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Broadband shock-associated noise of moderately imperfectly expanded supersonic jets
Journal of Sound and Vibration (1990) 140(l), 55-71
BROADBAND
SHOCK-ASSOCIATED
MODERATELY
IMPERFECTLY
SUPERSONIC
NOISE OF
EXPANDED
JETS
C. K. W. TAM Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027,
U.S.A.
(Received 6 April 1989, and in revised form 7 September 1989)
Recently a stochastic model theory of broadband shock-associated noise from supersonic jets was developed in reference [l]. The theory is capable of predicting the near and far field noise spectra of slightly imperfectly expanded jets. In this paper the noise prediction formulas are extended to the moderately imperfectly expanded Mach number range. This is done by first observing that in this wider Mach number range the shock cell structure in the jet plume usually begins with a strong shock followed by a quasi-periodic shock cell structure. The quasi-periodic shock cell structure resembles that of a slightly imperfectly expanded jet. It is found that the shock noise intensity depends only on the amplitude of the quasi-periodic shock cell structure and not on the strong first shock. This allows one to extend the noise prediction formulas of reference [l] by simply modifying the shock cell strength in the original formulas. Here this step is implemented semi-empirically and a set of shock noise prediction formulas are developed. Extensive comparisons between the predicted far field noise spectra and experimental measurements have been carried
out. Favorable agreements are found. 1. INTRODUCTION
In a recent paper [l] a stochastic model theory of broadband shock-associated noise from supersonic jets was developed. The model is based on the large turbulence structures/instability waves shock cell interaction noise generation mechanism proposed earlier by Tam and Tanna [2]. The large turbulence structures/instability waves are the dominant component of the turbulence in the mixing layer of a supersonic jet. As these large turbulence structures/instability waves propagate downstream they inevitably interact with the quasi-periodic shock cells in the plume of the imperfectly expanded jet. According to the theory, a part of the unsteady disturbances generated by this interaction radiates to the far field. This is the observed broadband shock-associated noise of the jet. The model theory is capable of predicting the far and near field acoustic spectra of this noise component. Extensive comparisons with the far field spectral measurements of Norum and Seiner [3] and the near field sound pressure level contour data of Yu [4] at slightly off-design jet operating conditions show favorable overall agreements. These noise prediction formulas, however, are valid only for slightly imperfectly expanded cold jets; namely, jets with stagnation temperature equal to the ambient temperature. The primary objective of the present investigation is to extend the range of applicability of the theory to jets which are moderately imperfectly expanded. ‘The difference between overexpanded and underexpanded jets will also be accounted for. In Figure 1 is shown the noise intensity of a supersonic jet from a Mach 1.5 convergentdivergent nozzle measured by Seiner and Yu [5] at different fully expanded jet Mach number Mj. The microphone was placed in the direction at 150” to the jet axis. Plotted
55 0022460X/90/
130055 + 17 %03.00/O
@ 1990 Academic
Press Limited
C’. K. W. TAM
56
901 0
I
0.5
I I.0
I
I
I.5
2.0
1
2.5
by%)‘”
Figure 1. Overall sound pressure level at 150” to the jet axis. Nozzle design Mach number expanded jet; 0, perfectly expanded jet. Data from reference [S].
= 1.5.0,
Imperfectly
in this figure also (black circles) are the noise intensities of perfectly expanded supersonic jets with the same nozzle exit area. Since there is no shock noise when a jet is perfectly expanded this curve (black circles) represents the minimum noise level (consisting of the turbulent mixing noise) of the jet at a given fully expanded jet Mach number. The difference in noise levels between the open circles and black circles is the contribution of broadband shock noise. The point A with it4j = 1.5 represents the noise level of the jet at perfectly expanded condition. When the jet is underexpanded the noise intensity increases with an increase in the fully expanded jet Mach number following the curve AB. Beyond the point B the noise level first decreases and then levels off. When the jet is overexpanded the noise intensity follows the curve AC as the jet Mach number decreases. The noise level increases even though the jet velocity and Mach number decrease. However, further decrease in the jet Mach number beyond the point C causes the overall noise level to decrease. One obvious fact emerges from this set of data and similar data measured in reference [2] is that the dependence of broadband shock-associated noise on jet Mach number for underexpanded jets is quite different from that for overexpanded jets. The only exception is for jets operating near the perfectly expanded condition. For slightly imperfectly expanded jets the theory of reference [l] predicts that the broadband shockassociated noise intensity, I,, is approximately proportional to (Mf - MS)*, where Md is the nozzle design Mach number. In other words, underexpanded and overexpanded jets with the same Mach number deviation from the nozzle design Mach number would generate nearly equal noise intensities. This is in agreement with experimental measurements (see reference [2]) and is consistent with the scaling formula 1, a p” = (Mj - l)* for supersonic jets from convergent nozzles first established semi-empirically by HarperBourne and Fisher [6]. However, from Figure 1 this type of dependence is valid only over a very limited range of off-design Mach numbers. In this paper the physical process which controls the intensity of broadband shock noise over a wider range of off-design jet Mach number will first be investigated. In section 3 modified noise prediction formulas based on the results of this investigation and the theory of reference [l] will be established for moderately imperfectly expanded jets. The intended range of applicability of these formulas includes Mach numbers slightly less than that of the local maximum point C
SHOCK
NOISE
FROM
IMPERFECT
57
JETS
in Figure 1 to those slightly higher than the absolute maximum point at B. Extensive comparisons between the calculated far field noise spectra and the measurements of Norum and Seiner [3] will be provided. Favorable agreements are found. 2. SHOCK CELL
STRUCTURES
OF IMPERFECTLY
EXPANDED
JETS
The intensity of broadband shock-associated noise depends critically on the strength of the shock cells inside the jet plume. The shock cell strength in turn depends critically on the nozzle design Mach number, M,, and the fully expanded jet Mach number, Mi. For moderately imperfectly expanded jets this relationship is highly non-linear. Here the dependence of the shock cell structure on these parameters will be examined. When a supersonic jet is operating at a slightly off-design condition the shock cell structure is weak. Under this condition the entire shock cell structure can be calculated by a linear multiple-scales model as proposed by Tam, Jackson and Seiner [7]. In this approach the shock cells are decomposed into a superposition of linear wave guide modes of the mean flow of the jet. At the nozzle exit the shear layer is thin so that the linear vortex sheet shock cell model of Prandtl [8] and Pack [9] is applicable. This vortex sheet solution provides a starting condition for the multiple scales expansion. Tam, Jackson and Seiner introduced one important modification to the Prandtl-Pack vortex sheet shock cell solution. They suggested using two slightly different shock cell starting conditions at the nozzle exit, one for underexpanded jets and the other for overexpanded jets, Their reason is that the effective jet diameter of an underexpanded jet is larger than the exit diameter of the nozzle, whereas it is the other way around for an overexpanded jet. They found that the amplitude of the mth waveguide mode of the shock cells is proportional to
(2.1)
for underexpanded
jets and (M&M;)
1+- y-1 2
1
~,nJ,tc+,)
M2
(2.2)
d
for overexpanded jets where D is the nozzle exit diameter and Dj is the fully expanded jet diameter (see reference [2]). J1 is the first order Bessel function and a,,, is the mth zero of the zeroth order Bessel function. On restricting their consideration to slightly imperfectly expanded jets, Tam, Jackson and Seiner [7] demonstrated that their calculated axial pressure distributions of the shock cells agree very favorably with the measurements of Norum and Seiner [3]. Even the measured fine structures in the first three or four shock cells are reproduced by the theory. Equations (2.1) and (2.2) give monotonically increasing amplitude as the operating jet Mach number deviates increasingly from the nozzle design Mach number. This, of course, cannot be correct. To see how the shock cell amplitude changes with jet Mach number one can examine the axial static pressure distribution of imperfectly expanded jets measured by Norum and Seiner [3]. The axial pressure distribution of underexpanded supersonic jets from a Mach 1.5 C-D nozzle is shown in Figure 2. The pressure distribution at Mj = 1.67 is given in Figure 2(a). At this jet operating condition the shocks are weak but regular and quasi-periodic. The pressure distribution can be calculated by the linear
58
C. K. W. TAM
0.6
Figure 2. Axial static pressure distribution of underexpanded supersonic jets from a Mach 1.5 C-D nozzle. Data from reference [3]. (a) M, = 1.67, r/D = 0.45; (b) Mj = 1.80, r/D = 0.45; (c) M, = 1.99, r/D = 0.42; (d) Mj=2.10, r/D=0*45; (e) Mj=2*24, r/D=0.51.
