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LETTER TO THE EDITOR: AN EXPERIMENTAL STUDY OF UNDEREXPANDED SQUARE JET NOISE
Journal of Sound and Vibration (1996) 197(2), 255–261
AN EXPERIMENTAL STUDY OF UNDEREXPANDED SQUARE JET NOISE M. B†, R.-H. C L. C Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, U.S.A. (Received 7 February 1996)
1. This work was motivated by the effect of screech tone noise on fuel-oxidizer mixing and the noise of jets associated with high speed civil transport (HSCT). Previous studies include both theoretical and experimental investigations for axisymmetric and asymmetric (elliptical and rectangular) supersonic jets [1–3]. It was revealed that noise associated with shock structure and having well defined frequencies (screech tone) could enhance mixing. The frequencies and amplitudes of these discrete noise tones can also be theoretically predicted, given the jet nozzle geometry and pressure ratio. A recent review that concentrated on circular jets [4] summarizes the important aspects of supersonic jet noise and outlines the challenges of further studies on asymmetric jets. Future generations of HSCT are likely to have a non-circular nozzle geometry. For the purpose of varying, or possibly modifying or extending, the existing theories, this letter is aimed at underexpanded square jets. Emphasis will be on their screech tones for which, to the authors’ knowledge, no acoustic measurements have been carried out. A square nozzle is an intuitive combination of circular and two-dimensional rectangular geometries. Its jet noise characteristics also demonstrate some interesting composite features, as will be reported below. First, it will serve the purpose of comparison to briefly mention key theoretical results. For circular jets, whether overexpanded or underexpanded, the screech tone frequency is related to a Strouhal number defined by Tam [5]:
$
0
fs Dj (g − 1) 2 = 0·67(Mj2 − 1)−1/2 1 + 0·7Mj 1+ Mj Uj 2
1 0 1% −1/2
T0 Ta
1/2
−1
,
(1)
where fs is the fundamental screech tone frequency and Dj , Uj and Mj (Uj normalized by the acoustic speed of the surroundings) are the jet diameter, velocity and Mach number, respectively, under the fully expanded conditions, and T0 and Ta are the total temperature of the jet and the temperature of the surroundings, respectively. Dj can be calculated as follows [5]:
$
Dj 1+(g − 1)Mj2 /2 = Dd 1+(g − 1)Mj2 /2
%
(g + 1)/4(g − 1)
0 1 Md Mj
1/2
,
(2)
where the subscript d denotes the design conditions at the nozzle exit.
255 0022–460X/96/420255 + 07 $25.00/0
7 1996 Academic Press Limited
256
For rectangular jets with large aspect ratios (AR q 4 for the jet to be two-dimensional), the screech tone frequency is determined by a Strouhal number defined by [6]:
fs hd = Uj
6 6$
760 1 7 0 1 76 7
Uc /Uj 2(1+Uc /aa )(Mj2 − 1)1/2
1+(g − 1)Mj2 /2 1+(g − 1)Md2 /2
%
(g + 1)/2(g − 1)
2
hj +1 bj
Md −1 Mj
1/2
,
(3)
bd +1 bd + hd
where h and b are the jet’s fully expanded thickness and width, respectively; subscripts j and d denote fully expanded and nozzle exit conditions, repectively, and Uc and aa are the convective velocity of the jet and the sonic speed of the ambient, respectively. Uc for both circular and rectangular jets can be taken to be 0·7Uj [4, 6]. The Strouhal number thus determined has been shown to agree well with experiments for Mj up to 1·7 and AR = 10 and 16·7 [6]. The experiment for the purpose of this letter was carried out as follows. 