1. 1 Chapter 6 Elements of Airplane Performance Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

2. Simple Mission Profile for an Airplane 1 Switch on + Worming + Taxi Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 2 Altitude 1 2 3 4 5 6 Takeoff Climb Un-accelerated level flight (Cruising flight) Descent Landing Simple mission profile

3. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 3 Airplane Performance Equations of Motions Static Performance (Zero acceleration Dynamic Performance (Finite acceleration) Thrust required Thrust available Maximum velocity Power required Power available Maximum velocity Rate of climb Gliding flight Takeoff Landing

4. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 4 Time to climb Maximum altitude Service ceiling Absolute ceiling Range and endurance Road map for Chapter 6

5. Study the airplane performance requires the derivation of the airplane equations of motion As we know the airplane is a rigid body has six degrees of freedom But in case of airplane performance we are deal with the calculation of velocities ( e.g.V max,V min..etc),distances (e.g. range, takeoff distance, landing distance, ceilings …etc), times (e.g. endurance, time to climb,…etc), angles (e.g.climb angle…etc) Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 5

6. So, the rotation of the airplane about its axes during flight in case of performance study is not necessary. Therefore, we can assume that the airplane is a point mass concentrated at its c.g. Also, the derivation of the airplane’s equations of motion requires the knowledge of the forces acting on the airplane The forces acting on an airplane are: Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 6

7. 1- Lift force L 2- Drag force D 3- Thrust force T Propulsive force 4- Weight W Gravity force Thrust T and weight W will be given But what about L and D? We are in the position that we can’t calculate L and D with our limited knowledge of the airplane aerodynamics Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 7 Components of the resultant aerodynamic force R

8. So, the relation between L and D will be given in the form of the so called drag polar But before write down the equation of the airplane drag polar it is necessary to know the airplane drag types Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 8

9. 9 ■ Drag Types [ Kinds of Drag ] Total Drag Skin Friction Drag Pressure Drag Form Drag (Drag Due to Flow separation)Induced Drag Wave Drag Note : Profile Drag = Skin Friction Drag + Form Drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

10. 10 ►Skin friction drag This is the drag due to shear stress at the surface. ►Pressure drag This is the drag that is generated by the resolved components of the forces due to pressure acting normal to the surface at all points and consists of [ form drag + induced drag + wave drag ]. ►Form drag This can be defined as the difference between profile drag and the skin-friction drag or the drag due to flow separation. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

11. 11 ►Profile Drag ● Profile drag is the sum of skin-friction and form drags. ● It is called profile drag because both skin-friction and form drag [ or drag due to flow separation ] are ramifications of the shape and size of the body, the “profile” of the body. ● It is the total drag on an aerodynamic shape due to viscous effects Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

12. 12 Skin-friction Form drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

13. 13 ►Induced drag ( or vortex drag ) This is the drag generated due to the wing tip vortices, depends on lift, does not depend on viscous effects, and can be estimated by assuming inviscid flow. Finite wing flow tendencies Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

14. 14 Formation of wing tip vortices Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

15. 15 Complete wing-vortex system Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

16. 16Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

17. 17 The origin of downwash The origin of induced drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

18. 18 ►Wave Drag This is the drag associated with the formation of shock waves in high-speed flight. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

19. 19 ■ Total Drag of Airplane ● An airplane is composed of many components and each will contribute to the total drag of its own. ● Possible airplane components drag include : 1. Drag of wing, wing flaps = D w 2. Drag of fuselage = D f 3. Drag of tail surfaces = D t 4. Drag of nacelles = D n 5. Drag of engines = D e 6. Drag of landing gear = D lg 7. Drag of wing fuel tanks and external stores = D wt 8. Drag of miscellaneous parts = D ms Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

20. 20 ● Total drag of an airplane is not simply the sum of the drag of the components. ● This is because when the components are combined into a complete airplane, one component can affect the flow field, and hence, the drag of another. ● these effects are called interference effects, and the change in the sum of the component drags is called interference drag. ● Thus, (Drag) 1+2 = (Drag) 1 + (Drag) 2 + (Drag) interference Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

21. 21 ■ Buid-up Technique of Airplae Drag D ● Using the build-up technique, the airplane total drag D is expressed as: D = D w + D f + D t + D n +D e + D lg + D wt + D ms + D interference ► Interference Drag ● An additional pressure drag caused by the mutual interaction of the flow fields around each component of the airplane. ● Interference drag can be minimized by proper fairing and filleting which induces smooth mixing of air past the components. ● The Figure shows an airplane with large degree of wing filleting. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

