Experimental study of turbulent-jet wave packets and their acoustic efficiency

This paper details the statistical and time-resolved analysis of the relationship between the near-field pressure fluctuations of unforced, subsonic free jets ($0.4\le M\le 0.6$) and their far-field sound emissions. Near-field and far-field microphone measurements were taken on a conical array close to the jets and an azimuthal ring at ${20}^{\circ }$ to the jet axis, respectively. Recent velocity and pressure measurements indicate the presence of linear wave packets in the near field by closely matching predictions from the linear homogenous parabolized stability equations, but the agreement breaks down both beyond the end of the potential core and when considering higher order statistical moments, such as the two-point coherence. Proper orthogonal decomposition (POD), interpreted in terms of inhomogeneous linear models using the resolvent framework allows us to understand these discrepancies. A new technique is developed for projecting time-domain pressure measurements onto a statistically obtained POD basis, yielding the time-resolved activity of each POD mode and its correlation with the far field. A single POD mode, interpreted as an optimal high-gain structure that arises due to turbulent forcing, captures the salient near-field–far-field correlation signature; further, the signatures of the next two modes, understood as suboptimally forced structures, suggest that these POD modes represent higher order, acoustically important near-field behavior. An existing Green's-function-based technique is used to make far-field predictions, and results are interpreted in terms of POD/resolvent modes, indicating the acoustic importance of this higher order behavior. The technique is extended to provide time-domain far-field predictions.

Figure 1

Comparison between linear PSE predictions (——) and measured $u^{\prime }$ ($\square$) for the axisymmetric mode on the jet centreline for $M=0.4$ and $0.4\le \text{St}\le 0.6$. Measured and PSE data from Cavalieri et al. [12]. (a) St = 0.4, (b) St = 0.5, and (c) St = 0.6.

Figure 2

Comparison between linear PSE predictions and measured $u^{\prime }$ for $M=0.4$ at three axial stations. $——$, PSE; $–\; –\; –$, experimental $u^{\prime }$; $......$, axisymmetric mode of $u^{\prime }$; $-·-·-·-·-·$, first POD mode of axisymmetric mode of $u^{\prime }$. Data from Cavalieri et al. [12]. (a) $x/D=3.5$, (b) $x/D=5$, and (c) $x/D=8$.

Figure 3

Overview of four-ring setup. (a) Overview and (b) Installed rig.

Figure 4

Seven-ring near-field azimuthal array setup (7-pt correlation).

Figure 5

Four-ring near-field azimuthal array setup (2-pt correlation).

Figure 6

Energy distribution (bars) and cumulative sum (lines) for the first 10 POD modes for $m=0$ from the four-ring array measurements. $\text{St}=0.2$ (black bars and solid line), $\text{St}=0.5$ (gray bars and dashed line), and $\text{St}=0.8$ (white bars and dotted line).

Figure 7

Comparison of PSE estimates and measured autospectra magnitudes for $m=0$ and $M=0.6$. —–, PSE; $\square$, four-ring full signal; $◯$, four-ring first POD mode; $*$, seven-ring full signal; $+$, seven-ring first POD mode. (PSE estimates multiplied by 10.) (a) St = 0.1, (b) St = 0.2, (c) St = 0.3, (d) St = 0.4, (e) St = 0.5, (f) St = 0.6, (g) St = 0.7, (h) St = 0.8, and (i) St = 0.9.

Figure 8

Comparison of PSE estimates and measured autospectra phases for $m=0$ and $M=0.6$. Phase reference is $x/D=2$. See Fig. 7 for legend. (a) St = 0.1, (b) St = 0.2, (c) St = 0.3, (d) St = 0.4, (e) St = 0.5, (f) St = 0.6, (g) St = 0.7, (h) St = 0.8, and (i) St = 0.9.

Figure 9

Amplitude envelope of CSM reconstructions [see Eq. (5)] for $m=0, M=0.6$, and $\text{St}=0.5$ from (a) individual and (b) cumulative POD modes.

Figure 10

Comparison of PSE estimates and measured autospectra magnitudes for $m=0, M=0.6$, and $\text{St}=0.5$ for the seven-ring data at the three radial array locations. ——, PSE; $◯$, full signal; $\times$, first POD mode. (PSE estimates not multiplied by 10.) $\left(\mathrm{a}\right)r_A/D=0.8, \left(\mathrm{b}\right)r_A/D=0.7$, and $\left(\mathrm{c}\right)r_A/D=0.6$.

Figure 11

Two-point coherence ($|{\mathsf{R}}_{m,\omega }(x,x_2){|}^2/{\mathsf{R}}_{m,\omega }(x,x){\mathsf{R}}_{m,\omega }(x_2,x_2)$) $m=0, M=0.6$, and $\text{St}=0.5$ based on cumulative reconstructions using POD modes. (a) POD 1, (b) POD 1-2, (c) POD 1-3, and (d) Full signal.

Figure 12

$m=0$ POD modes for four-ring (——) and seven-ring ($\times$ and $◯$) setups. In panels (b) and (c), every second St is skipped for clarity. Arbitrary scaling between $\text{St}$. (a) POD mode 1, (b) POD mode 2, and (c) POD mode 3.

