An experimental study on the coupling between adjacent Hartmann whistles

Abstract

An experimental study has been conducted to examine interactions between two Hartmann whistles placed in close proximity. Microphone measurements of acoustic emission from the whistles, in addition to high-speed flow field visualizations, were used to assess the physical mechanisms present. A strong coupling mode has been identified that results in frequency locking between the resonators. Observed for only one set of parameters in the experimental matrix, the frequency-locked coupling requires sufficient spacing between the nozzle exit and resonator cup of each whistle to establish the acoustic-driven feedback loop. The visualization results show this coupling to arise due to acoustic excitation from the adjacent resonator, inducing hydrodynamic perturbations that grow sufficiently to alter the flow field around the perturbed resonator. The study indicates that coupling dynamics can play a primary role in dual-whistle resonance and frequency output.

Keywords

Flow-acoustic resonance Hartmann whistle flow visualization experimental aeroacoustics jet resonance jet impingement

Introduction

Flow-driven acoustic resonance occurs in a number of applications, including cases of multiple source fields that may couple under certain circumstances. A common example is that of twin screeching jets, which has been studied theoretically, experimentally, and computationally.^1–4 Feedback occurs through acoustic perturbations which excite vortical flow disturbances that convect downstream and produce new acoustic emissions to continue the cycle. In twin jets exhibiting coupling, the mechanisms can be either through acoustics or hydrodynamic interactions.^5,6 In the present study, flow-driven resonance in a configuration commonly termed the Hartmann whistle, or Hartmann generator, is examined for understanding coupling between a pair of closely spaced jet/resonators. In contrast to the free jet coupling of Tam,¹ Tam et al.,² Seiner et al.,³ Kuo et al.,⁴ Bell et al.,⁵ and Gao et al.,⁶ additional coupling mechanisms are present with the jet/resonator configuration due to the impingement physics.^7,8

The Hartmann air-jet generator (termed the Hartmann whistle to follow) was first discovered and described by Hartmann and Trolle.⁹ The basic principle of operation is as follows: a circular jet of fluid, typically air, exits a nozzle and proceeds to enter a cavity following a short gap. The cavity is most often a round tube closed at the end opposite the jet. An acoustic tone is produced through one of three mechanisms, known as the jet instability mode, the jet screech mode, and the jet regurgitant mode.¹⁰ In the jet regurgitant mode, the fluid periodically fills the resonance tube and then is emptied, producing periodic fluctuations in the gap between nozzle and tube. The tonal sound can reach very high levels of sound pressure level (150+ dB, 20 µPa reference pressure).

Most research on air-jet generators has been focused on characterizing the physical parameters of the nozzle–resonator that give rise to different sound characteristics and deriving empirical relationships between the physical parameters and the resultant sound. Raman and Srinivasan¹⁰ compiled a comprehensive review of the history, development, and characterization of such devices. Studies have often focused on both acoustic measurements and flow visualizations, including schlieren images, to understand the shock structure and how the oscillations of the flow interact with the nozzle–resonator structure.

For the present study, several authors’ work was used to design a dual nozzle and resonator system that could produce a bi-tonal, high sound pressure level signal. In this article, the focus will be on understanding the interactions of the dual nozzle system and the impact on acoustic response. The design of the current system was informed by the literature. In particular, Narayanan et al.¹² and Brun and Boucher¹³ found that chamfer in the resonance tube affected frequency response and increased sound pressure level by several decibels. Raman and Srinivasan¹⁰ summarized several frequency models (section 5.1 of their work) that were used to predict an approximate frequency response given whistle design parameters. Brun and Boucher¹³ described a method for combining several chambered Hartmann nozzle–resonator devices into one ‘multiwhistle’ and creating a horn for the overall system. The dual nozzle–resonator system analysed herein stems from an effort to develop an alternative method of combining two Hartmann generators; it is necessary to understand the flow and acoustic response of this design to assess this approach.

Two interaction mechanisms between adjacent Hartmann whistles are identified from the present study based upon emitted acoustics and high-speed schlieren flow visualization results. A strong coupling mechanism is observed for a single case studied – a case with increased jet/resonator spacing and high pressure. In this case, coupling appears to be due to screech-like acoustic excitation from one whistle to the second whistle, inducing frequency-locked resonance. A much weaker interaction was also observed due to acoustic excitation of the radial fountain jet that exhausts from the edges of the resonator. This particular coupling is apparent when one whistle is in resonance while the second is not resonating.

