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

OUTLINE1. 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

2. 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?

3. 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?

4. Flow Over Wings

– Impact of wing tips? How do you model, how do you proceed?

– What are implications for design?

5. 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

1. Conservation of Mass (integral and control volume form)

2. Conservation of Momentum (integral form)

3. 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?)

1. Uniform Flow2. Source / Sink Flow3. Doublet Flow4. 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 and 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: 2a

– S+C: Lift slope: 2– S: Moment Coefficient, c/4 = 0

– C: Moment Coefficient, c/4 = /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 --M diagram

– Don’t waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

BASIC CONCEPTSCHAPTERS 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

1. Conservation of Mass (integral and control volume form)

2. Conservation of Momentum (integral form)

3. 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

VdVdp

12

BERNOULLI’S EQUATION

2

222

21

1

22

2

Vp

Vp

Vp

• If flow is irrotational p+1/2V2 = constant everywhere

• Remember:

– Bernoulli’s equation holds only for inviscid (frictionless) and incompressible (=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 ( 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

Constant along a streamline

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, w, (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, 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• 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

cl

Cambered airfoil haslift at =0At negative airfoilwill have zero lift

16

AIRFOIL DATA (APPENDIX D)NACA 23012 WING SECTION

c l

c m,c

/4

Re dependenceat high

cl

c dc m

,a.c

.

cl vs. Independent of Re

cd vs. clDependent on Re

cm,a.c. vs. cl very flat

17

HOW DOES AN AIRFOIL GENERATE LIFT?1. 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 AV=constant, velocity must increase

Streamtube A is squashedmost in nose region(ahead of maximum thickness)

AB

18

HOW DOES AN AIRFOIL GENERATE LIFT?2. As velocity increases pressure decreases

– 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

VdVdp

Vp

constant2

1 2

Most of lift is producedin 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 L=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

INVISCID, INCOMPRESSIBLE FLOWCHAPTER 3

22

KET CONCEPTS• Elementary Flows (Building Blocks, why such a name?)

1. Uniform Flow2. Source / Sink Flow3. Doublet Flow4. 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 and 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 FUNCTIONSTABLE 3.1

24

LIFTING FLOW OVER A CYLINDER

VL

R

r

r

RrV

ln2

1sin2

2

Kutta-Joukowski Theorem

FLOW OVER AIRFOILSINCOMPRESSIBLE: CHAPTER 4COMPRESSIBLE: 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 = /4(A2 - A1)

• Role of thickness

dx

dzV

x

dc

02

1

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

– Thin Airfoil Theory:

• Symmetric Airfoil:

• Cambered Airfoil:

2114

4

AAc

cx

cx

lcp

cp

4

4

..

..

cx

cx

CA

CA

28

22

0,

1

5.0

1

MM

CC p

p

For M∞ < 0.3, ~ constCp = Cp,0 = 0.5 = const

Effect of compressibility(M∞ > 0.3) is to increaseabsolute magnitude of Cp and M∞ increasesCalled: Prandtl-Glauert Rule

Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

(Why not M∞ = 0.99?)

PREVIEW: COMPRESSIBILITY CORRECTIONEFFECT OF M∞ ON CP

SoundBarrier ?

29

RESULT

Velocity Potential Equation: Nonlinear EquationCompressible, Steady, Inviscid and Irrotational Flows

Note: This is one equation, with one unknown, a0 (as well as T0, P0, 0, h0) are known constants of the flow

021

11

12

22

22

22

22

2

yxyxayyaxxa

02

Velocity Potential Equation: Linear EquationIncompressible, 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 19-22 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

0ˆˆ

1

0ˆˆ

1

2

2

2

22

2

yxM

y

v

x

uM

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 havesonic regions even thoughfreestream M∞ < 1

32

DESIGN OPTIONS: SWEEP, AERA RULE, SUPERCRITICAL AIRFOILS

• Sweep:– 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

FLOW OVER WINGSCHAPTER 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

• Finite Wing Consequences:

1. Tilted lift vector contributes a drag component, called induced drag (drag due to lift) → CL < cl and CD > cd

2. Lift slope is reduced relative to infinite wing (a < a0)

Chord line

: Geometric Angle of Attacki: Induced Angle of Attackeff: Effective Angle of Attack

ii

ii

LD

LD

sin

inducedeffectivegeometric

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: = eff + i

• Only unknown is (y)

– V∞, c, , L=0 are known for a finite wing of given design at a given a

– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span

2

20

00

00 4

1b

bL dy

yy

dyd

VycV

yy

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

eAR

CC L

iD

2

,

Span Efficiency Factor

Key Points:Goes with square of CL

Inversely related to AR

Also called drag due to liftAR

CC

AR

C

Sq

D

AR

CSq

AR

CLLD

AR

C

LiD

Li

LLii

Li

2

,

2

2

For Elliptical Planforms

Arbitrary Finite Wing

38

SUMMARY: TOTAL DRAG ON SUBSONIC WING

eAR

Cc

Sq

DcC

DDD

DDDD

Lprofiled

iprofiledD

inducedprofile

inducedpressurefriction

2

,,

Also called drag due to lift

Profile DragProfile Drag coefficient relatively constant with M∞ at subsonic speeds

