Aerodynamics.pdf

 FOREWORD 1. The field of military aviation is expanding at a rapid pace, therefore it is essential that the wealth of knowledge is continuously updated, amongst our operators. 2. In Indian Air Force, a young trainee pilot is groomed from the nascent stage, eithe...

 FOREWORD 1. The field of military aviation is expanding at a rapid pace, therefore it is essential that the wealth of knowledge is continuously updated, amongst our operators. 2. In Indian Air Force, a young trainee pilot is groomed from the nascent stage, either at the National Defence Academy (NDA) or the Air Force Academy (AFA), well before taking the first flight. In order to achieve the desired quality of training amongst pilots, there is a need to have a single, authentic and comprehensive, source of knowledge. 3. After extensive efforts, the Flying Instructors School (FIS) has prepared four volumes of quality training notes. These books are aimed to cater to the knowledge requirements, of not only the ab- initio, but also to those of an instructor and beyond. 4. In order to cater to the future advancements in the field of aviation there would be a requirement to update this knowledge. I am confident that these books would ensure standardization in training and enrichment of knowledge at all levels. I compliment FIS for an excellent contribution in this regard. -sd- (S Bhojwani) Air Marshal Date : Nov 05 AOC-in-C  FIS BOOK 1 AERODYNAMICS INDEX Chapter Title Page No. 1. Mechanics 1 2. The Atmosphere and Airspeed 17 3. Theories of Development of Aerodynamic Forces 31 4. Viscosity and Boundary Layer 57 5. Aerofoils 69 6. Lift 83 7. Drag 91 8. Stalling 113 9. Spinning 129 10. Wing Planform 147 11. Lift Augmentation 177 12. Flight Controls 191 13. Trimming and Balance Tabs 209 14. Stability 215 15. Level Flight 245 16. Climbing and Gliding 249 17. Manoeuvres 261 17 (a). Operating Strength Limitations 279 18. Range and Endurance 291 19. Transonic and Supersonic Aerodynamics 325 20. Design for High Speed Flight 363 21. Propeller Theory 391 22. Application of Aerodynamics to Specific Problems of Flight 413 23. Basics of Rotor dynamics 445 24. Hovering and Horizontal Movement 451 25. Control in Forward Flight 461 26. Power Requirements 477 27 Autorotation 483  CHAPTER 1 MECHANICS Scalars and Vectors 1. Scalar. A quantity that has only magnitude and no direction is called a scalar quantity, e.g. length, area, volume, mass, density, speed, work, energy etc. 2. Vector. A quantity that has both magnitude and direction is called a vector quantity, e.g. displacement, velocity, acceleration, force, momentum etc. 3. A vector quantity is usually represented by a line with arrowhead. The magnitude of the vector is shown by the length of the line and the direction of the arrow represents the direction. To find the resultant of scalar quantities, they are simply added algebraically. To find the resultant of two or more vectors, the triangle law, parallelogram law or polygon law is applied. A single Fig 1-1: Triangle Law of Vectors vector may also be resolved into two or more components. 4. Triangle Law of Vectors. If two vectors (a & b) are represented in magnitude and direction by two sides of a triangle (BC & CA) taken in order, the resultant is represented by vector c in magnitude and direction by the third side of the triangle (BA) taken in the opposite Fig 1-2: Parallelogram Law of Vectors order (Fig 1-1). 5. Parallelogram Law of Vectors. If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram (AB & AD) drawn from a point, the resultant vector r is represented in magnitude and direction by the diagonal (AC) passing through that point (Fig 1-2). Fig 1-3: Polygon Law of Vectors 6. Polygon Law of Vectors. If the vectors are represented by the sides of a polygon, taken in order, the resultant r is represented by the closing side of a polygon (AF) taken in opposite order (Fig 1-3). FIS Book 1: Aerodynamics 2 Basic Definitions 7. Speed. The rate at which a body is displaced in position is called its speed. Speed is a scalar quantity. 8. Velocity. The rate of displacement of a body in a given direction is called its velocity. It is a vector quantity. Unit of both speed and velocity is m/s. 9. Acceleration. It is the rate of change of velocity. The change may be in speed or direction 2 or both. Unit: m/s. 10. Mass. The mass of a body is the quantity of matter contained in it. Unit: kilogram (kg). 11. Inertia. There is a natural tendency for things to continue doing what they are already doing. A body that is at rest tends to remain at rest. A body that is moving tends to continue moving at the same speed in the same direction. This tendency of a body in equilibrium, to continue in the same state is due to its inertia. It is a property possessed by all bodies, which shows the reluctance to change their state of equilibrium. It is a quality and has no units. Inertia can be measured only in terms of mass, which is a scalar quantity. 12. Force. Application of force changes or tends to change the state of rest or uniform motion of a body in a straight line. In other words, force is that which changes or tends to change the momentum of a body. F = ma. Unit: Newton. 13. Momentum. The momentum of a body is the quantity of motion contained in a body and is the product of the mass and velocity of the body. Momentum is a vector quantity having the same direction as the velocity vector, and may be resolved into components, or combined with another momentum to give a resultant, like other vector quantities. Momentum should not be confused with inertia which is merely a quality related to the mass of the body, not its velocity. Momentum = mv. Unit: kg.m/s. 14. Laws of Motion. Sir Issac Newton propounded three laws of motion: (a) First Law. A body will continue in its state of rest or uniform motion in a straight line unless compelled to change that state by an external force. (b) Second Law. The rate of change of momentum is proportional to the applied force, and the change of momentum takes place in the direction of the applied force. (c) Third Law. Action and reaction are equal and opposite. (Note that action and reaction refer in this case to different bodies).  3 Mechanics 15. Newton’s second law is equivalent to defining force as that which causes acceleration. From the second law we can say that F ∝ rate of change of momentum. But momentum = mv, and if m is constant for a given body, F ∝ m X rate of change of v, or F ∝ ma. If the unit of force is chosen at that force which gives unit acceleration to unit mass, the equation can be written as F = ma. 16. Impulse of a Force. Suppose a force F acts on a body of mass m for a short time t (e.g. when a bat strikes a ball), such that the velocity of the body changes from u to v. As F = ma, The acceleration ‘a’ produced by the force will be F/m. Substituting ‘a’ in v = u + at. v = u + _F_ δt, m or F. δt = mv – mu. 17. The expression on the left (F.δt) is known as the impulse of the force, and is equal to the change of momentum caused by the force. 18. Conservation of Momentum. Consider two bodies travelling in the same direction which collide with each other, the duration of the collision being a short time say δt. Throughout the collision each will experience a force equal and opposite of that experienced by the other (Newton’s third law). The impulse of the force is F δ t and is the same for each body. Thus the change of momentum will be the same for each body. If at the time of collision, body A was overtaking body B, it is apparent that the effect of the impact will be to decrease the momentum of A and increase that of B, and the total momentum of the system of two bodies will remain unchanged. 19. The Law of Conservation of Momentum. The effects of the interaction of parts of a closed system are summarized in the law of conservation of momentum, which states that the total momentum in any given direction before impact is equal to the total momentum in that direction after impact. 20. Work. A force is said to do work on a body when it moves the body along its line of action. The amount of work done is equal to the product of the force and the distance moved (s) in the direction of the force. W = Fs. Unit: Nm or Joules. 21. Energy. The energy of a body is its capacity to do work. E = Fs. Unit: Joules. 22. Power. The power of an engine is its rate of doing work. FIS Book 1: Aerodynamics 4 P = Fs = Fv Where: F = Force T s = Distance the body has moved T = Time Unit: Watts or Joules / second. 23. Newton’s Universal Law of Gravitation. It states that any two bodies in the universe attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. F ∝ M1 X M2 (1.1) s2 24. Acceleration Due to Gravity ‘g’. It can be seen from the universal law of gravitation that any body of some mass will be attracted towards any other body. If ‘M1’ be the mass of the earth and ‘M2’ be the mass of a body then it will be attracted towards the earth (assuming this is the only body attracting it) with a force which is given by equation (1.1). Here ‘s’ is the distance between the given body and the centre of the earth. This implies that different bodies at the same distance from the centre of the earth, having different masses will have correspondingly different forces of gravity acting upon them. However all of them will experience the same acceleration due to gravity, provided that only the force of gravity acts upon them. This might sound a little complex however the following proof shall make the understanding easier. As per Newton’s law of gravitation we know that: F ∝ M1 X M2 s2 Also if a body of mass ‘M2’ is being accelerated then the force acting on that body is: F = M2 X a Where in this case ‘a’ is acceleration due to earth’s gravity. Hence ‘a’ may be replaced by ‘g’ which denotes acceleration due to gravity. Therefore, we have: M2 X g = M1 X M2 s2 Since M2 gets eliminated on both the sides of the equation, in all the cases where ‘S’ remains same the acceleration due to gravity ‘g’ shall remain same as the mass of the earth is constant. 