1. MAE 1202: AEROSPACE PRACTICUM Lecture 6: Compressible and Isentropic Flow 2 Introduction to Airfoils February 25, 2013 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. READING AND HOMEWORK ASSIGNMENTS Reading: Introduction to Flight, by John D. Anderson, Jr. –For this week’s lecture: Chapter 5, Sections 5.1 - 5.5 –Mid-Term Exam: Monday, March 18, 2013 Exam will be given during laboratory session Lecture material only (no MATLAB, CAD, etc.) Covers Chapter 4 and 5 (through 5.5) Mid-Term Exam Review: week after spring break during evening Lecture-Based Homework Assignment: –Problems: 5.2, 5.3, 5.4, 5.6 DUE: Friday, March 1, 2013 by 5 PM Turn in hard copy of homework –Also be sure to review and be familiar with textbook examples in Chapter 5

3. ANSWERS TO LECTURE HOMEWORK 5.2: L = 23.9 lb, D = 0.25 lb, M c/4 = -2.68 lb ft –Note 1: Two sets of lift and moment coefficient data are given for the NACA 1412 airfoil, with and without flap deflection. Make sure to read axis and legend properly, and use only flap retracted data. –Note 2: The scale for c m,c/4 is different than that for c l, so be careful when reading the data 5.3: L = 308 N, D = 2.77 N, M c/4 = - 0.925 N m 5.4:  = 2° 5.6: (L/D) max ~ 112

4. 1 st LAW OF THERMODYNAMICS (4.5) System (gas) composed of molecules moving in random motion Energy of all molecular motion is called internal energy per unit mass, e, of system Only two ways e can be increased (or decreased): 1.Heat,  q, added to (or removed from) system 2.Work,  w, is done on (or by) system SYSTEM (unit mass of gas) Boundary SURROUNDINGS qq e (J/kg)

5. 1 st LAW IN MORE USEFUL FORM (4.5) 1 st Law: de =  q +  w –Find more useful expression for  w, in terms of p and  (or v = 1/  ) When volume varies → work is done Work done on balloon, volume ↓ Work done by balloon, volume ↑ Change in Volume (-)

6. ENTHALPY: A USEFUL QUANTITY (4.5) Define a new quantity called enthalpy, h: (recall ideal gas law: pv = RT) Differentiate Substitute into 1 st law (from previous slide) Another version of 1 st law that uses enthalpy, h:

7. HEAT ADDITION AND SPECIFIC HEAT (4.5) Addition of  q will cause a small change in temperature dT of system Specific heat is heat added per unit change in temperature of system Different materials have different specific heats –Balloon filled with He, N 2, Ar, water, lead, uranium, etc… ALSO, for a fixed dq, resulting dT depends on type of process… qq dd

8. SPECIFIC HEAT: CONSTANT PRESSURE Addition of  q will cause a small change in temperature dT of system System pressure remains constant qq dd Extra Credit #1: Show this step

9. SPECIFIC HEAT: CONSTANT VOLUME Addition of  q will cause a small change in temperature dT of system System volume remains constant qq dd Extra Credit #2: Show this step

10. HEAT ADDITION AND SPECIFIC HEAT (4.5) Addition of  q will cause a small change in temperature dT of system Specific heat is heat added per unit change in temperature of system However, for a fixed dq, resulting dT depends on type of process: Specific heat ratio For air,  = 1.4 Constant PressureConstant Volume

11. ISENTROPIC FLOW (4.6) Goal: Relate Thermodynamics to Compressible Flow Adiabatic Process: No heat is added or removed from system –  q = 0 –Note: Temperature can still change because of changing density Reversible Process: No friction (or other dissipative effects) Isentropic Process: (1) Adiabatic + (2) Reversible –(1) No heat exchange + (2) no frictional losses –Relevant for compressible flows only –Provides important relationships among thermodynamic variables at two different points along a streamline  = ratio of specific heats  = c p /c v  air =1.4

12. DERIVATION: ENERGY EQUATION (4.7) Energy can neither be created nor destroyed Start with 1 st law Adiabatic,  q=0 1 st law in terms of enthalpy Recall Euler’s equation Combine Integrate Result: frictionless + adiabatic flow

13. ENERGY EQUATION SUMMARY (4.7) Energy can neither be created nor destroyed; can only change physical form –Same idea as 1 st law of thermodynamics Energy equation for frictionless, adiabatic flow (isentropic) h = enthalpy = e+p/  = e+RT h = c p T for an ideal gas Also energy equation for frictionless, adiabatic flow Relates T and V at two different points along a streamline

14. SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW Incompressible flow of fluid along a streamline or in a stream tube of varying area Most important variables: p and V T and  are constants throughout flow Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area T, p, , and V are all variables continuity Bernoulli continuity isentropic energy equation of state at any point

15. EXAMPLE: SPEED OF SOUND (4.9) Sound waves travel through air at a finite speed Sound speed (information speed) has an important role in aerodynamics Combine conservation of mass, Euler’s equation and isentropic relations: Speed of sound, a, in a perfect gas depends only on temperature of gas Mach number = flow velocity normalizes by speed of sound –If M < 1 flow is subsonic –If M = 1 flow is sonic –If M > flow is supersonic If M < 0.3 flow may be considered incompressible

16. KEY TERMS: CAN YOU DEFINE THEM? Streamline Stream tube Steady flow Unsteady flow Viscid flow Inviscid flow Compressible flow Incompressible flow Laminar flow Turbulent flow Constant pressure process Constant volume process Adiabatic Reversible Isentropic Enthalpy

17. EXAMPLES AND APPLICATIONS Measurement of Airspeed Shock Waves Supersonic Wind Tunnels and Rocket Nozzles

18. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW If M > 0.3, flow is compressible (density changes are important) Need to introduce energy equation and isentropic relations c p : specific heat at constant pressure M 1 =V 1 /a 1  air =1.4

19. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW So, how do we use these results to measure airspeed p 0 and p 1 give Flight Mach number Mach meter M 1 =V 1 /a 1 Actual Flight Speed using pressure difference What is T 1 and a 1 ? Again use sea-level conditions T s, a s, p s (a 1 =340.3 m/s)

20. EXAMPLE: TOTAL TEMPERATURE A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K What temperature does the nose of the rocket ‘feel’? T 0 = 200(1+ 0.2(36)) = 1,640 K! Total temperature Static temperature Vehicle flight Mach number

21. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW What can happen in supersonic flows? Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)

22. HOW AND WHY DOES A SHOCK WAVE FORM? Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed If M < 1 information available throughout flow field If M > 1 information confined to some region of flow field

23. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW Notice how different this expression is from previous expressions You will learn a lot more about shock wave in compressible flow course

24. SUMMARY OF AIR SPEED MEASUREMENT Subsonic, incompressible Subsonic, compressible Supersonic

25. HOW ARE ROCKET NOZZLES SHAPPED?

26. MORE ON SUPERSONIC FLOWS (4.13) Isentropic flow in a streamtube Differentiate Euler’s Equation Since flow is isentropic a 2 =dp/d  Area-Velocity Relation

27. CONSEQUENCES OF AREA-VELOCITY RELATION IF Flow is Subsonic (M < 1) –For V to increase (dV positive) area must decrease (dA negative) –Note that this is consistent with Euler’s equation for dV and dp IF Flow is Supersonic (M > 1) –For V to increase (dV positive) area must increase (dA positive) IF Flow is Sonic (M = 1) –M = 1 occurs at a minimum area of cross-section –Minimum area is called a throat (dA/A = 0)

28. TRENDS: CONTRACTION M 1 < 1 M 1 > 1 V 2 > V 1 V 2 < V 1 1: INLET 2: OUTLET

29. TRENDS: EXPANSION M 1 < 1 M 1 > 1 V 2 < V 1 V 2 > V 1 1: INLET 2: OUTLET

30. PUT IT TOGETHER: C-D NOZZLE 1: INLET 2: OUTLET

31. MORE ON SUPERSONIC FLOWS (4.13) A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest Supersonic wind tunnel section Rocket nozzle

32. Chapter 5 Overview

33. HOW DOES AN AIRFOIL GENERATE LIFT? Lift due to imbalance of pressure distribution over top and bottom surfaces of airfoil (or wing) –If pressure on top is lower than pressure on bottom surface, lift is generated –Why is pressure lower on top surface? We can understand answer from basic physics: –Continuity (Mass Conservation) –Newton’s 2 nd law (Euler or Bernoulli Equation) Lift = PA

34. HOW DOES AN AIRFOIL GENERATE LIFT? 1.Flow velocity over top of airfoil is faster than over bottom surface –Streamtube A senses upper portion of airfoil as an obstruction –Streamtube A is squashed to smaller cross-sectional area –Mass continuity  AV=constant: IF A↓ THEN V↑ Streamtube A is squashed most in nose region (ahead of maximum thickness) A B

35. HOW DOES AN AIRFOIL GENERATE LIFT? 2.As V ↑ p↓ –Incompressible: Bernoulli’s Equation –Compressible: Euler’s Equation –Called Bernoulli Effect 3.With lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction → Lift Most of lift is produced in first 20-30% of wing (just downstream of leading edge) Can you express these ideas in your own words?

