Conditional sampling analysis of high-speed schlieren movies of Mach wave radiation in a supersonic jet

For the establishment of an effective and efficient method to reduce aircraft noise or acoustic loads during the lift-off of a launch vehicle, an adequate understanding of the jet-noise generation mechanism is needed. For this reason, schlieren visualization has been used to observe the density gradient fluctuations of a jet flow and acoustic waves from the jet.¹ Performance improvements in high-speed cameras enable us to record longer visualization movies with higher frame rates and higher spatial resolution. However, the visualization movies usually contain many complicated gray-value fluctuations that are related to the turbulent jet flow and acoustic waves from the jet. This makes it difficult to obtain an adequate understanding of the noise generation mechanism from the visualization movies alone. Thus, to improve our understanding, an analysis method must be used to extract the fluctuations correlated to the acoustic waves from the complicated schlieren visualization movies.

Most of the noise from high-speed jets is broadband except some cases when, for example, a feedback loop is formed. Therefore, the visualized fluctuations are usually the mixture of multiple frequency components. This motivates us to decompose a visualization movie into each frequency component by using, e.g., the Fourier transform. However, as many studies^2–6 have reported, the broadband jet noise is intermittent, and the sound-pressure signals of microphones consist of many short-period, large-amplitude components. Because the Fourier transform decomposes the signal into infinitely extended Fourier modes, the details of such temporally localized events may be lost.⁵ Thus, an analysis method is required to treat this intermittency.

The objective of this study is to show that a conditional sampling analysis is useful in extracting the target fluctuations from high-speed schlieren movies. As an example of broadband noise, the Mach waves¹ from a supersonic jet are the focus. The Mach waves are generated due to the supersonic convection (relative to the surrounding atmosphere) of the wavepackets at the jet boundary, which are intermittent, large-scale coherent structures and are regarded as a Kelvin–Helmholtz instability wave.⁷ The high-speed schlieren visualization movies of a supersonic jet are recorded simultaneously with a sound-pressure measurement using a microphone in the propagation direction of the Mach waves. Then, the gray-value fluctuations correlated to the Mach waves are extracted from these movies by using conditional sampling analysis with intermittent triggers detected from the signal of the microphone. The extracted results are compared with the known characteristics of Mach waves.

Conditional sampling analysis is a commonly used method in studies on turbulent flow.⁸ This method extracts fluctuations correlated with a trigger event that occurs repeatedly but not always periodically. In jet noise research, this method has been used by previous authors, e.g., Moore⁹ and Guj et al.,¹⁰ to detect the trigger events from the raw signals of microphones. To detect the intermittent trigger events, Hileman et al.,² introduced the wavelet transform, which decomposes a signal into nonstationary multiple-frequency components. They revealed that the signal of the microphone consisted of many intermittent, short-period, large-amplitude components, and these were used as the trigger events.

However, Hileman et al.² used a real-valued wavelet function. Kœnig et al.⁵ suggested that the real-valued wavelet function tended to break a single event into multiple peaks and therefore, a complex-valued wavelet function (Morlet or Paul wavelets) was more appropriate in detecting the intermittent components. Because the Morlet wavelet was likely to have a better frequency resolution than the Paul wavelet, Akamine et al.^11,12 utilized it to detect the trigger events. In these studies, the authors applied this method to high-speed schlieren visualization movies of supersonic jets impinging on an inclined flat plate, and successfully extracted fluctuations correlated to the intermittent trigger events.

Figure 1 shows an overview of the conditional sampling analysis in this paper. The input data are a pair of high-speed schlieren movies and the sound-pressure signal of the microphone, which are simultaneously measured. First, from the sound-pressure signal $p(t)$ [Fig. 1(a)], the time variation of the amplitude at each frequency $|W̃|(t,f)$ is obtained by using a wavelet transform [Fig. 1(b)]. In this map, the trigger events are defined as the local maxima at a target frequency $ftarget$⁠. From the time of these local maxima $t̂k (k=1, …, K)$⁠, the trigger time $t̂k′$ is determined by checking the phase values of the wavelet coefficient [Fig. 1(c)]. Subsequently, the gray values at $t̂k′+τ$ (where $τ$ is the time lag) are sampled from the schlieren movie and averaged as follows [Figs. 1(d) and 1(e)]:

where $gij(t)$ is the raw gray value at pixel $(i,j)$⁠, and $g¯ij$ is its time average. $g̃ij(τ)$ is the conditional average of the gray values at pixel $(i,j)$ at $τ$ before the trigger times.

