2.1. Jet models

The jet is modelled using locally parallel linear stability theory. All variables are normalised by the nozzle diameter $D$, and the ambient density and speed of sound $\rho _\infty$ and $c_\infty$, respectively. The Reynolds decomposition

(2.1)\begin{equation} q(x,r,\theta,t)=\bar{q}(r)+q'(x,r,\theta,t) \end{equation}

is applied to the flow-state vector $q$, where the mean and fluctuating components are $\bar {q}$ and $q'$, respectively, and x, r, $\theta$ and t are the axial distance, the radial distance, the azimuthal angle and the time, respectively. We assume the normal-mode ansatz,

(2.2)\begin{equation} q'(x,r,\theta,t)=\hat{q}(r)\,\textrm{e}^{\textrm{i}(kx+m\theta-\omega t)}, \end{equation}

where $k$ is the streamwise wavenumber normalised by the nozzle diameter $D$, $m$ is the azimuthal mode and $\omega =2{\rm \pi} StM_a$ is the non-dimensional frequency, where $St=fD/U_j$ is the nozzle-diameter-based Strouhal number and $M_a=U_j/c_\infty$ is the acoustic Mach number. As discussed by Tam & Ahuja (Reference Tam and Ahuja1990) and Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), $k_p$ modes belong to a hierarchical family of waves identified by their azimuthal and radial orders $m$ and $n$, respectively, and exist as propagative waves only in a well-defined $St$ number range (see § 2.3). We here restrict attention to azimuthal mode $m=0$ due to the axisymmetry property of screech modes A1 and A2 and we let the radial order vary in the range $n=1,2$, which is sufficient for the frequency range of the screech modes of interest.

This theory is applied to two different models: a simplified cylindrical V-S and a finite-thickness flow model. Both are governed by the compressible linearised Euler equations (LEE).

2.1.1. Cylindrical V-S model

Following our previous work (Mancinelli et al. Reference Mancinelli, Jaunet, Jordan and Towne2019a), the linear wave dynamics is first modelled using a cylindrical V-S (Lessen, Fox & Zien Reference Lessen, Fox and Zien1965; Michalke Reference Michalke1970). The V-S dispersion relation is

(2.3)\begin{align} &D(k,\omega;M_a,T,m)\nonumber\\ &\quad =\frac{1}{\left(1-\dfrac{kM_a}{\omega}\right)^2} + \frac{1}{T}\frac{I_m\left(\dfrac{\gamma_i}{2}\right)\left(\dfrac{\gamma_o}{2}K_{m-1}\left(\dfrac{\gamma_o}{2}\right)+ mK_m\left(\dfrac{\gamma_o}{2}\right)\right)}{K_m\left(\dfrac{\gamma_o}{2}\right)\left(\dfrac{\gamma_i}{2}I_{m-1}\left(\dfrac{\gamma_i}{2}\right) - mI_m\left(\dfrac{\gamma_i}{2}\right)\right)}=0, \end{align}

with

(2.4a)\begin{gather} \gamma_i = \sqrt{k^2-\frac{1}{T}(\omega-M_ak)^2}, \end{gather}

(2.4b)\begin{gather}\gamma_o = \sqrt{k^2-\omega^2}, \end{gather}

where $I_m$ and $K_m$ are modified Bessel functions of the first and second kinds, respectively, and $T=T_j/T_\infty$ is the jet-to-ambient temperature ratio. The relation between the acoustic and jet Mach numbers is given by $M_j=U_j/c_j=M_a/\sqrt {T}$. The branch cut of the square root in (2.4a) and (2.4b) is chosen such that $-{\rm \pi} /2\leq \textrm {arg}(\gamma _{i,o})<{\rm \pi} /2$.

Frequency/wavenumber pairs $(\omega, k)$ that satisfy (2.3) define eigenmodes of the V-S for given values of $m$, $M_a$ and $T$. To find these pairs, we specify a real or complex frequency $\omega$ and compute the associated eigenvalues $k$ for the K–H and guided jet modes according to (2.3). By virtue of the normalisation adopted, the free-stream acoustic waves are simply defined by $k_a^\pm =\pm 2{\rm \pi} StM_a$.