25. Weight. The force with which a body is attracted towards the centre of the earth is called its weight. It is equal to the product of mass and acceleration due to gravity at that point. W = mg, where g is the acceleration due to gravity at that point. Since g is not constant over the earth’s surface (more at poles and lesser at equator), therefore the weight of a body is variable, unlike its mass which remains constant.  5 Mechanics 26. A spring balance measures weight. A see-saw type of balance or a steel yard or a weighbridge measures mass as in these cases force of gravity acts on both the sides and gets eliminated. 27. Density. Density is the mass per unit volume of a substance. It is denoted by the symbol ρ. Unit: kg/m³. 28. Pressure. Pressure is the force exerted per unit area. Unit: N/m2 or Pascals. 29. Stress. Stress is the force exerted between two contacting bodies or parts of a body. It is measured as the force per unit area. Unit: N/m2. 30. Strain. Strain is the deformation caused by stress. It is recorded as the change of size over the original size. Since it is a ratio, it does not have any unit. 31. Motion on Curved Path. (a) Refer Fig 1-4. Consider a body travelling along a circle of radius ‘r’ with velocity ‘v’. (b) In a small time say ‘t’ the body travels from point A to point B describing an arc S which subtends an angle ‘θ’ at the centre O. (c) Considering the radian theory, the relationship between the arc, angle subtended and the radius is:- Fig 1-4: Motion Along a Curved Path s=rθ Or θ = s / r (d) By definition we know that the rate of change of angle per unit time is angular velocity (ω), therefore:- ω = θ/t ω = s X 1 r t ω = s X 1 t r FIS Book 1: Aerodynamics 6 Now s / t is the rate of change of displacement i.e. the velocity ‘v’. Therefore, we have ω = vX1 r ω = v (1.2) r (e) Now ‘v’ is the linear velocity of the body. If θ is vary small, it can be considered, that the two vectors (AD and BC) representing motion at these two points (Pt A and Pt B) are of the same length. Though the magnitude has remained the same, the direction of the velocity has changed. Therefore, by definition, the body is subjected to acceleration. Vector DC represents the change in velocity between Pt A and Pt B in time ‘t’ and is represented by ‘∆v’. Therefore, the acceleration is given by DC/t. i.e. ∆ v / t, and this is the centripetal acceleration (CPA) ‘a’. (f) From the figure, using radian theory, it can be said that: DC = ∆ v = v X θ (1.3) CPA = a = ∆ v / t (1.4) Substituting the value of ∆ v from equation (1.3) into equation (1.4), we have: a = vXθ/t a = vXω (ω = θ / t) a = v X v /r (ω = v / r) (From equation 1.2) Therefore, a = v² (1.5) r 32. Centripetal Force. It was shown that a body travelling with uniform speed in a circle has an acceleration of v²/r towards the centre of the circle. The force producing this acceleration is termed as the centripetal force, and for a body of mass ‘m’ the centripetal force is mv²/r towards the centre of the circle. As force (F) = ma But in circular motion a = v2 therefore, r Centripetal Force (CPF) = mv² (1.6) r But m = w / g where w = weight and g = acceleration due to gravity Therefore, CPF = Wv2 (1.7) rg  7 Mechanics 33. Centrifugal Force. It should be noted that in the case of a body travelling in a circular path at the end of a string, while the mass is experiencing centripetal force towards the hand, there is an equal and opposite reaction on the hand holding the string, known as centrifugal force. It is the outward force on the body travelling along a curved path and represents the resistance of the mass to centripetal acceleration. Centrifugal Force is an inertia force and it exists only as an equal and opposite reaction to centripetal force. 34. Let us take the simple example of a stone on the end of a piece of string. If the stone is whirled round so as to make one revolution per second, and the length of the string is 1 meter, the distance travelled by the stone per second will be 2πr, i.e., 2π X 1 or 6.28 m. Therefore v = 6.28 m/s, r = 1 m. Therefore, acceleration towards centre = v²/r = (6.28 X 6.28) / 1 = 39.5 m/s² (approx) 35. Notice that this is nearly four times the acceleration due to gravity, or nearly 4g. Since we are only using this example as an illustration of principles, let us simplify matters by assuming that the answer is 4g i.e. 39.24 m/s². 36. This means that the velocity of the stone towards the centre is changing at a rate 4 times as great as that of a falling body. Yet it never gets any nearer to the centre! No, but what would have happened to the stone if it had not been attached to the string? It would have obeyed the tendency to go straight on, and in doing so would have departed farther and farther from the centre. The acceleration of 4g may, in a sense be taken as the rate at which it is being prevented from doing this. Now lets see what centripetal force will be required to produce this acceleration of 4g? CPF = The mass of the stone X 4g. So, if the mass is ½ kg, the centripetal force will be ½ X 4g = ½ X 4 X 9.81 = 19.62, say 20 Newtons. Therefore the pull in the string is 20 N in order to give the mass of ½ kg, an acceleration of 4g. Notice that the force is 20 Newtons, very roughly 2X9.81 m/s² or 2kgf, the acceleration is 4g. There is a tendency to talk about "g" as if it were a force, it is not, it is acceleration. 37. Now this is all very easy provided the centripetal force is the only force acting upon the mass of the stone - but is it? Unfortunately, no. There must, in the first place, be a force of gravity acting upon it. FIS Book 1: Aerodynamics 8 38. If the stone is rotating in a horizontal circle its weight will act at right angles to the pull in the string, and so will not affect the centripetal force. But of course a stone cannot rotate in a horizontal circle, with the string also horizontal, unless there is something to support it. So let us imagine the mass to be on a table, but it will have to be a smooth, frictionless table, or we shall introduce yet more forces. We now have, at least in imagination, the simple state of affairs as illustrated in Fig 1-5. Fig 1-5: Stone Rotating in Horizontal Circle Supported on a Table 39. Now suppose we rotate the stone in a vertical circle, like an aeroplane looping the loop, the situation is rather different (Fig 1-6). Even if the stone were not rotating, but just hanging on the end of the string, there would be a tension in the string, due to it’s weight, and this as near as matters would be ½ kgf, or very roughly 5 Newtons, for a mass of ½ kg. If it must rotate with an acceleration of 4 g, the string must also provide a centripetal force of 20 Newtons. So when the stone is at the bottom of the circle D, the total pull in the string will be 25 Newtons. When the stone is in the top position, C, it’s own weight will act towards the center Fig 1-6: Stone Rotating in a Vertical and this will provide 5 N, so the string need only pull Circle with an additional 15 N to produce the total of 20 N for the acceleration of 4g. At the side positions, A and B, the weight of the stone acts at right angles to the string and the pull in the string will be 20 N. 40. To sum up: The pull in the string varies between 15 N and 25 N, but the acceleration is all the time 4g and, of course, the centripetal force is all the time 20N. From the practical point of view what matters most is the pull in the string, which is obviously most likely to break when the stone is in position D and the tension is at the maximum value of 25 N. 41. Co-relation of Linear and Angular Motion. The following formulae for circular motion, similar to those for linear motion, may be derived for uniform angular velocity and uniform angular acceleration. Let’s assume: ω1 = Initial velocity in radians per sec. ω2 = Velocity in radians per sec after t sec. α = Angular acceleration in radians per sec². θ = Angle traversed, in radians.  9 Mechanics Using the basic equations of motion we have:- (a) ω2 = ω1 + αt (as v = u +at) (b) θ = ( ω1 + ω2 ) t (as θ = Average Velocity X Time) 2 (c) θ = ω1t + ½ α t² (as s = ut + ½at²) (d) ω1² - ω2²= 2αθ (as v² - u² = 2as) (e) Since ω = v/r, therefore angular acceleration α will be α = ω t α = v X 1 r t α = a r 42. Vector representation of Angular Quantities. Angular velocity and acceleration are vector quantities. By convention they are represented by vectors. The vector length represents the magnitude of the quantity and it’s direction is perpendicular to the plane of rotation, i.e. parallel to the axis of rotation. The direction of the arrow is such that, on looking in its direction, the rotation is clockwise. The convention is known as the ‘right handed screw law’. Such vectors can be combined and resolved according to the normal principles of vectors. 43. Centre of Gravity (C.G.). It is that point Fig 1-7: Vector Representation of Angular Moments through which the weight of the body can be considered to act. Moments and Couples 44. Moment of a Force. The moment of a force around a point is the product of the force and the perpendicular distance between that point and the line of action of the force. 