Chapter 3 Basic Aerodynamic Theory PDF

CHAPTER 3 Basic Aerodynamic Theory For Test Chapter 3 1 Outline Measuring the Dynamic Pressure The Principle of Continuity Bernoulli’s Theorem Streamlines and the Streamtube Summary Chapter 3 2 ...

CHAPTER 3 Basic Aerodynamic Theory For Test Chapter 3 1 Outline Measuring the Dynamic Pressure The Principle of Continuity Bernoulli’s Theorem Streamlines and the Streamtube Summary Chapter 3 2  Aerodynamics  Aerodynamics is the study of power of the air both when it is at rest and in motion.  The speed of sound at sea level, in ISA conditions, is approximately 660 knots. (The speed of the air flowing around a training aircraft in the cruise is subsonic: typically, in the order of 100kts.) Chapter 3 3  Dynamic Pressure Measuring Dynamic Pressure  All aerodynamic forces acting on an aircraft are determined by dynamic pressure, so it is essential to have some means of measuring dynamic pressure and presenting that information to the pilot.  For that purpose, a sealed tube(Pitot tube) , open at the forward end, is located where it will collect air when the aircraft is moving. The pressure in the tube is (Dynamic + Static) and, is called “Pitot” pressure (because the air passes through the Pitot tube). Chapter 3 4  Dynamic Pressure  We must find some way of ‘removing’ the static pressure from the pitot pressure. A hole (vent) in a surface parallel to the airflow will sense static pressure. Referring to the figure, if the pressure from the pitot tube is fed to one side of a diaphragm mounted in a sealed case, and static pressure is fed to the other side, the two static pressures will cancel each other, and the diaphragm movement will be influenced only by changes in dynamic pressure. Chapter 3 5  Dynamic Pressure  Movement of the diaphragm moves a pointer over a scale so that changes in dynamic pressure can be observed by the flight crew.  But the instrument is calibrated at ISA sea level density, so the instrument will only give a ‘true’ indication of the speed of the aircraft through the air when the air density is 1.225 kg/m3. Chapter 3 6  Indicated Air Speed  Let us assume that a certain aircraft needs to achieve a speed of 60 knots, in order to get airborne. As the aircraft accelerates, the Dynamic Pressure acting on it will increase because of the increasing airspeed. ( Dynamic Pressure = ½ ρ V² )  The pilot reads the Dynamic Pressure as an airspeed on his Airspeed Indicator not the true Airspeed (TAS).  Indicated Airspeed = True Airspeed in ISA sea level conditions only. Chapter 3 7 Relationships between Airspeeds Indicated Airspeed (IAS) The speed registered on the Airspeed Indicator. Calibrated Airspeed (CAS) An accurate measure of dynamic pressure when the aircraft is flying slowly. The position of the pitot tube(s) and static vent(s), together with the aircraft’s configuration (flaps, landing gear etc.) and attitude to the airflow (angle of attack and sideslip) will affect the pressures sensed, particularly the pressures sensed at the static vent(s). Under the influence of the above conditions a false dynamic pressure (IAS) will be displayed. When IAS is corrected for this ‘position’ or ‘pressure’ error, as it is called, the resultant is Calibrated Airspeed. (The airspeed corrections to be applied may be displayed on a placard on the flight-deck, or in the Flight Manual, and will include any instrument error). 8 Relationships between Airspeeds Equivalent Airspeed (EAS) An accurate measure of dynamic pressure when the aircraft is flying fast. Air entering the pitot tube(s) is compressed, which gives a false dynamic pressure (IAS) reading, but only becomes significant at higher speeds. At a given air density, the amount of compression depends on the speed of the aircraft through the air. When the IAS is corrected for ‘position’ AND ‘compressibility’ error, the resultant is Equivalent Airspeed. True Airspeed (TAS) or (V) The speed of the aircraft through the air. THE ONLY SPEED - All the other, so called, speeds are pressures. 𝑬𝑨𝑺 𝜌𝑠𝑒𝑎 𝑙𝑒𝑣𝑒𝑙 𝜌 𝑻𝑨𝑺 = 𝝈 = 𝑬𝑨𝑺 𝒙 𝜌 Where, 𝜎 = 𝜌 is the Relative Density 𝑠𝑒𝑎 𝑙𝑒𝑣𝑒𝑙 The Airspeed Indicator is calibrated for ‘standard’ sea level density, so it will only read TAS if the density of the air through which the aircraft is flying is 1.225 kg/m3. Thus at 40 000 ft. where the ‘standard’ density is one quarter of the sea-level value, to maintain the same EAS the aircraft will have to move through the air twice as fast! 9 Relationships between Airspeeds The Speed of Sound (a) Sound is ‘weak’ pressure waves which propagate spherically through the atmosphere from their source. The speed at which pressure waves propagate is proportional to the square root of the absolute temperature of the air. The lower the temperature, the lower the speed of propagation. On a ‘standard’ day at sea level the speed of sound is approximately 340 m/s (660 knots TAS). At higher aircraft True Airspeeds (TAS) and/or higher altitudes, it is essential to know the speed of the aircraft in relation to the local speed of sound. This speed relationship is known as the Mach Number (M). 𝑇𝐴𝑆 𝑀= Where (a) is the Local Speed of Sound (𝑎) Critical Mach Number (MCRIT) It is the Mach number of the aircraft when the speed of the airflow over some part of the aircraft (usually the point of maximum thickness on the airfoil) first reaches the speed of sound. 10  Airspeed Summary  The airspeed indicator is really a pressure gauge, the ‘needle’ of which responds to changes in dynamic pressure (½ ρ V2 ).  