3.1. Experimental set-up

The experiments were performed at the Bruit & Vent facility, located at the Prometée plateforme, in Chasseneuil-du-Poitou (France). The facility is composed of an anechoic room, measuring $12.6 \times 10.6 \times 7.85$ m $^3$ and a cutoff frequency of 200 Hz. The facility allows jet flows up to Mach number 1.4. Acoustic measurements were conducted with a circular microphone array with radius $R/D = 14.3$ , containing 18 B&K type 4944-B 1/4-inch pressure-field microphones, equally distributed in the azimuthal direction, with $\varDelta \varphi = 20^\circ$ azimuthal angle between neighbouring microphones (Amaral et al. Reference Amaral, Hasparyk, Lebedev, Eysseric, Cavalieri, Maia and Jordan2023a ).

Figure 4(a) shows a sketch of the microphones distribution on the circular antenna. The coordinate system, i.e. the radial and azimuthal directions ( $r$ and $\varphi$ , respectively), is shown in red colour. The streamwise direction ( $x$ ), not shown in the figure, points out of the page toward the reader.

Figure 4.

Sketch representing the microphones distribution over the circular antenna (a) and the cylindrical surface covered by the acoustic measurements (b). Coordinate system in red colour.

A side view of the experimental apparatus is shown in figure 4(b). The coordinate system, shown in red, has origin at the nozzle exit centre, i.e. $(x/D,r/D) = (0,0)$ . Moreover, $\theta = \tan ^{-1}(R/x) = \tan ^{-1}(14.3D/x)$ indicates the polar angle and $\varrho = R/\sin (\theta )$ is the polar radius. With the aid of a traverse system, the antenna, represented in green, can be moved in the streamwise direction (Amaral et al. Reference Amaral, Lebedev, Eysseric and Jordan2023b , Reference Amaral, Lebedev and Jordanc ; Hasparyk et al. Reference Hasparyk, Jordan, Lesshafft, Pickering and Colonius2023, Reference Hasparyk, Jordan, Martini and Lesshafft2024). Hence, in the current set-up, it is possible to measure the sound in a cylindrical surface of $R = 14.3 D = 715$ mm radius in the $0 \leqslant x/D \leqslant 39.29$ streamwise range, corresponding to a polar range of 20° $\leqslant \theta \leqslant$ 90°. The current measurements were discretised using a $5^\circ$ step in polar angle ( $\Delta \theta =5^\circ$ ). A projection of the cylindrical surface used in the acoustic experiments is denoted by the dashed rectangle in the figure, which contains relevant dimensions.

Acoustic signals were sampled at 200 kHz, with 10 s of acquisition time. Regarding jet Mach numbers, measurements were conducted in the range $0.4 \leqslant M_{\kern-1pt j} \leqslant 0.9$ with increments of $\Delta M_{\kern-1pt j} = 0.1$ . The flow was set at isothermal condition for all Mach numbers studied herein. Amaral et al. (Reference Amaral, Lebedev and Jordan2023c ) describes with more detail the control strategies employed to maintain the jet in isothermal condition and constant Mach number during the experiments. Table 1 displays the Reynolds number ${Re}_D = {U_{\kern-1pt j} D}/{\nu }$ , with $\nu$ the kinematic viscosity and $U_{\kern-1pt j}$ the jet exit speed for the Mach number of interest.

Table 1.

Values of Reynolds number ( ${Re}_D$ ) and jet flow speed at the nozzle exit ( $U_{\kern-1pt j}$ ) as a function of the jet Mach number ( $M_{\kern-1pt j}$ ).

3.2. Data post-processing

The first step to post-process the data began with the application of a Gaussian filter over the time series prior to downsampling the data from 200 kHz to 100 kHz. The pressure measurements $\boldsymbol{p}$ were then Fourier decomposed in the azimuthal direction as

(3.1) \begin{equation} \boldsymbol{p}\left (x,r=R/D,\theta ,t\right ) = \sum _{m} \boldsymbol{p}_m\left (x,r=R/D,t\right ) e^{i m \theta } + \mbox{c.c.} , \end{equation}

where $i = \sqrt {-1}$ , $t$ denotes time, $m$ denotes the azimuthal mode and $\mbox{c.c.}$ indicates the complex conjugate. Although the tabbed nozzle is not axisymmetric, we chose to show the acoustic results in terms of azimuthal modes in order to compare the tabbed nozzle results with the reference sound radiation by the baseline jet, whose azimuthal mode decomposition of the sound field is well documented (Jordan & Colonius Reference Jordan and Colonius2013).

The noise spectra were evaluated using Welch’s method (Welch Reference Welch1967), by segmenting the pressure time series into blocks of $N_b = 2048$ size, with 75 % overlap. Each block was windowed using a Hann function prior to computing the fast Fourier transform (FFT). The resulting spectra were obtained by averaging the power spectra of each segment. Correction factors associated with the windowing function energy loss as well as with the mirroring of spectra to negative frequencies owing to the FFT real-valued signals. The spectral representation of each azimuthal component of the data is given by

(3.2) \begin{equation} \hat {\boldsymbol{p}}_m\left (x,r=R/D,\omega \right ) = \sum _\omega \boldsymbol{p}_m\left (x,r=R/D\right ) e^{i \omega t} + \mbox{c.c.} , \end{equation}

where $\omega = 2 \pi f$ denotes the frequency and the hat indicates components transformed from a time to frequency domain.

The dimensionless power spectral density (PSD, in dB St⁻¹) was obtained for each azimuthal mode and Strouhal number following Brès et al. (Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018),

(3.3) \begin{equation} {\textit{PSD}} = 10 \log _{10}\left [\frac {\text{diag} ({\hat {\unicode{x1D64B}}}\,)}{{p_{\textit{ref}}}^2}\frac {U_{\kern-1pt j}}{D}\frac {1}{{M_{\kern-1pt j}}^4}\right ]\! , \end{equation}

where ${\hat {\unicode{x1D64B}}}=\langle\hat{p}_m \hat{p}_n\rangle$ indicates the pressure cross-spectrum density (CSD), in Pa $^2$ St⁻¹, $\text{diag} ({\cdots} )$ denotes that only the diagonal values are gathered (i.e. the pressure auto-spectrum) and $p_{\textit{ref}} = 2 \times 10^{-5}$ Pa is the reference acoustic pressure level. All the data will be presented as a function of the Strouhal number, which is given by

(3.4) \begin{equation} St = \frac {f D}{U_{\kern-1pt j}} \mbox{.} \end{equation}

Figure 5.

Acoustic spectra for polar angle $\theta =$ 90° (nozzle exit, sideline) for both tripped ( $-$ ) and untripped ( $\hbox{-}\hbox{-}$ ) configurations. (a,b,c) Axisymmetric mode ( $m =$ 0), (d,e,f) first helical mode ( $|m| =$ 1) and (g,h,i) second helical mode ( $|m| =$ 2). (a,d,g) Mach 0.4, (b,e,h) Mach 0.7 and (c,f,i) Mach 0.9. Panels (a,b,c) display the full sound radiation without azimuthal decomposition, shown as thick curves.