The remainder of the article is organized as follows: the experimental apparatus and methods are discussed in the next section; ‘Results’ section contains the analysis of key results and findings related to understanding coupling between adjacent Hartmann whistles; ‘Conclusions’ section summarizes the work, provides our conclusions, and includes needs for further research on this topic.

Methods

The experiments are performed using a custom-built apparatus with two Hartmann whistles, as depicted in Figure 1. There are a number of parameters in the setup; those parameters pertaining to a single whistle are well understood from the prior literature. The remaining dimensions are unique to the dual-whistle configuration. Starting with the resonators, D_c is the cavity inlet diameter at the end of the chamfer; L₁ and L₂ are the cavity depth for resonator 1 and 2, respectively; and Z is the spacing between the resonator pair. d_j represents the diameter of the nozzle at the exit, and OD is the outer diameter of the nozzle lip. Lastly, S is the standoff distance between the nozzle outlets and the resonators.

Figure 1.

Schematic of the two-whistle arrangement, along with definitions of geometric parameters, used for the study.

Table 1 provides the key relative dimensions of the apparatus scaled on d_j. To note, the two nozzles are identical, but the resonator depths have been intentionally chosen to be different, producing single-whistle resonances that are 10% different in frequency. This modification allows one to identify when coupling phenomena override the natural single-whistle resonance. To provide a basic notion of scale, the nozzle diameter Reynolds number for the experiments presented is approximately 100,000.

Table 1.

Dimensions of the dual Hartmann whistle.

$\frac{L_{1}}{d_{j}}, \frac{L_{2}}{d_{j}}$	∼0.9
$\frac{D_{c}}{d_{j}}$	1.2
$\frac{S}{d_{j}}$	Variable. Studied for values of 1 and 2
$\frac{Z}{d_{j}}$	2.5

Acoustic measurements and flow visualizations were acquired in separate experiments. Microphone measurements were obtained by placing the apparatus in a small anechoic chamber, while the high-speed schlieren imaging was done in a non-anechoic laboratory. While the results are not presented, microphone measurements were also obtained during schlieren acquisition in the non-anechoic arrangement for qualitative comparisons of the flow-field/tone relationships.

The schlieren imaging setup is depicted in Figure 2. A light source was positioned to focus on a pin hole to generate a point light source. This light expanded onto a collimating lens to form the beam used for illuminating the dual Hartmann whistle. After passing through the region of interest, the light was focused onto an adjustable knife edge by an identical lens. The knife edge enhances the sensitivity of the imaging to small density variations, producing the schlieren effect. High-speed imaging of the schlieren visualization was acquired with a Photron FASTCAM SA1.1. Referring to directions as depicted in Figure 2, the light source and camera were able to slide to adjust the focus of the image, whereas the Hartmann whistle could be moved vertically to adjust the region viewed by the camera.

Figure 2.

Schlieren imaging configuration for the dual Hartmann whistle experiment.

Acoustic measurements were taken in a simple anechoic chamber lined with convoluted acoustic foam, and all fixtures wrapped in foam, to reduce reflections. Pressure measurements were obtained using a 1/4-inch prepolarized pressure field microphone (PCB microphone model 378C10) and associated pre-amplifier and electronics. The microphone was placed at a 23° angle from the dual Hartmann system opposite the nozzle exit, 56 cm away from the resonator. The microphone was digitally sampled at 250 kHz for a duration of 16 s for each case.

The experiments were conducted on the dual Hartmann whistle at various supply pressures, $p_{supply}$ , for two different standoff distances, S. For both the standoff distances, a number of conditions for supply-to-ambient pressure ratio, $p_{supply} / p_{a}$ , were measured, in the range $3.6 \leq p_{supply} / p_{a} \leq 6.2$ . Schlieren data were recorded at frame rates of 180 and 20 kHz for each test condition. All schlieren images were recorded at an exposure time of 1 µs in order to freeze the motion. The 180 kHz rate, which was limited to a resolution of 128 × 128 pixels, was chosen to capture the time evolution of a single-whistle flow field. Alternatively, the 20 kHz rate enabled us to capture the entire schlieren region containing both whistles so that coupling interactions could be studied.

