III.
PHYSICAL ACOUSTICS
Academic Research Staff
Prof. K.
U.
Ingard
Graduate Students
V. K. Singhal
J. A, Tarvin
A..A. loretta
S. R. Rogers
A.
ATTENUATI\ON
OF' SOUND IN TURBULEUNT PIPE F'LOW
U. S. Navy - Office of Naval Research (Contract N00014-67 -A-0204-0019)
U. Ingard, V. K.
Singhal
In turbulent pipe flow there is a static pressure drop along the length of the pipe,
xwhich is caused by turbulent friction. This turbulent friction can also attenuate the fundamental acoustic
is presented here.
mode propagating in the duct.
A theoretical analysis of this effect
The variation of static density along the duct axis and the change
of friction factor with Reynolds number are included in the theoretical model.
Also,
some experimental data are compared with results of this analysis.
It has been found experimentally that the static pressure gradient dpo,/dx is almost
a constant along the pipe when the entrance and the exit sections are ignored.
present analysis dp /dx is taken from experiment.
dp
d
dx
In the
Let
-a = experimental constant.
(1)
Here a > 0 if x increases in the direction of flow.
"barotropic"; that is,
and also that the fluid is
ratio of specific heats.
We assume a one-dimensional model
p - p , where 1
<
f < y, with y the
The continuity and momentum equations for the steady basic
unperturbed flow in the duct are
pu
= constant
du
dx +
m o
(2)
m-
dpo
puo o =
(3)
dx
2
dp
pu
.
The term
-- a
where a is related to the friction factor f and is defined as
dx
2
1
2
to be
-( CP oU 0 represents the turbulent shear stress. The equation of state is assumed
Po
Po
QPR No.
p
Pref
r(4)
Pref
109
PHYSICAL ACOUSTICS)
(III.
where f = 1 for isothermal basic flow, and
e = y for adiabatic basic flow.
Equations 1-4
can be combined to give the basic flow.
Po
Pref-
PO
ref
(5)
ax
b
(6)
x
m
u
m
r
Pref
I
x
Pref
(1
\
1x ,
P
P
(7)
/ref
where
b Pref
apo
Pref
Po
b -P
(8)
and subscript ref denotes the conditions at x = 0.
We retain only first-order terms in the pressure gradient a oi
and ignore all second-order terms,
are neglected.
2
2
such as ab, b , a .
"density" gradient b,
2dx
Thus, d u odx 2 and d2
29
Also, a is assumed to be a function of u
alone.
o
The continuity and momentum equations for the sound field are
aP
8
1
(Pou+PlUo) = 0
at +
auu
Po -
+-
1
1
aul1
+ me
m
a
-o 5u
O
u
au
o
+ (Poul+uo)
o 1
+ 2
p c u
0ooI
1
+xOamoU1 +-
:-
2
ap uo
aP1
ax
(10)
Here pV is the attenuation of sound caused by viscosity and heat conduction.
To separate the effects of variations in static density, we redefine the acoustic variables.
p1(x, t)
po(x)
and
t)
V= - u (x,
u 0 (x)
(11)
The first-order acoustic flow is taken as adiabatic so that
2
P 1 = coPl.
QPR No.
109
(12)
(III.
PHYSICAL ACOUSTICS)
Equations 9 and 10 when transformed to new variables become
_)x_
(at + u
a
T-+ uo 8x
2
apo
a
+u
2
0
ax
c +
2P vo
ax
m
Sau
°
+
(13)
a
U
8xo 8x
o
+
2
o0
m
aax
ax
2
18
V
uo
--o U
y ap
mO ax
m
(14)
R = 0.
14 with respect to x, eliminating av/ax, using Eq. 13 and the
Differentiating Eq.
basic flow solution leads to the wave equation
82R
o axat
a2R
at2 + 2u
at 2
+ u
+ u 2o aax2 2
8x
aR
-
1 8a
2 au
o ax
2a
at
o
1
aa 2
- + 2p c +
2 au0 Uo
vo
m0
aR
+ -
2
au
2
o
2
a(y+ 1)7
m
2 82R
o ax2
(15)
ax
o0
This is a linear second-order partial differential equation with variable coefficients
2a
aa
2a
(y+1) are conand
for the density perturbation R. The coefficients p v, -, O a,
O
O
stants, but c
o
and u
o
are functions of x.
k(x) dx] Eq.
For a wave solution of the form exp[-it + if
15 gives a dispersion
relation for the wave number k.
k
1-2
ak
- i(1-2)
+ i
+ kMll
o0
OO
2
+i
c
where M
O
= u
/c
2a(y+1)
o----
o
W
2a
m c
o0
aa
a-f
2 av
1
+
+
v
2
aa
m
2
21
o
ao
2
o
(16)
=0,
o
is the Mach number of the basic flow.