2.1.2. Finite-thickness flow model

Writing the LEE equations in terms of the pressure, the compressible Rayleigh equation

(2.5)\begin{align} &\frac{\partial^2\hat{p}}{\partial r^2}+\left(\frac{1}{r}-\frac{2k}{\bar{u}_xk-\omega}\frac{\partial\bar{u}_x}{\partial r}-\frac{\gamma -1}{\gamma\bar{\rho}}\frac{\partial\bar{\rho}}{\partial r}+\frac{1}{\gamma\bar{T}}\frac{\partial\bar{T}}{\partial r}\right)\frac{\partial\hat{p}}{\partial r}\nonumber\\ &\quad -\left(k^2+\frac{m^2}{r^2}-\frac{\left(\bar{u}_xk-\omega\right)^2}{\left(\gamma -1\right)\bar{T}}\right)\hat{p}=0 \end{align}

is obtained, where $\gamma$ is the specific heat ratio for a perfect gas. The solution of the linear stability problem is obtained specifying a real or complex frequency $\omega$ and solving the resulting augmented eigenvalue problem $k=k(\omega )$, with $\hat {p}(r)$ the associated pressure eigenfunction. The eigenvalue problem is solved numerically by discretising (2.5) in the radial direction using Chebyshev polynomials. A mapping function is used to non-uniformly distribute the grid points such that they are dense in the region of shear (Trefethen Reference Trefethen2000).

With the aim of keeping the model as simple as possible and exploring the impact of the shear-layer thickness, we use the hyperbolic-tangent velocity profile (Lesshafft & Huerre Reference Lesshafft and Huerre2007)

(2.6)\begin{equation} \bar{u}_x(r)=\frac{1}{2}M_a\left(1+\tanh\left(\frac{R}{4\theta_R}\left(\frac{R}{r}-\frac{r}{R}\right)\right)\right), \end{equation}

where $\theta _R$ is the shear-layer momentum thickness and $R$ is the nozzle radius.

Different values of $R/\theta _R$ are considered to assess the impact of the shear-layer thickness on screech generation and, specifically, on the frequency range of existence of propagative guided modes. Consistent with results obtained from experimental particle image velocimetry (PIV) measurements (see Appendix A), we select a shear layer with $R/\theta _R=10$.

2.2. Resonance conditions

Resonance conditions are obtained by assuming that upstream- and downstream-travelling waves exchange energy at the nozzle exit and at the $s$th shock-cell location, where an interaction between the K–H instability wave and a representative source point in the shock-cell pattern occurs. Following Landau & Lifshitz (Reference Landau and Lifshitz1958), the condition to be satisfied for resonance to occur is

(2.7)\begin{equation} R_1R_2\,\textrm{e}^{\textrm{i}\Delta kL_s}=1, \end{equation}

where $R_1(St,M_j)\in \mathcal {C}$ and $R_2(St,M_j,L_s)\in \mathcal {C}$ are the reflection coefficients at the nozzle exit and shock-cell location, respectively, $\Delta k=k^+-k^-$ is the difference between the wavenumbers of the downstream- and upstream-travelling waves and $L_s$ is the distance between the nozzle exit and the $s$th shock cell. Following Jordan et al. (Reference Jordan, Jaunet, Towne, Cavalieri, Colonius, Schmidt and Agarwal2018), (2.7) can be re-written in terms of magnitude and phase constraints associated with the imaginary and real parts of the eigenvalues $k$, respectively,

(2.8a)\begin{gather} \textrm{e}^{\Delta k_i L_s}=\vert R_1R_2\vert, \end{gather}

(2.8b)\begin{gather}\Delta k_r L_s +\phi =2p{\rm \pi}, \end{gather}

where $\Delta k_i$ and $\Delta k_r$ are the imaginary and real parts of $\Delta k$, $\phi =\angle R_1R_2$ is the phase of the reflection-coefficient product and $p$ is an integer associated with the sum of oscillation peaks of the downstream- and upstream-travelling waves within the distance $L_s$. We show in Appendix B the equivalence between the phase criterion in (2.8b) and the wave-interaction model first proposed by Tam & Tanna (Reference Tam and Tanna1982).

