4.1.1. Mean optical phase map

The jet flow being axisymmetric, the map of mean optical phase $\overline {\delta \phi }$ is evaluated by ensemble and azimuthal averaging of 854 instantaneous maps acquired with the six holographic lines. This operation thus provides the mean zeroth-order azimuthal Fourier mode. The number of instantaneous optical phase maps considered is slightly reduced compared with the total number of acquired snapshots since some outliers for which the optical phase maps were not correctly unwrapped were discarded.

The resulting map is displayed in figure 9(a). In this figure, three particular axial locations are highlighted. First, the Mach disc is estimated to be located around $x=1.34D$. This value is in agreement with estimates reported by Addy (Reference Addy1981) and Nicolas et al. (Reference Nicolas, Donjat, Léon, Besnerais, Champagnat and Micheli2017) for a contoured converging nozzle. Similarly, a matching estimate of $0.35D$ for the Mach disc diameter is obtained. Second, the axial locations of the tips of the first two shock cells are evaluated to be around $1.83-D$ and $3.63-D$, providing a shock-cell spacing of approximately $1.8D$. Again, this estimate is consistent with the value generally obtained with choked jets at such a pressure ratio (Davies & Oldfield Reference Davies and Oldfield1962; Powell Reference Powell2010). The mean structure of the present choked under-expanded jet thus appears to be consistent with results reported in the literature.

Figure 9. (a) Map of mean unwrapped optical phase $\overline {\delta \phi }$ measured on a choked under-expanded jet at $\text {NPR}{}=5$ using the multi-view DHI set-up and (b) the mean density ratio $\bar {\rho }/\rho _{a}$ deduced by axisymmetric tomographic reconstruction, with $\rho _{a}$ the density of the quiescent atmosphere.

4.1.2. Mean density field

By construction, the mean optical phase map in figure 9(a) displays perfect reflection symmetry across the horizontal axis. As shown in figure 9(b), tomographic reconstruction of the associated density field using the methodology outlined in § 2.5 was then performed accordingly, assuming exact axisymmetry. This is achieved by considering slices of mean optical phase $\overline {\delta \phi }$ at all axial locations $x/D$ as input data for the tomographic process where it was chosen to use 64 repeated projections evenly distributed on half a circle. It can be noted that a more conventional approach relying on the inverse Abel transform could have been used for this task, as performed by Sugawara et al. (Reference Sugawara, Nakao, Miyazato, Ishino and Miki2020) for example. However, as emphasised subsequently, the purpose here is to provide validation results for the present tomographic methodology and to discuss the advantages of relying on a regularised tomographic approach.

A suitable value for the regularisation parameter $\mu$ in (2.17) is required for satisfactory density field reconstructions. The L-curve plots and the corresponding (log–log) graph curvature plots $\kappa (\mu )$ obtained for three different slices are displayed in figures 10(a) and 10(b), respectively. These L-curves give clear graph curvature maxima that are obtained for similar values of $\mu$. As a consequence, a single value $\mu =5\times 10^{-5}$ was considered for final reconstruction of the mean density field.

Figure 10. (a) L-curves $\log (\|\boldsymbol {\chi }\|_\text {TV})=f(\log (\|\boldsymbol{\mathsf{T}}\boldsymbol {\chi }-\boldsymbol {b}\|_{2}))$ obtained at three axial locations $x/D\in \{1,2,3\}$ for a logarithmically varying regularisation parameter $\mu$ in the reconstruction process of the mean density field $\bar {\rho }$. (b) The corresponding (log–log) graph curvature curves $\kappa (\mu )$ with $\kappa = f^{\prime \prime }/(1+f^{\prime 2})^{3/2}$.

A longitudinal cross-section of the reconstructed density field ratio $\bar {\rho }/\rho _a$ is shown in figure 9(b) and mean relative density profiles $(\bar {\rho }-\rho _{a})/\rho _{a}$ in this $(xz)$-plane are given in figure 11. The mean density field being axisymmetric, its 3-D structure can be easily pictured, particularly up to the first shock cell which appears to be very stable in time. Following the discussion provided by Panda & Seasholtz (Reference Panda and Seasholtz1999) for similar jets, different flow regions have been highlighted in figure 11 using dashed grey lines. The first line labeled (1) indicates approximately the curved boundary of the jet above which one finds the outer mixing layer. Line (2) indicates the presence of a weak shock generated by the coalescence of compression waves. This weak shock delimits the expansion region in the jet core and is known as the barrel shock or the intercepting shock. This expansion ends with a normal shock wave, before which the density falls well below the ambient one, forming the Mach disc identified by (4). From the point of intersection of the Mach disc and the barrel shock (known as the triple point) emanates a reflected shock labeled (3) and an embedded shear layer (5) which delimits a subsonic region downstream of the Mach disc and a supersonic flow region downstream of the reflected shock wave (3). As expected, a density larger than the ambient density (and than $\rho _{j}$) is found between this slip stream (5) and the reflected shock (3). The reflected shock (3) then intersects the external shear layer, giving rise to an expansion fan (6) which closes the first shock cell. Further downstream, a second expansion thus appears leading to a second, more diffuse shock cell. The structure of this density field and the density levels found appear to be in satisfactory agreement with the measurements obtained by Panda & Seasholtz (Reference Panda and Seasholtz1999) using Rayleigh scattering for jets at slightly lower and higher Mach numbers, i.e. $M_{j}=1.6$ and $1.8$, respectively.

Figure 11. Solid line: Radial profiles of mean relative density ratio $\bar {\rho }/\rho _{a}-1$ obtained by axisymmetric tomographic reconstruction at various axial locations $x/D$ overlayed on the map of mean density ratio shown in figure 9(b). Here $\rho _{a}$ refers to the ambient density. Each density profile has its origin horizontally shifted to coincide with the axial location of the slice in the mean density map displayed in the background. The scale of these profiles is given by the double-headed arrow along the $x$-axis. For conversion purpose, $\rho _{j}/\rho _{a}\approx 1.55$. The white dotted lines provide iso-contours of $\bar {\rho }/\rho _{a}-1$ at a level of $0.05$.