45. Addition of Moments. The moment (or turning effect) of a force can be either clockwise or anticlockwise. The net effect of a system of force can be calculated by adding all the moments algebraically. The clockwise moments by convention are taken as positive while the anticlockwise moments are designated negative. FIS Book 1: Aerodynamics 10 46. Couples. A system of two equal, unlike, parallel forces is called a couple. A couple can cause no motion in translation but causes only rotation. The moment of a couple is equal to the product of one force and the perpendicular distance between the two forces. The work done by a couple is the product of the moment of the couple and the angle in radians described by the arm. 47. Moment of Inertia. It was shown that a body travelling with uniform speed in a circle has an acceleration of v²/r towards the centre of the circle. The force producing this acceleration is termed as the centripetal force, and for a body of mass ‘m’ the centripetal force is mv²/r towards the centre of the circle. Now consider a rigid body, such as a flywheel, free to rotate about an axis through its centre, at an angular velocity ‘ω’. When the wheel is rotating, all the particles of the wheel have the same angular velocity, but their linear velocities will depend on their individual distances from the axis, those on the rim moving much faster than those near the axis. 48. A particle of mass ‘m’, at distance ‘r’ from the centre and with linear velocity ‘v’, has a kinetic energy equal to ½ m v², or ½ m r² ω². The total kinetic energy of all the particles is ½ ω² X Σ m r² (Σ = Epsilon which denotes summation). 49. The sum of the products mr² for all the particles in a rigid body rotating about a given axis is called the Moment of Inertia (I) and I = mr2. 50. Moment of inertia can be considered as the rotational equivalent of mass. As in linear motion, if a force is applied to a body then force = mass x acceleration, likewise in circular motion if a torque is applied to a wheel, then Torque = Moment of Inertia X Angular Acceleration. 51. Radius of Gyration. The radius of gyration is a useful concept whereby the mass M (= Σ m) of the wheel is considered to be concentrated in a ring of radius ‘k’ from the axis, such that Σ mr² = Mk². (k is then known as the radius of gyration and I = Mk²). 52. Angular Momentum. If the mass (m) of a body is concentrated at a radius (r) from the axis of rotation, the angular momentum is defined as the product of moment of inertia and angular velocity. Angular momentum = IXω = m r² ω Alternatively, angular momentum can also be calculated from the product of the instantaneous linear momentum (m v) and the radius.  11 Mechanics Angular momentum = m v r or m r² ω 53. Conservation of Angular Momentum. If no external forces are applied then the angular momentum of a system remains constant. As we know that angular momentum is the product of moment of inertia and angular velocity, therefore, if one is decreased, the other must increase to conserve the angular momentum. 54. Equilibrium. When two or more co-planar forces act on a rigid body, the body (or system of forces) is said to be in equilibrium if the following conditions are satisfied: (a) The algebraic sum of the resolved parts of all the forces in same fixed direction should be equal to zero. (b) The algebraic sum of the resolved parts of all the forces in the direction at right angles to the first direction should be equal to zero. (c) The algebraic sum of the moments of all the forces around a given point in their plane should be equal to zero. Equations of Motion: Kinematics 55. Kinematics is the study of the bodies in motion. The symbols and units used for the derivations shall be as follows: Time = t (sec) Distance = s (metres) Initial Velocity = u (m/s) Final Velocity = v (m/s) Acceleration = a (m/s²) Uniform Velocity If velocity is uniform at u meters per sec then clearly: Distance travelled = Velocity X time or s = ut Uniform Acceleration Final velocity = Initial velocity + Increase of velocity Increase in velocity will be the product of acceleration and time i.e. a X t and therefore, v = u + at (1.8) FIS Book 1: Aerodynamics 12 Distance travelled = Average Velocity X Time s = (v + u) X t (1.9) 2 Substituting the value of v from equation (1.8) we get s = (u + at + u) X t 2 i.e., s = u t + ½ a t² (1.10) Extracting the value of t form equation (1.8) we get t = v-u and putting this value into equation (1.9) we get a s = (v + u) (v -- u) = v² - u² 2a 2a or v² = u² + 2as (1.11) With the aid of these simple formulae it is easy to work out problems of uniform velocity or uniform acceleration. Forms of Energy 56. Energy can exist in various forms e.g. kinetic energy, potential energy, pressure energy, heat energy etc. The capacity to do work is called Energy. The amount of work the body can do, will be the measure of its energy. 57. Kinetic Energy. It is the energy a body possesses by virtue of its motion and is measured by the amount of work that the body can perform against the impressed force before its velocity is destroyed or if a body starts from rest then the energy required to accelerate it to the final velocity would be the measure of its kinetic energy. If a body of mass m starts from rest under a force F, then v² = u² + 2as v² = 0 + 2as s = v2 2a KE = Fs = ma.s = m a v2 2a KE =½ mv2 (1.12) 58. Potential Energy. It is the energy a body possesses by virtue of its position and is measured by the amount of work that the body can perform in moving from its initial position to some standard position called the zero position. For bodies moving under the influence of the earth's gravity  13 Mechanics the zero position is usually the earth’s surface. Work or Energy = Force X distance But F = m.a where a = g for a body under the influence of earth’s gravitational field or F = m.g = Weight h is the height of the body above the earth’s surface (Zero position). Therefore, PE = mgh (1.13) Also PE = Wh (∵ m g = W) 59. Pressure Energy. It is the energy a fluid possesses by virtue of its compression. Suppose a Pressure P is applied to a plate of area A causing it to move through a distance s then, Work or Energy = Force X distance ∵ Force = Pressure X Area = PA ∴ Energy = (PA). s ∵ Area X Length (Distance) = volume Energy = Pressure X Volume (∵ volume = m / ρ ) Energy = P X m / ρ where ρ is the density Pressure energy = PV or P X m (1.14) ρ Units and Dimensions 60. There are various systems in use for measuring physical quantities like length, mass and time, e.g., the CGS, FPS and metric systems. The most important system used now-a-days is the SI or System International. 61. As per the SI system there are three basic physical quantities i.e. mass, length and time. These have the dimensions M, L and T respectively. Many other quantities are merely combinations of these fundamental dimensions M, L and T. Base Units Quantity Name of Unit Symbol Dimension Length metre m L Mass kilogram kg M Time second s T Temperature Kelvin k - Plane angle radian rad 1 FIS Book 1: Aerodynamics 14 Derived Units Quantity Name of the unit Symbol Unit Equivalent to Dimension Speed / Velocity - - m/s LT-¹ Acceleration - - m/s² LT-² Density - - kg/m³ ML-³ Force Newton N Kg m/s² MLT-² Pressure/ Stress Pascal Pa N/m² M L-¹ T-² Work / Energy Joule J Nm ML²T-² Power Watt W Nm/s or J/s ML²T-³ Weight Kilogram force kg f kg m/s² MLT-² Strain - - - 1 Moment - - N-m ML²T-² Momentum - - Kg m/s MLT-¹ Angular Velocity - - Rad /s T-¹ Angular - - Rad / s² T- ² Acceleration Gas Laws 62. BoyIe’s Law. Under constant temperature conditions, the volume of a given quantity of a perfect gas is inversely proportional to its pressure. i.e. P ∝ 1/V or PV = constant 63. CharIe’s Law. If the pressure of a given mass of a perfect gas remains constant, then it’s volume increases by 1/273 of its value at 0º C for every degree centigrade rise in temperature. Thus, if temperature is measured in Kelvin then, V0 = V1 = V2 or V = constant (1.15) T0 T1 T2 T The law further states that if the volume of a given mass of a perfect gas remains constant, then its pressure increases by 1/273 of its value at 0ºC for every degree centigrade rise in temperature. If temperature is measured in Kelvin then, P0 = P1 = P2 or P = constant (1.16) T0 T1 T2 T However, when a gas is heated, P, V and T all change. Therefore, combining these laws, we get  15 Mechanics P1V1 = P2V2 or PV = constant (1.17) T1 T2 T ∵ Density, ρ = m/V So for a given mass of gas (m = 1), ρ ∝ 1 or V ∝ 1 V ρ Therefore, _P1 = _P2_ = constant (1.18) ρ1T1 ρ2T2 64. Gas Equation. Since PV = constant, it can be expressed as T PV = RT where R is called the gas constant. (1.19) The numerical value of R depends upon the molecular weight and quantity of the gas in question (e.g. R for air = 288 Jules per Kg Kelvin) 65. Laws of Fluid Pressure. The following are the laws of fluid motion:- (a) In a fluid at rest the pressure at any point acts equally in all directions. (b) In a fluid at rest the pressure acts at right angles to the surface with which the fluid is in contact. (c) In a fluid at rest the pressure increases with depth. If the fluid is incompressible the increase in pressure is directly proportional to depth. If the fluid is compressible the rate of increase of pressure increases with depth. 66. Adiabatic Changes. A change is said to be adiabatic if no heat is taken from or given to the surroundings during the change. 67. Isothermal Changes. A change is said to be isothermal if the temperature of the object under change remains the same as that of the surrounding throughout the change. 68. Friction. When two solid surfaces which are in contact move, or tend to move, relative to each other, a force acts in the plane of contact of the surfaces in a direction opposing the motion (Fig 1-8). This is known as friction. Various theories have been propounded to explain this phenomenon, but it will Fig 1-8: Friction between Two Bodies suffice here to summarise the observed effects. 69. Experiments have shown that: FIS Book 1: Aerodynamics 16 (a) If the two surfaces are not in relative motion, the force of friction balances the applied force and is known as static friction. As the applied force is increased so the force of static friction increases until it is at a maximum when the surfaces are just on the point of sliding. This is called the limit of static friction or limiting friction for the surfaces. (b) Once motion has started, the force needed to maintain motion without acceleration is less than that needed to overcome the static friction. The friction between the moving surfaces is called sliding or kinetic friction. (c) Different materials experience different friction forces, and friction is dependent on the smoothness of the surfaces. 