3.3. Results

Figures 5 and 6 show noise spectra at $90^\circ$ (nozzle exit plane, sideline) and $30^\circ$ (downstream) polar angles, respectively, for the baseline (red curves) and tabbed (blue curves) nozzles. The figures display the spectra for the axisymmetric mode, $m = 0$ , the first two helical modes, $|m| = 1$ and $|m| = 2$ (where $|{\cdots} |$ means that both the positive and negative counterparts of the helical modes are taken into account), and Mach numbers $M_{\kern-1pt j} =$ 0.4, 0.7 and 0.9. Results are shown for the nozzles with a boundary-layer trip (continuous lines) and without it (dashed lines), with the full sound radiation shown as thick curves. Results for $M_{\kern-1pt j} =$ 0.5, 0.6 and 0.8 follow the same trend, and are included in Appendix A. The acoustic spectra are presented in terms of azimuthal decomposition, following verification of far-field homogeneity in the present measurements – an observation that is also supported by previous studies (Tam & Zaman Reference Tam and Zaman2000). Appendix A provides an example of the acoustic spectra prior to azimuthal decomposition, where the homogeneity of the acoustic field can be verified.

Figure 5 shows the sideline results. Starting at the lowest Mach number studied ( $M_{\kern-1pt j}=0.4$ ), one can observe a noise reduction of up to 3 dB St⁻¹ in the axisymmetric component of the tripped jet, for a wide range of Strouhal numbers, up to $St \approx$ 1, with larger reductions at lower frequencies. The reduction is even more pronounced for the untripped configurations, with the tabbed nozzle presenting a noise reduction of up to 6 dB St⁻¹ for $M_{\kern-1pt j} =$ 0.4. Differences between tripped and untripped round jets are expected, since the resulting noise is dependent on the state of the boundary layer inside the nozzle (Brès et al. Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018). It should be noted that one can only ensure that the boundary layer has transitioned to turbulence upstream of the nozzle exit when the trip ring is employed, especially at lower Reynolds numbers. Interestingly, the presence of the boundary-layer trip has virtually no effect on the noise generated by the tabbed jets, indicating that the boundary-layer dynamics in this case might be dominated by the tab-generated streaks, potentially causing the flow to transition even without the presence of the trip. Increasing the Mach number reduces the gap between the tripped and untripped spectra, diminishing the noise reduction provided by the tabs to approximately 2 dB St⁻¹ for $M_{\kern-1pt j} =$ 0.7 and 1.5 dB St⁻¹ for $M_{\kern-1pt j} =$ 0.9. It is expected that the tabs will be more effective at $M_{\kern-1pt j} =$ 0.4, since their design was based on boundary-layer thickness obtained at this Mach number (§ 2). Similar trends are observed for the first two helical modes.

In most cases, for the sideline results, a small increase in the noise generated is observed at higher frequencies, around the roll-off region of the spectrum. However, the sound produced at these frequencies has lower amplitudes, which mitigates the impact of this amplification. In some cases ( $M_{\kern-1pt j}=0.9$ , $m=0$ for instance), small peaks are observed in the spectrum at high frequencies in the tabbed jets, which could be related to a secondary instability of the strong streaks generated by the tabs.

Figure 6.

Acoustic spectra for polar angle $\theta =30^\circ$ (downstream). See the comments in the caption of figure 5.

At the polar angle $\theta =30^\circ$ , the noise reduction is not as pronounced, as shown in figure 6. However, reductions are still observed in a wide range of frequencies, with benefits of up to 2 dB St⁻¹ for the tripped nozzle and 3 dB St⁻¹ for the untripped one (for the $M_{\kern-1pt j} =$ 0.4, $m =$ 0 case). As before, increasing the Mach number reduces the trip and tab effects, but reductions of up to 1.5 dB St⁻¹ are still attained for $M_{\kern-1pt j} =$ 0.9. As before, the boundary-layer trip is more effective for the baseline configuration, and has virtually no effect on the tabbed nozzles. As observed in Cavalieri et al. (Reference Cavalieri, Jordan, Colonius and Gervais2012), the axisymmetric mode shows higher intensity levels at lower frequencies and also dominates the spectra for lower polar angles (e.g. at $\theta =30^\circ$ ). The helical modes, on the other hand, tend to dominate the spectra at higher polar angles (e.g. at $\theta =90^\circ$ ).

The overall sound pressure level (OASPL) as a function of the polar angle for the same cases shown in figures 5 and 6 is shown in figure 7. To evaluate the OASPL as a function of polar angle, the microphone traverse was moved downstream to obtain measurements at the desired angular increment $\delta \theta$ , and the spectra were adjusted using a far-field assumption (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012). The noise spectra were integrated over the 0.1 $\leqslant St \leqslant$ 3.5 range. Noise reductions here follow roughly the same trends observed in the previous spectra; for instance, for the axisymmetric mode, noise reductions of up to 2 and 3 dB are observed when applying the tabs for both the tripped and untripped configurations, respectively, especially up to $\theta \approx 60^\circ$ . For the tripped case with $M_{\kern-1pt j}=0.4$ and $m=0$ , the noise benefit reduces for higher polar angles. Nonetheless, consistent reductions in OASPLs are observed in all analysed cases and across all polar angles, indicating that the reduction trends observed in the spectra are consistent. For instance, for higher azimuthal modes, untripped configuration and low Mach numbers, the tabbed nozzle provides noise reductions of up to 4 dB in these conditions. As in Cavalieri et al. (Reference Cavalieri, Jordan, Colonius and Gervais2012), the axisymmetric mode shows a marked directivity towards low polar angles, whereas the helical modes are primarily directed in the $35^\circ\!-\!55^\circ$ polar angle range. The axisymmetric mode exhibits higher levels at low polar angles, whereas the helical modes dominate for polar angles higher than $30^\circ$ . Further comparisons for other Mach numbers also exhibit the same behaviour, and can be found in Appendix A.

Figure 7.

Overall sound pressure level for polar angles in the $\theta =20^\circ\!-\!90^\circ$ interval for tripped (closed symbols) and untripped (open symbols) configurations. Power levels evaluated for 0.1 $\leqslant St \leqslant$ 3.5. (a,b,c) Axisymmetric mode ( $m =$ 0), (d,e,f) first helical mode ( $|m| =$ 1) and (g,h,i) second helical mode ( $|m| =$ 2). (a,d,g) Mach 0.4, (b,e,h) Mach 0.7 and (c,f,i) Mach 0.9.

