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

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

ANSWERS TO LECTURE HOMEWORK• 5.2: L = 23.9 lb, D = 0.25 lb, Mc/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 cm,c/4 is different than that for cl, so be careful when reading the data

• 5.3: L = 308 N, D = 2.77 N, Mc/4 = - 0.925 N m

• 5.4: = 2°

• 5.6: (L/D)max ~ 112

1st 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) system2. Work, w, is done on (or by) system

SYSTEM(unit mass of gas)

Boundary

SURROUNDINGS

q

wqde

e (J/kg)

1st LAW IN MORE USEFUL FORM (4.5)

• 1st 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 ↑

pdvqde

wqde

pdvw

sdAppsdAw

spdA

AA

ΔW

distanceforceΔW

Change inVolume (-)

ENTHALPY: A USEFUL QUANTITY (4.5)

vdpdhq

vdpdedhdeq

pdvdeq

vdppdvdedh

RTepveh

Define a new quantitycalled enthalpy, h:(recall ideal gas law: pv = RT)

Differentiate

Substitute into 1st law(from previous slide)

Another version of 1st lawthat uses enthalpy, h:

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, N2, Ar, water, lead, uranium, etc…

• ALSO, for a fixed dq, resulting dT depends on type of process…

Kkg

J

dT

qc

q

d

SPECIFIC HEAT: CONSTANT PRESSURE• Addition of q will cause a small change in temperature dT of system• System pressure remains constant

Tch

dTcdh

dTcq

dT

qc

p

p

p

p

pressureconstant

q

d

Kkg

J

dT

qc

Extra Credit #1:Show this step

SPECIFIC HEAT: CONSTANT VOLUME• Addition of q will cause a small change in temperature dT of system• System volume remains constant

Kkg

J

dT

qc

q

d

Tce

dTcde

dTcq

dT

qc

v

v

v

v

olumeconstant v

Extra Credit #2:Show this step

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

Tch

dTcdh

dTcq

dT

qc

p

p

p

p

pressureconstant

• However, for a fixed dq, resulting dT depends on type of process:

Tce

dTcde

dTcq

dT

qc

v

v

v

v

olumeconstant v

Kkg

J

dT

qc

v

p

c

c

Specific heat ratioFor air, = 1.4

Constant Pressure Constant Volume

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

1

1

2

1

2

1

2

T

T

p

p = ratio of specific heats= cp/cv

air=1.4

DERIVATION: ENERGY EQUATION (4.7)

022

0

0

0

0

0

21

22

12

2

1

2

1

VVhh

VdVdh

VdVdh

VdVvdh

VdVdp

vdpdhq

q

wqde

V

V

h

h

Energy can neither be created nor destroyedStart with 1st law

Adiabatic, q=01st law in terms of enthalpy

Recall Euler’s equation

Combine

Integrate

Result: frictionless + adiabatic flow

ENERGY EQUATION SUMMARY (4.7)• Energy can neither be created nor destroyed; can only change physical form

– Same idea as 1st law of thermodynamics

constant2

222

22

2

21

1

Vh

Vh

Vh

constant2

222

22

2

21

1

VTc

VTc

VTc

p

pp

Energy equation for frictionless,adiabatic flow (isentropic)

h = enthalpy = e+p/= e+RTh = cpT for an ideal gas

Also energy equation forfrictionless, adiabatic flow

Relates T and V at two different points along a streamline

SUMMARY OF GOVERNING EQUATIONS (4.8)STEADY AND INVISCID FLOW

222

211

2211

2

1

2

1VpVp

VAVA

222

111

222

211

1

2

1

2

1

2

1

222111

2

1

2

1

RTp

RTp

VTcVTc

T

T

p

p

VAVA

pp

• 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 stateat any point

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:

RTp

a

a

VM

• 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

d

dpa 2

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

EXAMPLES AND APPLICATIONS

Measurement of AirspeedShock Waves

Supersonic Wind Tunnels and Rocket Nozzles

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

21

1

0

1

21

1

0

02

11

2

11

21

2

1

MT

T

Tc

V

T

T

TcVTc

p

pp

11

21

1

0

12

11

0

2

11

2

11

M

Mp

p

cp: specific heat at constant pressureM1=V1/a1

air=1.4

MEASUREMENT OF AIRSPEED:SUBSONIC COMRESSIBLE FLOW

• So, how do we use these results to measure airspeed

111

2

111

2

11

2

11

2

1

102

2

1

1

10212

1

1

1

0212

1

1

1

021

s

scal p

ppaV

p

ppaV

p

paV

p

pM

p0 and p1 giveFlight Mach numberMach meter

M1=V1/a1

Actual Flight Speed

Actual Flight Speedusing pressure difference

What is T1 and a1?Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

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’?

