投影片 1 - National Cheng Kung University
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Transcript 投影片 1 - National Cheng Kung University
Chapter 7
Unsteady Wave Motion
7.1 Introduction
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
x, T T x, u ux
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
x, t ,T T x, t , u ux, t
An important application of unsteady wave motion is a shock tube
7.2 Moving Normal Shock Waves
u=0
up 0
u2 W u p
W
change
Coordinate system
2
1
x
u1 W
Shock Mach number
W
Ms
a1
1W 2 W u p
1u1 2u2
P1 1u12 P2 2u22
P1 1W 2 P2 2 W u p
2
W u p
W2
h1
h2
2
2
2
u12
u22
h1 h2
2
2
W 2 W u p
h2 h1
2
2
W2
P 1 P P
h1
e1 1 2 1 2
2
1 2 2 1 1
1 P2 P1 2 1
e2 e1
1 2 2 2 1 1 2
P1
e2 e1
P2
2
h2 h1 1
v1 v2
P2 P1 2
P1 P2
v1 v2
2
Hugoniot equation(identically the same
as eq(3.72) for a stationary shock)
As expected,it is a pure thermodynamic relation
which do not care abort the coordinate system
For a calorically perfect gas ,
T2 T1
e cvT ,
P1 P2 RT1 RT2
2Cv P1
P2
P
r 1
2
r 1 P1
T P
2 2
P2
T1 P1 1 r 1
r 1 P1
v
RT
R
, cv
P
r 1
1 1 T2
T2
PP
1 1 2 r 1
T1
2
P1 P2 T1
P
2 r 1 2 r 1
T2
P1
T1
P
2 r 1 1 r 1
P2
1 r 1
P
RT
P2
r 1 P1
2 P2 T1
P
1 P1 T2
r 1
2
r 1 P1
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)
P2
2r
M 12 1
1
P1
r 1
W a1
r 1 P2
1 1
2r P1
u p W 1 1
2
2r
a1 P2 r 1
u p 1
r P1
P2 r 1
P r 1
1
1
2
M2 supersonic or subsonic?
2r
u p u p a1
1 P2 r 1
1
a2 a1 a2
r P1 P2 r 1
P r 1
1
If P2/P1
1
2
r 1 P2
1
r
1
P1
r 1 P P 2
2 2
r 1 P1 P1
1
2
2r
u p 1 P2 r 1
a2 r P1 P2
P
1
So r=1.4 , up/a2
1
2
r 1 P
2
r 1 P1
P 2
2
P1
1
2
1.89 as P2/P1
M2 can be supersonic
up
2
a
r r 1
P
2
lim
P
2
1
Also
h02 h01 for a moving nomal shock
P02 P01
u12
h01 h1 h1
2
u 2p
u22
h02 h2 h2
2
2
So
Also
h02 h01
.
Dh0 P
q f V
Dt
t
7.3 Reflected Shock Wave
Unsteady,
h0 is not constant
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
up
WR
up
2
u5 0
5
uP WR
WR
2
5
Coordinates transform
R
Velocity jump Formula
2
W
u1
u2
5
1
( 2)
(1)
I
x
u u2 u1
Ms
W u1
a1
I : MS
1W u1 2 W u2
1a1M s
u u2 u1 W u1
u2 W
1a1M s
2
1
a1M s a1 (1 1 ) M s
2
2
2 r 1M S2
u a1 1
M S
2
r
1
M
S
1
2
u a1 1 1 M S a1
M s
M s
r 1
2
us
a1
u
I
u p u2
R:MR
1
2
u p u2 a1
M S
r
1
M
S
MR
uR u p
a2
u
II
u p u2
1
2
a2
M R
M R
r 1
1
a
1
1 M S
M R a2
M S
2 r 1M S2
MR
MS
2r
2
M S 1
1
2
M R2 1 M S2 1 r 1
r 1M S
7.4 Physical Picture of Wave Propagation
Note : The local wave velocity w
the
local velocity of a fluid element of the gas , u
Propagated by molecular collisions ,
which is a phenomenon superimposed
on top of the mass motion of the gas
In general , W
H
WT WP
∴ the shape of the pulse
continuously deforms as
it propagates along the x
axis
0
7.5 Elements of Acoustic Theory
V 0
t
DV
P
Dt
Ds
0
Dt
, u
perturbations ( in general , are not necessary small)
continuity
momentum
u
0
t
x
u
u
u
0
t
x
x
x
u
u
p
u
t
x
x
u
u
u
u
u
u
a 2
t
t
x
x
x
Non-linear but exact eqn for 1-D isentropic flow
Note : p p, s for a gas in equilibrium , any themodynamic state variable is uniquely by any
two other state variable.
