UNSTEADY WAVE MOTION
Transcript of UNSTEADY WAVE MOTION
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Fakultas Teknik Mesin dan Dirgantara
UNSTEADY WAVE MOTION
UNSTEADY WAVE MOTION
AE3110 Aerodynamics 1
Dr. -ing. Mochammad Agoes M. ST. MSc.Ema Amalia, ST., MT.
Pramudita Satria Palar, ST, MT, PhD
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Outline
➢ Reflected Shock Wave
➢ Element of Acoustic theory
➢ Wave Propagation (Physical Picture)
➢ Moving Normal Shock Waves
➢ Introduction
➢ Finite (Non Linier) Wave
➢ Incident and Reflected Shock Waves
➢ Finite Compression Waves
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Introduction
▪ A wave is described as a physical disturbance in the flow, where the wave is
propagated by molecular collision which has the definite propagation velocity
( sonic for sound wave and supersonic for shock wave)
▪ For stationary wave, the wave is propagating into a flow which it is moving in
the opposite direction at the same velocity magnitude as the wave velocity, u1
▪ For unsteady wave, the wave propagates through space to the right with a
propagation velocity, W
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Introduction
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Introduction (Shock Tube)
Shock tube is widely
used in:
• Study of high
temperature gases.
• Chemical kinetics.
• Inside the shock tube,
the temperature can
reach 500-4000 K
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Shock Tube model
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The driver and driven
gas typically have
different properties
Shock tube at initial condition
HIGH PRESSURE LOW PRESSURE
Diaphragm
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Moving waves
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Shock tube after the diagprahm is broken
Expansion wave will
be formed on the leftCompression wave will be
formed on the right
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What happens when shock reaches end of tube ?
WR
52
34
Tim
e 2
Reflected ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
End wall
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Reflected shock wave
HIGH PRESSURE LOW PRESSURE
Driver gas Driven gas
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Tim
e 0
Diaphragm
x
p
W1234
Tim
e 1
ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
WR
52
34
Tim
e 2
Reflected ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
Induced mass
motion velocity
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Wave Diagram
The Shock Tube
x
t
0
1
2
Shock Tube
x- t Diagram
t = 1up
t = 25
1
23
4
WR
52
34T
ime 2
Reflected ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
Wave diagram
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What we will analyze for moving waves
WR
52
34
Tim
e 2
Reflected ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
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▪ Unsteady shock wave
The governing equation for
moving SW wave :
w = u1
w-up = u2
Governing Equation Moving and stationary waves
stationary wave
Moving wave
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Moving Normal Shock Waves
The Pressure Collapse Problem
FUNDAMENTAL PHYSICAL SITUATION REPRESENTED BY
THE SHOCK TUBE
Moving Normal Shocks
IDENTICAL TO A STATIONARY SHOCK – ONLY DIFFERENCE IS FRAME OF REFERENCE
Fixed Frame of Reference (shock in rest) Frame of Reference Moving With Shock
u1’=Wu2’=W - upu2 u1
1’2’12
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Moving Normal Shock waves
For stationary shockwave
It relates the wave velocity of the moving shockwave
to the pressure ration across the wave and the speed
of sound of the gas
Mach number of moving shockwave
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Hugoniot EquationThe change of fluid properties across a normal shockwave is expressed
in terms of purely thermodynamic variables (without Mach number)
Continuity equation
Momentum equation
Energy equation
Density ratio across the
shock as a function of
pressure ratio
Moving Normal Shock Waves
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Moving Normal Shock Waves
Induced Mass motion velocity, Up
Frame of Reference Moving With Shock
u1=Wu2=W - up
p2 p1
Up = f(p1/p2) Density and temperature ratio across the
shock as a function of pressure ratio
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Reflected shock wave
WR
52
34
Tim
e 2
Reflected ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
Induced mass
motion velocity
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Moving Normal Shock waves
Mach number of reflected shockwave
Incident shock equations Reflected shock equations
From incident shock equations and reflected shock equations and specializing to
a calorically perfect gas, a relation between Mach number of incident Ms and
Mach number of reflected shockwave can be written as follows
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Expansion waves
In region 4
In the Shock tube in regions between 3 and 4 expansion wave occurs starting with Mach
line as a head of the expansion (in region 4) and end with tail Mach line (in region 3), the
process through the expansion is isentropic.
