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


Driver gas Driven gas

14

Tim

e 0

Diaphragm

x

p

W1234

Tim

e 1

ShockContact surfaceExpansion wave

upa4 a3 - up

x

p

WR

52

34

Tim

e 2


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


upa4 a3 - up

x

p

Wave diagram

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What we will analyze for moving waves

WR

52

34

Tim

e 2


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


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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


upa4 a3 - up

x

p

Induced mass

motion velocity

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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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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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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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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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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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1’2’12

stationary

Moving wave

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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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1’2’ 12

stationary

Impinge wave

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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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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


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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incident wave

Using below equation to calculate Mach number and the flow properties downstream moving shock

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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

UNSTEADY WAVE MOTION

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