus information, not to be learnt. 38  Adjusted drag polar More close-to-reality shape (shifted parabola) Due to : Minimal drag being targeted for a non-zero lift coefficient Corresponds to cambered airfoils (symmetric airfoils used mostly only for aerobatic planes) : Oswald factor : Minimum drag coef. : Optimal Lift coef. + Non-parabolic region (for high lift coef. values, due to big flow separation) E.Boniol 79 NOTE (previous slide) This slide shows the adjusted drag polar, which is the closer-to-reality shape to be used in real aircraft design and Flight Mechanics studies. For academic simplicity purposes, we shall only use the symmetrical drag polar. Keep in mind that for accurate studies for low angles of attack cases, the adjusted drag polar is the one to be used ! E.Boniol 80 Slides named NOTE are bonus information, not to be learnt. 39  Synthesis Back to a 2D airfoil  Aerodynamic force/resultant = Lift + Drag  Lift always acts perpendicular to the relative wind  Drag always acts parallel to the relative wind E.Boniol 81 Synthesis Back to a 2D airfoil Aerodynamic force/resultant = Pressure distribution (Lift & Pressure drag) + Shear stress (Friction drag) E.Boniol 82 Slides named NOTE are bonus information, not to be learnt. 40  Synthesis Back to a 2D airfoil  The aerodynamic force can be seen as acting at one point called the center of pressure  The location of the centre of pressure varies with the AoA : moves forward until the stall  Because of this, we rather use another point for the Flight Mechanics computations, called the aerodynamic center (to be seen in Aero 3) E.Boniol 83 Synthesis Drag of the airplane  Fuselage, empenage, engines, landing gears, etc. add parasitic drag. On a 3D wing or on the whole airplane (even though Dragwing 0,3))  Shockwave = Sudden compression : through the shockline, static pressure increases (NOT A SHOCKWAVE) E.Boniol 85 NOTE (previous slide) The white trailing cone on the above photograph is not a shockwave. Shockwaves are very hard to observe directly ; they need precise lighting conditions like in a dedicated supersonic wind tunnel, as they are a sudden compression of the air itself. These are Shockwaves ! What is shown on the previous slide are the effects of an expansion fan : - As a shockwave may occur when the air is forced to compress, an expansion fan occurs when the air is forced to expand - Along the expansion fan, static pressure decrease and may cause water dropplets condensation, which is what is shown on the previous slide picture. E.Boniol 86 Slides named NOTE are bonus information, not to be learnt. 42  Mach number From basic thermodynamics, it can be shown that :  Speed of sound a depends on temperature  Mach number = ratio between Airspeed V and Speed of sound a  Mach 1 is more quickly reached at high altitudes in the Troposphere. Speed of sound :  Thermodynamic constant for the air E.Boniol 87 NOTE (previous slide) From basic thermodynamics, it can be shown that :  Speed of sound a depends on temperature  Mach number is the ratio between Airspeed V and Speed of sound a Thus, for a same airspeed, Mach number is not the same in different temperature conditions (i.e at different altitudes for example (see Lecture 2))  For example, Mach 1 is more quickly reached at high altitudes in the Troposphere. E.Boniol 88 Slides named NOTE are bonus information, not to be learnt. 43  Mach cone, Mach angle  As a first approximation, we can consider an aircraft as a point emitting sound waves Subsonic Sonic Supersonic  In the supersonic domain (Mach > 1), soundwaves forming a cone called the Mach cone, with Mach angle 𝝁. E.Boniol 89 NOTE (previous slide)  As a first approximation, we can consider an aircraft as a point emitting sound waves  In the subsonic domain (Mach < 1), soundwaves are emitted both backward and forward  At the speed of sound (Mach = 1), forward sound waves pile up  In the supersonic domain (Mach > 1), the aircraft is in advance compared to the soundwaves, forming a cone called the Mach cone. With some trigonometry, the angle of this cone (Mach angle) can be easily shown to be : 𝝁 is also used for viscosity (nothing to do with the Mach angle) E.Boniol 90 Slides named NOTE are bonus information, not to be learnt. 44  Airplane geometry In reality, an airplane is not a point  Shockwaves will have different angles depending on the surrounding geometry The Mach angle 𝜇 : minimum angle a shockwave can have for a given Mach number. E.Boniol 91 NOTE (previous slide) The Mach angle 𝜇 : minimum angle a shockwave can have for a given Mach number.  Thus, knowing the Mach targeted for the operation of a given fighter aircraft, this Mach angle value is used to determine the angle between the aircraft axis and the axis between the nose and the wing tips (to ensure the wing tips cannot create additional shockwaves) E.Boniol 92 Slides named NOTE are bonus information, not to be learnt. 45  Oblique Shockwaves : Angle The shockwave angle can be determined with respect to the deflection angle of the airplane surfaces, using formulas (from basic geometry) or a graph.  Supersonic flow is overall easier to study (evolving through a discrete process, shockwave after shockwave) E.Boniol 93 Effects of shockwaves Wave Drag Shockwaves are the cause of a sharp increase in Pressure drag called : Wave drag  Wave drag appears at the Critical Mach number (see later), and increases drastically from the Drag Divergence Mach number, up to a maximum at around Mach 1. This abrupt increase is commonly refered as the Sound Barrier (Mur du Son) Wave Drag Coefficient 0 E.Boniol 94 Slides named NOTE are bonus information, not to be learnt. 