0
The real part of the wave number k is likely to be
0/C
for the right traveling
+
o
-C/c
wave, and
1 - NI
used in Eq. 16.
QPR No. 109
for the left traveling wave.
We shall use this to find ak/ax to
be
After obtaining the solution we can check to see whether this is a
(III.
PHYSICAL ACOUSTICS)
+
1 +M
a?
1
+
+~ M
+
0
16 yields
The solution of Eq.
consistent approximation.
°
a
m c
o
00
b
(1+2M )(1-1
0
4 -9o
-/C
1
0
1-
v
O0
4
a
o4
o
0)
(
o
4 "yM4
o
4
aC
0
+ 4
4
3
4
o
'o
4
o
1
(1-2 Mo )(1+MI ) + 4
0
4 a I
0
+
(17)
0
2
0
0
(18)
provided
3
2
c
2a
+
V
1+±I
2
)
1, aa
2
me C
o0 0
a-I
aC 0
M2
9
o
o
o
< 1
for
k,-
«<1
for
k
L
2
ya
2PoC2
0p 0
)
b(1+2A 0)( 1-l
2 1 p0
o
and
a
m c
v
Cc
O
(1+
O
2) + 21
c
aMIO
M 2
-
o
o0
ya
2
S2
2
We note that ak ,ax
b(1-2_I o)(1+Io
0o
o
)
is a second-order quantity in a, b ,
,
' so we were right in
approximating ak/ax. xWe note that
a \1
a
m
c
o o
4{9
4fp0
o
(19)
2
:8
8
f
o
(20)
o
Therefore
QPR No.
109
42
(III.
1
P
k. =
1+
(1+M
0
)
+
v
1
8a
1 aM
4 a x0
~ M
XI
2 + a lo
0 (2+M)
o
(1+31)
4
1
1-
i
o
+ IM (1+,)
- 2M
(21)
0
S0a
k
PHYSICAL ACOUSTICS)
0
8+ 4 o DAaa
+
o.
n
a-0
-2-Al0
(22)
(1(-M
Obviously the attenuations are a function of the Mach number.
the attenuation is
due to viscosity and heat conduction.
At zero -Mach number
It is interesting to note that the
attenuation at very low Mach numbers can be made smaller than that resulting from
viscosity and heat conduction if the following relations between
satisfied.
ak.
1
Only the first three terms in Eqs.
< 0
if
a i
3
2-A 2-M
-3
+
v 4(1-M )
MI(4-3AI 2-1) o
)
8
CZ+
4(1-ATo )
o
0
2
oiI+AI o )
a2a
2I
4
a nd
ak.
1
< 0
aI
o
I2
(2v
+ +
AM 3 )
2
4(1+M )
A(
2+ M 3
11
4-3
a +D
A 2 (1-M
+
4
4
o)
o
D2 a
2
o
QPR No. 109
aa
4(l+M )
o
a,
,,
and
nI0 are
21 and 22 are kept for this purpose.
<0.
(III.
PHYSICAL ACOUSTICS)
= 0 and a/'2 >
At high Mach numbers (at high Reynolds numbers) aa/MIo
0
P.V
Hence
attenuation increases with Mach number in that region.
For the sake of simplicity, we shall use the power law for the friction factor for
2a
2
aa
-1/4
are
L
M and M
. In this approximation M
smooth pipes. Then a = (const)M1
o M2
oM
o0
Then, to lowest order, aki
of the order of a.
1
/aM 0V
< 0 if
P + 21a/64 < 0,
which is
Therefore, upstream attenuation always increases with Mach number.
21a
< 0. This means
Similarly, for the downstream attenuation, if p > 64-, then 8ak /Mo
impossible.
then the downstream attenuation will decrease initially with
that if a < (const) NT,
fT, then by turning
If the pipe roughness is so controlled that a < (const)
Mach number.
on a very low speed flow the sound attenuation can be reduced to a level less than
This result is very interesting and may
that caused by viscosity and heat conduction.
have useful application in the transmission of sound signals through long pipes.
that
v is proportional to the square root of the acoustic frequency f.
the constant depends on the viscosity and thermal
conductivity
of
Note
The value of
the
gas
and the
hydraulic diameter of the pipe.
Now we correct Eqs.
21 and 22 for the change in static density.
We use the nota-
tion
(23a)
R ~ exp[ikx-it]
exp[ik'x-i"t]
pl
(23b)
to obtain
ki
k.