Look up(Infinite Wing)

May be calculated fromInviscid theory:Lifting line theory

39

IMPORTANT STATEMENTS

2

20

000

00

0

4

1

2

1

b

bL

c

dyyy

dyd

Vy

ycV

yy

dx

dzV

x

d

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 effectiveangle of attack plus the induced angle of attack”

40

GENERAL LIFT DISTRIBUTION (2/4)

N

nnL

N

nn

N

nn

L

N

nn

nnAnA

c

b

dnnA

nAc

b

1 0

000

10

00

0 0

100

10

00

sin

sinsin

2

coscos

cos1

sin2

Substitute expression for () and d/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 (0), 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 0 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

SUPERSONIC AIRFOILS AND WINGSREVIEW: 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 --M diagram

– Don’t waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

43

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10

Upstream Mach Number, M1

M2,

P02

/P01

0

2

4

6

8

10

12

14

16

18

20

2/

1, p

2/p

1, T

2/T

1

Downstream Mach Number, M2Total Pressure Ratio, P02/P01Density Ratio, Rho1/Rho2Static Pressure Ratio, P2/P1Static Temperature Ratio T2/T1

SUMMARY OF NORMAL SHOCK RELATIONS

44

MEASUREMENT OF AIRSPEED:SUPERSONIC FLOW (M > 1)

1

21

124

1

11

21

21

1

21

21

2

1

02

21

1

2

M

M

M

p

p

Mp

p

Rayleigh Pitot Tube Formula21

21

1

2

11

21

21

22

122

2

02

1

2

2

02

1

02

M

MM

Mp

p

p

p

p

p

p

p

45

SUMMARY OF SHOCK RELATIONS

sin2,

2nM

M

11

21

12

1

21

21

1

21,

1

2

21,

21,

1

2

21,

21,

22,

n

n

n

n

n

n

Mp

p

M

M

M

MM

11

21

12

1

21

21

1

21

1

2

21

21

1

2

21

21

22

Mp

p

M

M

M

MM

Normal Shocks Oblique Shocks

sin11, MM n

46

--M RELATION

Deflection Angle,

Sho

ck W

ave

Ang

le,

Weak

Strong

M2 > 1

M2 < 1

22cos

1sincot2tan

21

221

M

M

Detached, C

urved Shock

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

M

1sin 1

48

EXAMPLE OF SUPERSONIC AIRFOILS

http://odin.prohosting.com/~evgenik1/wing.htm

FLIGHT MECHANICS

50

WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT RATIO (b2/S)

AR

SW

CeqD

D

ARS

W

Sb

SW

Sb

W

SCqb

W

eqD

D

b

W

eqD

SCqD

AR

b

S

W

b

W

D

i

D

i

i

D

2

0,2

0

2

2

2

2

2

0,

2

0

2

0,0

1

11

1

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 squareof 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 drag2: 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?

21

0,

0,2

maxmaxmax

2

0,2

2

0,22

2

0,

2

0,

4

0

D

DAA

D

DD

L

LDD

C

eAR

C

WT

SW

SW

WT

V

eARS

WTqSCq

eARSq

WSCq

eARSq

WCSqT

Sq

WC

eAR

CCSqSCqTD

Steady, level flight: T = D

Steady, level flight: L = W

Substitute into drag equation

Turn this equation into a quadraticequation (by multiplying by q∞)and rearranging

Solve quadratic equation and setthrust, T, to maximum availablethrust, 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, eAR

• 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

21

0,

0,2

maxmaxmax

4

D

DAA

C

eAR

C

WT

SW

SW

WT

V

53

THRUST REQUIRED VS. FLIGHT VELOCITY

eAR

CSqSCqT

CCSqSCqDT

LDR

iDDDR

2

0,

,0,

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

iDL

D

DR

RR

DR

CeAR

CC

eARSq

WSC

dq

dT

dq

dV

dV

dT

dq

dT

eARSq

WSCqT

,

2

0,

2

2

0,

2

0,

0

At point of minimum TR, dTR/dV∞=0(or dTR/dq∞=0)

Zero-Lift Drag = Induced DragAt minimum TR and maximum L/D

55

POWER REQUIRED

eAR

CSVqVSCqP

VCCSqVSCqDVVTP

LDR

iDDDRR

2

0,

,0,

Zero-Lift PR Lift-Induced PR

Zero-Lift PR ~ V3

Lift-Induced PR ~ 1/V

56

POWER REQUIRED

03

1

2

3212

1

,0,2

2

0,3

iDDR

DR

CCSVdV

dP

eARSV

WSCVP

iDD CC ,0, 3

1

At point of minimum PR, PTR/dV∞=0

iDD CC ,0,