Subsonic Aerodynamics PDF

Subsonic Aerodynamics Basics, Laws and Definitions Bernoulli's equation: Total Pressure Constant = Dynamic Pressure (½ ρ V2) + Static Pressure The law of conservation of mass states that for any system closed, the mass of the system must remain constant over time. The mass can neither be created n...

Subsonic Aerodynamics Basics, Laws and Definitions Bernoulli's equation: Total Pressure Constant = Dynamic Pressure (½ ρ V2) + Static Pressure The law of conservation of mass states that for any system closed, the mass of the system must remain constant over time. The mass can neither be created nor destroyed, although it may be rearranged in space. Bernoulli's theorem states that: In a streamlined flow of fluid, the sum of all energies is a constant The sum of static and dynamic pressure is constant. Static pressure plus dynamic pressure is constant. The sum of pressure energy and dynamic energy is constant. Dynamic Pressure is egual to (½ ρ V2) q = (½ ρ V2) Bernoulli's equation can be written as PT = PS + q Bernoulli's equation can be written as PT - q = PS Bernoulli's equation can be written as PSTAT + ½rho × TAS2 = constant The dynamic pressure is zero when velocity is zero. The dynamic pressure increases as static pressure decreases. If temperature of a gas is kept constant and pressure increases, the density increase. If density is kept constant, the dynamic pressure increases proportionally with the square of the velocity. (½ ρ V2) Given: p = pressure rho = density T = absolute temperature p / (rho * T) = constant Considering subsonic incompressible airflow through a Venturi Undisturbed Throat Undisturbed The static pressure in the undisturbed airflow is higher than in the throat. The static pressure in the throat is lower than in the undisturbed airflow The dynamic pressure in the throat is higher than in the undisturbed airflow The dynamic pressure in the undisturbed airflow is lower than in the throat. The speed in the undisturbed airflow is lower than in the throat. The speed of the airflow in the throat is higher than in the undisturbed airflow. The total pressure in the undisturbed airflow and in the throat is the same. In the throat of a venturi the static pressure is lower and the speed is higher compared to the undisturbed airflow. In the throat of a venturi the dynamic pressure is higher and the total pressure is the same compared to the undisturbed airflow. In the undisturbed airflow the dynamic pressure is lower and the total pressure is the same compared to the throat. In the undisturbed airflow the static pressure is higher and the speed is lower compared to the throat. In a convergent tube with an incompressible sub-sonic airflow, PS decreases, PDYN increases, PTOT remains constant. The (subsonic) static pressure decreases in a flow in a tube when the diameter decreases. If the cross-sectional area of a tube decreases, the speed of the flow increase If the cross-sectional area of a tube increases, the speed of the flow decrease A subsonic incompressible flow, as air flows through a tube of increasing cross-sectional area the air density does not vary. (only pressure vary) If the continuity equation is applicable: if the cross sectional area of a tube changes (low speed, subsonic and incompressible flow) the air density (rho) is rho1 = rho2. (density entering is egual to density leaving a venturi, density is constant) Density of the atmosphere will decrease with increase in humidity. (more humidity, less air) An increase in the flow's temperature will decrease the mass flow. Static pressure acts in all directions. The lift and drag forces, acting on an Aerofoil cross section depend on the pressure distribution around the aerofoil cross section. In a fluid disregard horizontal plane (Lift always perpendicular to undisturbed airflow) Lift is the component of the total aerodynamic force, perpendicular to the undisturbed airflow. Lift is the force that act at right angle to relative airflow  Weight is a force. (weight = m x g10) Wing loading is the ratio of aircraft weight to wing area. Drag is acting in the direction of relative wind (airflow) and lift is perpendicular to the relative wind (airflow). The angle of attack is the angle between wing chord line and direction of the relative airflow. angle between chord line and the relative undisturbed airflow. angle between chord line and the undisturbed airflow angle between chord line and the free stream The aeroplane angle of attack is the angle between wing root chord line and longitudinal axis An aeroplane's angle of incidence is the angle between its speed vector and longitudinal axis. Relationship between force (F), acceleration (a) and mass (m) is F = m × a Weight is a Force (mass x g acceleration) The weight of an object depends on the acceleration due to gravity. The mass of an object is independent of the acceleration due to gravity. The weight of a body can be determined by multiplying its mass by the acceleration due to gravity. F = m × a The mass of a body can be determined by dividing its weight by the acceleration due to gravity. The units of wing loading are (W/S) N/m2 and dynamic pressure (q) N/m2 Dihedral of a wing is the angle between the wing plane and the horizontal with the aeroplane in an unbanked, level condition. Dihedral of the wing is the angle between the 0.25 chord line of the wing and the lateral axis. The construction feature of a wing called "wash out" (wing twist) is a decrease in the angle of incidence from