1
k.
i++
:
+
ki
-
= k.
1
=
a
2
2
A
l
1
I
o
1
S
1 + M o0
-
:
1
i - M 0O
0
+
4
2
o
v
-
M
(2+M )
o
4
2
(25)
- 2M2
(26)
3(1-C) + M (3-f) + 2
M
8 f
I
o
aM
8a
a
o
+
PV
+ 4
+
QPR No. 109
(24b)
i
1
+
k.
(24a)
m
0
+
o
4
(2-M )
o
3(-1) + M(3-
0
(III.
PHYSICAL ACOUSTICS)
is approximately equal to (1+M )/(1-M ). But
Note that for small Mach numbers k.i/k.
for large Mach numbers
k.
+
k.
1- M
1+M
k.
+
]
1
o
o
k.
o
For isothermal basic flow these relations give
k
__1
+
M2
o
1 + M4
i
k
=
M
1
1 - M
1
v
4
i
+
-
ki
k
1 -
o
I
O
4
v
v
o
+
aM
o
2
o
4
2)
( +yM +lyM]
(
(27)
(28)
-y o+y
basic flow they yield
M
1
1 + M
aM
2
O
whereas for adiabatic
=
aM
aa
5 I
O
-
k
o
8a
a
M o
aM
o
+
+
aM
0
o
a
M
am
a
aM
O
+
2
2
o
8
2
o
M 21
o a
4
aM
o
+
o
o
8
( 5 - 3 y) + M (3-h') + 21\12
(3-y-5)
+
M 0(3-y)
(29)
-
-
(30)
2
We note that the "attenuation" caused by change in static density is the same in both
directions, whereas the attenuation caused by turbulence is different.
Also note that
the increase in static density in the upstream direction tends to amplify the sound signal, hence to decrease its attenuation resulting from turbulence.
For isothermal basic flow we note from Eqs. 21,
22, and 24 that the ratio of atten-
uation resulting from turbulence and static density variation goes as follows.
aa/ao = 0,
we get for the upstream direction
0
2+
2y-)M - m
2yMo(1-mio
0
- Yio
and for the downstream direction,
QPR No.
109
When
(III.
PHYSICAL ACOUSTICS)
3
2
2 - (2y-1)MI
-
y ,
2y NIo(1I
o)
At low NMach numbers this ratio is fairly large,
ation may be ignored.
cannot be ignored.
so that the effect of static density vari-
lach numbers the effect of static density variation
But at large
As the Mach number increases,
the contribution of static density
completely overshadows the turbulent attenuation on the downstream side.
UPSTREAM MEASURED
UPSTREAM CALCULATED,
0
,0d
7
/
FE
M
MEASURED
0.2
0.1
Fig. III-i.
0.3
0.4
Mleasured and calcul2ated upstream and downstream
attenuation of sound in turbulent pipe flow.
The attenuation of an acoustic pulse xwas measured with and without flow in a duct
of square cross section with a 3,/4 in.
The speaker was pulsed at 1240 Hz.
by proper design.
side.
Two microphones were placed 6 ft apart.
Interference from the pipe termination was avoided
Figure III-1 shows the experimental results, as well as the values
calculated from Eqs. 27 and 28 with
a
QPlR No.
-0. 32
4
),
0. 0014 + 0. 0125 Rey
d
-d
109
(31)
PHYSICAL
(III.
ACOUSTICS)
where d is the side of the square cross section of the duct, and Rev is the Reynolds
number of the mean flow.
The attenuation increases slowly with Mach number at low Mach numbers but for
MIach numbers greater than 0. 25 the effect of turbulence is very large indeed.
The
downstream attenuation increases at a slower rate than does the upstream attenuation.
That the downstream attenuation can decrease with increasing Mach number is evident
at Al = 0. 1 on the calculated curve.
The downstream measured curve deviates significantly from the downstream calculated curve in the 1Mach number range 0. 1-0. 32.
The reason for this is that in
in the transition range from laminar to turbulent flow.
this region the basic flow is
This effect also appears on the upstream measured curve.
sound waves and high-frequency scattering cannot be detected
Turbulence scatters
quantitatively with the simple pulse technique that we used.
This sc'attering of
the
incident harmonic sound pulse train is neglected in the present analysis and mcv account
for some of the observed discrepancies.
The friction-factor
expression.
law, Eq. 31,
used for the calculations
Our measurements show higher friction factors.
it is very hard to get a good estimate of c.
is the commonly accepted
In the transition region
The errors in the constant a are obviously
reflected as errors on the calculated curves in Fig. III-1.