Following our previous results (Mancinelli et al. Reference Mancinelli, Jaunet, Jordan and Towne2019a) and in agreement with observations from experiments and numerical simulations (Panda Reference Panda1999; Umeda & Ishii Reference Umeda and Ishii2001; Gao & Li Reference Gao and Li2010; Mercier et al. Reference Mercier, Castelain and Bailly2017) that suggest stronger interaction in the vicinity of the fourth shock cell, i.e. $s=4$, we here construct our model on the basis that this is the dominant equivalent source dictating the screech-frequency selection. The distance $L_s$ between the nozzle exit and the $s$th shock cell is given by

(2.9)\begin{equation} L_s(M_j)=L_1((1-\alpha)s+\alpha), \end{equation}

where $L_1$ is the length of the first shock cell and $\alpha$ is the rate of decrease of the shock-cell length with the downstream distance, whose value was estimated as $0.06$ by Harper-Bourne (Reference Harper-Bourne1974). The length of the first shock cell can be obtained using the V-S model and by carrying out a zero-frequency analysis. When $\omega \to 0$ the dispersion relation in (2.3) reduces to

(2.10)\begin{equation} I_m\left(\frac{\gamma_i}{2}\right)=0, \end{equation}

which, exploiting the properties of the Bessel functions, can be written as

(2.11)\begin{equation} J_m\left(\frac{\textrm{i}\gamma_i}{2}\right)=0. \end{equation}

Since we intend to describe the mean-flow shock-cell structure of an axisymmetric jet, we consider $m=0$. The first zero of the Bessel function $J_0$ in (2.11) occurs when

(2.12)\begin{equation} \textrm{i}\frac{\gamma_i}{2}=2.4048. \end{equation}

Using (2.4a) to replace $\gamma _i$ in (2.12), we obtain the Prandtl–Pack model (Pack Reference Pack1950) for the length of the first shock cell,

(2.13)\begin{equation} L_1(M_j)=\frac{\rm \pi}{2.4048}\sqrt{M_j^2-1}. \end{equation}

The reflection coefficients $R_1$ and $R_2$ are unknown functions of frequency and flow conditions. Two modelling approaches may be pursued in which the frequency, $\omega$, is either real or complex, and in each approach the reflection-coefficient product is handled differently. Following Mancinelli et al. (Reference Mancinelli, Jaunet, Jordan and Towne2019a), we first use the real-frequency analysis to show that the feedback process in the generation of A1 and A2 screech modes is underpinned by guided-jet modes rather than free-stream acoustic waves considered in the closure mechanism proposed by Powell (Reference Powell1953) and most other studies since. We then explore both temperature and shear-layer thickness effects on screech. This allows us to identify the limitations of real-frequency-based models and explore improvements that may be obtained using the complex-frequency model.

2.3. Neutral-mode model

If the frequency is considered real, resonance involves a spatially unstable downstream-travelling K–H mode and a neutrally stable upstream-travelling wave: $\omega \in \mathcal {R}$, $k^+\in \mathcal {C}$ and $k^-\in \mathcal {R}$. The reflection coefficients $R_1$ and $R_2$ at the nozzle exit and shock-cell location, respectively, are such that resonance involves tones that neither grow nor decay in time. A schematic depiction of this scenario is provided in figure 1(a).

Figure 1.

Schematic representation of the resonance loop: (a) real-frequency analysis, (b) neutral-mode model. Solid blue and dashed red lines refer to downstream- and upstream-travelling waves, respectively.

A further widely used simplification involves assuming that both the downstream- and upstream-travelling waves are neutrally stable: $k^+$ and $k^-\in \mathcal {R}$ (see schematic representation in figure 1b). Since $\Delta k_i$ is then identically zero, this allows us to neglect the magnitude constraint of (2.8a). Resonance-frequency prediction can be obtained either from the dispersion relation (2.3) or from the Rayleigh equation (2.5) by specifying real values of $\omega$, computing $k$ and then imposing the phase constraint (2.8b) to obtain the resonance frequency. Given that the phase of the reflection-coefficient product is an unknown, we follow Jordan et al. (Reference Jordan, Jaunet, Towne, Cavalieri, Colonius, Schmidt and Agarwal2018) and Mancinelli et al. (Reference Mancinelli, Jaunet, Jordan and Towne2019a) by exploring different phase values: $\phi =0$, $-{\rm \pi} /4$, $-{\rm \pi} /2$ and $-{\rm \pi}$, which lead to the following resonance criteria:

(2.14a)\begin{gather} \phi=0\;\Longrightarrow\;Re[k^+{-}k^-]=\Delta k_r=\frac{2p{\rm \pi}}{L_s}, \end{gather}

(2.14b)\begin{gather}\phi={-}{\rm \pi}/4\;\Longrightarrow\; Re[k^+{-}k^-]=\Delta k_r=\frac{(2p+1/4){\rm \pi}}{L_s}, \end{gather}

