Attenuation of l o w Frequency Sound during Turbulent Flow of Air in a Tube Jan M. Beeckmans, Bernard Dudon, and Omprakash Tulsian Faculty of Engineering Science, University of Western Ontario, London, Ontario, Canada
Sound in the frequency range 200 to 900 Hz, a t intensities up to 159 dB, did not affect the pressure gradient in turbulent air flow in a 3/4-inch tube, a t Reynolds numbers up to 52,000. The sound attenuation coefficient, after correction for the effects of air flow velocity and pressure distribution, was approximately equal to the attenuation coefficient in the absence of flow, for all flow velocities a t a sound frequency of 900 Hz. The corresponding attenuation coefficients a t lower frequencies increased considerably above the predicted values a t the higher flow velocities. An attempt was made to predict these trends by solving the appropriate differential equation for sound attenuation in a tube, using an eddy-viscosity profile calculated from M a r tinelli’s velocity distribution equation for turbulent flow in a tube. This procedure correctly predicted a sharp increase in the attenuation coefficient beyond a certain critical flow rate, which depended on the sound frequency, but overestimated the magnitude of the effect a t all frequencies.
JACKSOX
and Purdy (1965) reported that a resonant acoustical field changed the local heat transfer coefficient in a pipe in turbulent air flow. Depending on the flow velocity, the sound frequency, and the position in the pipe with respect to the standing nave pattern, the local heat transfer coefficient was increased or decreased. The work reported here was undertaken to clarify some aspects of the interaction of low frequency sound with the flow field in the region of the wall of a pipe during turbulent flow. Sound attenuation and pressure drop measurements were made in a tube 112 feet long and 3/4 inch in diameter; a relatively long and narrow tube was chosen so as to reduce standing ~ ~ a effects v e as much as possible. An analysis of the attenuation of sound in flowless tubes based on viscous dissipation a t the tube wall was first published by Helmholtz (1863), and was later corrected by Kirchhoff (1868) to allow for thermal conduction a t the wall. The expression derived by Kirchhoff is
The effective viscosity, h’, is given by
where p is the true viscosity, y the ratio of specific heats, k the thermal conductivity, and C, the specific heat a t constant pressure. Thermal conduction effects increase the attenuation coefficient, cy, by 39% in air a t room temperature. The Kirchhoff expression for the attenuation coefficient has been confirmed experimentally by a number of jnvestigators, including F a y (1940), Waetzmann and Wenke (1939), and Lawley (1952)) although the experimental values found for a tend to be a few per cent higher than the theoretical values. The tube size and frequency range of the experiments reported here fall within the “wide tube” region defined by Weston (1953), in which the Helmholtz-Kirchhoff formula is valid. As far as is known by the present writers, there are no data in the literature on the attenuation of sound waves in a smooth tube in the presence of flow. 356 Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
Description of Apparatus
The test pipe consisted of nine 12-foot sections of 3/4-inch smooth-bore copper pipe. The sound was produced by a conventional electromagnetic type driver, and introduced concentrically to a slightly enlarged entrance section of copper pipe. The first pressure tap, and the first microphone entry port, were located 1.5 feet downstream from the entry section. Figure 1 shows a section of a microphone entry port. The microphone body was held by means of a clamped guide, and the transducer end was further positioned by a machined insert soldered inside a copper tee. The insert contained an O-ring which fitted around the base of the microphone to prevent loss of air, and the bottom of the insert was machined in the form of a cylindrical groove, so that the cross section normal to flow a t this point was very nearly a 3/4-inch circle. The second microphone entry port was located 112 feet downstream, followed by a horn which was added to reduce the reflected wave. A second pressure tap was located 12 feet downstream from the first pressure tap. The pressure a t the first pressure tap was measured with a mercury manometer, and the pressure difference across the two pressure taps was measured with an inclined manometer. Compressed air was passed through a coarse filter, and across a calibrated critical orifice, before introduction to the test pipe. At the horn end of the test pipe, provision was made for probing the reflected wave by means of a 3/16-inch 0.d. probe, attached via a fitting to the microphone. The microphone was a 1/4-inch Breul and Kjaer type, and the signal was passed through a narrow-band filter to eliminate the random noise caused by the flow. The sound intensity reading at each frequency and flow rate was corrected for sound originating from the turbulence itself by subtracting the intensity of the latter, as measured in the absence of a n impressed single frequency signal. Calculations
The equation for plane waves in the presence of moderate attenuation may be written as follows: p = A [ , - m 2 + 3 ( w t - h z ) + Ceo2z+J(wt+li2Z) 1 (3)
P is the complex pressure, A and AC are the complex pressure amplitudes of the forward and reflected waves, respectively, w is the pulsatance ( 2 times ~ frequency), t the time, z the distance (positive in the forward direction), cy the attenuation constant (subscripts 1 and 2 for forward and reverse wave),
microphone
Tcl
v
5
s
900
z
4
Iz
Y
70-
LL LL
W
0
G 3 Figure 1 .