In addition to the dual-whistle measurements, a baseline single-whistle case was acquired by removing resonator 2 (referring to the label in Figure 1). This case was conducted only at the optimal resonance pressure of $p_{supply} / p_{a} = 4.7$ and a standoff distance of 1 d_j.

Quantitative statistical analyses have been performed for both the microphone measurements and the schlieren images.

The acoustic results were processed to obtain narrowband power spectral densities. Welch’s fast Fourier transform-based spectral estimation method^14,15 was used with 80 ms blocks, Hanning windows, and 50% overlap between blocks. This processing yields a 95% statistical confidence in each power spectral density value of approximately $\pm 1.3$ dB.

Single-point and two-point statistical analyses have been performed on the intensity signals from the schlieren images. It should be noted that the absolute magnitudes of pixel intensities have not been calibrated to provide local flow measurements; however, it is well known that the schlieren signal is proportional to the density gradients in the flow.¹⁶ Instead, the relative distributions and fluctuations of intensity are of most interest. The mean and standard deviation of the images were computed for cases presented using the 20 kHz imaging. Nine hundred and ninety-nine images were used for the analyses, resulting in typical statistics errors at 95% confidence of $\pm 2 %$ of the full-scale in the mean, and $\pm 5 %$ in the standard deviation. Single-point (i.e. single-pixel) spectral analysis of the intensity fluctuations was conducted on the 180 kHz imaging using 9999 images. Again, using Welch’s method, these images were divided into eight blocks and windowed using a Hamming window and 50% overlap to produce the spectra presented in the ‘Results’ section. Each narrowband spectral estimate has a relative local error of $\pm 70 %$ at 95% confidence, though this value is deceptively large given the dynamic range of the spectra. Finally, two-point correlations were performed across the field of the 20 kHz imaging using 999 images and normalized using the local variance values to produce correlation coefficient maps. Due to slowly varying trends in the measured lighting intensity, the pixel time signals were de-trended by subtracting a 22-sample sliding mean from each time sample. This was observed to eliminate positive correlations that extended across large regions of the frame, while accentuating active flow-field and acoustic field correlations. For correlations $ρ (x, y) > 0.25$ , the 95% confidence statistical error is less than $\pm 25 %$ .

In addition to the statistical errors expected for the measurements, other experimental uncertainties are present. The positioning of the Hartmann apparatus was done using a micrometer stage with a Vernier dial, allowing resolution of approximately ±25 µm for repeatability of the positioning. Pressures were set to $δ p_{supply} / p_{a} = \pm 0.15$ . Some variability in the schlieren imaging was present due to the sensitivity of the images to repeatability in knife-edge positioning. Given the large magnitudes in the dynamics being identified for the present studies, firm conclusions may be obtained despite the presence of these uncertainties.

Results

Acoustic measurements and high-speed flow visualizations were obtained for two adjacent Hartmann whistles operating simultaneously at the same supply pressures. As noted prior, the adjacent resonators are slightly different in depth – thus, yielding different resonant frequencies in the single-resonator configuration – in order to more easily identify cases when coupling occurs. In the discussions to follow, these results are analysed to understand some of the mechanisms through which the acoustics and flow fields couple between the two whistles. The acoustics results are first discussed to benchmark the global behaviour of the system, particularly as a function of supply pressure. Significant variation is present in tones generated, so the details of the resonator flow-field are examined via quantitative analysis of high-speed schlieren flow visualization results.

Acoustics

The acoustics behaviour of single Hartmann whistles is well known, as is the physics leading to the intense tone generation of these devices. The baseline whistle configuration was specified using these past works, and single-whistle operation was verified for one of the two whistles by removing the adjacent resonator. An example of the spectrum for the lower frequency resonator (labelled resonator 1 in Figure 1) operating at a pressure ratio, $\frac{p_{supply}}{p_{a}} = 4.7$ is provided in Figure 3. The frequencies in Figure 3, along with all to follow, have been normalized using the nozzle exit diameter and the nozzle exit velocity computed using isentropic relations given the pressure ratio and supply temperature. The spectrum shows that strong resonance is achieved at this supply pressure and configuration. As expected given the non-sinusoidal waveform of the flow field unsteadiness in the resonator cavity, harmonics of the primary tone are observed, consistent with many past studies on Hartmann whistles.

