3.2. Characteristics of averaged shock structures

As shown in figure 3, underexpanded jets impinge on the inclined plate, and downstream of the impingement location, the deflected jet turns into a wall jet (Brehm et al. Reference Brehm, Housman and Kiris2016). Previous experiments of André, Castelain & Bailly (Reference André, Castelain and Bailly2014) determined that the potential core length of an underexpanded free jet is approximately $9D$, and the potential core becomes longer with the increase of $M_j$. In the current experimental configuration, the jet impingement occurs before the end of the potential core. Thus the inclined plate can influence the shock cell structures to some extent.

Integrated spatial characteristics of shock cell structures are required to explain the staging behaviour of screeching free jets (Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022; Nogueira et al. Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a). Therefore, the averaged shock structures in the underexpanded jets impinging on an inclined plate are analysed in detail. The averaged schlieren images of the underexpanded impinging jets at $M_j=1.12$, 1.28 and 1.40 are shown in figure 4, and the averaged schlieren images of their free counterparts are also displayed as a reference. These averaged schlieren images are obtained by using 2560 successive schlieren images. Hundreds of screech periods are contained in this duration. The averaged greyscale distributions along $Y/D = 0$, and their wavenumber spectra, are also presented in figure 4 to better demonstrate the similarities and differences between averaged shock structures in underexpanded free and impinging jets. The streamwise domain of spatial Fourier transforms for impinging jets is $0\leqslant X/D\leqslant 6$. For underexpanded free jets, the axial fluctuations of greyscales associated with shock cells are almost invisible at $10D$ downstream of the nozzle exit. The streamwise domains for spatial transforms of free jets are respectively set as $0\leqslant X/D\leqslant 10$ and $0\leqslant X/D\leqslant 6$, to analyse the effect of the domain lengths on the shock wavenumber spectra.

Figure 4. Averaged schlieren images, distributions and wavenumber spectra of averaged greyscales along $Y/D=0$ of underexpanded free and impinging jets at (a) $M_j=1.12$, (b) $M_j=1.28$, and (c) $M_j=1.40$. The primary shock wavenumber peak $k_{s1}$ is indicated by the red square. The first and second suboptimal shock wavenumber peaks ($k_{s2}$ and $k_{s3}$) are respectively indicated by the magenta and cyan squares.

The wavenumber spectra of underexpanded free and impinging jets that are shown in figure 4 have similar features. There is a primary peak in each wavenumber spectrum that is associated with the first several shock cells whose shock spacings are relatively constant. On the right of this primary peak, a series of suboptimal peaks represent the axial variation in shock spacing that occurs further downstream (Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022). The primary shock wavenumber peak is denoted as $k_{s1}$ and is indicated by the red squares in figure 4; the first and second suboptimal shock wavenumber peaks are denoted as $k_{s2}$ and $k_{s3}$, indicated by the magenta and cyan squares, respectively. As shown in figure 4, the spacings of the first several shock cells of impinging jets have good agreement with those of the corresponding free jets. Thus the $k_{s1}$ values of impinging jets are very close to those of the corresponding free jets. However, the shock cells of impinging jets, which are located just upstream of the impinging plate, are obviously shorter. By comparing with the corresponding free jets, the sixth and seventh shock cells of the $M_j=1.12$ impinging jet, the fifth shock cell of the $M_j=1.28$ impinging jet, and the fourth shock cell of the $M_j=1.40$ impinging jet have shorter spacings. The axial variation of shock spacings of impinging jets is more drastic because of the existence of the impinging plate. Correspondingly, at $M_j =1.12$ and 1.28, the $k_{s2}$ values of impinging and free jets are very close to each other, and the $k_{s3}$ values of impinging jets have higher wavenumbers. In the case $M_j=1.40$, the wavenumbers of both $k_{s2}$ and $k_{s3}$ of the impinging jet are higher than those of the corresponding free jet. It is indicated that the effect of the impinging plate on jet shock cell structures becomes more obvious with the increase of $M_j$. Furthermore, in each case of figure 4, the peaks in the wavenumber spectra of the free jet based on different domain lengths are close to each other. This implies that the influence of domain lengths for spatial Fourier transforms on the shock spectra of free jets is limited, and the above-mentioned differences in the shock wavenumber spectra between the free and impinging jets are not due to the different domain lengths.

