3.1 Vorticity and pressure snapshots

Snapshots of the vorticity norm and pressure fluctuations obtained in the $(z,r)$ plane for the jets at $Re_{D}=3125$ and $Re_{D}=12\,500$ are represented, respectively, in figure 2 at times $tu_{j}/r_{0}=15$ , 25, 35 and 45 and in figure 3 at $tu_{j}/r_{0}=10$ , 20, 30 and 40, and in the corresponding movies movie_fig2 and movie_fig3 available at https://doi.org/10.1017/jfm.2018.864.

Figure 2.

Representation of vorticity norm for $r\leqslant 3r_{0}$ and of pressure fluctuations otherwise, obtained for $Re_{D}=3125$ at $tu_{j}/r_{0}=15$ , 25, 35 and 45, from top to bottom. The colour scales range up to the level of $5u_{j}/r_{0}$ for vorticity, and from $-250$ to $250~\text{Pa}$ for pressure, from blue to red.

Figure 3.

Representation of vorticity norm for $r\leqslant 3r_{0}$ and of pressure fluctuations otherwise, obtained for $Re_{D}=12\,500$ at $tu_{j}/r_{0}=10$ , 20, 30 and 40, from top to bottom. The colour scales range up to the level of $5u_{j}/r_{0}$ for vorticity, and from $-200$ to $200~\text{Pa}$ for pressure, from blue to red.

For both Reynolds numbers, two mixing layers can be seen on both sides of the jet axis in figures 2(a) and 3(a). They have just rolled up following the growth of instability waves in the hyperbolic–tangent velocity profile (Michalke Reference Michalke1964) for $Re_{D}=3125$ , whereas they appear to be turbulent for $Re_{D}=12\,500$ . In figures 2(b) and 3(b), the mixing layers, which have developed thanks to the vortex-pairing mechanism (Winant & Browand Reference Winant and Browand1974), merge on the jet axis, resulting in the disappearance of the potential core. Finally, in figures 2(c,d) and 3(c,d), the jets are fully developed, and contain vortical scales of intensity decreasing with time. Furthermore, more fine turbulent scales are observed in the higher Reynolds number jet, as expected.

For $Re_{D}=3125$ , in figure 2, alternatively positive and negative fluctuations are found at $tu_{j}/r_{0}=15$ in the pressure field in the close vicinity of the jet. They are most likely the footprints of the coherent turbulent structures of the jet flow in the near field (Arndt, Long & Glauser Reference Arndt, Long and Glauser1997; Coiffet et al. Reference Coiffet, Jordan, Delville, Gervais and Ricaud2005). At $tu_{j}/r_{0}=25$ , as the potential core closes, high-amplitude waves are emitted in the downstream direction. These waves are well visible at later times $tu_{j}/r_{0}=35$ and 45. They appear to be symmetric with respect to the jet centreline and to have a typical wavelength of approximately $15r_{0}$ . Moreover, they propagate mainly in the downstream direction, and have a very large spatial extent along the wave front direction. Interestingly, they look like the waves emitted at shallow angles by spatially developing subsonic jets at both low and high Reynolds numbers; see for instance the simulation results in Bogey & Bailly (Reference Bogey and Bailly2006) and Bogey (Reference Bogey2018).

For $Re_{D}=12\,500$ , in figure 3, the pressure field exhibits, unsurprisingly, more waves of short wavelength than previously for $Re_{D}=3125$ , but also additional noise components, whose contribution will be later quantified in the pressure spectra of figure 14. In particular, sound waves generated by vortex pairings in the mixing layers are noted at $tu_{j}/r_{0}=10$ and 20. In addition, upstream-propagating high-frequency waves are detected at $tu_{j}/r_{0}=30$ and 40 after the potential-core closing, which was not the case for the first jet. At these two times, more importantly given the aim of the present study, low-frequency waves are found to propagate in the downstream direction. Even at a lower level, they resemble those obtained for the jet at $Re_{D}=3125$ . Similar acoustic waves of comparable amplitude also appear to be radiated by a temporal jet at $Re_{D}=50\,000$ , computed in an extra simulation reported in appendix A. Therefore, these downstream-propagating sound waves, as well as their corresponding noise generation mechanism, persist when the Reynolds number increases, despite the richer spectral content of the jet turbulence, hence the relative weakening of the flow large-scale structures, in that case. Furthermore, they seem not to depend much on the shear-layer initial momentum thickness given that at $t=0$ , one sets $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=0.0358r_{0}$ for $Re_{D}=3125$ , $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=0.0179r_{0}$ for $Re_{D}=12\,500$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=0.0089r_{0}$ for $Re_{D}=50\,000$ .

3.2 Properties of the velocity fields

3.2.1 Velocity field statistics

The mean and r.m.s. values of axial velocity calculated for the jets are represented in figure 4 using $(t,r)$  coordinates. The results bear striking similarities to the flow fields measured in the $(z,r)$ plane of spatially developing subsonic jets, by Zaman (Reference Zaman1986) for example. The mean axial velocity fields of figure 4(a,b) show the jet development with time, and indicates that the potential core closes at $tu_{j}/r_{0}\simeq 21$ for $Re_{D}=3135$ and earlier at $tu_{j}/r_{0}\simeq 16$ for $Re_{D}=12\,500$ . In parallel, the axial turbulence intensities are found in figure 4(c,d) to grow in the jet shear layer, to reach values slightly higher than 20 %, and then to decrease. The peak values are obtained around $tu_{j}/r_{0}=21$ for $Re_{D}=3135$ and $tu_{j}/r_{0}=8$ for $Re_{D}=12\,500$ , that is near the time of potential-core closing in the first case, but significantly before in the second case.

Figure 4.

Space–time representation of (a,b) mean and (c,d) r.m.s. fields of axial velocity for (a,c) $Re_{D}=3125$ and (b,d) $Re_{D}=12\,500$ using the contour line values: (a,b)  $\langle u_{z}\rangle /u_{j}=0.05$ , 0.35, 0.65 and 0.95, (c,d)  $\langle u_{z}^{\prime }u_{z}^{\prime }\rangle ^{1/2}/u_{j}=0.02$ , 0.08, 0.14 and 0.20.

The time variations of the shear-layer momentum thickness, calculated here as

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=\int _{0}^{r_{0.001}}\frac{\langle u_{z}\rangle }{\langle u_{z}\rangle (r=0)}\left(1-\frac{\langle u_{z}\rangle }{\langle u_{z}\rangle (r=0)}\right)\,\text{d}r,\end{eqnarray}$$

where $r_{0.001}$ is such that $\langle u_{z}\rangle (r=r_{0.001})=0.001\langle u_{z}\rangle (r=0)$ , and of the mean centreline axial velocity are plotted in figure 5. In figure 5(a), the shear layers appear to spread slowly when the jet flows are laminar or turbulent, but more rapidly during the laminar–turbulent transitional periods, which happen between $tu_{j}/r_{0}\simeq 12.5$ and $25$ for $Re_{D}=3125$ , and between $tu_{j}/r_{0}\simeq 5$ and $12.5$ for $Re_{D}=12\,500$ . In figure 5(b), the velocity decays after the potential-core closing starts at $t_{c}u_{j}/r_{0}=21.6$ for $Re_{D}=3125$ and at $t_{c}u_{j}/r_{0}=16$ for $Re_{D}=12\,500$ , where $t_{c}$ is arbitrarily defined by $\langle u_{z}(r=0,t=t_{c})\rangle =0.95u_{j}$ . Afterwards, the decay occurs at a higher rate in the lower Reynolds number case.