2. The nozzle has a contraction from 50 mm × 50 mm down to a square exit of 5 mm × 5 mm. The ‘‘reservoir’’ pressure (Pr ) was measured on the wall of the 50 mm × 50 mm section. The air was supplied from a larger reservoir (at a nominal 298 K) and exited the nozzle into room air (at 298 K and 1 atmosphere). The fully expanded jet Mach number (Mj ) can easily be determined from the pressure ratio (Pr /Pa ) using the one-dimensional isentropic relation. The pressure ratio (R) range was 3·04, 3·31, 3·59, 3·86, 4·13 and 4·44, straddling 3·89, for which the maximum jet shear layer growth was a maximum for rectangular supersonic jets [3]. These pressure ratios corresponds to Mj equal to 1·37, 1·43, 1·48, 1·53, 1·58 and 1·62, respectively. The stagnation temperature of all of the jets studied is nominally 300 K. The near-field sound pressure level (SPL) was measured with a 12 in Bruel & Kjaer type 4133 microphone, having a dynamic range of 0–40 kHz, in which the frequency response was within 1 dB. The microphone was always aimed at the flow axis at the nozzle exit and was placed on a circular circumference (radius = 3 cm) at inlet angles x of 45°, 90° and 135° to the flow direction. Measurements for x = 0° were not carried out due to difficulty with the nozzle construction. The signals were recorded on a computer hard disk and were later analyzed using a fast Fourier transfer program [7]. The frequency resolution was 20 Hz. Schlieren flow visualization was also conducted, to reveal the shock cell structure and provide data for the determination of shock spacing. 3. The three components of the supersonic noise were observed. The results are shown in Figure 1. It is noted that the SPL values for x = 90° and 135° are shifted purposely to avoid overlapping with that of x = 45°. For the present Mj , the screech tones (in the frequency range of 17·12–24·14 kHz) dominate over the broadband and turbulent mixing noise. The relative magnitudes of turbulent mixing and broadband noises are similar to those in circular jets, depending on the inlet angle [4]. Both toroidal and helical (or flapping) modes exist above Mj e 1·48, as shown in Figure 1, with the latter having higher frequencies than the former. It is known that circular jets switch modes randomly, with the two modes never coexisting [2]. As Mj increases, the helical mode (not seen for Mj Q 1·48) becomes more dominant, as shown in Figure 2. This is even more evident than the inlet angle passes beyond 90°, indicating that, unlike the toroidal mode, the helical
257
Figure 1. The near-field sound pressure levels (SPL) of square jets: (a) Mj = 1·37; (b) Mj = 1·43; (c) Mj = 1·48; (d) Mj = 1·53; (e) Mj = 1·58; (f) Mj = 1·62. The vertical axes for (a)–(f) are in arbitary units.
mode propagates in all directions. Both the toroidal and helical mode frequencies decreased with increasing Mj (i.e., increasing R), similar to circular and rectangular jets [4]. Thus the mode switching occurred for Mj e 1·48, compared to Mj 1 1·2 for circular jets [2]. Smaller peaks exist near 40 kHz for Mj e 1·53 (shown in Figure 1). Their frequencies are smaller than twice the helical mode, but greater than twice the toroidal mode. They appear to have preferred directions in x e 90°, reminiscent of the characteristics of first harmonics [4]. However, their frequencies increased with Mj , opposite to the fundamental screech tone frequencies. A first test of Tam’s theory for non-axisymmetric jets could be the shock spacing. The largest shock spacing was predicted to be (equation 2.23 of reference [6]) LS = 2(Mj2 − 1)1/2 hj /[1+hj2 /bj2 ]1/2 .
(4)
From the Schlieren visualization (not shown here), the largest shock spacing is that between those of the first and the second shock (from tip to tip of the diamond shock cells). The comparison between the experimental and theoretical values is shown in Table 1, which reveals excellent agreement.
258
Figure 2. Near-field sound pressure levels (in arbitrary units) of toroidal and helical mode screech tones of square jets, measured at three inlet angles: (a) x = 45°; (b) x = 90°; (c) x = 135°. Q, Experimental toroidal mode; W, experimental helical mode.
