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Multiple tone operation of edgetones
J. SoundVib.(1970)12(3), 281-284
MULTIPLE
TONE
OPERATION G.
R.
OF EDGETONES
STEGEN~
Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, U.S.A. AND
K. KARAMCHETI Department of Aeronautics and Astronautics, Stanford University, Stmford, California, U.S.A.
Multiple tone operation of an edgetone is examined experimentally. It is shown that during stage 2 operation two discrete modes of operation may exist simultaneously. The second mode of operation is proven to be a persistence of stage 1 operation. 1. INTRODUCTION An edgetone is the tone of sound that is generated when a thin jet of gas impinges on a wedge-shaped edge placed a short distance from the jet exit. For a particular edgetone generator with a fixed jet exit speed l_J,the frequencyfof the edgetone is controlled by the wedge height h (distance from jet exit). An operating curve for a typical edgetone generator is shown in Figure 1. Initially, the frequency is inversely proportional to the wedge height. At a critical wedge height the frequency jumps up from the original operation curve, referred to
IOOO-
‘\ -i;
400200
3.0
1 4-O
\.
5.0
6.0
I 7.0
Wedge location (h/d)
Figure 1. Edgetone operation diagram.Stage 1, ---; ihdicate direction of frequency jumps.
stage
2, -;
stage 1 during stage 2, ---.
Arrows
as stage 1, to a new operation curve, stage 2. As the wedge height is subsequently reduced, the downward frequency jump from stage 2 to stage 1 occurs at a lower wedge height than the jump up, so that there is a hysteresis region in the edgetone operation curve. During the stage 2 operation in the hysteresis region, multiple edgetones are often observed [ 1,2]. Stage 2 seems to be operating simultaneously with stage 1. However, the amplitude of stage 1 is greatly reduced. In addition, its frequency may lie as much as 15 y0 below the usual stage 1 operating curve. Nevertheless, it is usually assumed that this secondary tone is a t This work was conducted while the first author was at Department of the Aerospace and Mechanical Engineering Sciences, University of California, San Diego, La Jolla, California, U.S.A. 19 281
G.
282
R. STEGEN AND K. KARAMCHETI
continuation of stage I operation, and consequently is labeled stage 1 during stage 2. The purpose of this note is to present physical evidence substantiating this contention. 2. EXPERIMENTAL
ARRANGEMENT
The velocity fluctuations present in the edgetone flow field can be characterized by their frequency5 amplitude, and phase angle 0 (measured relative to the fluctuations at the jet exit). Stegen and Karamcheti [3, 41 have previously discussed the measurement of these quantities in an edgetone flow field. Briefly, the amplitude and phase of the velocity fluctuations were measured using linearized constant temperature hot-wire anemometry (Transmetrics Model 6402/4402). An “x”-wire probe was placed on the center-line of the edgetone flow field, and oriented to respond to the longitudinal (u) and transverse (0) velocities of the jet. An electronic sum/difference unit provided a direct readout of u and U.The u signal was bandpass filtered to isolate the frequency of interest. The amplitude of v was then measured with an a.c. voltmeter. To measure the phase of a(x,t) in the laboratory, a reference signal synchronized to the edgetone fluctuations is needed. For this purpose, a second hot-wire probe was held fixed in the flow field at a point where it would not interfere with the “x” probe. The phase of v relative to the reference signal was measured with a phase meter (Ad-Yu Model 505). The “x“ probe could be moved along the flow field center-line with a micrometer drive. In this way, the amplitude and phase of the transverse velocity fluctuations could be measured everywhere of interest. The fixed operating parameters of the edgetone generator were as follows: center-line speed at jet exit U = 9.0 m/set; slit width d = O-163cm; jet breadth b = 4.57 cm; wedge angle CL= 20”. A single wedge height h = 5*58d was chosen for comparison purposes. By increasing h slowly to this value, purely stage 1 operation could be maintained, whereas, by approaching this value from a larger height, multiple edgetone operation could be maintained. 3. EXPERIMENTAL
RESULTS
3.1. PHASE MEASUREMENTS The results of the phase measurements are presented in Figures 2 and 3 as phase angle (8/27r) vs. center-line distance (x//i) from the slit. Comparison of the phase measurements for 2.0 k ? a
. 1,5-
. .
5 I.O0 : g 0,5-
. . .
0
.
.
. .
.
.
. . . ,* ?? , 0.6 0.8 0.2 0.4 Centerline distance (x/h)
.