z W
Detail of microphone port assembly
s2 I-
and kl and ICz are the wavelength constants (equal to Z.rr/X, where is the wavelength). By taking the modulus of Equation 3, it may be shown that
p(2 =
jA/2[e-2a’z
+
/c,zea2z + 21&(a2--1)z
sin(k1z
X
+ h x + $11
(4)
IC
= ($
- I)/($ + 1)
=
.rr/(kl
+ kz)
15,000
30,000
45,000
60p0(
REYNOLDS NUMBER
Sound attenuation coefficient vs. N R ~at ,
200 Hz
0
Calculated values, based on estimated eddy viscosity given b y Martinelli’s equation
p
6
0
6
1
U
(5)
where $ equals the ratio of the sound pressure level (S.P.L.) a t the last pressure antinode, t o the S.P.L. a t the last pressure node. I t can also be shown that 6
34 0
Figure 2.
m
(-4 was taken to be wholly real, which may be done without loss of generalization by suitable selection of the origin for t.) If the origin of x is taken as the point a t which the last pressure antinode occurs, it is easy to show that
50i-..=--i--.-:;
$
401
w
tI
(6)
1
, -
I
Now, since the wavelength of the sound in the forward direction is increased by a factor (v c)/c, where v is the mean flow velocity in the pipe, and c is the velocity of sound, and since the wavelength of the returning wave is decreased b y the factor (c - v ) / c , we have
+
s=
X(c2
-
v2)/4c2
(7)
I n general, u . (Since the products alz and azx were small for the small distance between the last microphone station and the location of the last pressure antinode, even a comparatively large error in a1and a2 a t this stage did not appreciably affect the value of ‘Ai.)Finally, a n accurate value was found for al, using a measurement of IPl a t the first microphone station by considering only the first term inside the brackets. At the upstream microphone, the value for z was large and negative, making the second and third terms inside the brackets negligible in relation to the first term. The attenuation coefficient of the reverse wave, cyz, was not calculated, but was assumed to be equal to a1 for the purpose of making the reflected wave correction. The correction to the S.P.L. a t the second microphone station due to the reflected wave depended only on the ratio of the S.P.L.’s nt the node and the antinode, and on the distance of the antinode from the micro-
b a
15,000
30,000
REYNOLDS
Figure 4.
I
I
I
0
45,000 60,000
NUMBER
Sound attenuation coefficient vs. NneI a t 400 Hz
phone station, and not on the absolute value of the S.P.L. a t these nodal positions. I t was therefore unnecessary to correct the probe readings for attenuation down the probe. Results
Experimental results on sound attenuation were obtained a t 200,’ 250, 300, 400, 600, TOO, and 900 Hz. The results a t 250 and 700 Hz are not shomn, but they were entirely consistent with the results obtained a t the other frequencies, given in Figures 2 to 6. The solid curve below each experiInd. Eng. Chem. Fundam., Vol. 9 , No. 3, 1970
357
j
l
I-
2
,
,
l
i
50 0
15,000
30,000
45,000
60,000
REYNOLDS NUMBER
Figure 5.
N R ~a, t 600 Hz
Sound attenuation coefficient V S .
m
9
991
T
95
c
75 1
0
Figure
6.
I
I
15,000 30QOG 45,000 REYNOLDS NUMBER
Sound attenuation coefficient vs.
