MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

Review of Basic Aerodynamics

February 28, 2011

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

AIRFOIL DATAFROM: INTRODUCTION TO FLIGHT, APPENDIX D

JOHN D. ANDERSON, JR.

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

LECTURE OUTLINE• Review of Euler’s Equation

– Euler’s equation for incompressible flow → Bernoulli’s Equation

• Basic Definitions– Airfoils, Wings and Other Objects– Airfoil Nomenclature– Lift, Drag, Moments

• Aerodynamics– How does an airfoil or wing generate lift?– What are effects of viscosity?– Why does an airfoil stall?

• Summary

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 we made:– Steady flow– Neglected 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

• Valid for Incompressible and Compressible flows• Valid for Irrotational and Rotational flows

VdVdp

INVISCID FLOW ALONG STREAMLINES

022

0

0

21

22

12

2

1

2

1

VVpp

VdVdp

VdVdpV

V

p

p

Relate p1 and V1 at point 1 to p2 and V2 at point 2Integrate Euler’s equation from point 1 to point 2 taking =constant

BERNOULLI’S EQUATION

2

222

21

1

22

2

Vp

VpVp

• 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

AIRFOILS VERSUS WINGS

AIRFOILS VERSUS FINITE WINGS

High AR

Low ARSbAR

2

Aspect Ratio

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

NACA FOUR-DIGIT SERIES• First set of airfoils designed using this approach was 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

AIRFOIL THICKNESS: WWI AIRPLANES

English Sopwith Camel

German Fokker Dr-1

Higher maximum CL

Internal wing structureHigher rates of climbImproved maneuverability

Thin wing, lower maximum CL

Bracing wires required – high drag

“HOW IS THIS USEFUL TO ME?”

02

22

waxyv

yxau

xyxa

yyxa

23

32

3

3

STREAMLINE FLOW PATTERNS

• Uniform flow + source produces a shape that looks something like the leading edge of an airfoil

• Concept of vortex sheet• Uniform flow + vortex sheet

can create an airfoil shape of interest

• Mathematical model mimics that shape of airfoil in flow field

STREAMLINES OVER AN AIRFOIL

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”

RESOLVING THE AERODYNAMIC FORCE• 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

RESOLVING THE AERODYNAMIC FORCE• Aerodynamic force, R, may also be resolved into components perpendicular and

parallel to chord line– Normal Force, N: Perpendicular to chord line– Axial Force, A: Parallel to chord line

• L and D are easily related to N and A

• For airfoils and wings, L and D most common• For rockets, missiles, bullets, etc. N and A more useful

cossinsincos

ANDANL

AERODYNAMIC MOMENT• Total aerodynamic force on airfoil is summation of F1 and F2

• Lift is obtained when F2 > F1

• Misalignment of F1 and F2 creates Moments, M, which tend to rotate airfoil/wing• Value of induced moment depends on point about which moments are taken

– Moments about leading edge, MLE or quarter-chord point, c/4, Mc/4

– In general MLE ≠ Mc/4 F1

F2

VARIATION OF L, D, AND M WITH • Lift, Drag and M 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 – Mac=constant (independent of )– For low speed airfoils aerodynamic center is near quarter-chord point

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 2nd law (Euler or Bernoulli Equation)

Lift = PA

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 squashedmost in nose region(ahead of maximum thickness)

AB

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

VdVdp

Vp

constant21 2

Most of lift is producedin first 20-30% of wing(just downstream of leading edge)

Can you express these ideas in your own words?

EVEN A FLAT PLATE WILL GENERATE LIFT• Curved surface of an airfoil is not necessary to produce lift

A

B

LIFT, DRAG, AND MOMENT COEFFICIENTS• 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 (cl, cd, cm)

Re,,21

21

2

2

Mfc

SqL

SV

Lc

ScVL

l

l

l

Matching Mach and Reynolds(called similarity parameters)

M∞, Re

M∞, Re

cl, cd, cm identical

LIFT, DRAG, AND MOMENT COEFFICIENTS• 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 (cl, cd, cm)

Re,,21

21

3

2

2

Mfc

ScqL

ScV

Mc

SccVM

m

m

m

Re,,21

21

2

2

2

Mfc

SqD

SV

Dc

ScVD

d

d

d

Re,,21

21

1

2

2

Mfc

SqL

SV

Lc

ScVL

l

l

l

Note on Notation:We use lower case, cl, cd, and cm for infinite wings (airfoils)We use upper case, CL, CD, and CM for finite wings

