Analysis of Expansion Waves P M V Subbarao Associate Professor Mechanical Engineering Department I I...
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Transcript of Analysis of Expansion Waves P M V Subbarao Associate Professor Mechanical Engineering Department I I...
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