Figure 5.

Time variations of (a) shear-layer momentum thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}/r_{0}$ and (b) mean axial velocity $\langle u_{z}\rangle /u_{j}$ at $r=0$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ .

The time variations of the r.m.s. values and of the skewness factors of axial velocity fluctuations at $r=0$ and $r=r_{0}$ are displayed in figure 6. In figure 6(a), strong humps are obtained in the profiles of turbulence intensity, leading to maximum values of 17 % at $r=0$ and 20 % at $r=r_{0}$ for $Re_{D}=3125$ , and of 12.6 % at $r=0$ and 21.3 % at $r=r_{0}$ for $Re_{D}=12\,500$ . They result from the merging of vortical structures in the shear layers and on the jet centreline, respectively (Bogey & Bailly Reference Bogey and Bailly2010). As expected, due to the difference in mean flow development, they occur more rapidly in the $Re_{D}=12\,500$ case. Moreover, the peak value at $r=r_{0}$ is reached much earlier than the peak value at $r=0$ for $Re_{D}=12\,500$ , but that is not true for $Re_{D}=3125$ . This suggests the presence of vortex pairings in the mixing layers only in the higher Reynolds number jet.

In figure 6(b), values of skewness around  $-1.1$ are found at $r=0$ close to the time of potential-core closing for both Reynolds numbers. This indicates intermittent occurrence of velocity deficits on the jet centreline at that time. These deficits follow the intrusion of shear-layer turbulent structures in the potential core of the temporal jet, as it happens at the end of the potential core of spatially developing jets (Bogey et al. Reference Bogey, Bailly and Juvé2003; Bogey & Bailly Reference Bogey and Bailly2007). At $r=r_{0}$ , the results are quite different from those at $r=0$ . In this case, the skewness factor is slightly positive, suggesting possible bursts of high-velocity flow structures in the mixing layers.

Figure 6.

Time variations of (a) axial turbulence intensity $\langle u_{z}^{\prime }u_{z}^{\prime }\rangle ^{1/2}/u_{j}$ and (b) the skewness factor of axial velocity fluctuations $u_{z}^{\prime }$ : at $r=0$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , and at $r=r_{0}$ for – – –  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ .

3.2.2 Velocity spectra

Spectra of axial velocity fluctuations computed in the jet flows at $r=0$ and $r=r_{0}$ are represented in figure 7 as a function of normalized axial wavenumber $k_{z}r_{0}$ , up to the value of $k_{z}r_{0}=60$ which is close to the filtering cutoff wavenumber of $k_{z}r_{0}=62.8$ obtained for four points per wavelength. The spectra calculated on the jet axis at the times of peak turbulence intensity level $tu_{j}/r_{0}=26.2$ for $Re_{D}=3125$ and $tu_{j}/r_{0}=22.8$ for $Re_{D}=12\,500$ , just after the potential-core closing, are depicted in figure 7(a). They have similar shapes and are consistent with the $k_{z}^{-5/3}$ power law for $k_{z}r_{0}\geqslant 2$ . However, the spectrum for $Re_{D}=3125$ contains stronger low-frequency components than that for $Re_{D}=12\,500$ . It seems also to be dominated by components around $k_{z}r_{0}\simeq 0.4$ , corresponding to a wavelength of $\unicode[STIX]{x1D706}_{z}\simeq 15r_{0}$ , which is not the case for the higher Reynolds number jet.

The velocity spectra estimated at $r=r_{0}$ are shown in figure 7(b). At the early times of $tu_{j}/r_{0}=2$ for $Re_{D}=3125$ and $tu_{j}/r_{0}=1$ for $Re_{D}=12\,500$ , components are found to emerge around wavenumbers $k_{z}r_{0}=4.8$ and $10.2$ , respectively. These peak wavenumbers are slightly lower than those given by the inviscid linear stability theory of Michalke (Reference Michalke1964), predicting that the amplification of temporally growing instability waves is maximum at $k_{z}r_{0}=6.2$ and $12.4$ in the two hyperbolic–tangent velocity profiles imposed at $t=0$ . The discrepancy between computations and theory is likely due to the initial conditions, but also to the low Reynolds numbers of $Re_{\unicode[STIX]{x1D703}}=56$ and 112 in the simulated shear layers. For such values, indeed, the growth rates and the peak wavenumbers of instability waves can be expected to be reduced compared to the inviscid case, as reported in the theoretical analyses conducted by Ragab & Wu (Reference Ragab and Wu1989) for two-dimensional mixing layers and by Morris (Reference Morris1976) for axisymmetric jets.

As for the velocity spectra obtained at $r=r_{0}$ at the times of peak turbulence intensity level $tu_{j}/r_{0}=21.6$ for $Re_{D}=3125$ and $tu_{j}/r_{0}=7.3$ for $Re_{D}=12\,500$ , also plotted in figure 7(b), they are relatively broadband, and show no clear agreement with the $k_{z}^{-5/3}$ power law. In this case, the dominant components are noted around $k_{z}r_{0}=1$ for $Re_{D}=3125$ and $k_{z}r_{0}=2.4$ for $Re_{D}=12\,500$ . The difference in peak wavenumber can be explained by the fact that the maximum turbulence level is reached at the time of potential-core closing for the first jet, but much earlier, in the developing mixing layers, for the other jet.

Figure 7.

Representation of the power spectral densities of axial velocity fluctuations $u_{z}^{\prime }/u_{j}$ as a function of axial wavenumber $k_{z}r_{0}$ : (a) at $r=0$ at time of peak level for —— $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , (b) at $r=r_{0}$ at time of peak level for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , and at $tu_{j}/r_{0}=2$ for – – –  $Re_{D}=3125$ and $tu_{j}/r_{0}=1$ for

  $Re_{D}=12\,500$ ; –  $\cdot$  –  $\cdot$   $k_{z}^{-5/3}$ slope.

In order to explore the coherence of the flow field in the azimuthal direction, the velocity spectra obtained at $r=r_{0}$ at the times of peak turbulence intensity level are displayed in figure 8(a) as a function of azimuthal mode $n_{\unicode[STIX]{x1D703}}$ . The time variations of the magnitudes of modes $n_{\unicode[STIX]{x1D703}}=0$ , $1$ and $2$ at $r=r_{0}$ are also represented in figure 8(b). For both jets, at all times, the contribution of the axisymmetric mode is dominant, closely followed by the contribution of mode $n_{\unicode[STIX]{x1D703}}=1$ . This is in line with the results of the literature about the amplification rates of spatially growing linear instability waves in jet flows; see for example Michalke (Reference Michalke1984) for the inviscid incompressible case and Keiderling, Kleiser & Bogey (Reference Keiderling, Kleiser and Bogey2009) for a viscous jet at a Mach number of 0.9.

Figure 8.

