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High amplitude acoustic transmission through duct terminations: Theory
Journal of Sound and
Vibration (1983) 91(4), 503-518
HIGH AMPLITUDE ACOUSTIC TRANSMISSION THROUGH DUCT TERMINATIONS: THEORY? A. CUMMINGS
AND
W. EVERSMAN
Department of Mechanical and Aerospace Engineering, University of Missouri-Rolla, Roila, Missouri 65401, U.S.A. (Received 30 September 1982, and in revised form
16 February 1983)
Recent experimental measurements have demonstrated that net acoustic energy dissipation can occur when sound waves interact with free shear layers, which are produced either by boundary layer separation in mean fluid flow at sharp edges, or by separation of the boundary layer in the acoustic flow at an edge in the absence of mean flow. This paper presents theoretical results which are offered in an attempt to explain these observations quantitatively. Comparison is made between the predicted and measured net energy loss which occurs upon transmission of high amplitude impulsive acoustic waves through various duct terminations, and also between calculated and measured reflection coefficients in the duct. The agreement is generally at least qualitatively good, and would appear to justify the physical assumptions on which the theoretical arguments are based. 1. INTRODUCTION In the past few years, there has been a growth of interest
in acoustic energy effects associated with the interaction between sound fields and free shear layers in separated flows. In particular, the net acoustic energy absorption which is observed to occur when an acoustic wave is transmitted out through an air jet issuing from a pipe has been studied by several workers. Bechert’s experimental and theoretical results [l] and Howe’s theoretical analyses [2, 31 for example, offer persuasive evidence that the absorption mechanism is the conversion of acoustic energy into vertical energy at a sharp edge, and the subsequent dissipation of this into heat, without further substantial interaction with the acoustic field. It has also been noted that, in the absence of mean flow, energy can be dissipated by the same mechanism if the acoustic velocity is high enough to cause flow separation, and recent work at Lockheed-Georgia has shown this effect well in a comprehensive study of noise transmission through duct terminations. Salikuddin and Ahuja [4] have reported substantial low-frequency acoustic losses at duct terminations both with and without mean flow. Salikuddin and Plumblee [5] also reported similar data in the absence of mean flow. Reference [5] shows, additionally, that the acoustic reflection coefficient at a duct termination can be considerably reduced (below its small amplitude value) if the amplitude of the incident wave is large. The purpose of this paper is to offer a theoretical explanation of the aforementioned experimental data, both with and without mean flow, and to make comparisons between experimental and theoretical data. The paper has a companion [6], in which experimental results are described. t The work on which this article is based was carried out while the authors were under contract to the Lockheed-Georgia Company, Marietta, Georgia, U.S.A. A paper, containing much of the material in this article, was presented at the AIAA 7th Aeroacoustics Conference, Palo Alto, California, U.S.A. 5-7 October, 1981. 503 @ 1983 Academic Press Inc. (London) Limited 0022-460X/83/240503+ 16 $03.00/O
504
A
(‘I;MMIN<;S
AND
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EVt_KSMAN
2. THEOR)’
A simple theoretical treatment, which should be applicable at low frequencies to high amplitude acoustic transmission through a nozzle flow, will first be described, and this will then be discussed in the special cases of low amplitude transmission with flow, and high amplitude transmission without flow. TRANSMISSION THROUGH A NOZZLE FL.OW 2.1. HIGH AMPLITUDE In the first place, attention is focussed on a converging nozzle (see Figure 1) at the end of a circular duct. A uniform mean flow and a superimposed plane acoustic wave travel along the duct, from left to right. A straight pipe end and an orifice plate would represent special cases of this convergent geometry. Except in the neighbourhood of the termination, non-linear effects are considered to be negligible. The flow is assumed to form a jet upon leaving the nozzle.
Figure
1. Flow through
a converging
nozzle.
A region in the flow, between planes 1 and 2, upstream and downstream of the nozzle exit plane, is considered (plane 2 coincides with the Vena confructu, which is incorporated here to account completely for the hydrodynamic losses). Suppose the flow in this region may be taken to be irrotational and incompressible, the fluid is inviscid, and the acoustic wavelength is much greater than the dimensions of the region and the diameter of the duct. It may reasonably be expected that the major effect of this contraction will be hydrodynamic in nature, and Bernoulli’s equation for unsteady flow yields? (P2-Pl)+po(V~-V:)/2+po(alac)(~2-~1)=o.
(1)
In equation (l), p, O, p. and C$ represent pressure, fluid velocity, density and velocity potential, respectively (a list of symbols is given in the Appendix). The second term on the left-hand side of this equation is the pressure drop contributed by the fluid acceleration, and the third term represents the effect of the “mass end correction” at the upstream side of the nozzle. This is the only frequency dependent part of the total pressure drop, caused by the concentration of kinetic energy in the reactive acoustic field just upstream of the nozzle exit. if C$aexp (iot) (o being radian frequency) then the pressure drop associated with the acoustic reactance is equal to -p. a(& - ~$i)/at (if one takes v = Vr$). Expressed in terms of an end correction Si, this becomes iwpOSiu, where poSi is the acoustic inductance. t The fact that equation (1) is only valid for two points on the same streamline pressures in planes 1 and 2 are uniform, the streamlines being parallel.
is not restrictive
here, since
DUCT
TERMINATIONS
AT
HIGH
505
AMPLITUDES
One can define a nozzle/duct area ratio u as Ao/A, and a contraction coefficient C, as A,/Ao, A,, being the area of the uena contracta. No velocity coefficient is used in the representation of the nozzle flow. While this would represent a loss mechanism caused by non-ideal flow, it is negligible in comparison to the effects already considered and is considered unimportant within the framework of the present analysis. If the nozzle flow velocity is represented as V + u,V being the time averaged component, then equation (1) becomes p1-p2=p0[Vz)(1-~*C~)/C~]+p0[u2(1-~*C~)/2C~]+iwp0S~v,
(2)
since ul = v2aC, (from continuity of mass flow at low frequencies), where p1 and p2 are now pressure perturbations and the time independent terms in equation (1) have been discarded. Although equation (1) is only valid, strictly speaking, for an inviscid fluid, separation of the boundary layer at the nozzle lip, and associated jet formation, are phenomena which clearly must be incorporated in any model purporting to embody the principal features of acoustic transmission. It is tacitly assumed here that the boundary layer is thin and plays little direct part (except, of course, where it separates from the solid boundaries) in the acoustic processes; away from the boundary layer, the flow may be taken to be inviscid. The radiation impedance of the duct termination may now be introduced. This would presumably have some sort of mean flow dependence in practice, but since this dependence is unknown at present, and also because any moderately small variations in impedance (more especially the reactive part) make very little difference to the low frequency behaviour of the duct termination the zero flow result of Levine and Schwinger [7] will be used here. At low frequencies, the radiation resistance is equal to poco(koa)*/4, where co is the adiabatic acoustic speed, /co = w/co and a is the radius of the nozzle exit. This low frequency approximation is unlikely to change significantly in the presence of small mean flow, and represents essentially the radiative properties of an acoustic monopole in a free field. One may express the radiation reactance in terms of an exterior end correction S2, as iopoS2. The Levine and Schwinger result at low frequency yields a2 = 0.61~. It is also assumed here that p. and co are the same inside and outside the duct. The interior end correction, Si, may be estimated from results of Ingard [Sj, who has presented data on end corrections associated with (amongst other geometries) a circular piston radiating into a circular pipe. The graphical data presented as a function of area ratio u have been approximated by line segments, the equations of which have been used in the calculations in this investigation. One may write an approximate expression for the radiation impedance of the orifice as 2, =poco(k0u)*/4
+ iwp0S2.
(3)
This is now taken to represent p2/u, with the implication that the acoustic pressure in the nozzle exit plane is the same as that in the venu confrucfu. Equation (2) may now be used to express the impedance seen in the duct at plane 1 as poco&
Juq 01
M
fl
.I
1’
(1-a2C:)+v(l-a2Cf)+ik (sl+s2)+(kou)2 4 2coCf Cf 0
(4)
where [I is the non-dimensional impedance and MJ is the mean flow Mach number in the nozzle exit (assumed here to be reasonably small). Note the inclusion of the area ratio u which arises from the continuity equation u1 = o-v. One could also include a resistive term in equation (4) to account for viscous losses in the acoustic boundary layer
506
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C’UMMINGS
AND
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LVERSMAN
in the neighbourhood of the nozzle lip, but this is generally so small that it has a negligible effect on the results. Equation (4) contains two terms in addition to those expected on the basis of linear acoustics; these result from the fact that the pressure decrease engendered by the acceleration of the fluid into the orifice is not matched by a corresponding pressure recovery on the downstream side of the nozzle, since the flow does not expand. (The largely incompressible flow perturbations in the vortex layer are assumed to have only a secondary effect in this process.) These two additional terms are, first, a resistive term containing the mean flow Mach number, and second, a non-linear resistance involving the velocity perturbation in the nozzle. It is these two terms, dependent (respectively) on mean flow and relatively high acoustic amplitude, which account for the aforementioned acoustic energy imbalance across the nozzle. In the case of zero mean flow there is still a non-linear impedance term, while in the presence of flow, provided 2h4, D IU(/CO(which would normally be the case), the non-linear contribution to the impedance would be swamped by the mean flow dependent term. The cases with and without mean flow merit somewhat different detailed treatment, and will be discussed separately in the next two sections. 2.2. THE CASE OF MODERATE MACH NUMBER When MI >>]Ul/2c0, the impedance l1 is linear and one may readily determine acoustic power relationships. WT is defined as the net power flow in the duct, and WF as the power radiated into the (assumed) acoustic free field surrounding the nozzle. Accounting for a uniform flow in the duct, with Mach number MD, one has WT =AIPil*[(l +~~)2-/r/2(I-M~)2]/2p~c~,
(5)
where Pi is the acoustic pressure amplitude in the sound wave incident on the nozzle and r is the pressure reflection coefficient at the nozzle, given by r = (5i- I)/(51 + 1).
