1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Final Exam Review and Closing Comments
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. OVERVIEW OF ACCOMPLISHMENT
“This book is designed for a 1-year course in aerodynamics. Chapters 1 to 6 constitute a solid semester [bold, italics added for emphasis] emphasizing inviscid, incompressible flow. Chapters 7 to 14 occupy a second semester dealing with inviscid, compressible flow.” – John D. Anderson, Jr.
What we did:
Chapters 1-5
Why not Chapter 6? → 3-D incompressible flow (sources, doublet, etc.)
Chapters 7-9, 11 and 12
Why not Chapter 10? → Fluids II material (nozzles, diffusers, etc.)
Multiple examples of applications to flight and projectile mechanics
What would we do if we had more time:
Viscous flow
Laminar and turbulent boundary layer models for drag prediction
Exact solutions, Faulkner-Skan equations and Thwaites method

3. OUTLINE
Basic Ideas
Can you convey basic ideas in aerodynamics in simple terms: lift, stall, streamline, Kutta-condition, camber, lifting line, separation, etc.
Explain in words or pictures what complicated equations are trying to say
Stream and Potential Functions: Inviscid, Incompressible Flow
What is the point? What is the utility? What is weakness?
How do you set-up and use these simple models?
Flow Over Airfoils
Incompressible flow: Theory vs. experiment
Compressible flow (why so complicated?): Theory vs. experiment
Supersonic flow: Why does shape of airfoil want to be so different?
Flow Over Wings
Impact of wing tips? How do you model, how do you proceed?
What are implications for design?
Flight Mechanics
What do (1)-(4) imply about aerodynamic design and performance impacts?

4. KEY CONCEPTS: CHAPTERS 1 and 2
Aerodynamic forces and moments (center of pressure)
Where do they come from, why do we care?
Mach and Reynolds number matching guarantee flow similarity
Types of flows
Inviscid vs. Viscous
Incompressible vs. Compressible
Mach number regimes
Fundamentals Principles
Conservation of Mass (integral and control volume form)
Conservation of Momentum (integral form)
Conservation of Energy (algebraic form)
Angular velocity, vorticity and circulation (why do we care about these concepts?)
Stream Function and Velocity Potential (how are these related?)

5. KET CONCEPTS: CHAPTER 3
Elementary Flows (Building Blocks, why such a name?)
Uniform Flow
Source / Sink Flow
Doublet Flow
Vortex Flow
What is the purpose? → Simulate real shapes in a simple manner
Combine (1) + (2) → flow over half-body or oval
Combine (1) + (3) → flow over a cylinder
Combine with (4) → flow over a lifting cylinder
Kutta-Joukowski Theorem
Combinations of sources, vortex, uniform flow, tornados, ground effect, etc.
Why can we combine so easily (simply add)?
Know how to set up y and f for all cases and combined flows (no time to solve)
Know how to get velocity components u and v
How would you model some basic shapes using these tools?
Homework #4 has many practice problems (nothing more difficult than these)

6. KEY CONCEPTS: CHAPTER 4, 11 and 12
Model an airfoil as a vortex sheet
What does this mean, why can we do this, why would we want to do this?
Thin airfoil theory: Mean camber line is a streamline of the flow
Symmetric vs. Cambered Airfoils
S+C: Lift coefficient: 2pa
S+C: Lift slope: 2p
S: Moment Coefficient, c/4 = 0
C: Moment Coefficient, c/4 = p/4(A2 - A1)
Role of airfoil thickness (incompressible, subsonic, supersonic)
What are added complexities (physics and math) associated with compressibility?
How can we correct for compressibility (what are strengths and weaknesses)?
Also see key concepts/comments for Chapters 7, 8, and 9
Chapter 12: §12.1- §12.3

7. KEY CONCEPTS: CHAPTER 5 Airfoils vs. Wings
What is different about these situations
Why should we care? When is it important to care?
How do we model a wing? Is it accurate?
What is lifting line theory
Key results
Elliptical Wings
Other Wings
Why do we taper a wing?
Why do we sweep wing?
Why do we vary AR (or span) as designers
Why do modern commercial airplane wings (A320, B757, etc.) look way they do?
Why do modern fighter wings not look like this?