Overall, the acoustic data show that the application of tabs consistently reduces the acoustic radiation of all the jets studied for a wide range of subsonic Mach numbers, even though tabs were only designed to induce significant streak growth at the lowest Mach number studied. Reductions are observed at all angles, and for all azimuthal wavenumbers, without the usual high-frequency penalty observed in chevron jets (Callender et al. Reference Callender, Gutmark and Martens2005; Zaman et al. Reference Zaman, Bridges and Huff2011). Considering the link between coherent structures and noise radiation (Mollo-Christensen Reference Mollo-Christensen1967; Crighton Reference Crighton1975; Jordan & Colonius Reference Jordan and Colonius2013; Cavalieri et al. Reference Cavalieri, Jordan and Lesshafft2019), a physical explanation for these reductions requires detailed analysis of the flow hydrodynamics and the changes in coherent structures induced by the presence of streak-generating tabs. This is addressed in the following.

4.1. Experimental set-up

Time-resolved PIV measurements were based on LaVision hardware, using two Phantom 2640 CMOS cameras and a Continuum Mesa solid Nd-YAG laser. The optical set-up was designed to obtain a laser sheet parallel to the nozzle plane and the seeding particles made of atomised glycerin and a heavy fog fluid. The laser sheet thickness and the mean size of the seeding particles are approximately 3 mm and 1.5 $\mu$ m, respectively. As in the acoustic experiments, the flow was set at isothermal condition, and the Mach number was chosen as $M_{\kern-1pt j} = 0.7$ (see table 1). LaVision hardware and software (Davis) were employed to perform the TR-PIV measurements, with an acquisition frequency of 12.5 kHz (which yields a Nyquist Strouhal number of approximately $\textit{St}_{\textit{Nyquist}} \approx 1.3$ ), and the time between two laser exposures was set at 2 $\mu$ s. Measurements were performed in cross-flow planes for streamwise stations from near the nozzle exit up to $x/D = 10$ , with the positions specified in table 2. These were chosen so as to capture the initial growth and saturation of the K–H mode, the end of the potential core and to evaluate the recircularisation of the jet further downstream. The three velocity components were measured for each cross-flow plane. After processing the raw data, the snapshots contain $(N_y,\,N_z) = (194,\,142)$ grid points in Cartesian coordinates and the field of view has approximately $250 \times 200$ mm $^2$ ( $5 D \times 4 D$ ).

Table 2.

Streamwise positions measured in the TR-PIV campaign.

Figure 8 shows a sketch of the stereo PIV set-up, which includes the coordinate system used in this study. The streamwise direction is denoted by $x$ , whereas $(y,\,z)$ indicate the cross-flow directions in Cartesian coordinates and $(r,\,\varphi )$ are the radial and azimuthal directions in polar coordinates, i.e. $r = \sqrt {y^2+z^2}$ , $y = r \cos (\varphi )$ and $z = -r \sin (\varphi )$ . A traverse system enables the movement of the cameras and laser in the streamwise direction keeping the cameras and laser relative position fixed for all measurements.

Figure 8.

Sketch representing the TR-PIV set-up, containing the nozzle, cameras, laser, field of view, coordinate system and all the relevant dimensions.

4.2. Data post-processing

The first step of the data post-processing is the interpolation from the Cartesian PIV grid to a polar coordinate system. Prior to this, it is necessary to find the jet axis $(y_c,\,z_c)$ for each measurement plane ( $x$ ). The mean streamwise velocity component ( $u$ ) was used to determine the geometric centre of the jet axis, following the procedures by Gudmundsson & Colonius (Reference Gudmundsson and Colonius2011), i.e.

(4.1a) \begin{align} y_c &= \displaystyle\frac {\sum _{n=1}^{N_y} \bar {u}_n y_n}{\sum _{n=1}^{N_y} \bar {u}_n}, \\[-12pt] \nonumber \end{align}

(4.1b) \begin{align} z_c &= \displaystyle\frac {\sum _{n=1}^{N_z} \bar {u}_n z_n}{\sum _{n=1}^{N_z} \bar {u}_n}, \end{align}

where $n$ denotes a grid point index. A bicubic interpolation scheme was employed to obtain the snapshots in polar coordinates (Jaunet, Jordan & Cavalieri Reference Jaunet, Jordan and Cavalieri2017; Lesshafft et al. Reference Lesshafft, Semeraro, Jaunet, Cavalieri and Jordan2019; Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019), using a grid composed of $(N_\varphi ,\,N_r) = (144,\,75)$ points in the azimuthal and radial directions, respectively.

The jet flow issuing from the tabbed nozzle exhibits a discrete rotational symmetry due to the presence of $L = 12$ tabs, i.e. the flow is invariant under rotations of $2\pi /L$ . Consequently, the flow is not axisymmetric, but periodic in the azimuthal direction with period $2\pi /L$ . This periodicity allows the flow to be decomposed using a Floquet expansion in the azimuthal direction. In this decomposition, the flow state can be represented as a superposition of azimuthal harmonics grouped according to their Floquet coefficient $\mu \in (-L/2,L/2]$ , which defines a symmetry group of the solution. Each Floquet coefficient corresponds to a family of azimuthal wavenumbers $m$ , given by the relation

(4.2) \begin{equation} m = \mu - jL , \end{equation}

where $j$ is an integer indexing the repetitions of the pattern every $L$ harmonics. Thus, the flow field can be expressed as (Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016; Lajús et al. Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019; Rigas et al. Reference Rigas, Pickering, Schmidt, Nogueira, Cavalieri, Bres and Colonius2019)

(4.3) \begin{equation} \boldsymbol{q}(x,r,\varphi ,t) = \sum _{\omega } \sum _{\mu } \sum _{j} \boldsymbol{\tilde {q}}(x,r) e^{i \omega t + i(\mu - jL)\varphi } + \mbox{c.c.} , \end{equation}

where $\boldsymbol{q}$ is the physical space solution and $\boldsymbol{\tilde {q}}$ represents the spatial structure of the mode associated with the Floquet coefficient $\mu$ , frequency $\omega = 2 \pi f$ and azimuthal wavenumber $m = \mu - jL$ .

This formulation makes explicit the fact that in the presence of azimuthal periodicity, the wavenumbers are grouped: each Floquet mode $\mu$ contains contributions from an infinite set of azimuthal harmonics spaced by $L$ . For instance, for $L = 12$ and an azimuthal resolution of $N_\varphi = 144$ points, one obtains $j \in (-N_\varphi /2L,N_\varphi /2L] = (-6,6]$ , resulting in twelve harmonics per Floquet group.

For clarity, the correspondence between the Floquet coefficient $\mu$ and the set of azimuthal wavenumbers $m$ is summarised in table 3. In this work, both the baseline (axisymmetric) and tabbed nozzle configurations are decomposed using this Floquet-based approach to ensure a consistent comparison, even though for the baseline case a standard Fourier transform would suffice.

Table 3.

Floquet coefficients $\mu$ and the associated azimuthal wavenumbers $m$ for $L =$ 12 and $N_\varphi =$ 144.

Spectral proper orthogonal decomposition of the velocity data is performed to extract the most energetic, coherent structures. Spectral proper orthogonal decomposition is a technique that decomposes the CSD of the flow state components into an orthonormal basis that optimally represent the data ensemble in terms of energy for a given inner product (Taira et al. Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017). For the present study, the three velocity components are used to build the CSD and the inner product corresponds to the TKE norm, accounting for numerical quadrature weights on the discrete grid. Hence, for each frequency, the SPOD modes are evaluated after the eigenvalue decomposition of the CSD, i.e.

