If the shock wave tries to move to right with velocity u1 relative to the upstream and the gas motion upstream with velocity u1 to the left

the shock wave is stationary for observers fixed in the laboratory

If the gas motion upstream is turned off . i.e We are watching a normal shock wave propagate with velocity W (crelative to the laboratory) into a quiescent gas

Induced velocity up behind the moving shock

xuuxTTx ,,

txuutxTTtx ,,,,,

Chapter 7 Unsteady Wave Motion

7.1 Introduction

0pu W

7.2 Moving Normal Shock Waves

2 1

x

change

Coordinate system

puWu 2 Wu 1

An important application of unsteady wave motion is a shock tube

u=0

Shock Mach number

22

2

22

2

11

2

222

2

111

2211

uh

uh

uPuP

uu

2121

12 2vv

PPee

1

2

12

12

1

11

2

1 2

1

2

PPP

eW

h

21

12

12

12

2

2

1

112 2

1

PPPP

ee

22

2

2

2

1

2

22

2

11

21

p

p

p

uWh

Wh

uWPWP

uWW

Hugoniot equation(identically the same as eq(3.72) for a stationary shock)

21

12

12

2

1vv

PP

hh

As expected,it is a pure thermodynamic relation which do not care abort the coordinate system

1a

WM s

22

22

12

puWWhh

1

2

21

21

1

2 111

21

T

T

PPr

PP

T

T

1

2

1

2

1

2

1

2

111

11

PP

rr

PP

rr

P

P

T

T

For a calorically perfect gas , , , Tce vP

RTv

1r

Rcv

RT

P

1

2

1

2

2

1

1

2

1

2

11

111

PP

rr

PP

rr

T

T

P

P

Note : for a moving shock wave it becomes convenient to think of P2/P1 as a major parameter governing change across the wave (instead of Ms)

11

21 2

1

1

2

Mr

r

P

P 11

2

1

1

21

P

P

r

raW

2

2

1

12112 2 P

RT

P

RT

C

PPTT

v

112

112

2

1

1

2

1

2

r

PP

r

rPP

r

T

T

2

11

Wu p

If P2/P1

So r=1.4 , up/a2 1.89 as P2/P1

M2 can be supersonic

21

1

21

21

11

12

1

rr

PPrr

P

P

r

au p

2

1

12 a

a

a

u

a

u pp

21

2

1

2

1

2

1

22

1

1

21

2

11

11

1

11

12

11

PP

PP

rr

PP

rr

rr

PPrr

P

P

r

21

2

1

2

1

22

1

1

21

2

2

11

12

1

PP

PP

rr

PPrr

P

P

ra

u p

12

lim21

2

rra

u p

PP

M2 supersonic or subsonic?

Also for a moving nomal shock

0102

0102

PP

hh

22

22

2

2

2202

1

2

1101

puhu

hh

hu

hh

So 0102 hh

Also Vfqt

P

Dt

Dh

.

0 0h is not constant

7.3 Reflected Shock Wave

Unsteady,

Note : a general characteristic of reflectted shock , WR<W

So : in x-t diagram the reflected shock path is more steeply Inclined than the incident shock path

pu

pu RW05 u

2 5

Coordinates transformRWRP Wu

2 5

Velocity jump Formula

W

1u2u

2 1

1

1

12

a

uWM

uuu

s

S

S

S MMr

Mrau

2

2

1 1

121

s

sS MM

raMau

1

1

21 1

2

11

x

5

)2(

)1(I

R

1

:a

uMI s

S 2

:a

uuMR pR

R

S

Sp MM

rauu

1

1

212

R

R MM

ra

1

1

22

S

S

R

R MM

a

a

MM

11

2

1

2

22

22 1

121

1

21

11 S

SS

S

S

R

R

Mr

MrM

r

r

M

M

M

M

2211 uWuW

sMa112

112

sMaWu

ss MaMauWuuu )1(2

111

2

1112

2uuu pI 2uuu pII

Note : The local wave velocity w the local velocity of a fluid element of the gas , u