70. The generally accepted laws of friction are: (a) When the surfaces are at rest, friction can take any value up to a certain maximum called the limiting friction. (b) The friction force F, when the surfaces are about to slide or are sliding, is within limits, proportional to the normal reaction (R) between the surfaces. Thus it can be stated that F = µ R (Fig 1-8) where, µ is a constant (over a wide range of values of R) called the coefficient of friction for the surfaces. If at rest but just at the verge of moving, the constant of friction considered is the called coefficient of static friction. When a body is in motion then the constant of friction is called the coefficient of sliding friction. (c) The force of friction is independent of the area of contact between the surfaces. (d) The sliding force is independent of the sliding velocity. The force of friction depends on the nature of the surfaces. Machines 71. A machine is any device which enables energy to be used in a convenient way to perform work. Typical examples of simple machines are levers, winches, inclined planes, pulleys and screws. Efficiency = Work done on load Work done by effort = Out put Input Efficiency is usually expressed in percentage. As friction must be overcome and as a part of the applied effort is expended in making parts of a machine move, the efficiency of a machine is always less than 100%.  17 CHAPTER 2 THE ATMOSPHERE AND AIRSPEEDS Fig 2-1: The Atmosphere FIS Book 1: Aerodynamics 18 THE ATMOSPHERE Introduction 1. Atmosphere is the term given to the layer of air, which surrounds the Earth and extends upwards from its surface to about 500 miles (800 km). The flight of all objects using fixed or moving wings to sustain them, or air breathing engines to propel them, is confined to the lower layers of atmosphere. The properties of atmosphere are therefore of great importance to all forms of flight. 2. The Earth's atmosphere can be said to consist of four concentric gaseous layers. The layer nearest to the surface is known as the troposphere, above which are the stratosphere, the mesosphere and the thermosphere. The boundary of troposphere, known as the tropopause, is not at a constant height but varies from an average of about 25,000 ft at the poles to 54,000 ft at the equator. Above the tropopause, stratosphere extends to approximately 30 miles. Ionosphere is the region of our atmosphere extending from roughly 40 to 250 miles altitude in which there is appreciable ionisation. The presence of charged particles in this region, which starts in the mesosphere and runs into the thermosphere, profoundly affects the propagation of electromagnetic radiations of long wavelengths (radio and radar waves). 3. Through these layers the atmosphere undergoes a gradual transition from its characteristics at sea level to those at the fringe of the thermosphere, which merges with space. The weight of the atmosphere is about one millionth of that of Earth, and an air column one square metre in section extending vertically through the atmosphere weighs 9800 kg. Since air is compressible, the troposphere contains a greater part (over three quarters in middle latitudes) of the whole mass of the atmosphere, while the remaining fraction is spread out with ever-increasing rarity over a height range of some hundred times that of the troposphere. 4. Average representative values of atmospheric characteristics are shown in Fig 2-1. It will be noted that the pressure falls steadily with height, but that temperature falls steadily upto the tropopause, where it then remains constant through the stratosphere, but increasing for a while in the warm upper layers. Temperature falls again in the mesosphere and eventually increases rapidly in the thermosphere. The mean free path (M) in Fig 2-1 is an indication of the distance of one molecule of gas from its neighbours. Therefore in the thermosphere, although the individual air molecules have the temperatures shown, their extremely rarefied nature results in a negligible heat transfer to any body present. Physical Properties of Air 5. Air is a compressible fluid and as such it is able to flow or change its shape when subjected even to minute pressures. At normal temperatures, metals such as iron and copper are highly  19 The Atmosphere and Airspeeds resistant to deformation by pressure, but in liquid form they flow readily. In solids the molecules adhere so strongly that large forces are needed to change their position with respect to other molecules. In fluids, however, the degree of cohesion of the molecules is so small that very small forces suffice to move them in relation to each other. A fluid in which there is no cohesion between the molecules and therefore no internal friction i.e. inviscid, and which is incompressible would be an "ideal" fluid - if it were obtainable. 6. Fluid Pressure. At any point in a fluid the pressure is same in all directions, and if a body is immersed in a stationary fluid the pressure on any point of the body acts at right angles to the surface at that point irrespective of the shape or position of the body. 7. Composition of Air. Since air is a fluid having a very low internal friction it can be considered, within limits, to be an ideal fluid. Air is a mixture of a number of separate gases, the proportions of which are: Element By Volume % By Weight % Nitrogen 78.08 75.5 Oxygen 20.94 23.1 Argon 0.93 1.3 Carbon dioxide 0.03 0.05 Hydrogen Neon Helium Krypton Traces Only Xenon Ozone Radon Table 2-1: Composition of Atmosphere It can be seen from table 2-1 that for all practical purposes the atmosphere can be regarded as consisting of 21% oxygen and 78% nitrogen by volume. Up to a height of some five to six miles water vapour is found in varying quantities. The amount of water vapour in a given mass of air depends on the temperature and whether the air is, or has recently been, over large areas of water. Higher the temperature, greater is the amount of water vapour that the air can hold. Standard Atmosphere 8. The values of temperature, pressure, density, viscosity etc. are never constant in any given layer of the atmosphere, in fact, they are all constantly changing. Experience has shown that there is a requirement for a standard atmosphere for the comparison of aircraft performances, calibration of FIS Book 1: Aerodynamics 20 altimeters and other practical uses. Some standard values have been established by the International Civil Aviation Organization (ICAO), and this set of figures, giving variation of physical properties of air with altitude, is known as the International Standard Atmosphere (ISA). 9. In the ISA sea level values are denoted by the suffix 0. It is also useful to refer to relative pressure, relative temperature and relative density, by which is implied in each case the local value divided by the sea level value. The manner in which these parameters vary in the standard atmosphere is given in the succeeding paragraphs. Temperature 10. Measurement of Temperature. The Celsius scale is normally used for recording atmospheric temperatures and working temperatures of engines and other equipment. On this scale water freezes at 0° and boils at 100°, at sea level. In scientific measurement of temperature absolute zero has a special significance. At this temperature a body is said to have no heat whatsoever. Temperatures relative to absolute zero are measured in Kelvin (also known before 1967 as degrees Kelvin) and are used in all formulae dealing with density and pressures. Kelvin (K) zero occurs at - 273.15° Celsius (C). 11. Conversion Factors. Occasionally the Fahrenheit (F) scale may still be encountered. It is not suitable for scientific purposes, as Fahrenheit zero has no particular significance. Water at sea level freezes at 32° F and boils at 212° F. To convert °F to °C, subtract 32° and multiply by 5/9. To convert °C to °F, multiply by 9/5 and add 32°. To convert °C to Kelvin or °K (Absolute) add 273° (or more precisely 273.15°). 12. The standard sea level temperature of air, T0, is 15°C or 288.15 K. The temperature falls steadily upto the tropopause, where it then remains constant through the stratosphere, but increasing for a while in the warm upper layers. Temperature falls again in the mesosphere and eventually increases rapidly in the thermosphere. The relative temperature θ (theta) = T / T0, where T and T0 are both expressed in K. Pressure 13. The standard sea level pressure is 1013.25 mb. The pressure at any point in a stationary fluid is determined by the weight of fluid in question. Same is the case with air in the atmosphere. It follows that the pressure must fall steadily with increasing altitude. However, because the density also falls, the rate at which the pressure falls must decline as the altitude increases. The relative pressure δ (delta) = p/p0.  21 The Atmosphere and Airspeeds 14. Pressure Altitude. If the sea level pressure should be other than 1013.25 mb, then there is a requirement to relate the actual pressure experienced at a given point to a height (above or below MSL) in the ISA at which the same pressure will be found. This is pressure altitude, and is widely used in Operating Data Manuals and performance charts. Pressure altitude can be found either by calculating a height change of 30 ft for every 1 mb difference in pressure away from 1013.25 mb at the surface, or by setting the sub-scale of an ICAN calibrated altimeter to 1013.25 mb and reading pressure altitude directly from the instrument. Density 15. Density (ρ) is mass per unit volume. The unit of density used in this book is kg per m³. The relationship of density to temperature and pressure can be expressed as: p / ρT = R (2.1) Where p = pressure in mb T = absolute temperature R = gas constant (= 8313 J/Mol °K for any gas and for air R = 287 J/kg °K). The relative density σ = ρ / ρ0. 