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 .