As displayed in figure 11, the density profiles are slightly staggered, which is a direct consequence of the use of TV regularisation because it promotes solutions for which $\boldsymbol {\nabla } \boldsymbol {\rho }$ is sparse. Although it can then be expected that the resulting density levels are not perfectly accurate, this drawback is compensated by a satisfactory capture of density jumps across the shock waves present in the flow. In particular, it can be highlighted that a sharp density jump across the reflected shock (3) is captured (see figure 11, $x/D=1.5$). It is interesting to note that, for comparison, Sugawara et al. (Reference Sugawara, Nakao, Miyazato, Ishino and Miki2020) did not retrieve such a feature in their case of study, probably because of their reconstruction process based on inverse Abel transform. Similarly, sharper density jumps across the embedded annular shear layer are obtained with a progressive thickening in the streamwise direction. As observed in figures 9(b) and 11, a second advantage over other tomographic reconstruction approaches is the absence of numerical artifacts in regions of uniform density such as the external region of the jet or within its core.

Finally, a quantitative comparison of the mean density ratio $\bar {\rho }/\rho _{j}$ estimated along the jet axis with other measurements reported in the literature is provided in figure 12. The present DHI results are displayed with a $\pm 5\,\%$ band to quantitatively illustrate the data dispersion associated with such an arbitrary but plausible uncertainty level. First, compared with tomographic BOS results obtained by Nicolas et al. (Reference Nicolas, Donjat, Léon, Besnerais, Champagnat and Micheli2017), similar density levels are observed but with a better capture of the density profile across the Mach disc. Indeed, tomographic BOS results probably suffer from a larger spatial resolution associated with the BOS optical set-up and a more complex tomographic reconstruction process relying on a $\ell _{2}$ regularisation which probably oversmooths the gradients. Second, compared with the density profile obtained by Panda & Seasholtz (Reference Panda and Seasholtz1999) at a higher Mach number $M_{j}=1.8$ ($\text {NPR}{}\approx 5.75$), a very similar shape is obtained, supporting the relevance of the reconstructed field. In the present case, the location of the Mach disc is closer to the jet exit plane and larger density levels (close to $\rho _{j}$) in the shock-cell regions are measured. This is expected because the higher the Mach number, the greater the excessive heating due to non-isentropic compression associated with the normal shock wave and thus the lower the density in these regions. Finally, it can be highlighted that the density profile measured after the first Mach disc is relatively flat, showing a plateau value over an extent in the streamwise direction corresponding to the internal subsonic region delimited by the embedded axisymmetric shear layer (from $x/D\approx 1.34$ to $x/D\approx 2$). A slightly different behaviour is observed in the second shock cell, with a progressive increase in density ($x/D\approx 3.5$), in agreement with the measurements reported by Panda & Seasholtz (Reference Panda and Seasholtz1999).

Figure 12. Mean density ratio $\bar {\rho }/\rho _{j}$ along the jet axis at $\text {NPR}{}=5$ ($M_{j}=1.71$) obtained with the present experimental apparatus (DHI) and with tomographic BOS (Nicolas et al. Reference Nicolas, Donjat, Léon, Besnerais, Champagnat and Micheli2017), with $\rho _{j}$ the fully expanded isentropic jet density. The axial profile measured by Panda & Seasholtz (Reference Panda and Seasholtz1999) using Rayleigh scattering at $M_{j}=1.8$ is also given for qualitative comparison purpose. The isentropic value at the nozzle exit is given by a red circle.

4.2.1. Phase fluctuation map

An instantaneous optical phase distribution obtained with one holographic line can be decomposed into a mean component $\overline {\delta \phi }$, discussed in the previous § 4.1, and a fluctuating component $\widetilde {\delta \phi }$, such that

(4.1)\begin{equation} \delta\phi(t, \eta, \xi, \theta_{k}) = \overline{\delta\phi}(\eta, \xi, \theta_{k}) + \widetilde{\delta\phi}(t, \eta, \xi, \theta_{k}) , \end{equation}

where $t$ refers to a discrete time, $(\eta, \xi )$ to the coordinates in the observation space and $\theta _{k}$ to the $k$th projection angle. Similarly, an instantaneous 3-D density field $\rho$ is decomposed into a mean field $\bar {\rho }$ and a fluctuating field $\tilde {\rho }$, writing

(4.2)\begin{equation} \rho(t, x, y, z) = \bar{\rho}(x, y, z) + \tilde{\rho}(t, x, y, z) , \end{equation}

where $(x, y, z)$ are the coordinates in physical space. By linearity of the operators in (2.3) and (2.4), the two optical phase terms $\overline {\delta \phi }$ and $\widetilde {\delta \phi }$ are the projected results at an angle $\theta _k$ of $\bar {\rho }$ and $\tilde {\rho }$, respectively. Analysing the statistical properties of $\widetilde {\delta \phi }$ thus provides information on the structure of $\tilde {\rho }$.

The azimuthally averaged field of root mean square (r.m.s.) values of $\widetilde {\delta \phi }$, referred to as $\delta \phi ^{\prime }$, is shown in figure 13 and highlights the projected regions of the flow with intense density variations. The reflected shock wave in the first shock cell appears to sustain some local but intense fluctuations, particularly at the junction with the external shear layer. Downstream, significant fluctuations in the jet shear layer and in the second shock cell can also be observed, as well as in the embedded shear layers to a lesser extent. Finally, this figure shows the well-known presence of a spatially stationary wave envelope outside the jet shear layer, which is the interfering result of downstream and upstream propagating waves that are building blocks in the screech feedback mechanism (Panda & Seasholtz Reference Panda and Seasholtz1999; Edgington-Mitchell Reference Edgington-Mitchell2019). The resulting wave is generally referred to as the jet near-field ‘standing wave’.

Figure 13. Map of root mean square values of temporal optical phase fluctuations $\delta \phi ^{\prime }= \text {r.m.s.}(\widetilde {\delta \phi })$.

4.2.2. Azimuthal structure of the observed projections

As emphasised previously, the six-view DHI set-up was designed to provide instantaneous optical phase maps with observation lines regularly distributed in the jet azimuthal direction, for projection angles $\theta _k\in \{0^{\circ }, 30^{\circ },\ldots, 150^{\circ }\}$. As the observed optical phases result from integration through the entire flow field, an augmented observation state for the optical phase fluctuations can be constructed by rotational symmetry around the jet axis, such that

(4.3)\begin{gather} \widetilde{\delta\phi}^a(t, \eta, \xi, \theta_k) = \widetilde{\delta\phi}(t, \eta, \xi, \theta_k), \end{gather}

(4.4)\begin{gather}\widetilde{\delta\phi}^a(t, \eta, \xi, \theta_k + 180^{{\circ}}) = \widetilde{\delta\phi}(t, \eta, -\xi, \theta_k) . \end{gather}

This augmented observation state is then defined for projection angles homogeneously distributed all around the jet, every $30^{\circ }$, and can be decomposed into azimuthal Fourier modes $\hat {\delta \phi }^a_{m}$ with azimuthal wavenumbers $m\in \{-N_{\theta },\ldots, N_{\theta }-1\}$, writing

(4.5)\begin{equation} \widetilde{\delta\phi}^a(t, \eta, \xi, \theta_k) = \frac{1}{2N_\theta} \sum_{m={-}N_{\theta}}^{N_{\theta}-1}\hat{\delta\phi}^a_{m}(t, \eta, \xi) \mathrm{e}^{\mathrm{i}m\theta_k} , \end{equation}

where $N_\theta =6$ in the present study. The instantaneous and average azimuthal structure of optical phase fluctuations can then be obtained, providing insights into the azimuthal structure of the flow density field.

