Download - Compressible Flow - TME085 - Lecture 8
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Compressible Flow - TME085
Lecture 8
Niklas Andersson
Chalmers University of Technology
Department of Mechanics and Maritime Sciences
Division of Fluid Mechanics
Gothenburg, Sweden
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Chapter 5 - Quasi-One-Dimensional Flow
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Overview
Compressible flow
Basic
Concepts
Com-
pressibility
flow
regimes
speed of
sound
Thermo-
dynamics
thermally
perfect
gas
calorically
perfect
gas
entropy1:st and
2:nd law
High tem-
perature
effects
molecular
motioninternal
energy
Boltzmann
distribution
equilibrium
gas
CFDSpatial
dis-
cretization
Numerical
schemes
Time
integration
Shock
handling
Boundary
conditions
PDE:s
traveling
waves
method
of char-
acteristics
finite
non-linear
waves
acoustic
waves
shock
reflection
moving
shocks
governing
equationsCrocco’s
equation
entropy
equation
substantial
derivativenoncon-
servation
form
conser-
vation
form
Conservation
laws
integral form
Quasi
1D Flowdiffusers
nozzles
governing
equations
2D Flow
shock
expansion
theory
expansion
fansshock
reflection
oblique
shocks
1D Flow
friction
heat
addition
normal
shocks
isentropic
flow
governing
equationsenergy
mo-
mentumcontinuity
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Learning Outcomes
4 Present at least two different formulations of the governing equations for
compressible flows and explain what basic conservation principles they are
based on
6 Define the special cases of calorically perfect gas, thermally perfect gas and real
gas and explain the implication of each of these special cases
7 Explain why entropy is important for flow discontinuities
8 Derive (marked) and apply (all) of the presented mathematical formulae forclassical gas dynamics
a 1D isentropic flow*
b normal shocks*
i detached blunt body shocks, nozzle flows
9 Solve engineering problems involving the above-mentioned phenomena (8a-8k)
what does quasi-1D mean? either the flow is 1D or not, or?
Niklas Andersson - Chalmers 4 / 52
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Roadmap - Quasi-One-Dimensional Flow
Basic concepts
Governing equations
Area-velocity relation
Nozzles
Diffusers
Numerical simulation
Free boundary reflection
Nozzle pressure ratio
Nozzle relations
Niklas Andersson - Chalmers 5 / 52
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Nozzle Flow with Varying Pressure Ratio
xAt
po pe
A(x) area function
At min{A(x)}po inlet total pressure
pe outlet static pressure
(ambient pressure)
po/pe pressure ratio
Niklas Andersson - Chalmers 6 / 52
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Nozzle Flow with Varying Pressure Ratio
x
M
1
throat
increasing po/pe
supersonic branch
subsonic branch
critical po/pe
For critical po/pe, a jump to supersonic solution will occur
Niklas Andersson - Chalmers 7 / 52
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Nozzle Flow with Varying Pressure Ratio
x
p/po
throat
1
p∗
po=
(2
γ + 1
) γγ−1 increasing po/pe
supersonic branch
subsonic branch
critical po/pe
As the flow jumps to the supersonic branch downstream of the throat, a normal
shock will appear in order to match the ambient pressure at the nozzle exit
Niklas Andersson - Chalmers 8 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio
throat
throat
x
M
1
x
p/po
p∗/po
1
pe/po
m/mchoked
1
1
00
Niklas Andersson - Chalmers 9 / 52
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Nozzle Flow with Varying Pressure Ratio (Summary)
(po/pe) < (po/pe)crI the flow remains entirely subsonicI the mass flow depends on pe, i.e. the flow is not chokedI no shock is formed, therefore the flow is isentropic throughout the nozzle
(po/pe) = (po/pe)crI the flow just achieves M = 1 at the throatI the flow will then suddenly flip to the supersonic solution downstream of the throat,
for an infinitesimally small increase in (po/pe)
(po/pe) > (po/pe)crI the flow is choked (fixed mass flow), i.e. it does not depend on pe
I a normal shock will appear downstream of the throat, with strength and position
depending on (po/pe)
Niklas Andersson - Chalmers 10 / 52
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Nozzle Flow with Varying Pressure Ratio
x
p/po
throat nozzle exit
1
p∗
po=
(2
γ + 1
) γγ−1
critical po/pe
supercritical po/peshock strength
po/pe
Niklas Andersson - Chalmers 11 / 52
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Nozzle Flow with Varying Pressure Ratio
Effects of changing the pressure ratio (po/pe) (where pe is the back pressure and pois the total pressure at the nozzle inlet)
I critical value: po/pe = (po/pe)cI nozzle flow reaches M = 1 at throat, flow becomes choked