SHOCK
NOISE
FROM
IMPERFECT
JETS
59
multiple-scales expansion method of Tam, Jackson and Seiner [7]. The pressure distribution at Mj = 1.80 is shown in Figure 2(b). This corresponds to the operating point B, the point of maximum broadband shock noise, in Figure 1. The shock cell amplitude is much higher than that of Figure 2(a). As jet Mach number increases further the shock cell structure undergoes a drastic change. This is shown in Figures 2(c)-(e). At these highly off-design conditions the first’shock becomes extremely strong. The subsequent shock cells are, however, regular and quasi-periodic. Dowstream of the first shock cell the entire structure appears to be very similar to that of the slightly off-design case (see Figure 2(a)). If only the quasi-periodic shock cells are considered, Figures 2(c)-(e) indicate that the amplitude now decreases with an increase in jet Mach number. In summary, over the range of moderately off-design jet operating condition the amplitude of the quasi-periodic shock cell structure of an underexpanded supersonic jet first increases with an increase in jet Mach number. The amplitude attains a maximum at the same jet Mach number for which the intensity of shock noise is maximum. Further increase in jet Mach number causes a gradual decrease in the shock cell amplitude. Since it is known that the dominant part of broadband shock-associated noise is generated in the region near the end of the potential core of the jet, the first shock shown in Figures 2(c)-(e), although is exceptionally strong, plays no role in influencing the shock noise intensity. With this in mind it is clear that the shape of the noise intensity curve AB of Figure 2 is determined entirely by the amplitude of the quasi-periodic shock cells in the jet plume. The behavior of the shock cells over the overexpanded range of off-design condition is illustrated in Figures 3(a)-(d). Figure 3(a) is for a slightly overexpanded jet. As can be seen the shocks are weak but quasi-periodic. By decreasing the jet Mach number the shock cell amplitude increases. In Figure 3(b) is shown the pressure distribution associated with the shock cell structure at the jet operating condition corresponding to that of point C, where the shock noise attains a local maximum, in Figure 1. Further decrease in jet Mach number causes the first shock of the shock cell structure to become particularly strong. It is then followed by a series of quasi-periodic shock cells resembling those at a slightly off-design condition. Just as in the case of underexpanded jets the amplitude of the quasi-periodic shock cells first increase then decreases as the operating jet Mach number deviates increasingly from the perfectly expanded condition. Again it is easy to see that the shape of the noise intensity curve AC in Figure 1 over the overexpanded Mach number range follows the behavior of the amplitude of the quasi-periodic shock cell structure just as in the case of underexpanded jets. 3. NOISE PREDICTION FORMULAS AND COMPARISONS WITH EXPERIMENTS In the model broadband shock noise theory of reference [ 1] it is assumed that the large turbulence structures/instability waves and the shock cells are independent entities. They are modelled separately. The theory involves principally an analysis of their interaction as the former propagate through the latter. The resulting acoustic radiation is then determined. In view of the observations and discussion of section 2, it appears that the basic approach and noise prediction formulas of reference [l] can be used for moderately imperfectly expanded jets provided that the appropriate shock cell waveguide mode amplitudes are used. Since the shock cell wavelength and the turbulence (or waves) spectrum are determined by the mean flow of the jet they are not greatly affected by how imperfect the supersonic jet is underexpanded or overexpanded. That is, they are not dependent on the parameter (Mf - Mi). This implies that the shape of the radiated noise spectrum is also relatively independent of how imperfect the jet is expanded. The shock cell strength which is an indication of the degree of imperfect expansion of the jet,
60
C.
K. W. TAM
14
,9” Q 1.4 I .2
I.o
0.6
L_JII_, 0
5
181I *
IO
15
x/D
Figure 3. Axial static pressure distribution of overexpanded supersonic jets from a Mach 1.5 C-D nozzle. Data from reference [3]. (a) M, = 1.37, r/D=0.3; (b) M, = 1.28, r/D=0.25; (c) M, = 1.22, r/D=0.23; (d) Mj=1.17, r/D=0.25.
therefore, affects primarily the noise level alone. In order to incorporate the needed changes into the theory of reference [l] the first step is to identify the parameter group which best characterizes the shock cell amplitude. This parameter must work for all nozzles regardless of their design Mach number. The analysis of reference [2] suggests
to be such a parameter. The next step is to generalize expressions (2.1) and (2.2) so that the amplitude formula would be valid for moderately imperfectly expanded jets. This involves replacing the linear dependence on
SHOCK
NOISE
FROM
IMPERFECT
61
JETS
in these expressions by suitable functions of this parameter. In this work this is carried out semi-empirically. It has been found that excellent results could be obtained by replacing this parameter in (2.1) and (2.2) by the following functions: /2
(3.1)
for underexpanded
jets, and (M:-MMf)
y-1 If2
~
MZ * >(
/2
(MS-M;) I+Y-l - 2
1+6
(3.2)
M’d
for overexpanded jets. The forms of these functions provide a Mach number dependence similar to that of the shock cell strength data given in Figures 2 and 3. The numerical constants of these expressions are, however, chosen to provide the best fit to the acoustic data. In developing the similarity source model (see equation (3.14) of reference [I]) it was assumed that the instability wave spectrum has the form A nm ei(“,,-ym)/fO:(Uj/a,)w’/2
e -(lnZ)(wx/u,-X,,,~‘/L’+i(k,-k”,)l;
(3.3)
It was argued that the multiplicative factor w”* should be included in this expression so that the x-integrated total source strength is bounded as w + 0. Subsequent analysis indicates, however, that this factor is not necessary at high frequency. As a matter of fact the inclusion of this factor results in a formula for the noise intensity which is divergent at high frequency; behaving like log (f) as f+ a. Here it is proposed to restrict the use of this factor empirically to Strouhal numbers, fDj/ Uj, below 0.35. For Strouhal numbers higher than 0.35 it will be taken as a constant equal to O-35. Upon incorporating the above modifications to the similarity source model of reference [ 1] it is straightforward to find that the broadband shock noise power spectrum, S, at a point (R, (Ir = inlet angle = 7~- 8; (R, 0, 4) are the spherical co-ordinates) from a cold supersonic jet is given by
(3.4) where the factor (foi/ Uj) in the denominator is to be replaced by S,, = 0.35 whenever (Dj/Uj) is less than S,,. J, is the first order Bessel function and 1?=2*65x 10m4S0. The quantity A2 which characterizes the shock cell strength is defined by l+~!~~;)2(~)2]/[l+3(
1+~~$~5)3],
for underexpanded
jets
AZ_=
for overexpanded
jets (3.5)
All the parameters except fm of equation (3.4) are the same as those of the far field noise spectrum formula (3.17) of reference [l]. Extensive computations and comparisons with experiments reveal that for best results fm should be taken to be 1.1 times the value defined in formula (3.19) of reference [l]. This slight change will be regarded as due to
62
C. K. W. TAM
an increase in k,, the wavenumber of the shock cell waveguide modes, and will be adopted throughout this paper. Equation (3.4) replaces the far field noise power spectrum formula (3.17) of reference [l]. On proceeding as above it is easy to find that the near field broadband shock noise power spectrum for moderately imperfectly expanded (cold) jets at a point (r, x, 4) in cylindrical coordinates is given by
S(r, x, f) =
CL2AjA2p&akujz 4.fC..Djl
-(q-k
’In’ II -k
uj)
)*(~*L*/f*~(16x~ln2~-~
2
x
HS)(i(
772 _
02/a2
‘x
)I/zr)
ei(X-U,xq,/(2d))?