The Strouhal number, based on the nozzle exit width and the fundamental toroidal and helical mode frequencies, was calculated according to both theories for circular and rectangular jets. The experimental and theoretical results, along with some previous results [1, 2], are compared in Figure 3. The Strouhal number of the fundamental toroidal mode frequencies was found to agree with Tam’s theory for rectangular jets within 6%, although the theory was developed for large aspect ratios. For comparison with the circular jet theory, equation (1), the hydraulic/equivalent diameter (=2h/zp) was used for helical modes. This was because, to the authors’ knowledge, no helical modes have been reported in large aspect ratio jets. The Strouhal number was then found to agree with the circular jet theory within 5%, as shown in Figure 3. These finding are worth noting because the theoretical Strouhal numbers and screech tone frequencies using circular jet theory, based on the equivalent diameter, and for rectangular jets, differ by a factor of nearly two. This is outside the range of experimental error. Similar observations can be made for the screech tone frequencies (see Figure 4). In light of the Strouhal number, the square jet noise characteristics for Mj e 1·48 switch between those of a rectangular jets with large aspect ratio (the troidal mode), and a circular T 1 Comparison of experimental and theoretical largest shock spacing 3·04 (Mj = 1·37)
3·31 (Mj = 1·43)
3·59 (Mj = 1·48)
3·89 (Mj = 1·53)
4·13 (Mj = 1·58)
4·44 (Mj = 1·62)
Current study 6·0 Equation (4) 6·62
6·75 7·23
7·75 7·71
7·75 8·19
8·50 8·65
8·75 9·01
259
Figure 3. Comparisons of experimental toroidal (Q) and helical, (W) screech tones and theoretical Strouhal numbers. —, Theoretical square jet, equation (3); –––, theoretical circular jet, equation (1).
jet (the helical mode). A word of caution may be appropriate regarding the use of equation (1) for helical modes. It is noted that equation (1) was derived for the screech tone propagating toward the nozzle exit, while the helical mode appears to propagate in all directions, as noted above. The frequencies of the toroidal and helical modes were both found to scale with pressure ratio following fS 1 R n, with the exponent n 1 −0·6, compared to −3/2 for rectangular jets with AR = 10 and 16·7 over the R range of 2·7–5·4 [3]. In theory, two-dimensionality is expected for AR q 4 [6]. Some authors based the Strouhal number on the nozzle exit conditions. Comparisons of such Strouhal numbers of screech tones of the current square jets with those of circular and rectangular jets are listed in Table 2. The pressure ratio chosen was about 3·8, for which the mixing was most significantly enhanced by screech tones. They show that for R13·0–4.0, for which the Strouhal numbers for square and circular jets fall within the same range, which is nearly twice that of rectangular jets having AR as small as 1·7. It should be noted that Strouhal numbers here is defined based on the nozzle exit velocity and
Figure 4. Comparisons of experimental toroidal and helical screech tones and theoretical screech tone frequencies. Key as in Figure 3.
260
T 2 Comparison of Strouhal number of square jets and circuluar and rectangular jets Geometry
Pressure ratio
Axisymmetric
3·67 4·00
0·24 0·275
Square
3·04 3·59 3·86 4·13 3·59 3·86 4·13
0·278 0.228 0·212 0·199 0·283 0·263 0·246
3·17 3·67 3·50 3·67 3·80
0·14 0·135 0·12 0·11 0·12
Rectangular AR = 1·7 AR = 5·83 AR = 10 AR = 12·75 AR = 16·7
Strouhal number†
Source of data Powell‡ Sherman‡
(toroidal) (toroidal) (toroidal) (toridal) (helical) (helical) (helical)
Present Present Present Present Present Present Present
results results results results results results results
Powell‡ Powell‡ Krothapolli [3] Hamitt‡ Krothapolli [3]
† This Strouhal number is defined based on nozzle exit diameter and velocity. ‡ As quoted in Krothapolli et al. [3].