In order to analyse the sensitivities of $k_{s1}$, $k_{s2}$ and $k_{s3}$ to the choice of radial position, spatial Fourier transforms are conducted at different radial positions, and the resultant wavenumber spectra are represented in figure 5. As shown in figure 4, the averaged shock structures of free jets are symmetric with respect to the jet axis. The high-amplitude regions corresponding to $k_{s1}$, $k_{s2}$ and $k_{s3}$ in the shock wavenumber spectra of free jets are concentrated mainly at the jet axis, as displayed in figure 5. Thus the determinations of the peak wavenumbers of averaged shock cells for free jets are based on the wavenumber spectra along $Y/D=0$. For impinging jets, as shown in figure 5, the high-amplitude regions corresponding to $k_{s1}$ and $k_{s2}$ are also located mainly at the jet axis. The $k_{s3}$ values can still be determined at the jet axis in figures 5(a), 5(b) and 5(c). However, for the impinging jet at $M_j= 1.51$, there are two spots of higher amplitudes near $Y/D= \pm 0.25D$ on the right of the $k_{s2}$, as shown in figure 5(d). This is because of the asymmetry of averaged shock structures under the influence of the inclined impinging plate at higher $M_j$. In view of the approximate wavenumbers of these two high-amplitude spots, $k_{s3}$ is defined as the average of the wavenumbers corresponding to the maximum amplitude points of these two spots for these cases, to represent the axial variation in shock spacing.

Figure 5. Normalized wavenumber spectra of averaged schlieren images for underexpanded free and impinging jets at (a) $M_j=1.12$, (b) $M_j=1.28$, (c) $M_j=1.40$, and (d) $M_j=1.51$. In each panel, the top image is the spectrum of the free jet and the bottom image is the spectrum of the impinging jet. The primary shock wavenumber peak $k_{s1}$ is indicated by the red square. The first and second suboptimal shock wavenumber peaks ($k_{s2}$ and $k_{s3}$) are respectively indicated by the magenta and cyan squares.

The full spectrum representations of averaged shock structures for underexpanded free and impinging jets are displayed in figures 6(a) and 6(b), respectively. The range of $M_j$ is from 1.09 to 1.56. The wavenumber spectra of 20 cases are displayed in each figure. The wavenumber spectrum in $-0.5 \leqslant Y/D \leqslant 0.5$ of the time-averaged schlieren image is plotted for each case. Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a) suggested that the A1 and A2 screech modes are respectively closed by the guided jet modes that are energized by the interaction between the K–H wavepacket and the first two shock wavenumber peaks. The $k_{s1}$ and $k_{s2}$ values of free and impinging jets are extracted from figures 6(a) and 6(b), and are compared in figure 6(c). In the present $M_j$ range, the $k_{s1}$ value of impinging jets has good agreement with that of free jets, and the specific values of $k_{s2}$ of impinging jets are slightly higher than those of free jets at some $M_j$. The $k_{s3}$ value is more noticeable in figure 6(b). For the shock wavenumber of impinging jets, $k_{s3}$ corresponds to the change of shock cell spacings caused by the appearance of the impinging plate. The values of $k_{s3}$ for impinging jets are also extracted from figure 6(b) and displayed in figure 6(c) for further research.

Figure 6. Normalized wavenumber spectra of averaged schlieren images as a function of $M_j$ for (a) underexpanded free jets and (b) impinging jets. The red, magenta and cyan dashed lines respectively display the trends of $k_{s1}$, $k_{s2}$ and $k_{s3}$. The $k_{s1}$, $k_{s2}$ and $k_{s3}$ values of underexpanded free and impinging jets are extracted from (a,b) and plotted in (c). The shock wavenumbers of free and impinging jets are respectively indicated by the crosses and squares in (c). The red, magenta and cyan symbols in (c) are respectively corresponded to $k_{s1}$, $k_{s2}$ and $k_{s3}$.

3.3. Eigenvalues and modes of SPOD

SPOD is applied on the schlieren image sequences in order to obtain the coherent flow structures that are associated with the generation of screech tones. In the present analyses, $M=2560$ snapshots are used in each case. For the current $M_j$ range, with the increase of $M_j$, the non-dimensional time step between consecutive snapshots $\Delta t\,U_j/ D_j$ ranges from 0.42 to 0.50. The range of the corresponding maximum resolvable non-dimensional frequency is from 1.19 to 1.00. Here, $N_f$ (size of each block) and $N_o$ (overlap between two consecutive blocks) are respectively set as 256 and 128 ($0.5N_f$), resulting in a total of $N_b = 19$ SPOD modes at each frequency. The non-dimensional frequencies $St_j$ of the acoustic resonances that are investigated in the following subsections are in the range 0.40 to 0.70. The non-dimensional time span of one block is more than 100, which corresponds to tens of periods of acoustic resonance.