d rl
(3.6)
,
where the factor (foj/ Uj) in the denominator is again to be replaced by S, = 0.35 whenever it is less than S,,. Equation (3.6) supercedes formula (3.20) of reference [l]. Formulas (3.4) and (3.6) are for jets with stagnation temperature equal to the ambient temperature (cold jets). For hot jets the jet density is less than that of the ambient gas. In these cases the amplitude of the large turbulence structures and hence the strength of the shock noise source is reduced. To take this factor into consideration a resealing of the density parameter in these formulas appear to be necessary. It has been found that most of the high temperature effects on the shock noise intensity can be adequately accounted for by multiplying the right sides of equations (3.4) and (3.6) by the factor -1
(Pjlpco)
(l+y”T )
9
where pj is the gas density of the jet. This factor is equal to unity for cold jets. For completeness, the far field broadband shock noise spectrum formula can now be written out in full, including all the above proposed extensions. This is done by multiplying formula (3.4) by the hot jet correction factor and making use of the equality pjaT = p,a& (aj and am are the jet and ambient speeds of sound). This equality is simply a statement of equal static pressure inside and outside the jet. The formula is
S (Rh_ff)=
R-02
EL’AjA’p&az
M,’
Rzf(fojluj)(l+t(r-1)Mi2)
x[i, cT:Jj(c7 m e
-~.~,,//-1)~(l+M~COSg~~L~U~/(Uj21”2)
)
(3.7)
, I
where the factor (pj/uj) in the denominator is to be replaced by L&=0*35 whenever (fD,/uj) is less than S,. The parameters of equation (3.7) can be calculated as follows (see reference [ 11): E = 2.65 x 10-4S0 (S, = 0.35);
)I
(Y+1)/4(Y--1)
y-1 I+TM:
(Mj/Mc,)“*;
D is the nozzle exit diameter, Dj is the effective jet diameter; x,(core length of jet)/Dj,=
1
4.3+1 4. 3 +
.2M;+l.2(1-Tj/T,), 1 . 2Mf J,
T,/T,
1I
where Tj and T, are the jet and ambient temperature respectively; L (half width of similarity noise source) = 3 - 3[x,( Mj)/xC( 1.67) J; Aj (effective area of jet) = $rDj; Jo and J1 are the Bessel functions of order 0 and l;a, are the zeros of JO(a,,,), M = 1,2,3,. . . ;
SHOCK
NOISE
FROM
IMPERFECT
63
JETS
A’ (shock cell strength function) is given by equation (3.5); U,/Uj (convection velocity of large turbulence structures to jet velocity) = 0.7 -O-025( T,/ T, - 1); T, is the stagnation temperature of the jet; one omits the last term if Tr/ T, < 1; A4, = ~,/a,; k,~,,~exshect~ (wavenumber of vortex sheet shock cell structure) = 2Um/(Djw), m = 1,2,3,. . . ; k, = (1.596-0.1773M,)l.lk,,,,,,,,,,,,,;
b=
~~~~%zw,~~~Iw;
S, = l~lkm~vo~exs~eet~,m =
3,4,5,. . . ; fm = u,lc,J21r( 1 + A4, cos +), m = 1,2,3,. . . , N. For frequencies with Strouhal number (~j/Uj) greater than three the wave model of large turbulence structures is no longer meaningful. It is, therefore, suggested that the sum in formula (3.7) be limited to N terms, where N is the largest integer for which fNL?/ uj s 3. To test the accuracy of formula (3.4) or (3.7) comparisons between calculated noise power spectra and the measurements of Norum and Seiner [3] have been made over the moderately imperfectly expanded Mach number range. Figures 4(a)-(c) show comparisons between calculated noise spectra and data at inlet angle $ = 30” to 120” at 15” intervals for underexpanded cold supersonic jets issued from a convergent nozzle, l& = 1.O. Figure 4(a) is for Mj = 1.22. This Mach number being close to Md = 1.0 is in the slightly imperfectly expanded Mach number range. Therefore, as expected the agreement is good for nearly all angles and frequencies. The comparisons at Mj = I.67 are shown in Figure
lI..~.I..:.I..~~I~.~.1.~~~I.~~~I 0
5
IO
15
Frequency
20
25
30
(kHd
Measured (Norum and Seiner [3]); -, Figure4. Far field noise spectra. Md = 1 .O. -, (3.4)). Nozzle exit diameter=3.982 cm. (a) M, = 1.22; (b) M, = 1.67; (c) Mj = 1.80.
calculated
(equation
64
C.
K.
W.
TAM
(b)
0
5
IO
15
Frequency
Figure
20
25
(kHz)
4-continued.
4(b). At this Mach number the broadband shock noise attains nearly its peak intensity. This operating condition corresponds to the maximum shock noise condition of point B in Figure 1. Figure 4(c) is at a still higher Mach number of Mj = 1.80, and the broadband shock noise intensity begins to decrease relative to that of Figure 4(b). In both Figures 4(b) and 4(c) the agreement between calculated and measured noise spectra is uniformly good over almost all measured inlet angle directions and frequencies. The exceptions are at low inlet angles. It was noted in reference [l] that a dip exists in the calculated noise spectrum near the peak noise frequency at these directions. This discrepancy is carried over to the present calculation since only shock cell strength is corrected in this work. In Figures 5(a), (b) and 6 are shown further comparisons between the calculated results of equation (3.4) or (3.7) and the measurements of Norum and Seiner [3] for C-D nozzles of Mach number 1.5 and 2.0 operating in the underexpanded mode. The noise power spectrum at lWj= 1.80 and Md = 1.50 is given in Figure 5(a). This is at point B of Figure 1. As can be seen there is favorable agreement (except for the dips in the calculated spectra as noted above) between computed spectra and data in every direction. In Figure 5(b) the jet Mach number is equal to 1.99. This operating Mach number lies beyond the maximum shock noise condition. Again good agreement is evident. The noise spectra of an underexpanded jet at Mj = 2.33 from a C-D nozzle of Mach number 2.0 is shown in Figure 6. Except at the lowest inlet angle the calculated spectra match those of the
SHOCK
NOISE
FROM
IMPERFECT
$
JETS
65
max dB (measured)
Frequency (kHz)
Figure rlcontinued.