diameter. The tabulated values of square jet helical modes are different from those of Figure 3, based on fully expanded conditions. The nearly simultaneous existence of the maximum SPL and the maximum spreading rate was reported for both circular jets and rectangular jets having AR = 10 and 16·7 [2, 3]. It is therefore useful to know at what value of Mj , or of R, there is a difference between the magnitudes of the toroidal and the helical mode. Since it is the upstream propagating screech tone that amplifies the instability wave at nozzle exit and enhances mixing, the results of Figure 2(a) (x = 45°) should be considered. The sound pressure level of the helical mode of the square jet appears to reach a maximum at Mj 1 1·58, or R 1 4·13. These values of Mj and R compare to those of rectangular jets having AR = 16·7, for which R = 3·8 [3], and that of circular jets, for which Mj 1 1·3 [2]. 4. The following conclusions can be drawn from the results of this study of underexpanded square jets. (1) Switching between the toroidal and helical screech modes were observed for the fully expanded Mach number (Mj ) greater than 1·48. The frequencies of both the toroidal and helical modes scale with pressure ratio following St 1 R−0·6. The amplitude of the helical mode reaches a maximum at Mj 1 1·58 (R = 4·13), in the similar range of Mj and R as circular jets and rectangular jets with large aspect ratios. The helical mode propagate in all directions, unlike the toroidal mode which propagates predominantly toward the nozzle exit. (2) Tam’s theory for large aspect ratio rectangular jets [6] predicts the shock spacing and the toroidal mode Strouhal number of the present square jets within 10% and 6%, respectively. His circular jet theory [5] predicts the helical mode Strouhal number within 5% when the hydraulic diameter of the square nozzle is used.
261
1. E. G, K. C. S, K. J. W and C. J. B 1988 Physics of Fluids 31, 2524–2532. Near-field pressure radiation and flow characteristics in low supersonic circular and elliptic jets. 2. E. G, K. C. S and C. J. B 1989 Physics of Fluids A1, 868–873. Mode switching in supersonic circular jets. 3. A. K, Y. H, D. B and K. K 1986 Journal of Sound and Vibration 106, 119–143. The role of screech tones in mixing of an underexpanded rectangular jet. 4. C. K. W. T 1995 Annual Review of Fluid Mechanics 27, 17–43. Supersonic jet noise. 5. C. K. W. T, J. S and J.-C. Y 1986 Journal of Sound and Vibration 110, 309–321. Proposed relationship between broadband shock associated noise and screech tones. 6. C. K. W. T 1988 Journal of Sound and Vibration 121, 135–147. The shock-cell structures and screech tone frequencies of rectangular and nonaxisymmetric supersonic jets. 7. J. W 1991 Fast Fourier Transform, Boca Raton, FL: CRC Press.
AN EXPERIMENTAL STUDY OF UNDEREXPANDED SQUARE JET NOISE M. B†, R.-H. C L. C Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, U.S.A. (Received 7 February 1996)
1. This work was motivated by the effect of screech tone noise on fuel-oxidizer mixing and the noise of jets associated with high speed civil transport (HSCT). Previous studies include both theoretical and experimental investigations for axisymmetric and asymmetric (elliptical and rectangular) supersonic jets [1–3]. It was revealed that noise associated with shock structure and having well defined frequencies (screech tone) could enhance mixing. The frequencies and amplitudes of these discrete noise tones can also be theoretically predicted, given the jet nozzle geometry and pressure ratio. A recent review that concentrated on circular jets [4] summarizes the important aspects of supersonic jet noise and outlines the challenges of further studies on asymmetric jets. Future generations of HSCT are likely to have a non-circular nozzle geometry. For the purpose of varying, or possibly modifying or extending, the existing theories, this letter is aimed at underexpanded square jets. Emphasis will be on their screech tones for which, to the authors’ knowledge, no acoustic measurements have been carried out. A square nozzle is an intuitive combination of circular and two-dimensional rectangular geometries. Its jet noise characteristics also demonstrate some interesting composite features, as will be reported below. First, it will serve the purpose of comparison to briefly mention key theoretical results. For circular jets, whether overexpanded or underexpanded, the screech tone frequency is related to a Strouhal number defined by Tam [5]:
$
0
fs Dj (g − 1) 2 = 0·67(Mj2 − 1)−1/2 1 + 0·7Mj 1+ Mj Uj 2
1 0 1% −1/2
T0 Ta
1/2
−1
,
(1)
where fs is the fundamental screech tone frequency and Dj , Uj and Mj (Uj normalized by the acoustic speed of the surroundings) are the jet diameter, velocity and Mach number, respectively, under the fully expanded conditions, and T0 and Ta are the total temperature of the jet and the temperature of the surroundings, respectively. Dj can be calculated as follows [5]:
$
Dj 1+(g − 1)Mj2 /2 = Dd 1+(g − 1)Mj2 /2
%
(g + 1)/4(g − 1)
0 1 Md Mj
1/2
,
(2)
where the subscript d denotes the design conditions at the nozzle exit.
