hsa[8] hspeed notes
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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