36. AIRFOILS VERSUS WINGS Why do airfoils have such a shape? How are lift and drag produced? NACA airfoil performance data How do we design? What is limit of behavior?

37. AIRFOIL THICKNESS: WWI AIRPLANES English Sopwith Camel German Fokker Dr-1 Higher maximum C L Internal wing structure Higher rates of climb Improved maneuverability Thin wing, lower maximum C L Bracing wires required – high drag

38. AIRFOIL NOMENCLATURE Mean Chamber Line: Set of points halfway between upper and lower surfaces –Measured perpendicular to mean chamber line itself Leading Edge: Most forward point of mean chamber line Trailing Edge: Most reward point of mean chamber line Chord Line: Straight line connecting the leading and trailing edges Chord, c: Distance along the chord line from leading to trailing edge Chamber: Maximum distance between mean chamber line and chord line –Measured perpendicular to chord line

39. NACA FOUR-DIGIT SERIES First digit specifies maximum camber in percentage of chord Second digit indicates position of maximum camber in tenths of chord Last two digits provide maximum thickness of airfoil in percentage of chord Example: NACA 2415 Airfoil has maximum thickness of 15% of chord (0.15c) Camber of 2% (0.02c) located 40% back from airfoil leading edge (0.4c) NACA 2415

40. WHAT CREATES AERODYNAMIC FORCES? (2.2) Aerodynamic forces exerted by airflow comes from only two sources: 1.Pressure, p, distribution on surface Acts normal to surface 2.Shear stress,  w, (friction) on surface Acts tangentially to surface Pressure and shear are in units of force per unit area (N/m 2 ) Net unbalance creates an aerodynamic force “No matter how complex the flow field, and no matter how complex the shape of the body, the only way nature has of communicating an aerodynamic force to a solid object or surface is through the pressure and shear stress distributions that exist on the surface.” “The pressure and shear stress distributions are the two hands of nature that reach out and grab the body, exerting a force on the body – the aerodynamic force”

41. RESOLVING THE AERODYNAMIC FORCE Relative Wind: Direction of V ∞ –We use subscript ∞ to indicate far upstream conditions Angle of Attack,  Angle between relative wind (V ∞ ) and chord line Total aerodynamic force, R, can be resolved into two force components –Lift, L: Component of aerodynamic force perpendicular to relative wind –Drag, D: Component of aerodynamic force parallel to relative wind

42. MORE DEFINITIONS Total aerodynamic force on airfoil is summation of F 1 and F 2 Lift is obtained when F 2 > F 1 Misalignment of F 1 and F 2 creates Moments, M, which tend to rotate airfoil/wing –A moment (torque) is a force times a distance Value of induced moment depends on point about which moments are taken –Moments about leading edge, M LE, or quarter-chord point, c/4, M c/4 –In general M LE ≠ M c/4 F1F1 F2F2

43. VARIATION OF L, D, AND M WITH  Lift, Drag, and Moments on a airfoil or wing will change as  changes Variations of these quantities are some of most important information that an airplane designer needs to know Aerodynamic Center –Point about which moments essentially do not vary with  –M ac =constant (independent of  ) –For low speed airfoils aerodynamic center is near quarter-chord point, c/4

44. AOA = 2°

45. AOA = 3°

46. AOA = 6°

47. AOA = 9°

48. AOA = 12°

49. AOA = 20°

50. AOA = 60°

51. AOA = 90°

52. SAMPLE DATA: SYMMETRIC AIRFOIL Lift (for now) Angle of Attack,  A symmetric airfoil generates zero lift at zero 

53. SAMPLE DATA: CAMBERED AIRFOIL Lift (for now) Angle of Attack,  A cambered airfoil generates positive lift at zero 

54. SAMPLE DATA Lift coefficient (or lift) linear variation with angle of attack, a –Cambered airfoils have positive lift when  = 0 –Symmetric airfoils have zero lift when  = 0 At high enough angle of attack, the performance of the airfoil rapidly degrades → stall Lift (for now) Cambered airfoil has lift at  =0 At negative  airfoil will have zero lift

55. SAMPLE DATA: STALL BEHAVIOR Lift (for now) What is really going on here What is stall? Can we predict it? Can we design for it?