The azimuthal mean energy distribution of the optical phase fluctuations is assessed by considering mean square values of the azimuthal Fourier modes amplitudes $|\hat {\delta \phi }^{a}_{m}|$. Indeed, time-averaging Parseval's relation allows to define a mean distribution of azimuthal optical phase fluctuation energy $\bar {E}_{\theta }(\eta, \xi )$ such that

(4.6)\begin{equation} \bar{E}_{\theta} = \sum_{k=0}^{2N_{\theta}-1} \text{var}(\widetilde{\delta\phi}^{a}(t, \theta_{k})) = \frac{1}{2N_{\theta}} \sum_{m={-}N_{\theta}}^{N_{\theta}-1}\langle|\hat{\delta\phi}^{a}_{m}(t)|^{2}\rangle , \end{equation}

where $\langle\,\cdot \,\rangle$ refers to the ensemble mean, $\text {var}(\,\cdot \,)$ to the variance operator along the time dimension and where the coordinates in the observation space have been omitted for clarity. The maps of normalised values of $\langle |\hat {\delta \phi }^{a}_{m}(t)|^{2}\rangle$ are displayed in figures 14(a)–14(d) for four azimuthal wavenumbers, the remaining modes having low energy content. Relying on (4.6), the normalisation term used is $\mathcal {M}=\max (2N_{\theta }\bar {E}_{\theta })$, permitting the identification of regions in the azimuthal Fourier modes contributing the most to the observed optical phase fluctuations. It is emphasised that in these figures, the contributions of positive and negative azimuthal wavenumbers ${\pm }m$ are summed, resulting in a factor of two on the maps of non-axisymmetric modes ($m>0$) because $\hat {\delta \phi }^{a}_{\pm m}$ are complex conjugates. Figure 14(b) highlights that most of the optical phase fluctuations are associated with the $m=\pm 1$ helical modes, with local values reaching up to 75 % of the observed maximum mean optical phase fluctuation energy. These large fluctuations are identified in the region of the flow where the second shock cell interacts with the external jet shear layer. Significant fluctuations are also present upstream, in the outer shear layer found between the two shock cells and in the reflected shock of the first cell. As displayed in figures 14(a), 14(c) and 14(d), the other azimuthal modes have approximately five times smaller maximum fluctuation energy amplitudes. Interestingly, all of these azimuthal modes capture significant fluctuations in the oblique shock of the first shock cell, particularly in the region where it intersects the external shear layer. This suggests that shock oscillation in the first shock cell cannot be explained primarily by the dynamics of a single pair of azimuthal modes, in contrast to the second shock cell whose motion appears to be largely driven by the first helical modes $m=\pm 1$.

Figure 14. Maps of relative energy distribution in the first four azimuthal Fourier modes of observed optical phase fluctuations: (a) $m = 0$, (b) $m = {\pm }1$, (c) $m = {\pm }2$ and (d) $m = {\pm }3$. For clarity only, the aerodynamic frame is used instead of the observation one. The normalisation term is $\mathcal {M}=\max (2N_\theta \bar {E}_\theta )$, see (4.6). As emphasised using coloured labels, (b) displays maximum values five times larger compared with (a), (c) and (d).

The dominance of the first pair of helical modes is made more evident in figure 15 which provides a measure of the relative magnitudes of the total energy contained in the maps $\langle |\hat {\delta \phi }^{a}_{m}|^{2}\rangle$ using the Frobenius norm denoted by $\|\,\cdot \,\|_{F}$. Although the first pair of helical modes at $m=\pm 1$ provides approximately 58 % of the total energy, the other azimuthal modes all have contributions of less than 10 %.

Figure 15. Relative magnitudes of phase fluctuation energy contained in the 2-D maps of mean square azimuthal Fourier modes amplitudes $\langle |\hat {\delta \phi }^{a}_{m}|^{2}\rangle$ as a function of the azimuthal wavenumber $m$, with summed contributions for complex conjugate pairs.

Finally, the map of r.m.s. values of optical phase fluctuations $\delta \phi ^{\prime }$ previously shown in figure 13 is decomposed in figures 16(a) and 16(b) by isolating and discarding the contributions of the first pair of helical modes, respectively. As observed in figure 16(a), the dominant pair of azimuthal modes at $m=\pm 1$ captures the footprint of the standing wave present in the external near-field of the jet, this feature being almost totally absent from the map associated with the remaining modes in figure 16(b). As previously stated, this standing wave can be associated with the screech feedback loop. This observation, along with the large optical phase fluctuations observed, suggest that the salient features of screech mode C characterised by a helical instability wave are captured by the present azimuthally filtered measurements.

Figure 16. Maps of r.m.s. values of temporal optical phase fluctuations $\delta \phi ^{\prime }$ obtained (a) by isolating the contributions of the first pair of complex-conjugate azimuthal modes ($m=\pm 1$) and (b) by considering the contributions of all the other azimuthal modes ($m\neq \pm 1$).

4.2.3. POD of the dominant helical Fourier mode

The previous azimuthal Fourier decomposition provided insights into the time-average azimuthal energy distribution of the observed optical phase fluctuations, highlighting that screech-related dynamical structures were embedded in the isolated complex-conjugate modes at $m=\pm 1$. Yet, because of the unsteady nature of the system, which is furthermore only observed at discrete and distant time intervals, exposing an average 3-D density picture of these structures is not straightforward.