This section supplements the explanation of the trigger detection using a wavelet transform [Figs. 1(b) and 1(c)]. According to Torrence and Compo,¹³ a continuous wavelet transform is formulated for the discrete and finite-length signal as

where $f$ is the frequency, $tn=nδt (n=0, …,N−1)$ is the discrete time with an interval $δt$⁠, and $ψ*(t)$ indicates the complex conjugate of the wavelet function. In this study, the wavelet function was the Morlet wavelet,

where $ω0=6$⁠. In Eq. (2), $s$ represents the scale factor, which modulates the frequency of the wavelet function. For the Morlet wavelet, the scale factor has a relation with the frequency¹³ 

The absolute value $| W(tn, f) |$ indicates how the sound-pressure signal around $tn$ resembles the Morlet wavelet at frequency $f$⁠. Note that if the signal $p$ includes the same amplitude components at different frequencies, the value $| W |$ of the lower-frequency component becomes lower owing to the normalization of the wavelet functions [e.g., $| W |≈s/δt$ if $p(tn)=cos 2πftn$]. Therefore, in this study the following value was used to compare the amplitudes of the different frequency components and detect the local maxima:

The local maxima in the map of $| W̃ |$ were found by a maximum filter, which returns the maximum value in the neighborhood for each point $(tn, f)$⁠. If the filtered value is identical to the original value, that point is a local maximum. In this detection, the window size of the maximum filter is important. Through trial and error, the window size was determined to be 0.125 ms on the time axis and 10 kHz on the frequency axis. This setting reduced the number of duplicated or failed detections.

The final step of the trigger detection, i.e., the phase adjustment, is the key to successful extraction. The phase values at the local maxima $t̂k$ differ from each other. This can be observed in Fig. 1(c), which plots the phase values $ϕ(tn)=tan−1{ Im W(tn, ftarget) / Re W(tn, ftarget) }$⁠. This phase difference leads to the cancellation of the fluctuations that correlate to the trigger events by averaging. Therefore, the trigger time is defined as time $t̂k′$ in Fig. 1(c), where $ϕ(t̂k′)$ is equal to an arbitrary reference phase value $ϕref$ (⁠$ϕref=−π$ in this study). This is calculated as follows:

because around the local maxima, $ϕ(tn)$ changes linearly, and its time derivative is almost equal to $2πftarget$⁠.

The present experiment was carried out at the hypersonic/high-enthalpy wind tunnel at the Kashiwa campus of the University of Tokyo. An unheated Mach 1.8 ideally expanded jet was generated with a stagnation pressure of $0.575±0.01$ MPa and a stagnation temperature of 291–305 K. The Reynolds number was approximately $1.5×106$ for the characteristic length of the nozzle-exit diameter $D=20$ mm. Details of the facility and nozzle were described by Akamine et al.^12,14

Figure 1(f) shows an overview of the experimental setup. Three high-speed schlieren visualization movies were recorded separately to cover the region in or around the supersonic jet from the nozzle exit to approximately $22D$ downstream. These movies were recorded using an optical system consisting of a mercury lamp, two concave mirrors (200 mm in diameter and 2 m in focal length), knife edge, camera lens (AF 70-300 mm f4-5.6 manufactured by Tamron, Saitama, Japan; the focal length was set to approximately 110 mm), and high-speed camera (FASTCAM SA-Z, Photron, Tokyo, Japan). The knife edge was perpendicular to the jet axis. The frame rate was 100 000 fps, the exposure time was approximately 0.347 μs, the resolution was 512 × 328 pixels, and the movie length was approximately 1 s.