I.0
Figure 2. Transverse velocity fluctuations; phase variation along center-line of edgetone generator. Comparison of stage 1 and stage 2. Stage 1, . ; stage 2, m. stage 1 and stage 2 (Figure 2) reveals a marked difference between the amplitude and spatial dependence for the two cases. During stage 1 the phase varies non-linearly with distance, in contrast to the linear variation during stage 2. Bauer [2] has previously presented similar results.
MULTIPLE TONE OPERATION OF EDGETONES
283
The phase characteristics of the sub-harmonic, stage 1 during stage 2, are compared with those of stage 1 in Figure 3. The phase characteristics are essentially identical, strong evidence that stage 1 during stage 2 is a continuation of stage 1 operation during stage 2 operation. In Figures 2 and 3 we have only presented a few data points for clarity. It was found that the measurements (37 data points at each condition) could be correlated by a simple power law: i.e. 8/27r = A(x/~)~. In this form, A represents the value of the phase at the wedge, an
Centerline
distance
(x/h
)
Figure 3. Transverse velocity fluctuations, phase variation along center-line of edgetone generator. Comparison of stage 1 and stage 1 during stage 2. Stage 1, ??; stage 1 during stage 2, 0.
important variable in edgetone theory [5, 61. The power law form is useful in deducing the phase speed C(x) of the fluctuations, since a (x)‘-“. Therefore, the exponent B is a measure of the spatial variation of the phase speed of the fluctuations. Table 1 below summarizes the results of this correlation. The differences between stage 1 and stage 2 shown in Figure 2 are clearly evident in the correlation. The phase angle at the wedge in stage 2 is twice that of stage 1. Also, the phase speed during stage 2 (inferred from Table 1) is almost constant, while during stage 1 it varies as xAo”j3.Stage 1 and stage 1 during stage 2 operate at different frequencies, but with identical phase characteristics. TABLE 1
Phase measurements
Stage 1 Stage 2 Stage 1 during stage 2
3.2.
Frequency (Hz)
e/2rr
370 815 316
0*90 (x//2)1*63 l-81 (_x//#.~~ 0.92 (x//z)‘*~~
AMPLITUDE MEASUREMENTS
The amplitude of the transverse velocity fluctuations was measured simultaneously with the phase. The spatial variation of the amplitude is summarized in Figure 4. Stage 1 and stage 2 are once again seen to be quite different in their mode of operation. Stage 1 grows in an approximately exponential fashion. In contrast, stage 2 has an initial value one-eighth that of stage 1 and grows briefly at the same rate as stage 1. Next, stage 2 experiences a rapid growth; and, finally, levels off at a value comparable to the maximum of stage 1.
284
G. R. STEGEN AND K. KARAMCHETI
Examining stage 1 during stage 2, we see that its amplitude is greatly reduced relative to stage 1. Apparently, the presence of stage 2 suppresses the initial amplitude of stage 1 during stage 2. The growth of stage 1 during stage 2 is very similar to the growth of stage 1, further evidence of their common origin. 0.1 .
0.002
0
0.2
0.4 Centerline
0.6 distance
0.0
I.0
(x/h)
Figure 4. Transverse velocity fluctuations, amplitude variation along center-line of edgetone generator. Stage 1, 0 ; stage 2, m; stage 1 during stage 2, 0.
4. CONCLUSIONS The phase characteristics of stage 1 operation and stage 1 during stage 2 operation have been shown to be identical. While the amplitude of the two modes of operation are different, the growth rates are quite similar. In contrast, the phase and growth characteristics of both stage 1 and stage 1 during stage 2 are distinctly different from those of stage 2. In view of the evidence, it is clear that stage 1 during stage 2 is indeed a persistence of stage 1 operation. ACKNOWLEDGMENTS This work was supported by the National Science Foundation at Stanford University and by Project THEMIS at the University of California, San Diego. REFERENCES 1. G. B. BROWN 1937Proc. phys. Sot. Lond. 49,493. The vortex motion causing edge tones. 2. A. BAUER 1963 Dissertation, Stanford University. Edgetone generation. 3. G. R. STEGEN1967 Dissertation, Stanford University. On the structure of an edgetone flow field. 4. G. R. STEGENand K. KARAMCHETI1967 Stanford University SUDAAR 303. On the structure of an edgetone flow field. 5. N. C~RLE 1953 Proc. R. Sot. London A 216,412. The mechanics 6. A. POWELL 1961 J. acoust. Sot. Am. 33,395. On the edgetone.
of edgetones.