In this work, it is proposed to rely on POD (Holmes et al. Reference Holmes, Lumley, Berkooz and Rowley2012) to efficiently isolate these coherent structures and to provide a simple modelling framework for their dynamics. POD aims at decomposing an ensemble of space–time-varying fields $\{f(t_{j}, \boldsymbol {x})\}_{j=1..P}$ into a linear combination of spatial orthogonal modes $\varPsi _{i}^{(f)}(\boldsymbol {x})$ optimally capturing its temporal variance or energy. Here, $\boldsymbol {x}\in \mathbb {R}^{3}$ refers to spatial coordinates in the frame of interest, which can be in the physical or in the observation space. It is emphasised that $f$ is considered in this work as a zero-mean complex-valued function and that, consequently, the spatial modes are also complex. The decomposition writes

(4.7)\begin{equation} f(t_{j}, \boldsymbol{x}) = \sum_{i=1}^{P}a_{i}(t_{j})\varPsi_{i}^{(f)}(\boldsymbol{x}) , \end{equation}

with $a_{i}(t_{j})$ the discrete temporal complex coefficients animating the spatial modes. In discrete form, data functions and spatial modes are replaced by $N_{xyz}$-vectors, with $N_{xyz}$ the number of spatial points considered, such that (4.7) may be written as

(4.8)\begin{equation} \boldsymbol{f}_{j} = \sum_{i=1}^{P}a_{i,j}\boldsymbol{\varPsi}_{i}^{(f)} . \end{equation}

Following Sirovich (Reference Sirovich1987), when the observations $\{\boldsymbol {f}_{j}\}$ are independent, this decomposition can be retrieved by considering the eigenvalue problem associated with the $P\times P$ temporal covariance matrix $\boldsymbol{\mathsf{C}}^{(f)}$ defined by elements $c^{(f)}_{m,n}=\boldsymbol {f}_{n}^{\operatorname {H}} \boldsymbol {f}_{m}/P$ where ${\cdot }^{\operatorname {H}}$ refers to the conjugate transpose operator. The eigenvectors of $\boldsymbol{\mathsf{C}}^{(f)}$ provide the temporal coefficients $a_{i,j}$ and the spatial modes are obtained by linear combination, such that

(4.9)\begin{equation} \boldsymbol{\varPsi}_{i}^{(f)} = \sum_{j=1}^{P} a_{i,j} \boldsymbol{f}_{j} . \end{equation}

Furthermore, $\boldsymbol{\mathsf{C}}^{(f)}$ being Hermitian, its eigenvalues $\lambda _{i}$ are real and give the energy associated with the spatial modes $\boldsymbol {\varPsi }_{i}^{(f)}$. This approach is computationally interesting when $P\ll N_{xyz}$ and is commonly referred to as the method of snapshots.

In this work, the $P$ augmented snapshots of optical phase $\widetilde {\delta \phi }^{a}$ are arranged as $QN_{\theta }^{a}$-dimensional vectors and stacked in a matrix $\boldsymbol {\varPhi }\in \mathbb {R}^{QN_{\theta }^{a}\times P}$, with $Q=MN_{x}$ the number of data points in one phase map and $N^{a}_{\theta }=2N_{\theta }$ the number of augmented projections. Snapshot POD is then applied to the deduced $P$ maps of helical Fourier modes at $m=1$ arranged as $Q$-dimensional vectors and stacked to form the snapshot matrix of Fourier modes $\hat {\boldsymbol {\varPhi }}_{m=1}\in \mathbb {C}^{Q\times P}$. The temporal covariance matrix can then be written $\boldsymbol{\mathsf{C}}^{(\phi )}=\hat {\boldsymbol {\varPhi }}_{m}^{\operatorname {H}}\hat {\boldsymbol {\varPhi }}_{m}/P$. Solving the associated eigenproblem provides the real eigenvalues $\lambda _{i}$ and the associated POD modes $\hat {\boldsymbol {\varPsi }}_{i}^{(\phi )}\in \mathbb {C}^{Q}$ for $i\in [1..P]$, which can be rearranged to form complex spatial maps of size $M\times N_{x}$. Selecting a limited number of main modes, an inverse discrete azimuthal Fourier transform may then be applied to obtain an arbitrary number $N^{\prime }_{\theta }>N_{\theta }$ of projections in the observation space of optical phase fluctuations, thus allowing an accurate reconstruction of the associated density fields to be performed relying on the tomographic reconstruction approach described in § 2.5.

This way of proceeding to estimate the density field of screech-related spatio-temporal modes is summarised in the flowchart shown in figure 17 using solid black arrows, where azimuthal Fourier transform and snapshot POD are directly performed on the entire set of observations. Other paths may however be imagined and it is important to clarify whether the present process yields main modes similar to those that would be obtained with azimuthal Fourier transform and POD directly applied to snapshots of density fields. Indeed, such an approach may be considered more physically relevant because the decomposition would then be performed in physical space and not in observation space. This way of proceeding is also illustrated in the flowchart shown in figure 17, but using dashed grey arrows. First, the inverse tomographic problem for the ensemble of acquired projections would have to be solved, providing the matrix $\boldsymbol{\mathsf{X}}\in \mathbb {R}^{N_{xyz}\times P}$ gathering the $P$ temporal snapshots of 3-D fields of $\tilde {\rho }$ obtained on a Cartesian grid of size $N_{xyz}=N_x M^{2}$. An azimuthal Fourier transform would then directly provide the azimuthal Fourier modes $\widehat {\boldsymbol{\mathsf{X}}}_{m}$ and the deduced covariance matrix would be $\boldsymbol{\mathsf{C}}^{(\rho )}=\widehat {\boldsymbol{\mathsf{X}}}_{m}^{\operatorname {H}}\widehat {\boldsymbol{\mathsf{X}}}_{m}/P$. Applying an inverse Fourier transform to the selected main POD modes would finally provide the main 3-D structures of density fluctuations associated with screech. Clearly, the major drawback with this approach lies in its first step, because it requires tomographic inversion of the entire set of projections. In practice, this represents a non-negligible computational burden and it cannot yield satisfactory results in the present work because of the limited number of available projection angles. Nonetheless, arguments can be provided to support the approximate equivalence of the two paths.

Figure 17. Flowchart illustrating possible approaches to evaluate the main screech-related coherent structures in the observed instantaneous density fields using snapshot POD. The dashed grey path indicates an idealised direct approach where azimuthal Fourier transform (FT) and POD would be performed on the reconstructed 3-D density fields snapshots. The solid black path describes the method that was followed in this work. The blue node highlights the starting point, the available observations, which are the sinograms gathering the variations of optical phase for each observation angle. The magenta dashed line indicates the possibility of applying a Radon transform between azimuthal modes.