255 0022–460X/96/420255 + 07 $25.00/0
7 1996 Academic Press Limited
256
For rectangular jets with large aspect ratios (AR q 4 for the jet to be two-dimensional), the screech tone frequency is determined by a Strouhal number defined by [6]:
fs hd = Uj
6 6$
760 1 7 0 1 76 7
Uc /Uj 2(1+Uc /aa )(Mj2 − 1)1/2
1+(g − 1)Mj2 /2 1+(g − 1)Md2 /2
%
(g + 1)/2(g − 1)
2
hj +1 bj
Md −1 Mj
1/2
,
(3)
bd +1 bd + hd
where h and b are the jet’s fully expanded thickness and width, respectively; subscripts j and d denote fully expanded and nozzle exit conditions, repectively, and Uc and aa are the convective velocity of the jet and the sonic speed of the ambient, respectively. Uc for both circular and rectangular jets can be taken to be 0·7Uj [4, 6]. The Strouhal number thus determined has been shown to agree well with experiments for Mj up to 1·7 and AR = 10 and 16·7 [6]. The experiment for the purpose of this letter was carried out as follows. 2. The nozzle has a contraction from 50 mm × 50 mm down to a square exit of 5 mm × 5 mm. The ‘‘reservoir’’ pressure (Pr ) was measured on the wall of the 50 mm × 50 mm section. The air was supplied from a larger reservoir (at a nominal 298 K) and exited the nozzle into room air (at 298 K and 1 atmosphere). The fully expanded jet Mach number (Mj ) can easily be determined from the pressure ratio (Pr /Pa ) using the one-dimensional isentropic relation. The pressure ratio (R) range was 3·04, 3·31, 3·59, 3·86, 4·13 and 4·44, straddling 3·89, for which the maximum jet shear layer growth was a maximum for rectangular supersonic jets [3]. These pressure ratios corresponds to Mj equal to 1·37, 1·43, 1·48, 1·53, 1·58 and 1·62, respectively. The stagnation temperature of all of the jets studied is nominally 300 K. The near-field sound pressure level (SPL) was measured with a 12 in Bruel & Kjaer type 4133 microphone, having a dynamic range of 0–40 kHz, in which the frequency response was within 1 dB. The microphone was always aimed at the flow axis at the nozzle exit and was placed on a circular circumference (radius = 3 cm) at inlet angles x of 45°, 90° and 135° to the flow direction. Measurements for x = 0° were not carried out due to difficulty with the nozzle construction. The signals were recorded on a computer hard disk and were later analyzed using a fast Fourier transfer program [7]. The frequency resolution was 20 Hz. Schlieren flow visualization was also conducted, to reveal the shock cell structure and provide data for the determination of shock spacing. 3. The three components of the supersonic noise were observed. The results are shown in Figure 1. It is noted that the SPL values for x = 90° and 135° are shifted purposely to avoid overlapping with that of x = 45°. For the present Mj , the screech tones (in the frequency range of 17·12–24·14 kHz) dominate over the broadband and turbulent mixing noise. The relative magnitudes of turbulent mixing and broadband noises are similar to those in circular jets, depending on the inlet angle [4]. Both toroidal and helical (or flapping) modes exist above Mj e 1·48, as shown in Figure 1, with the latter having higher frequencies than the former. It is known that circular jets switch modes randomly, with the two modes never coexisting [2]. As Mj increases, the helical mode (not seen for Mj Q 1·48) becomes more dominant, as shown in Figure 2. This is even more evident than the inlet angle passes beyond 90°, indicating that, unlike the toroidal mode, the helical
257
Figure 1. The near-field sound pressure levels (SPL) of square jets: (a) Mj = 1·37; (b) Mj = 1·43; (c) Mj = 1·48; (d) Mj = 1·53; (e) Mj = 1·58; (f) Mj = 1·62. The vertical axes for (a)–(f) are in arbitary units.