46  NOTE (previous slide)  Wave Drag is the drastic increase in pressure drag caused by Shockwaves and the Boundary Layer separation they induce passed the Drag Divergence Mach number  Wave drag coefficient then decreases once the transonic range is passed (there is no utility flying around Mach 1, this region must be crossed quickly) In the transonic and supersonic domains, total Drag coefficient may be expressed as : Total Drag Induced Drag coef. Skin friction Pressure Drag Wave Drag coef. Drag coef. coef. at Mcrit coef. E.Boniol 95 Effects of shockwaves Boundary layer separation  Interaction of the shockwave with the Boundary Layer may cause BL separation (equivalent to a Stall), this is why subsonic airfoils cannot sustain transonic /supersonic regime.  To mitigate the separation, slender airfoils must be used: fighter jet wings (thin wings, pointy leading and trailing edges), but poor characteristics at low speeds. E.Boniol 96 Slides named NOTE are bonus information, not to be learnt. 47  Transonic What is it ?  As the flow accelerates on the wing upper surface to create the lift, local supersonic flow can be encountered at this location, even if the upstream flow is subsonic.  The Mach number at which local supersonic flow appears is called the Critical Mach number (𝑴𝒄𝒓𝒊𝒕 ) E.Boniol 97 Increase the Drag Divergence Mach number Supercritical airfoils Commercial airliners often fly around or after the Critical Mach number To mitigate its effects, they use Supercritical airfoils (with a flatter upper surface and rear camber) This shape enables to weaken the shockwave, thus minimizing the BL separation and wave drag.  But, supercritical airfoils have poor characteristics at low speeds E.Boniol 98 Slides named NOTE are bonus information, not to be learnt. 48  Increase the Critical Mach number Swept wings Sweeping the wings enable to increase the critical Mach number :  Airspeed being : Sum of a spanwise component (not accelerating), and a chordwise component (accelerating, but smaller than the total relative wind)  Refining the airfoil compared to an unswept wing E.Boniol 99 Airplane maximum speed (VMO / MMO) Airplane maximum speed can be determined by : - Structural limitations (high dynamic pressure 𝟏𝟐 𝝆 𝑽𝟐 , Flutter) - Shockwaves formation (separated BL, Buffet, Flutter) VMO : Max Operating Airspeed  Dominant at Lower altitudes MMO : Max Operating Mach number  Dominant at Higher altitudes E.Boniol 100 Slides named NOTE are bonus information, not to be learnt. 49  NOTE (previous slide) The airplane maximum speed can be determined by : 𝟏 - Structural limitations (the airplane structure not able to sustain the high dynamic pressure 𝝆 𝑽𝟐 , and Structure resonance with 𝟐 aerodynamic loads (Flutter)) - Shockwaves formation (the position of the shockwave is unstable in the transonic domain, leading to strong vibrations (Buffeting), and may finally lead to structural damage or loss of control ; transonic flutter may also occure) - For supersonic jets, temperature increase due to friction within the Boundary Layer at high supersonic Mach numbers may also be the limiting factor  VMO / MMO are maximum operating airspeed and Mach number, meaning taking into account some margin with the values at which real hazard is triggered. The airspeed and the Mach number at which the phenomena listed above may occure and endanger the aircraft are respectively called Design Diving Airspeed (VD) and Design Diving Mach number (MD). E.Boniol 101 Coffin corner From this lecture and the next one : - Stall speed (low speed limitation) increases with altitude - Maximum speed based on Mach number decreases with altitude (Temperature decreases) The point where those two limitations cross is called the Coffin corner (not possible to fly above that point)  There is an altitude above which, whatever the thrust, the airplane will not be able to fly (Lift ceiling) E.Boniol 102 Slides named NOTE are bonus information, not to be learnt. 50  Aerodynamic forces on an Airfoil (2D) Roadmap Pressure Distribution Friction Shear stress Lift ⊥ to the upstream flow // to the upstream flow Profile Drag Pressure Drag Friction Drag Form Drag Wave Drag Flow separation due to Post-Shockwave recompression Friction on the airfoil wetted surface adverse pressure + Flow separation due to BL- (Energy loss in the fluid within the BL) gradient Shockwave interactions Only above airfoil Mcrit E.Boniol 103 Aerodynamic forces on an Aircraft (3D) Roadmap Pressure Distribution Friction Shear stress Lift ⊥ to the upstream flow // to the upstream flow Parasitic Drag Pressure Drag Friction Drag Form Drag Interference Drag Wave Drag Flow separation due to Flow separation due to BL Post-Shockwave recompression + Friction on the aircraft wetted adverse pressure gradient interactions between adjacent Flow separation due to BL-Shockwave airplane components interactions surface (Energy loss in the fluid within the BL) Only above aircraft Mcrit // to the upstream flow Induced Drag Backward tilt of the Lift vector because of effective AoA decrease due to finite wing effects (vortices) E.Boniol 104 Slides named NOTE are bonus information, not to be learnt. 51  Any question ? https://www.youtube.com/watch? https://www.youtube.com/watch? v=V1oCDR3DBbo&ab_channel=Uni v=jfic0f0eJm4&ab_channel=Nation versityofIowa alAerospaceLibrary To go further… E.Boniol 105 Slides named NOTE are bonus information, not to be learnt. 52