Figure 3.

Narrowband acoustic spectrum for resonator 1 at the condition $p_{supply} / p_{a} = 4.7$ .

Table 2 provides information regarding the tones observed for a number of cases. The single-resonator case from Figure 3 is listed, in addition to eight others with adjacent whistles operating simultaneously. The narrowband acoustic spectra for cases 4 and 6 are provided in Figure 4 for comparison of two different supply pressure cases. There are two clearly identifiable primary tones in the dual-resonance case of Figure 4(a), with the frequency $f_{1}$ corresponding closely to the single-resonator frequency reported in Table 2. Perhaps not surprisingly, visualizations discussed to follow indicate that resonator 1 has produced this similar value of $f_{1}$ at the same supply pressure. In Figure 4(b), the spectrum for case 6, both whistles are in resonance, although it was observed that resonator 1 would chaotically become disorganized and go out of resonance. The visualizations of case 6 discussed in the following section concentrate upon a range of time when resonator 1 is not in the resonant regurgitating mode. A third spectrum is plotted for case 8, which differs from all other cases in that the nozzle–resonator gap $S$ has been doubled to twice the nozzle exit diameter. This case exhibits only a single primary tone, but at a much lower frequency. As will be shown from flow visualization, this is the only case with clear coupling of the regurgitant mode across the resonators. To follow, analyses of the flow visualizations will be presented to shed light on the observations from the acoustics.

Table 2.

Summary of acoustic results.

Case	Pressure ratio	$\frac{S}{d_{j}}$	$f_{1} \frac{d_{j}}{V}$	$f_{1}, Peak d B_{SPL}$	$f_{2} \frac{d_{j}}{V}$	$f_{2}, Peak d B_{SPL}$
Single resonator	4.7	1	0.103	111	Single resonator	Single resonator
1	3.6	1	0.111	80	N/A	N/A
2	4.0	1	0.107	100	0.112	107
3	4.3	1	0.106	105	0.114	107
4	4.7	1	0.106	101	0.118	115
5	5.1	1	0.108	95	0.1243	113
6	5.4	1	0.108	101	0.1337	104
7	6.2	1	N/A	N/A	N/A	N/A
8	6.2	2	0.076	114	N/A	N/A

Figure 4.

Narrowband acoustic spectra measured for (a) case 4, (b) case 6, and (c) case 8.

Flow visualization

Flow visualizations were obtained using high-speed schlieren imaging of the compressible flow fields around the whistles. While interpreting the results, it is important to recall that schlieren imaging is proportional to the line-of-sight integrated density gradient of the flow field.¹⁶ Given the global nature of flow-field instabilities being studied, in addition to the radial configuration of the fountain jet, the line-of-sight results are relatively easily interpreted. All images were acquired at the same exposure setting of 1 µs. Two acquisition modes were used to balance resolution and frame rate demands considering the camera capabilities. High resolution images, showing both whistles, were taken with 20 kHz imaging, while a narrow region around resonator 1 was obtained at 180 kHz. Despite time-domain aliasing in the 20 kHz case, the two-point correlation fields to be presented provide proper evaluations of this statistics given the short exposure time used.

Sample instantaneous and mean flow-field visualizations for the cases discussed above, i.e. single resonator and cases 4, 6, and 8, are provided in Figure 5. Several instantaneous and mean flow features are apparent from these images, labelled for clarity in Figure 6; most of these features are present in the single-resonator case that is well known in the literature. An underexpanded jet issues towards the resonator cup, with clear expansion waves present in the mean flow visualization images for each case. There is a radially ejected, shock-containing fountain jet along the lip of the resonator (labelled I in Figure 6). An impingement shock is visible upstream of the fountain jet (II in Figure 6). Unique to the dual Hartmann whistle configuration, an unsteady interaction region is visible between the whistles (III in Figure 6). Blurring in the mean images is a product of flow field unsteadiness, which will be further examined through data analyses to follow. Also visible in the instantaneous images are intense coherent acoustic waves. These appear as arcs centred on one of the resonators (good examples visible in Figure 5(a) and (d) on the right side of the instantaneous images).