mode propagates in all directions. Both the toroidal and helical mode frequencies decreased with increasing Mj (i.e., increasing R), similar to circular and rectangular jets [4]. Thus the mode switching occurred for Mj e 1·48, compared to Mj 1 1·2 for circular jets [2]. Smaller peaks exist near 40 kHz for Mj e 1·53 (shown in Figure 1). Their frequencies are smaller than twice the helical mode, but greater than twice the toroidal mode. They appear to have preferred directions in x e 90°, reminiscent of the characteristics of first harmonics [4]. However, their frequencies increased with Mj , opposite to the fundamental screech tone frequencies. A first test of Tam’s theory for non-axisymmetric jets could be the shock spacing. The largest shock spacing was predicted to be (equation 2.23 of reference [6]) LS = 2(Mj2 − 1)1/2 hj /[1+hj2 /bj2 ]1/2 .
(4)
From the Schlieren visualization (not shown here), the largest shock spacing is that between those of the first and the second shock (from tip to tip of the diamond shock cells). The comparison between the experimental and theoretical values is shown in Table 1, which reveals excellent agreement.
258
Figure 2. Near-field sound pressure levels (in arbitrary units) of toroidal and helical mode screech tones of square jets, measured at three inlet angles: (a) x = 45°; (b) x = 90°; (c) x = 135°. Q, Experimental toroidal mode; W, experimental helical mode.
The Strouhal number, based on the nozzle exit width and the fundamental toroidal and helical mode frequencies, was calculated according to both theories for circular and rectangular jets. The experimental and theoretical results, along with some previous results [1, 2], are compared in Figure 3. The Strouhal number of the fundamental toroidal mode frequencies was found to agree with Tam’s theory for rectangular jets within 6%, although the theory was developed for large aspect ratios. For comparison with the circular jet theory, equation (1), the hydraulic/equivalent diameter (=2h/zp) was used for helical modes. This was because, to the authors’ knowledge, no helical modes have been reported in large aspect ratio jets. The Strouhal number was then found to agree with the circular jet theory within 5%, as shown in Figure 3. These finding are worth noting because the theoretical Strouhal numbers and screech tone frequencies using circular jet theory, based on the equivalent diameter, and for rectangular jets, differ by a factor of nearly two. This is outside the range of experimental error. Similar observations can be made for the screech tone frequencies (see Figure 4). In light of the Strouhal number, the square jet noise characteristics for Mj e 1·48 switch between those of a rectangular jets with large aspect ratio (the troidal mode), and a circular T 1 Comparison of experimental and theoretical largest shock spacing 3·04 (Mj = 1·37)
3·31 (Mj = 1·43)
3·59 (Mj = 1·48)
3·89 (Mj = 1·53)
4·13 (Mj = 1·58)
4·44 (Mj = 1·62)
Current study 6·0 Equation (4) 6·62
6·75 7·23
7·75 7·71
7·75 8·19
8·50 8·65
8·75 9·01
259
Figure 3. Comparisons of experimental toroidal (Q) and helical, (W) screech tones and theoretical Strouhal numbers. —, Theoretical square jet, equation (3); –––, theoretical circular jet, equation (1).
jet (the helical mode). A word of caution may be appropriate regarding the use of equation (1) for helical modes. It is noted that equation (1) was derived for the screech tone propagating toward the nozzle exit, while the helical mode appears to propagate in all directions, as noted above. The frequencies of the toroidal and helical modes were both found to scale with pressure ratio following fS 1 R n, with the exponent n 1 −0·6, compared to −3/2 for rectangular jets with AR = 10 and 16·7 over the R range of 2·7–5·4 [3]. In theory, two-dimensionality is expected for AR q 4 [6]. Some authors based the Strouhal number on the nozzle exit conditions. Comparisons of such Strouhal numbers of screech tones of the current square jets with those of circular and rectangular jets are listed in Table 2. The pressure ratio chosen was about 3·8, for which the mixing was most significantly enhanced by screech tones. They show that for R13·0–4.0, for which the Strouhal numbers for square and circular jets fall within the same range, which is nearly twice that of rectangular jets having AR as small as 1·7. It should be noted that Strouhal numbers here is defined based on the nozzle exit velocity and
Figure 4. Comparisons of experimental toroidal and helical screech tones and theoretical screech tone frequencies. Key as in Figure 3.