Representation of the power spectral densities of axial velocity fluctuations $u_{z}^{\prime }/u_{j}$ obtained at $r=r_{0}$ : (a) at time of peak level for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ as a function of azimuthal mode $n_{\unicode[STIX]{x1D703}}$ , (b) time variations of the magnitudes of modes ——  $n_{\unicode[STIX]{x1D703}}=0$ , – – –  $n_{\unicode[STIX]{x1D703}}=1$ and –  $\cdot$  –  $\cdot$   $n_{\unicode[STIX]{x1D703}}=2$ for $Re_{D}=3125$ , and

  $n_{\unicode[STIX]{x1D703}}=0$ , 

  $n_{\unicode[STIX]{x1D703}}=1$ and

  $n_{\unicode[STIX]{x1D703}}=2$ for $Re_{D}=12\,500$ .

3.2.3 Velocity autocorrelations

In the jet flow field, the space–time autocorrelations of axial velocity fluctuations are computed at radial position $r$ and time $t$ in the following way:

(3.2) $$\begin{eqnarray}{\mathcal{R}}_{u_{z}^{\prime }u_{z}^{\prime }}(\unicode[STIX]{x1D6FF}z,\unicode[STIX]{x1D6FF}t)=\frac{\langle u_{z}^{\prime }(r,\unicode[STIX]{x1D703},z,t)u_{z}^{\prime }(r,\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t+\unicode[STIX]{x1D6FF}t)\rangle }{\langle u_{z}^{\prime 2}(r,\unicode[STIX]{x1D703},z,t)\rangle ^{1/2}\langle u_{z}^{\prime 2}(r,\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t+\unicode[STIX]{x1D6FF}t)\rangle ^{1/2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}z$ is the spatial separation in the axial direction and $\unicode[STIX]{x1D6FF}t$ is the time delay.

The correlation maps obtained on the jet centreline are represented in figure 9 at times $t=t_{c}-6r_{0}/u_{j}$ , $t=t_{c}$ and $t=t_{c}+6r_{0}/u_{j}$ , and in the corresponding movies movie_fig9a and movie_fig9b. The solid and dashed lines show the inclinations expected for convection velocities of $0.6u_{j}$ and $u_{j}$ . The results for the two jets are similar. In figure 9(a,d), the correlations at $t=t_{c}-6r_{0}/u_{j}$ before the potential-core closing have an oscillating shape in the axial direction, and remain significant for very large time delays. In addition, they are well aligned with the trajectory for a convection velocity of $0.6u_{j}$ . In figure 9(b,e), the correlations at $t=t_{c}$ are comparable to the previous ones for negative time delays, but differ for positive time delays. In the latter case, the correlations are only positive and their inclination gets closer to that corresponding to a convection velocity of $u_{j}$ . This is due to the arrival of turbulent structures on the centreline when the potential core closes. Finally, the correlations at later times become less inclined with increasing time, as the jets develop and the velocity decays on the centreline.

Figure 9.

Representation of the space–time correlations of axial velocity fluctuations obtained at $r=0$ at (a,d)  $t=t_{c}-6r_{0}/u_{j}$ , (b,e)  $t=t_{c}$ and (c,f)  $t=t_{c}+6r_{0}/u_{j}$ for (a–c)  $Re_{D}=3125$ and (d–f) $Re_{D}=12\,500$ ; ——  $\unicode[STIX]{x1D6FF}t=\unicode[STIX]{x1D6FF}z/(0.6u_{j})$ ; – – –  $\unicode[STIX]{x1D6FF}t=\unicode[STIX]{x1D6FF}z/u_{j}$ . The colour scale ranges from $-1$ to 1, from blue to red.

For comparison, the correlation maps obtained at $r=r_{0}$ are provided in figure 10. They resemble the correlations at $r=0$ given in figure 9 at the same times. Slight differences however appear. First, before the potential-core closing, significant correlation levels persist for shorter spatial separations and time delays. This may result from the fact that for $t<t_{c}$ , the correlations on the centreline are calculated from the footprints of the coherent structures developing in the mixing layers, whereas those at $r=r_{0}$ are computed from a wider variety of turbulent scales. Second, the trajectories of the correlation spots are less inclined. This is particularly true at $t=t_{c}$ , where they indicate a convection velocity lower than $0.6u_{j}$ .

Figure 10.

Representation of the space–time correlations of axial velocity fluctuations obtained at $r=r_{0}$ at (a,d)  $t=t_{c}-6r_{0}/u_{j}$ , (b,e)  $t=t_{c}$ and (c,f)  $t=t_{c}+6r_{0}/u_{j}$ for (a–c)  $Re_{D}=3125$ and (d–f)  $Re_{D}=12\,500$ ; ——  $\unicode[STIX]{x1D6FF}t=\unicode[STIX]{x1D6FF}z/(0.6u_{j})$ ; – – –  $\unicode[STIX]{x1D6FF}t=\unicode[STIX]{x1D6FF}z/u_{j}$ . The colour scale ranges from $-1$ to 1, from blue to red.

Some flow properties estimated from the velocity correlation functions, namely the integral length scale $L_{uu}^{(z)}$ and the convection velocity $u_{c}$ , are shown in figure 11. In figure 11(a), the integral length scales are found to increase with time. The increase occurs suddenly around the time of potential-core closing on the jet axis, but progressively at $r=r_{0}$ . Furthermore, at $t>t_{c}$ , the integral length scales are larger for $Re_{D}=3125$ than for $Re_{D}=12\,500$ , and on the jet axis than at $r=r_{0}$ . In figure 11(b), the convection velocities at $r=r_{0}$ decrease monotonically with time from $u_{c}\simeq 0.6u_{j}$ at $tu_{j}/r_{0}=10$ down to $u_{c}\simeq 0.3u_{j}$ at $tu_{j}/r_{0}=50$ . On the jet axis, on the contrary, they first increase, then reach peak values of $0.86u_{j}$ for $Re_{D}=3125$ and of $0.95u_{j}$ for $Re_{D}=12500$ near the time of potential-core closing, and finally diminish. This suggests that the turbulent structures that enter in the potential core are convected at a velocity close to the jet velocity, as in spatially developing jets (Bogey & Bailly Reference Bogey and Bailly2007).

Figure 11.

Time variations of (a) the integral length scale $L_{uu}^{(z)}/r_{0}$ and (b) the convection velocity  $u_{c}/u_{j}$ , obtained at $r=0$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , and at $r=r_{0}$ for – – –  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ .

3.3 Properties of the pressure fields

3.3.1 Pressure field statistics

The r.m.s. values of pressure fluctuations obtained for the jets are represented in figure 12 using  $(t,r)$ coordinates. As previously for the velocity, the results resemble the r.m.s. pressure fields acquired in the $(z,r)$ plane of spatially developing subsonic jets (Zaman Reference Zaman1986; Ukeiley & Ponton Reference Ukeiley and Ponton2004; Bogey et al. Reference Bogey, Barré, Fleury, Bailly and Juvé2007). For both Reynolds numbers, the maximum values are located in the jet flows, approximately at the same position as the maximum r.m.s. values of axial velocity in figure 5(c,d). Outside the jets, lobes, whose origins seem to be the peak levels in the flow (Zaman Reference Zaman1986), are found. They reveal the production of noise in the jet and its propagation outside with increasing time. The r.m.s. pressure fields moreover suggest that the strongest acoustic waves are generated close to the time of potential-core closing for $Re_{D}=3125$ , but well before for $Re_{D}=12\,500$ , and that their main radiation angle with respect to the flow direction is larger in the latter case than in the former.

Figure 12.