16. Effect of Pressure on Density. When air is compressed, a greater amount can occupy a given volume, i.e. the mass, and therefore the density, has increased. Conversely, when air is expanded less mass occupies the original volume and the density decreases. From equation (2.1), it can be seen that, provided the temperature remains constant, density is directly proportional to pressure, i.e. if the pressure is halved, so is the density, and vice versa. 17. Effect of Temperature on Density. When air is heated it expands so that a smaller mass will occupy a given volume, therefore the density will have decreased, assuming that the pressure remains constant. The converse will also apply. Thus the density of the air will vary inversely as the absolute temperature, as indicated by equation (2.1). In the atmosphere the fairly rapid drop in pressure as altitude is increased has the dominating effect on density, as against the effect of the fall in temperature, which tends to increase the density. 18. Effect of Humidity on Density. The preceding paragraphs have assumed that the air is perfectly dry. In the atmosphere some water vapour is invariably present. This may be almost negligible in certain conditions but in others the humidity may become an important factor in the performance of an aircraft. The density of water vapour under standard sea level conditions is 0.760 kg per m³. Therefore water vapour can be seen to weigh 0.760/1.225 as much as air, roughly 5/8 as much as air at sea level. This means that under standard sea level conditions the portion of a mass of air, which holds water vapour, weighs (1 – 5/8), or 3/8 less than it would if it were dry. Therefore air FIS Book 1: Aerodynamics 22 is least dense when it contains a maximum amount of water vapour and most dense when it is perfectly dry. 19. Density Altitude. For aircraft operations air density is usually expressed as a density altitude. Density Altitude is defined as that height (above or below mean sea level) in the standard atmosphere to which the actual density at any particular point corresponds. For standard conditions of temperature and pressure, density altitude is the same as pressure altitude. Density Altitude equals pressure altitude ± (120 X t), where t is the difference between local air temperature at pressure altitude and the standard temperature for the same pressure altitude. If the air temperature is higher than standard, then (120 X t) is added to pressure altitude, if it is lower, subtracted. Viscosity 20. Coefficient of Viscosity (µ). The value of coefficient of viscosity of air at sea level in the -5 standard atmosphere, µ0, is 1.79 X 10 kg / ms. The coefficient, µ, for any given gas, depends directly on its temperature (µ is proportional to T¾). Thus µ falls steadily with increasing altitude until the tropopause is reached and remains constant in the stratosphere. 21. Kinematic Viscosity. An alternative measure of the viscosity of a gas is its kinematic –5 viscosity, ν, defined by the formula ν = µ / ρ. Its standard sea level value is 1.46 X 10 m²/s. In the troposphere µ & ρ both reduce with increasing altitude but ρ does so more rapidly. In the stratosphere ρ falls while µ is constant. Therefore ν increases with altitude in both regions. Altitude (ft) Tempe- Pressure Pressure Density Relative rature (°C) (mb) (psi) (Kg per m³) Density (%) 0 +15.0 1013.25 14.7 1.225 100.0 5,000 +5.1 843.1 12.22 1.056 86.2 10,000 -4.8 696.8 10.11 0.905 73.8 15,000 -14.7 571.8 8.29 0.771 62.9 20,000 -24.6 465.6 6.75 0.653 53.3 25,000 -34.5 376.0 5.45 0.549 44.8 30,000 -44.4 300.9 4.36 0.458 37.4 35,000 -54.3 238.4 3.46 0.386 31.0 40,000 -56.5 187.6 2.72 0.302 24.6 45,000 -56.5 147.5 2.15 0.237 19.4 50,000 -56.5 116.0 1.68 0.186 15.2 Table 2-2: ICAO Standard Atmosphere  23 The Atmosphere and Airspeeds 22. The ISA. To summarise the main parameters, the ISA assumes a mean sea level temperature of +15°C, a pressure of 1013.25 mb (101.325 KN / m2 or 760 mm of Mercury) and a density of 1.225 kg per m3. The temperature lapse rate is assumed to be uniform at the rate of 6.5° C per kilometre (1.98° C per 1,000 ft) up to a height of 11 km (36,090 ft) above which height it remains constant at -56.5° C (Table 2-2). 23. Speed of Sound. The speed of sound in a gas, a, is an important parameter which will be formally defined in the chapters on high speed flight of this book. In a given gas a = α0 √T. Its standard sea level value, a0, is 340 m/s or 661.2 kts. Dynamic Pressure 24. Because it possesses density, air in motion must possess energy and therefore exerts a pressure on any object in its path. This dynamic pressure is proportional to the density and the square of the speed. The energy due to movement, the kinetic energy (KE), of one cubic metre of air moving at a stated speed is given by the formula: KE = ½ ρV2 joules Where: ρ is the local air density in kg per m³ V is the speed in metres per second. One Joule = the work done when one Newton force displaces the point of its application through a distance of 1 metre in the direction of the force. If this volume of moving air is completely trapped and brought to rest by means of an open-ended tube, the total energy remains constant. In being brought to rest the kinetic energy becomes pressure energy (small losses are incurred because air is not an ideal fluid) which, for all practical purposes, is equal to ½ ρ V2 Newton per m2, or if the area of the tube is S square metres, then: Total Pressure (dynamic + static) = ½ ρ V2 S Newton. One Newton = that force which applied to a mass of 1 kilogram, gives it an acceleration of 1 metre per second per second (1 kg force = 9.81 Newton). 25. The term ½ρV2 is common to all aerodynamic forces and fundamentally determines the air loads imposed on an object moving through the air. It is often modified to include a correction factor or coefficient. The term stands for the dynamic pressure imposed by air of a certain density moving at a given speed and which is brought completely to rest. The abbreviation for the term ½ ρ V2 is the symbol "q". Note: Dynamic pressure cannot be measured on its own, as the ambient pressure of the atmosphere (known as static) is always present. This total pressure is also known as stagnation or pitot pressure. It can be seen that (Dynamic + Static) - Static = Dynamic. FIS Book 1: Aerodynamics 24 MEASUREMENT OF SPEED Air Speed and Ground Speed 26. When we speak of the speed on an aeroplane we mean its speed relative to the air, or air speed as it is usually termed. The existence of a wind simply means that portions of the air are in motion relative to the earth, and although the wind will affect the speed of the aeroplane relative to the earth, i.e. its ground speed, it will not affect its speed relative to the air. 27. For instance, suppose that an aeroplane is flying from A to B (60 km apart), and that the normal speed of the aeroplane (i.e. its air speed) is 100 kmph. If there is a wind of 40 kmph blowing from B towards A, the ground speed of the aeroplane as it travels from A to B will be 60 kmph, and it will take one hour to reach B, but the air speed will be 100 kmph. If, when the aeroplane reaches B, it turns and flies back to A, the ground speed on the return journey will be 140 kmph. The time to regain A will be less than half an hour, but the air speed will still remain 100 kmph, that is, the wind will strike the aeroplane at the same speed as on the outward journey. Similarly, if the wind had been blowing across the path, the pilot would have inclined his aeroplane several degrees towards the wind on both journeys so that it would have travelled crabwise, but again, on both outward and homeward journeys the air speed would have been 100 kmph and the wind would have been a head wind straight from the front as far as the aeroplane was concerned. 28. An aeroplane which encounters a head wind equal to its own air speed will appear to an observer on the ground to stay still, yet in reality it is moving with considerable velocity. A free balloon flying in a wind travels over the ground, yet it has no airspeed (a flag on the balloon will hang vertically downwards). Method of Measuring Air Speed 29. We have talked about air speed, but so far we have not explained the method by which it is usually measured, since it is essential that an aircraft have some means of measuring the speed at which it is passing through the air. The method of doing this is by comparing the total pressure (static + dynamic) with static pressure (Para 24). Let’s see how and why. The resistance of a body placed in an air stream depends, among other things, on the velocity of the stream and there is a connection between the resistance and the velocity. If we can eliminate the “other things” we should be able to devise an instrument, which by measuring resistance would give, if suitably calibrated, readings of velocity. 30. If we were to place a flat plate at right angles to the airflow and behind this plate we fitted a compression spring balance, then there would be a direct connection between the velocity of the airflow and the reading of the spring balance, provided that the area of the plate and the density of the  25 The Atmosphere and Airspeeds air remained constant. As regards the area, there is not much difficulty except that we must remember that it is the “frontal” area facing the airflow which matters, and therefore we must ensure that the plate is at all times at right angles to the airflow. The question of the density, however, is not quite so simple, in fact, the ordinary method of measuring airspeed is dependent on the density of the air and therefore readings of air speed can only be correct when the density is of some definite value. Pitot Static Tube 31. The system, which is almost universally employed, is the pitot-static head. This consists of two metal tubes fixed to some exposed part of the aeroplane and facing directly into the airflow. One of these, the pitot (or pressure) tube, has an open end facing the wind, while the other tube, called the static, is closed at the end but pierced with small holds or slits farther back. On modern aircraft the tubes are usually concentric, the pitot tube forming the centre and being surrounded by the static. The other end of each tube is connected by tubing to the air speed indicator in the cockpit. 