(6)
The power radiated to the far field can be approximated by arguments similar to those used by Bechert [l]. He replaced the duct termination by a monopole representing volumetric flow rate perturbations and a dipole arising from linearized momentum fluctuations. In the present analysis the use of a termination reflection coefficient different from - 1 suggests a further dipole contribution due to pressure fluctuation at the termination. In the end only the source contribution proves to be important for the relatively low Mach numbers to which the rest of the model is restricted. In this case the radiated power can be written directly in terms of the particle velocity at the termination by using the radiation resistance of poco(koa)2/4. The result is WF = AoPoco(koa )*I0I’/s.
(7)
In equation (7), v may be expressed in terms of Pi and
=Pi(1+r)/~WWo,
(8)
and inserting the expression for r from equation (6), to obtain (9) From equations (5)-(7) and (9)
DUCT
TERMINATIONS
AT
HIGH
507
AMPLITUDES
and discarding the non-linear term in equation (4) enables one to find WF/ WT with no difficulty. Equation (10) together with the definition of I1 in equation (4) provides a scheme for the calculation of the ratio of radiated power to transmitted power. When the condition A& >>]v]/2c0 is met, l1 is calculated directly. The case when this condition is not met leads to a non-linear impedance (depending explicitly on u) and is treated as in the high amplitude case with no flow in section 2.3. When equation (4) is substituted in equation (lo), with the non-linear component of [I neglected, and if terms of order higher than (&Kz)~and A4, are discarded (consistent with the low frequency, low Mach number assumption) the result for the ratio W,l WT is WFl wr = (k
I+ k3a I219
)‘lC(4N/C,2
(11)
which is precisely Howe’s result [12], with the inclusion of a factor l/C: in the term containing h4, in the denominator. In reference [2], Howe did not include the effects of a vena contracta in the jet flow. His theory may be modified to include this, briefly, as follows. The power absorbed by the jet flow has been shown by Howe to be given by %bs
=
PO
c
(12)
wxv*udr,
where o is the vorticity in the jet vortex layer, v is the convection velocity of the vorticity and II is the acoustic particle velocity. The integration volume r encloses the region where the acoustic field interacts with the vorticity. Following Howe, one has o=(V2+U2)S(r-R)Ct,
v=3V2+uJi,
(lh
b)
where V2 and u2 are the time averaged and unsteady flow velocity components downstream of the menu contructu (which is assumed to be much less than a nozzle diameter downstream of the nozzle exit plane), r is a radial co-ordinate and R is the radius of the uenu contrucfu; i$ is a triplet of unit vectors with f in the flow direction, j in the r direction and k = i x j. Equations (12) and (13) give w&s = p (v, + ?J$ II. j dS, L Js
(14)
where S is a surface just outside the jet vortex layer. From the continuity equation, IS
u. jdS=7ru2v
(15)
for small Strouhal numbers, where u is, a priori, the unsteady velocity in the orifice. Since VJco is equal to M,/C, and 02 = u/C,, equations (14) and (15) yield, for the time averaged absorbed power, m&S = ,,oc‘,,r‘r 2k&?/C,2,
(16)
where the zero order and quadratic terms in v have been discarded. Since w&s = WT - W.E, equations (7) and (16) may be combined to give an expression for W,l WT. The result is identical to that in equation (11). Concerning the experimental data presented in reference [4], it appears [9] that the nozzle geometry was such that no uenu contructu existed in the jet, so that one may effectively put C, equal to unity.
508
A.
CUMMINGS
2.3.THE CASE OF ZERO MEAN FLOW When A4, = 0, equation (4) gives i, = [v(l -(T’C:
AND AT
W.
HIGH
EVERSMAN AMPLITUDE
)/2c0C: +iko(S1 +SZ) + (koa)‘/4]/a,
(17)
and under conditions where IUI/ co >>(koa), the non-linear resistance dominates the nozzle impedance. If one imposes a sinusoidal incident pressure signal, then obviously the orifice velocity will not be sinusoidal. One would then have (from equation (2)) PI -P2’pov2(1
-a’c;
)/LX;,
(18)
so that, if pl -p2 = P sin wt (for example), ZI= *v^lsin wtl”2,
(19)
where v^ is the peak value of U. Ingard [lo] also noted this fact, in an investigation concerned with the non-linear impedance of orifices at high velocity amplitudes, and gave a Fourier series for the orifice velocity. Following from this, one may write equation (19) as v=6(1~11sinwt-0~159sin3wt+0~072sin5wt-O~043sin7wt+~~~).
(20)
Despite the distortion of the acoustic signal, the fundamental component of the orifice velocity is still 17 dB higher than the third, and clearly will carry most of the acoustic power. This fact enables the difficulties in reconciling equation (20) with the equations governing the linear acoustic field to be overcome (subject to approximations) as follows, by replacing v by its first Fourier component u(l), where u(l)=
V(l)sinwt=l*llv*sinwt.
(21)
Equations (18) and (19) give P = poG2(1 -q2c:
)/2CP,
(22)
and a non-linear orifice resistance may be defined in term of v (1): hL
= (pl
-p2)lutl).
(23)
Now one has, from equations (21), (22) and (23), rNL=poV(l)(l
--g2Cz )/2*464Cf.
This resistance may be incorporated into an (otherwise) linear theory describing acoustic transmission through the orifice, provided non-linear effects away from the orifice are unimportant. t One apparently rather embarrassing feature which has so far been ignored is that without mean flow there is alternating flow through the orifice during successive acoustic half-cycles, and that this will not be quite symmetrical, since the duct geometry does not correspondingly alternate in sympathy with the acoustic flow! So, except for rather small orifices, the contraction coefficient C, might be expected to vary somewhat between acoustic inflow and outflow. If (+<<1, however, this variation would be negligible. Otherwise, some sort of “average” value may be used. t Clearly, if p1 -pz is not sinusoidal, the reasoning which leads to equation (24) no longer applies. In this case, rNL may be defined in terms of the fundamental comqonent ofp 1- p2 as well as u(1). For example, if o=Vsinor,pr-ps~fsin~otandthenr NL=paV(l -/C, )/2.3&Z,. Although this result is close to that in equation (24), it demonstrates that the factor of 1.11 in equation (21) can vary. More generally, this variation will depend on the frequency makeup of p, -p2. This question is discussed further in the following section.
DUCT
TERMINATIONS
AT
HIGH
AMPLITUDES
509
Similarly, of course, the reactive component of the impedance in equation (17) might be expected to vary over an acoustic cycle. Again, however, an average value may successfully be used if necessary. Proceeding with the analysis, one can write expressions for the incident and reflected acoustic power in the duct, respectively, as W =AIPi12/2poCo,
W,=-AIP~121~~-1/2/2p~~~I~~+112,
WT= Wi+ W,.
(25a, b, 26)
In equations (25), one uses the expression in equation (17) for li, incorporating the non-linear resistance expression from equation (24) in terms of V( 1). Similarly, W, may be written as WF = Aop&(koa)2
V2( 1)/8.
(27)
Since l1 is a function of V(l), one cannot explicitly express W,/ W,, W,/ Wi, et ceferu, as functions of IPiJ and must first find V(1) before inserting it into the expressions for t1 and WF. Equation (8) gives a relationship between V(l), which replaces U, and [PiI, ~~OCOV(1)11+~~1_21PiI=O.
(28)
This equation may easily be solved numerically for V(l), if IPi/ is specified, and the Newton-Raphson iterative method was used here. The foregoing theory may also be applied in the case of ducts terminated with orifice plates, either singly or multiply perforated. In the latter case, the radiation resistance term needs to be multiplied by the number of perforations, to account for the additional radiation resistance, over and above that for a single orifice. Equation (17) would now read 5i =[TNLIPOcg+ik0(81+S2)+N(koa)2/41/~,
(29)
where N is the number of holes; rNL is given by equation (24). To calculate W,l WT, one simply finds WF from equation (27) and WT as the sum of Wi and W,, from equations (25). The power reflection coefficient, (WJ Wily may be found from equations (25) and is simply lr12= 1li - 112/1[1+ l12.
(30)
Here again, of course, one needs to find V(1) by the method previously outlined. One can now denote the orifice impedance inside the duct termination by 5, with [ = ~[i (by continuity of volume flow), and put l=9+ix,
(31)
8 and x being resistance and reactance respectively. Now lrj2 = j5-a12/lt
+a12 = [(e -a)2+x2]/[(e
+(r)2+x2].
(32)
The only way in which lrj2 can become very small, in practice, is when e=u
and
x <<(0 +u).
(33a, b)
One may easily show that the low amplitude (that is, effectively linear)value of 8 cannot simultaneously satisfy both of these requirements. If, however, 8 is dominated by its non-linearpart, then it is certainly possible that the denominator in equation (32) can become considerably larger than the numerator, corresponding to a low reflection coefficient.