Space–time representation of the r.m.s. fields of pressure fluctuations for (a)  $Re_{D}=3125$ and (b)  $Re_{D}=12\,500$ using the contour line values  $\langle p^{\prime }p^{\prime }\rangle ^{1/2}=45$ , $60$ , 80, 140, 1000, 3000 and 5000 Pa.

The time variations of the r.m.s. values of pressure fluctuations at $r=10r_{0}$ are shown in figure 13(a). The profiles reach maximum values of 83 Pa at $tu_{j}/r_{0}=35.4$ for $Re_{D}=3125$ and of 68 Pa at $tu_{j}/r_{0}=19$ for $Re_{D}=12\,500$ . Given that $t_{c}u_{j}/r_{0}=21.6$ in the first case and $t_{c}u_{j}/r_{0}=16$ in the second case, the peak for $Re_{D}=3125$ may be related to the potential-core closing, but it cannot be true for $Re_{D}=12\,500$ . In this case, it most likely results from vortex pairings in the shear layers. For completeness, the r.m.s. pressure levels obtained for $Re_{D}=3125$ at the three radial positions $r=5r_{0}$ , $10r_{0}$ and $20r_{0}$ , normalized at $r=10r_{0}$ using the $1/r$ law, are represented in figure 13(b). Peak values of 88, 83 and 85 Pa are noted at $tu_{j}/r_{0}=28.6$ , $35.4$ and $48$ , respectively. The similarity in amplitude supports that the spreading of the sound waves radiated by the jet is spherical. Moreover, the time delays between the peaks indicate a radiation angle of approximately 45 degrees with respect to the axial direction.

Figure 13.

Time variations of the r.m.s. values of pressure fluctuations $p^{\prime }$ : (a) at $r=10r_{0}$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , (b) at ——  $r=5r_{0}$ , – – –  $r=10r_{0}$ and –  $\cdot$  –  $\cdot$   $r=20r_{0}$ for $Re_{D}=3125$ normalized using a $1/r$ law.

3.3.2 Pressure spectra

The spectra of pressure fluctuations computed at $r=10r_{0}$ at $tu_{j}/r_{0}=35$ , 45 and 55 for $Re_{D}=3125$ and at $tu_{j}/r_{0}=30$ , 40 and 50 for $Re_{D}=12\,500$ are represented in figure 14 as a function of axial wavenumber $k_{z}r_{0}$ . At those times, the low-frequency acoustic waves generated around $t=t_{c}$ propagate at the radial position considered, as shown in figures 2 and 3. The three spectra for $Re_{D}=3125$ in figure 14(a) display very similar shapes. They contain strong components centred around a wavenumber of $k_{z}r_{0}\simeq 0.4$ , and decrease sharply for $k_{z}r_{0}>1.6$ . These components can be related to the presence of sound waves of wavelength of $\unicode[STIX]{x1D706}_{z}\simeq 15r_{0}$ in the pressure fields of figure 2. In the same way, the shape of the spectra for $Re_{D}=12\,500$ in figure 14(b) does not depend much on the time considered. With respect to the spectra for the jet at $Re_{D}=3125$ , they are comparable for $k_{z}r_{0}<1$ , but show additional high-frequency components typically between $k_{z}r_{0}=1.6$ and 6.4. Therefore, two contributions seem to emerge in the present sound spectra: a low-frequency contribution, noticeable at both Reynolds numbers, and a higher-frequency one, disappearing when the Reynolds number is reduced down to $Re_{D}=3125$ . This observation is consistent with those made based on the spectra of spatially developing jets, used as evidence for the presence of two jet noise components (Tam Reference Tam1998; Bogey & Bailly Reference Bogey and Bailly2006; Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008). It is also supported by the comparison of the present spectra with the large-scale and fine-scale similarity spectra found empirically by Tam, Golebiowski & Seiner (Reference Tam, Golebiowski and Seiner1996) and Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008) to fit the far-field spectra measured, respectively, in the downstream direction and in the upstream and sideline directions of both high-subsonic and supersonic jets. Indeed, the spectra for $Re_{D}=3125$ do not differ much from the large-scale similarity spectrum when the peak of the latter is placed at $(k_{z}r_{0})_{peak}=0.44$ , whereas the spectra for $Re_{D}=12\,500$ visibly coincide with a combination of the two similarity spectra with $(k_{z}r_{0})_{peak}=0.44$ and 0.55.

Figure 14.

Representation of the power spectral densities of pressure fluctuations $p^{\prime }/p_{a}$ as a function of wavenumber $k_{z}r_{0}$ , obtained at $r=10r_{0}$ : (a) for $Re_{D}=3125$ at ——  $tu_{j}/r_{0}=35$ , – – –  $tu_{j}/r_{0}=45$ and –  $\cdot$  –  $\cdot$   $tu_{j}/r_{0}=55$ , (b) for $Re_{D}=12\,500$ at ——  $tu_{j}/r_{0}=30$ , – – –  $tu_{j}/r_{0}=40$ and –  $\cdot$  –  $\cdot$   $tu_{j}/r_{0}=50$ ; 

 large-scale and 

 fine-scale similarity spectra of Tam et al. (Reference Tam, Golebiowski and Seiner1996).

The spectra of pressure fluctuations obtained at $r=20r_{0}$ at $tu_{j}/r_{0}=45$ , 55 and 65 for $Re_{D}=3125$ and at $tu_{j}/r_{0}=40$ , 50 and 60 for $Re_{D}=12\,500$ are plotted in figure 15 as a function of the azimuthal mode. In all cases, the energy is maximum for the axisymmetric mode, decreases for higher modes, and is small or negligible for modes $n_{\unicode[STIX]{x1D703}}>4$ . This azimuthal distribution of the acoustic pressure is close to that measured at low polar angles for spatially developing subsonic jets (Juvé, Sunyach & Comte-Bellot Reference Juvé, Sunyach and Comte-Bellot1979). For the present jets, in addition, the predominance of the axisymmetric mode is more marked as time passes, such as when lower axial angles are considered in the far field of spatial jets. That predominance is one important feature associated with the downstream subsonic jet noise component (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012).

Figure 15.

Representation of the power spectral densities of pressure fluctuations $p^{\prime }/p_{a}$ as a function of azimuthal mode $n_{\unicode[STIX]{x1D703}}$ , obtained at $r=20r_{0}$ : (a) for $Re_{D}=3125$ at ——  $tu_{j}/r_{0}=45$ , – – –  $tu_{j}/r_{0}=55$ and –  $\cdot$  –  $\cdot$   $tu_{j}/r_{0}=65$ , (b) for $Re_{D}=12\,500$ at ——  $tu_{j}/r_{0}=40$ , – – –  $tu_{j}/r_{0}=50$ and –  $\cdot$  –  $\cdot$   $tu_{j}/r_{0}=60$ .

3.3.3 Pressure autocorrelations

In the jet near field, the two-dimensional spatial autocorrelations of pressure fluctuations at radial position $r$ and time $t$ are calculated as

(3.3) $$\begin{eqnarray}{\mathcal{R}}_{p^{\prime }p^{\prime }}(\unicode[STIX]{x1D6FF}r,\unicode[STIX]{x1D6FF}z)=\frac{\langle p^{\prime }(r,\unicode[STIX]{x1D703},z,t)p^{\prime }(r+\unicode[STIX]{x1D6FF}r,\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t)\rangle }{\langle p^{\prime 2}(r,\unicode[STIX]{x1D703},z,t)\rangle ^{1/2}\langle p^{\prime 2}(r+\unicode[STIX]{x1D6FF}r,\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t)\rangle ^{1/2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}r$ and $\unicode[STIX]{x1D6FF}z$ are the spatial separations in the radial and axial directions.