During each schlieren movie, the sound-pressure signal was measured to detect the intermittent components of the Mach waves as the trigger events. A 1/4 in. free-field microphone (Type 4939, Brüel & Kjær, Nærum, Denmark) was set at $r/D=55$ and $θ=30°$⁠. This location was included in the overall sound pressure level (SPL) peak directivity range in Fig. 2(b) by Gee et al.,¹⁵ which was measured using the same facility and nozzle under the same jet condition as the present study. The signals were recorded at a sampling frequency of 400 kHz using the PCI-6133 data acquisition module and LabVIEW (National Instruments, Austin, TX).

In addition to the microphone location, the target frequency $ftarget$ is the other important parameter in the present analysis. Because the Mach waves are broadband noise, there are many candidate frequencies where the SPLs are high. At the present microphone location, the SPL spectrum has a broadband peak centered at approximately 6 kHz (Strouhal number $St=fD/Uj≈0.25$⁠). As a preliminary test, the analysis was performed at several target frequencies from 6 to 16 kHz. Then, it was found that higher frequency (i.e., shorter wavelength) components were better examples to describe the whole behavior of extracted fluctuations in the limited field of view. Therefore, this paper only shows the results at $ftarget=16±1$ kHz (⁠$St≈0.67$⁠) and the other frequencies will be examined in a future work. Note that the number of samples for averaging was 300.

The angle of arrival for the trigger events is not considered in this analysis. Therefore, there is no guarantee that the trigger events originate from the same source. If there are multiple sources, they would also be extracted. However, this is not a problem for the present analysis, because the Mach waves are dominant in the downstream of the supersonic jet.

Figure 2(a) shows an example frame of the raw schlieren movies, and Figs. 2(b)–2(h) are images extracted by a conditional sampling analysis at time lag $τ=−2.9, …,−2.3$ ms. There are many gray-value fluctuations in Fig. 2(a), whereas a wavelike structure can be observed clearly in Figs. 2(b)–2(h). This structure appears near the nozzle exit first [Fig. 2(b)], and then propagates in a downstream direction. The gray boxes in Fig. 2 represent the visualization areas at different runs. In Figs. 2(c) and 2(h), a wavelike structure consistently appears across the two adjacent boxes. This suggests that the same phenomenon was extracted from the schlieren movies recorded separately during different runs.

To confirm that the present analysis captures the Mach wave radiation as expected, the characteristics of the extracted wavelike structure are compared with the following known features of the Mach waves: (1) The Mach waves are generated due to the convection of the wavepacket of the jet.^1,7 (2) For the Mach 1.8 jet at $St=0.7$⁠, the convection speed is $Uc=0.77Uj$ at $x/D=3,z/D=0.5$ and $Uc=0.80Uj$ at $x/D=12, z/D=0$ (⁠$Uj$ is the jet velocity at the nozzle exit), according to the Rayleigh scattering measurement by Panda.¹⁶ (3) The inclination angle of the Mach waves is the Mach angle $sin−1a/Uc$ corresponding to the convection speed $Uc$⁠, where $a$ is the speed of sound in the surrounding atmosphere.¹ (4) The Mach waves propagates to the far-field by the speed of sound $a$⁠.

The dashed lines in Fig. 2 show the approximate location of the jet boundary. The angle of the jet boundary was determined as approximately $3.5°$ to the jet centerline axis from Fig. 2(a). In Figs. 2(b)–2(e), the wavelike structure appears between the dashed lines, where the jet flow exists. This corresponds to the wavepacket of the jet. To determine the convection speed of the wavepacket, the three largest gray-value peaks (W and the other two on the left side of W in Fig. 3) were tracked on a line at a constant $z$ (⁠$z/D=0.1, 0.2, 0.3,$ and $0.4$⁠). The peaks move from $x/D≈4$ to $x/D≈12$ at a nearly constant convection speed of approximately 373–385 m/s (⁠$0.78−0.80Uj$⁠, where $Uj=481$ m/s). This convection speed agrees with that obtained by Panda.¹⁶ Thus, features (1) and (2) are confirmed.

