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HIGH-SPEED AERODYNAMICSMACE 31321

Lecture 8 Expansion Waves

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OBJECTIVES OF THIS LECTURE

• To derive the Prandtl-Meyer function for expansion waves

• To learn how to calculate the flow properties for expansion waves

• To learn how to apply shock wave and expansion wave theory to supersonic aerofoils

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EXPANSION WAVE

When a supersonic flow is “turning away from itself”, an expansion fan forms.

Isentropic process: ∆s=0

Across the waves, M increases CONTINOUSLY, whereas p, ρ and T decreases CONTINOUSLY.

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PRANDTL-MEYER EXPANSION WAVES

1

11

1sinM

−=µ

2

12

1sinM

−=µ

An expansion fan can be visualised as an infinite number of Mach waves.

Centered expansion waves are commonly denoted as Prandtl-Meyer expansion waves.

The problem is to calculate the downstream flow for a given upstream flow and the deflection angle

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PRANDTL-MEYER EXPANSION WAVES• Consider a very weak expansion wave produced by

an infinitesimally small deflection dθ

µ

Mach wave

Vdθ

V+dV

w2w1

µ

µcos1 Vw =

( ) ( )θµ ddVVw ++= cos2

21 ww = ( ) ( )θµµ ddVVV ++= coscos

( )θµµdV

dVV+

=+

coscos

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PRANDTL-MEYER EXPANSION WAVES• Velocity changes across a very weak wave

( )θµµdV

dVV+

=+

coscos

θµθµµ

ddVdV

sinsincoscoscos1

−=+

1cos,sin ≈≈ θθθ dddFor small dθ:

µθµθµµ

tan11

sincoscos1

ddVdV

−=

−=+

)1(...11

1 2 <+++=−

xxxx

Since

µθ tan11 dVdV

+=+

µθ

tan/VdVd =

Let x=dθ tanµ and Neglecting 2nd order and higher:

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PRANDTL-MEYER EXPANSION WAVES• Velocity changes across a very weak wave

µθ

tan/VdVd =

1

M

12 −M

µ

11tan2 −

=M

µ

VV12 dMd −=θ ∫∫ −==

2

1VV12

0

M

M

dMdθ

θθ

Relation between V and M has to be found first.

M1sin =µ

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RELATION BETWEEN V AND M• From

• We get

• Substituting

∫∫ −==2

1VV12

0

M

M

dMdθ

θθ

MaV =

22

211 M

TT

aa oo −

+==⎟⎠⎞

⎜⎝⎛ γ

( ) MdM

MVdV

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+=

21211

1

γ

( ) MdM

M

MM

M∫

−+

−=

2

12

2

1211

1

γθ

ada

MdM

VdV

+=

2/12

211

−

⎟⎠⎞

⎜⎝⎛ −

+= Maa oγ M

ada ~

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• Re-arranging

• ν(M) is called the Prandtl-Meyer function. It is tabulated in Table C.

• Given M1, M2 can be found by knowing

PRANDTL-MEYER FUNCTION

( ) MdM

M

MM

M∫

−+

−=

2

12

2

1211

1

γθ

( ) ( ) MdM

M

MM

dM

M

M MM

∫∫−+

−−

−+

−=

12

1 2

2

1 2

2

1211

1

1211

1

γγ

( ) ( )12 MM ννθ −=

( ) ( ) θνν += 12 MM10

CALCULATION PROCEDURES

• Obtain the flow properties downstream of a convex corner for given flow properties upstream and the deflection angle θ.– For a given M1, obtain ν(M1) from Table C– Calculate ν(M2) from . – Obtain M2 from ν(M2) using Table C– Use Table A to find po/p, To/T at M2 and M1

respectively.– Since the expansion is isentropic, po,2= po,1,

To,2=To,1, T2/T1 and p2/p1 can then be found.