The correlation maps obtained at $r=10r_{0}$ at $tu_{j}/r_{0}=35$ and 55 for $Re_{D}=3125$ and at $tu_{j}/r_{0}=30$ and 50 for $Re_{D}=12\,500$ are shown in figure 16. For the two jets, the orientation of the correlation spot changes with time, indicating a direction of propagation getting closer to the jet direction. Its spatial extent also becomes larger. This is true both along the wave front, where correlation is strong over a very large distance, and normal to the wave front. For $Re_{D}=12\,500$ , the correlation levels are lower than for $Re_{D}=3125$ for large separation distances, which is consistent with the richer frequency content of the pressure spectra reported above. Furthermore, weak additional correlation spots aligned in a direction opposite to that of the main spots are visible in figure 16(c–d). They result from the acoustic waves propagating upstream in the jet near field, such as those noted in figure 3(d).

Figure 16.

Representation of the two-dimensional spatial correlations of pressure fluctuations obtained at $r=10r_{0}$ for $Re_{D}=3125$ at (a)  $tu_{j}/r_{0}=35$ and (b)  $tu_{j}/r_{0}=55$ , and for $Re_{D}=12\,500$ at (c)  $tu_{j}/r_{0}=30$ and (d)  $tu_{j}/r_{0}=50$ ; lines – – – aligned with and –  $\cdot$  –  $\cdot$  perpendicular to the average direction of acoustic propagation. The colour scale ranges from $-1$ to 1, from blue to red.

The radiation angles $\unicode[STIX]{x1D719}$ and the integral length scales $L_{\unicode[STIX]{x1D719}}$ computed along the average direction of propagation found from the spatial correlations of pressure at $r=10r_{0}$ are represented in figure 17. From $tu_{j}/r_{0}\simeq 30$ for $Re_{D}=3125$ and $tu_{j}/r_{0}\simeq 20$ for $Re_{D}=12\,500$ onwards, that is significantly after the jet potential-core closing, the radiation angles decrease with time in figure 17(a), from $\unicode[STIX]{x1D719}\simeq 60^{\circ }$ down to $\unicode[STIX]{x1D719}\simeq 30^{\circ }$ at $tu_{j}/r_{0}=50$ . During the same period, the integral length scales grow and reach values of approximately $0.9r_{0}$ at $tu_{j}/r_{0}=60$ . The first result is expected given that, by construction, the waves remaining at $r=10r_{0}$ long after their emission times can only be those propagating at low axial angles. The second one suggests an increase of the wavelength. This wavelength increase as the radiation angle decreases is consistent with the trend observed for the downstream noise component of spatial jets (Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008). It can also be explained by a wave packet modelling of the jet noise source, as illustrated by Papamoschou (Reference Papamoschou2018), for instance.

Figure 17.

Time variations of (a) the radiation angle $\unicode[STIX]{x1D719}$ and (b) the integral length scale $L_{\unicode[STIX]{x1D719}}/r_{0}$ along the average direction of acoustic propagation at $r=10r_{0}$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ .

3.4 Flow–noise cross-correlations

In order to identify links between the flow and sound fields of the jets, it is interesting to compute cross-correlations between flow quantities and acoustic pressure, as was done in several recent experimental and numerical investigations for spatially developing jets (Seiner Reference Seiner1974; Schaffar Reference Schaffar1979; Panda et al. Reference Panda, Seasholtz and Elam2005; Bogey & Bailly Reference Bogey and Bailly2007; Panda Reference Panda2007; Grizzi & Camussi Reference Grizzi and Camussi2012). In the present work, the axial velocity fluctuations $u_{z}^{\prime }$ , the product $u_{z}^{\prime }u_{z}^{\prime }$ and the vorticity fluctuations $|\unicode[STIX]{x1D714}|^{\prime }$ obtained at $r=0$ and $r=r_{0}$ , are correlated with the pressure fluctuations $p^{\prime }$ at $r=10r_{0}$ . These flow quantities are considered because they have been found in the previous studies mentioned above to correlate with far-field jet noise. Moreover, they appear in the source terms of acoustical analogies, namely $u_{z}^{\prime }$ and $u_{z}^{\prime }u_{z}^{\prime }$ in the shear-noise and self-noise source terms of Lighthill’s analogy (Ribner Reference Ribner1964), and vorticity in the vortex sound theory (Powell Reference Powell1964). The correlations are denoted by ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , ${\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }}$ and ${\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }}$ . For $u_{z}^{\prime }$ , for example, they are given by

(3.4) $$\begin{eqnarray}{\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}(\unicode[STIX]{x1D6FF}z,t_{1})=\frac{\langle u_{z}^{\prime }(r_{1},\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t_{1})p^{\prime }(r_{2},\unicode[STIX]{x1D703},z,t_{2})\rangle }{\langle u_{z}^{\prime 2}(r_{1},\unicode[STIX]{x1D703},z+\unicode[STIX]{x1D6FF}z,t_{1})\rangle ^{1/2}\langle p^{\prime 2}(r_{2},\unicode[STIX]{x1D703},z,t_{2})\rangle ^{1/2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}z$ is the separation distance in the axial direction, $r_{1}=0$ or $r_{1}=r_{0}$ , and $r_{2}=10r_{0}$ . In simple terms, the flow quantities at position $(r=r_{1},z+\unicode[STIX]{x1D6FF}z)$ at time $t=t_{1}$ are correlated with the pressure fluctuations at position $(r=10r_{0},z)$ at $t=t_{2}$ .

The correlations computed between the flow quantities at $r=0$ and pressure at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and at $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ are represented in figure 18 as functions of $\unicode[STIX]{x1D6FF}z$ and $t_{1}$ . The velocity–pressure correlations obtained between $t_{2}u_{j}/r_{0}=30$ and $50$ for $Re_{D}=3125$ and between $t_{2}u_{j}/r_{0}=30$ and $45$ for $Re_{D}=12\,500$ are displayed in the movies movie_fig18a and movie_fig18b. The dashed line represents the time of potential-core closing, and the solid line indicates a propagation at the ambient speed of sound between the centreline and near-field points. High levels of correlations, close to or higher than 0.2 for $Re_{D}=3125$ and 0.1 for $Re_{D}=12\,500$ , are found near the intersection of the two lines for negative separation distances $\unicode[STIX]{x1D6FF}z$ . This supports the presence of a sound source on the axis of both jets, radiating in the downstream direction, when the potential core disappears. The correlations are negative for $u_{z}^{\prime }$ and positive for $u_{z}^{\prime }u_{z}^{\prime }$ and $|\unicode[STIX]{x1D714}|^{\prime }$ . They can be related to the intermittent arrival of low-velocity vortical structures in the jet core. Strong correlations also appear before the time of potential-core closing for vorticity, and afterward for $u_{z}^{\prime }$ and $u_{z}^{\prime }u_{z}^{\prime }$ . They are due to the convection of the flow fluctuations in the jets, which will be illustrated later. These results are remarkably similar to those reported in Bogey & Bailly (Reference Bogey and Bailly2007) for Mach number 0.9, spatially developing jets at low and high Reynolds numbers.