260
T 2 Comparison of Strouhal number of square jets and circuluar and rectangular jets Geometry
Pressure ratio
Axisymmetric
3·67 4·00
0·24 0·275
Square
3·04 3·59 3·86 4·13 3·59 3·86 4·13
0·278 0.228 0·212 0·199 0·283 0·263 0·246
3·17 3·67 3·50 3·67 3·80
0·14 0·135 0·12 0·11 0·12
Rectangular AR = 1·7 AR = 5·83 AR = 10 AR = 12·75 AR = 16·7
Strouhal number†
Source of data Powell‡ Sherman‡
(toroidal) (toroidal) (toroidal) (toridal) (helical) (helical) (helical)
Present Present Present Present Present Present Present
results results results results results results results
Powell‡ Powell‡ Krothapolli [3] Hamitt‡ Krothapolli [3]
† This Strouhal number is defined based on nozzle exit diameter and velocity. ‡ As quoted in Krothapolli et al. [3].
diameter. The tabulated values of square jet helical modes are different from those of Figure 3, based on fully expanded conditions. The nearly simultaneous existence of the maximum SPL and the maximum spreading rate was reported for both circular jets and rectangular jets having AR = 10 and 16·7 [2, 3]. It is therefore useful to know at what value of Mj , or of R, there is a difference between the magnitudes of the toroidal and the helical mode. Since it is the upstream propagating screech tone that amplifies the instability wave at nozzle exit and enhances mixing, the results of Figure 2(a) (x = 45°) should be considered. The sound pressure level of the helical mode of the square jet appears to reach a maximum at Mj 1 1·58, or R 1 4·13. These values of Mj and R compare to those of rectangular jets having AR = 16·7, for which R = 3·8 [3], and that of circular jets, for which Mj 1 1·3 [2]. 4. The following conclusions can be drawn from the results of this study of underexpanded square jets. (1) Switching between the toroidal and helical screech modes were observed for the fully expanded Mach number (Mj ) greater than 1·48. The frequencies of both the toroidal and helical modes scale with pressure ratio following St 1 R−0·6. The amplitude of the helical mode reaches a maximum at Mj 1 1·58 (R = 4·13), in the similar range of Mj and R as circular jets and rectangular jets with large aspect ratios. The helical mode propagate in all directions, unlike the toroidal mode which propagates predominantly toward the nozzle exit. (2) Tam’s theory for large aspect ratio rectangular jets [6] predicts the shock spacing and the toroidal mode Strouhal number of the present square jets within 10% and 6%, respectively. His circular jet theory [5] predicts the helical mode Strouhal number within 5% when the hydraulic diameter of the square nozzle is used.
261
1. E. G, K. C. S, K. J. W and C. J. B 1988 Physics of Fluids 31, 2524–2532. Near-field pressure radiation and flow characteristics in low supersonic circular and elliptic jets. 2. E. G, K. C. S and C. J. B 1989 Physics of Fluids A1, 868–873. Mode switching in supersonic circular jets. 3. A. K, Y. H, D. B and K. K 1986 Journal of Sound and Vibration 106, 119–143. The role of screech tones in mixing of an underexpanded rectangular jet. 4. C. K. W. T 1995 Annual Review of Fluid Mechanics 27, 17–43. Supersonic jet noise. 5. C. K. W. T, J. S and J.-C. Y 1986 Journal of Sound and Vibration 110, 309–321. Proposed relationship between broadband shock associated noise and screech tones. 6. C. K. W. T 1988 Journal of Sound and Vibration 121, 135–147. The shock-cell structures and screech tone frequencies of rectangular and nonaxisymmetric supersonic jets. 7. J. W 1991 Fast Fourier Transform, Boca Raton, FL: CRC Press.