School of Aerospace Engineering
Reflected Waves
• Already examined what happens
when normal shock “hits” a
boundary
vs
– if incident shock hits solid wall, get
reflected (normal) shock - required
v=0 v
to satisfy velocity (bc) boundary
rs
condition (v=0)
pa
– if it hits open end, get reflected
vs
expansion waves satisfy pressure bc (p=pa<p2)
• Wave reflections “impose” bc
(pressure or velocity) on flow
Reflected Waves -1
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
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AE3450
School of Aerospace Engineering
Oblique Shock Reflection From Wall
• Consider “weak” (M2>1)
oblique shock wave
impinging on a flat wall
M1
θi
δ1
M2
M3
θr
– incident shock wave turns flow toward the lower wall
– flow can not pass through boundary, must turn back
parallel to lower wall - velocity boundary condition
– flow turns back on itself ⇒ compression ⇒in this case,
reflected wave is oblique shock
• Reflected shock weaker than incident shock
– M2<M1
Reflected Waves -2
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
Example: Oblique Shock Reflection
θ1
δ1
M1
• Given: Mach 3.2 flow with static
M2
p1
pressure of 25 psia approaching a
θi
θr
17° (δ1) turn produces oblique
shock wave at 33° (θ1). Oblique shock then
“hits” bottom wall, producing reflected oblique shock.
• Find:
θi , θr , M2, M3, p2, p3
• Assume: TPG/CPG with γ=1.4, steady, adiabatic, no
work, inviscid except shocks, ….
Reflected Waves -3
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
M3
School of Aerospace Engineering
Example Oblique Shock Reflection
• Analysis:
– θi
θi = θ1 = 33o
– M2, p2
33°=θ1
M1=3.2
p1=25 psia
o
M1n = M1 sin θi = 3.2 sin(33 ) = 1.742
with M1n,
B.1 or eqs. ⇒ M 2 n
= 0.630; p 2 p1 = 3.37
( )
M 2 = M 2 n sin (θi − δ ) = 0.630 sin 16 = 2.29
o
– θr
M2=2.29
δ1=17°
M3
M2
θi
θr
δ2
θ2
p2
p 2 = p1 = 84.3 psia
p1
θr = θ2 − δ2 flow must be turned horizontal again, so δ2=δδ1 =17°
⇒ θ2 = 42.2 o ; θr = 42.2o − 17o = 25.2o ≠ θi
– M3, p3 M 2 n = M 2 sin θ2 = 2.29 sin(42.2o ) = 1.54
with M2n, B.1 or eqs. ⇒ M 3n = 0.688; p3 p 2 = 2.59
p3 = 2.59(84.3 psia ) = 218 psia
M 3 = 0.688 sin 25.2o = 1.615
C.1 or VII.26
Reflected Waves -4
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
Mach Reflection
• If M2 low enough, required
turning angle for reflected wave
may exceed maximum oblique
shock angle
M1
I
R
M>1
M2
O
M<1
W
– no simple reflected wave possible, get
something like detached shock
– IO: incident oblique shock
– OW: strong curved shock, normal at wall
– OR: weak oblique shock
Reflected Waves -5
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
Oblique Shock and Pressure BC
• If oblique shock “hits” a pressure boundary
condition, reflected wave must adjust flow
pressure to match boundary pressure
• Type of reflected wave will depend on whether
pressure must drop or rise
– pressure rise⇒compression:
flow will “turn back on itself”
– pressure drop⇒ expansion:
flow will “open up”
Reflected Waves -6
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
Reflection From Expansion on Wall
• Consider PM fan impinging
on a flat wall
δ1=ν2-ν1
M1
M2
– incident expansion waves tend to turn flow away the
lower wall
– can not create vacuum, flow must be turned back
parallel to lower wall - velocity boundary condition
– flow “opens up”⇒ expansion⇒ in this case,
reflected waves are expansions (Mach waves)
• For case shown above (flow returning to original angle)
– ν3=δ2+ν2=δ2+(δ1+ν1)=2δ+ν1 (use to get M3)
Reflected Waves -7
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
M3
School of Aerospace Engineering
Non-Simple Region
• In region where incident and reflected
waves interact, can not use our simple
quasi-1D theory
M2
M3
M1
• In this non-simple region,
– get curved waves
– flow still isentropic
Non-simple Region
(magnified here)
• Outside this region, our quasi-1D methods still valid
Reflected Waves -8
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
Summary of Reflected Waves
• “Reflections” from supersonic waves represent
information from a boundary being transmitted into
supersonic flow
– reflections “impose” boundary condition on flow
• Generally, pressure or velocity boundary conditions
• Type of reflected wave will depend on whether
compression or expansion is needed to meet
boundary conditions
Reflected Waves -9
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
AE3450