The Mach angle corresponding to $Uc=0.80Uj$ is $sin−1a/Uc≈63°$⁠. Figure 3 shows that the inclination angles of the wavefronts outside of the jet are almost equal to this Mach angle. Moreover, this part propagates over a distance almost equal to $aΔτ$ between these two frames with an interval of $Δτ=0.2$ ms (⁠$τ=−2.9$ ms and $τ=−2.7$ ms), i.e., its propagation speed almost agrees with the speed of sound. These facts indicate that these wavelike structures can be regarded as Mach waves and their sources, i.e., the wavepacket in the jet. Thus, feature (3) is confirmed.

The final focus is the time lag between the trigger time and the appearance time of the wavelike structure. The wavelike structure appears around $x/D=5,z/D=1$ [indicated by the dotted lines in Fig. 2(b)] at $τ=−2.9$ ms, and then propagates in a direction close to the trigger-detecting microphone (⁠$r/D=55, θ=30°$⁠). The distance between this location and the microphone is 1.0 m. It takes approximately 2.9 ms to propagate over this distance at the speed of sound (343 m/s). Therefore, the wavelike structure reaches the microphone at $τ≈0$ ms, i.e., the trigger time. This indicates that the wavelike structure outside of the jet corresponds to the trigger event, i.e., the intermittent component of the Mach waves. This corresponds to feature (4). Thus, the present analysis properly captures the correlation between the wavepacket in the jet, the near-field Mach waves, and the far-field intermittent events.

Note that the wavelike structure only appears on the upper side of the jet in Fig. 2. In this experiment, the trigger microphone was set on the upper side of the jet. Therefore, this suggests that the Mach waves at the lower side of the jet have a low correlation to the trigger events detected on the upper side of the jet. This is consistent with the azimuthal pressure correlation from the simulation data of the Mach 1.47 jet obtained by Vold et al.,¹⁷ where the correlation coefficient was below 0.4 when the azimuthal spacing exceeded $90°$⁠.

In this paper, a conditional sampling analysis successfully extracted the Mach waves and their sources (the large-scale turbulent structures) from high-speed schlieren visualization movies by detecting the intermittent components of the acoustic signals measured with a microphone in the propagating region of the Mach waves. This indicated that the conditional sampling analysis is useful in two ways. One was that this analysis extracted the noise-related fluctuations from the complicated data of the flow or acoustic near-field. The other was that this analysis overcame the limitation of the visualization region; the same phenomena were extracted from movies recorded separately, as long as the trigger conditions were the same.

This analysis is potentially applicable not only to Mach waves but also other broadband turbulent noises, which may be intermittent similar to Mach waves. In addition, this analysis is applicable not only to schlieren movies but also any signals such as pressure and velocity. This method will lead to better observations of noise generation processes and thus contribute to a deeper understanding of their mechanisms.

Future work will include discussions regarding the effects of the target frequency or mother wavelets on the results, detailed comparisons with the results of other analysis methods (e.g., the Fourier decomposition), and more quantitative discussions such as a comparison between the extracted results and acoustic intensity vectors. Gee et al.¹⁵ measured acoustic intensity vectors in the same facility under the same conditions as the present study, and discussed the characteristics of the Mach waves such as the location of the source and propagating regions and their frequency dependency. The acoustic intensity vectors were calculated from the sound-pressure signal measured with microphones. Therefore, the extracted results will be more quantitatively examined by investigating their consistency with the acoustic intensity vectors.

A portion of this study was supported by a Grant-in-Aid for JSPS Research Fellows (Grant No. 16J08627) and for JSPS Scientific Research (B) (Grant No. JP18H01621). The authors would like to thank Kazuya Fukatsu, Yuya Sekiguchi, and Yuta Kondo of the University of Tokyo and George Kuwabara of Photron Limited.