8. KEY CONCEPTS: CHAPTER 7, 8, and 9
What are isentropic relations?
When can we use them?
Why would we use them? (replace energy equation, simple, algebraic)
When do they break down?
If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)
How do flow properties across normal and oblique shock waves change?
Is it important to capture these effects?
Expansion processes
Make use of Appendix A, B, and C as well as q-b-M diagram
Don’t waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

9. BASIC CONCEPTS CHAPTERS 1-2

10. KEY CONCEPTS Aerodynamic forces and moments (center of pressure)
Where do they come from, why do we care?
Mach and Reynolds number matching guarantee flow similarity
Types of flows
Inviscid vs. Viscous
Incompressible vs. Compressible
Mach number regimes
Fundamentals Principles
Conservation of Mass (integral and control volume form)
Conservation of Momentum (integral form)
Conservation of Energy (algebraic form)
Angular velocity, vorticity and circulation (why do we care about these concepts?)
Stream Function and Velocity Potential (how are these related?)

11. WHAT DOES EULER’S EQUATION TELL US?
Euler’s Equation (Differential Equation)
Relates changes in momentum to changes in force (momentum equation)
Relates a change in pressure (dp) to a chance in velocity (dV)
Assumptions:
Steady flow and no friction (inviscid flow), body forces, and external forces
dp and dV are of opposite sign
IF dp increases dV goes down → flow slows down
IF dp decreases dV goes up → flow speeds up
Incompressible and Compressible flows, Irrotational and Rotational flows

12. Constant along a streamline
BERNOULLI’S EQUATION
Constant along a streamline
If flow is irrotational p+1/2rV2 = constant everywhere
Remember:
Bernoulli’s equation holds only for inviscid (frictionless) and incompressible (r=constant) flows
Relates properties between different points along a streamline or entire flow field if irrotational
For a compressible flow Euler’s equation must be used (r is a variable)
Both Euler’s and Bernoulli’s equations are expressions of F=ma expressed in a useful form for fluid flows and aerodynamics

13. WHAT CREATES AERODYNAMIC FORCES?
Aerodynamic forces exerted by airflow comes from only two sources
Pressure, p, distribution on surface
Acts normal to surface
Shear stress, tw, (friction) on surface
Acts tangentially to surface
Pressure and shear are in units of force per unit area (N/m2)
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”

14. SOME DEFINITIONS Relative Wind: Direction of V∞
We used subscript ∞ to indicate far upstream conditions
Angle of Attack, a: 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
Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero
Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack

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

16. AIRFOIL DATA (APPENDIX D) NACA 23012 WING SECTION
Re dependence
at high a
cd vs. cl
Dependent on Re cl
cl vs. a
Independent of Re cd
cm,a.c. vs. cl very flat
cm,a.c.
cm,c/4 a cl

17. HOW DOES AN AIRFOIL GENERATE LIFT?
Flow velocity over the 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 rAV=constant, velocity must increase
Streamtube A is squashed
most in nose region
(ahead of maximum thickness) A B

18. HOW DOES AN AIRFOIL GENERATE LIFT?
As velocity increases pressure decreases
Incompressible: Bernoulli’s Equation
Compressible: Euler’s Equation
Called Bernoulli Effect
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)

19. WHY DOES AN AIRFOIL STALL?
Key to understanding
Friction causes flow separation within boundary layer
Separation then creates another form of drag called pressure drag due to separation

20. STALL CHARACTER: NACA 4412 VERSUS NACA 4421
Both NACA 4412 and NACA 4421 have same shape of mean camber line
Thin airfoil theory predict that linear lift slope and aL=0 should be the same for both
Leading edge stall shows rapid drop of lift curve near maximum lift
Trailing edge stall shows gradual bending-over of lift curve at maximum lift, “soft stall”
High cl,max for airfoils with leading edge stall
Flat plate stall exhibits poorest behavior, early stalling
Thickness has major effect on cl,max

21. INVISCID, INCOMPRESSIBLE FLOW CHAPTER 3

22. KET CONCEPTS Elementary Flows (Building Blocks, why such a name?)
Uniform Flow
Source / Sink Flow
Doublet Flow
Vortex Flow
What is the purpose? → Simulate real shapes in a simple manner
Combine (1) + (2) → flow over half-body or oval
Combine (1) + (3) → flow over a cylinder
Combine with (4) → flow over a lifting cylinder
Kutta-Joukowski Theorem
Combinations of sources, vortex, uniform flow, tornados, ground effect, etc.
Why can we combine so easily (simply add)?
Know how to set up y and f for all cases and combined flows (no time to solve)
Know how to get velocity components u and v
How would you model some basic shapes using these tools?
Homework #4 has many practice problems (nothing more difficult than these)