root to tip. Geometric washout (wing twist) means that the tip of the wing has TWIST/WASH OUT lower angle of attack than the root. The wings are normally with a 4° angle of incidence, but angle is different from tip to root by twisting the wing down near the root. Taper Ratio = Tip / Root A wing is said to be tapered if the chord at the wing tip is less than the chord at the root. Taper ratio is the ratio of the tip chord to the root chord. Aspect Ratio = wingspan ÷ mean chord Aspect Ratio = (wingspan)2 ÷ wing area The aspect ratio of the wing is the ratio between wingspan and mean geometric chord (MAC). Aspect ratio of a wing is the ratio between wingspan squared and wing area. The Mean geometric chord is the wing area divided by the wingspan. The Mean Aerodynamic/geometric Chord (MAC) is the chord of an equivalent untwisted, rectangular wing with the same pitching moment and lift characteristics as the actual wing. Wing sweep angle is the angle between the quarter-chord line of the wing and the lateral axis. (misured at ¼ cord) A wing would be said to be swept back if the quarter chord line was inclined backwards from the lateral axis.  The thickness to chord ratio of an aerofoil is the ratio of the maximum thickness of an aerofoil section to its chord. thickness ÷ chord The relative thickness of an aerofoil is expressed in % chord. The chord line of an aerofoil section is the line, drawn from the leading edge to the trailing edge. In a symmetrical Aerofoil the mean camber line is a straight line co- incident with the chord line. An aerofoil is cambered when the line, which connects the centres of all inscribed circles, is curved. Camber of an aerofoil section is the largest distance between the chord line and the camber line. The mean camber line of an aerofoil section is a line from the leading to the trailing edge equidistant from the upper and lower surfaces. The camber line is a line connecting the leading and trailing edge midway between the upper and lower surface of a aerofoil Indicated airspeed (IAS) = the direct instrument reading obtained from the ASI Calibrated airspeed (CAS) = IAS corrected for position (pressure) and instrument errors Equivalent airspeed (EAS) = the speed at sea level, is CAS corrected for compressibility error True airspeed (TAS) = EAS corrected from Density error Summary Errors correction: IAS CAS = Instrument | Position EAS = CAS | Compressibility TAS = EAS | Density Temp IP CD Temp Eat Chicken Tomato Marsala The true airspeed (TAS) is lower than the indicated airspeed (IAS) at altitudes below sea level, under ISA conditions. (in red sea there is a depression) At a constant TAS the dynamic pressure will be greater at sea level than at high altitude. Two-dimensional Airflow Around an Aerofoil Lift theory Fluid dynamic: The lift force, acting on an aerofoil is mainly caused by reduced pressure on the upper side of the aerofoil. (suction) Lift theory Mechanical: The Newton 3 law principal of action and reaction say that lift in created from an Aerofoil with an angle of attack. The air is deflected downward by the action (angle) of the Aerofoil, and in reaction the wing is pushed upward. Lift is generated when: the flow direction of a certain mass of air is changed. Of the total lift produced by the wing the upper surface produces the greater proportion at all speeds. Typically the CL/CD ratio is maximum at an angle of attack of 4°. (CL/CD ratio = Lift/Drag at AoA of 4° there is max lift) The profile drag of a wing section or blade element is proportional to: the square of the relative velocity of the air. V2 the drag coefficient. CD the air density. ½ ρ The total drag of an aerofoil in two dimensional flow comprises form/profile/Pressure drag and skin friction drag. The total drag of an aerofoil in three dimensional flow comprises form/profile/Pressure drag, skin friction drag and induced drag Induced drag, vortices in three dimensional flow A flat plate, when positioned in the airflow at a small angle of attack, will produce both lift and drag. Every object in airflow produce some lift and drag  As the air flows over the upper surface of an aerofoil its speed increases, static pressure decreases, total pressure is constant. In an aerofoil with a positive camber and a positive angle of attack the highest flow velocities occur in the upper side With increasing downwash, lift generated by the aerofoil increases. An aerofoil which is producing lift will have upwash ahead of the wing and downwash behind it. Stagnation point is where flow is separated by the presence of an object and speed is zero. Stagnation point is the point where the velocity of the relative airflow is reduced to zero. Consider a flat plate with zero angle of attack. The boundary layer is the region in which the velocity varies between the speed of the undisturbed airflow and zero. At the stagnation point the static pressure reaches a maximum value. as the angle of attack decreases The point of lowest static pressure moves aft. The stagnation point moves up. Moves forward as the angle of attack increases The point of lowest static pressure moves forward. The stagnation point moves down. Moves backward As the angle of attack increases, the stagnation point on the aerofoil's profile moves downwards on the profiles leading edge. Static pressure and stagnation point move along the surface of the wing according to AoA The speed at stagnation point is zero and in upper part is faster: V1 = 0 and V2 > V  The aerodynamic centre is the point