I supercritical value: po/pe = (po/pe)scI nozzle flow is supersonic from throat to exit, without any interior normal shock -
isentropic flow
I normal shock at exit: (po/pe) = (po/pe)ne < (po/pe)scI normal shock is still present but is located just at exit - isentropic flow inside nozzle
Niklas Andersson - Chalmers 12 / 52
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Nozzle Flow with Varying Pressure Ratio
Normal shock at exit
x
p/po
throat nozzle exit
1 (po/pe)c
(po/pe)sc
(po/pe)ne
Niklas Andersson - Chalmers 13 / 52
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Nozzle Flow with Varying Pressure Ratio
normal shockpo/pe = (po/pe)nenormal shock at nozzle exit
oblique shock(po/pe)ne < po/pe < (po/pe)scoverexpanded nozzle flow
pressure matchedpo/pe = (po/pe)scpressure matched nozzle flow
expansion fanpo/pe > (po/pe)scunderexpanded nozzle flow
Niklas Andersson - Chalmers 14 / 52
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Nozzle Flow with Varying Pressure Ratio
Quasi-one-dimensional theory
I When the interior normal shock is ”pushed out” through the exit (by increasing
(po/pe), i.e. lowering the back pressure), it disappears completely.I The flow through the nozzle is then shock free (and thus also isentropic since we
neglect viscosity).
Three-dimensional nozzle flow
I When the interior normal shock is ”pushed out” through the exit (by increasing
(po/pe)), an oblique shock is formed outside of the nozzle exit.I For the exact supercritical value of (po/pe) this oblique shock disappears.I For (po/pe) above the supercritical value an expansion fan is formed at the nozzle
exit.
Niklas Andersson - Chalmers 15 / 52
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Roadmap - Quasi-One-Dimensional Flow
Basic concepts
Governing equations
Area-velocity relation
Nozzles
Diffusers
Numerical simulation
Free boundary reflection
Nozzle pressure ratio
Nozzle relations
Niklas Andersson - Chalmers 16 / 52
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Chapter 5.6
Wave Reflection From a Free
Boundary
Niklas Andersson - Chalmers 17 / 52
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Free-Boundary Reflection
Free boundary - shear layer, interface between different fluids, etc
Niklas Andersson - Chalmers 18 / 52
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Free-Boundary Reflection - Shock Reflection
free boundary (p∞)
incident shock
reflected expansion
I No jump in pressure at the free boundary possible
I Incident shock reflects as expansion waves at the free boundary
I Reflection results in net turning of the flow
Niklas Andersson - Chalmers 19 / 52
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Free-Boundary Reflection - Expansion Wave Reflection
free boundary (p∞)
incident expansion wave reflected shock
I No jump in pressure at the free boundary possible
I Incident expansion waves reflects as compression waves at the free boundary
I Finite compression waves coalesces into a shock
I Reflection results in net turning of the flow
Niklas Andersson - Chalmers 20 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
overexpanded nozzle flow
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
shock reflection at jet centerline
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
shock reflection at free boundary
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
expansion wave reflection at jet centerline
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
expansion wave reflection at free boundary
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
repeated shock/expansion system
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
free boundary
free boundary
shock diamonds
Niklas Andersson - Chalmers 21 / 52
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Free-Boundary Reflection - System of Reflections
oblique shock
normal shock
oblique shock
overexpanded jet
Niklas Andersson - Chalmers 22 / 52
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Free-Boundary Reflection - Summary
Solid-wall reflection
Compression waves reflects as compression waves
Expansion waves reflects as expansion waves
Free-boundary reflection
Compression waves reflects as expansion waves
Expansion waves reflects as compression waves
Niklas Andersson - Chalmers 23 / 52
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Roadmap - Quasi-One-Dimensional Flow
Basic concepts
Governing equations
Area-velocity relation
Nozzles
Diffusers
Numerical simulation
Free boundary reflection
Nozzle pressure ratio
Nozzle relations
Niklas Andersson - Chalmers 24 / 52
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Chapter 5.5
Diffusers
Niklas Andersson - Chalmers 25 / 52
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Supersonic Wind Tunnel
po
M > 1
test section
(open)
pe = pamb
po/pe = (po/pe)sc
M = 3.0 in test section ⇒ po/pe = 36.7 !!!