510
A. (‘UMMINGS 3.
COMPARISON
AND
WITH
Vv
t-LkKSMAN
MEASUREMENTS
In this section, comparisons are made between some of the Lockheed experimental data and predictions based on the theory described in this paper. The Lockheed data presented here are not, in general, the same data as those which appear in reference [4], but are taken from a Lockheed Engineering Report [ll]: both sets of data form only part of a larger body of information on the same topic. One major point to note in these comparisons is as follows. The foregoing theoretical arguments are based on the assumption of simple harmonic time variation. In the obtaining of the experimental data, however, high amplitude impulsive signals were used, and Fourier transforms performed to obtain the frequency spectra for each signal. (In this way, a single test could yield information at all frequencies of interest.) In a linear system, this method may validly be used to yield the transfer function uersus frequency. In the case of tests on the transmission characteristics of nozzles with mean flow, the impedance is essentially linear, and the impulse test method should be applicable. With high amplitude impulse testing in the absence of mean flow, however, the situation is different, and the transmission characteristics of the duct termination will be non-linear because of the non-linear orifice impedance. It would not appear therefore that the impulse test method should give results which would be comparable to those expected on the basis of single frequency excitation. It will be seen, however, that the comparison between experiment and theory is actually quite favourable, and this may indicate that the effects of non-linear distortion of the signal are less severe than might seem to be the case at first sight. This has been supported to some extent by isolated cases where experimental data are available for both impulse testing and harmonic testing. It is, however, not within the scope of this article to discuss in detail the problems involved in applying an impulse test method to a non-linear system, and hence the experimental and theoretical data will be disussed here without consideration of the aforementioned problem. It should be pointed out that testing has been carried out, and data are available, for ranges of kOa and MJ in excess of what one would consider small in the context of the physical model presented here. Mach numbers as high as A& = O-4 are reported here and the impulse testing technique automatically provides frequency response data over a wide bandwidth. It will be seen that comparison of theory and experiment is at least qualitatively in agreement over a broader range than the theoretical limitations would suggest. Two quantities are used here to characterize the transmission properties of the duct termination: the “sound power transfer function” AFT, equal to 10 log (W,/ W,), and the “sound power reflection coefficient” A ~1, equal to 10 log (W,/ Wi). The power transfer function is a measure of the net acoustic power loss occurring upon transmission at the duct termination, either with or without mean flow. The reflection coefficient is only discussed in the absence of mean flow. The phase of the pressure reflection coefficient is also of some interest. 3.1. DUCT TERMINATIONS WITH MEAN FLOW AT HIGH AMPLITUDES: AFT. Comparison between the Lockheed experimental data and theoretical predictions of the transmission characteristics of duct terminations, made either on the basis of equation (10) or equation (11) (the differences are minimal except for very low mean flow Mach numbers), follows much the same pattern as comparison between the experimental data of Bechert [l] and the theoretical results of Howe [2,3] and Bechert [l]: agreement
DUCT
TERMINATIONS
AT
HIGH
511
AMPLITUDES
between experiment and theory is generally quite good, and certainly good enough to lend credibility to the assumptions made in the theory. One feature of particular interest which does emerge from some of the experimental data, however, is that there are data on duct terminations with the same value of u, at the same values of AI,, but with totally different geometries. Specifically, there are measurements of W,/ W, (amongst other quantities) for both a conical nozzle and a “daisy lobe” suppressor nozzle reported in reference [4]. 101
I
I
I
1
I
f
cut-onfor 1st rodlo
m 2
d
mode
-40. (b)
T
cut-onfor 1st rodlal mode
4
5
Figure 2. Sound power transfer function for daisy lobe and conical nozzles. K = 0.581 (a = 0.367), (a) M, = 0.2; (b) MI = 0.4: 0, Experiment, conical nozzle; & experiment, daisy lobe nozzle; -, theory (equation (11) with C, = 1.0).
(R, being the Figures 2(a) and (b) show measurements of A FT, plotted against k&, radius of the duct) for both the conical and daisy lobe nozzles. Theoretical curves from equation (ll), with C, = 1.0, are also shown. The jet flow Mach numbers (Mr) are 0.2 and 0.4, respectively, in these two cases. The open/solid area ratio, K, is equal to V/(1 -a). One immediately sees that the conical and daisy lobe nozzles have almost identical characteristics. This is certainly to be expected on the basis of equation (1 l), if an equivalent value of a, equal to m, is used.t
t At low frequencies, such that the maximum dimension of the duct outlet is much less than an acoustic wavelength the radiation resistance of an unflanged duct termination of any shape is equal to j+,A0w2/41rc,.
512
A.
C‘IJMMINGS
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I:\‘k:KSMAN
Rather surprisingly, there is very little difference between the measured values of AFT, for M, = 0.2 and for MJ = 0.4. Equation (11) indicates a 3 dB increase in absorption with a doubling of M,, where 4A4, >>(koa )‘, but this difference is not observed in the measured data. In both Figures 2(a) and (b), the measured and theoretical data are in fairly good agreement. Both sets of data have the expected trend of decreasing power absorption at increasing frequency. This occurs because the radiation resistance of the nozzle
increases in proportion to tkoaj2; because of this. a greater proportion of the energy present in the unsteady nozzle flow escapes as sound at the higher frequencies. 3.2.
DUCT
Figures
TERMINATIONS
3 and 4(a)-(c)
WITH
show
multiply and singly perforated impulsive incident waves.
ZERO
FLOW,
AT
HIGH
AMPLITUDES:
AFT
consisting of both plates, for zero mean flow but with high amplitude
data
on A m for duct terminations
Figure 3. Sound power transfer function for perforated plate. (a) K = 0.0323 (v = 0.0313). hole diameter = 3.2 mm, number of holes = 32; (b) K = 0.311 (a = 0.237), hole diameter = 1.6 mm, number of holes = 968: 0, Experiment; -, theory, C, = 0.61.
In calculating the theoretical curves, representative values of IPil had to be estimated. This was, of course, difficult since the incident waves were transient in nature. The peak pressures were in the neighbourhood of 155 dB, and this would indicate that levels of 150 to 160 dB might be appropriate. Accordingly, curves were plotted corresponding to incident sound pressure levels (Lb) of both 150 dB and 160 dB. In Figures 3 and 4(a)-(c), one sees that in all cases there is a marked sound power deficit, particularly at low frequencies.
DUCT
0-
TERMINATIONS
AT
HIGH
513
AMPLITUDES
I ib) t cut-on for 1st radial mode
10
(d) 0
o-
o_o_o_o-o-o=o-~-o-o
0 0
t cut-on for 1st radial mode
Figure 4. Sound power transfer function for single orifice. (a) K =0*0159 (a=0.0156); (b) K =0.125 ((r=O.lll); (c) K =0.5 (a=0.333); (d) K +~(a+ 1). 0, Experiment; -, theory. For (a)-(c) theory C, = 0.61; for (d) theory C, = 0.246.
514
A. CUMMINGS
AND
W. EVtlRSMAN
Figures 3(a) and (b) show the effect, on A FT,of varying u and K for multiply perforated plates. With (+ = 0.0313, K = 0.0323, the measured value of AFT is about -31 dB for koRD = 0.125. A tenfold increase in K reduces the low frequency loss by about 10 dB. In calculating the theoretical curves, a contraction coefficient, C,., of 0.61 has been used. There is strong evidence that, in the case of orifice plates with zero mean flow, a uena contracta does exist in the orifice outflow. In the absence of quantitative data, the typical steady flow value of 0.61 was used here. The theoretical curves are in surprisingly good agreement with the measurements. A 10 dB increase in Lk does not produce a corresponding change in An; and the latter quantity typically only increases (in a negative sense) by about 2-3 dB. Figures 4(a)-(c) show data on single orifices, with large variations in cr and K. A 30-fold increase in K reduces the measured low frequency loss by about 15 dB. Again, theoretical curves with C, = 0.61, for Lb = 150 dB and 160 dB, are shown. As before, agreement with the measurements is generally quite good. At the higher values of koRD, some discrepancies are evident, and almost certainly some of these are attributable to higher order acoustic mode transmission. Also, the non-linear distortion of the acoustic signal may play a part in this. Figure 4(d) shows data on a “straight pipe” termination, with (7 -+ 1 and K + CO. Acoustic losses are still evident, though less pronounced, for a given value of kJ3,, than they are for smaller values of cr and K. The experimental results show the same general trend as those in Figures 3 and 4(a)-(c) with an increase in Am as the frequency decreases. The present theory is not applicable to this geometry, which violates the assumptions on which the theory is based (see section 2.3). Attempts to estimate an average value of C, from loss coefficients for steady flow into, and out of, a straight pipe were unsuccessful. Accordingly, a purely empirical value of C,, equal to 0.25, was used to give “best fit” theoretical curves, for comparison with the measurements. The theoretical curves are in good agreement with the experimental data at all values of k,& up to 4, but of course it is only the consistency of this agreement-with frequency-which may be used as a basis for quantitative comparison between theory and measurement, and not agreement in absolute terms. 3.3,
DUCT
TERMINATIONS
WITH
ZERO
FLOW
AT
HIGH
AMPLITUDES:
REFLECTION
COEFFICIENT
Figures 5(a)-(c) show data on the reflection coefficient of single orifice duct terminations, for zero mean flow at high pressure amplitude. The quantity plotted is the power reflection coefficient AR1 = 10 log (W,/ Wi), and the independent variable is log K. Essentially, these curves show the effect, on A RI, of the open/solid area ratio. Three different frequencies are represented, with koRD = O-1, 0.2 and 1.0. The measurements immediately reveal a substantial low frequency reflection loss for a range of log K values. This loss becomes smaller as the frequency rises. In Figure 5(a), the curve of ARI, predicted on the basis solely of linear acoustic theory, is also plotted. It is clear this by no means accounts for the observed large reflection loss. The 150 dB and 160 dB theoretical curves, derived from equation (30), are plotted; a value of C, = 0.61 is again used. These are qualitatively in reasonable agreement with the measurements. We infer from this that it is non-linear effects which account for the relatively large reflection loss at low frequencies. The theoretical curves display a minimum value of A RI at a particular value of log K (or equivalently, v). As Lb increases, the value of u at this minimum increases, and the value of AR1 decreases. This behaviour is consistent with the predictions of equations (32) and (33): if Lb is larger, then 0 is larger. Accordingly, a larger value of u is required
DUCT TERMINATIONS
515
AT HIGH AMPLITUDES
J
-20
-
I
(b) Oh
0
0
I60 dB
150dB
-10 -
I/ -20
0 -2
-I
-2
0 log
-I
0 log
K
K
(c i
(a)
Figure 5. Sound power reflection coefficient for single orifice. (a) -, Theory, k,R, theory, k,R, = 1.0. C, = 0.61; 0, measurement.