Figure 18.

Space–time correlations obtained between centreline quantities at $t=t_{1}$ and pressure fluctuations at $r=10r_{0}$ (a–c) at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and (d–f) at $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ : (a,d)  ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , (b,e)  ${\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }}$ and (c,f)  ${\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }}$ . The colour scale ranges from $-0.20$ to $0.20$ , from blue to red; —— propagation at the ambient speed of sound; – – –  $t_{1}=t_{c}$ .

For comparison, the correlations obtained between the flow quantities at $r=r_{0}$ and pressure at $r=10r_{0}$ are represented in figure 19 in the same way as in figure 18. The correlation levels are significantly lower than those evaluated from the centreline fluctuations as in spatial jets (Bogey & Bailly Reference Bogey and Bailly2007). They do not exceed values of 0.07 for $Re_{D}=3125$ and 0.06 for $Re_{D}=12\,500$ . They even fail to emerge from the background noise in most cases, except for ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ in figure 19(a) where an elongated correlation pattern similar, albeit very attenuated, to that visible in figure 18(a) appears for negative spatial separation distances. These results suggest that the sound source radiating in the downstream direction is not located in the centre of the mixing layers.

Figure 19.

Space–time correlations obtained between quantities at $r=r_{0}$ and $t=t_{1}$ and pressure fluctuations at $r=10r_{0}$ (a–c) at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and (d–f) at $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ : (a,d)  ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , (b,e)  ${\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }}$ and (c,f)  ${\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }}$ . The colour scale ranges from $-0.20$ to $0.20$ , from blue to red; —— propagation at the ambient speed of sound; – – –  $t_{1}=t_{c}$ .

Coming back to the correlations computed from centreline flow quantities, the correlations ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ are presented again in figure 20, together with mixed lines built from the convection velocities on the jet axis given in figure 11(b). The correlation spots coloured in blue turn out to follow these lines, revealing that their orientations result from the convection of the flow fluctuations in the jets (Bogey & Bailly Reference Bogey and Bailly2007). The time variations of these orientations are clearly seen in the corresponding movies.

Figure 20.

Space–time correlations ${\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ obtained between velocity fluctuations at $r=0$ and $t=t_{1}$ and pressure fluctuations at $r=10r_{0}$ (a) at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and (b) at $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ . The colour scale ranges from $-0.30$ to $0.30$ , from blue to red; —— propagation at the ambient speed of sound; – – –  $t_{1}=t_{c}$ ; –  $\cdot$  –  $\cdot$  trajectory based on the convection velocity on the jet axis.

In order to provide more quantitative results, the peak values of correlations determined from centreline fluctuations at time $t_{1}$ and pressure fluctuations at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ are plotted in figure 21 as a function of $t_{1}$ . Maximum values equal to 0.234, 0.132 and 0.180 for $Re_{D}=3125$ and to 1.122, 0.890 and 0.108 for $Re_{D}=12\,500$ are reached for $-{\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , ${\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }}$ and ${\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }}$ , respectively, slightly after or at the time of potential-core closing. Significant levels are also obtained before $t=t_{c}$ for ${\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }}$ in figure 21(a). After the maximum values at $t\simeq t_{c}$ , moreover, the levels decrease rather slowly for $-{\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , supporting that the flow velocity feature correlating with the emitted sound waves persists long after the time of noise generation.

Figure 21.

Representation of peak values of correlations between centreline quantities at $t=t_{1}$ and pressure fluctuations at $r=10r_{0}$ (a) at $t_{2}u_{j}/r_{0}=40$ for $Re_{D}=3125$ and (b) at $t_{2}u_{j}/r_{0}=35$ for $Re_{D}=12\,500$ : ——  $|\text{min}({\mathcal{C}}_{u_{z}^{\prime }p^{\prime }})|$ , – – –  $\max ({\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }})$ , –  $\cdot$  –  $\cdot$   $\max ({\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }})$ , as a function of $t_{1}$ ; $\cdots \cdots$   $t_{1}=t_{c}$ .

Finally, in the corresponding movies, the flow–noise correlation maps do not seem to depend much on the time of arrival of the acoustic waves at $r=10r_{0}$ . However, the correlation peak values gradually increase with time $t_{2}$ , as highlighted in figure 22. For $-{\mathcal{C}}_{u_{z}^{\prime }p^{\prime }}$ , for instance, they grow from 0.17 up to 0.27 for $Re_{D}=3125$ and from 0.10 up to 0.15 for $Re_{D}=12\,500$ between $t_{2}u_{j}/r_{0}=30$ and 45. Therefore, the flow–noise correlation levels are higher as the angle of radiation of the sound waves with respect to the flow direction is lower, which is in agreement with the experimental trends for spatial jets (Panda et al. Reference Panda, Seasholtz and Elam2005).

Figure 22.

Representation of peak values of correlations between centreline quantities at $t=t_{1}$ and pressure fluctuations at $r=10r_{0}$ at $t=t_{2}$ for (a)  $Re_{D}=3125$ and (b)  $Re_{D}=12\,500$ : ——  $|\text{min}({\mathcal{C}}_{u_{z}^{\prime }p^{\prime }})|$ , – – –  $\max ({\mathcal{C}}_{u_{z}^{\prime }u_{z}^{\prime }p^{\prime }})$ , –  $\cdot$  –  $\cdot$   $\max ({\mathcal{C}}_{|\unicode[STIX]{x1D714}|^{\prime }p^{\prime }})$ , as a function of $t_{2}$ .

3.5 Conditional averaging of the jet flow and pressure fields

In order to visualize and better understand the noise generation mechanism related to the intermittent intrusion of turbulent structures in the core of the present jets, a conditional averaging method is applied in this section to the flow and acoustic fields provided by the simulations. The method consists in averaging a limited number of samples, selected using a trigger condition, in a signal or in a movie. It has been employed in the past in many studies to isolate a specific phenomenon from the background noise. For aeroacoustic purposes, it has enabled researchers, for example, to extract coherent structures and velocity events in mixing layers and jets (Dahan et al. Reference Dahan, Elias, Maulard and Perulli1978; Hileman et al. Reference Hileman, Thurow, Caraballo and Samimy2005; Camussi & Grizzi Reference Camussi and Grizzi2014), and acoustic components in the pressure fields of free and impinging subsonic and supersonic jets (Kearney-Fisher, Sinha & Samimy Reference Kearney-Fisher, Sinha and Samimy2013; Akamine et al. Reference Akamine, Okamoto, Teramoto, Okunuki and Tsutsumi2016). Thanks to this method, Kastner et al. (Reference Kastner, Samimy, Hileman and Freund2006), in particular, found that the rapid breakdown of the flow large-scale structures plays an important role in the generation of subsonic jet noise.