First, it is highlighted that Radon and azimuthal Fourier transforms can be intertwined. Indeed, as demonstrated in Appendix A, this is the consequence of conservation of azimuthal symmetries by the Radon transform and of its linearity. The azimuthal Fourier modes in the observation space can thus be alternatively obtained writing $\hat {\boldsymbol {\varPhi }}_{m}=\boldsymbol{\mathsf{T}}_{m}\widehat {\boldsymbol{\mathsf{X}}}_{m}$ where $\boldsymbol{\mathsf{T}}_{m}$ refers to the Radon transform of an azimuthal Fourier mode. This relation is displayed in figure 17 using a magenta arrow. Given this result, the question then actually amounts to knowing if the temporal covariance matrices $\boldsymbol{\mathsf{C}}^{(\rho )}$ and $\boldsymbol{\mathsf{C}}^{(\phi )}$ provide similar eigenfunctions for the dominant modes. Noting that $\boldsymbol{\mathsf{C}}^{(\phi )}=\widehat {\boldsymbol{\mathsf{X}}}_{m}^{\operatorname {H}}(\boldsymbol{\mathsf{T}}_{m}^{\operatorname {H}}\boldsymbol{\mathsf{T}}_{m})\widehat {\boldsymbol{\mathsf{X}}}_{m}/P$, the two covariance matrices would be similar if the projection operator $\boldsymbol{\mathsf{T}}_{m}$ was orthogonal. This is not the case for the Radon transform and the projection–backprojection pair is known to produce blurred, spectrally filtered images (Epstein Reference Epstein2008; Paleo Reference Paleo2017). This spectral feature appears for example in the filtered-backprojection approach where a frequency filter is used to obtain a practical inversion algorithm. Here $\boldsymbol{\mathsf{T}}_{m}^{\operatorname {H}}\boldsymbol{\mathsf{T}}_{m}$ acts as a low-pass frequency filter in the image domain and thus promotes large-scale density variations in the covariance matrix $\boldsymbol{\mathsf{C}}^{(\phi )}$. Although a detailed quantitative analysis of this filtering process is out of the scope of the present work, these observations suggest that the main eigenfunctions obtained from $\boldsymbol{\mathsf{C}}^{(\phi )}$ and $\boldsymbol{\mathsf{C}}^{(\rho )}$ are likely to be similar and the two paths shown in figure 17 should yield comparable modes, thus allowing the main coherent large-scale density structures to be isolated.

4.2.4. A reduced dynamical model

Snapshot POD described in the previous section was applied to the $P=854$ instantaneous Fourier modes of optical phase fluctuations at the main azimuthal wavenumber $m=1$. The normalised eigenvalues of $\boldsymbol{\mathsf{C}}^{(\phi )}$ are shown in figure 18 in descending order and in logarithmic scale for only a few modes, together with the associated cumulative sum in linear scale. Two POD modes concentrate approximately 83 % of the optical phase fluctuation energy for this azimuthal mode, with the eigenvalues $\lambda _{1}$ and $\lambda _{2}$ being an order of magnitude larger than the others.

Figure 18. Normalised eigenvalues (vertical grey bars) in logarithmic scale of the temporal covariance matrix $\boldsymbol{\mathsf{C}}^{(\phi )}$ deduced from the snapshots of azimuthal Fourier modes at an azimuthal wavenumber $m=1$, and their cumulative sum (blue cross markers) in linear scale for a reduced number of POD modes.

To study the phase relationship that may exist between these two main POD modes, scatter plots of the real and imaginary parts of the complex POD mode coefficients $a_{1}(t_{j})$ and $a_{2}(t_{j})$ are displayed in figure 19. First, in figure 19(a), $\operatorname {Im}(a_{1}(t_{j}))$ is plotted as a function of $\operatorname {Re}(a_{1}(t_{j}))$ using a normalisation amplitude $\bar {r}$. The resulting distribution can be described by a circular orbit with radial dispersion, suggesting that the temporal coefficients of the first POD mode embed relatively coherent temporal dynamics subject to amplitude fluctuations. Without considering the observed dispersion, the evolution of amplitude and phase of the first POD mode can then be approximated by a reduced dynamical model $\widehat {\mathcal {M}}_{1}$ taking the form of a simple harmonic oscillator written as

(4.10)\begin{equation} \widehat{\mathcal{M}}_{1}(t_{j}, \boldsymbol{x}) = \bar{r} \, \widehat{\varPsi}_{1}^{(\phi)}(\boldsymbol{x})\operatorname{e}^{\mathrm{i} \varOmega(t_{j})}, \end{equation}

where $\widehat {\varPsi }_{1}$ is the first spatial POD mode, $\boldsymbol {x}=(\eta, \xi )$ refers to the spatial coordinates in the observation space and $\varOmega (t_{j})$ is a real-valued function providing the instantaneous phase of the mode. For example, assuming stationary cyclic dynamics for screech, it could be modelled as $\varOmega (t_{j})=\pm \omega _{s}t_{j}$ with $\omega _{s}$ the screech angular frequency, such that the black orbit shown in figure 19(a) would be followed at constant angular velocity in one direction of rotation or the other.

Figure 19. Normalised scatter plots obtained from the collection of the first two complex POD modes coefficients $\{a_i(t_j)\}_{i=1..2}$. Data points are colour-coded to highlight phase links across the plots. (a) Scatter plot of $\operatorname {Re}(a_1(t_{j}))/\bar {r}$ and $\operatorname {Im}(a_1(t_{j}))/\bar {r}$ with $\bar {r}$ a normalising constant; the marker colours encode the phase of $a_1(t_j)$, denoted as $\varphi _{1}(t_{j})$; the large black circle represents a model orbit or phase portrait in the POD phase space. (b) Scatter plot of $\operatorname {Re}(a_2(t_{j}))/\bar {r}$ and $\operatorname {Im}(a_2(t_{j}))/\bar {r}$; marker colours are still a function of $\varphi _{1}(t_{j})$; two different phase portraits are identified and modelled by blue and red circular orbits. (c,d) Separated scatter plots deduced from (b) highlighting the two different superposed distributions and the associated modelled orbits.

