MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Overview of Compressible Flows:Critical Mach Number and Wing Sweep

April 25, 2011

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

EXAMPLES: INFLUENCE OF COMPRESSIBILITY

M∞ < 1

M∞ > 1

M∞ ~ 0.85

WHEN IS FLOW COMPRESSIBLE?

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3

Mach Number

Sta

gn

atio

n t

o S

tati

c D

ensi

ty R

atio

Cp/Cv=1.4

1

1

20

2

11

M

WHEN IS FLOW COMPRESSIBLE?

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3

Mach Number

Sta

gn

atio

n t

o S

tati

c D

ensi

ty R

atio

Cp/Cv=1.4

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2 0.3 0.4 0.5

Mach Number

Sta

gn

atio

n t

o S

tati

c D

en

sity

Rat

ioCp/Cv=1.4

1

1

20

2

11

M

EXAMPLES: COMPRESSIBLE INTERNAL FLOW

77*

A

A

A

A e

throat

exit

EXAMPLE: H2 VARIABLE SPECIFIC HEAT, CP

COMPRESSIBILITY SENSITIVITY WITH

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3

Mach Number

Sta

gn

atio

n t

o S

tati

c D

ensi

ty R

atio

Cp/Cv=1.4Cp/Cv=1.2

PRESSURE COEFFICIENT, CP

• Use non-dimensional description instead of plotting actual values of pressure

• Pressure distribution in aerodynamic literature often given as Cp

• So why do we care?

– Distribution of Cp leads to value of cl

– Easy to get pressure data in wind tunnel

– Shows effect of M∞ on cl

2

21

V

pp

q

ppC p

EXAMPLE: CP CALCULATION

See §4.10

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

COMPRESSIBILITY CORRECTION:EFFECT OF M∞ ON CP

20,

21

V

pp

q

ppC p

Flight Mach Number, M∞

Cp

at a

poi

nt o

n an

air

foil

of

fixe

d sh

ape

and

fixe

d an

gle

of a

ttac

k

22

0,

1

5.0

1

MM

CC pp

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

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

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

COMPRESSIBILITY CORRECTION:EFFECT OF M∞ ON CP

M∞

EXAMPLE: SUPERSONIC WAVE DRAG

F-104 Starfighter

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∞ < 1INCREASED DRAG!

CRITICAL FLOW AND SHOCK WAVES

MCR

0.1 DivergenceDragCR MM

• Sharp increase in cd is combined effect of shock waves and flow separation

• Freestream Mach number at which cd begins to increase rapidly called Drag-Divergence Mach number

CRITICAL FLOW AND SHOCK WAVES

‘bubble’ of supersonic flow

CRITICAL FLOW AND SHOCK WAVES

MCR

EXAMPLE: IMPACT ON AIRFOIL / WING DRAG

wdpdfdd

wavepressurefriction

cccc

DDDD

,,,

Only at transonic andsupersonic speedsDwave= 0 for subsonic speedsbelow Mdrag-divergence

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

AIRFOIL THICKNESS SUMMARY

• Which creates most lift?– Thicker airfoil

• Which has higher critical Mach number?– Thinner airfoil

• Which is better?– Application dependent!

Note: thickness is relativeto chord in all casesEx. NACA 0012 → 12 %

CAN WE PREDICT MCR?

1

2

2

0

0

2,

21

1

21

1

12

A

A

A

AAp

M

M

ppp

p

p

p

p

p

MC

A

• Pressure coefficient defined in terms of Mach number (instead of velocity)

PROVE THIS FOR CONCEPT QUIZ

• In an isentropic flow total pressure, p0, is constant

• May be related to freestream pressure, p∞, and static pressure at A, pA

CAN WE PREDICT MCR?

