Analysis of Expansion Waves

P M V SubbaraoAssociate Professor

Mechanical Engineering DepartmentI I T Delhi

Another Zero Cost Nozzle …..

Theory of Extrapolation of Physics

So if > 0 .. Compression around a concave corner

M1

M2

So if = 0 .. No Compression

Expansion Wave : Another Shock !!?!!

Consider the scenario shown in the adjacent figure. As a supersonic flow turns, the normal component of the velocity increases (w2 > w1).

The tangential component remains constant (v2 = v1).

The corresponding change is the entropy (Δs = s2 − s1) can be expressed

as follows,

1

2

1

222 lnln1 p

p

T

T

R

ss

As this is an expansion wave : T2 < T1 & p2 < p1

2

1

1

1

2

12 ln

pp

TT

R

ss

1

2

1

1

1

2

pp

TT

0ln

2

1

1

1

2

12

pp

TT

R

ss

Expanding Shock is Impossible !!!

1. A Finite Expansion wave shows Δs < 0.

2. Since this is not possible it means that it is impossible to turn a flow through a single shock wave.

0lnlimlim

2

1

1

1

2

12

1212

pp

TT

R

sspppp

3. The argument may be further extended to show that such an expansion process can occur only if we consider a turn through infinite number of expansion waves in the limit.

4. Accordingly an expansion process is an isentropic process.

Pressure and Temperature Change Across Expansion Fan

• Because each mach wave is infinitesimal, expansion is isentropic

- P02 = P01

- T02 = T01

p2

p1

P01

p1

p2

P02

1

1

2M1

2

1 1

2M2

2

1

T2

T1

T01

T1

T2

T 02

1

1

2M1

2

1 1

2M2

2

• Then it follows that < 0 .. We get an expansion wave

Prandtl-Meyer Expansion Waves

• Flow accelerates around corner.• Continuous flow region … sometimes called “expansion fan” consisting of a series of Mach waves.• Each Mach wave is infinitesimally weak isentropic flow region.• Flow stream lines are curved and smooth through fan.

Analysis of Prandtl-Meyer Expansion

• Consider flow expansion around an infinitesimal corner

Infinitesimal Expansion Fan Flow Geometry

V

d

Mach Wave

dd

V

V+dV

• From Law of Sines

V

sin2

d

V dV

sin2

Infinitesimal Expansion Fan Flow Geometry

V

V+dV d

Mach Wave

dd

V

• Using the trigonometric identities

sin2

sin2

cos sin cos2

cos

sin2

sin2

cos sin cos2

cos

sin2

d

sin2

cos d cos2

sin d

cos cos d cos2

cos sin sin 2

sin d

cos cos d sin sin d

&

• Substitution gives

• Since d is considered to be infinitesimal

cos d 1

sin d d

V

cos cos d sin sin dV dV

cos

1 dV

V cos

cos cos d sin sin d

• and the equation reduces to

1dV

V

cos cos sin d

1

1 tan d

• Exploiting the form of the power series (expanded about x=0)

1

1 x 1 x |x0

1

1 x 2

|x0

( 1)

x 0 ....O x2

xxx

1

1

1lim

0

• Since dV is infinitesimal … truncate after first order term

1

1 dVV

1 dV

V

1

1 dVV

1

1 tan d

• Solve for d in terms of dV/V

1 tan d 1 dV

V d 1

tan dV

V

• Using Mach Wave Relations:

sin 1

M

• Performing some algebraic and trigonometric voodoo

sin 1

M sin2

1

M 2

sin2 cos2 M 2

M 2 sin2 cos2

sin2 11

tan2 1

tan2 M 2 1

1

tan M 2 1

• and ….

d 1

tan dV

V M 2 1

dV

V• Valid forReal and ideal gas

• For a finite deflection the O.D.E is integrated over the complete expansion fan

M 2 1dV

VM1

M2

• Write in terms of mach by …

V M c dV dM c M dc dV

V

dM c M dc

M c

dM

M

dc

c

• Substituting in

M 2 1dV

VM1

M2

M 2 1dM

M

dc

c

M1

M2

• For a calorically perfect adiabatic gas flow

And T0 is constant

c0 RgT0 c0

c

T0

T

1 1

2M 2

2

0

21

1 M

cc

MdM

M

cdc 1

21

12

12/3

2

0

2

0

2/32

0

21

1

1

21

121

M

c

MdM

M

c

c

dc

0

2

2/32

0 21

1

1

21

12

1

c

M

MdM

M

c

c

dc

2

21

12

1

M

MdM

c

dc

• Returning to the integral for

V M2 1d

VM1

M2

M2 1dM

M

( 1)2

M dM

1 1

2M2

M1

M2

• Simplification gives

M 2 1dM

M1

( 1)