(4.4) \begin{equation} \unicode{x1D63E} \unicode{x1D652} \boldsymbol{\varPhi } = \boldsymbol{\varPhi } \boldsymbol{\varLambda } , \end{equation}

where $\unicode{x1D652}$ accounts for both the TKE norm and the numerical quadrature weights, $\boldsymbol{\varPhi }$ denotes the SPOD modes, $\boldsymbol{\varLambda }$ is a diagonal matrix containing real eigenvalues and $\unicode{x1D63E}$ is the CSD of the velocity field, with

(4.5) \begin{equation} \unicode{x1D63E} = \big \langle \boldsymbol{\tilde {q}} \boldsymbol{\tilde {q}}^{\dagger } \big \rangle , \end{equation}

where $\langle \ldots \rangle$ denotes the expected-value operator and $\dagger$ indicates the trans-conjugate (hermitian) operator. The CSD is computed frequency-by-frequency for each Floquet coefficient $\mu$ . The contribution of each SPOD mode to the CSD representation is characterised by its associated eigenvalue, which is ranked according to the energetic content from the most to the least energetic, i.e. $\lambda _1 \geqslant \lambda _2 \geqslant {\cdots} \lambda _{n}$ . In this work we employ the snapshot version of the SPOD algorithm (Towne et al. Reference Towne, Schmidt and Colonius2018; Schmidt & Colonius Reference Schmidt and Colonius2020) to extract the modes and the CSD is evaluated using Welch’s method (Welch Reference Welch1967), blocks of $N_{\textit{fft}} = 128$ size and 50 % overlap, yielding a total of 421 blocks. To mitigate spectral leakage, a Hamming tapering window was applied over all blocks. These parameters, together with the jet diameter and the jet flow speed, provide a Strouhal number resolution of $\Delta St \approx 0.02$ .

Figure 9.

Mean streamwise velocity component ( $\bar {u}/U_{\kern-1pt j}$ ). The lipline is indicated by a continuous circle. (a,b,c) Baseline nozzle. (d,e,f) Tabbed nozzle. (a,d) $x/D =1$ , (b,e) $x/D =3$ and (c,f) $x/D =10$ .

4.3. Mean fields and fluctuation statistics

The mean streamwise velocity component, interpolated in the polar grid, is displayed in figure 9 for the baseline (a,b,c) and tabbed (d,e,f) nozzles. The lipline is shown in the figure as a continuous line circle. Near the nozzle exit ( $x/D = 1$ , panels a,d), the influence of the tabs on the flow field is made clear by the presence of regions of low- and high-flow velocity appearing periodically in azimuth. In this region, the velocity distribution exhibits twelve vertices (equivalent to the number of tabs) in the region around the jet shear layer. Further downstream, the jet flows from the baseline and tabbed nozzles are almost indistinguishable, especially far from the nozzle exit ( $x/D = 10$ , panels c,f). Note that all results presented in § 4.3 were obtained before the application of Floquet decomposition.

The streamwise velocity profiles as a function of radius for the baseline and tabbed nozzles for several streamwise positions are shown in figure 10. For the tabbed nozzle, the profiles are shown for two positions: between two tabs (closed symbols) and directly behind a tab (open symbols). Overall, the jet development has the expected trends, with the development of the shear layer in the downstream direction leading to the formation of the potential core. Clear differences between the shear layers of the baseline and the tabbed case are observed in the region close to the nozzle, with the tabs tending to thicken the shear layer. For instance, at $x/D = 1$ , the shear layer is strongly modulated by well-developed streaks, leading to distinct profiles behind and between tabs. The differences are progressively less prominent in positions further downstream, possibly due to the small characteristic length of the streaks for this high value of $L$ (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019). This difference diminishes as the streaks decay, and the profiles become virtually identical beyond $x/D = 3$ , indicating a recircularisation of the jet. After the end of the potential core (which is around $x/D = 5.8$ in the present case), the profile assumes a Gaussian-like profile. At $x/D = 10$ , the profiles are only slightly different, and the induced streaks seem to have little effect on the mean flow in that region.

Figure 10.

Radial profiles at different streamwise positions. Lines denote baseline nozzle and symbols tabbed nozzle. Closed symbols denote profiles taken in between two tabs and open symbols indicate profiles taken behind a tab.

The streamwise velocity root-mean-square (r.m.s.) distributions for the baseline and tabbed nozzles are shown in figure 11. As expected, the peak r.m.s. values for these jets start at the centre of the shear layer and spread radially; this holds for both baseline and tabbed nozzles. The tabbed jet, however, displays a series of spots of high- and low-intensity r.m.s. distributions in the jet shear layer region, creating a corrugated pattern in the distribution. This is in agreement with what is expected for flows containing streaky structures, where the development of the K–H mode is modulated by the streaks. Such behaviour is observed at least up to $x/D = 3$ , and is in contrast with the baseline case, which shows an approximately homogeneous ring structure in the shear layer region. Interestingly, the r.m.s. intensity is higher for the tabbed case near the nozzle, but decays more rapidly downstream than the baseline nozzle. This suggests that, although coherent structures modulated by streaks grow more rapidly near the nozzle, they stabilise more quickly than those in the baseline case.

Figure 11.

Root mean square of the streamwise velocity component ( $u_{\textit{rms}}/U_{\kern-1pt j}$ ). The lipline is indicated by a continuous circle. See the comments in the caption of figure 9.

For completeness, figure 12 shows the r.m.s. profiles. The reductions in r.m.s. due to the tabs is significant. At $x/D = 3$ and 10, the r.m.s. peak values are noticeably higher for the baseline configuration in comparison with the tabbed configuration. Closer to the nozzle, the peak r.m.s. magnitudes are similar, further supporting the fact that the main changes in the flow from the application of tabs are in fact associated with the development of coherent structures further downstream. This aspect will be investigated in the following.

Figure 12.

Root -mean-square profiles. See the comments in the caption of figure 10.

Note that the r.m.s. profile for the baseline configuration at $x/D = 1$ (see panel (a) in figure 11, green curves in figure 12) is noisy. However, this will not hamper the coherent structures analysis, as the SPOD also acts as filter, as will be seen in the following section.

Since thrust cannot be measured directly in our facility, Appendix D presents momentum flux estimates derived from the TR-PIV measurements. Within the margin of error, the impact of the tabs appears to be small; however, it should be emphasised that an accurate assessment of thrust penalty would require a dedicated measurement instrument.

4.4. Coherent structures

The signature coherent structures present in the turbulent flow field were obtained through SPOD of the velocity field for each Floquet exponent. Modal energy maps of the leading SPOD modes are exhibited in figures 13 to 16 as a function of the Floquet coefficient  $\mu$ . Only positive frequencies and Floquet coefficients are shown. Fourier decomposition symmetry rules apply for negative frequencies and Floquet coefficients. Most of the current analysis will be focused on the $\mu =0$ component, which is expected to contain the coherent structures most relevant to downstream-sound radiation. The convergence of the SPOD modes is discussed in Appendix C.