measured spectra reasonably well. In summary, comparisons between the predicted noise spectra of equation (3.4) or (3.7) for underexpanded jets from three C-D nozzles of design Mach number 1.0, l-5 and 2-O with the measurements of Norum and Seiner [3] have been carried out over the moderately imperfectly expanded Mach number range. Good agreements are found at nearly every microphone direction over a wide frequency band. Similar comparisons between calculated results and measurements for overexpanded jets are shown in Figures 7 and 8. Figures 7(a) and 7(b) are for jets from Mach 2.0 C-D nozzles, while Figure 8 is for a Mach l-5 C-D nozzle. As can be seen the agreements are as good as for underexpanded jets. The jet operating condition of Figures 7(a) and 8 corresponds to the maximum shock noise point C of Figure 1. In Figure 7(b) at Mj = 1.49 are shown the noise spectra at a further overexpanded condition. Again, good overall agreements are found between the calculated and the measured spectra at all microphone directions both in spectral level and in peak noise frequency. These good agreements should provide confidence in the use of the above noise prediction formulas.
4.
SUMMARY
AND DISCUSSION
In this paper the near and far field broadband shock-associated noise prediction formulas of reference [l] which were developed for slightly imperfectly expanded jets
66
C. K. W. TAM
(EP) P”Oq ZH Ob “’ 7d.9
SHOCK
NOISE
IO
FROM
IMPERFECT
15 Frequency
67
JETS
20
25
30
(kHz)
Figure 6. Far field noise spectra. Md = 2.0, M, = 2.33. -, calculated (equation (3.4)). Nozzle exit diameter = 4.989 cm.
Measured
(Norum
and
Seiner
[3]);
--,
are extended to moderately imperfectly expanded jets. In carrying out the extension it was first observed that the shock system in the plume of a moderately imperfectly expanded jet generally consists of a strong first shock followed by a quasi-periodic shock cell structure. The quasi-periodic shock structure resembles that of a slightly imperfectly expanded jet. The intensity of broadband shock noise, on the other hand, appears to be dictated primarily by the amplitude of the quasi-periodic shock cells. These observations allow one to broaden the range of applicability of the noise prediction formulas of reference [l] by modifying the shock cell amplitudes in these formulas appropriately. This is done by first recognizing that there is an intrinsic difference between overexpanded and underexpanded supersonic jets. For these two classes of jets a suitable shock cell amplitude parameter is found. By means of this parameter a set of shock cell amplitude formulas has been determined semi-empirically. Extensive comparisons between the extended theory and measured broadband shock noise spectra have been carried out for jets from C-D nozzles with design Mach number 2.0, 1.5 and 1.0. Good agreements are found over the entire moderately imperfectly expanded Mach number range. It is well known that imperfectly expanded supersonic jets invariably emit discrete frequency sound called screech tones. The screech tones are generated by a self-excited feedback cycle [ 10, 111. Experimental observations [ 121 indicate that the feedback cycle
15 (kliz)
20
Md = 2-O. -,
Frequency
Figure 7. Far field noise spectra.
IO
$I
Measured
25
I
(Norum
30
max dB (measured)(
(a)
and Seiner [3]); -,
0
calculated
5
15
20
25
(3.4)). (a) A4, = 1.80; (b) M, = 1.49.
Frequency (kHz)
(equation
IO
SHOCK
NOISE
FROM
IMPERFECT
69
JETS
max dB (measured)
Frequency (kHz)
Figure 8. Far field noise calculated (equation (3.4)).
spectra.
Md = 1.5, h4, = 1.28. -,
Measured
(Norum
and
Seiner
[3]);
---,
has the tendency of suppressing the broadband shock noise component. A careful examination of the axial static pressure distribution associated with the shock cell structure of imperfectly expanded jets measured by Norum and Seiner [3] suggests that when strong screech tones are present the feedback loop causes a rapid disintegration of the periodic shock cell structure. It is believed, therefore, that the reduction in broadband shock noise is the direct result of a decrease in shock cell strength. A comparison of the calculated and measured noise spectra at Md = 2-O and Mj = l-67 is shown in Figure 9. At this jet operating condition strong screech tones exist, as can be readily seen in the measured noise spectra. The agreement between the calculated and the measured spectra is not as good as before. The measured spectra are generally lower in level consistent with the observations of reference [12]. Thus the prediction formulas developed in this paper would only provide an upper bound estimate of broadband shock noise when strong screech tones are present. ACKNOWLEDGMENT
This work was supported
by NASA Langley Research Center Grant NAG 1-421.
70
C. K. W. TAM
J/
IO
20
15 Freqwncy
max dB (measured)
25
30
(kHz)
Figure 9. Far field noise spectra with intense screech tones. Mj = 1.67, Md = 2.0. -, calculated (equation (3.4)). and Seiner [3]); -,
Measured (Norum
REFERENCES 1. C. K. W. TAM 1987 .ZoumaZ of Sound and Vibration 116, 265-302. Stochastic model theory of broadband shock associated noise from supersonic jets. 2. C. K. W. TAM and H. K. TANNA 1982Journal of Sound and Vibration 81, 337-358. Shock associated noise of supersonic jets from convergent-divergent nozzles. 3. T. D. NORUM and J. M. SEINER 1982 NASA TM 84521. Measurements of mean static pressure and far field acoustics of shock-containing supersonic jets. 4. J. C. Yu 1971 Ph.D. thesis, Syracuse University. Investigation of the noise fields of supersonic axisymmetric jet Rows. 5. J. M. SEINER and J. C. Yu 1984 American Znstitute of Aeronautics and Astronautics Journal 22, 1207-1215. Acoustic near field properties associated with broadband shock noise. 6. M. HARPER-B• URNE and M. J.FISHER 1973 Proceedings (No. 131) ofthe AGARD Conference on Noise Mechanisms, Brussels, Belgium. The noise from shock waves in supersonic jets. 7. C. K. W. TAM, J.A.JACKSON and J.M. SEINER 1985Journal of Fluid Mechanics 153,123-149. A multiple-scales model of the shock cell structure of imperfectly expanded supersonic jets. 8. L.PRANDTL 1904PhysikZeitschrif $599-601. Uber die Stationiiren Wellen in einem Gasstrahl.
SHOCK NOISE FROM IMPERFECT JETS
71
9. D. C. PACK 1950 Quarterly Journal of Mechanics and Applied Mathematics 3, 173-181. A note on Prandtl’s formula for the wavelength of a supersonic gas jet. 10. A. POWELL 1953 Proceedings of the Physical Society 66, 1039-1056. On the mechanism of choked jet noise. 11. C. K. W. TAM, J. M. SEINER and J. C. Yu 1986 Journal of Sound and Vibration 110,309-321. Proposed relationship between broadband shock associated noise and screech tones. 12. R. T. NAGEL, J. W. DENHAM and A. G. PAPATHANASIOU 1983 American Institute of Aeronautics and Astronautics Journal 21, 1541-1545. Supersonic jet screech tone cancellation.