1

21

1

21

12

1

21

1

21

12

12

2,

1

2

2

2,

CR

CRCRP

A

Ap

M

MC

M

M

MC • Combined result

– Relates local value of CP to local Mach number

– Can think of this as compressible flow version of Bernoulli’s equation

• Set MA = 1 (onset of supersonic flow)

• Relates CP,CR to MCR

HOW DO WE USE THIS?1. Plot curve of CP,CR vs. M∞

2. Obtain incompressible value of CP at minimum pressure point on given airfoil

3. Use any compressibility correction (such as P-G) and plot CP vs. M∞

– Intersection of these two curves represents point corresponding to sonic flow at minimum pressure location on airfoil

– Value of M∞ at this intersection is MCR

2

0,

1

M

CC pp

1

3

2

1

21

1

21

12

12

2,

CR

CRCRP

M

MC

IMPLICATIONS: AIRFOIL THICKNESS

• Thick airfoils have a lower critical Mach number than thin airfoils

• Desirable to have MCR as high as possible

• Implication for design → high speed wings usually design with thin airfoils

– Supercritical airfoil is somewhat thicker

Note: thickness is relativeto chord in all casesEx. NACA 0012 → 12 %

THICKNESS-TO-CHORD RATIO TRENDSA-10Root: NACA 6716TIP: NACA 6713

F-15Root: NACA 64A(.055)5.9TIP: NACA 64A203

Flight Mach Number, M∞

Thi

ckne

ss to

cho

rd r

atio

, %

ROOT TO TIP AIRFOIL THICKNESS TRENDS

http://www.nasg.com/afdb/list-airfoil-e.phtml

Root Mid-Span Tip

Boeing 737

SWEPT WINGS• All modern high-speed aircraft have swept wings: WHY?

WHY WING SWEEP?

V∞V∞

Wing sees component of flow normal to leading edge

WHY WING SWEEP?

V∞

Wing sees component of flow normal to leading edge

V∞

V∞,n

V∞,n < V∞

SWEPT WINGS: SUBSONIC FLIGHT• Recall MCR

• If M∞ > MCR large increase in drag

• Wing sees component of flow normal to leading edge

• Can increase M∞

• By sweeping wings of subsonic aircraft, drag divergence is delayed to higher Mach numbers

SWEPT WINGS: SUBSONIC FLIGHT• Alternate Explanation:

– Airfoil has same thickness but longer effective chord

– Effective airfoil section is thinner

– Making airfoil thinner increases critical Mach number

• Sweeping wing usually reduces lift for subsonic flight

SWEPT WINGS: SUPERSONIC FLIGHT

M

1sin 1

• 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

WING SWEEP COMPARISONF-100D English Lightning

SWEPT WINGS: SUPERSONIC FLIGHT

º(M=1.2) ~ 56º(M=2.2) ~ 27º

SU-27M∞ < 1

M∞ > 1

WING SWEEP DISADVANTAGE

• Wing sweep beneficial in that it increases drag-divergences Mach number• Increasing wing sweep reduces the lift coefficient

• At M ~ 0.6, severely reduced L/D

• Benefit of this design is at M > 1, to sweep wings inside Mach cone

TRANSONIC AREA RULE• Drag created related to change in cross-sectional area of vehicle from nose to tail

• Shape itself is not as critical in creation of drag, but rate of change in shape

– Wave drag related to 2nd derivative of volume distribution of vehicle

EXAMPLE: YF-102A vs. F-102A

EXAMPLE: YF-102A vs. F-102A

CURRENT EXAMPLES• No longer as relevant today – more

powerful engines

• F-5 Fighter

• Partial upper deck on 747 tapers off cross-sectional area of fuselage, smoothing transition in total cross-sectional area as wing starts adding in

• Not as effective as true ‘waisting’ but does yield some benefit.