4.4.1. Modal energy maps

Near the nozzle exit (figure 13), the spectrum for the first SPOD mode of the tabbed nozzle displays a clear peak for $\textit{St} \rightarrow 0$ . This is the signature of the tab-generated streaks, which are expected to be dominant at $m=12$ , and should thus be included in the SPOD modes for $\mu =0$ . Apart from the peak, energy amplifications occur over a broad region of the $\mu$ -St spectrum with respect to the baseline case. The region exhibiting low-rank behaviour (indicated by high amplitudes in the $\lambda _1/\lambda _2$ maps) is also broader for the tabbed nozzle.

Figure 13.

Modal energy maps for $x/D =$ 1. Here $\mu$ stands for Floquet coefficient. (a,c) Baseline nozzle. (b,d) Tabbed nozzle. (a,b) Leading eigenvalue. (c,d) Ratio between the leading and first suboptimal eigenvalues.

Results for $x/D = 3$ (further downstream, but before the end of the potential core) are shown in figure 14. In this region, coherent structures generated by the jet shear are expected to be more developed, especially the K–H mode. At this location, the baseline spectrum exhibits a double-peak feature, concentrating energy at high $\mu$ and low $St$ , and at very low $\mu$ and moderate $St$ ( $0.4 \leqslant St \leqslant 1$ ). As shown in a series of previous works (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019; Rigas et al. Reference Rigas, Pickering, Schmidt, Nogueira, Cavalieri, Bres and Colonius2019; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020; Nogueira & Cavalieri Reference Nogueira and Cavalieri2021; Maia et al. Reference Maia, Brès, Lesshafft and Jordan2023), the first peak is associated with the growth of streaks via the lift-up effect in this free-shear flow, while the second is associated with the presence of the K–H wavepacket. The K–H-associated peak is significantly attenuated by the tabbed nozzle, whereas the streak-related peak is accentuated and broadened. This is consistent with the expected damping of K–H wavepackets by induced streaks. Furthermore, the decrease in eigenvalue separation observed in the tabbed nozzle in the low- $\mu$ zone indicates a deterioration of the low-rank behaviour of the jet (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brés2018; Lesshafft et al. Reference Lesshafft, Semeraro, Jaunet, Cavalieri and Jordan2019) in this spectral region. This is also consistent with the weakening of the K–H mechanism.

Figure 14.

Modal energy maps for $x/D =$ 3. See the comments in the caption of figure 13.

Figure 15 shows the modal energy maps at $x/D = 5.8$ . In this case, the energy is displayed on a linear scale rather than a logarithmic one, to facilitate direct comparison of energy levels between the baseline and tabbed nozzles. At this streamwise location, the energy levels begin to approach the same order of magnitude. Additionally, higher energy levels and greater gain separation are observed around $\mu = 4$ and as $\textit{St} \rightarrow 0$ .

Figure 15.

Modal energy maps for $x/D =$ 5.8. See the comments in the caption of figure 13.

Finally, far away from the nozzle exit, at $x/D = 10$ (figure 16), the energy maps of both nozzles are remarkably similar, although a slight attenuation can still be observed for the tabbed nozzle, with higher energy levels and eigenvalue separation around $\mu = 3$ and $\textit{St} \rightarrow 0$ . This behaviour is consistent with previous literature, which shows the presence of large-scale streaks at lower $\mu$ (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). The similarities between the two spectra are also consistent with the resemblance between the r.m.s. plots, indicating that the tabs have a minor influence on the turbulence dynamics in downstream stations.

Figure 16.

Modal energy maps for $x/D =$ 10. See the comments in the caption of figure 13.

4.4.2. Eigenfunctions

We now proceed to the analysis of selected eigenfunctions for $\mu = 0$ at several stations of the flow. Starting at $x/D = 1$ , as expected from the SPOD spectra shown in figure 13, the first mode of the tabbed nozzle has much more energy in the low Strouhal region, compared with the baseline case. The mode associated with this high-energy region is shown in figure 17, which is compared with the first mode at the same frequency for the baseline nozzle. One-fourth of the domain is exhibited in the figure, with the streamwise velocity component represented by the contours (red-to-blue colour maps) and the cross-flow velocity components given by the arrows. In this figure the modes are weighted by the corresponding eigenvalue to give a better sense of the difference in amplitude between the baseline and the tabbed nozzle – the contour levels are kept constant between the baseline and tabbed cases, enabling direct comparison. As expected, the mode in the tabbed nozzle displays streak-like structures similar to Nogueira et al. (Reference Nogueira, Cavalieri, Jordan and Jaunet2019) – note that the $\mu = 0$ Floquet coefficient includes the wavenumbers $m = [0,\,12,\,24,\ldots ]$ .

Figure 17.

First SPOD mode for $x/D =$ 1, $\mu =$ 0 and $\textit{St} \rightarrow$ 0. Contours, weighted by the eigenvalues, denote the real value of the streamwise velocity component, with the red to blue colour map denoting positive to negative values. The arrows indicate the cross-flow velocity components. Only one-fourth of the domain is shown.

Past studies have demonstrated the importance of the lift-up mechanism in shear flow (Ellingsen & Palm Reference Ellingsen and Palm1975; Landahl Reference Landahl1980; Brandt Reference Brandt2014) and in the dynamics and energy spectrum of turbulent jets (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020; Maia et al. Reference Maia, Heidt, Pickering, Colonius, Jordan and Brès2024). The lift-up mechanism is activated by streamwise vorticity that generates flow structures characterised by alternating regions of high and low streamwise velocity (streaks). The fact that distinctive streamwise vortex and streak features are observed in both the baseline (where there is clearly no wake dynamics) and tabbed configurations suggests that the tabs have indeed enhanced the lift-up mechanism already present in the baseline case. This conjecture is supported by figure 17.

Figure 18 displays the first SPOD mode associated with $\textit{St} \approx 0.8$ . Differently from figure 17, now only the streamwise velocity component mode is shown and for the full domain. The baseline configuration clearly exhibits an axisymmetric K–H mode, whereas the tabbed case has a much more complex structure. The dominant mode for the tabbed nozzle may be associated either with an azimuthal modulation of the original axisymmetric mode, or to modes of higher dominant azimuthal wavenumber ( $m = 12$ , in this case), as shown in Lajús et al. (Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019). At this position, it is unclear if this mode is directly associated with the sound-generating K–H wavepacket.

Figure 18.

First SPOD mode for $x/D =$ 1, $\mu =$ 0 and $\textit{St} \approx$ 0.8. Contours denote the real value of the streamwise velocity component, with the red to blue colour map denoting positive to negative values.