23. SUMMARY OF STREAM AND POTENTIAL FUNCTIONS TABLE 3.1

24. LIFTING FLOW OVER A CYLINDER
Kutta-Joukowski Theorem

25. FLOW OVER AIRFOILS INCOMPRESSIBLE: CHAPTER 4 COMPRESSIBLE: CHAPTER 11

26. KEY CONCEPTS Model an airfoil as a vortex sheet
What does this mean, why can we do this, why would we want to do this?
Thin airfoil theory: Mean camber line is a streamline of the flow
Symmetric vs. Cambered Airfoils
S+C: Lift coefficient: 2pa
S+C: Lift slope: 2p
S: Moment Coefficient, c/4 = 0
C: Moment Coefficient, c/4 = p/4(A2 - A1)
Role of thickness

27. CENTER OF PRESSURE AND AERODYNAMIC CENTER
Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero
Thin Airfoil Theory:
Symmetric Airfoil:
Cambered Airfoil:
Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack

28. PREVIEW: COMPRESSIBILITY CORRECTION EFFECT OF M∞ ON CP
Sound
Barrier ?
Effect of compressibility
(M∞ > 0.3) is to increase
absolute magnitude of Cp and M∞ increases
Called: Prandtl-Glauert Rule
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const
Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7 (Why not M∞ = 0.99?)

29. RESULT Velocity Potential Equation: Nonlinear Equation
Compressible, Steady, Inviscid and Irrotational Flows
Note: This is one equation, with one unknown, f
a0 (as well as T0, P0, r0, h0) are known constants of the flow
Velocity Potential Equation: Linear Equation
Incompressible, Steady, Inviscid and Irrotational Flows

30. RESULT After order of magnitude analysis, we have following results
May also be written in terms of perturbation velocity potential
Equation is a linear PDE and is rather easy to solve (see slides for technique)
Recall:
Equation is no longer exact
Valid for small perturbations
Slender bodies
Small angles of attack
Subsonic and Supersonic Mach numbers
Keeping in mind these assumptions equation is good approximation

31. CRITICAL MACH NUMBER, MCR
As air expands around top surface near leading edge, velocity and M will increase
Local M > M∞
Flow over airfoil may have
sonic regions even though
freestream M∞ < 1

32. DESIGN OPTIONS: SWEEP, AERA RULE, SUPERCRITICAL AIRFOILS
Makes airfoil ‘thinner’ → increases critical Mach number
Sweeping wing usually reduces lift for subsonic flight
Area Rule: Drag created related to change in cross-sectional area of vehicle from nose to tail
Supercritical Airfoils: Designed to delay and reduce transonic drag rise, due to both strong normal shock and shock-induced boundary layer separation

33. FLOW OVER WINGS CHAPTER 5

34. KEY CONCEPTS Airfoils vs. Wings
What is different about these situations
Why should we care? When is it important to care?
How do we model a wing? Is it accurate?
What is lifting line theory
Key results
Elliptical Wings
Other Wings
Why do we taper a wing?
Why do we vary AR (or span) as designers
Why do modern commercial airplane wings (A320, B757, etc.) look the way they do?
Why do modern fighter wings not look like this?

35. PHYSICAL INTERPRETATION
a: Geometric Angle of Attack
ai: Induced Angle of Attack
aeff: Effective Angle of Attack
Chord line
Finite Wing Consequences:
Tilted lift vector contributes a drag component, called induced drag (drag due to lift) → CL < cl and CD > cd
Lift slope is reduced relative to infinite wing (a < a0)

36. PRANDTL’S LIFTING LINE EQUATION
Fundamental Equation of Prandtl’s Lifting Line Theory
In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack
Mathematically: a = aeff + ai
Only unknown is G(y)
V∞, c, a, aL=0 are known for a finite wing of given design at a given a
Solution gives G(y0), where –b/2 ≤ y0 ≤ b/2 along span

37. KEY RESULT True for all finite wings in general
Define a span efficiency factor, e (also called span efficiency factor)
Elliptical planforms, e = 1
For all other planforms, e < 1
0.60 < e < 0.99
Arbitrary Finite Wing
For Elliptical Planforms
Span Efficiency Factor
Key Points:
Goes with square of CL
Inversely related to AR
Also called drag due to lift

38. SUMMARY: TOTAL DRAG ON SUBSONIC WING
Profile Drag
Profile Drag coefficient relatively constant with M∞ at subsonic speeds
Also called drag due to lift
May be calculated from
Inviscid theory:
Lifting line theory
Look up
(Infinite Wing)