60,000
N R ~ a, t 900 Hz
0
Calculated values, based on estimated eddy viscosity given by Martinelli’s equation
sure p 2 ; p z is the pressure a t x = L,Q equals W R T / c , and D = 2WZMjRT/ag,. Equation 11 was used to calculate the lower predicted curves in Figures 2 to 6. I t appears to predict satisfactorily the trend in attenuation coefficients a t the higher frequencies for all flow rates, and a t the lower frequencies for the lower flow rates; all investigators have found that the HelmholtzKirchhoff formula predicts slightly low values in the flowless case. The probable cause of the deviation of the experimental data from the curve calculated by Equation 11 is discussed below. No effect of sound on the pressure drop across the first 12 feet of pipe was observed, within the range of experiments reported in this paper. The inclined manometer would have detected a change of approximately 0.01 inch of water. Predicted Attenuation in Presence of Eddy Viscosity
The approach outlined is simply to solve the differential equation for the absorption of sound by viscous dissipation in the vicinity of the wall of the pipe, using an eddy viscosity function derived from a standard turbulent velocity profile equation] in place of the true fluid viscosity. While it would be exceedingly difficult to justify this procedure from a fundamental standpoint] the authors believed that the idea was worth carrying through, and proceeded accordingly. I n the flowless case, Helmholtz showed that the sound energy is absorbed in a thin film near the wall, typically of the order of t = (211/p0)~/~,referred to subsequently as the characteristic length. Since the characteristic length was always much smaller than the tube radius in the present study, the problem of the attenuation of sound energy traveling parallel to an infinite gas-solid plane surface will give essentially the same results as the equivalent tube problem, with somewhat simplified equations. The hydrodynamic equation of sound motion in the vicinity of a plane may be written as bp bu b2u = P--
Z mental curve shows the expected attenuation coefficient curve calculated on the basis of the Kirchhoff-Helmholtz formula, corrected for the Doppler shift in the pulsatance, and for the variation in air density along the pipe. The latter was calculated from the equation for isothermal flow of a compressible fluid along a pipe :
*dyz
at
Here p and u are the complex sound pressure and sound velocity, and y is the distance from the plane. Equation 12 may be transformed into the following equation, using equations relating pressure amplitude vectors with velocity amplitude vectors for plane sound motion.
V =u+j-
p being the mean density a t point zl (pressure equals p ) and a t point x2 (pressure equals p z ) . The Doppler correction t o the pulsatance was w’ = w c / ( v
+ c)
(9)
(The Doppler shift was required because attenuation takes place within the boundary of a moving gas stream.) Using the defining relation for the attenuation coefficient, c y =
V is the velocity amplitude vector a t a considerable distance from the plane, and may be taken as having no imaginary component, by suitable choice of the time coordinate. Equation 13 may, after separation into real and imaginary parts, in turn be transformed into two simultaneous equations: I = a - - - t 2 d2b 2 by2
d In I -~
O=b+--
dx
it may be shown that the mean value of length L , with exit pressure p2, is given by
cy
over a pipe of
358
cy0
is the attenuation coefficient without flow, a t pres-
Ind. Eng. Chem. Fundam., Vol.
9, No. 3, 1970
z 2 b2a
2 by2
a represents the real part of u / V , and b the imaginary part
of the same ratio. If p, and hence is
t, is
constant, the solution to Equations 14
a = 1
where
b2u 2 by2
22
b
=
- e - d z COS(Y/t) sin(y/z)
(15)
Solutions were also obtained, by numerical methods, in the case where p was a function of y. Specifically, Martinelli’s (1947) velocity profile for turbulent flow in a pipe was used to calculate p as a function of y. The profiles for a and b obtained in this way were used to calculate the attenuation coefficient in a pipe under the stated conditions of flow, and using a mean air velocity. The sound energy absorbed per unit time per unit area of surface was calculated by means of the equation
as the flow stresses, whereas in our experirneiits the sonic shearing stresses were much lower than the flow stresses. It is probable that sound will have no appreciable effects on heat, mass, or momentum transfer unless the sonic stresses are comparable to the flow stresses, and probably this condition will have to hold into the transition layer. Since sonic shear stresses decay very rapidly away from the wall, whereas the flow stress remains practically constant within this region, sonic effects on the flow regime will probably occur only a t low frequencies (large characteristic length), a t relatively low flow velocities, and a t high acoustic intensities.