The second scatter plot shown in figure 19(b) displays $\operatorname {Im}(a_{2}(t_{j}))$ as a function of $\operatorname {Re}(a_{2}(t_{j}))$, normalised by the same scalar $\bar {r}$ as for the first POD mode coefficient. This graph is more complex to describe and, to facilitate its reading, each data point has been coloured according to the instantaneous phase of the first POD mode coefficient, i.e. $\varphi _{1}(t_{j})=\arg (a_{1}(t_{j}))$. This colour-coding can be observed in figure 19(a). It serves to highlight the phase relationships that exist between the two POD modes and allows for a simple cluster analysis to be performed visually. Two different distributions can be identified in figure 19(b), with more explicit cluster partitions separately drawn in figures 19(c) and 19(d). The first distribution highlighted in figure 19(c) shows a collection of points scattered around an approximate circular orbit (displayed in blue), with an average radius around $0.6\bar {r}$ and with markers colours indicating an approximate phase-match with the first POD mode coefficients. The second isolated distribution of points shown in figure 19(d) also describes an approximate circular orbit (drawn in red), but with a larger average radius around $0.75\bar {r}$ and an approximate antiphase relationship with the first POD mode coefficients. Consequently, for one state of the dynamical system described by the first POD mode, two different orbits may be exclusively followed in the phase space of the second POD mode. As shown thereafter, this behaviour captures the two possible azimuthal directions of rotation of the helical instability here observed. The probabilities of following either orbit in the phase space of the second POD mode are almost equal ($50\,\%\pm 0.5\,\%$), because a similar number of snapshots were found in the scatter plots figure 19(c,d). It is concluded that the helical structure subsequently presented has no preferred azimuthal direction of rotation about the jet axis. Finally, this state decomposition indicates that the studied jet can only support one of the two identified structures over a period of time. More particularly, it is evaluated that the transition probability from one state to the other over the time interval separating two snapshots, i.e. 0.1 s corresponding to approximately $460$ screech cycles, is approximately 7 % for the considered NPR.

This switching in the direction of rotation observed during continuous operation appears to be facility-dependent. For example, Powell et al. (Reference Powell, Umeda and Ishii1992) reported that in their experiments ‘the sense of rotation remained the same during continuous operation of the jet, but was liable to be different when the jet has been turned off and then on again’. In the present case, the observed intermittency suggests that the feedback loop has been stochastically perturbed during operation, allowing the dynamical system to switch between the two observed limit cycles. At least two possible mechanisms can be imagined. First, stochastic variations in upstream flow conditions (pressure variations, flow turbulence,$\ldots$) could have acted as forcing terms in the jet receptivity process. Second, the present experiment has a significantly larger nozzle lip thickness compared with Powell et al. (Reference Powell, Umeda and Ishii1992). This difference in boundary conditions may be of importance because the nozzle lip geometry plays a major role in receptivity and has been shown to have a significant effect on the feedback loop (Raman Reference Raman1999; Edgington-Mitchell Reference Edgington-Mitchell2019). Although a thicker nozzle lip is known to increase the strength of the screech noise and thus the feedback loop, an increase in the energy of the present stochastic dynamical system could also promote intermittent escapes from the two potential wells in which the two observed limit cycles evolve. Notably, a more stable helical mode has been observed by the present authors with the same jet facility but with part of the nozzle lip covered with acoustic dampening foam. The nature of this intermittency therefore needs to be clarified by complementary experimental studies of these two hypotheses.

The exact phase relationships between the two POD modes for the two possible orbits evidenced in figure 19(b) are made more explicit in figure 20, where the phase of the second POD mode, noted $\varphi _{2}$ is plotted as a function of $\varphi _{1}$. Linear relationships provide satisfactory models for the two observed trends with a phase difference of ${\rm \pi}$. As a consequence, extending the reduced model for the first POD mode given by (4.10), a simple linear model describing the main jet dynamics captured by these two POD modes in the azimuthal Fourier domain of the observation space can be proposed using two phase-coupled harmonic oscillators, writing

(4.11)\begin{equation} \widehat{\mathcal{M}}^{{\pm}}(t_{j}, \boldsymbol{x}) = \bar{r}\left(\widehat{\varPsi}_{1}(\boldsymbol{x}) + c_{2}^{{\pm}}\widehat{\varPsi}_{2}(\boldsymbol{x})\operatorname{e}^{\mathrm{i} p_{2}^{{\pm}}}\right)\operatorname{e}^{\mathrm{i}\varOmega(t_{j})}, \end{equation}

where $c_{2}^{\pm }$ and $p_{2}^{\pm }$ are parameters accounting for the reduced radii and the phase shifts associated with the two circular orbits identified in the phase space of the second POD mode. Note that the exponent ${\cdot}^{(\phi )}$ was omitted for clarity. Relying on figures 19 and 20, the two orbits are more explicitly described by $c_{2}^{+}\approx 0.6$ and $p_{2}^{+}\approx {\rm \pi}/10$ for the orbit almost in phase with the first POD mode (blue in figures 19b and 19c), and $c_{2}^{-}\approx 0.75$ and $p_{2}^{-}\approx 11{\rm \pi} /10$ for the second one associated with antiphase dynamics (red in figures 19b and 19d). In the context of dynamical systems theory, this model (4.11) is expected to approximate the attractor of the system whose limit cycle is here described by periodic orbits in POD phase space.

Figure 20. Scatter plot of $\varphi _{1}(t_{j})$ and $\varphi _{2}(t_{j})$, which are the phases of the complex POD modes coefficients $a_{1}(t_j)$ and $a_{2}(t_j)$ respectively. The markers are colour-coded by $\varphi _{1}$ as in figure 19. The blue and red solid lines represent linear phase relationships modelling the phase links observed in figure 19(b) for the two orbits displayed with the same two colours.

It is highlighted that the observed difference in orbit radii in the phase space of the second POD mode has not found a definite explanation. It has been verified that this result does not depend on the number of snapshots considered and that could have led to covariance convergence issues, nor apparently on the optical set-up because halving the number of projections gave similar results. It is conjectured that this difference in the mode amplitudes between the two observed states has a physical origin and actually captures slightly different density fluctuations between the two possible azimuthal directions of rotation of the observed helical instability wave. This could be induced by some subtle non-axisymmetric experimental conditions (jet swirl, nozzle geometry, jet surrounding, etc.) that could slightly favour the growth of the instability in one azimuthal direction, without significantly favoring the probability of rotation in one of the two directions.