(2.14c)\begin{gather}\phi={-}{\rm \pi}/2\;\Longrightarrow\; Re[k^+{-}k^-]=\Delta k_r=\frac{(2p+1/2){\rm \pi}}{L_s}, \end{gather}

(2.14d)\begin{gather}\phi={-}{\rm \pi}\;\Longrightarrow\; Re[k^+{-}k^-]=\Delta k_r=\frac{(2p+1){\rm \pi}}{L_s}. \end{gather}

Predictions obtained using $\phi =-{\rm \pi} /4$ (2.14b) and in-phase reflection conditions (2.14a) provide best agreement with the experimental data when using the V-S and finite-thickness dispersion relations, respectively. This discrepancy in phase between the two flow models is expected since the phase speed of the K–H mode changes significantly between the V-S and the finite-thickness model (Michalke Reference Michalke1984).

Figure 2(a) shows the eigenvalue trajectories for azimuthal mode $m=0$, as a function of frequency, in the $k_r$–$k_i$ and $k_r$–$St$ planes for the V-S dispersion relation. Shown are the trajectories of the K–H and guided-jet modes, and of the downstream- and upstream-travelling free-stream acoustic waves. The flow considered is cold and fully expanded, with temperature ratio $T=T_j/T_\infty \approx 0.81$ and jet Mach number $M_j=1.08$. A non-dispersive wave with a phase speed equal to $0.8\,U_j$ frequently adopted to approximate the K–H mode (Schmidt et al. Reference Schmidt, Towne, Colonius, Cavalieri, Jordan and Brès2017) has been added to the plot. The $k_{KH}$ mode is slightly dispersive with phase speed greater than the usually adopted value of $0.8\,U_j$ as shown by the comparison with the constant phase speed line, $St=(0.8Uj)k$. We note the following features for the guided-jet modes. They are characterised by a negative phase speed and, following Tam & Ahuja (Reference Tam and Ahuja1990) and Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), we distinguish between guided-jet modes with positive and negative group velocities. Only the $k_p^-$ modes may close the resonance loop. Unlike free-stream acoustic waves, they are dispersive, with subsonic phase speed very close to the value of the ambient speed of sound, particularly in the low-frequency part of the branch. They furthermore exist as propagative waves only in a well-defined Strouhal-number range delimited by the branch and saddle points, $B(m,n)$ and $S(m,n)$, respectively (Tam & Hu Reference Tam and Hu1989). Outside this frequency band, the guided modes are evanescent and therefore not included in the modelling. For the first radial order, the branch point coincides with the origin in the $k_r$–$St$ plane (Tam & Ahuja Reference Tam and Ahuja1990).

Figure 2.

Eigenvalues $k_{KH}^+$, $k_p^\pm$ and $k_a^\pm$ in the case of a real-frequency analysis for azimuthal mode $m=0$, cold jet and fully expanded jet Mach number $M_j=1.08$: (a) V-S model, (b) finite-thickness model with $R/\theta _R=10$; (i) $k_r$–$k_i$ plane, (ii) $k_r$–$St$ plane. Blue $\diamond$ corresponds to $k_{KH}^+$, red $\circ$ to $k_p^-$ for $n=1$, green $\Box$ to $k_p^+$ for $n=1$, magenta $\triangle$ to $k_p^-$ for $n=2$, cyan $\times$ to $k_p^+$ for $n=2$, dash-dotted black line to a non-dispersive approximation of K–H mode whose phase speed is $0.8\,U_j$, dashed black lines to $k_a^\pm$. The branch and saddle points of the guided-jet modes for each pair of $(m,n)$ orders are indicated with letters $B$ and $S$, respectively.

The trajectories of the K–H and guided-jet modes using the finite-thickness dispersion relation with $R/\theta _R=10$ for the same flow conditions considered above are shown in figure 2(b). The $k_{KH}$ wave shows a higher dispersive nature with a much lower phase speed than that computed using the V-S, as discussed above. We also note that, contrary to the V-S, the amplitude of the growth rate of the $k_{KH}$ mode is not monotonically increasing with the frequency.