Figure 13

Comparison between near-field and far-field $m=0$ pressure mode signals for both full signal and POD reconstruction. (a) Far-field signal plotted with $\pm 2\sigma$ (dashed line) to show peaks, (b) full seven-ring near-field signal, (c) projection to POD mode 1 of the four-ring basis truncated to seven-ring axial extent, and (d) projection to POD mode 1 of the four-ring basis. The dotted horizontal lines highlight the alignment between selected far-field peaks and near-field features. Far-field pressure is plotted in retarded time.

Figure 14

Normalized retarded correlation between near-field ${\tilde{p}}_0$ and far-field ${\tilde{p}}_0$. The absolute maximum is 15.1%.

Figure 15

Individual $m=0$ POD mode contributions to near-field–far-field correlation ($R_{{\tilde{p}}_{\text{NF}}{\tilde{p}}_{\text{FF}}}/{\sigma }_{{\tilde{p}}_{\text{NF}}}{\sigma }_{{\tilde{p}}_{\text{FF}}}$) for 7-ring data. The absolute maximum for each subplot is indicated in brackets. (a) POD 1 (max. 13.7%), (b) POD 2 (max. 13.5%), (c) POD 3 (max. 14.0%), (d) POD 4 (max. 9.7%), (e) POD 5 (max. 8.7%), (f) POD 6 (max. 8.4%), and (g) POD 7 (max. 6.1%).

Figure 16

Cumulative $m=0$ POD mode near-field–far-field correlations ($R_{{\tilde{p}}_{\text{NF}}{\tilde{p}}_{\text{FF}}}/{\sigma }_{{\tilde{p}}_{\text{NF}}}{\sigma }_{{\tilde{p}}_{\text{FF}}}$) for 7-ring data. The absolute maximum for each subplot is indicated in brackets. (a) POD 1-1 (max. 13.7%), (b) POD 1-2 (max. 16.5%), (c) POD 1-3 (max. 17.2%), (d) POD 1-4 (max. 16.4%), (e) POD 1-5 (max. 16.0%), (f) POD 1-6 (max. 15.9%), and (g) POD 1-7 (max. 15.8%).

Figure 17

Individual $m=0$ POD mode contributions to near-field–far-field correlation ($R_{{\tilde{p}}_{\text{NF}}{\tilde{p}}_{\text{FF}}}/{\sigma }_{{\tilde{p}}_{\text{NF}}}{\sigma }_{{\tilde{p}}_{\text{FF}}}$) for the reduced-order projection using Eq. (8). The absolute maximum for each subplot is indicated in brackets. (a) POD 1 (max. 15.1%), (b) POD 2 (max. 14.0%), and (c) POD 3 (max. 14.2%).

Figure 18

Cumulative $m=0$ POD mode near-field–far-field correlations ($R_{{\tilde{p}}_{\text{NF}}{\tilde{p}}_{\text{FF}}}/{\sigma }_{{\tilde{p}}_{\text{NF}}}{\sigma }_{{\tilde{p}}_{\text{FF}}}$) for the reduced-order projection using Eq. (8). The absolute maximum for each subplot is indicated in brackets. (a) POD 1-1 (max. 15.1%), (b) POD 1-2 (max. 15.1%), and (c) POD 1-3 (max. 17.0%).

Figure 19

Geometric considerations for time-domain predictions with a finite-window Fourier transform.

Figure 20

Comparison of Green's-function-predicted directivities from the current four-ring measurements for the $M=0.6$ jet. $\square$, Full prediction; $◯$, POD mode 1 prediction; – – –, PSE prediction; $*$, truncated PSE prediction ($0.5\le x/D\le 8.9$); and ——–, Cavalieri et al.'s [2] experimental data. (a) St = 0.2, (b) St = 0.5, and (c) St = 0.8.

Figure 21

CSM plots for $\text{St}=0.5$ plotted by wave number (obtained by spatial Fourier transform). The central square section, delimited by $k=\pm k_{\infty }$ and $k_2=\pm k_{\infty }$, where $k_{\infty }=\omega /c_0$ encloses radiation that “leaks” to the far field. Contour levels are logarithmic. (a) POD 1-1, (b) POD 1-2, and (c) POD 1-3.

Figure 22

Comparison of PSE estimates (lines; from Ref. [48]) and first POD mode of LES (symbols; from Ref. [20], every fifth data point shown) for $m=0, M=0.9$, and $\text{St}=0.61$ using the array locations from the current seven-ring experiment. —— and $\square ,r_A/D=0.6$; – – – and $◯,r_A/D=0.7$; ...... and $*,r_A/D=0.8$. (a) Velocity and (b) Pressure.

Figure 23

Radial comparison of PSE estimates (lines, from Ref. [48]) and first POD mode of LES (symbols; from Ref. [20]) for $m=0, M=0.9$, and $\text{St}=0.61$ at $x/D=2$. (a) Velocity and (b) Pressure.

Figure 24

Analysis procedure for time-domain far-field signal prediction.

Figure 25

Validation results for the far-field Green's function predictions from the near-field array with $X_0\le x/D\le 70$. —, Lighthill solution; $◯$, Green's function solution. (a) Statistical prediction (St = 0.2), (b) Statistical prediction (St = 0.5), (c) Statistical prediction (St = 0.8), and (d) Time-domain prediction at $\theta ={20}^{\circ }$ (Every fifth data point of the Green's function result is shown.).