BROADBAND
SHOCK-ASSOCIATED
MODERATELY
IMPERFECTLY
SUPERSONIC
NOISE OF
EXPANDED
JETS
C. K. W. TAM Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027,
U.S.A.
(Received 6 April 1989, and in revised form 7 September 1989)
Recently a stochastic model theory of broadband shock-associated noise from supersonic jets was developed in reference [l]. The theory is capable of predicting the near and far field noise spectra of slightly imperfectly expanded jets. In this paper the noise prediction formulas are extended to the moderately imperfectly expanded Mach number range. This is done by first observing that in this wider Mach number range the shock cell structure in the jet plume usually begins with a strong shock followed by a quasi-periodic shock cell structure. The quasi-periodic shock cell structure resembles that of a slightly imperfectly expanded jet. It is found that the shock noise intensity depends only on the amplitude of the quasi-periodic shock cell structure and not on the strong first shock. This allows one to extend the noise prediction formulas of reference [l] by simply modifying the shock cell strength in the original formulas. Here this step is implemented semi-empirically and a set of shock noise prediction formulas are developed. Extensive comparisons between the predicted far field noise spectra and experimental measurements have been carried
out. Favorable agreements are found. 1. INTRODUCTION
In a recent paper [l] a stochastic model theory of broadband shock-associated noise from supersonic jets was developed. The model is based on the large turbulence structures/instability waves shock cell interaction noise generation mechanism proposed earlier by Tam and Tanna [2]. The large turbulence structures/instability waves are the dominant component of the turbulence in the mixing layer of a supersonic jet. As these large turbulence structures/instability waves propagate downstream they inevitably interact with the quasi-periodic shock cells in the plume of the imperfectly expanded jet. According to the theory, a part of the unsteady disturbances generated by this interaction radiates to the far field. This is the observed broadband shock-associated noise of the jet. The model theory is capable of predicting the far and near field acoustic spectra of this noise component. Extensive comparisons with the far field spectral measurements of Norum and Seiner [3] and the near field sound pressure level contour data of Yu [4] at slightly off-design jet operating conditions show favorable overall agreements. These noise prediction formulas, however, are valid only for slightly imperfectly expanded cold jets; namely, jets with stagnation temperature equal to the ambient temperature. The primary objective of the present investigation is to extend the range of applicability of the theory to jets which are moderately imperfectly expanded. ‘The difference between overexpanded and underexpanded jets will also be accounted for. In Figure 1 is shown the noise intensity of a supersonic jet from a Mach 1.5 convergentdivergent nozzle measured by Seiner and Yu [5] at different fully expanded jet Mach number Mj. The microphone was placed in the direction at 150” to the jet axis. Plotted
55 0022460X/90/
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@ 1990 Academic
Press Limited
C’. K. W. TAM
56
901 0
I
0.5
I I.0
I
I
I.5
2.0
1
2.5
by%)‘”
Figure 1. Overall sound pressure level at 150” to the jet axis. Nozzle design Mach number expanded jet; 0, perfectly expanded jet. Data from reference [S].
= 1.5.0,
Imperfectly
in this figure also (black circles) are the noise intensities of perfectly expanded supersonic jets with the same nozzle exit area. Since there is no shock noise when a jet is perfectly expanded this curve (black circles) represents the minimum noise level (consisting of the turbulent mixing noise) of the jet at a given fully expanded jet Mach number. The difference in noise levels between the open circles and black circles is the contribution of broadband shock noise. The point A with it4j = 1.5 represents the noise level of the jet at perfectly expanded condition. When the jet is underexpanded the noise intensity increases with an increase in the fully expanded jet Mach number following the curve AB. Beyond the point B the noise level first decreases and then levels off. When the jet is overexpanded the noise intensity follows the curve AC as the jet Mach number decreases. The noise level increases even though the jet velocity and Mach number decrease. However, further decrease in the jet Mach number beyond the point C causes the overall noise level to decrease. One obvious fact emerges from this set of data and similar data measured in reference [2] is that the dependence of broadband shock-associated noise on jet Mach number for underexpanded jets is quite different from that for overexpanded jets. The only exception is for jets operating near the perfectly expanded condition. For slightly imperfectly expanded jets the theory of reference [l] predicts that the broadband shockassociated noise intensity, I,, is approximately proportional to (Mf - MS)*, where Md is the nozzle design Mach number. In other words, underexpanded and overexpanded jets with the same Mach number deviation from the nozzle design Mach number would generate nearly equal noise intensities. This is in agreement with experimental measurements (see reference [2]) and is consistent with the scaling formula 1, a p” = (Mj - l)* for supersonic jets from convergent nozzles first established semi-empirically by HarperBourne and Fisher [6]. However, from Figure 1 this type of dependence is valid only over a very limited range of off-design Mach numbers. In this paper the physical process which controls the intensity of broadband shock noise over a wider range of off-design jet Mach number will first be investigated. In section 3 modified noise prediction formulas based on the results of this investigation and the theory of reference [l] will be established for moderately imperfectly expanded jets. The intended range of applicability of these formulas includes Mach numbers slightly less than that of the local maximum point C
SHOCK
NOISE
FROM
IMPERFECT
57
JETS
in Figure 1 to those slightly higher than the absolute maximum point at B. Extensive comparisons between the calculated far field noise spectra and the measurements of Norum and Seiner [3] will be provided. Favorable agreements are found. 2. SHOCK CELL
STRUCTURES
OF IMPERFECTLY
EXPANDED
JETS
The intensity of broadband shock-associated noise depends critically on the strength of the shock cells inside the jet plume. The shock cell strength in turn depends critically on the nozzle design Mach number, M,, and the fully expanded jet Mach number, Mi. For moderately imperfectly expanded jets this relationship is highly non-linear. Here the dependence of the shock cell structure on these parameters will be examined. When a supersonic jet is operating at a slightly off-design condition the shock cell structure is weak. Under this condition the entire shock cell structure can be calculated by a linear multiple-scales model as proposed by Tam, Jackson and Seiner [7]. In this approach the shock cells are decomposed into a superposition of linear wave guide modes of the mean flow of the jet. At the nozzle exit the shear layer is thin so that the linear vortex sheet shock cell model of Prandtl [8] and Pack [9] is applicable. This vortex sheet solution provides a starting condition for the multiple scales expansion. Tam, Jackson and Seiner introduced one important modification to the Prandtl-Pack vortex sheet shock cell solution. They suggested using two slightly different shock cell starting conditions at the nozzle exit, one for underexpanded jets and the other for overexpanded jets, Their reason is that the effective jet diameter of an underexpanded jet is larger than the exit diameter of the nozzle, whereas it is the other way around for an overexpanded jet. They found that the amplitude of the mth waveguide mode of the shock cells is proportional to
(2.1)
for underexpanded
jets and (M&M;)
1+- y-1 2
1
~,nJ,tc+,)
M2
(2.2)
d
for overexpanded jets where D is the nozzle exit diameter and Dj is the fully expanded jet diameter (see reference [2]). J1 is the first order Bessel function and a,,, is the mth zero of the zeroth order Bessel function. On restricting their consideration to slightly imperfectly expanded jets, Tam, Jackson and Seiner [7] demonstrated that their calculated axial pressure distributions of the shock cells agree very favorably with the measurements of Norum and Seiner [3]. Even the measured fine structures in the first three or four shock cells are reproduced by the theory. Equations (2.1) and (2.2) give monotonically increasing amplitude as the operating jet Mach number deviates increasingly from the nozzle design Mach number. This, of course, cannot be correct. To see how the shock cell amplitude changes with jet Mach number one can examine the axial static pressure distribution of imperfectly expanded jets measured by Norum and Seiner [3]. The axial pressure distribution of underexpanded supersonic jets from a Mach 1.5 C-D nozzle is shown in Figure 2. The pressure distribution at Mj = 1.67 is given in Figure 2(a). At this jet operating condition the shocks are weak but regular and quasi-periodic. The pressure distribution can be calculated by the linear
58
C. K. W. TAM
0.6
Figure 2. Axial static pressure distribution of underexpanded supersonic jets from a Mach 1.5 C-D nozzle. Data from reference [3]. (a) M, = 1.67, r/D = 0.45; (b) Mj = 1.80, r/D = 0.45; (c) M, = 1.99, r/D = 0.42; (d) Mj=2.10, r/D=0*45; (e) Mj=2*24, r/D=0.51.