4.2.5. POD spatial modes and projection in the observation space

The reduced model proposed in (4.11) is composed of two complex spatial modes $\widehat {\varPsi }_{1}$ and $\widehat {\varPsi }_{2}$ in the azimuthal Fourier domain of the observation space. Their spatial structure in terms of amplitude and phase are displayed in figure 21. Focusing first on the amplitudes shown in figures 21(a) and 21(c), it can be observed that both modes capture coherent optical phase fluctuations in different regions of the jet. For example, $\widehat {\varPsi }_{1}$ captures greater variance downstream and slightly upstream of the second shock cell, whereas $\widehat {\varPsi }_{2}$ displays maximum variations in the external shear layer just upstream of the second shock-cell and in the reflected shock of the first shock cell. It may also be noted that both modes embed the footprint of the external stationary wave, but that it is more pronounced in $\widehat {\varPsi }_{1}$. Focusing now on the phase of the two complex modes displayed in figures 21(b) and 21(d), well-defined patches of approximately constant phase can be observed, providing indications about the spatial coherence of the isolated fluctuations. For both modes, antiphase distributions are obtained with respect to the jet axis, which is expected because the helical mode at $m=1$ is studied. The first mode exhibits patches of constant phase around $-{\rm \pi} /2$ and ${\rm \pi} /2$, while the second mode displays patch values around $0$ and $-{\rm \pi}$, indicating that both modes are in azimuthal phase quadrature. These observations can be interpreted as follows: taken individually, the two modes $\widehat {\varPsi }_{1}$ and $\widehat {\varPsi }_{2}$ describe purely azimuthal fluctuations at $m=1$ and are not able to render features such as axial convection or helicity. These can only be rendered in the observed space by streamwise and azimuthal quadrature between two modes, which is a known feature of POD applied to convective instabilities (see, for example, Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011), Schmid, Violato & Scarano (Reference Schmid, Violato and Scarano2012) and Edgington-Mitchell et al. (Reference Edgington-Mitchell, Oberleithner, Honnery and Soria2014b) among others). As shown in the following section, summation of the two modes does indeed recover convective structures in the observation space and rotating helical density structures in the physical space.

Figure 21. (a,c) Amplitude and (b,d) phase maps of the two main complex POD modes $\widehat {\varPsi }_1(\boldsymbol {x})$ and $\widehat {\varPsi }_2(\boldsymbol {x})$ identified in the first azimuthal Fourier mode ($m=1$) of observed instantaneous optical phase fluctuations $\widetilde {\delta \phi }^a$.

The reduced model (4.11) is expressed in the azimuthal Fourier domain of the observation space. For the purpose of reconstructing the associated density field, an inverse azimuthal Fourier transform is applied to obtain complex and real-valued model projections in the observation space, referred to as $\mathcal {M}^{\pm }(t_j, \boldsymbol {x},\theta _k^\prime )$ and $\mathcal {P}^{\pm }(t_j, \boldsymbol {x},\theta _k^\prime )$ respectively. Here, owing to the known azimuthal symmetry of the model, $\{\theta _k^\prime \}_{k=1..N^\prime _{\theta }}$ can be arbitrary projection angles, particularly such that $N^{\prime }_{\theta }>N_{\theta }=6$, the actual number of holographic lines. This operation may be written as

(4.12)\begin{equation} \mathcal{P}^{{\pm}}(t_j, \boldsymbol{x},\theta_k^{\prime})=\underbrace{\frac{1}{2N_{\theta}}\widehat{\mathcal{M}}^\pm(t_j, \boldsymbol{x})\operatorname{e}^{\mathrm{i}\theta_k^\prime}}_{\mathcal{M}^{{\pm}}} + {\rm c.c.} \end{equation}

and equally spaced model projections on a half-circle were obtained with $\theta ^\prime _{k}=k{\rm \pi} /N^\prime _{\theta }$. Similarly to tomographic reconstruction of the mean density field in § 4.1, $N^{\prime }_{\theta }=64$ was observed to provide well-resolved density reconstructions.

4.2.6. Density reconstruction of the helical structure

The 3-D tomographic reconstructions obtained for $\mathcal {P}^{+}$ and $\mathcal {P}^{-}$ with two arbitrary values of $\varOmega$ are illustrated using 3-D contour plots in figures 22(a) and 22(b), respectively. Two laterally translated median longitudinal cross-sections extracted at $y=0$ and $z=0$ (grey scales) and an axially translated transverse cross-section at $x/D=3.85$ (at the edge of the reconstructed domain, coloured scales) are also displayed in each figure to further illustrate the internal density distribution of these large-scale structures. An animation is provided as additional material for better 3-D visualisation.

Figure 22. Illustrations of the 3-D density fields obtained by tomographic reconstruction (denoted $T^{-1}$) of the reduced dynamical models for two arbitrary values of model phase $\varOmega$: (a) $T^{-1}(\mathcal{P}^+)$ and (b) $T^{-1}(\mathcal{P}^-)$.

As indicated previously, the projected reduced models $\mathcal {P}^{\pm }$ clearly describe a helical structure, particularly visible in the external shear layer between the two shock cells, with two possible azimuthal directions of rotation. In the longitudinal cross-sectional planes, this dynamical model yields apparent convection and growth of density fluctuations as would be expected from an azimuthal instability wave. A particular feature of this result is that the density fluctuations associated with this reconstructed helical structure extend into the interior of the jet. Large density fluctuations can be observed in the embedded shear layer of the first and second shock cells. As seen in the transverse cross-sections showing coloured contour plots, these internal fluctuations are out-of-phase with the fluctuations observed in the external jet mixing layer. Similar observations have been reported by Edgington-Mitchell et al. (Reference Edgington-Mitchell, Honnery and Soria2014a) who performed planar PIV measurements in an under-expanded jet at a NPR of $4.2$ characterised by helical dynamics associated with a screech mode C. These authors suggested that the helical instability observed in the external jet shear layer was driving large velocity fluctuations in the jet embedded shear layers. Based on qualitative observations, they furthermore hypothesised that the motion of the reflected shocks may play an important role in the development of these inner shear layer fluctuations.

4.2.7. Analysis of the reconstructed density fluctuations

To investigate this point further, a more quantitative description of the density fluctuations captured by the present reduced model is provided in figure 23. The complex-valued version of the model $\mathcal {M}^{\pm }$ is considered instead of its real part $\mathcal {P}^{\pm }$ because the former intrinsically contains information about the spatial correlations of the density fluctuations. The amplitudes of the fluctuations normalised by $\rho _{j}$ in a longitudinal plane of the jet are first displayed in figure 23(a). These amplitudes are furthermore scaled by $2/\sqrt {2}$ to account for the contribution of the complex conjugate term in (4.12) to the overall density fluctuations and to provide r.m.s. estimates. With this scaling, this reduced model is shown to capture maximum r.m.s. density fluctuations of approximately $0.1\rho _{j}$ in the second shock cell, an order of magnitude that is in agreement with values reported by Panda & Seasholtz (Reference Panda and Seasholtz1999) using Rayleigh scattering measurements with phase-averaging on the screech noise. In this figure, the growth of the helical instability in the outer jet shear layer can be observed with a streamwise increase of the density fluctuations in the profiles located between the two shock cells. Significant fluctuations are also found in the vicinity of the oblique reflected shock of the first shock cell. As discussed by Panda (Reference Panda1998) and André et al. (Reference André, Castelain and Bailly2011), these fluctuations are the footprint of oblique shock oscillations that have been proposed to be induced either by upstream acoustic waves associated with the screech feedback loop or by instability waves growing in the outer mixing layer. The present quantitative results, and particularly the visualisations shown in figure 22, provide support for the presence of azimuthal shock oscillations for screech mode C, as suggested by Panda (Reference Panda1998). Similarly, the density fluctuations observed around the reflected shock of the second shock cell are also to be related to azimuthal shock oscillations, but with greater spatial dispersion due to larger cell motions.