MULTIPLE
TONE
OPERATION G.
R.
OF EDGETONES
STEGEN~
Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, U.S.A. AND
K. KARAMCHETI Department of Aeronautics and Astronautics, Stanford University, Stmford, California, U.S.A.
Multiple tone operation of an edgetone is examined experimentally. It is shown that during stage 2 operation two discrete modes of operation may exist simultaneously. The second mode of operation is proven to be a persistence of stage 1 operation. 1. INTRODUCTION An edgetone is the tone of sound that is generated when a thin jet of gas impinges on a wedge-shaped edge placed a short distance from the jet exit. For a particular edgetone generator with a fixed jet exit speed l_J,the frequencyfof the edgetone is controlled by the wedge height h (distance from jet exit). An operating curve for a typical edgetone generator is shown in Figure 1. Initially, the frequency is inversely proportional to the wedge height. At a critical wedge height the frequency jumps up from the original operation curve, referred to
IOOO-
‘\ -i;
400200
3.0
1 4-O
\.
5.0
6.0
I 7.0
Wedge location (h/d)
Figure 1. Edgetone operation diagram.Stage 1, ---; ihdicate direction of frequency jumps.
stage
2, -;
stage 1 during stage 2, ---.
Arrows
as stage 1, to a new operation curve, stage 2. As the wedge height is subsequently reduced, the downward frequency jump from stage 2 to stage 1 occurs at a lower wedge height than the jump up, so that there is a hysteresis region in the edgetone operation curve. During the stage 2 operation in the hysteresis region, multiple edgetones are often observed [ 1,2]. Stage 2 seems to be operating simultaneously with stage 1. However, the amplitude of stage 1 is greatly reduced. In addition, its frequency may lie as much as 15 y0 below the usual stage 1 operating curve. Nevertheless, it is usually assumed that this secondary tone is a t This work was conducted while the first author was at Department of the Aerospace and Mechanical Engineering Sciences, University of California, San Diego, La Jolla, California, U.S.A. 19 281
G.
282
R. STEGEN AND K. KARAMCHETI
continuation of stage I operation, and consequently is labeled stage 1 during stage 2. The purpose of this note is to present physical evidence substantiating this contention. 2. EXPERIMENTAL
ARRANGEMENT
The velocity fluctuations present in the edgetone flow field can be characterized by their frequency5 amplitude, and phase angle 0 (measured relative to the fluctuations at the jet exit). Stegen and Karamcheti [3, 41 have previously discussed the measurement of these quantities in an edgetone flow field. Briefly, the amplitude and phase of the velocity fluctuations were measured using linearized constant temperature hot-wire anemometry (Transmetrics Model 6402/4402). An “x”-wire probe was placed on the center-line of the edgetone flow field, and oriented to respond to the longitudinal (u) and transverse (0) velocities of the jet. An electronic sum/difference unit provided a direct readout of u and U.The u signal was bandpass filtered to isolate the frequency of interest. The amplitude of v was then measured with an a.c. voltmeter. To measure the phase of a(x,t) in the laboratory, a reference signal synchronized to the edgetone fluctuations is needed. For this purpose, a second hot-wire probe was held fixed in the flow field at a point where it would not interfere with the “x” probe. The phase of v relative to the reference signal was measured with a phase meter (Ad-Yu Model 505). The “x“ probe could be moved along the flow field center-line with a micrometer drive. In this way, the amplitude and phase of the transverse velocity fluctuations could be measured everywhere of interest. The fixed operating parameters of the edgetone generator were as follows: center-line speed at jet exit U = 9.0 m/set; slit width d = O-163cm; jet breadth b = 4.57 cm; wedge angle CL= 20”. A single wedge height h = 5*58d was chosen for comparison purposes. By increasing h slowly to this value, purely stage 1 operation could be maintained, whereas, by approaching this value from a larger height, multiple edgetone operation could be maintained. 3. EXPERIMENTAL
RESULTS
3.1. PHASE MEASUREMENTS The results of the phase measurements are presented in Figures 2 and 3 as phase angle (8/27r) vs. center-line distance (x//i) from the slit. Comparison of the phase measurements for 2.0 k ? a
. 1,5-
. .
5 I.O0 : g 0,5-
. . .
0
.
.
. .
.
.
. . . ,* ?? , 0.6 0.8 0.2 0.4 Centerline distance (x/h)
.