From this, the attenuation coefficient could be calculated by using the relation
Nomenclature
E apcV
b
All solutions were fully converged, although in most cases this proved to be a very lengthy process, requiring many hours of computer time per point. I t is believed that the apparent lack of smoothness in the calculated curve was due to the discontinuous nature of the Martinelli equations. All attenuation coefficients calculated in this manner were multiplied by the factor 1.39, to account for heat conduction effects. This was to some extent arbitrary, but i t does not affect the conclusions, since a t the very least heat conduction effects amount to 39% of the viscous dissipation effects calculated using a constant viscosity. If the latter quantity is added to the value for the attenuation coefficient calculated from frictional dissipation in the nonconstant viscosity case, the points would all be lowered slightly by an amount proportional to the difference between the “turbulent” value of a, and its “corrected no-flow” value.
D
a=-
ft = pressure amplitude vector of forward wave, ft see-1
= tube radius,
a
A c
c c, g b
Bc .I
k L
,21
P
P
Q
R
T
t U U
Conclusions
The calculated effect of turbulence on the sound attenuation coefficient is clearly too large, but it appears to be in the right direction. It was much greater a t 200 than a t 900 Hz, and increased as the flow velocity increased; both effects are in agreement with experiment. The discrepancy between experiment and theory probably arises because the eddy viscosity cannot become completely effective a t the small sound displacement amplitudes used in these experiments. *41so, it is not surprising that the eddy viscosity values found for steady forward flow should not necessarily be appropriate for calculating frictional effects in rapid oscillatory flow. Turbulence also increases sound attenuation in the absence of wall effects, but this mechanism could not account for the rapid increase found for a (hleyer et al., 1958). The absence of an effect of sound on the friction factor is not necessarily inconsistent with the results of Jackson and Purdy, because their work was done with a 3.86-inchi.d. pipe. The flow velocity and frictional stresses a t a stated Reynolds number in the experiments of these workers were lou er by a t least an order of magnitude than the corresponding values in our experiments a t an equal K R ~A. rough calculation shows that the shearing stresses a t the wall due to sound in Jackson and Purdy’s experiments were of the same order
I.[‘ X
Y 2
a
Y 6 h $ @ I* P’ W
P
= dimensionless velocity amplitude = velocity of sound, ft see-1 = ratio of reverse to forward wave pressure amplitude = specific heat of air, Btu lb-1 OR-’ = constant in Equation 11, lb? f t - 5 = rate of absorption of sound energy per unit surface area of tube, lbr sec-’ ft-l = function factor - units conversion factor, lbi sec* lbm-l ft-1 = sound intensity, lbi sec-1 ft-’ = square root of -1 = thermal conductivity of air, Btu 1bm-l OR-1 f t P wavelength constants, ft-’ length of test section of pipe, ft molecular weight of air, lb,,/lb mole pressure, lbi ft-2 sound pressure amplitude vector, lbi ft -2 constant in Equation 11, lbt ft-* gas constant, lbi ft OR-’ lb mole-l absolute temperature, O R time velocity amplitude vector in vicinity of wall, ft sec-l velocity amplitude vector in center of tube, ft sec-l gas flow rate in pipe, lb mole it-2 see-1 distance along pipe, ft distance from wall, ft characteristic length, ft sound attenuation constant, dB/cm ratio of specific heats of air distance between node and antinode. ft wavelength, ft ratio of S.P.L. a t antinode to S.P.L. a t node rjhase anale in Eauatiori 4. radian i+cosity-of air, 16, qec-1 ft-1 effective viscosity of air in Equation 1, lb, sec-’ ft--l pulsatance, 27r times frequency density of medium
literature Cited
Fay, R. D., J . Acoust. SOC.Amer. 12,62 (1940). Helmholtz, H., Verh. n’aturhzst. Med. Ver.Heidelberg 3, 16 (1863 j. Jackson. T. W.. Purdv. K. It.. A.S.M.E. nauer 65-HT-30. A.S.Ri.E.-A.I.Ch.E. Heat Transfer ConfereAc;, Los Aiigeles,’ 1965.
Kirchhoff, G., Pogg. Ann. Phys. 134, 177 (1868). Lawley, L. E., Proc. Phys. SOC.(London)Ser. B 65, 181 (1952). RIartinelli, R . C., Trans. A.S.M.E. 69, 947 (1947). Meyer, E., Michel, F., Knrtze, G., J . Acoust. SOC.Amer., 30,
165 (1958). Waetzmann,’E., Wenke, W,, iikust. Z. 4, 1 (1939). Westori, D. E., Proc. Phys. SOC.(London)Ser. B 6 6 , 695 (1953). RECEIVED for review June 16, 1969 ACCEP’TED February 19, 1970
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
359