SHOCK
NOISE
FROM
IMPERFECT
JETS
59
multiple-scales expansion method of Tam, Jackson and Seiner [7]. The pressure distribution at Mj = 1.80 is shown in Figure 2(b). This corresponds to the operating point B, the point of maximum broadband shock noise, in Figure 1. The shock cell amplitude is much higher than that of Figure 2(a). As jet Mach number increases further the shock cell structure undergoes a drastic change. This is shown in Figures 2(c)-(e). At these highly off-design conditions the first’shock becomes extremely strong. The subsequent shock cells are, however, regular and quasi-periodic. Dowstream of the first shock cell the entire structure appears to be very similar to that of the slightly off-design case (see Figure 2(a)). If only the quasi-periodic shock cells are considered, Figures 2(c)-(e) indicate that the amplitude now decreases with an increase in jet Mach number. In summary, over the range of moderately off-design jet operating condition the amplitude of the quasi-periodic shock cell structure of an underexpanded supersonic jet first increases with an increase in jet Mach number. The amplitude attains a maximum at the same jet Mach number for which the intensity of shock noise is maximum. Further increase in jet Mach number causes a gradual decrease in the shock cell amplitude. Since it is known that the dominant part of broadband shock-associated noise is generated in the region near the end of the potential core of the jet, the first shock shown in Figures 2(c)-(e), although is exceptionally strong, plays no role in influencing the shock noise intensity. With this in mind it is clear that the shape of the noise intensity curve AB of Figure 2 is determined entirely by the amplitude of the quasi-periodic shock cells in the jet plume. The behavior of the shock cells over the overexpanded range of off-design condition is illustrated in Figures 3(a)-(d). Figure 3(a) is for a slightly overexpanded jet. As can be seen the shocks are weak but quasi-periodic. By decreasing the jet Mach number the shock cell amplitude increases. In Figure 3(b) is shown the pressure distribution associated with the shock cell structure at the jet operating condition corresponding to that of point C, where the shock noise attains a local maximum, in Figure 1. Further decrease in jet Mach number causes the first shock of the shock cell structure to become particularly strong. It is then followed by a series of quasi-periodic shock cells resembling those at a slightly off-design condition. Just as in the case of underexpanded jets the amplitude of the quasi-periodic shock cells first increase then decreases as the operating jet Mach number deviates increasingly from the perfectly expanded condition. Again it is easy to see that the shape of the noise intensity curve AC in Figure 1 over the overexpanded Mach number range follows the behavior of the amplitude of the quasi-periodic shock cell structure just as in the case of underexpanded jets. 3. NOISE PREDICTION FORMULAS AND COMPARISONS WITH EXPERIMENTS In the model broadband shock noise theory of reference [ 1] it is assumed that the large turbulence structures/instability waves and the shock cells are independent entities. They are modelled separately. The theory involves principally an analysis of their interaction as the former propagate through the latter. The resulting acoustic radiation is then determined. In view of the observations and discussion of section 2, it appears that the basic approach and noise prediction formulas of reference [l] can be used for moderately imperfectly expanded jets provided that the appropriate shock cell waveguide mode amplitudes are used. Since the shock cell wavelength and the turbulence (or waves) spectrum are determined by the mean flow of the jet they are not greatly affected by how imperfect the supersonic jet is underexpanded or overexpanded. That is, they are not dependent on the parameter (Mf - Mi). This implies that the shape of the radiated noise spectrum is also relatively independent of how imperfect the jet is expanded. The shock cell strength which is an indication of the degree of imperfect expansion of the jet,
60
C.
K. W. TAM
14
,9” Q 1.4 I .2
I.o
0.6
L_JII_, 0
5
181I *
IO
15
x/D
Figure 3. Axial static pressure distribution of overexpanded supersonic jets from a Mach 1.5 C-D nozzle. Data from reference [3]. (a) M, = 1.37, r/D=0.3; (b) M, = 1.28, r/D=0.25; (c) M, = 1.22, r/D=0.23; (d) Mj=1.17, r/D=0.25.
therefore, affects primarily the noise level alone. In order to incorporate the needed changes into the theory of reference [l] the first step is to identify the parameter group which best characterizes the shock cell amplitude. This parameter must work for all nozzles regardless of their design Mach number. The analysis of reference [2] suggests
to be such a parameter. The next step is to generalize expressions (2.1) and (2.2) so that the amplitude formula would be valid for moderately imperfectly expanded jets. This involves replacing the linear dependence on
SHOCK
NOISE
FROM
IMPERFECT
61
JETS
in these expressions by suitable functions of this parameter. In this work this is carried out semi-empirically. It has been found that excellent results could be obtained by replacing this parameter in (2.1) and (2.2) by the following functions: /2
(3.1)
for underexpanded
jets, and (M:-MMf)
y-1 If2
~
MZ * >(
/2
(MS-M;) I+Y-l - 2
1+6
(3.2)
M’d
for overexpanded jets. The forms of these functions provide a Mach number dependence similar to that of the shock cell strength data given in Figures 2 and 3. The numerical constants of these expressions are, however, chosen to provide the best fit to the acoustic data. In developing the similarity source model (see equation (3.14) of reference [I]) it was assumed that the instability wave spectrum has the form A nm ei(“,,-ym)/fO:(Uj/a,)w’/2
e -(lnZ)(wx/u,-X,,,~‘/L’+i(k,-k”,)l;
(3.3)
It was argued that the multiplicative factor w”* should be included in this expression so that the x-integrated total source strength is bounded as w + 0. Subsequent analysis indicates, however, that this factor is not necessary at high frequency. As a matter of fact the inclusion of this factor results in a formula for the noise intensity which is divergent at high frequency; behaving like log (f) as f+ a. Here it is proposed to restrict the use of this factor empirically to Strouhal numbers, fDj/ Uj, below 0.35. For Strouhal numbers higher than 0.35 it will be taken as a constant equal to O-35. Upon incorporating the above modifications to the similarity source model of reference [ 1] it is straightforward to find that the broadband shock noise power spectrum, S, at a point (R, (Ir = inlet angle = 7~- 8; (R, 0, 4) are the spherical co-ordinates) from a cold supersonic jet is given by
(3.4) where the factor (foi/ Uj) in the denominator is to be replaced by S,, = 0.35 whenever (Dj/Uj) is less than S,,. J, is the first order Bessel function and 1?=2*65x 10m4S0. The quantity A2 which characterizes the shock cell strength is defined by l+~!~~;)2(~)2]/[l+3(
1+~~$~5)3],
for underexpanded
jets
AZ_=
for overexpanded
jets (3.5)
All the parameters except fm of equation (3.4) are the same as those of the far field noise spectrum formula (3.17) of reference [l]. Extensive computations and comparisons with experiments reveal that for best results fm should be taken to be 1.1 times the value defined in formula (3.19) of reference [l]. This slight change will be regarded as due to
62
C. K. W. TAM
an increase in k,, the wavenumber of the shock cell waveguide modes, and will be adopted throughout this paper. Equation (3.4) replaces the far field noise power spectrum formula (3.17) of reference [l]. On proceeding as above it is easy to find that the near field broadband shock noise power spectrum for moderately imperfectly expanded (cold) jets at a point (r, x, 4) in cylindrical coordinates is given by
S(r, x, f) =
CL2AjA2p&akujz 4.fC..Djl
-(q-k
’In’ II -k
uj)
)*(~*L*/f*~(16x~ln2~-~
2
x
HS)(i(
772 _
02/a2
‘x
)I/zr)
ei(X-U,xq,/(2d))?