which the pitching moment coefficient does not change with varying angle of attack. The location of the aerodynamic centre of an aerofoil section is at approximately 25% chord irrespective of angle of attack. The point, where the single resultant aerodynamic force acts on an aerofoil, is called centre of pressure. The point, where the aerodynamic lift acts on a wing is the centre of pressure. The Center of Pressure of an aerofoil section is the point where the resultant aerodynamic force is applied. The Center of Pressure is the point on the chord line through which the resultant of all aerodynamic forces acts. When angle of attack is decreased the centre of pressure will move backward When speed is increased in straight and level flight (lift constant) the magnitude of the total lift force remain the same centre of pressure move aft When angle of attack is increased the centre of pressure will move forward When speed is decreased in straight and level flight (lift constant) centre of pressure move forward the magnitude of the total lift force remain constant The location of the centre of pressure of a positively cambered aerofoil section at increasing angle of attack will shift forward until approaching the critical angle of attack. In a symmetric Aerofoil center of pressure & aerodynamic centre are at approximately 25% chord irrespective of angle of attack. They doesn’t change with angle of attack Body B body will have the highest pressure drag/form drag Body C body will have the lowest pressure drag/form drag (the wind come from left, B have more pressure drag because the back is not rounded and create more drag) Order of decreasing drag is B, A, D, C Order of increasing drag is C, D, A, B  If in a flow the streamlines converge, the static pressure in the flow will decrease and velocity will increase The static pressure decreases as the streamlines converge. The velocity increases as the streamlines converge. Ice is most likely to form on the leading edge of the wing where the stagnation point is located. Coefficients V2=TAS2 |q d pressure| Area | Chamber/AoA LIFT = (½ρ V2) S CL DRAG = (½ρ V2) S CD The terms q and S in the lift formula are dynamic pressure and the area of the wing. CL varies with angle of attack The correct drag formula is DRAG = CD × ½ RHO × V2 × S If the lift generated by a given wing is 1 000 kN, if the wing area is doubled lift is doubled 2000 kM Lift is a function of velocity, wing area, CL and density. At a given TAS, increase of air density Lift and drag will increase. For a given angle of attack the lift/drag ratio is unaffected by density changes. (density act egually on Clift/Cdrag ratio LIFT = +(½ρ V2) S CL / DRAG = +(½ρ V2) S CD The frontal area of a body, placed in a certain airstream is increased by a factor 3. If the shape does not alter, the form drag will increase by a factor of 3 ( if density, speed or area change, the Lift or Drag change, but L/D ratio is always the same +LIFT = (½ρ V2) S CL/+DRAG = (½ρ V2) S CD) Comparing the lift coefficient and drag coefficient for conventional aeroplanes at normal angle of attack CL is much greater than CD. (airplane is made to have more lift than drag at 4° AoA) If the wing area is increased, lift will increase because it is directly proportional to wing area. A body is placed in a certain airstream. If the density of the airstream decreases to half (/2) its original value, the parasite drag will decrease by a factor of 2 ( if density is halved q2=(½ρ)/2 means that Drag is /2  (½ρ/2 V2 S CD) Regarding the lift formula, if density doubles, lift will also double. 2 x (½ρ) = LIFT2 Regarding the lift formula, if airspeed doubles, lift will be 4 times greater. 2 x (V2) = LIFT4 Both lift and drag of an aerofoil are proportional to the square of the velocity of the relative airflow. Variables required to calculate drag using the drag formula are dynamic pressure, drag coefficient and wing area. DRAG = (½ρ V2) S CD Variables are required to calculate lift from the lift formula Dynamic pressure, lift coefficient and wing area. LIFT = (½ρ V2) S CL Minimum drag of an aeroplane in straight and level flight occurs at the maximum CL / CD ratio. At the highest value of the lift/drag ratio the total drag is lowest. (Maximum Lift respect Drag means drag is less) The lift coefficient (CL) is a function of Camber of the aerofoil section. Angle of attack of the aerofoil section. For aircraft of the same weight, flying at the same IAS the angle of attack will be the same at altitude as at sea level. (IAS = dynamic pressure, if pressure is constant, angle is constant) If an aeroplane maintains straight and level flight at the same angle of attack at two different altitudes, the TAS is higher at the higher altitude. If TAS is kept constant, to maintain straight and level flight with reduced air density the angle of attack of an aircraft's wings must be increased. Increasing air pressure (angle of attack, OAT and TAS are constant) the drag increases. +(½ρ V2) Increasing air density (angle of attack and TAS are constant) the drag increases. +(½ρ) The lift coefficient of an aerofoil section increases with an increase in angle of attack up to the stall. In a positively cambered aerofoil section the angle of attack has a negative value when the lift coefficient equals zero. There is a nose down pitching moment about a positively cambered aerofoil when the lift coefficient equals zero. Considering a positively cambered aerofoil section, the pitching moment when the lift coefficient CL=