• Full double-decker does not glean this wave drag benefit (no different than any single-deck airliner with a truly constant cross-section through entire cabin area)

EXAMPLE OF SUPERSONIC AIRFOILS

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

SUPERSONIC AIRFOIL MODELS• Supersonic airfoil modeled as a

flat plate

• Combination of oblique shock waves and expansion fans acting at leading and trailing edges

– R’=(p3-p2)c

– L’=(p3-p2)c(cos

– D’=(p3-p2)c(sin

• Supersonic airfoil modeled as double diamond

• Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner

– D’=(p2-p3)t

APPROXIMATE RELATIONS FOR LIFT AND DRAG COEFFICIENTS

1

4

1

4

2

2

,

2

Mc

Mc

wd

l

http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/home.htm

CASE 1: =0°

Shock waves Expansion

CASE 1: =0°

CASE 2: =4°

Aerodynamic Force Vector

Note large L/D=5.57 at =4°

CASE 3: =8°

CASE 5: =20°

At around =30°, a detached shock begins to form before bottom leading edge

CASE 6: =30°

DESIGN OF ASYMMETRIC AIRFOILS

QUESTION 9.14• Consider a diamond-wedge airfoil as shown in Figure 9.36, with half angle =10°

• Airfoil is at an angle of attack =15° in a Mach 3 flow.

• Calculate the lift and wave-drag coefficients for the airfoil.

Compare with your solution

EXAMPLE: MEASUREMENT OF AIRSPEED• Pitot tubes are used on aircraft as speedometers (point measurement)

SubsonicM < 0.3

SubsonicM > 0.3

SupersonicM > 1

M < 0.3 and M > 0.3: Flows are qualitativelysimilar but quantitatively different

M < 1 and M > 1: Flows arequalitatively and quantitatively different

MEASUREMENT OF AIRSPEED:INCOMPRESSIBLE FLOW (M < 0.3)

• May apply Bernoulli Equation with relatively small error since compressibility effects may be neglected

• To find velocity all that is needed is pressure sensed by Pitot tube (total or stagnation pressure) and static pressure

Comment: What is value of ?

• If is measured in actual air around airplane (difficult to do)

– V is called true airspeed, Vtrue

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

– V is called equivalent airspeed, Ve

02

12

1pVp

pp

V

01

2

Staticpressure

Dynamicpressure

Totalpressure

Incompressible Flow

MEASUREMENT OF AIRSPEED:SUBSONIC COMRESSIBLE FLOW (0.3 < M < 1.0)

• 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

MEASUREMENT OF AIRSPEED:SUBSONIC COMRESSIBLE FLOW (0.3 < M < 1.0)

• 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 give flight Mach number

• Instrument called Mach meter

• M1 = V1/a1

• V1 is actual flight speed

• Actual flight speed using pressure difference

• What are T1 and a1?

• Again use sea-level conditions Ts, as, ps (a1 = (RT)½ = 340.3 m/s)

• V is called Calibrated Velocity, Vcal

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

EXAMPLE: SUBSONIC AND SUPERSONIC FLIGHT• Flight at four different speeds, pitot measures p0 = 1.05, 1.2, 3 and 10 atm

• What is flight speed if flying in 1 atm static pressure and Tambient = 288 K (a = 340 m/s)?

• Determine which measurements are in subsonic or supersonic flow

– p0/p = 1.893 is boundary between subsonic and sonic flows

• 1.05 atm → p0/p = 1.05 → subsonic

– Use compressible flow form, M = 0.265, V ~ 90 m/s ~ 200 MPH– Could use Bernoulli which will provide small error (~ 1%) and give V directly– Compressible form requires knowledge of speed of sound (temperature)

– Apply Bernoulli safely? p0/p < 1.065

• 1.2 atm → p0/p = 1.2 → subsonic

– M = 0.52, V ~ 177 m/s ~ 396 MPH– Use of compressible subsonic form justified (Bernoulli ~ 3% error)

• 3 atm → p02/p1 = 3 → supersonic

– M1 = 1.39, V ~ 473 m/s ~ 1057 MPH (Bernoulli ~ 22% error)

• 10 atm → p02/p1 = 10 → supersonic

– M1 = 2.73, V ~ 928 m/s ~ 2076 MPH (MCO → LAX in 1 hour 30 minutes)