Figure 3 shows the resonance-frequency selection in the case of a resonance between $k_{KH}^+$ and $k_p^-$ modes of first and second radial orders for azimuthal mode $m=0$, $M_j=1.08$, $T_j/T_\infty \approx 0.81$ and in-phase reflection conditions. For the sake of brevity, we only report the results obtained using the finite-thickness dispersion relation. Resonance can only exist in a frequency range where both downstream- and upstream-travelling waves exist and are propagative. It is thus clear that when $k_p^-$ modes are considered, eligible resonance frequencies lie in the band delimited by the branch and saddle points. The resonance frequency is found by imposing the phase constraint, which implies computing the intersection between the $\Delta k_r$ and the value on the right hand-side of the equation of the in-phase resonance criterion (2.14a).

Figure 3.

Value of $\Delta k_r$ between the $k_{KH}^+$ and $k_p^-$ modes computed using the finite-thickness model and identification of the resonance frequency in the case of neutral-mode assumption for azimuthal mode $m=0$, $M_j=1.08$ and $T\approx 0.81$. Solid red line refers to $k_p^-$ for $n=1$, dash-dotted magenta line to $k_p^-$ for $n=2$, horizontal dashed black lines to resonance criteria in the case of in-phase reflection conditions (2.14a).

2.4. Complex-mode model

In the complex-frequency analysis both $k^+$ and $k^-\in \mathcal {C}$. The screech loop involves an unstable downstream-travelling K–H mode and an evanescent upstream-travelling wave. The complex-frequency analysis provides, as an additional result, information on the temporal amplification of the resonance loop: for $\omega _i>0$ the screech loop is amplified in time and resonance is sustained; for $\omega _i<0$ the screech loop is attenuated in time and resonance is damped. A schematic depiction of these scenarios is provided in figure 4.

Figure 4.

Schematic representation of the resonance loop in the complex-frequency analysis. Solid blue lines refer to the downstream-travelling wave, dashed red lines to the upstream-travelling wave and dash-dotted green lines to the fed downstream-travelling wave in a new cycle of the resonance loop.

Resonance-frequency prediction involves finding triplets $[k^+,k^-,\omega ]\in \mathcal {C}$ simultaneously satisfying either the dispersion relation (2.3) or the Rayleigh equation (2.5) and both the magnitude and phase constraints (2.8a) and (2.8b), respectively. This implies knowledge of the reflection-coefficient product $R_1R_2$ as a function of frequency and jet Mach number. As this function is unknown, we first use the reflection-coefficient product as a parameter with a prescribed amplitude. We then identify a frequency-Mach-number-dependent model for the reflection-coefficient product. Prediction involves solving (2.3) or (2.5) and (2.8a) to find complex frequencies and eigenvalues that satisfy both the V-S dispersion relation or the Rayleigh equation and the magnitude constraint. The phase constraint (2.8b) is then used to select resonance frequencies.

For the following discussion, we consider the eigenspectrum of the V-S. This allows us to highlight certain features of the complex-mode model in comparison with the neutral-mode one. However, the complex-mode resonance conditions are used after to compute resonance frequency using the finite-thickness dispersion relation. Figure 5 shows the V-S solutions $\omega (k)$ for $k_{KH}^+$ and $k_p^-$ modes of second radial order for azimuthal mode $m=0$, Mach number $M_j=1.08$, $T\approx 0.81$ and with the fourth shock-cell location taken as the downstream reflection point. Eigenvalues obtained from both real- and complex-frequency analyses are represented. Eigenvalues for $\omega \in \mathcal {C}$ are computed for two values of the reflection-coefficient product amplitude: $\vert R_1R_2\vert =10^{-3}$ and $10^{-2}$. As pointed out above, both downstream- and upstream-travelling waves are characterised by $k_i<0$ for $\omega \in \mathcal {C}$, implying that $k_{KH}^+$ and $k_p^-$ are, respectively, spatially unstable and spatially evanescent. Consistent with sustained resonance, the eigenvalues for $\omega \in \mathcal {C}$ move in the $k_r$–$k_i$ plane towards regions of positive imaginary frequency. Specifically, for a given value of the reflection-coefficient product, only $\omega _i\geq 0$ satisfies the amplitude condition in (2.8a). This feature of the complex-mode model allows us to identify frequency-Mach-number combinations for which resonance is sustained.

Figure 5.

