School of Aerospace Engineering
Ae
• Recall previous analysis of
CD nozzles, e.g., as back
pressure is reduced
• Have looked at range of pb
that produce
– isentropic solutions or
– shocks in nozzle
• What happens when
– pb< pes,sup (U)
– pes,sup<pb<pe, sh (O)
Under/Overexpanded Nozzles -1
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
At
po
To
pe
pb
M
Mes,sup
1
Me,sh
Mes,sub
p/po
x
1
M<1, isen.
pes,sub
p*/po
x
M<1, shock
pe,sh
O
pes,supU
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School of Aerospace Engineering
• Start with pb< pes,sup
p/po
1
– underexpanded case
– pe>pb (not enough
expansion has occurred)
M<1, isen.
pes,sub
M<1, shock
pe,sh
pes,sup
p*/po
x U
• So boundary condition at exit
requires a supersonic expansion process
Þ (2-d) Prandtl-Meyer flow
– flow turns, pressure drops to
back pressure
– but flow must not cross
centerline (symmetry line;
like a wall)
Under/Overexpanded Nozzles -2
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
CL
p1=pe
p2=pb
pb
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nonsimple region
• Centerline
1
2
3
– flow along centerline CL
p1>pb
p3<pb
must be parallel to
p2=pb
centerline
– to get flow to turn inward again requires another
pb
PM expansion
– but second expansion means pressure in region 3
drops below back pressure (p3<p2=pb)
– so now we will need to do something to match our
pressure boundary condition at the edge of the jet
Under/Overexpanded Nozzles -3
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
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School of Aerospace Engineering
• Jet boundary
– along “edge” of C
jet must satisfy L
pressure b.c.
1
slip line
2
p1>pb
3
p3<pb
p2=pb
4
5
p5>pb
p4=pb
to match pressure, p must rise Þ compression
pb
can get PM compression waves
combine to form oblique shocks, turn flow inward
centerline (velocity) boundary condition requires
reflected compression waves
– now pressure exceeds surrounding pressure (p5>pb),
way we started, so process/cycle repeats itself
–
–
–
–
Under/Overexpanded Nozzles -4
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
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School of Aerospace Engineering
1
CL
2
p1>pb
3
p3<pb
p2=pb
4
5
p5>pb
p4=pb
• For pe>pb, nozzle produces a jet flow consisting of
regions of decreasing and increasing p (or density)
– inviscid flow (ideal), process would continue endlessly
– viscous case (real), viscous losses and turbulent mixing
with surroundings causes wave pattern to decay after
small number of cycles
Under/Overexpanded Nozzles -5
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
AE3450
School of Aerospace Engineering
• pes,sup>pb>pe, sh
– overexpanded case
– too much expansion has
occurred
p/po
1
M<1, isen.
pes,sub
M<1, shock
pe,sh
O
pes,sup
p*/po
x
• So boundary condition at exit
requires a supersonic compression process
Þ oblique shock (2-d)
– flow turns, pressure rises to
CL
back pressure
– flow must not cross centerline,
so get reflected shocks
– then expansion to match p, etc.
Under/Overexpanded Nozzles -6
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
p1<pb
p3>pb
p2=pb
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pb
School of Aerospace Engineering
1
CL
p1<pb
4
2
5
3
p2=pb
p5<pb
p3>pb
p4=pb
• For pe<pb, nozzle produces jet flow consisting of regions
of increasing and decreasing pressure (or density)
– same pattern as for underexpanded case, just “out-ofphase” (compressions first, then expansions for
overexpanded vs. expansions then compressions for
underexpanded)
Under/Overexpanded Nozzles -7
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
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School of Aerospace Engineering
• Regions of high pressure, also have high density (and
temperature)
– these hot, dense gases emit light (radiation)
1
CL
2
p1>pb
4
3
p3<pb
p2=pb
5
p5>pb
p4=pb
SR-71 at
takeoff
Under/Overexpanded Nozzles -8
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
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