2M 2

1 1

2M 2

M1

M2

M 2 1dM

M

1 1

2M 2

( 1)

2M 2

1 1

2M 2

M1

M2

M 2 1

dM

M

1 1

2M 2

M1

M2

• Evaluate integral by performing substitution

dM

Mdu,M2 e2u

M2 1dMM

1 1

2M2

e2u 1

1 1

2e2u

du

Let

• Standard Integral Table Form

• From tables (math handbook)

e2u 1

1 1

2e2u

du emx 1

1 bemx dux

emx 1

1 bemx du 2

mtan 1 emx 1

2 b 1 m

tan 1 b

b 1emx 1

b 1 bmx

• Substituting m 2,b 1

2,emx M 2

M2 1dM

M

1 1

2M2

2

2tan 1 M2 1 2

2

1

21

1

2

tan 1

1

2 1

21

M2 1

1 1

tan 1 1 1

M2 1

tan 1 M2 1

(M ) 1 1

tan1 1

1M 2 1

tan

1 M 2 1Let

• More simply

(M)“Prandtl-Meyer Function”

Implicit function … more Newton!

12 MM

1

1tan 1 1

1M 2

2 1

tan 1 M 2

2 1

1

1tan 1 1

1M1

2 1

tan 1 M1

2 1

M2 versus M1,

M1= 5

M1= 3

M1= 1

Pressure and Temperature Change Across Expansion Fan

• Because each mach wave is infinitesimal, expansion is isentropic

- P02 = P01

- T02 = T01

p2

p1

P01

p1

p2

P02

1

1

2M1

2

1 1

2M 2

2

1

T2

T1

T 01

T1

T2

T 02

1

1

2M1

2

1 1

2M 2

2

Maximum Turning Angle

•How much a supersonic flow can turn through.• A flow has to turn so that it can satisfy the boundary conditions. •In an ideal flow, there are two kinds of boundary condition that the flow has to satisfy,

•Velocity boundary condition, which dictates that the component of the flow velocity normal to the wall be zero. •It is also known as no-penetration boundary condition. •Pressure boundary condition, which states that there cannot be a discontinuity in the static pressure inside the flow.

• If the flow turns enough so that it becomes parallel to the wall, we do not need to worry about this boundary condition.

• However, as the flow turns, its static pressure decreases. • If there is not enough pressure to start with, the flow won't be

able to complete the turn and will not be parallel to the wall. • This shows up as the maximum angle though which a flow can

turn. Lower to Mach number to start with (i.e. small M1), greater the maximum angle though which the flow can turn.

• The streamline which separates the final flow direction and the wall is known as a slipstream.

• Across this line there is a jump in the temperature, density and tangential component of the velocity (normal component being zero).

• Beyond the slipstream the flow is stagnant (which automatically satisfies the velocity boundary condition at the wall).

• In case of real flow, a shear layer is observed instead of a slipstream, because of the additional no-slip boundary condition.

Maximum Turning Angle

p2

0 M2

() 1 1

tan 1 1 1

2 1

tan 1 2 1

1 1

1

2

max 1 1

1

2

1 1

tan 1 1 1

M12 1

tan 1 M1

2 1

p2

p1

P01

p1

p2

P02

1

1

2M1

2

1 1

2M 2

2

1

• Plotting as a max function of Mach number

• {T2, p2} = 0

Highest Value for Maximum Turning Angles

Anatomy of Prandtl – Meyer Expansion Wave

Combination of Shock & Expansion Wave

An Important Product !!!

Supersonic Flow Over Flat Plates at Angle of Attack

Review: Oblique Shock Wave Angle

tan 2 tan M1 sin

2 1

tan2 2 M12 cos 2

2 M1

2 sin2 1 tan 2 M1

2 cos 2

Prandtl-Meyer Expansion Waves

<0 .. We get an expansion wave (Prandtl-Meyer)

(M 2 ) (M1) (M ) 1

1tan 1 1

1M 2 1

tan 1 M 2 1