32. The pitot tube is connected to the inside of a capsule (a flat circular box of a corrugated metal such as is used in an altimeter of aneroid barometer), while the static tube is connected to the casing of the instrument. When there is no air speed the normal atmospheric pressure will be communicated to both sides of the diaphragm or capsule, but when the air blows up against the open pitot tube the extra pressure due to the air velocity plus the normal atmospheric pressure will act on one side, while the static tube still conveys the ordinary atmospheric pressure to the other. The difference in pressure causes a deflection of the capsule, which, by suitable gearing, makes a pointer move round the scale, which is marked off in air speeds. 33. It might be expected that the open pitot tube would effectively bring the air to rest. The extra force on the open end of the pitot tube is ½ ρV2S, where S is the area of the opening. Therefore the pressure increases on the open end due to the airflow will be force divided by the area, i.e. ½ ρV2. 34. This dynamic pressure is used to indicate the speed using an instrument known as an Air Speed Indicator (ASI). It should be noted that both the dynamic and static pressures act on the pitot tube while only the static pressure acts on the static tube. The ASI therefore reads the difference between the two i.e. the dynamic pressure and gives an indication of air speed. Relationship Between Air Speeds 35. The general term air speed is further qualified as: (a) Indicated Air Speed (IAS) (Vi). The reading on the ASI is the indicated air speed. Note: In practice, any instrument error is usually very small and can, for all practical purposes, be ignored. FIS Book 1: Aerodynamics 26 (b) Calibrated Air Speed (CAS) (Vc). When the IAS has been corrected by the application of pressure error correction (PEC) and instrument error correction, the result is known as Calibrated Air Speed (CAS). PEC can be obtained from the Aircrew Manual or ODM for the type. When the PEC figures for individual instruments are displayed on the cockpit, they include the instrument error correction for that particular ASI. CAS is also known as RAS (Rectified Air Speed). (c) Equivalent Air Speed (EAS) (Ve). Equivalent air speed is obtained by adding the compressibility error correction (CEC) to CAS. (d) True Air Speed (TAS) (V). The TAS is obtained by dividing the EAS by the square root of the relative air density. 36. The ASI can be calibrated to read correctly for only one density / altitude. All ASIs are calibrated for ICAO atmosphere conditions. Under these conditions and at the standard sea level density (ρ0), the EAS (Ve) is equal to the TAS (V). At any other altitude where the density is ρ then, Ve = V √ (ρ / ρ0) = V √ σ. Thus, at 40,000 ft where the standard density is one quarter of the sea level value, the EAS will be half the TAS. 37. From the preceding paras it can be seen that there are two important speeds. The TAS is significant because it gives a measure of the speed of a body relative to the undisturbed air and the EAS is significant because the aerodynamic forces acting on an aircraft are directly proportional to the dynamic pressure and thus the EAS. The Venturi Tube 38. Having talked much about airspeeds let us see exactly which part of the aircraft this indicated air speed is correct for. Towards this let us consider a venturi and the flow through it. As per Bernoulli’s theorem, as the cross section area of the venturi towards the throat reduces, the flow velocity increases, considering the flow density remaining constant. Therefore, if a complete pitot - static tube is connected to the air speed indicator and inserted at various parts of the venturi, it will be clear that the air Fig 2-2: The Venturi Tube speed increases from mouth to throat and  27 The Atmosphere and Airspeeds then decreases again to the exit. The air speed increases very nearly in the same proportion as the area of cross-section of the venturi decreases (Fig 2-2). 39. Let us illustrate our points by taking a numerical example. Suppose the area of the throat of a venturi tube is a quarter of the area of the mouth, and suppose the speed of airflow at the mouth is 60 m/s. If we neglect any change of density of the air as it flows through the venturi, the speed of flow at the throat will be 240 m/s. Now if a complete pitot-static head is placed at the mouth, and is connected to an air speed indicator, the reading will, of course, be 60 m/s, which is the air speed at the mouth. If the complete head is placed at the throat, the reading will be 240 m/s. 40. But suppose a pitot tube is placed at the throat, where the speed is 240 m/s and the static side of the air speed indicator is connected to a static tube at the mouth then what will the air speed indicator read? 41. In order to determine this let us call: The static pressure at the mouth, i.e. the atmospheric pressure, Pa Velocity at mouth, Va Static pressure at throat, Pb Velocity at throat, Vb The air density will be the same at both, i.e. ρa = ρb = ρ. Then on the pitot tube at the throat the total pressure will be the sum of the static and dynamic pressures. i.e. Pb + ½ ρ Vb2 Which, by Bernoulli’s Theorem, is the same as Pa + ½ ρ Va2 On the static tube at mouth, the pressure will be just Pa Therefore pressure on diaphragm in the air speed indicator will be the difference between these two, i.e. Pb + ½ ρ Vb2 – Pa Or Pa + ½ ρ Va2 – Pa which equals ½ ρVa2 Now when the net pressure on an air speed indicator is ½ ρVa2, ρ being the standard sea- level density, the instrument reads Va. Therefore, in this example, the reading will be 60 m/s. FIS Book 1: Aerodynamics 28 Notice that the speed recorded is that of the airflow where the static is, and not where the pitot is. 42. Let us reverse the positions of pitot and static and see what happens. Place the pitot at the mouth were the air speed is 60 m/s and the Static at the throat where it is 240 m/s. Total pressure on pitot = Pa + ½ ρVa2 = Pb + ½ ρVb2 (By Bernoulli’s theorem) Pressure on static = Pb Therefore difference between the two = Pa + ½ ρ Va2 – Pb = Pb + ½ ρVb2 – Pb = ½ ρVb2 Therefore the indicator reads Vb i.e. 240 m/s, again a speed corresponding to the position of the static and not the pitot. 43. The explanation is that since the total pressure in the pitot tube is the sum of the dynamic and the static pressure it remains the same at any part of the air stream, in accordance to the Bernoulli’s theorem. Therefore so long as the pitot tube is in the stream, it makes no difference at what part of the stream it is. The static pressure, on the other hand, must be correct for the place where we wish to measure the air speed. 44. It should be noted that the pitot tube must be in the smooth air stream whose velocity we wish to measure. It is no good putting it behind an obstruction, nor in the slipstream from the propeller where there will be greater energy, nor in any place where there is turbulence, because Bernoulli’s Theorem is only true for streamline flow. But it can be in a position where the air is flowing at a higher or lower speed than that of the aeroplane, and, provided the static gives the true atmospheric pressure, the air speed indicator will read the correct speed of the aeroplane. 45. Mach Number. Small disturbances generated by a body passing through the atmosphere are transmitted as pressure waves, which are in effect sound waves, whether audible or not. In considering the motion of the body it is frequently found convenient to express its velocity relative to the velocity of these pressure waves. This ratio is Mach number (M). M = V/a where V = TAS and a = local speed of sound in air. TAS Calculations 46. Given below are useful methods of converting RAS to TAS by simple mental processes.  29 The Atmosphere and Airspeeds (a) TAS = RAS + RAS X altitude (in thousands of feet) (2.1) 60 or, more conveniently : TAS = RAS (kmph) + RAS (km per min) X altitude (thousands of feet) Usually, for pilot navigation, IAS may be substituted for RAS. Example RAS 240 kmph (4 km per min), altitude 15,000 ft. TAS = 240 + (4 X 15) = 300 kmph It should be noted that this method is valid only below 25,000 ft and is increasingly inaccurate above 550 kmph (300 kt) TAS because of compressibility effects. (acceptably accurate upto M 1.0). (b) An alternative method gives the air speed correction as a percentage, or fraction of RAS to be added to the RAS. The figures have been ‘rounded-out’ for convenience and are based on ISA conditions only, but are sufficiently accurate for use in pilot navigation (Table 2-3). Height (ft) Percent (%) Aide de Memoir Fraction 5,000 7½ 3² 1/12 10,000 15 4² 1/6 15,000 25 5² 1/4 20,000 35 6² 1/3 25,000 50 7² 1/2 30,000 60 8² 3/5 35,000 75 9² 3/4 40,000 100 10² 1 45,000 120 11² 6/5 or 5/4 50,000 150 12² 1½ Table 2-3: RAS to TAS Conversion These figures are also subject to increasing compressibility errors above 550 kmph (300 kt) TAS. FIS Book 1: Aerodynamics 30  31 CHAPTER 3 THEORIES OF DEVELOPMENT OF AERODYNAMIC FORCES 1. Definitions (a) Total Reaction (TR). The resultant of all the aerodynamic forces acting on the wing or aerofoil section. (b) Lift. That component of the TR, which is perpendicular to the flight path or Relative Air Flow (RAF). (c) Drag. That component of TR which is tangential to the flight path i.e. parallel to the RAF. (d) Chord Line. A straight line joining the centers of curvature of the leading and trailing edges of an aerofoil. (e) Chord (c). The distance between the leading and trailing edge measured