SAMPLE DATA: NACA 23012 AIRFOIL

Lift Coefficientcl

Moment Coefficientcm, c/4

Flow separationStall

AIRFOIL DATA (5.4 AND APPENDIX D)NACA 23012 WING SECTION

c lc m

,c/4

Re dependence at high Separation and Stall

cl

c dc m

,a.c

.

cl vs. Independent of Re

cd vs. Dependent on Re

cm,a.c. vs. cl very flat

R=Re

EXAMPLE: SLATS AND FLAPS

Flap extended

Flap retracted

AIRFOIL DATA (5.4 AND APPENDIX D)NACA 1408 WING SECTION

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

Lift

(for

now

)

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

SAMPLE DATA: STALL BEHAVIORLi

ft (f

or n

ow)

What is really going on here

What is stall?

Can we predict it?

Can we design for it?

AIRFOIL DATA (APPENDIX D)NACA 23012 WING SECTION

c lc 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

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!– Called d’Alembert’s paradox (Must include friction in theory)

REAL EFFECTS: VISCOSITY ()• 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”

THE REYNOLDS NUMBER• 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 can be neglected

cVRe

Within B.L. flowhighly viscous(low Re)

Outside B.L. flowInviscid (high Re)

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

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

WHY DOES BOUNDARY LAYER SEPARATE?• Adverse pressure gradient interacting with velocity profile through B.L.• High speed flow near upper edge of B.L. has enough speed to keep moving

through adverse pressure gradient• Lower speed fluid (which has been retarded by friction) is exposed to same

adverse pressure gradient is stopped and direction of flow can be reversed• This reversal of flow direction causes flow to separate

– Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile

– To help prevent flow separation we desire a turbulent B.L.

WHY DOES AN AIRFOIL STALL?• Two major consequences of separated flow over airfoil

– Dramatic loss of lift (stalling)• Separated flow causes higher pressure on upper surface of airfoil

– Major increase in drag• Separation causes lower pressure on trailing edge• Unbalance of pressure force causes pressure drag due to separation

AOA = 2°

AOA = 3°

AOA = 6°

AOA = 9°

AOA = 12°

AOA = 20°

AOA = 60°

AOA = 90°

SUMMARY OF VISCOUS EFFECTS ON DRAG• Friction has two effects:

– Skin friction due to shear stress at wall– Pressure drag due to flow separation

pressurefriction DDD

Total drag due toviscous effectsCalled Profile Drag

Drag due toskin friction

Drag due toseparation= +

Less for laminarMore for turbulent

More for laminarLess for turbulent

So how do you design?Depends on case by case basis, no definitive answer!

COMPARISON OF DRAG FORCES

GOLF BALL AERODYNAMICS

Drag dominated by pressure drag behind sphere

LAMINAR VERSUS TURBULENT FLOW• Reynolds number also tells you about 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

LAMINAR VERSUS TURBULENT FLOW

All B.L.’s transition from laminar to turbulent

cf,turb > cf,lam

Turbulent velocityprofiles are ‘fuller’

GOLF BALL AERODYNAMICS

Large Wake of Separated Flow, High Pressure DragLaminar B.L. Separation Point

Reduced Size Wake of Separated Flow, Lower Pressure DragTurbulent B.L. Separation Point

GOLF BALL AERODYNAMICS

• Pressure drag dominates sphere• Dimples encourage formation of

turbulent B.L.• Turbulent B.L. less susceptible

to separation• Delayed separation → Less drag

Laminar B.L. Turbulent B.L.

Lam

inar

B.L

.Tu

rbul

ent B

.L.

EXAMPLE: BOEING 727

• Designed in 1960’s to operate out of airports with relatively short runways• Desire to minimize take-off and landing distances• Maximum CL = 3.0

• For W = 160,000 lb, Wing Area = 1,650 ft2, Vstall ~ 113 MPH

EXAMPLE: F-104 LOCKHEED STARFIGHTER

• First airplane designed for sustained flight at Mach 2• Very sharp leading edge on wings (razor sharp leading edges, thickness 3.4 %)• Designed to minimize wave drag at supersonic speeds• Very poor low-speed aerodynamic performance• Such wings tend to stall at low angles of attack, CLmax is only about 1.15

• Vstall (full of fuel) ~ 198 MPH

• Vstall (fuel empty) ~ 152 MPH

• Vstall proportional to W1/2

INFINITE VERSUS FINITE WINGS

SbAR

2

Aspect Ratiob: wingspanS: wing area

High AR

Low AR