wind tunnel with supersonic test section
open test section
Niklas Andersson - Chalmers 26 / 52
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Supersonic Wind Tunnel
po
M > 1
test section
(closed)
peM < 1
pamb
normal shock
po/pamb = (po/pe)(pe/pamb) < (po/pe)sc
M = 3.0 in test section ⇒po/pamb = 36.7/10.33 = 3.55
wind tunnel with supersonic test section
enclosed test section, normal shock at exit
Niklas Andersson - Chalmers 26 / 52
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Supersonic Wind Tunnel
po
M > 1
test section
(closed)
peM < 1
p2
(po2 = pamb
)
normal shock
po/pamb = (po/pe)(pe/p2)(p2/po2 )
M = 3.0 in test section ⇒po/pamb = 36.7/10.33/1.17 = 3.04
Note! this corresponds exactly to total pressure
loss across normal shock
wind tunnel with supersonic test section
add subsonic diffuser after normal shock
Niklas Andersson - Chalmers 26 / 52
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Supersonic Wind Tunnel
po
M > 1
test section
(closed)
pe
normal shock
oblique shocks
M < 1
p2
(po2 = pamb
)well-designed supersonic + subsonic diffuser ⇒
1. decreased total pressure loss
2. decreased po and power to drive wind tunnel
wind tunnel with supersonic test section
add supersonic diffuser before normal shock
Niklas Andersson - Chalmers 26 / 52
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Supersonic Wind Tunnel
Main problems:
1. Design is extremely difficult due to complex 3D flow in diffuser
I viscous effectsI oblique shocksI separations
2. Starting requirements: second throat must be significantly larger than first throat
solution:
I variable geometry diffuserI second throat larger during startup procedureI decreased second throat to optimum value after flow is established
Niklas Andersson - Chalmers 27 / 52
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Supersonic Wind Tunnel
pressureloss
throat area (second throat)optimum
startconditions
Niklas Andersson - Chalmers 28 / 52
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Roadmap - Quasi-One-Dimensional Flow
Basic concepts
Governing equations
Area-velocity relation
Nozzles
Diffusers
Numerical simulation
Free boundary reflection
Nozzle pressure ratio
Nozzle relations
Niklas Andersson - Chalmers 29 / 52
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Quasi-One-Dimensional Euler
Equations
Niklas Andersson - Chalmers 30 / 52
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Quasi-One-Dimensional Euler Equations
Example: choked flow through a convergent-divergent nozzle
x
A(x)
Assumptions: inviscid, Q = Q(x, t)
Niklas Andersson - Chalmers 31 / 52
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Quasi-One-Dimensional Euler Equations
A(x)∂
∂tQ+
∂
∂x[A(x)E] = A′(x)H
where A(x) is the cross section area and
Q =
ρρuρeo
, E(Q) =
ρu
ρu2 + p
ρhou
, H(Q) =
0p0
Niklas Andersson - Chalmers 32 / 52
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Numerical Approach
I Finite-Volume Method
I Method of lines, three-stage Runge-Kutta time stepping
I 3rd-order characteristic upwinding scheme
I Subsonic inflow boundary condition at min(x)
I To, po given
I Subsonic outflow boundary condition at max(x)
I p given
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Finite-Volume Spatial Discretization
xcell j
xj− 3
2xj− 1
2xj+1
2xj+3
2
(∆xj = x
j+12
− xj− 1
2
)
Integration over cell j gives:
1
2
[A(xj− 1
2) + A(xj+ 1
2)]∆xj
d
dtQj+[
A(xj+ 12)Ej+ 1
2− A(xj− 1
2)Ej− 1
2
]=[
A(xj+ 12)− A(xj− 1
2)]Hj
Niklas Andersson - Chalmers 34 / 52
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Finite-Volume Spatial Discretization
Qj =
ˆ xj+1
2
xj− 1
2
QA(x)dx
/ˆ xj+1
2
xj− 1
2
A(x)dx
Ej+ 12≈ E
(Q(xj+ 1
2
))
Hj ≈
ˆ xj+1
2
xj− 1
2
HA′(x)dx
/ˆ xj+1
2
xj− 1
2
A′(x)dx
Niklas Andersson - Chalmers 35 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
x
r
Nozzle geometry
Niklas Andersson - Chalmers 36 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.18 [bar]
po/pe 1.02
m 81.61 [kg/s]
Mmax 0.35
Niklas Andersson - Chalmers 37 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.16 [bar]
po/pe 1.03
m 106.27 [kg/s]
Mmax 0.49
Niklas Andersson - Chalmers 38 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.14 [bar]
po/pe 1.05
m 131.45 [kg/s]
Mmax 0.69
Niklas Andersson - Chalmers 39 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.13 [bar]
po/pe 1.06
m 144.88 [kg/s]
Mmax 0.93
Niklas Andersson - Chalmers 40 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.10 [bar]
po/pe 1.09
m 145.62 [kg/s]
Mmax 1.31
Niklas Andersson - Chalmers 41 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 1.00 [bar]
po/pe 1.20
m 145.6 [kg/s]
Mmax 1.58
Niklas Andersson - Chalmers 42 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 0.90 [bar]
po/pe 1.33
m 145.6 [kg/s]
Mmax 1.77
Niklas Andersson - Chalmers 43 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 0.80 [bar]
po/pe 1.50
m 145.6 [kg/s]
Mmax 1.94
Niklas Andersson - Chalmers 44 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 0.70 [bar]
po/pe 1.71
m 145.6 [kg/s]
Mmax 2.10
Niklas Andersson - Chalmers 45 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 0.60 [bar]
po/pe 2.00
m 145.6 [kg/s]
Mmax 2.24
Niklas Andersson - Chalmers 46 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
x
M
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
x
p/po
1 1.2 1.4 1.6 1.8 2 2.2 2.4
80
100
120
140
po/pe
m
po 1.20 [bar]
pe 0.50 [bar]
po/pe 11.8
m 145.6 [kg/s]
Mmax 2.26
Niklas Andersson - Chalmers 47 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 3
1.14
1.16
1.18
1.2
·105
x
po
po 1.20 [bar]
pe 1.10 [bar]
po/pe 1.09
m 145.62 [kg/s]
Mmax 1.31
Niklas Andersson - Chalmers 48 / 52
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Nozzle Simulation - Back Pressure Sweep
0 0.5 1 1.5 2 2.5 3
2.97
2.98
2.99
3
3.01
3.02·105
x
ho
po 1.20 [bar]
pe 1.10 [bar]
po/pe 1.09
m 145.62 [kg/s]
Mmax 1.31
Niklas Andersson - Chalmers 49 / 52
![Page 71: Compressible Flow - TME085 - Lecture 8](https://reader034.vdocuments.mx/reader034/viewer/2022043020/626c1193c015696cbc6ec127/html5/thumbnails/71.jpg)
Modern Compressible Flow
Niklas Andersson - Chalmers 50 / 52
![Page 72: Compressible Flow - TME085 - Lecture 8](https://reader034.vdocuments.mx/reader034/viewer/2022043020/626c1193c015696cbc6ec127/html5/thumbnails/72.jpg)
Roadmap - Quasi-One-Dimensional Flow
Basic concepts
Governing equations
Area-velocity relation
Nozzles
Diffusers
Numerical simulation
Free boundary reflection
Nozzle pressure ratio
Nozzle relations
Niklas Andersson - Chalmers 51 / 52
![Page 73: Compressible Flow - TME085 - Lecture 8](https://reader034.vdocuments.mx/reader034/viewer/2022043020/626c1193c015696cbc6ec127/html5/thumbnails/73.jpg)