AOE 2104--Aerospace and Ocean Engineering Fall 2009

1. AOE 2104--Aerospace and Ocean EngineeringFall 2009 AOE 2104 Introduction to Aerospace Engineering Lecture 2 Basic Aerodynamics Virginia Tech Lecture 2 1September 2009

2. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 email. Virginia Tech Lecture 2 1September 2009

3. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Standard Atmosphere Any questions ? 2 equations used to construct the standard atmosphere model ? 2 types of regions found in the temperature variations with altitude and their characteristics ? 3 steps to determine p, r, and T at any altitude ? Name and define the different types of altitudes. Virginia Tech Lecture 2 1September 2009

4. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Basic Aerodynamics Virginia Tech Lecture 2 1September 2009

5. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 ? Virginia Tech Lecture 2 1September 2009

6. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 Virginia Tech Lecture 2 1September 2009

7. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 (§§ 4.1-4.2) • Newton’s 2nd Law (and Conservation of Momentum) • Euler’s and Bernoulli’s Equations (§§ 4.3-4.4) • Conservation of Energy • Energy Equation (§§ 4.5-4.7) Virginia Tech Lecture 2 1September 2009

8. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Conservation of Mass – The Continuity Equation Physical Principle: Mass can neither be created nor destroyed (in other words, input = output). Eq.(4.2) Virginia Tech Lecture 2 1September 2009

9. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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. Virginia Tech Lecture 2 1September 2009

10. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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. Virginia Tech Lecture 2 1September 2009

11. AOE 2104--Aerospace and Ocean EngineeringFall 2007 AOE 2104--Aerospace and Ocean EngineeringFall 2009 Compressible Versus Incompressible Flows Virginia Tech Lecture 2 1September 2009

12. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 A2 < A1then V2 > V1. Virginia Tech Lecture 2 1September 2009

13. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Continuity – Sample Problem 1 A convergent duct was found in the basement of Randolph. The inlet and exit areas are measured to be Ai = 5m2 and Ae = 2m2. Assuming we use this duct with an inlet velocity of Vi = 9 mi/h, find the exit velocity. First, we need to be consistent with the unit system. Let’s work in SI units. Vi = 9 mi/h = 9x1609/3600 m/s  Vi = 4 m/s. Vi << 100 m/s so the flow is considered incompressible. From Incompressible Continuity, Therefore, the exit velocity will be 10 m/s. Virginia Tech Lecture 2 1September 2009

14. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Continuity – Sample Problem 2 Virginia Tech Lecture 2 1September 2009

15. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 2nd 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 2nd 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. Virginia Tech Lecture 2 1September 2009

16. Streamline v O dy p F = ma dz dx AOE 2104--Aerospace and Ocean EngineeringFall 2009 Momentum Equation – Free Body Diagram • 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) tw • Gravity r dxdydz g • Ignore gravity (smaller than other forces) and assume inviscid flow (non-viscous i.e. no friction), balance of the forces on x. Virginia Tech Lecture 2 1September 2009

17. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Momentum Equation – Force Balance Virginia Tech Lecture 2 1September 2009

18. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Momentum for Incompressible Flows – Bernoulli’s Equation • For incompressible flows, r= const. • Integrating Euler’s equation between 2 points along a streamline gives: • This equation is known as Bernoulli’s Equation. Virginia Tech Lecture 2 1September 2009

19. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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. Virginia Tech Lecture 2 1September 2009

20. AOE 2104--Aerospace and Ocean EngineeringFall 2009 3 New Tools – Continuity, Euler, and Bernoulli’s Equations • Continuity Equation • r A V = const • Assumptions:steady flow. • Euler’s Equation • dp = - r 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 2nd Law to fluid dynamics. Virginia Tech Lecture 2 1September 2009

21. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Momentum Equations - Sample Problem 1 Virginia Tech Lecture 2 1September 2009

22. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Momentum Equations - Sample Problem 2 Virginia Tech Lecture 2 1September 2009

23. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 Virginia Tech Lecture 2 1September 2009

24. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Low-Speed Subsonic Wind Tunnels (§4.10) Virginia Tech Lecture 2 1September 2009

25. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 (A2/A1) is fixed for wind tunnel and r is constant for low-speed (incompressible) flows, the quantity driving the tunnel is p1-p2. • But how can we determine p1-p2 ??? Virginia Tech Lecture 2 1September 2009

26. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Manometer Virginia Tech Lecture 2 1September 2009

27. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Wind Tunnels – Sample Problem 1 Virginia Tech Lecture 2 1September 2009

28. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Wind Tunnels – Sample Problem 1 Solution • Height of liquid: Dh = 10cm = 0.1m • Specific weight of liquid mercury: w = (1.36x104)x9.8 = 1.33x105 N/m2 • Actual pressure difference: p1-p2 = w Dh = 1.33x104 N/m2. • To find V2 from Bernoulli, use • We computed p1-p2, A1/A2 = 15 is given, so we need to find r. • Since we are in a low-speed wind tunnel, flow is incompressible, so r = const, which means we can compute it at any point in the tunnel. Since p1 and T1 are given, use Equation of State to find r = r1: • Combining all the results we get V2 = 144 m/s (slightly over the incompressible velocity limit, which means compressibility effects should be taken into account). Virginia Tech Lecture 2 1September 2009

29. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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 p0 Virginia Tech Lecture 2 1September 2009

30. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Total (stagnation) Pressure (p0 ) 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) Virginia Tech Lecture 2 1September 2009

31. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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” Virginia Tech Lecture 2 1September 2009

32. AOE 2104--Aerospace and Ocean EngineeringFall 2009 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” Virginia Tech Lecture 2 1September 2009

33. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Airspeed Indicator If the only known density is at sea level, “Indicated or Equivalent Airspeed” Virginia Tech Lecture 2 1September 2009

34. AOE 2104--Aerospace and Ocean EngineeringFall 2009 True Airspeed and Therefore, the relationship between true and indicated airspeed is: Virginia Tech Lecture 2 1September 2009

35. AOE 2104--Aerospace and Ocean EngineeringFall 2009 www.aeromech.usyd.edu.au/ aero/instruments/ http://www.tech.purdue.edu/at/courses/aeml/airframeimages/pitottube.jpg Virginia Tech Lecture 2 1September 2009

36. AOE 2104--Aerospace and Ocean EngineeringFall 2009 www.aeromech.usyd.edu.au/ aero/instruments/ http://home4.highway.ne.jp/t-park/tp/image/seventh/s-port.jpg Virginia Tech Lecture 2 1September 2009

37. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Measurement of Airspeed – Sample Problem Virginia Tech Lecture 2 1September 2009

38. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Measurement of Airspeed – Sample Problem Solution • From Standard Atmosphere (App. B), at 5000ft, p = 1761 lb/ft2. • Pitot tube measures stagnation pressure so p0 = 1818 lb/ft2. • Density is found from measured temperature and tabulated pressure • = p/(RT) = 1761/(1716*505)  r= 2.03x10-3 slug/ft3.   7.6% difference Virginia Tech Lecture 2 1September 2009

39. AOE 2104--Aerospace and Ocean EngineeringFall 2009 Virginia Tech Lecture 2 1September 2009

40. AOE 2104--Aerospace and Ocean EngineeringFall 2009 For Next Class: Review Chapter 4 and let me know what questions you have Thursday: HW 1 due. Stay Tuned for HW 2. Virginia Tech Lecture 2 1September 2009