PTH WWW

Propagated by molecular collisions , which is a phenomenon superimposed on top of the mass motion of the gas

7.4 Physical Picture of Wave Propagation

In general ,

∴ the shape of the pulse continuously deforms as it propagates along the x axis

0

7.5 Elements of Acoustic Theory

0

0

Dt

Ds

PDt

VD

Vt

0

0

xu

x

u

x

u

t

x

u

t

continuity momentum

xa

x

uu

x

uu

t

u

t

ux

p

x

uu

t

u

2

Note : for a gas in equilibrium , any themodynamic state variable is uniquely by any tw

o other state variable. spp ,

dss

pdp

pdp

s

dadp

dp

ds

s

2

0

perturbations ( in general , are not necessary small)u ,

Non-linear but exact eqn for 1-D isentropic flow

Now consider acoustic waves => & are very small perturbations

s

pa

2

au

222 a

aa

u

=>

Momention eqn becomes

x

uu

x

uu

t

u

t

u

x

aa

22

Acoustic equations

0

x

u

t

xa

t

u

2

=>

Linear . Approximate eqs for small perturbations . Not exactMore and more accurate as the perturbation become smaller and smaller

1

2

0

1 2

2

2

tx

u

tt

2

22

22

xa

tx

u

x

=>2

22

2

2

xa

t

1-D form of the classic wave equ

Linearized Small Perturbation Theory

Let =>0G taxF

u

If constconsttax

Note that & are not independent

Let =>0g taxfu

'fx

u

'fa

t

u

t

u

ax

u

10

t

u

at

au

0

x

u

t

)()( taxGtaxF

similarly

)()( taxgtaxfu

F, G , f , g , are arbitrary functions of their argument

The other way to derive the above equation :

u a

0, u

ua

a

uaa

au

2

ap

s

a

Pau

Summary :

a

Pau + : right – running waves

– : left – running waves

Note : 1. (+) => particles move in the positive x direction

(–) => particles move in the negative x direction 2. In acoustic terminology , that part of a sound wave where >0 => condensation => in the same direction as the wave motion <0 => rarefaction => in the opppsite direction as the fashion

u

u

uu

7.6 FINITE WAVES – Δρ and Δu are not small

rr cCRTP cp r

1 rT

In contrast to the linearized sound wave , different parts of the finite wave propagate at different velocities relative to the laboratory . Consider a fluid element located at x2 which is moving to the right with velocity u2

Wave speed relative to the laboratory .

Physically , the propagation of a local part of the finite wave is the local speed of sound superimposed ontop of the local gas motion .

Point

2

222 ua

1x

111 ua 21 aa & 1u moving to the left

21 The wave shape will distort

In fact, of u1 > a1 → W1 moves to the left

The compression wave will continually steepen until it coalesces into a shock wave , whereas the distortionof the wave form is illustrated in Fig. 7.9

Governing equation for a finite wave :

Continuity : 0 VDt

D Dt

Dp

aDt

D2

1

01

2 V

Dt

Dp

a

For 1-D flow

01

2

x

u

x

pu

t

p

a 1

Momention : pDt

VD

For 1-D flow

01

x

p

x

uu

t

u

2

2

,

a

dpds

sdp

pd

sp

ps

21

01

x

pau

t

p

ax

uau

t

u

01

x

pau

t

p

ax

uau

t

u

12

01

01

Dt

pD

aDt

uDDt

pD

aDt

uD

),( txuu ),( txpp

dxx

udt

t

udu

Consider a specific path so that dtaudx

dtx

uau

t

udu

Similarly

dtx

pau

t

pdp

0a

dpdu

The methed of characteristics – along specific paths , the P.D.E reduces to O.D.E

C+ characteristic

C- characteristic

dtaudx )(

dtaudx )(

0a

dpdu

0a

dpdu

= (along C+ characteristic)

= (along C- characteristic)