Now proceeding to $x/D = 3$ and Floquet coefficient $\mu = 0$ , the first mode of the baseline case is more energetic than the tabbed case (see figure 14), suggesting a damping of the dominant physical mechanism generating the coherent structures (in this case, the K–H instability), in line with previous results (Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016; Marant & Cossu Reference Marant and Cossu2018; Lajús et al. Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019; Wang et al. Reference Wang, Lesshafft, Cavalieri and Jordan2021). A sharp decay in energy is observed at a lower Strouhal number for the tabbed case, leading to a shift in the peak frequency for this case. This shift leads to strong reductions in the energy of the first mode around $0.3 \leqslant \textit{St} \leqslant 1$ .

Shapes of the resulting eigenfunctions are also compared in figures 19, 20 and 21 for $\textit{St} \approx 0.3$ , 0.5 and 0.7, respectively. The baseline case has the expected K–H shape, peaking close to the centre of the shear layer, and with the usual phase shift across it. Very small modulations following the L-fold coupling are also observed throughout the domain. This may result from SPOD’s slow convergence in order to separate independent azimuthal modes $m$ under these conditions, as the flow is expected to be axisymmetric. Regarding the tabbed nozzle, a strong modulation following the azimuthal wavenumber of the streaks ( $m = 12$ ) is observed throughout the domain, being more clearly identified at higher radial positions. This result is in line with previous linear stability results (Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016; Lajús et al. Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019; Wang et al. Reference Wang, Lesshafft, Cavalieri and Jordan2021), which indicated that the presence of corrugations around the shear layer (in this case, induced by the tab-generated streaks). This would change the shape of the K–H mode.

Figure 19.

First SPOD mode for $x/D =$ 3, $\mu =$ 0 and $\textit{St} \approx$ 0.3. See the comments in the caption of figure 18.

Figure 20.

First SPOD mode for $x/D =$ 3, $\mu =$ 0 and $\textit{St} \approx$ 0.5. See the comments in the caption of figure 18.

Figure 21.

First SPOD mode for $x/D =$ 3, $\mu =$ 0 and $\textit{St} \approx$ 0.7. See the comments in the caption of figure 18.

Finally, let us analyse the structures for $x/D = 10$ and $\mu =3$ , which is the most energetic region of the modal energy map at this location (figure 16). The spectra for both baseline and tabbed nozzles are very similar. The peak in the spectrum is, once again, found at $\textit{St} \rightarrow 0$ , and associated with large-scale streaky structures, as seen in Nogueira et al. (Reference Nogueira, Cavalieri, Jordan and Jaunet2019), Pickering et al. (Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020), Maia et al. (Reference Maia, Brès, Lesshafft and Jordan2023). Mode shapes at this peak frequency for both baseline and tabbed nozzles are virtually identical and associated with streaks of dominant azimuthal wavenumber $m = 3$ , as seen in figure 22. As shown in Nogueira et al. (Reference Nogueira, Cavalieri, Jordan and Jaunet2019), low-wavenumber streaks extend further in the streamwise direction, peaking further downstream in the domain.

Figure 22.

First SPOD mode for $x/D =$ 10, $\mu =$ 3 and $\textit{St} \rightarrow$ 0. See the comments in the caption of figure 18.

5.1. One-way Navier–Stokes

The current analysis is based on the linearisation of the compressible viscous Navier–Stokes equations, written in a cylindrical polar coordinate system and normalised by the ambient specific volume, the jet diameter and the ambient sound speed (Nogueira et al. Reference Nogueira, Weightman and Edgington-Mitchell2023, Reference Nogueira, Beekman, Weightman and Edgington-Mitchell2024). Similar to previous studies, the Reynolds number selected for the analysis was ${Re}=3\times 10^4$ , which is lower than the experimental value to account for the effects of turbulent viscosity in the structures (Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021a ); no optimisation of this parameter was attempted in this study, but it is well known that the effect of this viscosity on the overall shape of the K–H instability waves is usually minor for sufficiently high Reynolds numbers (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021). After linearisation, the vector of flow disturbances $\boldsymbol{\hat {q}}$ is represented using the Floquet ansatz in azimuth (Lajús et al. Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019; Nogueira & Edgington-Mitchell Reference Nogueira and Edgington-Mitchell2021), and in the frequency domain as

(5.1) \begin{equation} \boldsymbol{\hat {q}}(x,r,\phi ,\omega ) = \boldsymbol{q}(x,r,\phi ) {\textrm{e}}^{-\textrm{i}\omega t + \textrm{i}\mu \phi }, \end{equation}

where $\omega =2\pi St M_{\kern-1pt j}$ is the angular frequency and $\mu$ is the Floquet coefficient. Note that this approach takes advantage of the symmetry of the problem in the same way the post-processing of the experimental data did, which will facilitate the comparison between SPOD and OWNS results. The linearised system of equations may then be written in operator form as

(5.2) \begin{equation} -\textrm{i} \omega \boldsymbol{q}+\boldsymbol{A}\frac {\partial \boldsymbol{q}}{\partial x}-\boldsymbol{L}\boldsymbol{q}=0, \end{equation}

where the explicit form of operators $\boldsymbol{A}$ and $\boldsymbol{L}$ can be found in Appendix B. The system of equation (5.2) may be marched spatially by discretising the streamwise derivatives using an appropriate method. As in Zhu & Towne (Reference Zhu and Towne2023), an implicit midpoint method was chosen as it provides a good balance between accuracy and computational cost. Since the Navier–Stokes operators contain both downstream- and upstream-propagating modes, the spatial march is generally unstable; this problem can be avoided by either rewriting the formulation so that the resulting system is approximately parabolic (Bertolotti & Herbert Reference Bertolotti and Herbert1991; Herbert Reference Herbert1994) or by explicitly filtering the upstream modes at each step of the march (Towne & Colonius Reference Towne and Colonius2015; Towne, Rigas & Colonius Reference Towne, Rigas and Colonius2019). One-way Navier–Stokes follows the latter approach, with filtering in the current formulation performed in a recursive manner (Zhu & Towne Reference Zhu and Towne2023) as

(5.3) \begin{align} \boldsymbol{q}^0_n &= \frac {1}{h}\boldsymbol{q}_n, \end{align}

(5.4) \begin{align} \big(\boldsymbol{L}+\textrm{i}\omega -\textrm{i}\boldsymbol{A}\beta ^*_j \big)\boldsymbol{q}^j_n &= \big(\boldsymbol{L}+\textrm{i}\omega -\textrm{i}\boldsymbol{A}\beta ^-_j \big)\boldsymbol{q}^{j-1}_n, \quad j=1,2,\ldots ,N_\beta , \end{align}

(5.5) \begin{align} \boldsymbol{q}_n &= \boldsymbol{q}^{N_\beta }_n, \end{align}

where $\boldsymbol{q}_n$ is the solution at the position $x_n$ in the flow. From the initial solution $\boldsymbol{q}^0_n$ , obtained from the spatial march, the field is filtered $N_\beta$ times to remove the signature of upstream waves in the march. The definition of the recursion parameters $\beta ^*, \beta ^-$ and the factor $h$ follows previous works and can be found in Zhu & Towne (Reference Zhu and Towne2023).