Figure 5.

Instantaneous (top row) and mean (lower row) schlieren images for the single-resonator case (a) along with cases 4 (b), 6 (c), and 8 (d).

Figure 6.

Diagram of key features in the dual Hartmann whistle mean flow field. I: Shock-containing fountain jet; II: impingement normal shock; III: fountain jet interaction region.

The unsteadiness of the schlieren images may be statistically examined in much the same way as commonly done for quantitative physical measurements. In this particular case, our objective is to link large-scale unsteady features to acoustic tone observations. To first establish the relevance of such statistical analyses, consider the single-resonator case. In Figure 7, a sample sequence of instantaneous images acquired at 180 kHz and covering one resonance cycle is shown. For statistical understanding of the behaviours, the schlieren image standard deviation distribution (Figure 7(b)) and power spectral density of a point (Figure 7(c)) in the jet/resonator visualization is presented. Figure 7(c) is analogous to the acoustic spectrum of Figure 3 for the same case, indicating the direct relationship between hydrodynamic unsteadiness in the nozzle–resonator gap and the emitted acoustics. The unsteadiness present is indicative of the regurgitant mode observed many times in the literature; this mode results in the 0.103 $V / d_{j}$ primary resonant frequency (see Figure 7(c)) that has already been reported in Table 1. The standard deviation field (Figure 7(b)) indicates that fluctuations are most intense in in the jet/resonator zone, but also that the radial fountain jet spreads rapidly due to unsteadiness.

Figure 7.

Analysis of the unsteady data from the single-resonator case. (a) Time series of a typical resonant cycle (increasing time frames, upper left to lower right); (b) normalized standard deviations, $σ / {[σ]}_{\max}$ of the visualization intensities; (c) power spectral density at the point indicated by the ‘x’ symbol in (b).

Standard deviation and power spectral density plots for cases 4, 6, and 8 are provided in Figure 8. In case 4, we can verify from the spectral analysis that the same whistle as the single-resonator case has resulted in nearly the same resonance frequency and similar behaviour in the regurgitant mode. It was mentioned earlier that case 6 exhibited intermittent resonance of resonator 1 during the experiments. The results in Figure 8(b), and further analysed to follow, depict a period over which only resonator 2 is in resonance. Acoustics results obtained in the non-anechoic arrangement for the flow visualizations (not shown) confirm that the tones emitted lacked the resonator 1 tone. The schlieren standard deviation distribution for case 6 also indicates that only resonator 2 is in regurgitant-mode resonance; however, both whistles exhibit unsteadiness of the fountain jets. Note that the smaller, less distinct peak in the spectrum of Figure 8(b), right, arises from the acoustics of resonator 2 propagating through the depth of the schlieren image. This peak is also visible in the acoustic spectrum of Figure 4(b). The variance field for case 8 (Figure 8(c)) indicates that both whistles have large amplitude unsteadiness in the nozzle–resonator gap, while producing only a single acoustic tone. Cases 6 and 8, in particular, need further analysis to better understand the mechanisms leading to these observations.

Figure 8.

Normalized standard deviation, $σ / {[σ]}_{\max}$ (left), and power spectral density (right) at the points indicated by the ‘x’ symbol in the standard deviation plots for case 4 (a), case 6 (b), and case 8 (c).

The question of interaction between resonators may be addressed from two-point cross-correlation analysis of the visualization data. High correlation across regions in the visualization can indicate coupling or cause/effect relationships. For instance, significant correlation coefficients across the two whistles would indicate that they are operating at the same frequency, and thus, coupled. (Note that since the resonators are slightly different depths, the single-resonator frequencies have been intentionally made different to better identify this coupling.) Additionally, by selecting the auto-correlation point in the acoustic field and obtaining cross-correlations elsewhere, including within the whistle flow fields, the source for these acoustic waves may be identified.