= 0.1; (b) -,
theory,
koRD = 0.2; (c) -,
to make AR1 minimum (see equation (33a)). This, coupled with the larger 8 value, implies a smaller value of lrj*, so that A RI becomes larger in a negative sense. The measurements in Figure 5(a) also seem to show a trend to reach a minimum in the region of log K = -0.8 (in keeping with LL = 155 dB). This lends further support to the idea that the peak pressure in the impulsive signal might be respresentative of the equivalent discrete frequency value. Salikuddin and Plumblee [5] have shown pressure/time histories at a location inside the duct, and these indicate the passage of the incident and reflected pulses. Evidently, from these traces, the incident and reflected waves are roughly in phase for small values of K: but in antiphase for large values of K. In the region -1.0
AND
CONCLUSIONS
The object of this paper is to present physical arguments and associated quantitative theoretical data in an attempt to explain certain phenomena observed in the Lockheed measurements. These effects are twofold: first a net acoustic power imbalance across duct terminations, both with mean airflow, at high pressure amplitudes, and without flow
516
A. C‘LJMMINGS
AND
W.
EVEKSMAN
at high amplitudes; and secondly, a considerable reduction in reflection coefficients at high amplitudes, without mean flow. Simple theoretical models have been devised in order to represent the effects of mean flow and high amplitude on acoustic transmission through a duct termination. These have been based on the assumption of simple harmonic time variation, although the measurements were made on the basis of Fourier transformed transient pressure signals. Comparisons between the theoretical and experimental data have been encouragingly good, and this fact is taken to support the physical arguments on which the theory is based. With mean flow, the net power deficit is explained by a conversion of acoustic energy into vertical energy, which is simply convected away by the mean flow and eventually dissipated as heat. The presence of a sharp edge (in this case, the nozzle lip) causes a fluctuating vorticity field in the wake of the nozzle, and the interaction between this and the acoustic velocity field causes energy to be extracted from the sound field and fed to the vorticity field. Howe’s work [2,3] elucidates this effect. The measurements and theory discussed here, relating to the situation where mean flow is present, are generally in keeping with the measurements and theory of Bechert [l] and Howe [2,3] on the same topic. Additionally. it has been shown here that the detailed geometry of the nozzle appears to be somewhat unimportant by comparison with other factors such as open/solid area ratio, frequency and jet flow Mach number. In the absence of mean flow, the acoustic energy loss still occurs by the conversion of acoustic into vertical energy. In the theory described here, it is only implicit that this is the case, and the details of the vorticity field are not considered. It is clear, however, from the work of Ingard and Labate [12] that ring vortices are shed from either side of an orifice plate at the end of a tube in which high amplitude plane waves are transmitted. Salikuddin and Ahuja [4] have shown a schlieren photograph which demonstrates ring vortex production as an acoustic wave is transmitting through a converging nozzle. There is little doubt that the generation of ring vortices is responsible for the acoustic energy losses evident in the results discussed in this article. The measurements and theory presented here, on the net acoustic power loss at duct terminations at high amplitudes and in the absence of mean flow, are in astonishingly good agreement, both quantitatively and qualitatively, considering the theory is based on sinusoidal time varation, the measurements were made with transient signals, and the orifice impedance is non-linear at high amplitudes. Cummings [13] presented data on the transmission of high amplitude tonebursts through orifice plates, and found that a theoretical approach similar to that described here also gave good predictions for a type of signal which must be considered to be a “halfway house” between a short duration impulsive signal and a continuous sinusoidal signal. The experimental and theoretical data on the reflection coefficient at high amplitude and without flow, are in good qualitative agreement. It has been demonstrated that non-linear effects are responsible for the considerable reduction in the reflection coefficient which is observed to occur at certain open/solid area ratios, at low frequencies and at high amplitudes. Although the somewhat limited objectives of this piece of work have, one feels, largely been fulfilled, there is clearly scope for a great deal of additional work in the general area of acoustic absorption in shear layers. The non-linear interaction which may occur between a complex acoustic signal and vorticity fields would merit investigation in a number of contexts in addition to the present case. Also, more specifically, the validity of test methods involving use of transient acoustic signals in such non-linear systems would merit further study.
DUCT TERMINATIONS AT HIGH AMPLITUDES
517
REFERENCES 1. D. W. BECHERT 1979 American Institute of Aeronautics and Astronautics 5th Aeroacoustics Conference Paper No. 79-0575. Sound absorption caused by vorticity shedding demonstrated
with a jet flow. 2. M. S. HOWE 1980 American
Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-0972. The dissipation of sound at sharp edges. 3. M. S. HOWE 1979 Journal of Fluid Mechanics 91, 202-229. Attenuation of sound in a low
Mach number nozzle flow. 4. M. SALIKUDDIN and K. K. AHUJA 1981 American Institute of Aeronautics and Astronautics 7th Aeroacoustics Conference Paper No. 81- 1978. Acoustic power dissipation on radiation through duct terminations-experiments. 5. M. SALIKUDDIN and H. E. PLUMBLEE 1980 American Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-0991. Low frequency sound absorption of orifice plates, perforated plates, and nozzles. 6. M. SALIKUDDIN and K. K. AHUJA 1983 Journal of Sound and Vibration 91, 479-502. Acoustic power dissipation through duct terminations: experiments. 7. H. LEVINE and J. SCHWINGER 1948 Physical Review 73, 383-406. On the radiation of sound from an unflanged circular pipe. 8. U. INGARD 1953 Journal of the Acoustical Society of America 25, 1037-1061. On the theory and design of acoustic resonators. 9. K. K. AHUJA 1981. Private communication. 10. K. U. INGARD 1970 Journal of the Acoustical Society of America 48, 32-33. Nonlinear distortion of sound transmitted through an orifice. Engineering Report 11. A. CUMMINGS and W. EVERSMAN 1980 Lockheed-Georgia LG80ER0154. An investigation of acoustic energy loss in radiation from ducts to the far field at low frequencies, low Mach numbers, and high sound pressure levels. 12. U. INGARD and S. LABATE 1950 Journal of the Acoustical Society of America 22, 211-218. Acoustic circulation effects and the nonlinear impedance of orifices. 13. A. CUMMINGS 1980 American Society of Mechanical Engineers Winter Annual Meeting Paper No. 81-WA/NCA10. High-amplitude acoustic power losses in perforated materials.
APPENDIX: A a C, co
i i$ K k, L; M N P P R RD r rNL
S t U V V V
W
LIST OF SYMBOLS
cross sectional area radius of nozzle exit contraction coefficient adiabatic speed of sound J--T triad of unit vectors open/solid area ratio =&J/co sound pressure level in an incident wave mean flow Mach number number of holes in a perforated plate acoustic pressure amplitude thermodynamic pressure, or pressure perturbation radius of the vena contracta radius of the duct pressure reflection coefficient at the duct exit; or a radial co-ordinate a non-linear resistance a surface just outside the jet vortex layer time acoustic particle velocity time averaged velocity component velocity perturbation convection velocity of vorticity in the jet vortex layer time averaged acoustic power acoustic power absorbed by the jet flow
518 2, AFT A SP’) 61,& 5 6 P (T iJ X w w
A. C‘LJMMINGS
radiation impedance in the nozzle sound power transfer function (in sound power reflection coefficient the Dirac delta function inner, outer mass end corrections non-dimensional impedance real part of [ density nozzle/duct area ratio an integration volume velocity potential, or argument of imaginary part of [ radian frequency vorticity in the jet vortex layer
Subscripts D in the duct F in the far field of an incident wave ; at the jet nozzle exit 0 of orifice r of a reflected wave T denotes transmitted power vc of the vena contracta 0 time averaged value 1 at position 1 2 at position 2
AND
u’.