22. 22 Wing fillets Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

23. 23 ● No theoretical method can predict interference drag, thus, it is obtained from wind-tunnel or flight-test measurements. ● For rough drag calculations a figure of 5% to 10% can be attributed to interference drag on a total drag, i.e, D interference ≈ [ 5% – 10% ] of components total drag ■ The Airplane Drag Polar ● For every airplane, there is a relation between C D and C L that can be expressed as an equation or plotted on a graph. ● The equation and the graph are called the drag polar. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

24. 24 For the complete airplane, the drag coefficient is written as C D = C Do + K C L 2 This equation is the drag polar for an airplane. Where: C Do drag coefficient at zero lift ( or parasite drag coefficient ) K C L 2 = drag coefficient due to lift ( or induced drag coefficient C Di ) K = 1/ π e AR Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

25. 25 Schematic of the drag polar Prof. Galal Bahgat Salem Aerospace Dept. Cairo University e Oswald efficiency factor = 0.75 – 0.9 (sometimes known as the airplane efficiency factor) AR wing aspect ratio = b 2 /S, b wing span and S wing planform area

26. Airplane Equations of Motion Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 26

27. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 27

28. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 28

29. Apply Newton’s 2 nd low of motion: In the direction of the flight path Perpendicular to the flight path Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 29

30. Un-accelerated Level Flight Performance (Cruising Flight) Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 30

31. Thrust Required for Level Un-accelerated Flight (Drag) Thrust required T R for a given airplane to fly at V ∞ is given as : T R = D Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 31

32. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 32 ● T R as a function of V ∞ can be obtained by tow methods or approaches graphical/analytical ■ Graphical Approach

33. 1- Choose a value of V ∞ 2 - For the chosen V ∞ calculate C L L = W = ½ρ ∞ V 2 ∞ S C L C L = 2W/ ρ ∞ V 2 ∞ S 3- Calculate C D from the drag polar C D = C Do = K C L 2 4- Calculate drag, hence TR, from T R = D = ½ρ ∞ V 2 ∞ S C D 5- Repeat for different values of V ∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 33

34. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 34 V∞V∞ CLCL CDCD C L /C D W/[C L /C D ] 6- Tabulate the results

35. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 35 (T R ) min occurs at (C L /C D ) max

36. ■ Analytical Approach It is required to obtain an equation for T R as a function of V ∞ T R = D Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 36 Required equation

37. Parasite and induced drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 37 T R /D V∞V∞ C Do =C Di

38. Note that T R is minimum at the point of intersection of the parasite drag D o and induced drag D i Thus D o = D i at [T R ] min or C Do = C Di = KC L 2 Then [C L ] (TR)min = √C Do /K And [C Do ] (TR)min = 2C Do Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 38

39. Finally, (L/D) max = (C L /C D ) max = √C Do /K /2C Do (C L /C D ) max = 1/ √4KC Do Also,[V ∞ ] (TR)min = [V ∞ ] (CL/CD)max is obtained from: W = L = ½ρ ∞[V] 2 ( TR)min S [C L ] (TR)min Thus: [V] ( TR)min = {2(W/S)( √K/C Do )/ ρ ∞ } ½ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 39

40. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 40 L/D as function of angle of attack α L/D as function of velocity V ∞

41. L/D as function of V ∞ : Since, But L=W Then or Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 41

42. Flight Velocity for a Given T R T R = D In terms of q ∞ = ½ρ ∞ V 2 ∞ we obtain Multiplying by q ∞ and rearranging, we have This is quadratic equation in q ∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 42

43. Solving for q ∞ By replacing q ∞ = ½ρ ∞ V 2 ∞ we get Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 43

44. Let Where (T R /W) is the thrust-to-weight-ratio (W/S) is the wing loading The final expression for velocity is This equation has two roots as shown in figure corresponding to point 1 an 2 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 44

45. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 45 ● When the discriminant equals zero,then only one solution for V ∞ is obtained ● This corresponds to point 3 in the figure, namely at (T R ) min

46. Or, (T R /W) min = √4C Do K Then the velocity V 3 =V (TR)min is Substituting for (T R /W) min = √4C Do K we have Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 46

47. Effect of Altitude on (TR)min We know that (T R /W) min = √4C Do K This means that (T R ) min is independent of altitude as show in Figure Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 47

48. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 48 Thrust Available T A

49. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 49 Sonic speed

50. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 50 Thrust Available T A and Maximum Velocity V max

51. Power Required P R Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 51

52. Variation of P R with V ∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 52

53. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 53 PRPR

54. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 54

55. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 55

56. Power Available P A Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 56

57. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University 57