( ) ( ) θνν += 12 MM

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QUESTION

• A supersonic flow with M1=1.5, p1=1atm, and T1=288K is expanded around a sharp corner through a deflection angle of 15o,. Calculate – M2 Answer: 2.0– p2 Answer: 0.469atm– T2 Answer: 232K– p02 Answer: 3.671atm– T0,2 Answer: 417.6K– The angle that forward and rearward Mach lines make

with respect to the upstream flow direction. Answer: 41.81o, 15o

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SOLUTIONS

• From Table C, for M1=1.5, ν(Μ1)=11.91o.

• Hence ν(Μ2)= ν(Μ1)+θ=11.91+15=26.91o

• From Table C, M2=2.0 (rounding to the nearest entry in table)

• From Table A,

45.1,671.31

1,0

1

1,0 ==TT

pp

8.1,824.72

2,0

2

2,0 ==TT

pp

For M1=1.5,

For M2=2.0,

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SOLUTIONS• Since the flow is isentropic, po,2= po,1, To,2=To,1, Thus

• From Table C

atmppp

pp

ppp 469.01671.31

824.71

11

1,0

1,0

2,0

2,0

22 =×××==

KTTT

TT

TTT 23228845.11

8.11

11

1,0

1,0

2,0

2,0

22 =×××==

atmppp

pp 671.31671.311

1,01,02,0 =×===

KTTT

TT 6.41728845.111

1,01,02,0 =×===

Angle of forward Mach line = µ1=41.8o

Angle of rearward Mach line = µ2 – θ = 30 – 15 = 15o

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SUPERSONIC AEROFOILS• Consider a flat plate of length c at an angle of

attack of α in a supersonic flow.

– Sketch the pattern of the shock wave and expansion waves on the plate.

– How is the level of static pressure acting on the upper and lower surfaces of the plate compared to p1?

– Does the plate experience a lift and a drag?

M1>1

p1

α

p3

p2

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• Consider a flat plate of length c at an angle of attack of α in supersonic flow.

– Sketch the pattern of the shock wave and expansion waves on the plate.

– How is the level of static pressure acting on the upper and lower surfaces of the plate compared to p1?

– Does the plate experience a lift and a drag?

SUPERSONIC AEROFOILS

M1>1

p1

α

p3 > p1

p2< p1

RL

D

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• In supersonic inviscid flow over a 2D body, the drag is always finite.

• The drag is produced by the presence of shock waves and is called the wave drag.

SUPERSONIC AEROFOILS

RL

D

( )( )( ) α

αsincos

23

23

23

ppcDppcL

cppR

−=

−=

−=M1>1

p1

α

p3

p2

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HOMEWORK

• Consider a wedge with a θ=15o half angle in a Mach 5 flow. Assume the pressure at the base is equal to the freestream static pressure. • Sketch the wave pattern on the wedge• Derive the relation between the drag coefficient and

the pressure acting on each side of the wedge.• Calculate the drag coefficient. (CD=0.114)

θM1=5

l

c

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SOLUTION• From the sketch, the drag force

• The drag coefficient• Since

θM1=5

p2

p2

p1

[ ]1212 sin2sin2sin2 ppllplpD −=−= θθθ

θcoscl = [ ]12tan2 ppcD −= θ

2115.0 Vc

DCD ρ=

21

211

211 5.05.0 aMV ρρ ==

211

1211

5.0

5.0

MpRTM

γ

γρ

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −==

1

1221

211

tan45.0 p

ppMVc

DCD γθ

ρ

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SOLUTION• From the θ-β-M chart, for M1=5 and θ=15o, β=24.2o,

the normal Mach number ahead of the shock is

• From the normal shock table, for Mn,1=2.05, we have

• Hence

05.22.24sin5sin11, =×== on MM β

736.41

2 =pp

θM1=5

p2

p2

p1

( )

114.0

1736.454.115tan4

tan4

2

1

1221

=

−×

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

o

D ppp

MC

γθ

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REFERENCES

• In “Fundamentals of Aerodynamics” by Andersons, 2nd edition.–§9.6–§9.7