2 Lift Equation Lift Coefficient of Lift, Cl Direction of Flight Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameLift EquationLiftDirection of FlightCoefficient of Lift, ClDetermined experimentallyCombines several factorsShapeAngle of attack𝐶 𝑙 =𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝐿𝑖𝑓𝑡𝐷=𝐷𝑟𝑎𝑔 𝑁𝐶 𝑙 = 2𝐿 𝐴𝜌 𝑣 2𝐶 𝑙 = 𝐿 𝑞𝐴𝐴=𝑊𝑖𝑛𝑔 𝐴𝑟𝑒𝑎 𝑚 2Rearranging the coefficient of lift equation shows that lift is increased by wing area, air density, and velocity. Velocity is a squared function, giving it a more significant impact on lift.𝜌=𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑘𝑔 𝑚 3Alternate format𝑣=𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑚 𝑠𝑞=𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑃𝑎
3 Applying the Lift Equation Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameApplying the Lift EquationThe Cessna 172 from Activity step #2 takes off successfully from Denver, CO during an average day in May (22 OC) with a standard pressure (101.3 kPa). Assume that the take-off speed is 55 knots (102 kph). What is the minimum coefficient of lift needed at the point where the aircraft just lifts off the ground? The Cessna wing area is 18.2 m2 and weight is 2,328 lb (1,056 kg).Average temperature source =
4 Applying the Lift Equation Convert mass into weightConvert velocity𝑤=𝑚𝑔𝑤=(1,056 𝑘𝑔) 𝑚 𝑠 2𝑤=10,359 𝑁𝑉= 102 𝑘𝑝ℎ 𝑚 𝑘𝑚 𝑚𝑖𝑛 ℎ𝑟 𝑠 𝑚𝑖𝑛𝑉=28.3 𝑚 𝑠
7 Boundary Layer Fluid molecules stick to object’s surface Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameBoundary LayerFluid molecules stick to object’s surfaceCreates boundary layer of slower moving fluidBoundary layer is crucial to wing performanceMore information is available through the NASA Reynolds Number webpage:
8 Boundary Layer and Lift Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameBoundary Layer and LiftAirflow over object is slower close to object surfaceAir flow remains smooth until critical airflow velocityAirflow close to object becomes turbulent
9 Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameReynolds Number, ReRepresentative value to compare different fluid flow systemsObject moving through fluid disturbs moleculesMotion generates aerodynamic forcesAirfoil1Airfoil2More information is available through the NASA Reynolds Number webpage:ComparabletowhenRe1=Re2
10 Angle of Attack (AOA) Affects Lift Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameAngle of Attack (AOA) Affects LiftLift increases with AOA up to stall angleLiftDirection of FlightAirflowLiftDirection of FlightAirflowAirflow becomes turbulent at the critical angle of attack. Airflow separates from airfoil, and lift decreases dramatically. NASA developed an applet to show how the angle of attack impacts lift. It can be accessed through this link:StallLiftAngle of Attack
11 Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameReynolds NumberRatio of inertial (resistant to change) forces to viscous (sticky) forcesDimensionless number𝑅 𝑒 = 𝜌v𝑙 𝜇𝑅 𝑒 = v𝑙 νν= 𝜇 𝜌or𝑙=𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓𝐹𝑙𝑢𝑖𝑑 𝑇𝑟𝑎𝑣𝑒𝑙 𝑚𝑅 𝑒 =𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑁𝑢𝑚𝑏𝑒𝑟More information is available through the NASA Reynolds Number webpage:𝜌=𝐹𝑙𝑢𝑖𝑑 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑘𝑔 𝑚 3𝜇=𝐹𝑙𝑢𝑖𝑑 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑁𝑠 𝑚 2v=𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑚 𝑠ν=𝐾𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑚 2 𝑠
12 Applying Reynolds Number A P-3 Orion is cruising at 820 kph (509 mph) at an altitude of 4,023 m (13,198 ft). Assume a fluid viscosity coefficient of 1.65x10-5 N(s)/m3. What is the average Reynolds Number along a wing cross section measuring 1.1 m (3.6 ft) from leading edge to trailing edge? Need components to calculate Re𝑅 𝑒 = 𝜌v𝑙 𝜇