p
p
dp dp ds
s
s
ds 0
p
dp d a 2 d
s
Now consider acoustic waves => & u are very small perturbations
Momention eqn becomes
p
a2
s
=>
a 2
a a
2
u a
2
u
u
u
u
u
u
t
t
x
x
2 a 2
a
x
Acoustic equations
u
0
t
x
u
a2
t
x
1
=>
2
1
2
2u
2
0
t
t
xt
2
2u
2
a2
x
xt
x 2
Linear . Approximate eqs for small perturbations . Not exact
More and more accurate as the perturbation become smaller
and smaller
Linearized Small Perturbation Theory
=>
2
2
2
a
t 2
x 2
1-D form of the classic wave equ
F ( x at ) G( x at )
similarly
u f ( x at ) g ( x at )
F, G , f , g , are arbitrary functions of their argument
Let G 0 => F x at
If x at const const
Note that u & are not independent
Let
g 0 => u f x at
u
a f '
t
u
f'
x
u
1 u
x
a t
u
a
u
0
t
x
u
0
t
a t
The other way to derive the above equation :
a
u
, u 0
a u
a
a a u
u
a
p
a2
s
u
Summary :
a
P
u
a
a
P
a
+ : right – running waves
– : left – running waves
Note : 1. u (+) => particles move in the positive x direction
u (–) => particles move in the negative x direction
2. In acoustic terminology , that part of a sound wave where
>0 => condensation => u in the same direction as the wave motion
<0 => rarefaction => u in the opppsite direction as the fashion
7.6 FINITE WAVES – Δρ
and Δu are not small
p r c
P RT C r c r
T r 1
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 2 relative to the laboratory .
2 a2 u2
Physically , the propagation of a local part of the finite wave is the local speed of sound superimposed
on
x1 local gas motion .
top of the
a u
Point 1 1 1
1 2
a1 a2 & u1 moving to the left
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 distortion
of the wave form is illustrated in Fig. 7.9
Governing equation for a finite wave :
Continuity
:
D
V 0
Dt
D 1 Dp
Dt a 2 Dt
1 Dp
V 0
a 2 Dt
For 1-D flow
1 p
p
u
u
0
a 2 t
x
x
Momention :
1
DV
p
Dt
For 1-D flow
u
u 1 p
u
0
t
x x
2
p, s
dp
d dp ds 2
a
s p
p s
1 2
2 1
u
u
u a
u 1 p
p
u
u
a
u
a
0
t
x a t
x
Du 1
Dt a
D u 1
Dt a
u u( x, t )
du
t
x
1 p
p
u a 0
a t
x
D p
0
Dt
D p
0
Dt
p p( x, t )
u
u
dt dx
t
x
Consider a specific path so that
dx u a dt
u
u
du u a dt
x
t
Similarly
p
p
dp u a dt
x
t
du
dp
0
a
The methed of characteristics – along specific paths , the P.D.E reduces to O.D.E
C+
characteristic
dx (u a)dt
C-
characteristic
dx (u a)dt
du
dp
0
a
du
dp
0
a
dp
const
a
J =
u
J =
dp
u
const
a
(along
C+ characteristic)
Riemann Invaruants
(along
For a clalorically perfect gas
a2
rp
isentropic
dp a 2 d
rp
a2
C- characteristic)
RT p r
T r 1
dT r 1 r 2 d
d
r 1
r 1
T
d
dT 2da
T
a
2 da
r 1 a
2a
const
r 1
2a
J u
const
r 1
J u
a
a 2 RT ,
r 1
J J
4
(along a C+ charcteristic)
(along a C- charcteristic)
u
1
J J
2
7.7 Incident and Reflected Expansion Waves
Prove theat the C- characteristics are straight lines
In the constant – property region 4 , u4 0 and a4 is a constant
C+ characteristics have the same slope & J+ is the same everywhere in region 4
J a J b
J a J c J e
J J J
J J
alsoJ J
b
d
e
a
dx
ua
dt
f
e
f
r 1
J J
4
ae a f
u
f
1
J J
2
ue u f
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 .
