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### Theme 1

1. 1. THEME 1. INTRODUCTION Aerodynamics of an aircraft is the science on the general laws of air motion andits specific features at flow around an aircraft and its parts, on forces and momentswhich are affecting the plane and its parts, about thermal effect of a flow aboard theplane. It is based on the laws of physics, mechanics and thermodynamics. The knowledge of aerodynamics of an aircraft is a necessary condition forconsequent study of such course, as «Flight dynamics», «Aircraft structure» and«Designing of aircraft», «Production technology of aircraft». The following example shows a role of aerodynamics in aircraft creation. Thefivefold increase of the expenses on aerodynamic researches is profitable, if it results inincrease of lift-to-drag ratio at 1 %. The large role in research of aerodynamics of an aircraft and its parts is played byexperimental researches (wind-tunnel tests and flight experiment). The especiallyimportant role is played by experiment as a way of checking the theoretical data. 1.1. Sections of aerodynamics. The division of aerodynamics into sections is made on speeds and altitudes offlight. Such division is conditional, since the basic criteria are the limitations orassumptions introduced during the studies or research of the aerodynamic characteristicsof an aircraft. The division of aerodynamics on speeds is conducted depending on a Machnumber, which is a measure of the compressibility of air flow. The Mach number isnon-dimensional value equal to ratio of body velocity to a local velocity of a sound: V∞M∞ = , where V∞ is the speed of motion of an aircraft; a is the speed of a sound. a So, the first section of aerodynamics studies motion of bodies at Mach numberslying in limits M ∞ ≤ 0 .4 . M ∞ ≤ 0 .4 - but again it is conditional number up to whichliquid (air) is possible to be considered as the incompressible medium. Air behaves just 3
2. 2. as a liquid, thus the compressibility of the environment does not influence theaerodynamic characteristics. The error at M ∞ = 0 .4 can be no more than 2% . This section is named «Aerodynamics of the incompressible environments». The second section is named «Subsonic aerodynamics». In this section themotion of an aircraft is studied at Mach numbers lying in limits 0 .4 ≤ M ∞ ≤ M∗ (fromM ∞ ≥ 0 .4 up to M ∞ ≤ M* ). The number M ∞ (undisturbed subsonic flow), at whichsomewhere on a surface of a streamlined body for the first time local velocity of a flowreaches speed of a sound (V∞ = a ), is named critical and is marked as M* . The criticalMach number M* depends on the shape of a streamlined body. In an airfoil point wherethe gas flow speed is maximum, according to a Bernoullis relation speed of a sound isdetermined as a = a min . Therefore, maximum value of a local Mach numberM = M max is reached there where stream cross-section is the least. To this point therecorresponds also minimum value of a coefficient of pressure C p = C p min . The Machnumber M* is always less than or equal to one ( M* ≤ 1 ). The number M ∞ = M* is thehighest limit of numbers M ∞ , at which the ratios obtained for completely subsonic floware fair. The third section is named «Transonic aerodynamics». At Mach numbersM ∞ > M* on a streamlined surface both subsonic and supersonic zones of flow takeplace. The zones with subsonic speeds do not fade away at once at reaching supersonicspeed of flight. Depending on the shape of an airfoil it occurs at Mach numbersM ∞ ≈ 1.1 ÷ 1.2 . Such flow mode is also named transonic. Section «Transonicaerodynamics» studies speed range M* ≤ M ∞ ≤ 1.1 ÷ 1.2 . The upper Mach numberM ∞ ≈ 1.2 also is selected conditionally, and in some books it is possible to meet valuesM ∞ ≈ 1.25 . The fourth section «Supersonic aerodynamics» is limited by range of Machnumbers 1.1 ÷ 1.2 ≤ M ∞ ≤ 4 ÷ 5 . 4
3. 3. The fifth section is named «Hypersonic aerodynamics». It corresponds to Machnumbers more than M ∞ ≤ 4 ÷ 5 . For example, "Shuttle" or "Buran" enter intoatmosphere with Mach numbers M ∞ ≈ 20 ÷ 25 . Besides, aerodynamics is divided on altitudes of flight H . The main criterion ofdivision is the Knudsen number. λ Knudsen number k n = , where λ is free length of molecules run, l is reference lsize of liquid flow. In standard conditions λ ≈ 10 − 6 m , at t oC = 15 , P ≈ 760 mm . Hg . Depending on Knudsen number aerodynamics is divided onto the followingsections: «Aerodynamics of continuum». Values of number k n ≤ 0 .1 and altitude offlight H ≤ 80 Km correspond to this section. «Aerodynamics of the strongly rarefied environment». Values of numberk n > 10 and altitude of flight H > 120 Km correspond to this section. It is possible to consider air as continuum at k n ≤ 0 .01 in the given course. Formodern aircraft which are flying at altitudes up to H < 40 Km , this condition isperformed. 1.2. Aircraft and its main structural members Lets proceed to consideration of an aircraft and its parts. We will introducegeneral concepts and aerodynamic characteristics, we will show on examples a role ofthe aerodynamic characteristics in formation of flight properties of an aircraft. An airplane is an aircraft heavier than air having a power plant for obtainingthrust and wings for creating lift. Describing the shape of an aircraft, these concepts: are used a base plane of anaircraft and base system of coordinates. 5