13 Applying Reynolds Number Calculate Air TemperatureCalculate Air Pressure𝑇=15.04℃− ℃ 𝑚 ℎ𝑇=15.04℃− ℃ 𝑚 (4,023 𝑚)𝑇=−11.1℃𝑝=101.29𝑘𝑃𝑎 −11.1℃+273.1℃ 𝐾 ℃ 𝐾𝑝=61.5 𝑘𝑃𝑎
17 Drag Equation Drag Coefficient of drag, Cd Direction of Flight Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameDrag EquationDragDirection of FlightCoefficient of drag, CdDetermined experimentallyCombines several factorsShapeAngle of attack𝐶 𝑑 =𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝐷𝑟𝑎𝑔𝐷=𝐷𝑟𝑎𝑔 𝑁𝐶 𝑑 = 2×𝐷 𝐴×𝜌× 𝑣 2𝐶 𝑑 = 𝐷 𝑞 ×𝐴𝐴=𝑊𝑖𝑛𝑔 𝐴𝑟𝑒𝑎 𝑚 2The area referenced with the coefficient of drag varies depending on what Cd is compared with. Drag typically refers to total surface area, frontal area, or wing area – all of these are proportional to each other. Our equation refers to wing area to more directly compare Cd to Cl. More information about reference area is available through NASA at𝜌=𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑘𝑔 𝑚 3Alternate format𝑣=𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑚 𝑠𝑞=𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑃𝑎
18 Coefficient of Drag (Cd) Object shape affects Cd
19 Applying the Drag Equation Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameApplying the Drag EquationThe same Cessna 172 from Activity step #2 takes off under the same conditions as described earlier in this presentation. How much drag is produced when the wing is configured such that the coefficient of drag is 0.05?Average temperature source =
21 Downwash and Wingtip Vortices Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameDownwash and Wingtip VorticesPressure difference at wing tipsAir to spill over wingtip perpendicular to main airflowAir flows both upward and rearward, forming a vortexDecreases liftIncreases dragThe pressure difference above and below the wing causes air flow that is perpendicular to the main airflow over the wing. This causes a flow that is both upward and rearward, causing the air to form a vortex. This can be seen on some aircraft at slow airspeeds in high humidity conditions (e.g., take off and landing).
22 Wingtip Vortices Air flows both upward and rearward, forming a vortex Presentation NameCourse NameUnit # – Lesson #.# – Lesson NameWingtip VorticesAir flows both upward and rearward, forming a vortexWinglets are vertical airfoils that limit vortices and improve fuel efficiencyMore information about winglets is available from NASA atThe aspect ratio is the square of the span, s, divided by the wing area, A. AR = s2 / AFor a rectangular wing, this reduces to the ratio of the span to the chord, c. AR = s / cLong, slender, high aspect ratio wings have lower induced drag than short, thick, low aspect ratio wings. Induced drag is a three dimensional effect related to the wing tips. The longer the wing, the farther the tips are from the main portion of the wing, and the lower the induced drag. Lifting line theory shows that the optimum (lowest) induced drag occurs for an elliptic distribution of lift from tip to tip. The efficiency factor, e, is equal to 1.0 for an elliptic distribution and is some value less than 1.0 for any other lift distribution.The outstanding aerodynamic performance of the British Spitfire of World War II is partially attributable to its elliptical wing, which gave the aircraft a very low amount of induced drag. A more typical value of e = .7 exists for a rectangular wing. The total drag coefficient Cd is equal to the base drag coefficient at zero lift Cd0 plus the induced drag coefficient Cdi.
23 ReferenceNational Aeronautics and Space Administration (2011). Aerodynamic forces. Retrieved fromNational Aeronautics and Space Administration (2011). Reynolds number. Retrieved fromNational Aeronautics and Space Administration (2011). Winglets. Retrieved fromRaymer, P. (2006). Aircraft design: A conceptual approach. Reston, VA: American Institute of Aeronautics and Astronautics.