L.L’~KSMAN
exit plane decibels) (in decibels) at the duct termination
r
Vibration (1983) 91(4), 503-518
HIGH AMPLITUDE ACOUSTIC TRANSMISSION THROUGH DUCT TERMINATIONS: THEORY? A. CUMMINGS
AND
W. EVERSMAN
Department of Mechanical and Aerospace Engineering, University of Missouri-Rolla, Roila, Missouri 65401, U.S.A. (Received 30 September 1982, and in revised form
16 February 1983)
Recent experimental measurements have demonstrated that net acoustic energy dissipation can occur when sound waves interact with free shear layers, which are produced either by boundary layer separation in mean fluid flow at sharp edges, or by separation of the boundary layer in the acoustic flow at an edge in the absence of mean flow. This paper presents theoretical results which are offered in an attempt to explain these observations quantitatively. Comparison is made between the predicted and measured net energy loss which occurs upon transmission of high amplitude impulsive acoustic waves through various duct terminations, and also between calculated and measured reflection coefficients in the duct. The agreement is generally at least qualitatively good, and would appear to justify the physical assumptions on which the theoretical arguments are based. 1. INTRODUCTION In the past few years, there has been a growth of interest
in acoustic energy effects associated with the interaction between sound fields and free shear layers in separated flows. In particular, the net acoustic energy absorption which is observed to occur when an acoustic wave is transmitted out through an air jet issuing from a pipe has been studied by several workers. Bechert’s experimental and theoretical results [l] and Howe’s theoretical analyses [2, 31 for example, offer persuasive evidence that the absorption mechanism is the conversion of acoustic energy into vertical energy at a sharp edge, and the subsequent dissipation of this into heat, without further substantial interaction with the acoustic field. It has also been noted that, in the absence of mean flow, energy can be dissipated by the same mechanism if the acoustic velocity is high enough to cause flow separation, and recent work at Lockheed-Georgia has shown this effect well in a comprehensive study of noise transmission through duct terminations. Salikuddin and Ahuja [4] have reported substantial low-frequency acoustic losses at duct terminations both with and without mean flow. Salikuddin and Plumblee [5] also reported similar data in the absence of mean flow. Reference [5] shows, additionally, that the acoustic reflection coefficient at a duct termination can be considerably reduced (below its small amplitude value) if the amplitude of the incident wave is large. The purpose of this paper is to offer a theoretical explanation of the aforementioned experimental data, both with and without mean flow, and to make comparisons between experimental and theoretical data. The paper has a companion [6], in which experimental results are described. t The work on which this article is based was carried out while the authors were under contract to the Lockheed-Georgia Company, Marietta, Georgia, U.S.A. A paper, containing much of the material in this article, was presented at the AIAA 7th Aeroacoustics Conference, Palo Alto, California, U.S.A. 5-7 October, 1981. 503 @ 1983 Academic Press Inc. (London) Limited 0022-460X/83/240503+ 16 $03.00/O
504
A
(‘I;MMIN<;S
AND
W.
EVt_KSMAN
2. THEOR)’
A simple theoretical treatment, which should be applicable at low frequencies to high amplitude acoustic transmission through a nozzle flow, will first be described, and this will then be discussed in the special cases of low amplitude transmission with flow, and high amplitude transmission without flow. TRANSMISSION THROUGH A NOZZLE FL.OW 2.1. HIGH AMPLITUDE In the first place, attention is focussed on a converging nozzle (see Figure 1) at the end of a circular duct. A uniform mean flow and a superimposed plane acoustic wave travel along the duct, from left to right. A straight pipe end and an orifice plate would represent special cases of this convergent geometry. Except in the neighbourhood of the termination, non-linear effects are considered to be negligible. The flow is assumed to form a jet upon leaving the nozzle.
Figure
1. Flow through
a converging
nozzle.
A region in the flow, between planes 1 and 2, upstream and downstream of the nozzle exit plane, is considered (plane 2 coincides with the Vena confructu, which is incorporated here to account completely for the hydrodynamic losses). Suppose the flow in this region may be taken to be irrotational and incompressible, the fluid is inviscid, and the acoustic wavelength is much greater than the dimensions of the region and the diameter of the duct. It may reasonably be expected that the major effect of this contraction will be hydrodynamic in nature, and Bernoulli’s equation for unsteady flow yields? (P2-Pl)+po(V~-V:)/2+po(alac)(~2-~1)=o.
(1)
In equation (l), p, O, p. and C$ represent pressure, fluid velocity, density and velocity potential, respectively (a list of symbols is given in the Appendix). The second term on the left-hand side of this equation is the pressure drop contributed by the fluid acceleration, and the third term represents the effect of the “mass end correction” at the upstream side of the nozzle. This is the only frequency dependent part of the total pressure drop, caused by the concentration of kinetic energy in the reactive acoustic field just upstream of the nozzle exit. if C$aexp (iot) (o being radian frequency) then the pressure drop associated with the acoustic reactance is equal to -p. a(& - ~$i)/at (if one takes v = Vr$). Expressed in terms of an end correction Si, this becomes iwpOSiu, where poSi is the acoustic inductance. t The fact that equation (1) is only valid for two points on the same streamline pressures in planes 1 and 2 are uniform, the streamlines being parallel.
is not restrictive
here, since
DUCT
TERMINATIONS
AT
HIGH
505
AMPLITUDES
One can define a nozzle/duct area ratio u as Ao/A, and a contraction coefficient C, as A,/Ao, A,, being the area of the uena contracta. No velocity coefficient is used in the representation of the nozzle flow. While this would represent a loss mechanism caused by non-ideal flow, it is negligible in comparison to the effects already considered and is considered unimportant within the framework of the present analysis. If the nozzle flow velocity is represented as V + u,V being the time averaged component, then equation (1) becomes p1-p2=p0[Vz)(1-~*C~)/C~]+p0[u2(1-~*C~)/2C~]+iwp0S~v,
(2)
since ul = v2aC, (from continuity of mass flow at low frequencies), where p1 and p2 are now pressure perturbations and the time independent terms in equation (1) have been discarded. Although equation (1) is only valid, strictly speaking, for an inviscid fluid, separation of the boundary layer at the nozzle lip, and associated jet formation, are phenomena which clearly must be incorporated in any model purporting to embody the principal features of acoustic transmission. It is tacitly assumed here that the boundary layer is thin and plays little direct part (except, of course, where it separates from the solid boundaries) in the acoustic processes; away from the boundary layer, the flow may be taken to be inviscid. The radiation impedance of the duct termination may now be introduced. This would presumably have some sort of mean flow dependence in practice, but since this dependence is unknown at present, and also because any moderately small variations in impedance (more especially the reactive part) make very little difference to the low frequency behaviour of the duct termination the zero flow result of Levine and Schwinger [7] will be used here. At low frequencies, the radiation resistance is equal to poco(koa)*/4, where co is the adiabatic acoustic speed, /co = w/co and a is the radius of the nozzle exit. This low frequency approximation is unlikely to change significantly in the presence of small mean flow, and represents essentially the radiative properties of an acoustic monopole in a free field. One may express the radiation reactance in terms of an exterior end correction S2, as iopoS2. The Levine and Schwinger result at low frequency yields a2 = 0.61~. It is also assumed here that p. and co are the same inside and outside the duct. The interior end correction, Si, may be estimated from results of Ingard [Sj, who has presented data on end corrections associated with (amongst other geometries) a circular piston radiating into a circular pipe. The graphical data presented as a function of area ratio u have been approximated by line segments, the equations of which have been used in the calculations in this investigation. One may write an approximate expression for the radiation impedance of the orifice as 2, =poco(k0u)*/4
+ iwp0S2.
(3)
This is now taken to represent p2/u, with the implication that the acoustic pressure in the nozzle exit plane is the same as that in the venu confrucfu. Equation (2) may now be used to express the impedance seen in the duct at plane 1 as poco&
Juq 01
M
fl
.I
1’
(1-a2C:)+v(l-a2Cf)+ik (sl+s2)+(kou)2 4 2coCf Cf 0
(4)
where [I is the non-dimensional impedance and MJ is the mean flow Mach number in the nozzle exit (assumed here to be reasonably small). Note the inclusion of the area ratio u which arises from the continuity equation u1 = o-v. One could also include a resistive term in equation (4) to account for viscous losses in the acoustic boundary layer
506
A.
C’UMMINGS
AND
h
LVERSMAN
in the neighbourhood of the nozzle lip, but this is generally so small that it has a negligible effect on the results. Equation (4) contains two terms in addition to those expected on the basis of linear acoustics; these result from the fact that the pressure decrease engendered by the acceleration of the fluid into the orifice is not matched by a corresponding pressure recovery on the downstream side of the nozzle, since the flow does not expand. (The largely incompressible flow perturbations in the vortex layer are assumed to have only a secondary effect in this process.) These two additional terms are, first, a resistive term containing the mean flow Mach number, and second, a non-linear resistance involving the velocity perturbation in the nozzle. It is these two terms, dependent (respectively) on mean flow and relatively high acoustic amplitude, which account for the aforementioned acoustic energy imbalance across the nozzle. In the case of zero mean flow there is still a non-linear impedance term, while in the presence of flow, provided 2h4, D IU(/CO(which would normally be the case), the non-linear contribution to the impedance would be swamped by the mean flow dependent term. The cases with and without mean flow merit somewhat different detailed treatment, and will be discussed separately in the next two sections. 2.2. THE CASE OF MODERATE MACH NUMBER When MI >>]Ul/2c0, the impedance l1 is linear and one may readily determine acoustic power relationships. WT is defined as the net power flow in the duct, and WF as the power radiated into the (assumed) acoustic free field surrounding the nozzle. Accounting for a uniform flow in the duct, with Mach number MD, one has WT =AIPil*[(l +~~)2-/r/2(I-M~)2]/2p~c~,
(5)
where Pi is the acoustic pressure amplitude in the sound wave incident on the nozzle and r is the pressure reflection coefficient at the nozzle, given by r = (5i- I)/(51 + 1).