In the present work, the signal for the sample selection is the signal of axial velocity on the jet axis at the time $t_{c}$ of potential-core closing, and the trigger condition relies on the use of a lower threshold. These choices are motivated by the fact that centreline axial velocity strongly correlates with the near-field pressure fluctuations long after $t_{c}$ , and that it is negatively skewed, as shown in previous sections. The corresponding signals obtained in the first simulation runs of the two temporal jets are depicted in figure 23. For both Reynolds numbers, as expected from the profiles of skewness factors of velocity of figure 6(b), intermittent negative spikes of velocity much lower than the jet velocity, even below $0.5u_{j}$ in certain cases, can be seen. In the conditional averaging procedure, the trigger events are identified by detecting the minimum values of velocity below a threshold of $(2/3)u_{j}$ , yielding a difference of approximately three standard deviations below the mean value for the signals considered. This threshold value is chosen arbitrarily, but the use of other velocity thresholds, such as $0.60u_{j}$ and $0.75u_{j}$ , for example, does not change the results qualitatively but only quantitatively, as verified in appendix D. The trigger events separated by a distance of less than $10r_{0}$ are discarded to avoid sample overlapping. This reduces for instance the number of samples to 5 in the two signals of figure 23, and leads to a total number equal to 45 for $Re_{D}=3125$ and 43 for $Re_{D}=12\,500$ . The flow and pressure fields are then recorded over windows centred around the position $z_{c}$ of the trigger events, and the resulting snapshots are ensemble averaged. This is carried out at time $t_{c}$ , but also at previous and subsequent times using the same synchronization with the negative spikes of axial centreline velocity at time $t_{c}$ , in order to get access to the time evolution of the noise generation mechanism. The results obtained in the four azimuthal planes at $\unicode[STIX]{x1D703}=0$ , $\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x03C0}$ and $3\unicode[STIX]{x03C0}/2$ are averaged, which allows us to increase convergence, but also to focus on the generation of the axisymmetric sound waves dominating in the jet near field, as shown by the spectra of figure 15. Despite the limited number of events selected, the averaged fields displayed in what follows clearly exhibit coherent features. This may be explained by the fact that these events are by far the most energetic ones in the flow.

Figure 23.

Axial velocity obtained at $r=0$ and $t=t_{c}$ for (a)  $Re_{D}=3125$ and (b)  $Re_{D}=12\,500$ ; – – –  $u_{z}/u_{j}=2/3$ ; $\triangle$ events selected for the conditional averaging.

The conditional fields of velocity fluctuations in the jet flow and of the pressure fluctuations outside, obtained at six times distributed every $5r_{0}/u_{j}$ between $t_{c}-10r_{0}/u_{j}$ and $t_{c}+15r_{0}/u_{j}$ , are represented in figure 24 for $Re_{D}=3125$ and in figure 26 for $Re_{D}=12\,500$ ; see also the corresponding movies movie_fig24 and movie_fig26. The conditional fields of vorticity fluctuations at  $t_{c}-5r_{0}/u_{j}$ , $t_{c}$ and $t_{c}+5r_{0}/u_{j}$ are also given in figures 25 and 27. In the figures, velocity contours are drawn for the values of one third and two thirds of the mean centreline velocity in order to make the extent of the mixing layer visible. Similar features are observed for the two jets despite the difference in Reynolds number. The amplitudes are however lower in the pressure field for the higher Reynolds number case, in agreement with the results of § 3.3.

Figure 24.

Representation of the fluctuations of axial velocity for $r\leqslant 1.75r_{0}$ and of pressure otherwise, obtained for $Re_{D}=3125$ at (a)  $t_{c}-10r_{0}/u_{j}$ , (b)  $t_{c}-5r_{0}/u_{j}$ , (c)  $t_{c}$ , (d)  $t_{c}+5r_{0}/u_{j}$ , (e)  $t_{c}+10r_{0}/u_{j}$ and (f)  $t_{c}+15r_{0}/u_{j}$ using conditional averaging; ——  $u_{z}=(1/3)\langle u_{z}(r=0)\rangle$ and $u_{z}=(2/3)\langle u_{z}(r=0)\rangle$ . The colour scales range from $-0.075u_{j}$ up to $0.075u_{j}$ and from $-90$ up to $90~\text{Pa}$ , from blue to red.

Figure 25.

Representation of the fluctuations of vorticity norm for $r\leqslant 1.75r_{0}$ and of pressure otherwise, obtained for $Re_{D}=3125$ at (a)  $t_{c}-5r_{0}/u_{j}$ , (b)  $t_{c}$ and (c)  $t_{c}+5r_{0}/u_{j}$ using conditional averaging; ——  $u_{z}=(1/3)\langle u_{z}(r=0)\rangle$ and $u_{z}=(2/3)\langle u_{z}(r=0)\rangle$ . The colour scales range from $-0.75u_{j}/r_{0}$ up to $0.75u_{j}/r_{0}$ and from $-90$ up to $90~\text{Pa}$ , from blue to red.

Figure 26.

Representation of the fluctuations of axial velocity for $r\leqslant 1.75r_{0}$ and of pressure otherwise, obtained for $Re_{D}=12\,500$ at (a)  $t_{c}-10r_{0}/u_{j}$ , (b)  $t_{c}-5r_{0}/u_{j}$ , (c)  $t_{c}$ , (d)  $t_{c}+5r_{0}/u_{j}$ , (e)  $t_{c}+10r_{0}/u_{j}$ and (f)  $t_{c}+15r_{0}/u_{j}$ using conditional averaging; ——  $u_{z}=(1/3)\langle u_{z}(r=0)\rangle$ and $u_{z}=(2/3)\langle u_{z}(r=0)\rangle$ . The colour scales range from $-0.075u_{j}$ up to $0.075u_{j}$ and from $-40$ up to $40~\text{Pa}$ , from blue to red.

Figure 27.

Representation of the fluctuations of vorticity norm for $r\leqslant 1.75r_{0}$ and of pressure otherwise, obtained for $Re_{D}=12\,500$ at (a)  $t_{c}-5r_{0}/u_{j}$ , (b)  $t_{c}$ and (c)  $t_{c}+5r_{0}/u_{j}$ using conditional averaging; ——  $u_{z}=(1/3)\langle u_{z}(r=0)\rangle$ and $u_{z}=(2/3)\langle u_{z}(r=0)\rangle$ . The colour scales range from $-0.75u_{j}/r_{0}$ up to $0.75u_{j}/r_{0}$ and from $-40$ up to $40~\text{Pa}$ , from blue to red.

At the time of potential-core closing, in figures 24(c) and 25(b) for $Re_{D}=3125$ , and in figures 26(c) and 27(b) for $Re_{D}=12\,500$ , spots of strong negative and positive values are found, respectively, in the velocity and vorticity fields on the jet axis around $z=z_{c}$ , as expected due to the trigger condition used in the averaging procedure. In the pressure fields, lobes of positive fluctuations located at $z\simeq z_{c}$ and surrounded by two smaller lobes of negative fluctuations are noticed. They can be regarded as the traces of the turbulent structures in the jet near field (Arndt et al. Reference Arndt, Long and Glauser1997).

The axial profiles of velocity and vorticity fluctuations at $r=0$ and of pressure fluctuations at $r=2.5r_{0}$ at $t=t_{c}$ are plotted in figure 28. For both jets, negative and positive spikes emerge for velocity and vorticity. They are centred around $z=z_{c}$ , are of length $0.6r_{0}$ , and exhibit extremum values of $-0.38u_{j}$ and $2.5u_{j}/r_{0}$ for $Re_{D}=3125$ and of $-0.36u_{j}$ and $3.5u_{j}/r_{0}$ for $Re_{D}=12\,500$ . At the same axial position, a hump of positive values peaking at 324 Pa for $Re_{D}=3125$ and at 215 Pa for $Re_{D}=12\,500$ is seen in the pressure profiles. It is the central part of the instability wave, of typical wavelength $12r_{0}$ , appearing in the pressure fields of figures 24(c) and 26(c). Therefore, when the potential core closes, the eruption of shear-layer vortical structures through the centre of the jet results in a velocity deficit and a vorticity excess in the flow, and in hydrodynamic pressure waves in its immediate vicinity.

