PHYSICAL ACOUSTICS
IV.
Academic and Research Staff
Prof. K. U. Ingard
Dr. C. Krischer
Graduate Students
P. A. Montgomery
R. M. Spitzberg
A. G. Galaitsis
G. F. Mazenko
JETS
STUDIES OF SUPERSONIC
A.
Our study of supersonic jets continues using a modified version of the corona
1
The following modifications have been incorprobe described in a previous report.
A
and the geometry of the surrounding system.
rent
and
age
drop
In
prolonging
the
center
line of the
probe
jet is
jet is
of the
structure
and the shock cell
variation along the jet,
of distance
function
represents
curve
This
shown.
a
as
output
and ground.
collector
current
the
between
of the
trace
a longitudinal
Fig. IV-1
along
1000-~2 resistor
a
across
now measured as the volt-
The probe current is
life.
probe
cur-
discharge
the
decreasing
thereby
wire,
in series with the central discharge
placed
been
has
resistor
10-M2
flow velocity
on the
dependent
and less
stable
wire.
discharge
central
the
around
been placed
has
more
discharge
the
makes
This
collector
A current
porated:
density
the
clearly
notice-
able.
as
well
As
this
trum of the flctuations in the probe
lations
shown
no
Of particular
is
with
in
Fig. IV-2,
interesting
detectable
While
note
to
oscillations
engaged
density
oscillation
from
for
of the
Fig. IV-4,
some
probe
the
occur
in
have
the
under
of these
This work was supported by the U.S.
Contract N00014-67-A-0204-0019.
QPR No.
101
oscillation
is
and
second
variation
Ao
certain
frequency
in
conditions
shock
to
of the
third
the
is
Hz.
of 8. 8
cell,
of the
and
probe
characterisoutput
shown in Fig. IV-3.
depends
Navy (Office
variation
cells.
conditions.
oscillations
a
such
some
studied
probe bias voltages is
will occur
frequency
also
of
example
amplitude
first
the
For example,
different
an
to
maximum
that the
this work we
in
tics of the probe itself.
with
A typical
which corresponds
spec-
the
the amplitude variation of these oscil-
interest is
along the jet.
distance
operating
correspond
which undoubtedly
lines
spectrum contains
oscillations.
cell
It
this
jets
of the
certain
Under
current.
investigated
also
we have
response,
steady-state
As
can be
on both
of Naval
voltage
Selfseen
operating
Research) under
60
0
D
0
c
40
20
0
5
10
DISTANCE
Fig. IV- 1.
15
20
25
FROM JET NOZZEL (cm)
Probe output voltage measured along the axis of the jet as a function of
distance from the jet nozzle for a pressure ratio of 18. 6 and a probe
bias of 900 V.
0.4
8.7
0.3
kHz
8.8 kHz
E
0.2
Z
0.1
5
10
DISTANCE
Fig. IV-2.
QPR No.
15
FROM JET NOZZEL (cm)
20
The rms jet noise along the jet axis as measured by the fluctuations of
the probe output voltage. The bandwidth of the spectrum analyzer is ±6 Hz
and the probe bias is 900 V. The curves have not been corrected for
changes in the small-signal gain of the probe, which are due to changes of
the average gas density. Note that the higher frequencies appear closer
to the jet nozzle in each shock cell.
101
*
1000 V
O
900 V
A
800 V
A
700 V
60
E
40
0
O
0
C
20
I
PRESSURE (Torr)
Fig. IV-3. Probe output voltage as a function of ambient
pressure under low flow-rate conditions, for
These curves
various probe bias voltages.
are used to calibrate the probe for measuring
the gas density in the supersonic argon jet.
I
I
I
I
I
I
I
PRESSURE
(Torr)
Fig. IV-4.
Probe self-oscillation frequency as a
function of pressure for various probe
bias voltages.
I
PHYSICAL ACOUSTICS)
(IV.
voltage and gas
the probe
These oscillations
density.
do not appear
A more detailed analysis
operates in the jet.
to be present when
of this matter will be
presented in a future report.
R. H. Price
References
1.
R. H. Price and U. Ingard, "Techniques for Measuring Density Fluctuations
Jets," Quarterly Progress Report No. 100, January 15, 1971, pp. 70-72.
B.