The method requires an initial field to be marched in the downstream direction. This initial solution is obtained by solving the locally parallel problem close to the nozzle plane ( $x/D = 0.1$ ), which can be written as

(5.6) \begin{equation} \textrm{i} \alpha \boldsymbol{A}\boldsymbol{q}_0=(\textrm{i} \omega \boldsymbol{I} + \boldsymbol{L})\boldsymbol{q}_0, \end{equation}

where $\alpha$ is the complex eigenvalue whose real and imaginary parts represent the wavenumber and growth rate of the disturbances, respectively. This value is also used in the definition of recursion parameters to ensure that the desired mode is well captured in the spatial march. The eigenspectrum for the current problem contains several modes associated with different wavenumbers following the Floquet ansatz (Lajús et al. Reference Lajús, Sinha, Cavalieri, Deschamps and Colonius2019), and care must be taken to select the most appropriate mode for the march. Here, the most unstable mode whose eigenfunction is dominated by $m = 0$ disturbances for each frequency were chosen. This mode is expected to be dominant further downstream, and to be responsible for most of the downstream radiation in these jets (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012).

The computational domain is discretised using Chebyshev polynomials in the radial direction and Fourier in azimuth (Trefethen Reference Trefethen2000; Weideman & Reddy Reference Weideman and Reddy2000), as in previous studies. A radial mapping following Lesshafft & Huerre (Reference Lesshafft and Huerre2007) was used in order to provide an appropriate point density in the region of interest, around the shear layer. The mean flow is defined based on the experimental mean velocity slices, and is further described below. A mesh of $(N_r,N_\phi ) = (100,32)$ was used, considering that only a sector of $\Delta \phi = 2\pi /12$ needed to be discretised due to the application of the Floquet ansatz. The streamwise step size was also chosen as $\Delta x/D = 0.1$ , which was sufficient to converge the main features of the coherent structures evaluated herein.

5.2. Comparisons with experimental data

5.2.1. Definition of base flow

The current experimental dataset contains time-resolved velocity fields at selected streamwise stations, which allows for the azimuthal characterisation of the dominant coherent structures in the flow. On the modelling side, the cross-plane slices also allow evaluation of locally supported waves using locally parallel linear stability analysis, but this approach cannot be used to evaluate the streamwise development of the structures. Here we use OWNS to overcome that limitation. The method requires a 3-D mean flow as input; here we attempt to define an appropriate mean flow based on the limited planar measurements available. It is important to keep in mind the limitations of this approach, as no information about the exact state of the shear layers at each spatial point is available, and the spacing between streamwise slices is too large for reliable interpolation. Nevertheless, the mean flow characteristics follow the same trends observed in previous results in the literature, which provides some level of confidence in the approach, despite these limitations.

The mean flows at each streamwise station are Fourier transformed in azimuth to separate the streak component from the axisymmetric component of the mean. The latter is then fitted to the expression

(5.7) \begin{equation} U_0(x,r)= \begin{cases} U_c(x) &\quad \text{if } r \leqslant r_c(x), \\ U_c(x) \ {\textrm{e}}^{-\left ( \frac {r-r_c(x)}{\delta _s(x)} \right )^2} &\quad \text{if } r \gt r_c(x),\\ \end{cases} \end{equation}

where $U_c$ is the centreline velocity, $r_c$ controls the position of the shear layer and $\delta _s$ controls its thickness. As shown in Rodríguez et al. (Reference Rodríguez, Cavalieri, Colonius and Jordan2015), all these parameters are slowly varying in the streamwise direction, which will lead to a well-behaved interpolation process. This interpolation is performed using a modified Akima cubic Hermite method in Matlab (Akima Reference Akima1974), with special attention to ensuring the non-negativity of the $r_c$ parameter. The same process is followed for the $m = 12$ component of the mean flow, where streaks will be present in the tabbed nozzle; for this component, a Gaussian profile is assumed in radius, and the resulting coefficients are smoothly interpolated in the streamwise direction. The resulting behaviour of all three coefficients in expression (5.7) is shown in figure 23 for both nozzles. At the initial streamwise station, both nozzles have roughly the same coefficients, but changes in the shear layer development (parameters $r_c$ and $\delta _s$ ) are observed very close to the nozzle. These changes also appear to cause a shortening of the potential core, contrasting with previous observations for tabs and chevron nozzles (Zaman et al. Reference Zaman, Bridges and Huff2011; Wernet & Bridges Reference Wernet and Bridges2021). Such differences may be related to the lack of data near the end of the potential core or the high number of tabs used here (Bridges & Brown Reference Bridges and Brown2004). These features are confirmed in the reconstructed mean flows in the $(x,r)$ plane, shown in figure 24. Generally, the shear layer of the tabbed nozzle develops more rapidly and is thicker throughout most of the domain, becoming very similar near the domain’s end. The $m = 12$ component of the mean flow has higher amplitudes at the initial three diameters and decays afterwards, in line with the behaviour observed in Nogueira et al. (Reference Nogueira, Cavalieri, Jordan and Jaunet2019). The radial and azimuthal velocity components of the mean velocity were not considered, as no clear model exists for non-axisymmetric jets.

Figure 23.

Streamwise variation of the mean flow parameters for both round and tabbed nozzles.

Figure 24.

Azimuthal components of mean flow for both round and tabbed nozzles.

Thus, the reduced-order mean flow used in this work is based on a reconstruction using only the $m=0$ and 12 components of the mean flow, which are then curve-fitted into analytical functions (known to capture the overall behaviour of these components), and interpolated in the streamwise direction. The 3-D flow reconstruction is therefore composed by only two azimuthal components, which are expected to capture most of the effects of the streaks in the flow. A comparison between the original profiles extracted from the experimental data and the fitted curves is shown in figure 25, confirming that the current approach successfully approximates the mean flow characteristics at the stations where data are available. A reconstruction using both the $m=0$ and 12 components of the mean flow at some of the most relevant streamwise stations in the flow is shown in figure 26. These plots confirm that the resulting 3-D mean flow retains some of the most important characteristics of the flow, including the shear layer modulation induced by the streaks, especially in the near-nozzle region; this level of fidelity is essential for faithful predictions using linear models.

Figure 25.

Comparison between the axisymmetric component of the mean flow from experimental data and the resulting fitted curve at the streamwise stations used in the analysis for both round (a) and tabbed nozzles (b).

Figure 26.

Cross-plane reconstructions of the mean flow (right-hand side of the plot) compared with experimental data (left-hand side of the plot) for $x/D = 0.1$ , 1 and 3.