4. 4. The aircraft base plane is the plane, concerning which the majority of structuralmembers of an aircraft are located symmetrically on the left and on the right. This planeis often named as a plane of symmetry. Base system of coordinates is the right rectangular coordinate system0 R x R y R z R , fixed concerning an aircraft. An origin 0 R is named as an aircraft basepoint, Axis 0 R x R - aircraft base axis. The base point is in a base plane of an aircraft.Its position is determined from task to be solved. The axes 0 R x R and 0 R y R are also inan aircraft base plane. The first is directed forwards, the second - upwards. The axis0 R z R is directed along the right half wing. The main parts of an aircraft (fig. 1.1) are: a wing, fuselage (body), tail unit,landing gears and power plant. 1 - wing; 2 - fuselage; 3 - power plant; 4 - horizontal tail; 5 - elevator; 6 - stabilizer; 7 - vertical tail; 8 - rudder; 9 - fin; 10 - flaps; 11 - aileron. Fig. 1.1. Main parts of an aircraft Wing is the main lifting surface of an aircraft. The wing is designed to createlifting force necessary for aircraft gravity balance. The wing usually has a plane ofsymmetry. Fuselage is designed for accommodation of the crew, passengers, equipment,fuel, freights and power plant. Usually fuselage creates small lift and considerable drag. Power plant consists of engines with devices and systems providing theiroperation, air intakes, propellers and nozzles. The power plant is intended for thrustcreation. 6
5. 5. Tail unit of an aircraft consists of horizontal tails and vertical tails and isdesigned for maintenance of stability and controllability in longitudinal and lateralmotion. Landing gears consist of a landing gear, high-lift devices, accelerating andbraking devices. 1.3. Coordinate system While studing aircraft aerodynamics body 0 xyz and wind 0 xa ya za coordinatesystems (Fig. 1.2) are more often used. Both coordinate systems are right rectangular. The body coordinate system is fixed relatively to an aircraft and moves together with it. Its origin 0 is usually placed in a center of mass. The axes 0x , 0 y , 0 z are named as longitudinal, normal and transverse axes. The axes 0x and 0 y are located in a base plane of an aircraft. The axis Fig. 1.2. Coordinate systems 0x is directed from an aircraft tail section to thenose part, the axis 0 y is directed towards top part of an aircraft. The axis 0 z goesperpendicularly to an aircraft base plane and is directed to the right side of an aircraft. Beginning of wind coordinate system 0 xa ya za usually is also placed in the centerof mass. There distinguish a wind axis 0 xa , lift axis 0 ya and lateral axis 0 za . The windaxis 0 xa is directed posigrade of an aircraft. The lift axis 0 ya lies in a base plane of anaircraft (or in a plane parallel it) and is directed to an aircraft top. The lateral axis 0 zapasses so that it has supplemented axes 0 xa and 0 ya up to the right coordinate system.The wind system is not rigidly connected with an aircraft and can change the orientationin relation to it during the flight. The orientation of an aircraft relatively to the velocity vector is determined byangle of attack α and angle of slip β . An angle of attack α is an angle between aprojection of velocity vector to a vehicle plane of symmetry (base plane of an aircraft) 7
6. 6. 0 xy and centerline 0x . A slip angle β is an angle between velocity vector and plane ofsymmetry 0 xy . In some cases normal coordinate system 0 x g y g zg (Fig 1.3) is used. It is the mobile right system. Its beginning 0 is combined with the beginning of body coordinate system. The axis 0 y g is directed upwards along a local vertical, and directions of axes 0 x g and 0 z g are selected according to the task to be solved. The plane 0 x g zg is always located horizontally in this Fig. 1.3. Normal Coordinate system coordinate system. The angle between the axis 0 x g andprojection of a centerline to a horizontal plane is named as yaw angle and designated asψ . The angle between the aircraft centerline 0x and horizontal plane 0 x g zg is namedas the pitch angle and designated as ϑ . The angle between the transverse axis 0 z andaxis 0 z g of normal coordinate system, displaced in the position at which yaw angle isequal to zero ( ψ = 0 ), is named as the bank angle and designated as γ . 1.4. Aerodynamic forces and moments. Coefficients of aerodynamic forces and moments. The main vector of forces system which affect onto a flight vehicle at its motionfrom the air, is named as full aerodynamic force and is designated as R A . The conceptof aerodynamic force is usable not only to an aircraft as a whole, but also to its parts: awing, a fuselage and so on. 8