NONLINEAR TRANSMISSION OF SOUND THROUGH
in
AN ORIFICE
Experimental data have already been reported on nonlinear transmission of sound
At high enough levels of sound (i. e. , >130 dB) sep-
through a sharp-edged orifice.
and the average velocity u at the orifice becomes a non-
aration of flow occurs
linear function of Ap (the pressure drop across the orifice) 3
Ap
P
2 ufu,
2C 2
(1)
where p is the density of the air, and C 1 is the orifice coefficient. If A and A
D
°
are the duct and orifice cross section, respectively, then (1) can be written in terms
of the average duct velocity U.
Ap -
2c
2
(
A o
juU.
(2)
(2)
U U.
1
To be thorough, we include in the right-hand side of (2) a resistive, as well as a reactive, term to account for viscous and inertial effects in the vicinity of the orifice.
The equation of motion then becomes
Ap =
Q U U - pcOU +
Q
p/2C
dU
for x = 0,
(3)
where
=
(AD/Ao
with C the speed of sound,
QPR No. 101
t m (AD/Ao),
'
t
m
the effective
thickness
of the orifice,
4
and 0 a
(IV.
PHYSICAL ACOUSTICS)
resistive coefficient that can be expressed in terms of the geometry of the orifice
and the coefficient of viscosity.
As a result of the nonlinearity the transmitted wave will contain a line spectrum
representing the fundamental and overtones of the driving frequency.
attempt to explain the experimentally obtained spectrum
be infinite in both directions from the plate.
5
In a previous
the tube was assumed to
The theoretically obtained spectrum
then contained only odd harmonics of the frequency of the incident wave. Although
these harmonics were indeed predominant in the experimental data,
even harmonics
did occur.
using a more
In the following analysis we shall re-examine this matter,
realistic model of the experimental arrangement.
In the experiment1 the orifice is at x = 0, while a highimpedance piston located
at x = -L
u
p
moves according to
= u
O
cos wt,
thereby generating a wave that is partially transmitted through the orifice.
In the
steady state the incident, reflected, and transmitted waves have the form
PI (x, t)
PR(X, t)
=
sin n
I n cos n
+A
Rn cos n
+ U
sin n
Tn cos n
+V
sin n
n
n=1
PT(x,t)
and the corresponding velocities are
UI (x, t)
I
cos n
T
T
cos nl - U
V
n +
+V
cos
cos n4
-R
UR(x, t)
n= 1
UT(x, t))
= kx -
where
wt, i = kx + wt.
n
+ An sin n
sin ni
n'
sin
sin n4)
(6)
n
The coefficients of (5) will be obtained by substituting
(5) and (6) in (3), as well as in the two boundary conditions that express continuity of
velocity:
UI(-L, t) + UR(-L
,
t) = u
(7)
U (0, t) + UR(0, t) = UT( 0 , t) - U.
The calculations are
QPR No. 101
straightforward
(8)
and yield
c
(IV.
PHYSICAL ACOUSTICS)
A +U
n
n
=V
n
(9)
I
-R
=T
(In-Rn)cn + (Un-An )s n = pcuo 5n
(10)
(I +R )s + (U +A )c = 0
n nnl
n
nnl
n-1
In+ Rn + (-1)Tn +q V
= B
k
[1- 6n][T j
.T-V
. V j ] + 2B
j=1
[Tn+j T+V n+Vj]
n+
n+] ]]
j=1
(11)
nA
n
-U
n
+ (0-1)V
n
- q T
nin
= B
[1-6n] [TnjVj+TjV n j
]
+ 2B i [Tn+j Vjj= 1
j n+j],
where
c
- cos nKL,
n
q
qn
nKt
s
n
= sin nKL
B-
(AD/A),
mDo
Q
2 Z2'
2p c
This system of equations has been simplified further, and computer work is
carried out to obtain approximate values of the coefficients
eters.
being
for a given set of param-
The results will be forthcoming.
A. G. Galaitsis
References
1. U. Ingard and H. Ising, J.
2. U. Ingard and S. Labate,
Acoust. Soc. Am. 42,
J.
Acoust. Soc. Am.
6-17 (1967).
22, 211-218 (1950).
3.
S. Whitaker, Introduction to Fluid Mechanics (Prentice-Hall, Inc. , Englewood Cliffs,
N. J. , 1968).
4.
U. Ingard, J.
Acoust. Soc. Am.
5.
U.
Acoust. Soc. Am. 48,
Ingard, J.
QPR No. 101
25,
1037-1061 (1953).
32-33 (1970).