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Expansion waves
Velocity between the head and tail
of the centered expansion
Each Mach line in expansion wave has unique value
which called characteristic line, mathematically
written as
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Expansion waves
The local speed of expansion waves changes linearly
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Expansion waves
Velocity in region 2
Velocity in region 3
Relations between region 2 and region 3
𝑢3 = 𝑢2= 𝑢𝑝 𝑝3 = 𝑝2
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Moving Normal Shock waves
A Normal Shock wave travels at a constant speed of 552 m/s into still air
at a pressure of 1 atm and a temperature of 280 K.
▪ Find the velocity(up), pressure (pm2), and temperature (Tm2), of the air
following wave
▪ What are the stagnation pressure and the stagnation temperature after
the passage of the wave
Hint:
Example 1 moving shock wave
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Moving Normal Shock Waves
A Normal Shock wave travels towards end of a tube at velocity of 750
m/s . The air in the tube is at pressure of 1 atm and a temperature of 283
K
▪ Determine the speed of the reflected wave and pressure, and
temperature of ahead reflected shock wave
Example 2. reflected shockwave
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Moving Normal Shock Waves
The driver section of air-air shock tube is pressurized to 8 atm and the
expansion section is evacuated to 0.05 atm. If the initial temperature of
the air in both sections is 300 K
▪ Determine the temperature pressure and velocity downstream of the
shockwaves and downstream of the rarefaction waves
Example 3. Shock tube
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Driven Flow in a Shock Tube
Schlieren shows density ratio of 3.0 across shock.
• Find shock Mach number mass motion velocity and p and T after shock
▪ Time until shock strikes probe?
▪ What happens when the shock reaches end of tube
▪ Calculate reflected Mach number and reflected velocity
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Example
Driven Flow in a Shock Tube
p2= 566.7 kPa, T2= 566.7K
M1'=2.24
M2'=0.542
up=519m/sW=776m/s
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WEEK 2
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Moving Normal Shock waves
A Normal Shock wave travels at a constant speed of 552 m/s into still air
at a pressure of 1 atm and a temperature of 280 K.
▪ Find the velocity(up), pressure (pm2), and temperature (Tm2), of the air
following wave
▪ What are the stagnation pressure and the stagnation temperature after
the passage of the wave
Hint:
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Moving Normal Shock waves
1’2’12
stationary
Moving wave
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Moving Normal Shock Waves
A Normal Shock wave travels towards end of a tube at velocity of 750
m/s . The air in the tube is at pressure of 1 atm and a temperature of 283
K
▪ Determine the speed of the reflected wave and pressure, and
temperature of ahead reflected shock wave
Hint:
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Moving Normal Shock waves
1’2’ 12
stationary
Impinge wave
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Moving Normal Shock waves
2 1
Reflected Shockwave stationary
By iterating process
𝑀𝑠 = 2.22
𝑀𝑅 = 1.84
𝑀𝑅 =𝑊𝑅 + 𝑢𝑝
𝑎2
𝑎2 = 𝛾𝑅𝑇2 = 462.14
𝑊𝑅 = 𝑀𝑅𝑎2 − 𝑢𝑝 = 350.66 𝑚/𝑠
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Moving Normal Shock Waves
The driver section of air-air shock tube is pressurized to 8 atm and the
expansion section is evacuated to 0.05 atm. If the initial temperature of
the air in both sections is 300 K
▪ Determine the temperature pressure and velocity downstream of the
shockwaves and downstream of the rarefaction waves
Example 3. Shock tube
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Moving Normal Shock wavesImpinge wave
HIGH PRESSURE LOW PRESSURE
Driver gas Driven gas
14
Tim
e 0
Diaphragm
x
p
W1234
Tim
e 1
ShockContact surfaceExpansion wave
upa4 a3 - up
x
p
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Moving Normal Shock waves
incident wave
Using below equation to calculate Mach number and the flow properties downstream moving shock
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Moving Normal Shock waves
Region 1 to region 2
Region 2 to region 3
Isentropic relation
Region 3 to region 4 expansion wave
𝑀2 =𝑉2𝑎2
=𝑊 − 𝑢𝑝
𝑎2=𝑊
𝑎1
𝑎2𝑎2
−𝑢𝑝
𝑎2= 𝑀𝑠
𝑎2𝑎2
−𝑀2′ = 1.21𝑉2 = 𝑊 − 𝑢𝑝