(6)
The power radiated to the far field can be approximated by arguments similar to those used by Bechert [l]. He replaced the duct termination by a monopole representing volumetric flow rate perturbations and a dipole arising from linearized momentum fluctuations. In the present analysis the use of a termination reflection coefficient different from - 1 suggests a further dipole contribution due to pressure fluctuation at the termination. In the end only the source contribution proves to be important for the relatively low Mach numbers to which the rest of the model is restricted. In this case the radiated power can be written directly in terms of the particle velocity at the termination by using the radiation resistance of poco(koa)2/4. The result is WF = AoPoco(koa )*I0I’/s.
(7)
In equation (7), v may be expressed in terms of Pi and
=Pi(1+r)/~WWo,
(8)
and inserting the expression for r from equation (6), to obtain (9) From equations (5)-(7) and (9)
DUCT
TERMINATIONS
AT
HIGH
507
AMPLITUDES
and discarding the non-linear term in equation (4) enables one to find WF/ WT with no difficulty. Equation (10) together with the definition of I1 in equation (4) provides a scheme for the calculation of the ratio of radiated power to transmitted power. When the condition A& >>]v]/2c0 is met, l1 is calculated directly. The case when this condition is not met leads to a non-linear impedance (depending explicitly on u) and is treated as in the high amplitude case with no flow in section 2.3. When equation (4) is substituted in equation (lo), with the non-linear component of [I neglected, and if terms of order higher than (&Kz)~and A4, are discarded (consistent with the low frequency, low Mach number assumption) the result for the ratio W,l WT is WFl wr = (k
I+ k3a I219
)‘lC(4N/C,2
(11)
which is precisely Howe’s result [12], with the inclusion of a factor l/C: in the term containing h4, in the denominator. In reference [2], Howe did not include the effects of a vena contracta in the jet flow. His theory may be modified to include this, briefly, as follows. The power absorbed by the jet flow has been shown by Howe to be given by %bs
=
PO
c
(12)
wxv*udr,
where o is the vorticity in the jet vortex layer, v is the convection velocity of the vorticity and II is the acoustic particle velocity. The integration volume r encloses the region where the acoustic field interacts with the vorticity. Following Howe, one has o=(V2+U2)S(r-R)Ct,
v=3V2+uJi,
(lh
b)
where V2 and u2 are the time averaged and unsteady flow velocity components downstream of the menu contructu (which is assumed to be much less than a nozzle diameter downstream of the nozzle exit plane), r is a radial co-ordinate and R is the radius of the uenu contrucfu; i$ is a triplet of unit vectors with f in the flow direction, j in the r direction and k = i x j. Equations (12) and (13) give w&s = p (v, + ?J$ II. j dS, L Js
(14)
where S is a surface just outside the jet vortex layer. From the continuity equation, IS
u. jdS=7ru2v
(15)
for small Strouhal numbers, where u is, a priori, the unsteady velocity in the orifice. Since VJco is equal to M,/C, and 02 = u/C,, equations (14) and (15) yield, for the time averaged absorbed power, m&S = ,,oc‘,,r‘r 2k&?/C,2,
(16)
where the zero order and quadratic terms in v have been discarded. Since w&s = WT - W.E, equations (7) and (16) may be combined to give an expression for W,l WT. The result is identical to that in equation (11). Concerning the experimental data presented in reference [4], it appears [9] that the nozzle geometry was such that no uenu contructu existed in the jet, so that one may effectively put C, equal to unity.
508
A.
CUMMINGS
2.3.THE CASE OF ZERO MEAN FLOW When A4, = 0, equation (4) gives i, = [v(l -(T’C:
AND AT
W.
HIGH
EVERSMAN AMPLITUDE
)/2c0C: +iko(S1 +SZ) + (koa)‘/4]/a,
(17)
and under conditions where IUI/ co >>(koa), the non-linear resistance dominates the nozzle impedance. If one imposes a sinusoidal incident pressure signal, then obviously the orifice velocity will not be sinusoidal. One would then have (from equation (2)) PI -P2’pov2(1
-a’c;
)/LX;,
(18)
so that, if pl -p2 = P sin wt (for example), ZI= *v^lsin wtl”2,
(19)
where v^ is the peak value of U. Ingard [lo] also noted this fact, in an investigation concerned with the non-linear impedance of orifices at high velocity amplitudes, and gave a Fourier series for the orifice velocity. Following from this, one may write equation (19) as v=6(1~11sinwt-0~159sin3wt+0~072sin5wt-O~043sin7wt+~~~).
(20)
Despite the distortion of the acoustic signal, the fundamental component of the orifice velocity is still 17 dB higher than the third, and clearly will carry most of the acoustic power. This fact enables the difficulties in reconciling equation (20) with the equations governing the linear acoustic field to be overcome (subject to approximations) as follows, by replacing v by its first Fourier component u(l), where u(l)=
V(l)sinwt=l*llv*sinwt.
(21)
Equations (18) and (19) give P = poG2(1 -q2c:
)/2CP,
(22)
and a non-linear orifice resistance may be defined in term of v (1): hL
= (pl
-p2)lutl).
(23)
Now one has, from equations (21), (22) and (23), rNL=poV(l)(l
--g2Cz )/2*464Cf.
This resistance may be incorporated into an (otherwise) linear theory describing acoustic transmission through the orifice, provided non-linear effects away from the orifice are unimportant. t One apparently rather embarrassing feature which has so far been ignored is that without mean flow there is alternating flow through the orifice during successive acoustic half-cycles, and that this will not be quite symmetrical, since the duct geometry does not correspondingly alternate in sympathy with the acoustic flow! So, except for rather small orifices, the contraction coefficient C, might be expected to vary somewhat between acoustic inflow and outflow. If (+<<1, however, this variation would be negligible. Otherwise, some sort of “average” value may be used. t Clearly, if p1 -pz is not sinusoidal, the reasoning which leads to equation (24) no longer applies. In this case, rNL may be defined in terms of the fundamental comqonent ofp 1- p2 as well as u(1). For example, if o=Vsinor,pr-ps~fsin~otandthenr NL=paV(l -/C, )/2.3&Z,. Although this result is close to that in equation (24), it demonstrates that the factor of 1.11 in equation (21) can vary. More generally, this variation will depend on the frequency makeup of p, -p2. This question is discussed further in the following section.
DUCT
TERMINATIONS
AT
HIGH
AMPLITUDES
509
Similarly, of course, the reactive component of the impedance in equation (17) might be expected to vary over an acoustic cycle. Again, however, an average value may successfully be used if necessary. Proceeding with the analysis, one can write expressions for the incident and reflected acoustic power in the duct, respectively, as W =AIPi12/2poCo,
W,=-AIP~121~~-1/2/2p~~~I~~+112,
WT= Wi+ W,.
(25a, b, 26)
In equations (25), one uses the expression in equation (17) for li, incorporating the non-linear resistance expression from equation (24) in terms of V( 1). Similarly, W, may be written as WF = Aop&(koa)2
V2( 1)/8.
(27)
Since l1 is a function of V(l), one cannot explicitly express W,/ W,, W,/ Wi, et ceferu, as functions of IPiJ and must first find V(1) before inserting it into the expressions for t1 and WF. Equation (8) gives a relationship between V(l), which replaces U, and [PiI, ~~OCOV(1)11+~~1_21PiI=O.
(28)
This equation may easily be solved numerically for V(l), if IPi/ is specified, and the Newton-Raphson iterative method was used here. The foregoing theory may also be applied in the case of ducts terminated with orifice plates, either singly or multiply perforated. In the latter case, the radiation resistance term needs to be multiplied by the number of perforations, to account for the additional radiation resistance, over and above that for a single orifice. Equation (17) would now read 5i =[TNLIPOcg+ik0(81+S2)+N(koa)2/41/~,
(29)
where N is the number of holes; rNL is given by equation (24). To calculate W,l WT, one simply finds WF from equation (27) and WT as the sum of Wi and W,, from equations (25). The power reflection coefficient, (WJ Wily may be found from equations (25) and is simply lr12= 1li - 112/1[1+ l12.
(30)
Here again, of course, one needs to find V(1) by the method previously outlined. One can now denote the orifice impedance inside the duct termination by 5, with [ = ~[i (by continuity of volume flow), and put l=9+ix,
(31)
8 and x being resistance and reactance respectively. Now lrj2 = j5-a12/lt
+a12 = [(e -a)2+x2]/[(e
+(r)2+x2].
(32)
The only way in which lrj2 can become very small, in practice, is when e=u
and
x <<(0 +u).
(33a, b)
One may easily show that the low amplitude (that is, effectively linear)value of 8 cannot simultaneously satisfy both of these requirements. If, however, 8 is dominated by its non-linearpart, then it is certainly possible that the denominator in equation (32) can become considerably larger than the numerator, corresponding to a low reflection coefficient.