I.0
Figure 2. Transverse velocity fluctuations; phase variation along center-line of edgetone generator. Comparison of stage 1 and stage 2. Stage 1, . ; stage 2, m. stage 1 and stage 2 (Figure 2) reveals a marked difference between the amplitude and spatial dependence for the two cases. During stage 1 the phase varies non-linearly with distance, in contrast to the linear variation during stage 2. Bauer [2] has previously presented similar results.
MULTIPLE TONE OPERATION OF EDGETONES
283
The phase characteristics of the sub-harmonic, stage 1 during stage 2, are compared with those of stage 1 in Figure 3. The phase characteristics are essentially identical, strong evidence that stage 1 during stage 2 is a continuation of stage 1 operation during stage 2 operation. In Figures 2 and 3 we have only presented a few data points for clarity. It was found that the measurements (37 data points at each condition) could be correlated by a simple power law: i.e. 8/27r = A(x/~)~. In this form, A represents the value of the phase at the wedge, an
Centerline
distance
(x/h
)
Figure 3. Transverse velocity fluctuations, phase variation along center-line of edgetone generator. Comparison of stage 1 and stage 1 during stage 2. Stage 1, ??; stage 1 during stage 2, 0.
important variable in edgetone theory [5, 61. The power law form is useful in deducing the phase speed C(x) of the fluctuations, since a (x)‘-“. Therefore, the exponent B is a measure of the spatial variation of the phase speed of the fluctuations. Table 1 below summarizes the results of this correlation. The differences between stage 1 and stage 2 shown in Figure 2 are clearly evident in the correlation. The phase angle at the wedge in stage 2 is twice that of stage 1. Also, the phase speed during stage 2 (inferred from Table 1) is almost constant, while during stage 1 it varies as xAo”j3.Stage 1 and stage 1 during stage 2 operate at different frequencies, but with identical phase characteristics. TABLE 1
Phase measurements
Stage 1 Stage 2 Stage 1 during stage 2
3.2.
Frequency (Hz)
e/2rr
370 815 316
0*90 (x//2)1*63 l-81 (_x//#.~~ 0.92 (x//z)‘*~~
AMPLITUDE MEASUREMENTS
The amplitude of the transverse velocity fluctuations was measured simultaneously with the phase. The spatial variation of the amplitude is summarized in Figure 4. Stage 1 and stage 2 are once again seen to be quite different in their mode of operation. Stage 1 grows in an approximately exponential fashion. In contrast, stage 2 has an initial value one-eighth that of stage 1 and grows briefly at the same rate as stage 1. Next, stage 2 experiences a rapid growth; and, finally, levels off at a value comparable to the maximum of stage 1.
284
G. R. STEGEN AND K. KARAMCHETI
Examining stage 1 during stage 2, we see that its amplitude is greatly reduced relative to stage 1. Apparently, the presence of stage 2 suppresses the initial amplitude of stage 1 during stage 2. The growth of stage 1 during stage 2 is very similar to the growth of stage 1, further evidence of their common origin. 0.1 .
0.002
0
0.2
0.4 Centerline
0.6 distance
0.0
I.0
(x/h)
Figure 4. Transverse velocity fluctuations, amplitude variation along center-line of edgetone generator. Stage 1, 0 ; stage 2, m; stage 1 during stage 2, 0.
4. CONCLUSIONS The phase characteristics of stage 1 operation and stage 1 during stage 2 operation have been shown to be identical. While the amplitude of the two modes of operation are different, the growth rates are quite similar. In contrast, the phase and growth characteristics of both stage 1 and stage 1 during stage 2 are distinctly different from those of stage 2. In view of the evidence, it is clear that stage 1 during stage 2 is indeed a persistence of stage 1 operation. ACKNOWLEDGMENTS This work was supported by the National Science Foundation at Stanford University and by Project THEMIS at the University of California, San Diego. REFERENCES 1. G. B. BROWN 1937Proc. phys. Sot. Lond. 49,493. The vortex motion causing edge tones. 2. A. BAUER 1963 Dissertation, Stanford University. Edgetone generation. 3. G. R. STEGEN1967 Dissertation, Stanford University. On the structure of an edgetone flow field. 4. G. R. STEGENand K. KARAMCHETI1967 Stanford University SUDAAR 303. On the structure of an edgetone flow field. 5. N. C~RLE 1953 Proc. R. Sot. London A 216,412. The mechanics 6. A. POWELL 1961 J. acoust. Sot. Am. 33,395. On the edgetone.
of edgetones.