along the chord line. Mean chord is often used as a datum linear dimension in the same way that the wing area (S) is used as a datum area e.g. for the LCA the chord is about 4.5 m and it is about 5.6 m for the MiG 29 etc. (f) Span (b). Span of a wing is the distance between the two wing tips and is usually denoted as b, the examples are. 50.5 m for the IL 76, 11.36 m for the MiG 29 etc. (g) Wing Area (S). Area of the wing projected on a plane perpendicular to the normal axis e.g. about 40 m for the Mirage 2000 and 23 m2 for the MiG 21 etc. 2 (h) Mean Line or Camber Line. A line joining the leading and trailing edges of an aerofoil equidistant from the upper and lower surfaces. Maximum camber is usually expressed as a ratio of the maximum distance between the camber line and the chord line to chord length. Where the camber line lies above the chord line, the aerofoil is said to have positive camber. (j) Angle of Attack (α). Angle between the chord line and the flight path or RAF. In many old textbooks this was referred to as Incidence. (k) (Rigger’s) Angle of Incidence. Angle at which an aerofoil is attached to the fuselage. It is the angle between the mean chord line and the longitudinal fuselage datum. FIS Book 1: Aerodynamics 32 The term is often used erroneously instead of Angle of Attack. Examples of angle of 0 0 incidences are 1 for the hawk, 2.5 for the HPT 32 etc. (l) Thickness / Chord Ratio (t/c). Maximum thickness or depth of an aerofoil section expressed as a percentage of chord length e.g. 5.5% for the SU 30 and 10% for the Hawk aircraft. (m) Centre of Pressure (CP). A point, usually on the chord line, through which the TR may be considered to act. (n) Streamline. It is the path traced by a particle in a steady fluid flow. It can also be defined as an imaginary line drawn in the field of flow such that the velocity vector at any point on the line is always tangential to the line. It follows from this definition that through every point in the field there passes one, and only one, streamline. In general, no two streamlines can intersect each other, since this would imply that at the point of intersection the velocity vector points in two directions at once, and this is clearly inconsistent. There is one exceptional case where a streamline may intersect itself i.e. at a point in the flow where the local velocity is zero. Here there is no velocity vector, and so no inconsistency. Such a point is called 'stagnation point'. (p) Aspect Ratio (AR). = Span = b or ___Span2__ = b2 Chord c Wing Area S (q) Wing Loading. The weight per unit area of the wing := __Weight__ = W Wing Area S (r) Load Factor (g or n). = Total Lift Weight (s) Particle Path. This is simply the path through the field followed by an individual fluid particle. In a steady flow, the velocity vector is clearly always tangential to this path, so that in steady flow a particle path and a streamline are the same. (t) Stream Tube. Consider an imaginary closed curve drawn in the field of flow, as in Fig 3-1. If all the streamlines passing through the points on this curve are drawn, they generate a tubular surface, which is called a stream tube. Since the velocity vector is Fig 3-1: Stream Tube always tangential to the surface of such a tube, it  33 Theories of Development of AD Forces follows that there is no flow into or out of the tube through the 'walls', but only along the tube. (u) Stream Filament. This is a stream tube of infinitesimal cross-section. Types of Flow 2. Steady Streamlined Flow. In a steady streamline flow the flow parameters (e.g. speed, direction, pressure etc.) may vary from point to point in the flow but, at any point, are constant with respect to time. This flow can be represented by streamlines and is the type of flow, which it is hoped, will be found over the various components of an aircraft. Steady streamline flow may be divided into two types: (a) Classical Linear Flow. Fig 3-2 a illustrates the flow found over a conventional aerofoil at low angle of attack in which the streamlines, more or less, follow the contour of the body Fig 3-2 a: Classical Linear Flow and there is no separation of the flow from the surface. (b) Controlled Separated Flow or Leading Edge Vortex Flow. This is a halfway stage between steady streamline flow and unsteady flow described later. Due to boundary layer effects, generally at a sharp leading edge, the flow separates from the surface, not breaking down into a turbulent chaotic condition but instead, forming a strong vortex, which because of its stability and predictability, can be controlled and made to give a useful lift force. Such flows illustrated in Fig 3-2 b are found in swept and delta planforms, particularly at the higher angles of attack, and are dealt with in more detail in later chapters. Fig 3-2 b: Leading Edge Vortex Flow 3. Unsteady Flow. In this type of flow the flow parameters vary with time and the flow cannot be represented by streamlines. 4. Two-dimensional Flow. If a wing is of infinite span, or, if it completely spans a wind tunnel from wall to wall, then each section of the wing will have exactly the same flow pattern round it except near the tunnel walls. This type of flow is called two-dimensional flow since the motion is confined to a plane parallel to the free stream direction. FIS Book 1: Aerodynamics 34 5. Three-dimensional Flow. The wing on an aircraft has a finite length and, therefore, whenever it is producing lift the pressure differential tries to equalize around the wing tip. This induces a span-wise drift of the air flowing over the wing, inwards on the upper surface and outwards on the lower surface, producing a three-dimensional flow. 6. Vortices. Because the effect of the spilling at the wing tip is progressively less pronounced from tip to root, the amount of transverse flow reduces towards the fuselage. As the upper and lower airflows meet at the trailing edge they form vortices, small at the wing root and larger towards the tip. These form one large vortex in the vicinity of the wing tip, rotating clockwise on the port wing and anti- clockwise on the starboard wing, as viewed from the rear. Tip spillage means that an aircraft wing can never produce the same amount of lift as an infinite span wing. If the wing has a constant section and riggers angle of incidence from root to tip then the lift per unit span of the wing may be considered to be virtually constant until about 1.2 chord distance of the wing tip. 7. Vortex Influences. The overall size of the vortex at the trailing edge will depend on the amount of the transverse flow. Therefore, the greater the force (pressure difference), the larger it will be. The familiar pictures of wing-tip vortices showing them as thin white streaks (Fig 3-3), only show the low pressure central core and it should be appreciated that the influence on the airflow behind the trailing edge is considerable. The Fig 3-3: Wing Tip Vortices Visible as White Streaks number of accidents following loss of control by flying into wake vortex turbulence testifies to this. The vortex upsets the balance between the upwash and downwash of two-dimensional flow, reinforcing the downwash and reducing the effective angle of attack, also inclining the lift vector slightly backwards, as the effective relative airflow is now inclined downwards. The component of TR, parallel to the line of flight is increased. This increase is termed as the induced drag. The resolved lift vector perpendicular to the flight direction is reduced. 8. The pattern of the airflow round an aircraft at low speeds depends mainly on the shape of the aircraft and its attitude relative to the free stream flow. Other factors are the size of the aircraft, density & viscosity of air and speed of the airflow. These factors are usually combined to form a parameter known as Reynolds Number “RN” (discussed later in the chapter) and the airflow pattern is then dependent only on the shape, attitude and Reynolds Number.  35 Theories of Development of AD Forces 9. Reynolds Number (i.e. size, density, viscosity and speed) and condition of surface determines the characteristics of the boundary layer. This, in turn, modifies the pattern of the airflow and distribution of pressure around the aircraft. The effect of boundary layer on the lift produced by the wings may be considered insignificant throughout the normal operating range of angles of attack. In later chapters it will be shown that the behavior of boundary layer has a profound effect on the lift produced at high angles of attack. 10. When considering the velocity of the airflow it does not make any difference to the pattern whether the aircraft is moving through the air or the air is flowing past the aircraft. It is the relative velocity which is the important factor. AERODYNAMIC THEORIES General 11. Several methods or theories have been developed to predict the performance of a given wing/aerofoil shape. These can be used to explain the subtle changes in shape necessary to produce the required performance appropriate to the role of the aircraft. In practice, the appropriate wing shape is calculated from the performance criteria. 