39. IMPORTANT STATEMENTS Fundamental Equation of Thin Airfoil Theory
“The camber line is a streamline of the flow”
Fundamental Equation of Prandtl’s Lifting-Line Theory
“The geometric angle of attack is equal to the sum of the effective
angle of attack plus the induced angle of attack”

40. GENERAL LIFT DISTRIBUTION (2/4)
Substitute expression for G(q) and dG/dy into fundamental equation of Prandtl’s lifting line theory
Last term on the right (integral term) is a standard form and may be simplified as:
Equation is evaluated at a given spanwise location (q0), just as fundamental equation of Prandtl’s lifting line theory is evaluated at a given spanwise location (y0)
Only unknowns in equation are An’s
Written at q0 equation is 1 algebraic equation with N unknowns
Write equation at N spanwise locations to obtain a system of N independent algebraic equations with N unknowns

41. SUPERSONIC AIRFOILS AND WINGS REVIEW: CHAPTER 7 SHOCK WAVES / EXPANSIONS: CHAPTERS 8 AND 9

42. KEY CONCEPTS What are isentropic relations? When can we use them?
Why would we use them? (replace energy equation, simple, algebraic)
When do they break down?
If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)
How do flow properties across normal and oblique shock waves change?
Is it important to capture these effects?
Expansion processes
Make use of Appendix A, B, and C as well as q-b-M diagram
Don’t waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

43. SUMMARY OF NORMAL SHOCK RELATIONS

44. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW (M > 1)
Rayleigh Pitot Tube Formula

45. SUMMARY OF SHOCK RELATIONS
Normal Shocks
Oblique Shocks

46. q-b-M RELATION Shock Wave Angle, b Detached, Curved Shock
Strong
M2 < 1
Weak
Shock Wave Angle, b
M2 > 1
Detached, Curved Shock
Deflection Angle, q

47. SWEPT WINGS: SUPERSONIC FLIGHT
If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag
If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag
For supersonic flight, swept wings reduce wave drag

48. EXAMPLE OF SUPERSONIC AIRFOILS

49. FLIGHT MECHANICS

50. WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT RATIO (b2/S)
Span loading (W/b), wing loading (W/S)
and AR (b2/S) are related
Zero-lift drag, D0 is proportional to wing area
Induced drag, Di, is proportional to square
of span loading
Take ratio of these drags, Di/D0
Re-write W2/(b2S) in terms of AR and substitute into drag ratio Di/D0
1: For specified W/S (set by take-off or landing requirements) and CD,0 (airfoil choice), increasing AR will decrease drag due to lift relative to zero-lift drag
2: AR predominately controls ratio of induced drag to zero lift drag, whereas span loading controls actual value of induced drag

51. FURTHER IMPLICATIONS FOR DESIGN: VMAX
Maximum velocity at a given altitude is important specification for new airplane
To design airplane for given Vmax, what are most important design parameters?
Steady, level flight: T = D
Steady, level flight: L = W
Substitute into drag equation
Turn this equation into a quadratic
equation (by multiplying by q∞)
and rearranging
Solve quadratic equation and set
thrust, T, to maximum available
thrust, TA,max

52. FURTHER IMPLICATIONS FOR DESIGN: VMAX
TA,max does not appear alone, but only in ratio (TA/W)max
S does not appear alone, but only in ratio (W/S)
Vmax does not depend on thrust alone or weight alone, but rather on ratios
(TA/W)max: maximum thrust-to-weight ratio
W/S: wing loading
Vmax also depends on density (altitude), CD,0, peAR
We can increase Vmax by
Increase maximum thrust-to-weight ratio, (TA/W)max
Increasing wing loading, (W/S)
Decreasing zero-lift drag coefficient, CD,0

53. THRUST REQUIRED VS. FLIGHT VELOCITY
Zero-Lift TR
(Parasitic Drag)
Lift-Induced TR
(Induced Drag)
Zero-Lift TR ~ V2
(Parasitic Drag)
Lift-Induced TR ~ 1/V2
(Induced Drag)

54. THRUST REQUIRED VS. FLIGHT VELOCITY
At point of minimum TR, dTR/dV∞=0
(or dTR/dq∞=0)
Zero-Lift Drag = Induced Drag
At minimum TR and maximum L/D

55. POWER REQUIRED Zero-Lift PR Lift-Induced PR Zero-Lift PR ~ V3
Lift-Induced PR ~ 1/V

56. POWER REQUIRED
At point of minimum PR, PTR/dV∞=0