J is constant through the expansion wave .
u
2a
constant through the wave
r 1
2a
4
r 1
a
r 1 u
1
a4
2 a4
T r 1 u
1
T4
2 a4
p r 1 u
1
p4
2 a4
2r
r 1 u
1
4
2 a4
2
r 1
r 1
Consider the C- characteristics
dx
ua
dt
u
2
x
a4
r 1
t
x u at
for
a4 x u3 a3
t
2
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
J J
J J
5
u5
2a5
r 1
1
2
u2
2 a2
r 1
u1
u5 0
2
2a1
2a
u 2 2
r 1
r 1
u1 0
obtained from simple
wave solution
(for point 1 , 2 , 3 , 4)
J J , J J
6
3
6
5
The slopes of straight lines 3-6 & 5-6 are
1
dt 1
1
1
tan 1 tan 1
tan
dx 2
u a 3
u a 6
1
dt 1
1
1
tan 1 tan 1
tan
dx 2
u a 5
u a 6
for line 3-6
for line 5-6
7.8 Shock Tube Relations
Driver section
Driver section
High Pressure
Low Pressure
p1
p4 a4 r 4 T4 M 4
p4
p1
Diaphragm pressure ratio
T1
M1
a1
r1
Determines uniquely the strengths of the incident
Shock and expansion waves .
Contact surface
up
4
3
2
w
1
u3 u2 u p
p2 p3
2r1
a p
r1 1
u p u2 1 2 1
r1 p1 p2 r1 1
p r 1
1 1
p3 r4 1 u3
1
p4
2 a4
1
2
2 r4
r4 1
r4 1 a1 p2
a4 p1
p4 p2
1
p1 p1
p
2r1 2r1 r1 1 2 1
p1
2 r4
r4 1
P2
P1 are implicit function of
P4
P1
p4 p4 p3 p2 p4 p2
p1 p3 p2 p1 p3 p1
r 1 u3
1 4
2
a
4
1
2 r4
r4 1
2r1
2
1
M
1
s
r 1
1
2r1
M s 1
r1 1
p4
2r
p1 r 1 a M 2 1 r 1
4
1
s
1 r 1 a M
1
4
s
4
4
The incident shock streugth
p2
M
p
s
will be made stronger as
1
M T
a1
a r R
We want
4
a1
a4 is made smaller
r T M
1 1 1
a4
r4 T4 M 4
1 as small as possible
The driver gas should be a low – molecular – weight gas at high T
4
The driver gas 1 should be a high – molecular – weight gas at low T
7.9 Finite Compression Wave
After the breaking of the diaphragm , the incident
shock is not formed instantly . Rather , in the
immediate region downstream of the diaphragm
location , a series of finite compression waves are
first formed because the diaphragm breaking
process is a complex three – dimensional picture
requir a finite amount of time . These compression
wave quickly coalesce into incident shock wave .