Calibration of the airspeed indicator is based on standard sea level density (1.225 kg/m3). The “airspeed” recorded will be different from the actual speed of the aircraft through the air unless operating under standard sea level conditions (unlikely). The actual speed of the aircraft relative to the free stream is called true airspeed (TAS) and denoted by (V). The ‘speed’ recorded by the airspeed indicator calibrated as above, if there are no other errors, is called equivalent airspeed (EAS), when the aircraft is flying fast.  It may seem to be a drawback that the instrument records equivalent rather than true airspeed, but the true airspeed may always be determined from it. Also, many of the handling characteristics of an aircraft depend mainly on the dynamic pressure, i.e., on the equivalent airspeed, so it is often more useful to have a direct reading of EAS than TAS. Chapter 3 11  The Principle of Continuity  One of the fundamental laws of the universe is ENERGY and MASS can neither be created nor destroyed, only changed from one form to another.  To demonstrate the effect of this basic Principle of Continuity on aerodynamic theory, it is useful to study a streamline flow of air through a tube which has a reduced cross-sectional area in the middle.  The air mass flow (= mass per unit time), through the tube will be: cross-sectional area (A) x the airflow velocity (V) x the air density (ρ)  Mass flow will remain a constant value at all points along the tube.  The Equation of Continuity is: 𝑨 𝒙 𝑽 𝒙 𝝆 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 Chapter 3 12 The Principle of Continuity 𝑨 𝒙 𝑽 𝒙 𝝆 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 Chapter 3 13  The Principle of Continuity  Because air is a compressible fluid, any pressure change in the flow will affect the air density. However, at low subsonic speeds (< M 0.4) density changes will be insignificant and can be disregarded. The equation of continuity can now be simplified to: A × V = constant, or: Velocity (V) = Constant / Area (A)  Because the mass flow must remain constant, it can be seen from the equation of continuity that the reduction in the tube’s cross-sectional area results in an increase in velocity, and vice versa.  The equation of continuity enables the velocity changes of airflow around a given shape to be predicted mathematically, (< M 0.4). 14  The Principle of Continuity A Venturi tube has a reduced cross-sectional area in the middle. At opening of Venturi: Mass Flow = 1 m² x 52 m/s x 1 kg/m³ = 52 kg/s = constant An increase in area will reduce velocity. As the Venturi widens out again, the velocity of the air decreases because, while mass flow remains constant, the cross-sectional area has increased to 1 m². 15  Bernoulli’s Theorem  “In the steady flow of an ideal fluid the sum of the static pressure and the dynamic pressure remains constant”.  Note: An ideal fluid is both incompressible and has no viscosity.  This statement can be expressed as: Static pressure + Dynamic pressure = Constant or: p + 1/2 ρ V2 = Constant  Consider a mass of air:  Static Pressure 101 325 N/m2  Density 1.225 kg/m3  Velocity 52 m/s  Therefore, dynamic pressure Q = ½ × 1.225 × (52)2 = 1656 N/m2  Static pressure (101 325 N/m2 ) + Dynamic pressure (1656 N/m2 ) = Constant (102 981 N/m2 ) Chapter 3 16 Bernoulli’s Theorem Chapter 3 17  Bernoulli’s Theorem  Because the velocity of air at the throat has doubled, its dynamic pressure has risen by a value of four, and the static pressure has decreased. The significant point is that:  Static Pressure + Dynamic Pressure is a constant. This constant is referred to either as: TOTAL PRESSURE, STAGNATION PRESSURE or PITOT PRESSURE  Flow velocity is dependent on the shape of the object over which it flows. And from Bernoulli’s theorem, it is evident that an increase in velocity will cause a decrease in static pressure, and vice versa. Chapter 3 18  Bernoulli’s Theorem Pressure Difference Between Top and Bottom of a Curved Surface The continuity Equation and Bernoulli’s Theorem holds true even when air is flowing over an unenclosed curved surface in the open atmosphere. Air flowing over such a curved surface (called airflow) will increase in velocity and its Static Pressure will decrease. When air flows around a wing, or airfoil, the airflow over the upper surface will increase in velocity causing the Static Pressure above the wing to decrease. Underneath the wing, the Static Pressure remains at a higher value. 19  Streamlines and the Streamtube  A streamline: the path traced by a particle of air in a steady airflow. Streamlines cannot cross. Close together streamlines = increased velocity.  Diverging streamlines = decelerating airflow + increasing pressure and Converging streamlines = accelerating airflow + decreasing pressure.  A streamtube : imaginary tube made of streamlines. There is no flow into or out of the streamtube through the “walls”, only a flow along the tube.  With this concept it is possible to visualize the airflow around an airfoil being within a tube made up of streamlines. Chapter 3 20  Summary At flow speeds of less than about M=0.4, pressure changes will not affect air density.  Continuity: Air accelerates when the cross-sectional area of a streamline flow is reduced. Air decelerates when the cross-sectional area increases again.  Bernoulli: If a streamline flow of air accelerates, its kinetic energy will increase, and its static pressure will decrease. When air decelerates, the kinetic energy will decrease, and the static pressure will increase again.  By coupling the principle of continuity and Bernoulli’s theorem an aerodynamic force can be generated. Chapter 3 21 Thank you 22