• T0 = 200(1+ 0.2(36)) = 1,640 K!

21

1

0

2

11 M

T

T

Total temperature

Static temperature Vehicle flightMach number

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)

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

MEASUREMENT OF AIRSPEED:SUPERSONIC FLOW

1

21

124

1 21

1

21

21

2

1

02

M

M

M

p

p

Notice how different this expression is from previous expressionsYou will learn a lot more about shock wave in compressible flow course

SUMMARY OF AIR SPEED MEASUREMENT

• Subsonic, incompressible

• Subsonic, compressible

• Supersonic

1

21

124

1 21

1

21

21

2

1

02

M

M

M

p

p

111

21

102

2

s

scal p

ppaV

s

e

ppV

02

HOW ARE ROCKET NOZZLES SHAPPED?

MORE ON SUPERSONIC FLOWS (4.13)

V

dVM

A

dAV

dV

A

dA

a

VdV

V

dV

A

dA

dp

VdVd

VdVdp

V

dV

A

dAd

AV

1

0

0

0

constantlnlnVlnAln

constant

2

2

Isentropic flow in a streamtube

Differentiate

Euler’s Equation

Since flow is isentropica2=dp/d

Area-Velocity Relation

CONSEQUENCES OF AREA-VELOCITY RELATION

V

dVM

A

dA12

• 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)

TRENDS: CONTRACTION

M1 < 1

M1 > 1

V2 > V1

V2 < V1

1: INLET 2: OUTLET

TRENDS: EXPANSION

M1 < 1

M1 > 1

V2 < V1

V2 > V1

1: INLET 2: OUTLET

PUT IT TOGETHER: C-D NOZZLE

1: INLET 2: OUTLET

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

Chapter 5 Overview

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

constant2

1 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?

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?

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

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 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

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/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 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

MORE DEFINITIONS• 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

– 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, 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 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 – Mac=constant (independent of )

– For low speed airfoils aerodynamic center is near quarter-chord point, c/4

AOA = 2°

AOA = 3°

AOA = 6°

AOA = 9°

AOA = 12°

AOA = 20°

AOA = 60°

AOA = 90°

SAMPLE DATA: SYMMETRIC AIRFOIL

Lif

t (fo

r no

w)

Angle of Attack,

A symmetric airfoil generates zero lift at zero

SAMPLE DATA: CAMBERED AIRFOIL

Lif

t (fo

r no

w)

Angle of Attack,

A cambered airfoil generates positive lift at zero

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

Lif

t (fo

r no

w)

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

SAMPLE DATA: STALL BEHAVIORL

ift (

for

now

)

What is really going on here

What is stall?

Can we predict it?

Can we design for it?

WHY DOES LIFT CURVE BEND OVER?

http://www.soton.ac.uk/Racing/Greenpower/BoundaryLayers/

Low

Moderate

High

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

TYPES OF FLOWS: FRICTION VS. NO-FRICTION

Flow very close to surface of airfoil isInfluenced by friction and is viscous(boundary layer flow)Stall (separation) is a viscous phenomena

Flow away from airfoil is not influencedby friction and is wholly inviscid

COMMENTS ON VISCOUS FLOWS (4.15)

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

cVRe

Within B.L. flowhighly viscous(low Re)

Outside B.L. flowInviscid (high Re)

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

LAMINAR VS. TURBULENT FLOW

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

cf,turb > cf,lam

Turbulent velocityprofiles are ‘fuller’

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

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

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

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

d

d

Same total drag as airfoil

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

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 (cl, cd, cm)

Re,,21

2

1

2

2

Mfc

Sq

L

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 (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 (cl, cd, cm)

Re,,21

2

1

3

2

2

Mfc

Scq

L

ScV

Mc

SccVM

m

m

m

Re,,21

2

1

2

2

2

Mfc

Sq

D

SV

Dc

ScVD

d

d

d

Re,,21

2

1

1

2

2

Mfc

Sq

L

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 l

c 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