7. 7. Components of full aerodynamic force X * , Y , Z along axes of body coordinate system are determined by making projections of R A on these axes 0 xyz . The component X * taken with a converse sign is named as aerodynamic longitudinal force and designated as X. Aerodynamic force Fig. 1.4. Components of aerodynamic components Y, Z are named as force in body coordinate aerodynamic normal and aerodynamictransversal forces. Forces X , Y , Z can be both positive and negative depending onthe shape of an aircraft and the mode of flight (Fig. 1.4). Lets project force R A onto axes of wind coordinate system 0 xa ya za . Lets designate its projections as X* , Ya , Z a . a Taken with a converse sign the component X* is named as drag force and designated as a X a . The drag force is always positive. Aerodynamic force components Ya , Z a are named as aerodynamic lifting force and aerodynamic lateral force. They can be both positive, and negative (Fig. 1.5). Fig. 1.5. Aerodynamic force In aerodynamics it is accepted to workcomponents in wind coordinate system not with absolute forces values but withvalues of their coefficients. Having divided values of the aerodynamic forces ondynamic pressure q∞ = ρ∞ V∞ 2 (where ρ ∞ is the density of an undisturbed air flow, 2V∞ is undisturbed air flow velocity ran against the plane at versed motion) and on thereference area S , we get coefficients of aerodynamic forces: 9
8. 8. X Y Z Cx = ; Cy = ; Cz = ; (1.1) q∞ S q∞ S q∞ S Xa Y Z C xa = ; C ya = a ; C za = a . (1.2) q∞ S q∞ S q∞ S The coefficients C x , C y , C z , C ya , C za are named as coefficients ofaerodynamic longitudinal, normal, transversal, lifting and lateral force, and C xa is thedrag coefficient. As the reference area S it can be adopted for definition of coefficients ofaerodynamic forces: • Gross wing area while aircraft considering; • Area of wing formed by outer panels while considering a wing separately; • Mid-section area in case of considering a fuselage, engines, nacelles etc. Lets proceed to consideration of the aerodynamic moments. Lets put an origin ofa body system in the center of mass and we can assume this point as a point of reductionof aerodynamic forces. The moment M caused by these forces is named as theaerodynamic moment. The aerodynamic moment components along axes of body coordinate system are designated as M x , M y , M z and named as aerodynamic roll moment, aerodynamic yaw moment and aerodynamic pitch moment (Fig. 1.6). Lets introduce non-dimensional Fig. 1.6. Components of the coefficients of the moments: aerodynamic moment Mx My Mz mx = ; my = ; mz = , q∞ S l q∞ S l q∞ S b (1.3)where l is the reference length, usually it is a wing span; b is the chord of a wing,usually it is the length of the mean aerodynamic chord. In case of the aircraft parts under consideration the reference area and referencelinear dimensions of these parts are used as S , b , l in the reduced formulae. 10
9. 9. The coefficients mx , m y , mz are named as coefficients of aerodynamic roll, yawand pitch moments. While considering the aerodynamic forces both moments and their coefficientsthe word "aerodynamic" can be omitted if doesn’t cause an error explanation of theseterms. Till now we spoke only about summarized forces and moments. But in somecases it is necessary to know local forces which are affecting on unit area of an aircraftsurface or on its separate parts in specified point. Aerodynamic forces caused bypressure distribution along an aircraft surface are usually determined by overpressures.An overpressure is usually expressed in shares of undisturbed flow drag, i.e. as non-dimensional value which is named as coefficient of pressure: p − p∞ Cp = . (1.4) q∞ Lets write down also formulae determining proportions between forces coefficients in body and wind coordinate systems. Lets consider flow about the wing with infinite span by flat flow under some angle of attack (Fig. 1.7). Lets direct an axis xa along undisturbed stream velocity, axis Fig. 1.7. Aerodynamic forces in wind ya - perpendicularly to axis xa to the airfoil and body coordinate systems top outline. An axis x of body coordinatesystem will be directed along chord, axis y - perpendicularly to axis x to the upperoutline. We will place an origin of both systems in a center of pressure. Center ofpressure of an airfoil is the crosspoint of action line of resultant aerodynamic force ofthe airfoil with a chord or its prolongation. As it follows from fig. 1.7: Ya = Y cos α − X sinα ;⎫ ⎬ (1.5) X a = X cos α + Y sinα ⎭ 11
10. 10. Y = Ya cos α + X a sin α ;⎫ ⎬ (1.6) X = X a cos α − Ya sinα ⎭ Having forces substituted by their expressions under the formulae (1.1), (1.2) andhaving reduced the identical coefficients, we will receive: C y = C y cos α − C x sinα ;⎫ ⎪ a ⎬ (1.7) C x a = C x cos α + C y sin α ⎪ ⎭ C y = C ya cos α + C xa sinα ;⎫ ⎪ ⎬ (1.8) C x = C xa cos α − C ya sinα ⎪⎭ At small angles of attack α it is possible to assume cos α ≈ 1 , sinα ≈ α . Besidesit is possible to neglect C xa << C ya and addend C xa sin α . Therefore it is possible towrite down expressions (1.7) and (1.8) at small angles of attack as: Cy ≈ Cy ; ⎫ ⎪ a ⎬ (1.9) Cxa ≈ Cx + C y α ⎪ ⎭ C y ≈ C ya ; ⎫ ⎪ ⎬ (1.10) C x ≈ C x a − C ya α ⎪ ⎭ 12