Figure 23. (a) Amplitude and (b) phase maps of the density fluctuations captured by tomographic reconstruction of the reduced complex model $\mathcal {M}^{+}$ in the longitudinal plane of the jet. In (a), the black solid lines (black solid line) provide the profiles of fluctuation amplitude for $\mathcal {M}^{+}$ and the dark-grey dashed lines (grey dashed line) give the profiles for $\mathcal {M}^{-}$. In (b), the core region of the jet over the first two nozzle diameters is hashed because the phase estimates in this region are extremely uncertain.

Figure 23(a) further highlights the presence of intense density fluctuations in the embedded shear layers of both shock cells. In the first, the fluctuations in the internal layer appear to grow moderately in the streamwise direction, up to the end of the shock cell around $x=2-D$. This is in agreement with visualisations by Yip et al. (Reference Yip, Lyons, Long, Mungal, Barlow and Dibble1989), obtained using Rayleigh scattering imaging, who observed that the reduced growth in the streamwise direction could be explained by the large convective Mach numbers in this region and a dampening of internal Kelvin–Helmholtz (KH) instabilities due to compressibility effects. From there, the internal mixing layer interacts with the expansion fan generated by the intersection of the first reflected shock with the outer mixing layer, yielding a significant radial diffusion of density fluctuations toward the jet axis. The embedded shear layer of the second shock cell also displays intense but more diffuse density variations in the radial direction. Of particular interest, the amplitudes of density fluctuations appear to be very similar for $\mathcal {M}^{+}$ and $\mathcal {M}^{-}$, with only moderate differences in maximum amplitudes appearing downstream, for $x>3.5D$, with $\mathcal {M}^{-}$ being slightly more intense. The different radii of the two circular orbits identified in the two POD modes phase space in figure 19 thus have a limited effect and do not significantly change the structure of the two isolated helical modes.

Valuable insights are finally given by the phase map shown in figure 23(b) that allows to identify phase links in the field of density fluctuations captured by $\mathcal {M}^{\pm }$. It is noted that the upstream core region of the jet is hashed because no reliable phase estimates could be obtained in this part of the flow where almost no density fluctuations were captured (see figure 23a). To begin with, the phase pattern of the standing wave previously identified in figure 13 in the external near-field of the jet can be clearly identified, with an axial wavelength of approximately $2.5D$. A smooth phase-coupling between this standing wave and the perturbations developing in the outer mixing layer is observed upstream and downstream of the first shock cell, for $x>1.0D$. Furthermore, the oblique shock oscillations in the two shock cells are directly in phase with the density fluctuations captured in the region of interaction with the outer shear layer. This again indicates a strong coupling between shock oscillations and the helical KH instability wave developing in this outer mixing layer, in agreement with the main mechanism proposed in the literature (Panda Reference Panda1998; André et al. Reference André, Castelain and Bailly2011). The interior region of the jet, however, displays fluctuations that are generally out-of-phase with those found in the outer mixing layer and the oblique shock oscillations. First, the helical density fluctuations in the expansion fan of the first shock cell are in approximate quadrature phase-shift relative to the shock oscillations. Second, considering any of the two shock cells displayed, the density fluctuations associated with the oblique shock motions are in approximate antiphase relative to the upstream and downstream density fluctuations. Of particular interest is a continuous large-scale variation of phase inside the jet, in the streamwise direction and across the shock waves. This large-scale internal phase distribution exhibits a phase jump across the outer shear layer but shows uniform variations in the embedded shear layers.

Based on these observations, it is proposed to interpret this out-of-phase link between fluctuations found in the embedded shear-layers and in the outer mixing layer (discarding the near-field standing wave pattern) as the consequence of a phase-shift between large-scale waves developing inside the jet and in the outer mixing layer. In these two regions of the jet, a similar approximate large-scale axial wavelength of approximately $3.5D$ is estimated. This value corresponds to approximately two shock-cell spacings and is identified as the footprint of the outer-layer KH instability wave having support in the entire jet (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021). This suggests that the phase differences observed between the two regions can be related to the classical phase jump across shear layers observed for KH instability waves, a feature commonly found in axial velocity fluctuations of KH modes (see Nogueira et al. (Reference Nogueira, Jordan, Jaunet, Cavalieri, Towne and Edgington-Mitchell2021) for example), but which also appears in fields of density fluctuations. As an example, figure 24 displays the phase map of density fluctuations captured by parabolised stability equations (PSE) (Herbert Reference Herbert1997; Sinha et al. Reference Sinha, Rodrıguez, Brès and Colonius2014) applied to an analytic model of a perfectly expanded supersonic jet (which is thus free of shock cells and screech) at a similar Mach number $M_j=1.71$, for an azimuthal wavenumber $m=1$ and for a Strouhal number equivalent to that deduced from screech noise measurements obtained for the present under-expanded jet. The spatial distribution of phase values in the mixing layer and in the jet core is qualitatively in agreement with the large-scale variations identified in figure 23(b), displaying a similar phase decrease radially toward the jet axis and providing support for the proposition. More significant differences in phase pattern in the outer jet region can be observed and are associated with the absence of the screech-related standing wave in this simple example, which serves only to highlight the internal structure of the KH instability wave. From this perspective, the helical density fluctuations found in the embedded shear layers in figure 23 are interpreted as being directly driven by the outer KH instability wave, suggesting that reflected shocks oscillations only play a minor or indirect role in the associated dynamics. It is emphasised, however, that this conclusion is based on spatio-temporal correlations rendered by a reduced model and that further experimental or numerical work should thus be considered to clarify its validity.

Figure 24. Phase map of density fluctuations evaluated using PSE in the longitudinal plane of a perfectly expanded supersonic jet at $M_j=1.7$, for a Strouhal number ${St}_j=0.23$ and an azimuthal wavenumber $m=1$.

Finally, figure 23(b) suggests that a near-perfect phase match exists across the outer shear layer in the nozzle-lip region between the outer standing wave and the inner-jet large-scale fluctuations. Although a phase-match condition between KH modes and upstream-traveling waves is to be expected in the receptivity region of the mixing layer for the resonant mechanism at work to be closed, it is interesting to observe that this appears to be accompanied by a phase match of density fluctuations in the two regions surrounding the initial shear layer.