510
A. (‘UMMINGS 3.
COMPARISON
AND
WITH
Vv
t-LkKSMAN
MEASUREMENTS
In this section, comparisons are made between some of the Lockheed experimental data and predictions based on the theory described in this paper. The Lockheed data presented here are not, in general, the same data as those which appear in reference [4], but are taken from a Lockheed Engineering Report [ll]: both sets of data form only part of a larger body of information on the same topic. One major point to note in these comparisons is as follows. The foregoing theoretical arguments are based on the assumption of simple harmonic time variation. In the obtaining of the experimental data, however, high amplitude impulsive signals were used, and Fourier transforms performed to obtain the frequency spectra for each signal. (In this way, a single test could yield information at all frequencies of interest.) In a linear system, this method may validly be used to yield the transfer function uersus frequency. In the case of tests on the transmission characteristics of nozzles with mean flow, the impedance is essentially linear, and the impulse test method should be applicable. With high amplitude impulse testing in the absence of mean flow, however, the situation is different, and the transmission characteristics of the duct termination will be non-linear because of the non-linear orifice impedance. It would not appear therefore that the impulse test method should give results which would be comparable to those expected on the basis of single frequency excitation. It will be seen, however, that the comparison between experiment and theory is actually quite favourable, and this may indicate that the effects of non-linear distortion of the signal are less severe than might seem to be the case at first sight. This has been supported to some extent by isolated cases where experimental data are available for both impulse testing and harmonic testing. It is, however, not within the scope of this article to discuss in detail the problems involved in applying an impulse test method to a non-linear system, and hence the experimental and theoretical data will be disussed here without consideration of the aforementioned problem. It should be pointed out that testing has been carried out, and data are available, for ranges of kOa and MJ in excess of what one would consider small in the context of the physical model presented here. Mach numbers as high as A& = O-4 are reported here and the impulse testing technique automatically provides frequency response data over a wide bandwidth. It will be seen that comparison of theory and experiment is at least qualitatively in agreement over a broader range than the theoretical limitations would suggest. Two quantities are used here to characterize the transmission properties of the duct termination: the “sound power transfer function” AFT, equal to 10 log (W,/ W,), and the “sound power reflection coefficient” A ~1, equal to 10 log (W,/ Wi). The power transfer function is a measure of the net acoustic power loss occurring upon transmission at the duct termination, either with or without mean flow. The reflection coefficient is only discussed in the absence of mean flow. The phase of the pressure reflection coefficient is also of some interest. 3.1. DUCT TERMINATIONS WITH MEAN FLOW AT HIGH AMPLITUDES: AFT. Comparison between the Lockheed experimental data and theoretical predictions of the transmission characteristics of duct terminations, made either on the basis of equation (10) or equation (11) (the differences are minimal except for very low mean flow Mach numbers), follows much the same pattern as comparison between the experimental data of Bechert [l] and the theoretical results of Howe [2,3] and Bechert [l]: agreement
DUCT
TERMINATIONS
AT
HIGH
511
AMPLITUDES
between experiment and theory is generally quite good, and certainly good enough to lend credibility to the assumptions made in the theory. One feature of particular interest which does emerge from some of the experimental data, however, is that there are data on duct terminations with the same value of u, at the same values of AI,, but with totally different geometries. Specifically, there are measurements of W,/ W, (amongst other quantities) for both a conical nozzle and a “daisy lobe” suppressor nozzle reported in reference [4]. 101
I
I
I
1
I
f
cut-onfor 1st rodlo
m 2
d
mode
-40. (b)
T
cut-onfor 1st rodlal mode
4
5
Figure 2. Sound power transfer function for daisy lobe and conical nozzles. K = 0.581 (a = 0.367), (a) M, = 0.2; (b) MI = 0.4: 0, Experiment, conical nozzle; & experiment, daisy lobe nozzle; -, theory (equation (11) with C, = 1.0).
(R, being the Figures 2(a) and (b) show measurements of A FT, plotted against k&, radius of the duct) for both the conical and daisy lobe nozzles. Theoretical curves from equation (ll), with C, = 1.0, are also shown. The jet flow Mach numbers (Mr) are 0.2 and 0.4, respectively, in these two cases. The open/solid area ratio, K, is equal to V/(1 -a). One immediately sees that the conical and daisy lobe nozzles have almost identical characteristics. This is certainly to be expected on the basis of equation (1 l), if an equivalent value of a, equal to m, is used.t
t At low frequencies, such that the maximum dimension of the duct outlet is much less than an acoustic wavelength the radiation resistance of an unflanged duct termination of any shape is equal to j+,A0w2/41rc,.
512
A.
C‘IJMMINGS
AND
W.
I:\‘k:KSMAN
Rather surprisingly, there is very little difference between the measured values of AFT, for M, = 0.2 and for MJ = 0.4. Equation (11) indicates a 3 dB increase in absorption with a doubling of M,, where 4A4, >>(koa )‘, but this difference is not observed in the measured data. In both Figures 2(a) and (b), the measured and theoretical data are in fairly good agreement. Both sets of data have the expected trend of decreasing power absorption at increasing frequency. This occurs because the radiation resistance of the nozzle
increases in proportion to tkoaj2; because of this. a greater proportion of the energy present in the unsteady nozzle flow escapes as sound at the higher frequencies. 3.2.
DUCT
Figures
TERMINATIONS
3 and 4(a)-(c)
WITH
show
multiply and singly perforated impulsive incident waves.
ZERO
FLOW,
AT
HIGH
AMPLITUDES:
AFT
consisting of both plates, for zero mean flow but with high amplitude
data
on A m for duct terminations
Figure 3. Sound power transfer function for perforated plate. (a) K = 0.0323 (v = 0.0313). hole diameter = 3.2 mm, number of holes = 32; (b) K = 0.311 (a = 0.237), hole diameter = 1.6 mm, number of holes = 968: 0, Experiment; -, theory, C, = 0.61.
In calculating the theoretical curves, representative values of IPil had to be estimated. This was, of course, difficult since the incident waves were transient in nature. The peak pressures were in the neighbourhood of 155 dB, and this would indicate that levels of 150 to 160 dB might be appropriate. Accordingly, curves were plotted corresponding to incident sound pressure levels (Lb) of both 150 dB and 160 dB. In Figures 3 and 4(a)-(c), one sees that in all cases there is a marked sound power deficit, particularly at low frequencies.
DUCT
0-
TERMINATIONS
AT
HIGH
513
AMPLITUDES
I ib) t cut-on for 1st radial mode
10
(d) 0
o-
o_o_o_o-o-o=o-~-o-o
0 0
t cut-on for 1st radial mode
Figure 4. Sound power transfer function for single orifice. (a) K =0*0159 (a=0.0156); (b) K =0.125 ((r=O.lll); (c) K =0.5 (a=0.333); (d) K +~(a+ 1). 0, Experiment; -, theory. For (a)-(c) theory C, = 0.61; for (d) theory C, = 0.246.
514
A. CUMMINGS
AND
W. EVtlRSMAN
Figures 3(a) and (b) show the effect, on A FT,of varying u and K for multiply perforated plates. With (+ = 0.0313, K = 0.0323, the measured value of AFT is about -31 dB for koRD = 0.125. A tenfold increase in K reduces the low frequency loss by about 10 dB. In calculating the theoretical curves, a contraction coefficient, C,., of 0.61 has been used. There is strong evidence that, in the case of orifice plates with zero mean flow, a uena contracta does exist in the orifice outflow. In the absence of quantitative data, the typical steady flow value of 0.61 was used here. The theoretical curves are in surprisingly good agreement with the measurements. A 10 dB increase in Lk does not produce a corresponding change in An; and the latter quantity typically only increases (in a negative sense) by about 2-3 dB. Figures 4(a)-(c) show data on single orifices, with large variations in cr and K. A 30-fold increase in K reduces the measured low frequency loss by about 15 dB. Again, theoretical curves with C, = 0.61, for Lb = 150 dB and 160 dB, are shown. As before, agreement with the measurements is generally quite good. At the higher values of koRD, some discrepancies are evident, and almost certainly some of these are attributable to higher order acoustic mode transmission. Also, the non-linear distortion of the acoustic signal may play a part in this. Figure 4(d) shows data on a “straight pipe” termination, with (7 -+ 1 and K + CO. Acoustic losses are still evident, though less pronounced, for a given value of kJ3,, than they are for smaller values of cr and K. The experimental results show the same general trend as those in Figures 3 and 4(a)-(c) with an increase in Am as the frequency decreases. The present theory is not applicable to this geometry, which violates the assumptions on which the theory is based (see section 2.3). Attempts to estimate an average value of C, from loss coefficients for steady flow into, and out of, a straight pipe were unsuccessful. Accordingly, a purely empirical value of C,, equal to 0.25, was used to give “best fit” theoretical curves, for comparison with the measurements. The theoretical curves are in good agreement with the experimental data at all values of k,& up to 4, but of course it is only the consistency of this agreement-with frequency-which may be used as a basis for quantitative comparison between theory and measurement, and not agreement in absolute terms. 3.3,
DUCT
TERMINATIONS
WITH
ZERO
FLOW
AT
HIGH
AMPLITUDES:
REFLECTION
COEFFICIENT
Figures 5(a)-(c) show data on the reflection coefficient of single orifice duct terminations, for zero mean flow at high pressure amplitude. The quantity plotted is the power reflection coefficient AR1 = 10 log (W,/ Wi), and the independent variable is log K. Essentially, these curves show the effect, on A RI, of the open/solid area ratio. Three different frequencies are represented, with koRD = O-1, 0.2 and 1.0. The measurements immediately reveal a substantial low frequency reflection loss for a range of log K values. This loss becomes smaller as the frequency rises. In Figure 5(a), the curve of ARI, predicted on the basis solely of linear acoustic theory, is also plotted. It is clear this by no means accounts for the observed large reflection loss. The 150 dB and 160 dB theoretical curves, derived from equation (30), are plotted; a value of C, = 0.61 is again used. These are qualitatively in reasonable agreement with the measurements. We infer from this that it is non-linear effects which account for the relatively large reflection loss at low frequencies. The theoretical curves display a minimum value of A RI at a particular value of log K (or equivalently, v). As Lb increases, the value of u at this minimum increases, and the value of AR1 decreases. This behaviour is consistent with the predictions of equations (32) and (33): if Lb is larger, then 0 is larger. Accordingly, a larger value of u is required
DUCT TERMINATIONS
515
AT HIGH AMPLITUDES
J
-20
-
I
(b) Oh
0
0
I60 dB
150dB
-10 -
I/ -20
0 -2
-I
-2
0 log
-I
0 log
K
K
(c i
(a)
Figure 5. Sound power reflection coefficient for single orifice. (a) -, Theory, k,R, theory, k,R, = 1.0. C, = 0.61; 0, measurement.