As shown in figure 7, for the impinging jets at $M_j =1.12$, 1.28 and 1.40, the tonal frequencies of the noise spectra agree well with the peak frequencies in the distributions of the eigenvalues of the first SPOD modes; and at tonal frequencies, the first SPOD modes are obviously more energetic than the second and third SPOD modes. These results indicate that the flow structures at tonal frequencies possess more energy and show a low-rank dynamic feature. Therefore, in the following subsections, the structures of screech feedback loops are investigated based mainly on the first SPOD modes at tonal frequencies. As shown in figure 2(b), the impinging jets at $M_j=1.12$, 1.28 and 1.40 generate the screech tones of the A1, A2 and A3 modes, respectively. The frequencies of the chosen SPOD modes are indicated by blue circles in figure 7, and summarized in table 1.

Figure 7. SPOD energy spectra and acoustic spectra for the impinging jets at (a) $M_j=1.12$, (b) $M_j=1.28$, and (c) $M_j=1.40$. The black solid, dashed and dotted lines respectively indicate the SPOD eigenvalues of the first three SPOD modes. The red lines indicate the noise spectra. The blue circles indicate the chosen SPOD modes.

Table 1. Frequencies of screech tones and chosen SPOD modes for different jets.

As presented in figure 8, upstream of the impingement location, the real parts of the chosen SPOD modes are symmetric with respect to the jet axis, which agrees with the azimuthal feature of the A1 and A2 modes, and indicates that the A3 mode is another kind of axisymmetric screech mode. In figure 8, the normalized amplitude fields of the chosen SPOD modes exhibit standing wave patterns (Panda Reference Panda1999; Edgington-Mitchell Reference Edgington-Mitchell2019) along the jet shear layer. The presence of this standing wave pattern is due to the partial interference between the upstream- and downstream-propagating waves that constitute the screech feedback loop (Panda Reference Panda1999). Moreover, as presented in figures 8(c, f), the amplitudes on the lower side of the jet are slightly weaker than those on the jet's upper side. This phenomenon will be discussed further in the following subsections.

Figure 8. The first SPOD modes for jets with (a,d) $M_j = 1.12$ at $St_j=0.60$, (b,e) $M_j = 1.28$ at $St_j=0.49$, and (c, f) $M_j = 1.40$ at $St_j=0.47$. (a–c) Real part of the mode. The blue contours indicate negative values, and yellow contours indicate positive values. (d–f) Normalized amplitude field of the mode. The colour scales range from 0 (blue) to 1 (yellow).

3.4. Wavenumber spectra of SPOD modes

In a chosen SPOD mode, the flow structures oscillate at a specific screech frequency. However, the different components of the screech feedback loop are associated with a broad range of wavenumbers. Thus spatial Fourier transforms are conducted on the chosen SPOD modes using (2.5). The axial range for the spatial Fourier transform is from $X/D=0$ to $X/D=8$. The normalized wavenumber spectra of the current three SPOD modes are presented in figure 9. In the positive wavenumber domain, the majority of energy is associated with the downstream-propagating K–H wavepacket. The wavenumber of the K–H wavepacket ($k_h$), which is estimated as the wavenumber corresponding to the maximum amplitude in the positive wavenumber domain, is indicated by the yellow dashed lines in figure 9 and listed in table 2. Then the convective velocity of the K–H wavepacket is calculated as $U_c=\omega _s/ k_h$, where $\omega _s$ is the angle frequency of screech. As shown in table 2, the $U_c$ value for the chosen cases ranges from $0.60 U_j$ to $0.65 U_j$, which agrees with the results in Mercier, Castelain & Bailly (Reference Mercier, Castelain and Bailly2017) and Powell et al. (Reference Powell, Umeda and Ishii1992).