d rl
(3.6)
,
where the factor (foj/ Uj) in the denominator is again to be replaced by S, = 0.35 whenever it is less than S,,. Equation (3.6) supercedes formula (3.20) of reference [l]. Formulas (3.4) and (3.6) are for jets with stagnation temperature equal to the ambient temperature (cold jets). For hot jets the jet density is less than that of the ambient gas. In these cases the amplitude of the large turbulence structures and hence the strength of the shock noise source is reduced. To take this factor into consideration a resealing of the density parameter in these formulas appear to be necessary. It has been found that most of the high temperature effects on the shock noise intensity can be adequately accounted for by multiplying the right sides of equations (3.4) and (3.6) by the factor -1
(Pjlpco)
(l+y”T )
9
where pj is the gas density of the jet. This factor is equal to unity for cold jets. For completeness, the far field broadband shock noise spectrum formula can now be written out in full, including all the above proposed extensions. This is done by multiplying formula (3.4) by the hot jet correction factor and making use of the equality pjaT = p,a& (aj and am are the jet and ambient speeds of sound). This equality is simply a statement of equal static pressure inside and outside the jet. The formula is
S (Rh_ff)=
R-02
EL’AjA’p&az
M,’
Rzf(fojluj)(l+t(r-1)Mi2)
x[i, cT:Jj(c7 m e
-~.~,,//-1)~(l+M~COSg~~L~U~/(Uj21”2)
)
(3.7)
, I
where the factor (pj/uj) in the denominator is to be replaced by L&=0*35 whenever (fD,/uj) is less than S,. The parameters of equation (3.7) can be calculated as follows (see reference [ 11): E = 2.65 x 10-4S0 (S, = 0.35);
)I
(Y+1)/4(Y--1)
y-1 I+TM:
(Mj/Mc,)“*;
D is the nozzle exit diameter, Dj is the effective jet diameter; x,(core length of jet)/Dj,=
1
4.3+1 4. 3 +
.2M;+l.2(1-Tj/T,), 1 . 2Mf J,
T,/T,
1I
where Tj and T, are the jet and ambient temperature respectively; L (half width of similarity noise source) = 3 - 3[x,( Mj)/xC( 1.67) J; Aj (effective area of jet) = $rDj; Jo and J1 are the Bessel functions of order 0 and l;a, are the zeros of JO(a,,,), M = 1,2,3,. . . ;
SHOCK
NOISE
FROM
IMPERFECT
63
JETS
A’ (shock cell strength function) is given by equation (3.5); U,/Uj (convection velocity of large turbulence structures to jet velocity) = 0.7 -O-025( T,/ T, - 1); T, is the stagnation temperature of the jet; one omits the last term if Tr/ T, < 1; A4, = ~,/a,; k,~,,~exshect~ (wavenumber of vortex sheet shock cell structure) = 2Um/(Djw), m = 1,2,3,. . . ; k, = (1.596-0.1773M,)l.lk,,,,,,,,,,,,,;
b=
~~~~%zw,~~~Iw;
S, = l~lkm~vo~exs~eet~,m =
3,4,5,. . . ; fm = u,lc,J21r( 1 + A4, cos +), m = 1,2,3,. . . , N. For frequencies with Strouhal number (~j/Uj) greater than three the wave model of large turbulence structures is no longer meaningful. It is, therefore, suggested that the sum in formula (3.7) be limited to N terms, where N is the largest integer for which fNL?/ uj s 3. To test the accuracy of formula (3.4) or (3.7) comparisons between calculated noise power spectra and the measurements of Norum and Seiner [3] have been made over the moderately imperfectly expanded Mach number range. Figures 4(a)-(c) show comparisons between calculated noise spectra and data at inlet angle $ = 30” to 120” at 15” intervals for underexpanded cold supersonic jets issued from a convergent nozzle, l& = 1.O. Figure 4(a) is for Mj = 1.22. This Mach number being close to Md = 1.0 is in the slightly imperfectly expanded Mach number range. Therefore, as expected the agreement is good for nearly all angles and frequencies. The comparisons at Mj = I.67 are shown in Figure
lI..~.I..:.I..~~I~.~.1.~~~I.~~~I 0
5
IO
15
Frequency
20
25
30
(kHd
Measured (Norum and Seiner [3]); -, Figure4. Far field noise spectra. Md = 1 .O. -, (3.4)). Nozzle exit diameter=3.982 cm. (a) M, = 1.22; (b) M, = 1.67; (c) Mj = 1.80.
calculated
(equation
64
C.
K.
W.
TAM
(b)
0
5
IO
15
Frequency
Figure
20
25
(kHz)
4-continued.
4(b). At this Mach number the broadband shock noise attains nearly its peak intensity. This operating condition corresponds to the maximum shock noise condition of point B in Figure 1. Figure 4(c) is at a still higher Mach number of Mj = 1.80, and the broadband shock noise intensity begins to decrease relative to that of Figure 4(b). In both Figures 4(b) and 4(c) the agreement between calculated and measured noise spectra is uniformly good over almost all measured inlet angle directions and frequencies. The exceptions are at low inlet angles. It was noted in reference [l] that a dip exists in the calculated noise spectrum near the peak noise frequency at these directions. This discrepancy is carried over to the present calculation since only shock cell strength is corrected in this work. In Figures 5(a), (b) and 6 are shown further comparisons between the calculated results of equation (3.4) or (3.7) and the measurements of Norum and Seiner [3] for C-D nozzles of Mach number 1.5 and 2.0 operating in the underexpanded mode. The noise power spectrum at lWj= 1.80 and Md = 1.50 is given in Figure 5(a). This is at point B of Figure 1. As can be seen there is favorable agreement (except for the dips in the calculated spectra as noted above) between computed spectra and data in every direction. In Figure 5(b) the jet Mach number is equal to 1.99. This operating Mach number lies beyond the maximum shock noise condition. Again good agreement is evident. The noise spectra of an underexpanded jet at Mj = 2.33 from a C-D nozzle of Mach number 2.0 is shown in Figure 6. Except at the lowest inlet angle the calculated spectra match those of the
SHOCK
NOISE
FROM
IMPERFECT
$
JETS
65
max dB (measured)
Frequency (kHz)
Figure rlcontinued.