V-S solutions satisfying (2.3) in the case of complex-frequency analysis for azimuthal mode $m=0$, fully expanded jet Mach number $M_j=1.08$ and $T\approx 0.81$ for: (a) $k_p^-$ for $n=2$, (b) $k_{KH}^+$. The colour map shows $\omega _i$, whereas the white contour represents $St=\omega _r/2{\rm \pi} M_a$. Dashed black lines represent the eigenvalues in the case of real-frequency analysis. Markers represent eigenvalues satisfying both the dispersion relation (2.3) and the magnitude constraint (2.8a) for $s=4$ and different values of $\vert R_1R_2\vert$: magenta $\circ$ to $\vert R_1R_2\vert =10^{-3}$, green $\triangle$ to $\vert R_1R_2\vert =10^{-2}$.

Figure 6 illustrates the resonance-frequency selection when $\omega \in \mathcal {C}$ for azimuthal mode $m=0$, $M_j=1.08$ and $T\approx 0.81$. Consistent with what was found using finite-thickness neutral-mode model, best agreement is observed for in-phase reflection conditions (2.14a). In what follows, we compare model predictions with experiments.

Figure 6.

Value of $\Delta k_r$ between the $k_{KH}^+$ and $k_p^-$ modes computed using the finite-thickness model and identification of the resonance frequency in the case of complex frequency for azimuthal mode $m=0$, $M_j=1.08$, $T\neq 1$ and magnitude of the reflection-coefficient product $\vert R_1R_2\vert =0.3$. Solid red line refers to $k_p^-$ for $n=1$, dash-dotted magenta line to $k_p^-$ for $n=2$, horizontal dashed black lines to in-phase resonance criteria (2.14a).

3.1. Acoustic measurements

Pressure fluctuations were measured by GRAS 46BP microphones, whose frequency response is flat in the range 4 Hz–70 kHz. Data were acquired by a National Instruments PXIe-1071 acquisition card with a sampling frequency of 200 kHz, which provides a maximum resolved Strouhal-number range $[2, 3.2]$, well above the $St$ of interest in this paper. The acquisition time was set to 30 s, which is six orders of magnitude larger than the longest convective time, thus ensuring statistical convergence of the quantities presented in the paper. An azimuthal array of six microphones was placed in the nozzle exit plane at a radial distance $r/D=1$ from the jet centreline. It was thus possible to resolve the most energetic azimuthal Fourier modes: $m=0,\pm 1,\pm 2$. The accuracy of the azimuthal Fourier decomposition at the tone frequencies was checked by computing the coherence between neighbouring microphones (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012). Additional details on the jet facility and a schematic depiction of the experimental set-up and microphone disposition can be found in Mancinelli et al. (Reference Mancinelli, Jaunet, Jordan and Towne2019a,Reference Mancinelli, Jaunet, Jordan, Towne and Girardb).

3.2. PIV measurements

PIV measurements were performed for several jet Mach numbers $M_j=1.08$, $1.12$, $1.16$, $1.22$, $1.3$, $1.43$, $1.47$ and $1.55$ in order to provide a description of the mean flow to inform the finite-thickness model (see § 2.1.2 and Appendix A). The flow was seeded using Ondina oil particles before entering the stagnation chamber ensuring seeding homogeneity. The particles were illuminated by a $2\times 200$ mJ Nd-YAG laser and the images were recorded with a 4 Mpix CCD camera equipped with a Sigma DG Macro 105 mm allowing the measurement of the jet up to $x/D = 10$ in the downstream direction. The PIV image pairs were acquired at a sampling rate of 7.2 Hz with a $\Delta t$ of 1 $\mathrm {\mu }$s. For each configuration, a total of 10 000 image pairs were acquired in order to obtain well converged statistics. The images were processed using LaVision's Davis 8.0 software using a multi-pass iterative correlation algorithm (Willert & Gharib Reference Willert and Gharib1991; Soria Reference Soria1996) starting with an interrogation area of $64\times 64$ pixels and finishing with $16\times 16$ pixels with deforming windows. The overlap between neighbouring interrogation windows was 50 %, leading to a resolution of approximately 2.375 vectors per millimetre (i.e. 55 vectors per jet diameter) in the measured field. At each correlation pass, a peak validation criterion was used: vectors were rejected if the correlation peak was lower than 0.3. This value was selected as the minimum acceptable value ensuring validation in the potential regions of the flow while rejecting most of the evident erroneous vectors. Outliers were then further detected and replaced using universal outlier detection (Westerweel & Scarano Reference Westerweel and Scarano2005). Main results of the PIV measurements are given in Appendix A.