MAE 1202: AEROSPACE PRACTICUM

Lecture 5: Compressible and Isentropic Flow 1

February 11, 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 4, Sections 4.10 - 4.21, 4.27

– For next week’s lecture: Chapter 5, Sections 5.1 - 5.13

• Lecture-Based Homework Assignment:

– Problems: 4.7, 4.11, 4.18, 4.19, 4.20, 4.23, 4.27

• DUE: Friday, February 22, 2013 by 5 PM

– 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

CREO DESIGN CONTEST

• Create most elaborate, complex, stunning Aerospace Related project in Creo

• Criteria: Assembly and/or exploded view

• First place

– Either increase your grade by an entire letter (C → B), or

– Buy your most expensive textbook next semester

• Second place: +10 points on final exam

• Third place: +10 points on final exam

CAD DESIGN CONTEST

CAD DESIGN CONTEST

If you do the PRO|E challenge…

Do not let it consume you!

BERNOULLI’S EQUATION

2

222

21

1

22

2

Vp

Vp

Vp

• One of most fundamental and useful equations in aerospace engineering!

• Remember:

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

– Bernoulli’s equation relates properties between different points along a streamline

– For a compressible flow Euler’s equation must be used ( is 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

EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)• How do we measure an airplanes speed in flight?

• Pitot tubes are used on aircraft as speedometers (point measurement)

14

STATIC VS. TOTAL PRESSURE• In aerodynamics, 2 types of pressure: Static and Total (Stagnation)

• Static Pressure, p– Due to random motion of gas molecules– Pressure we would feel if moving along with flow– Strong function of altitude

• Total (or Stagnation) Pressure, p0 or pt

– Property associated with flow motion– Total pressure at a given point in flow is the pressure that would exist if flow were

slowed down isentropically to zero velocity

• p0 ≥ p

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW

02

12

1pVp

pp

V

01

2

Staticpressure

Dynamicpressure

Totalpressure

Incompressible Flow

Total and Static Ports

16

17

TOTAL PRESSURE MEASUREMENT (4.11)

• Measures total pressure

• Open at A, closed at B

• Gas stagnated (not moving) anywhere in tube

• Gas particle moving along streamline C will be isentropically brought to rest at point A, giving total pressure

18

EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)• Point A: Static Pressure, p

– Only random motion of gas is measured

• Point B: Total Pressure, p0

– Flow is isentropically decelerated to zero velocity

• A combination of p0 and p allows us to measure V1 at a given point

• Instrument is called a Pitot-static probe p0

p

19

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW

02

12

1pVp

02

12

1pVp

pp

V

01

2

Staticpressure

Dynamicpressure

Totalpressure

Incompressible Flow

20

TRUE VS. EQUIVALENT AIRSPEED• What is value of ?

• If is measured in actual air around the airplane

• Measurement is difficult to do

• Practically easier to use value at standard seal-level conditions, s

• This gives an expression called equivalent airspeed

pp

Vtrue

02

s

e

ppV

02

TRAGIC EXAMPLE: Air France Crash

• Aircraft crashed following an aerodynamic stall caused by inconsistent airspeed sensor readings, disengagement of autopilot, and pilot making nose-up inputs despite stall warnings

• Reason for faulty readings is unknown, but it is assumed by accident investigators to have been caused by formation of ice inside pitot tubes, depriving airspeed sensors of forward-facing air pressure.

• Pitot tube blockage has contributed to airliner crashes in the past 21

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 Force = 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?

Incorrect Lift Theory• http://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html

SUMMARY OF GOVERNING EQUATIONS (4.8)

222

211

2211

2

1

2

1VpVp

VAVA

• Steady, incompressible flow of an

inviscid (frictionless) fluid along a streamline or in a stream tube of varying area

• Most important variables: p and V

• T and are constants throughout flow

continuity

Bernoulli

What if flow is high speed, M > 0.3?

What if there are temperature effects?

How does density change?

1st LAW OF THERMODYNAMICS (4.5)

System

e (J/kg)

Boundary

Surroundings

• System (gas) composed of molecules moving in random motion

• Energy of molecular motion is 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) system

2. Work, w, is done on (or by) system

THOUGHT EXPERIMENT #1

• Do not allow size of balloon to change (hold volume constant)

• Turn on a heat lamp

• Heat (or q) is added to the system

• How does e (internal energy per unit mass) inside the balloon change?

THOUGHT EXPERIMENT #2

• *You* take balloon and squeeze it down to a small size

• When volume varies work is done

• Who did the work on the balloon?

• How does e (internal energy per unit mass) inside the balloon change?

• Where did this increased energy come from?

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

ddp

a 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

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

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