measured spectra reasonably well. In summary, comparisons between the predicted noise spectra of equation (3.4) or (3.7) for underexpanded jets from three C-D nozzles of design Mach number 1.0, l-5 and 2-O with the measurements of Norum and Seiner [3] have been carried out over the moderately imperfectly expanded Mach number range. Good agreements are found at nearly every microphone direction over a wide frequency band. Similar comparisons between calculated results and measurements for overexpanded jets are shown in Figures 7 and 8. Figures 7(a) and 7(b) are for jets from Mach 2.0 C-D nozzles, while Figure 8 is for a Mach l-5 C-D nozzle. As can be seen the agreements are as good as for underexpanded jets. The jet operating condition of Figures 7(a) and 8 corresponds to the maximum shock noise point C of Figure 1. In Figure 7(b) at Mj = 1.49 are shown the noise spectra at a further overexpanded condition. Again, good overall agreements are found between the calculated and the measured spectra at all microphone directions both in spectral level and in peak noise frequency. These good agreements should provide confidence in the use of the above noise prediction formulas.
4.
SUMMARY
AND DISCUSSION
In this paper the near and far field broadband shock-associated noise prediction formulas of reference [l] which were developed for slightly imperfectly expanded jets
66
C. K. W. TAM
(EP) P”Oq ZH Ob “’ 7d.9
SHOCK
NOISE
IO
FROM
IMPERFECT
15 Frequency
67
JETS
20
25
30
(kHz)
Figure 6. Far field noise spectra. Md = 2.0, M, = 2.33. -, calculated (equation (3.4)). Nozzle exit diameter = 4.989 cm.
Measured
(Norum
and
Seiner
[3]);
--,
are extended to moderately imperfectly expanded jets. In carrying out the extension it was first observed that the shock system in the plume of a moderately imperfectly expanded jet generally consists of a strong first shock followed by a quasi-periodic shock cell structure. The quasi-periodic shock structure resembles that of a slightly imperfectly expanded jet. The intensity of broadband shock noise, on the other hand, appears to be dictated primarily by the amplitude of the quasi-periodic shock cells. These observations allow one to broaden the range of applicability of the noise prediction formulas of reference [l] by modifying the shock cell amplitudes in these formulas appropriately. This is done by first recognizing that there is an intrinsic difference between overexpanded and underexpanded supersonic jets. For these two classes of jets a suitable shock cell amplitude parameter is found. By means of this parameter a set of shock cell amplitude formulas has been determined semi-empirically. Extensive comparisons between the extended theory and measured broadband shock noise spectra have been carried out for jets from C-D nozzles with design Mach number 2.0, 1.5 and 1.0. Good agreements are found over the entire moderately imperfectly expanded Mach number range. It is well known that imperfectly expanded supersonic jets invariably emit discrete frequency sound called screech tones. The screech tones are generated by a self-excited feedback cycle [ 10, 111. Experimental observations [ 121 indicate that the feedback cycle
15 (kliz)
20
Md = 2-O. -,
Frequency
Figure 7. Far field noise spectra.
IO
$I
Measured
25
I
(Norum
30
max dB (measured)(
(a)
and Seiner [3]); -,
0
calculated
5
15
20
25
(3.4)). (a) A4, = 1.80; (b) M, = 1.49.
Frequency (kHz)
(equation
IO
SHOCK
NOISE
FROM
IMPERFECT
69
JETS
max dB (measured)
Frequency (kHz)
Figure 8. Far field noise calculated (equation (3.4)).
spectra.
Md = 1.5, h4, = 1.28. -,
Measured
(Norum
and
Seiner
[3]);
---,
has the tendency of suppressing the broadband shock noise component. A careful examination of the axial static pressure distribution associated with the shock cell structure of imperfectly expanded jets measured by Norum and Seiner [3] suggests that when strong screech tones are present the feedback loop causes a rapid disintegration of the periodic shock cell structure. It is believed, therefore, that the reduction in broadband shock noise is the direct result of a decrease in shock cell strength. A comparison of the calculated and measured noise spectra at Md = 2-O and Mj = l-67 is shown in Figure 9. At this jet operating condition strong screech tones exist, as can be readily seen in the measured noise spectra. The agreement between the calculated and the measured spectra is not as good as before. The measured spectra are generally lower in level consistent with the observations of reference [12]. Thus the prediction formulas developed in this paper would only provide an upper bound estimate of broadband shock noise when strong screech tones are present. ACKNOWLEDGMENT
This work was supported
by NASA Langley Research Center Grant NAG 1-421.
70
C. K. W. TAM
J/
IO
20
15 Freqwncy
max dB (measured)
25
30
(kHz)
Figure 9. Far field noise spectra with intense screech tones. Mj = 1.67, Md = 2.0. -, calculated (equation (3.4)). and Seiner [3]); -,
Measured (Norum
REFERENCES 1. C. K. W. TAM 1987 .ZoumaZ of Sound and Vibration 116, 265-302. Stochastic model theory of broadband shock associated noise from supersonic jets. 2. C. K. W. TAM and H. K. TANNA 1982Journal of Sound and Vibration 81, 337-358. Shock associated noise of supersonic jets from convergent-divergent nozzles. 3. T. D. NORUM and J. M. SEINER 1982 NASA TM 84521. Measurements of mean static pressure and far field acoustics of shock-containing supersonic jets. 4. J. C. Yu 1971 Ph.D. thesis, Syracuse University. Investigation of the noise fields of supersonic axisymmetric jet Rows. 5. J. M. SEINER and J. C. Yu 1984 American Znstitute of Aeronautics and Astronautics Journal 22, 1207-1215. Acoustic near field properties associated with broadband shock noise. 6. M. HARPER-B• URNE and M. J.FISHER 1973 Proceedings (No. 131) ofthe AGARD Conference on Noise Mechanisms, Brussels, Belgium. The noise from shock waves in supersonic jets. 7. C. K. W. TAM, J.A.JACKSON and J.M. SEINER 1985Journal of Fluid Mechanics 153,123-149. A multiple-scales model of the shock cell structure of imperfectly expanded supersonic jets. 8. L.PRANDTL 1904PhysikZeitschrif $599-601. Uber die Stationiiren Wellen in einem Gasstrahl.
SHOCK NOISE FROM IMPERFECT JETS
71
9. D. C. PACK 1950 Quarterly Journal of Mechanics and Applied Mathematics 3, 173-181. A note on Prandtl’s formula for the wavelength of a supersonic gas jet. 10. A. POWELL 1953 Proceedings of the Physical Society 66, 1039-1056. On the mechanism of choked jet noise. 11. C. K. W. TAM, J. M. SEINER and J. C. Yu 1986 Journal of Sound and Vibration 110,309-321. Proposed relationship between broadband shock associated noise and screech tones. 12. R. T. NAGEL, J. W. DENHAM and A. G. PAPATHANASIOU 1983 American Institute of Aeronautics and Astronautics Journal 21, 1541-1545. Supersonic jet screech tone cancellation.