B.
ORIFICE
FLOW NOISE
Navy -- Office of Naval Research (Contract NO0014-67-A-0204-0019)
U. S.
U. Inpard
In one of our recent experiments in which air was sucked through an orifice in a
plate forming one of the walls of a suction chamber, we focused attention mainly on the
noise on the upstream side of the orifice plate.
Of particular interest was the depen-
dence of the noise emission on the static pressure ratio in the vicinity of the critical
value of this ratio.
In these experiments we also studied the sound emission into the suction chamber.
Figure III-2 shows the measured sound-pressure level (SIL) inside the chamber as
a function of the static pressure ratio across the orifice plate.
It is interesting to try to interpret the observed dependence of the SPL on the pressure ratio in terms of the expression
W
QPR No.
2
= C
109
n
c
2
oA
o
(1)
(III.
PHYSICAL ACOUSTICS)
for the acoustic power W 2 emitted from the flow on the downstream side of the orifice
plate.
In this expression C
n
is a numerical constant, V
o
is the velocity in the orifice,
The
P2 is the fluid density in the suction chamber, and A is the orifice area.
n
factor C n(Vo/C2) can be regarded as the efficiency of noise emission by the jet stream
discharging kinetic energy at the rate p 2 XVA /2 (the difference between the density
0
po
in the jet at the orifice and the density pc in the chamber is ignored). The exponent n
depends on the nature of the flow fluctuations in the stream.
For flow pulsations n = 1
lateral-flow fluctuations and corresponding pressure fluctuations
(monopole),
orifice plate correspond to n = 3 (dipole),
on the
and fully developed turbulence in the stream
In general, the emitted noise is contributed by all
(quadrupole) corresponds to n = 5.
three effects and we may set
V
W2=
C
Sm c2
V 3
+ Cd
5
q2
Cq
V
2
A
(2)
o
We shall show in this report that our experimental data can be fitted well to an expression of this form.
1.
Pressure Drop
In our experiments we did not measure directly the Mach number in the orifice but
rather the static pressures P1
and P
2
outside and inside the orifice plate.
We shall
relate these pressures by making the assumption that the flow is isentropic from the
outside of the chamber to the orifice and that there is very little "pressure recovery"
in the suction chamber on the downstream side of the orifice plate.
The pressure in the
jet, P , as it leaves the orifice is then assumed to be the same as the pressure P.
in the chamber. The local temperature To and the density po in the jet at the orifice
will be somewhat different from the corresponding quantities in the almost quiescent air
in the suction chamber.
Under these conditions, we obtain
2 - 2 °1 I2- (P
S
o y-1
P
where
c1 is
)7
2
-1
cC1 1 - 1 - AP
P -
the sound speed outside the chamber,
- P
1
o"
If we neglect pressure recovery, we have AP
chamber,
,3)
P1 the static pressure outside the
and AP the pressure drop P
sure inside the chamber.
QPR No. 109
- P 2 , where P
2
is the static pres-
Since the temperatures inside and outside the chamber are
the same, we may set cl = c 2 .
be written
P
The expression for the acoustic power in Eq.
2 can then
(III.
P2C1
2
C F(x) + C F3(x) + C F5(x) F 3 (x)
Finay,
if we set pCd=
(x)
we obtain
=
Finally, if we set p2
=
Pl
(l - x )
PHYSICAL ACOUSTICS)
we obtain
,
F 4 (x) +
+ C d4x
dF 6 (x)
+C
W2
= [C
x +C
m
/C/)
3
(P
F 8 (x) (1-x)
I0/
where x = (P
1 -P 2 )/P1 - APiP 1 .
In Fig. III-2 the functions represented by the three terms in Eq. 5 (including the
C d , and C
factor (1-x)) are shown, with the constants Cq,
q
d'
m
adjusted to produce a best
fit with the experimental data.
*/
/
0//
0//
-
*-- 7
/
-10--
-20
>
,
f
e" /
MONOPOLE
*
0
*
0
EXPERIMENTAL DATA
DIPOLE
/
QUADRUPOLE
I
SI
I
I
II
I
I
I
0.1
AP/P
Fig. 111-2.
QPR No.
109
I
I
I
1.
1
Orifice flow noise.
We find that the best fit is obtained if C
expression for W 2 becomes
i
m
140 C , C d
q'
d
19 C
q
so that our empirical
(III.
PHYSICAL ACOUSTICS)
z
C [140 F1' +19
6
+
8
](1-x.
(6)
Once our "reverberation" chamber (suction chamber) has been calibrated, the value of
C
q
can be determined from our measurements.
QPtR No.
109