5.2.2. Streak-modulated wavepackets

The remainder of the analysis focuses on the coherent structures predicted by OWNS using as input the base flows obtained from velocity measurements. We focus on three frequencies, $St = 0.3$ , $0.5$ and $0.7$ , which cover most of the region of interest for jet noise. It is important to note that the tabs lead to two significant changes in the problem. The first is the changes in the axisymmetric part of the mean flow induced by the presence of the streaks, as shown in figure 24. The modified $m = 0$ component of the tabbed jet clearly exhibits different shear layer characteristics, which also affects the potential core length and centreline decay; thus, it is expected that coherent structures generated from both nozzles already differ due to the significant changes in the mean flow. The second effect is the presence of strong steady streaks in the time-averaged flow, which cause the azimuthal modulation of the shear layer. We look at these two effects separately by performing OWNS calculations using only the $m=0$ component of the mean flow for both nozzles, and the reconstruction using $m=0$ and 12 components for the tabbed nozzle. Wavepackets are predicted for $\mu = 0$ , which is known to be responsible for most of the downstream acoustic radiation (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012).

Figure 27.

Isosurfaces at $30\,\%$ of the maximum amplitude of the streamwise velocity (a,c,e) and pressure (b,d,f) for $St = 0.3$ , 0.5 and 0.7 and $\mu = 0$ . Only the real part of the tabbed case modes are shown.

An overview of the resulting 3-D streamwise velocity and pressure fields obtained from OWNS is shown in figure 27 for the tabbed case, which highlights the qualitative effect of shear layer modulation in the coherent structures. The first interesting feature shown in these plots is the fact that the azimuthal modulation is observed more clearly in the velocity, while the pressure field remains mostly axisymmetric. All structures have the same overall wavepacket shape, with their envelopes and wavelengths decreasing in size for increasing frequency, in line with the literature (Sasaki et al. Reference Sasaki, Cavalieri, Jordan, Schmidt, Colonius and Brès2017). The azimuthal modulation in the streamwise velocity follows the same trend, but its phase compared with the axisymmetric part of the wavepacket seems to change as it evolves in $x$ . Interestingly, the azimuthal modulation remains strong for the entire extent of the structure, even after the mean flow has become essentially axisymmetric. This suggests that even high-wavenumber streaks (which have a small streamwise extent, as shown in Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019), can affect the entire wavepacket structure, especially if they are present in the region where the K–H instability is most sensitive.

Figure 28.

Centreline signature of $m = 0$ modes.

To evaluate the effect of the tabs in the actual growth of the wavepacket, the absolute value of the velocity disturbances predicted by OWNS at the centreline is plotted in figure 28. Here, the initial amplitude of the wavepackets are considered to be the same; this is consistent with the similar initial r.m.s. magnitudes observed in both round and tabbed nozzles, shown in figure 12. It is clear that the K–H mode grows more strongly in the round nozzle for all frequencies, when compared with the tabbed nozzles. This is in line with the fact that the tabbed nozzles have thicker shear layers at most streamwise positions in the flow, which explains the stabilisation observed in the SPOD results and the far-field noise reductions. By removing the streaks from the mean flow and recomputing the wavepackets using only the axisymmetric part, a strong damping is still observed at the centreline, confirming that most of the effect of the tabs in the wavepacket development is via changes in the axisymmetric part of the mean flow, which contains the azimuthally averaged effect of the streaks. That said, the presence of streaks in the 3-D calculations provides a further reduction in amplitude, suggesting that the interaction between the streaks and the K–H instability leads to a further stabilisation not directly related to changes in the $m = 0$ component of the mean flow particularly at higher frequencies.

Figure 29.

Saturated pressure fields (real part) associated with the wavepackets predicted by OWNS for $\mu =0$ . Contours shown for $St = 0.3$ (a), 0.5 (b) and 0.7 (c), and for calculations based on the round nozzle (a), tabbed nozzle (b) and the $m = 0$ component of the tabbed nozzle (c).

One of the advantages of OWNS is the fact that the method captures part of the acoustic field associated with the predicted coherent structures (Towne & Colonius Reference Towne and Colonius2015; Towne et al. Reference Towne, Rigas and Colonius2019). In figure 29, saturated contours of pressure are shown for the cases studied herein, with all structures forced to have the same initial amplitude at the centreline, as in figure 28. One limitation of these plots is the fact that the domain only ranges from $x/D = 0.1$ to $10$ , and the characteristic acoustic beam generated from the structures should be outside this range for the lower frequencies. Even so, some interesting trends are observed. First, it is clear that the near-field acoustic signature of the wavepackets at lower frequencies ( $St = 0.3$ and 0.5) is weaker for the tabbed cases, especially if the streaky component of the mean flow is considered in the calculations. The beginning of the acoustic beam for these frequencies is observed by the end of the domain, and has a significantly lower amplitude for the tabbed cases as well, suggesting that some level of sound reduction can be achieved directly by the damping of the wavepacket using streaks. For the higher frequency ( $St=0.7$ ), the acoustic signature is generally weaker (which is expected as the wavepacket is less acoustically matched, Cavalieri et al. Reference Cavalieri, Jordan and Lesshafft2019). The changes in the mean flow by the tabs leads to some changes in the near-field directivity, and the model predicts a shift in the equivalent sound source position, which could lead to more sound radiation at slightly higher polar angles. It is important to remember that OWNS does not predict the entire acoustic field, but only the downstream-travelling part associated with the linear wavepacket; the sound field is also determined by other factors, including coherence decay (Cavalieri et al. Reference Cavalieri, Jordan, Agarwal and Gervais2011; Cavalieri & Agarwal Reference Cavalieri and Agarwal2014; Baqui et al. Reference Baqui, Agarwal, Cavalieri and Sinayoko2015), which is essential to recover the actual far-field sound radiation. Despite these limitations, the observed trends of sound reduction at lower frequencies are in line with the acoustic measurements presented in § 3.3.

Figure 30.

Absolute value of radial velocity from SPOD and OWNS. Results for $x/D = 1$ (a,b,c) and 3 (d,e,f), and $St = 0.3$ (a,d). 0.5 (b,e) and 0.7 (c,f).

Finally, a comparison between the structures obtained from OWNS and the dominant SPOD mode is shown in figure 30. Depending on the streamwise station, structures extracted using SPOD may be associated with streaks, Orr mechanism or K–H wavepackets (Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020), and the lack of streamwise coherence complicates direct mode identification. In an attempt to overcome the issue, comparisons are restricted to positions where the K–H mode is expected to be dominant, namely $x/D=1$ and 3, and comparisons are based on the absolute value of the radial velocity component, as this component offers clearer comparisons with modelling approaches (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021; Nogueira & Edgington-Mitchell Reference Nogueira and Edgington-Mitchell2021). Good agreement is found at higher frequencies, where the azimuthal modulation predicted by the model agrees well with the experimental data, despite the interpolation process of the mean flow. Interestingly, the model predicts a significantly broader radial support for the radial velocity, which could also be related to the differences in mean flow profile or due to the fact that the mean radial velocity was not included in the calculations. The agreement deteriorates for $St = 0.3$ and $x/D = 1$ , but at the moment it is unclear if the structure obtained from SPOD at this station is the same as the one observed further downstream; in fact, the shape of the velocity field for this mode resembles other $m = 12$ -dominated modes (either K–H or streak-like), which may be responsible for the discrepancy.