12. The theories discussed in this volume are: Lift by Pressure Distribution, Momentum Theory, Dimensional Analysis and Circulation Theory. LIFT BY PRESSURE DISTRIBUTION 13. The most useful, non-mathematical method is an examination of the flow pattern and pressure distribution on the surface of a wing in flight. This approach will reveal the most important factors affecting the amount of lift produced, based on experimental (wind-tunnel) data. 14. Basic Aerodynamic Theory. The shape of the aircraft (and boundary layer) will determine the velocity changes and consequently the airflow pattern and pressure distribution. For a simplified explanation of why these changes occur it is necessary to consider: (a) The Equation of Continuity. (b) Bernoulli’s Theorem. Equation of Continuity 15. The Equation of continuity states that mass can neither be created nor destroyed, or simply stated, air mass flow remains constant. FIS Book 1: Aerodynamics 36 16. Consider the steady flow of an inviscid fluid through the stream filament shown in Fig 3-4. Let Al be the area of cross-section at any station 1, A2 at any other station 2, both small. Let ρ1, VI and ρ2, V2 be the fluid density and velocity at these stations respectively, each effectively constant over the given section since the area is, by definition, small. Since the fluid is a continuous medium, and there is no flow across the walls of the filament, and since the flow is steady, so that there is no build up of fluid in the region bounded by stations 1 and 2, it follows that the rate of mass flow into this region at station 1 must be equal to the rate of mass flow out of the region at Fig 3-4: Steady Flow of Inviscid station 2, i.e., Fluid Though a Stream Tube ρI A1 VI = ρ2 A2 V2 Since stations 1 and 2 are arbitrarily chosen, it follows that the expression ρ A V has the same value at any such station, i.e., ρ A V = Constant (along a stream filament) (3.1) 17. This equation is the simplest form of what is known as the Equation of Continuity. For incompressible flow, ρ is a constant, so that the equation reduces to AV = constant. In compressible flow theory it is convenient to assume that changes in fluid density will be insignificant at speeds below about 0.4 M. This is because the pressure changes are small and have little effect on the density. The equation of continuity may now be simplified to: AV = constant (3.2) This is the general equation of continuity and applies to both compressible and incompressible fluids. From this it may be seen that a reduction in the tube’s cross-sectional area will result in an increase in velocity and vice versa. This equation enables the velocity changes around a given shape to be predicted mathematically. 18. These equations remain valid for a stream tube of finite size provided that the density and velocity do not vary across the section. If they do vary, then the rate of mass flow across a section may be written as ∫ ρ V. dA, A so that the equation of continuity becomes ∫ ρ V. dA = constant, for compressible flow A ∫ V. dA = constant, for incompressible flow (3.3) A  37 Theories of Development of AD Forces BERNOULLI'S THEOREM FOR INCOMPRESSIBLE FLOW Bernoulli’s Theorem 19. Consider a gas in steady motion. It possesses the following types of energy: (a) Potential energy due to height. (b) Heat (Kinetic) energy. (c) Pressure (Potential) energy. (d) Kinetic energy due to motion, In addition work and heat may pass in or out of the system. 20. Daniel Bernoulli demonstrated that in the steady streamline flow of an ideal fluid, the sum of the energies present remained constant. It is emphasized that the words in bold represent the limitations of Bernoulli’s experiments. In low subsonic flow (< 0.4 M), it is convenient to regard air as being incompressible and inviscid (i.e. ideal) and predictions of the pressure changes round a given aerofoil section agree closely with measured values. Above 0.4 M, however, these simplifications would cause large errors in predicted values and are no longer permissible. 21. For a mathematical derivation of Bernoulli’s equation, consider again the flow of an incompressible, inviscid fluid through a stream filament, as shown in Fig 3-5. Consider in particular the mass of fluid bounded by stations I and 2. Let ρ1, A1, V1, ρ2, A 2, V2, have the same connotations as in paragraph 16. In addition, let p1 be the fluid pressure at station 1, which is located at height h1 above a fixed reference level. Let p2, h2 have similar connotations at station 2. Next consider the same mass of fluid a short Fig 3-5: Flow of Ideal Fluid through a time δt later. Station 1 will have moved through a Stream Tube distance Vl δt along the tube, station 2 a distance V2 δt. Since the fluid is incompressible, ρ1 = ρ2 = ρ (3.4) And by continuity Al Vl = A2 V2 = AV (3.5) The work done by the pressure forces at stations 1 and 2 will then be given by W = p1 A1 V l δ t – p2 A2 V2 δt = A V (pl – p2) δt (3.6) Since the fluid is inviscid, there are no tangential stresses at the wall of the tube, so that this is the total work done on the given mass of fluid. FIS Book 1: Aerodynamics 38 The gain in kinetic energy is ( ½ ρ 2 A 2 V 2 δt V22 ) – ( ½ ρ 1 A 1 V 1 δt V12 ) = { ½ ρ A V (V22 – V12 ) δt } (3.7) The gain in potential energy is (ρ2 V2 A2 δt g h2) – (ρ1 V1 A δt g h1 ) = { ρ A V δ t g (h2 – h1)} (3.8) where g is the acceleration due to gravity. The principle of conservation of energy gives work done = gain in kinetic energy + gain in potential energy. A V (p 1 – p 2) δ t = {½ ρAV (V22 – V12 ) δt } + { g ρ V A (h2 – h1) δt } pI – p 2 = { ½ ρ V22 } – { ½ ρ V12 } + { g ρ h2 } – { g ρ h1} p 1 + ½ ρ V12 + ρ g h 1 = p 2 + ½ ρ V22 + ρ g h 2 (3.9) 22. Again, since stations 1 and 2 are arbitrarily chosen, it follows that the expression p+½ρ 2 V + ρ g h has the same value at all stations along the stream filament. If we now consider the limiting case when the filament shrinks to a single streamline, we have: p + ½ ρ V2 + ρgh = constant, along a streamline. (3.10) 23. This equation, which is an energy equation, is known as Bernoulli's Equation. When considering the problem of an aircraft, or part of an aircraft moving through the atmosphere, the changes in the value of ρgh within the field of flow affected by the presence of the aircraft are generally very small compared with the changes in the values of the other two parameters in the equation. It is therefore a valid and useful approximation, in such problems, to write p + ½ ρ V2 = constant, along a streamline. 24. Further, at some distance upstream of the aircraft, the flow consists of a uniform stream. It follows that on any given streamline in this region the value of p + ½ ρV2 is the same as it is on any other streamline. Thus the value of this expression is constant, not simply along a given streamline, but throughout the whole field of flow. Bernoulli's Equation is then written as p + ½ ρ V2 = constant (3.11) and applies throughout the field. 25. Another simplified way of arriving at the expression of equation (3.11) is by assuming that in low subsonic flow, changes in potential energy and heat energy are insignificant and that there is no transfer of heat or work. For practical purposes therefore, in the streamline flow of air round a wing at low speed: Pressure energy + kinetic energy = constant  39 Theories of Development of AD Forces This simplified law can be expressed in terms of pressure, thus: p + ½ ρV2 = constant, Where: p = Static pressure ρ = Density and V = Flow velocity The significance of this law will be recognized if it is translated into words: static pressure + dynamic pressure is a constant. This constant is referred to as Total Head Pressure, Stagnation Pressure or Pitot Pressure. 26. One of the most interesting examples of Bernoulli’s theorem is provided by the venturi tube (Fig 3-6). In the diagram, at point A, B & C the mass flow remains constant, or in other words ρ 0 A0 V0 = ρ 1 A1 V1 = ρ 2 A2 V2 Or ρAV = K (3.12) 27. Before reaching the constriction, the flow has a certain velocity and pressure. As soon as the flow reaches the constriction or the venturi, the velocity increases and pressure decreases which can be marked by the streamlines becoming closer, and again past the venturi the flow pressure increases and the velocity decreases marked by the streamlines becoming further apart. This phenomenon is of great Fig 3-6: Flow Through a Venturi Tube importance later on in the development of an aerofoil. 28. It has already been stated that the flow velocity is governed by the shape of the aircraft. From Bernoulli’s Theorem (simplified) it is evident that an increase in velocity causes a decrease in static pressure and vice versa. 29. Velocity Indication. As the air flows round the aircraft its speed changes. In subsonic flow a reduction in the velocity of the streamline flow is indicated by an increased spacing of the streamlines whilst increasing velocity is indicated by decreased spacing of the streamlines. Associated with the velocity changes there will be corresponding pressure changes. 30. Pressure Differential. As the air flows towards an aerofoil it will be turned towards the low pressure (partial vacuum) at the upper surface, this being termed as ‘upwash’. After passing over the aerofoil the airflow returns to its original position and state. This is termed as ‘downwash’ and is FIS Book 1: Aerodynamics 40 shown in Fig 3-7. The reason for the pressure and velocity changes around an aerofoil is explained by the Bernoulli’s Theorem. The difference in pressure between the upper and lower surface of an aerofoil is usually expressed as relative pressure by ‘–’ and ‘+’. However, the pressure Fig 3-7: Two-dimensional Flow around an Aerofoil above is usually a lot lower than ambient pressure and the pressure below is usually slightly lower than ambient pressure (except at high angles of attack), i.e. both negative. Static, Total and Dynamic Pressure 31. It follows from Bernoulli's Equation that, in a given field of flow, any increase in velocity is accompanied by a reduction in pressure, and vice versa. The pressure is lower at points in the flow where the speed is higher. The greatest pressure is achieved when the speed is zero, i.e., at stagnation points, and the pressure at such points is called the stagnation pressure, usually denoted by po. The pressure and velocity at any given point in the flow are then related by the equation p + ½ ρV2 = constant = po. (3.13) 32. This shows that the stagnation pressure is a constant, i.e., that the pressure is the same at all stagnation points in a given flow, and further that the concept of stagnation pressure has significance even in a flow where there are no actual stagnation points. For this reason, it is often referred t