For a clalorically perfect gas

consta

dpu

consta

dpu

rp

a 2

2a

rp

isentropic

dadp 2

J

J

Riemann Invaruants

rpRT 1 rT

d

rdr

T

dTr

r

11

1

2

a

da

r

d

1

2

constr

auJ

1

2

constr

auJ

1

2

(along a C+ charcteristic)

(along a C- charcteristic)

JJr

a4

1 JJu2

1

7.7 Incident and Reflected Expansion Waves

a

da

T

dTRTa

2,2

Prove theat the C- characteristics are straight lines

04 uIn the constant – property region 4 , and is a constant C+ characteristics have the same slope & J+ is the same everywhere in region 4

4a

ba JJ

eca JJJ

fdb JJJ

fe JJ

feJJalso

JJr

a4

1 JJu

2

1

fe aa fe uu

audt

dx Is the same at all points → Straight line

Also p , , T are constant along the given straight – line C - characteristic

Note : 1. Such a wave is defined as a simple wave – a wave propagating into a constant – property region. Also , it is a centered wave – originetes at a given point . 2. C+ cheracteristics can be curved .

3.For a simple centered expansion wave , the solution can be obtained is a closed analytical form . is constant through the expansion wave . constant through the wave

J

1

2

r

au

1

2 4

r

a

44 2

11

a

ur

a

a2

44 2

11

a

ur

T

T

12

44 2

11

rr

a

ur

p

p

12

44 2

11

r

a

ur

Consider the C- characteristics

audt

dx taux

t

xa

ru 41

2for 334 aut

xa

4. In non – simple region , a numerical procedure is needed . The characteristic lines and the compatibility conditions are pieced together point by point .

Non – simple region

25 JJ

1

2 55 r

au

1

2 22 r

au

obtained from simple wave solution(for point 1 , 2 , 3 , 4)

21 JJ

1

2

1

2 22

11

r

au

r

au

The slopes of straight lines 3-6 & 5-6 are

6

1

3

11 1tan

1tan

2

1tan

auaudx

dt

6

1

5

11 1tan

1tan

2

1tan

auaudx

dt

for line 3-6

for line 5-6

05 u 01 u

5636

, JJJJ

1p 1T 1M 1a 1r

High Pressure

Driver section

Low Pressure

4p 4T4a 4r 4M

Driver section

1

4

p

pDiaphragm pressure ratio Determines uniquely the strengths of the incident

Shock and expansion waves .

4up w

3 2 1

Contact surface

7.8 Shock Tube Relations

12

1

2111

1

2

4

14

1

2

1

4

4

4

1122

11

rr

pprrr

pp

aar

p

p

p

p

puuu 23 32 pp

2

1

1

1

1

2

1

1

1

2

1

12

11

12

1

rr

pprr

p

p

r

auu p

1

2

4

34

4

34

4

2

11

r

r

a

ur

p

p

1

2

P

Pare implicit function of

1

4

P

P

1

2

3

4

1

2

2

3

3

4

1

4

p

p

p

p

p

p

p

p

p

p

p

p

11

21

2

11 2

1

1

12

4

344

4

s

rr

Mr

r

a

ur

12

2

4

1

1

4

1

1

1

4

4

4

111

1

11

21

rr

s

s

s

MM

aa

rr

Mrr

p

p

The incident shock streugth will be made stronger as is made smaller sMpp

1

2

4

1

aa

4

1

4

1

4

1

4

1

M

M

T

T

r

ra

a TMRra

We want as small as possible1

4

The driver gas should be a low – molecular – weight gas at high T

The driver gas should be a high – molecular – weight gas at low T

4

1

7.9 Finite Compression Wave

After the breaking of the diaphragm , the incidentshock is not formed instantly . Rather , in the immediate region downstream of the diaphragmlocation , a series of finite compression waves are first formed because the diaphragm breakingprocess is a complex three – dimensional picturerequir a finite amount of time . These compressionwave quickly coalesce into incident shock wave .