Figure 28.

Representation of the fluctuations of (a) axial velocity and (b) vorticity norm at $r=0$ , and of (c) pressure at $r=2.5r_{0}$ , obtained at $t=t_{c}$ using conditional averaging for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ .

The evolutions of these flow and pressure patterns are revealed in figures 24–27 and in the corresponding movies. In both jets, before the potential-core closing, a spot of lower velocity and higher level of vorticity relative to the background flow field develops in the interfacial region between the mixing layer and the potential core. It is discernible at a very early stage, at $t_{c}-10r_{0}/u_{j}$ around $z=z_{c}-7r_{0}$ in the velocity fields of figures 24(a) and 26(a), and at $t_{c}-5r_{0}/u_{j}$ around $z=z_{c}-3.5r_{0}$ in the vorticity fields of figures 25(b) and 27(b). Its strength is characterized in figure 29 showing the extremum values of velocity and vorticity fluctuations at $r=0$ and between $r=0$ and $r=(2/3)r_{0}$ as a function of time. In absolute values, they first increase relatively slowly, but rise sharply after $t_{c}-3r_{0}/u_{j}$ . They reach a peak at $t\simeq t_{c}$ when the vortical spot arrives on the jet axis, and thereafter collapse. For $t\geqslant t_{c}$ , the structure extracted by the conditional averaging also decelerates. It is thus located consecutively around $z=z_{c}$ , $z_{c}+4r_{0}$ , $z_{c}+7r_{0}$ and $z_{c}+9r_{0}$ in figure 26(c–f), for example. Moreover, the velocity deficit appears to persist over a long time period, and is still quite visible at $t_{c}+15r_{0}/u_{j}$ for both jets, with a large extent in the axial direction. On the contrary, the vorticity excess vanishes rapidly, and is hardly detectable at $t_{c}+5r_{0}/u_{j}$ in figures 25(c) and 27(c).

Figure 29.

Time variations of (a) the minimum values of axial velocity fluctuations and (b) the maximum values of vorticity norm fluctuations obtained at $r=0$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , and between $r=0$ and $r=(2/3)r_{0}$ for – – –  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ using conditional averaging.

The development and disappearance of the high-vorticity spot on the high-velocity side of the mixing layer is accompanied by the growth and decay of a hydrodynamic pressure wave in the jet near field. To quantify this phenomenon, the normalized maximum values of pressure fluctuations obtained at $r=2.5r_{0}$ , as well as the corresponding values averaged over $2r_{0}\leqslant r\leqslant 3r_{0}$ , are represented in figure 30(a). For both jets, the two profiles are very similar to each other, indicating that the shape of the signal envelope does not depend appreciably on the radial position. In addition, the peak values of pressure fluctuations are achieved at $t\simeq t_{c}+2r_{0}/u_{j}$ . Interestingly, this is roughly the time when the correlations between the centreline velocity and the radiated noise are the strongest (refer to figure 21), suggesting that noise generation occurs at that moment.

Outside the jets, the emission and propagation of acoustic waves in the downstream direction are then clearly visible for both jets. One dominant wave of positive pressure, and two weaker waves of negative pressure on both sides, are radiated. They are attached to the flow velocity deficit at $t=t_{c}$ , but progressively move further away at later times as they travel at the speed of sound, more rapidly than the flow structures. The noise generation mechanism highlighted here strongly resembles that associated with a wave packet, exemplified in Tam et al. (Reference Tam, Viswanathan, Ahuja and Panda2008), Suzuki (Reference Suzuki2010) and Jordan & Colonius (Reference Jordan and Colonius2013). In the present work, it is clearly identified in a subsonic jet flow without resorting to any wave packet modelling.

The axial positions of the wave packets in the present jets, given in practice by those of the pressure peaks in the near field, are presented in figure 30(b) as a function of time. They are nearly identical for the two jets, regardless of the Reynolds number. Furthermore, they are very close to the positions of the minimum values of velocity in the flow, also depicted in the figure. This demonstrates the connection between the structure developing at the edge of the potential core and the hydrodynamic wave outside the jet flow. Both appear to travel at a rather uniform velocity between $t_{c}-10r_{0}/u_{j}$ and $t_{c}+5r_{0}/u_{j}$ , but subsequently at a velocity decreasing with time.

Figure 30.

Time variations (a) of the maximum values of pressure fluctuations, normalized by their peak values, averaged over $2r_{0}\leqslant r\leqslant 3r_{0}$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ and at $r=2.5r_{0}$ for – – –  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ , (b) of the axial positions of the pressure peaks averaged over $2r_{0}\leqslant r\leqslant 3r_{0}$ for ——  $Re_{D}=3125$ and

  $Re_{D}=12\,500$ and of the minimum values of axial velocity between $r=0$ and $r=(2/3)r_{0}$ for ○  $Re_{D}=3125$ and ○  $Re_{D}=12\,500$ ; – – – propagation at $0.8u_{j}$ .

The velocities estimated from the axial positions of the wave packets over segments of duration $5u_{j}/r_{0}$ are shown in figure 31. For both jets, despite the presence of oscillations, they are close to 65 per cent of the initial jet velocity at $t_{c}-10r_{0}/u_{j}$ , then gradually increase up to approximately $0.75u_{j}$ and finally fall down for $t>t_{c}+5r_{0}/u_{j}$ . Compared to the convection velocities calculated at $r=0$ and $r=r_{0}$ from velocity cross-correlations, they differ significantly. First, they are always higher than the convection velocity at $r=r_{0}$ , which can be related to the fact that the structure developing in the flow is not located in the centre of the mixing layer, as mentioned above. Second, at $t\simeq t_{c}$ , they are lower than the convection velocity at $r=0$ suggesting that in average, the structure accelerates less than it could have been expected when the potential core closes. Inversely, it appears to slow down suddenly shortly after.

Figure 31.

Time variations of —— the velocities calculated from the positions of the near-field pressure peaks represented in figure 30(b) and of the convection velocities at – – –  $r=0$ and –  $\cdot$  –  $\cdot$   $r=r_{0}$ for (a)  $Re_{D}=3125$ and (b)  $Re_{D}=12\,500$ .

Finally, the present conditional fields provide explanations for the properties of the correlation maps obtained between flow and acoustic fluctuations for the temporal jets, but also for spatially developing jets (Schaffar Reference Schaffar1979; Bogey & Bailly Reference Bogey and Bailly2007). The absence of velocity deficit and vorticity excess in the mixing layer seems to be responsible for the negligible correlations computed from the flow signals at $r=r_{0}$ . The persistence of a velocity deficit over a long period and the rapid disappearance of a vorticity excess on the jet axis may also be the reasons for which the correlations based on the centreline signals remain significant long after the time of sound generation for velocity but not for vorticity. Lastly, several pressure waves of alternating sign are emitted during the noise generation process, which may be the cause of the presence of the multiple spots of notable level in the correlation fields.