= 0.1; (b) -,
theory,
koRD = 0.2; (c) -,
to make AR1 minimum (see equation (33a)). This, coupled with the larger 8 value, implies a smaller value of lrj*, so that A RI becomes larger in a negative sense. The measurements in Figure 5(a) also seem to show a trend to reach a minimum in the region of log K = -0.8 (in keeping with LL = 155 dB). This lends further support to the idea that the peak pressure in the impulsive signal might be respresentative of the equivalent discrete frequency value. Salikuddin and Plumblee [5] have shown pressure/time histories at a location inside the duct, and these indicate the passage of the incident and reflected pulses. Evidently, from these traces, the incident and reflected waves are roughly in phase for small values of K: but in antiphase for large values of K. In the region -1.0
AND
CONCLUSIONS
The object of this paper is to present physical arguments and associated quantitative theoretical data in an attempt to explain certain phenomena observed in the Lockheed measurements. These effects are twofold: first a net acoustic power imbalance across duct terminations, both with mean airflow, at high pressure amplitudes, and without flow
516
A. C‘LJMMINGS
AND
W.
EVEKSMAN
at high amplitudes; and secondly, a considerable reduction in reflection coefficients at high amplitudes, without mean flow. Simple theoretical models have been devised in order to represent the effects of mean flow and high amplitude on acoustic transmission through a duct termination. These have been based on the assumption of simple harmonic time variation, although the measurements were made on the basis of Fourier transformed transient pressure signals. Comparisons between the theoretical and experimental data have been encouragingly good, and this fact is taken to support the physical arguments on which the theory is based. With mean flow, the net power deficit is explained by a conversion of acoustic energy into vertical energy, which is simply convected away by the mean flow and eventually dissipated as heat. The presence of a sharp edge (in this case, the nozzle lip) causes a fluctuating vorticity field in the wake of the nozzle, and the interaction between this and the acoustic velocity field causes energy to be extracted from the sound field and fed to the vorticity field. Howe’s work [2,3] elucidates this effect. The measurements and theory discussed here, relating to the situation where mean flow is present, are generally in keeping with the measurements and theory of Bechert [l] and Howe [2,3] on the same topic. Additionally. it has been shown here that the detailed geometry of the nozzle appears to be somewhat unimportant by comparison with other factors such as open/solid area ratio, frequency and jet flow Mach number. In the absence of mean flow, the acoustic energy loss still occurs by the conversion of acoustic into vertical energy. In the theory described here, it is only implicit that this is the case, and the details of the vorticity field are not considered. It is clear, however, from the work of Ingard and Labate [12] that ring vortices are shed from either side of an orifice plate at the end of a tube in which high amplitude plane waves are transmitted. Salikuddin and Ahuja [4] have shown a schlieren photograph which demonstrates ring vortex production as an acoustic wave is transmitting through a converging nozzle. There is little doubt that the generation of ring vortices is responsible for the acoustic energy losses evident in the results discussed in this article. The measurements and theory presented here, on the net acoustic power loss at duct terminations at high amplitudes and in the absence of mean flow, are in astonishingly good agreement, both quantitatively and qualitatively, considering the theory is based on sinusoidal time varation, the measurements were made with transient signals, and the orifice impedance is non-linear at high amplitudes. Cummings [13] presented data on the transmission of high amplitude tonebursts through orifice plates, and found that a theoretical approach similar to that described here also gave good predictions for a type of signal which must be considered to be a “halfway house” between a short duration impulsive signal and a continuous sinusoidal signal. The experimental and theoretical data on the reflection coefficient at high amplitude and without flow, are in good qualitative agreement. It has been demonstrated that non-linear effects are responsible for the considerable reduction in the reflection coefficient which is observed to occur at certain open/solid area ratios, at low frequencies and at high amplitudes. Although the somewhat limited objectives of this piece of work have, one feels, largely been fulfilled, there is clearly scope for a great deal of additional work in the general area of acoustic absorption in shear layers. The non-linear interaction which may occur between a complex acoustic signal and vorticity fields would merit investigation in a number of contexts in addition to the present case. Also, more specifically, the validity of test methods involving use of transient acoustic signals in such non-linear systems would merit further study.
DUCT TERMINATIONS AT HIGH AMPLITUDES
517
REFERENCES 1. D. W. BECHERT 1979 American Institute of Aeronautics and Astronautics 5th Aeroacoustics Conference Paper No. 79-0575. Sound absorption caused by vorticity shedding demonstrated
with a jet flow. 2. M. S. HOWE 1980 American
Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-0972. The dissipation of sound at sharp edges. 3. M. S. HOWE 1979 Journal of Fluid Mechanics 91, 202-229. Attenuation of sound in a low
Mach number nozzle flow. 4. M. SALIKUDDIN and K. K. AHUJA 1981 American Institute of Aeronautics and Astronautics 7th Aeroacoustics Conference Paper No. 81- 1978. Acoustic power dissipation on radiation through duct terminations-experiments. 5. M. SALIKUDDIN and H. E. PLUMBLEE 1980 American Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-0991. Low frequency sound absorption of orifice plates, perforated plates, and nozzles. 6. M. SALIKUDDIN and K. K. AHUJA 1983 Journal of Sound and Vibration 91, 479-502. Acoustic power dissipation through duct terminations: experiments. 7. H. LEVINE and J. SCHWINGER 1948 Physical Review 73, 383-406. On the radiation of sound from an unflanged circular pipe. 8. U. INGARD 1953 Journal of the Acoustical Society of America 25, 1037-1061. On the theory and design of acoustic resonators. 9. K. K. AHUJA 1981. Private communication. 10. K. U. INGARD 1970 Journal of the Acoustical Society of America 48, 32-33. Nonlinear distortion of sound transmitted through an orifice. Engineering Report 11. A. CUMMINGS and W. EVERSMAN 1980 Lockheed-Georgia LG80ER0154. An investigation of acoustic energy loss in radiation from ducts to the far field at low frequencies, low Mach numbers, and high sound pressure levels. 12. U. INGARD and S. LABATE 1950 Journal of the Acoustical Society of America 22, 211-218. Acoustic circulation effects and the nonlinear impedance of orifices. 13. A. CUMMINGS 1980 American Society of Mechanical Engineers Winter Annual Meeting Paper No. 81-WA/NCA10. High-amplitude acoustic power losses in perforated materials.
APPENDIX: A a C, co
i i$ K k, L; M N P P R RD r rNL
S t U V V V
W
LIST OF SYMBOLS
cross sectional area radius of nozzle exit contraction coefficient adiabatic speed of sound J--T triad of unit vectors open/solid area ratio =&J/co sound pressure level in an incident wave mean flow Mach number number of holes in a perforated plate acoustic pressure amplitude thermodynamic pressure, or pressure perturbation radius of the vena contracta radius of the duct pressure reflection coefficient at the duct exit; or a radial co-ordinate a non-linear resistance a surface just outside the jet vortex layer time acoustic particle velocity time averaged velocity component velocity perturbation convection velocity of vorticity in the jet vortex layer time averaged acoustic power acoustic power absorbed by the jet flow
518 2, AFT A SP’) 61,& 5 6 P (T iJ X w w
A. C‘LJMMINGS
radiation impedance in the nozzle sound power transfer function (in sound power reflection coefficient the Dirac delta function inner, outer mass end corrections non-dimensional impedance real part of [ density nozzle/duct area ratio an integration volume velocity potential, or argument of imaginary part of [ radian frequency vorticity in the jet vortex layer
Subscripts D in the duct F in the far field of an incident wave ; at the jet nozzle exit 0 of orifice r of a reflected wave T denotes transmitted power vc of the vena contracta 0 time averaged value 1 at position 1 2 at position 2
AND
u’.
L.L’~KSMAN
exit plane decibels) (in decibels) at the duct termination
r