Two-point correlations in acoustics-dominated regions

Two-point cross-correlations of selected points dominated by acoustics are considered to establish a baseline understanding of cause/effect relationships between the hydrodynamic field unsteadiness and the acoustic field. Consider first the single-resonator case, with the auto-correlation point placed within the acoustic field (Figure 9). As expected, significant correlations between the acoustic waves, which exhibit a circular pattern centred on the whistle, and the jet/resonator flow field are present due to the regurgitant unsteadiness in the resonator cup acting as the acoustic source.

Figure 9.

Two-point correlation coefficient centred at a point indicated by the ‘x’ symbol in the acoustics-dominated region for the single-whistle case.

Correlations in regions dominated by acoustics are provided in Figure 10 for cases 4, 6, and 8. In case 4 (Figure 10(a)), which exhibits two tones, the acoustics correlate with both jets, though the dual tone operation and difference in peak tone magnitudes result in much less clarity in the interpretation than that seen for the single whistle. In case 6, which exhibited a single tone in the flow visualizations, radiation appears to have an origin in only one whistle, resonator 2 (Figure 10(b), see the bands of positive and negative correlations near resonator 2). The regurgitant mode of unsteadiness was not present in resonator 1, further confirmed by the cross-correlation analysis. In the interesting case 8, which exhibits a single tone but regurgitant mode unsteadiness in both resonators, the acoustics are correlated with both whistles, though the origin of visualized acoustic waves appears to centre on resonator 1 (Figure 10(c)). The latter observation is most likely due to experimental setup-related variations in the acoustic efficiency between the two whistles.

Figure 10.

Two-point correlation coefficient centred at a point indicated by the ‘x’ symbol in the acoustics-dominated region for (a) case 4, (b) case 6, and (c) case 8. The left column is centred on a point between the whistles and the right column is centred below the whistles.

Two-point correlations in hydrodynamic regions

Selected points in the hydrodynamics-dominated regions labelled in Figure 6 have been chosen for further examination via two-point correlation analyses (Figure 11). To follow, the results and associated physical implications for cases 4, 6, and 8 are discussed.

Figure 11.

Two-point correlation coefficient centred at a point indicated by the ‘x’ symbol in hydrodynamics-dominated regions for (a) case 4, (b) case 6, and (c) case 8. The left column is centred on a point in region I, middle column in region II, and right column in region III according to the labels in Figure 6.

Case 4

Recall from Figure 4(a) that case 4 exhibits two distinct tones in the far-field acoustics. This observation may indicate relatively independent operation of the two resonators, if this tone pair is due to linear addition of two uncorrelated sources. The correlation maps in Figure 11(a) support this interpretation. The correlation maps centred in the nozzle–resonator gap of both resonators 1 and 2 correlate with structures at acoustic wavelengths outside the hydrodynamic regions. Of particular note is the lack of correlation between the two whistle flow fields (Figure 11(a), maps in the centre). These results indicate that the parameters in case 4 have led to minimal coupling via any mechanism, hydrodynamic or acoustic.

The interaction region (III) exhibits correlations with both resonator flow fields, but these correlations are greater with resonator 1. The visualization variance shown in Figure 8(a) indicates that resonator 1 fluctuations were greater than resonator 2, likely owing to a slight imbalance of the regurgitant unsteadiness across the two resonators. The significant correlations between region (III) and the acoustics field seem to be due to consistency of time-scales with resonator regurgitant unsteadiness driving the large-scale hydrodynamic fluctuations in this region (resonator 1, in particular).

Case 6

Case 6 correlation maps are provided in Figure 11(b). The correlation field in region (II) of resonator 2 is similar to those seen for case 4. Although not shown, this same structure is not present in region (II) of resonator 1 the regurgitant unsteadiness is not present. Examining the fountain jet region (I) of resonator 1, some weak coupling with resonator 2 is detected. The compact cells of large correlation magnitudes are consistent with screech-like dynamics (Figure 11(b), left). Rather than a self-excited feedback loop as present in screech, however, the intense acoustics radiated by resonator 2 appears to excite flapping in the shock-containing fountain jet of resonator 1. The frequency spectrum of schlieren intensities in this region showed a tone at the same frequency as the regurgitant unsteadiness. This flapping will contribute to the acoustics radiated, but given the relative efficiency of this mechanism compared to the regurgitant whistle, it is not expected to make a significant contribution. The interaction zone is correlated with the regurgitant structures from resonator 2 and weakly with the acoustic field.

