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AOE Aerospace and Ocean Engineering Fall 2009 Virginia Tech1 September 2009Lecture 2 AOE 2104 Introduction to Aerospace Engineering Lecture 2 Basic Aerodynamics

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Virginia Tech Reminder: The first homework assignment (paper copy) is due AT THE BEGINNING OF NEXT CLASS!! Also I would appreciate any feedback on the class that you have. You are welcome to see me after class, tell me during class, or send me an . AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech 3 steps to determine p, , and T at any altitude ? 2 equations used to construct the standard atmosphere model ? Name and define the different types of altitudes. 2 types of regions found in the temperature variations with altitude and their characteristics ? Any questions ? Standard Atmosphere AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Basic Aerodynamics AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Basic Aero – Why? How? What do we have so far? Why are we looking into aerodynamics? To determine the forces acting on a vehicle in flight Remember aerodynamic forces arise from two natural phenomena How are we going to proceed ? Using Laws of Physics to quantify the interaction between the vehicle and the environment it is evolving in. What do we have so far ? AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Our Aerodynamic Tool Box Four aerodynamic quantities that define a flow field Steady vs unsteady flow Streamlines Sources of aerodynamic forces Equation of state for perfect gases Hydrostatic Equation Standard Atmosphere Model 6 different altitudes AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Aerodynamic Tools Needed: Governing Laws We are going to need the 3 following physical principles to describe the interaction between the vehicle and its associated flow field: Conservation of Mass Continuity Equation (§§ ) Newton’s 2nd Law (and Conservation of Momentum) Euler’s and Bernoulli’s Equations (§§ ) Conservation of Energy Energy Equation (§§ ) AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Conservation of Mass – The Continuity Equation Physical Principle: Mass can neither be created nor destroyed (in other words, input = output). Eq.(4.2) AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Streamline A streamline is a line that is tangent to the local velocity vector. If the flow is steady, the streamline is the path that a particle follows. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Remarks on Continuity The equation we just derived assumes that both velocities and densities are uniform across areas 1 and 2. In reality, both velocities and densities will vary across the area Continuity Equation is extensively used in the design and operation of wind tunnels and rocket nozzles (we will see how later). A stream tube is delimited by 2 streamlines and does not have to be bounded by a solid wall. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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AOE Aerospace and Ocean Engineering Fall 2007 Virginia Tech Compressible Versus Incompressible Flows AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Continuity for Incompressible Flows All fluids are compressible in reality. However, many flows are “incompressible enough” so that the incompressibility assumption holds. Incompressibility is an excellent model for Flows of liquids (e.g. water and oil) Air at low speed (V < 100 m/s or 225 mi/h) Equation of Continuity for Incompressible Flows reduces to So that if A 2 V 1. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Continuity – Sample Problem 1 A convergent duct was found in the basement of Randolph. The inlet and exit areas are measured to be A i = 5m 2 and A e = 2m 2. Assuming we use this duct with an inlet velocity of V i = 9 mi/h, find the exit velocity. First, we need to be consistent with the unit system. Let’s work in SI units. V i = 9 mi/h = 9x1609/3600 m/s V i = 4 m/s. V i << 100 m/s so the flow is considered incompressible. From Incompressible Continuity, Therefore, the exit velocity will be 10 m/s. First, we need to be consistent with the unit system. Let’s work in SI units. V i = 9 mi/h = 9x1609/3600 m/s V i = 4 m/s. V i << 100 m/s so the flow is considered incompressible. From Incompressible Continuity, Therefore, the exit velocity will be 10 m/s. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Continuity – Sample Problem 2 AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Momentum Equation Continuity is a great addition to our toolbox, however it says nothing about pressure. Why is pressure important? Let’s look at Newton’s 2 nd Law: Sum of the forces =Time rate of change of momentum F=d(mv)/dt F=m dV/dt assuming m = const. F=m a The pressure is going to translate into force, which by Newton’s 2 nd Law results in change of momentum. Assuming incompressibility (m = const), this will result in change of velocity (thus impacting performance for example). To find momentum, simply apply F = ma to an infinitesimally small fluid element moving along a streamline. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Assume fluid element is moving in the x-direction. 3 types of force act on the element: Pressure force (normal to the surface) p Shear stress (friction, parallel to the surface) w Gravity dxdydz g Ignore gravity (smaller than other forces) and assume inviscid flow (non-viscous i.e. no friction), balance of the forces on x. v O Streamline pF = ma dx dz dy Momentum Equation – Free Body Diagram AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Momentum Equation – Force Balance AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Momentum for Incompressible Flows – Bernoulli’s Equation For incompressible flows, = const. Integrating Euler’s equation between 2 points along a streamline gives: This equation is known as Bernoulli’s Equation. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Description of Bernoulli’s Equation Static Pressure Pressure felt by an object or person suspended in the fluid and moving with it. Can be thought of as internal energy. Dynamic Pressure Pressure due to the fluid motion. Can be thought of as kinetic energy. Total (stagnation) Pressure Pressure that would be felt if the fluid was brought isentropically to a stop. Can be thought of as total energy. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech 3 New Tools – Continuity, Euler, and Bernoulli’s Equations Continuity Equation A V = const Assumptions: steady flow. Euler’s Equation dp = - V dV Assumptions: steady, inviscid flow. Bernoulli’s Equation Assumptions: steady, inviscid, incompressible flow along a streamline. Euler and Bernoulli’s equations are essentially applications of Newton’s 2 nd Law to fluid dynamics. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Momentum Equations - Sample Problem 1 AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Momentum Equations - Sample Problem 2 AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Practical Applications By combining Continuity, Euler, and Bernoulli’s equation, one can obtain the velocity at any point on an aircraft assuming surrounding conditions are known (either through measurements or using Standard Atmosphere). Two major applications for this: Low-Speed Subsonic Wind Tunnel testing/designing Flight measurements of velocity AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Low-Speed Subsonic Wind Tunnels (§4.10) AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Wind Tunnel Calculations From Bernoulli, between points 1 and 2: Using Continuity: Combining the two, we get: Since the ratio of throat to reservoir area (A 2 /A 1 ) is fixed for wind tunnel and is constant for low-speed (incompressible) flows, the quantity driving the tunnel is p 1 -p 2. But how can we determine p 1 -p 2 ??? AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Manometer AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Wind Tunnels – Sample Problem 1 AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Wind Tunnels – Sample Problem 1 Solution Height of liquid: h = 10cm = 0.1m Specific weight of liquid mercury: w = (1.36x10 4 )x9.8 = 1.33x10 5 N/m 2 Actual pressure difference: p 1 -p 2 = w h = 1.33x10 4 N/m 2. To find V 2 from Bernoulli, use We computed p 1 -p 2, A 1 /A 2 = 15 is given, so we need to find . Since we are in a low-speed wind tunnel, flow is incompressible, so = const, which means we can compute it at any point in the tunnel. Since p 1 and T 1 are given, use Equation of State to find = 1 : Combining all the results we get V 2 = 144 m/s (slightly over the incompressible velocity limit, which means compressibility effects should be taken into account). Height of liquid: h = 10cm = 0.1m Specific weight of liquid mercury: w = (1.36x10 4 )x9.8 = 1.33x10 5 N/m 2 Actual pressure difference: p 1 -p 2 = w h = 1.33x10 4 N/m 2. To find V 2 from Bernoulli, use We computed p 1 -p 2, A 1 /A 2 = 15 is given, so we need to find . Since we are in a low-speed wind tunnel, flow is incompressible, so = const, which means we can compute it at any point in the tunnel. Since p 1 and T 1 are given, use Equation of State to find = 1 : Combining all the results we get V 2 = 144 m/s (slightly over the incompressible velocity limit, which means compressibility effects should be taken into account). AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Measurement of Airspeed (§4.11) Bernoulli’s equation provides an easy method for determining the velocity of any fluid Therefore, we need to know p and p 0 AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Total (stagnation) Pressure (p 0 ) Measurement The total pressure is easy to measure if the flow direction is known. An opened- end tube aligned with the flow direction is enough. This type of tube is called "Pitot probe” (named after Henri Pitot who invented it in 1732; see §4.3 for historical background) AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Static Pressure (P) Measurement The static pressure is also easy to measure using a tube with a close end and pressure taps around its circumference. “Static probe” AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Dynamic Pressure Measurement Finally, it is possible to measure directly the difference between stagnation and static pressure by combining the Pitot and static probes into a Pitot-static probe (!). “Pitot-Static probe” AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Airspeed Indicator If the only known density is at sea level, “Indicated or Equivalent Airspeed” AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech True Airspeed Therefore, the relationship between true and indicated airspeed is: and AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech aero/instruments/ aero/instruments/ aeml/airframeimages/pitottube.jpg AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech aero/instruments/ park/tp/image/seventh/s-port.jpg AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech Measurement of Airspeed – Sample Problem AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech From Standard Atmosphere (App. B), at 5000ft, p = 1761 lb/ft 2. Pitot tube measures stagnation pressure so p 0 = 1818 lb/ft 2. Density is found from measured temperature and tabulated pressure = p/(RT) = 1761/(1716*505) = 2.03x10 -3 slug/ft 3. 7.6% difference Measurement of Airspeed – Sample Problem Solution AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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Virginia Tech For Next Class: Review Chapter 4 and let me know what questions you have Thursday: HW 1 due. Stay Tuned for HW 2. AOE Aerospace and Ocean Engineering Fall September 2009Lecture 2

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