56. WHY DOES LIFT CURVE BEND OVER? http://www.soton.ac.uk/Racing/Greenpower/BoundaryLayers/ Low  Moderate  High    

57. REAL EFFECTS: VISCOSITY (  ) To understand drag and actual airfoil/wing behavior we need an understanding of viscous flows (all real flows have friction) Inviscid (frictionless) flow around a body will result in zero drag! –This is called d’Alembert’s paradox –Must include friction (viscosity,  ) in theory Flow adheres to surface because of friction between gas and solid boundary –At surface flow velocity is zero, called ‘No-Slip Condition’ –Thin region of retarded flow in vicinity of surface, called a ‘Boundary Layer’ At outer edge of B.L., V ∞ At solid boundary, V=0 “The presence of friction in the flow causes a shear stress at the surface of a body, which, in turn contributes to the aerodynamic drag of the body: skin friction drag” p.219, Section 4.20

58. TYPES OF FLOWS: FRICTION VS. NO-FRICTION Flow very close to surface of airfoil is Influenced by friction and is viscous (boundary layer flow) Stall (separation) is a viscous phenomena Flow away from airfoil is not influenced by friction and is wholly inviscid

59. COMMENTS ON VISCOUS FLOWS (4.15)

60. THE REYNOLDS NUMBER, Re One of most important dimensionless numbers in fluid mechanics/ aerodynamics Reynolds number is ratio of two forces: –Inertial Forces –Viscous Forces –c is length scale (chord) Reynolds number tells you when viscous forces are important and when viscosity may be neglected Within B.L. flow highly viscous (low Re) Outside B.L. flow Inviscid (high Re)

61. LAMINAR VS. TURBULENT FLOW Two types of viscous flows –Laminar: streamlines are smooth and regular and a fluid element moves smoothly along a streamline –Turbulent: streamlines break up and fluid elements move in a random, irregular, and chaotic fashion

62. LAMINAR VS. TURBULENT FLOW All B.L.’s transition from laminar to turbulent c f,turb > c f,lam Turbulent velocity profiles are ‘fuller’

63. FLOW SEPARATION Key to understanding: Friction causes flow separation within boundary layer Separation then creates another form of drag called pressure drag due to separation

64. REVIEW: AIRFOIL STALL (4.20, 5.4) Key to understanding: Friction causes flow separation within boundary layer 1.B.L. either laminar or turbulent 2.All laminar B.L. → turbulent B.L. 3.Turbulent B.L. ‘fuller’ than laminar B.L., more resistant to separation Separation creates another form of drag called pressure drag due to separation –Dramatic loss of lift and increase in drag

65. SUMMARY OF VISCOUS EFFECTS ON DRAG (4.21) Friction has two effects: 1.Skin friction due to shear stress at wall 2.Pressure drag due to flow separation Total drag due to viscous effects Called Profile Drag Drag due to skin friction Drag due to separation = + Less for laminar More for turbulent More for laminar Less for turbulent So how do you design? Depends on case by case basis, no definitive answer!

66. COMPARISON OF DRAG FORCES d d Same total drag as airfoil

67. TRUCK SPOILER EXAMPLE Note ‘messy’ or turbulent flow pattern High drag Lower fuel efficiency Spoiler angle increased by + 5° Flow behavior more closely resembles a laminar flow Tremendous savings (< $10,000/yr) on Miami-NYC route

68. LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3) Behavior of L, D, and M depend on , but also on velocity and altitude –V ∞,  ∞, Wing Area (S), Wing Shape,  ∞, compressibility Characterize behavior of L, D, M with coefficients (c l, c d, c m ) Matching Mach and Reynolds (called similarity parameters) M ∞, Re c l, c d, c m identical

69. LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3) Behavior of L, D, and M depend on , but also on velocity and altitude –V ∞,  ∞, Wing Area (S), Wing Shape,  ∞, compressibility Characterize behavior of L, D, M with coefficients (c l, c d, c m ) Note on Notation: We use lower case, c l, c d, and c m for infinite wings (airfoils) We use upper case, C L, C D, and C M for finite wings

70. SAMPLE DATA: NACA 23012 AIRFOIL Lift Coefficient c l Moment Coefficient c m, c/4  Flow separation Stall

71. AIRFOIL DATA (5.4 AND APPENDIX D) NACA 23012 WING SECTION clcl c m,c/4 Re dependence at high  Separation and Stall  clcl cdcd c m,a.c. c l vs.  Independent of Re c d vs.  Dependent on Re c m,a.c. vs. c l very flat R=Re

72. EXAMPLE: SLATS AND FLAPS

73. Flap extended Flap retracted AIRFOIL DATA (5.4 AND APPENDIX D) NACA 1408 WING SECTION