Case 8

Case 8 clearly exhibits that the regurgitation at the resonator cup is coupled across the two whistles (Figure 11(c), centre column). It is interesting to note that the correlations across the two jet/resonator fields are not at the analogous locations, rather shifted axially, providing a hint about the coupling mechanism. As with the other cases, the hydrodynamic interaction region (III) exhibits some correlation with the acoustic regions, along with significant correlations with both resonators.

To examine the coupling details further, consider a few instantaneous images of the mean-subtracted schlieren intensity fluctuations (Figure 12). The configuration difference between this case and all others is the standoff distance between the nozzle exit and the resonator cup. Increasing this distance provides more time and space for convective instabilities to grow before interacting with the resonator, driving larger perturbations in the jet. The visualizations show that intense acoustic waves from resonator 1 strike the jet of resonator 2 near the nozzle exit. Given the intensity of these pressure waves, it may be safely assumed that these acoustics create large perturbations in the jet that, given sufficient time and space and being in a receptive wavenumber for instability, will grow as the perturbation convects downstream. Indeed, evidence of this is present in the images in the left column of Figure 12. The acoustic mechanism of coupling will have a slight time delay due to the propagation time of the acoustic wave, resulting in the phase shift in correlation seen in Figure 11(c). In this case, the acoustic feedback mechanisms, similar to screech and closely related to the mechanisms shown by Mitchell et al.⁸ for impinging supersonic jets, have resulted in forcing of the regurgitant hydrodynamic instability. Although not shown, it was also observed that the intense acoustic waves from resonator 1 reflected off the lip of the nozzle for this whistle, traveling back downstream and exciting a large-scale perturbation in the jet. As this mechanism has a time-scale related to the distance between $S / c_{a}$ , where $c_{a}$ is the ambient speed of sound, greater than that of the self-excited regurgitant mode present in the single-whistle and non-interacting dual whistles, the frequency of this coupling is considerably different, 30–40% lower than all other observed frequencies.

Figure 12.

Examples of intense acoustic waves from the lower whistle interacting with the upper whistle.

Conclusions

An experiment was conducted to study the acoustic output and interactions between two adjacent Hartmann whistles placed in close proximity. A microphone was used to capture acoustic pressure, while high-speed schlieren visualization was processed to obtain quantitative information on the space-time structure of the flow and acoustics fields.

Two cross-resonator interactions have been observed in this work, one a strong coupling mechanism with major consequences on acoustic radiation and the second with only minor impact. Frequency-locked coupling between the two whistles was observed for a greater standoff distance between nozzle exit and resonator. Large correlations of time-resolved schlieren visualizations resulted between the two whistles in the nozzle–resonator gap flow region. Further, this region correlated with acoustic waves that intersect the flow field of the opposite resonator, exciting perturbations in the jet at the system resonance frequency. The second, less dramatic, interaction occurs when one whistle is in resonance and a second is steadier. In this case, intense acoustic waves propagate from the resonating whistle to the steadier whistle and excite the radial fountain jet on the steadier whistle. The visualizations show that the cross-section motion of the shock-containing radial jet is reminiscent of an axial screeching jet, but the motions are excited from the other whistle, i.e. not self-excited as in free-jet screech. It is worth noting that both these interactions observed are acoustics-excited interactions that alter the unsteady hydrodynamic fields. Despite the close proximity of the two resonators, no hydrodynamic/hydrodynamic interactions were identified that led to resonance.

Further work is needed to fully describe the regimes of two-whistle coupling. The present work represents a limited range of geometric parameters, and additional mechanisms, such as hydrodynamic/hydrodynamic coupling, could be present for some cases. In particular, studies involving the variation of the whistle spacing and nozzle–resonator spacing, and their combinations, would give the insights needed to explain the interactions that occur along with their impacts on whistle acoustic output.

Footnotes

Acknowledgements

The authors gratefully acknowledge Mr John Gillespie, propulsion research engineer at Virginia Tech, for his contributions to the experimental hardware design.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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