Figure 9. Normalized wavenumber spectra of the chosen SPOD modes for the jets at (a) $M_j=1.12$, (b) $M_j=1.28$, and (c) $M_j=1.40$. The dashed vertical yellow lines indicate the wavenumber of $k_h$. The dashed vertical red, magenta and cyan lines respectively indicate the wavenumbers of $k_h\pm k_{s1}$, $k_h\pm k_{s2}$ and $k_h\pm k_{s3}$. The dashed vertical green lines indicate the wavenumber of the upstream-propagating acoustic wave $k_a=-\omega _s/a_\infty$. The $k_h\pm$ spectrum of the averaged schlieren image is also shown in the bottom of each panel as a reference.

Table 2. Characteristic wavenumbers in the wavenumber spectra of SPOD modes.

At screech frequencies, the upstream-propagating guided jet mode can be stimulated by the interaction between the K–H wavepacket and shock cell structures (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021a); Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a) suggested that the A1 and A2 screech modes are respectively closed by the guided jet modes that are energized by the interaction between the $k_h$ and different shock wavenumbers. Thus, for the present three cases, $k_h\pm$, the spectrum of the averaged schlieren image, is also plotted in figure 9. It should be noted that there are some high-amplitude regions that are located mainly in the lower wavenumber domains in the wavenumber spectra of averaged schlieren images. The wavenumbers of these high-amplitude regions are obviously less than $k_{s1}$ and may correspond to the nozzle-to-plate distance. The low-wavenumber domains of the wavenumber spectra of averaged schlieren images are omitted in figure 9, which leads to the white gaps in figure 9. The wavenumbers of high-amplitude regions in the wavenumber spectra of SPOD modes have good agreement with $k_h\pm k_{s1}$, $k_h\pm k_{s2}$ and $k_h\pm k_{s3}$. These results suggest that the interactions between the K–H wavepacket and the shock cells distribute energy to all of these wavenumbers. However, the radial features of the jet modes corresponding to these energetic wavenumbers are different. In the positive wavenumber domain, the stimulated waves at $k_h + k_{s}$ are trapped mainly in the jet core. These are the duct-like modes that are identified by Edgington-Mitchell et al. (Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021a) via local stability analyses, and are similar to that identified in high-subsonic jets (Towne et al. Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017). For the case $M_j=1.12$, there is more energy at $k_h-k_{s1}$, $k_h-k_{s2}$ and $k_h-k_{s3}$ in the negative wavenumber domain. At the first interaction wavenumber, there are significant supports outside the shear layer, which agrees with the radial feature of the upstream-propagating guided jet mode (Tam & Ahuja Reference Tam and Ahuja1990), while at the latter two wavenumbers, the energy decays rapidly in the radial direction. This implies that the stimulated wave at $k_h-k_{s1}$ is the upstream-propagating guided jet mode, and the waves at $k_h-k_{s2}$ and $k_h-k_{s3}$ would be duct-like. At $M_j=1.28$ and 1.40, the energies associated with $k_h-k_{s2}$ and $k_h-k_{s3}$ are respectively no longer trapped within the jet core and have supports outside the jet shear layer. This indicates that at the screech frequencies of the A2 and A3 modes, the upstream-propagating guided jet modes are respectively energized by the interactions between the K–H wavepacket and the first and second suboptimal shock wavenumber peaks. Thus the present results support the conclusion of Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a) and indicate that the A3 screech mode of impinging jets is related to the interaction between $k_h$ and the $k_{s3}$. As discussed in previous subsections, $k_{s3}$ is associated with the change of shock cell spacings that is caused by the impinging plate. The external boundaries, such as the impinging plate, can affect the screech mode of jets by changing the axial sizes of shock cells.

In order to generalize the conclusions of figure 9, SPOD is conducted on the schlieren image sequences of the cases in the $M_j$ range from 1.09 to 1.43. Spatial Fourier transforms are conducted on the first SPOD modes at tonal frequencies of different axisymmetric screech modes. The wavenumber spectrum $Y/D \subseteq [0, 1]$ for each case is shown in figure 10. Overall, the results of figure 9 are further confirmed by the wavenumber spectra that are displayed in figure 10. The acoustic resonances of the A1, A2 and A3 screech modes are respectively attributed to the interactions between the K–H wavepacket and the shock wavenumbers $k_{s1}$, $k_{s2}$ and $k_{s3}$.

Figure 10. Normalized wavenumber spectra of the first SPOD modes at the screech frequencies of (a) the A1 mode, (b) the A2 mode, and (c) the A3 mode. The yellow symbols indicate the wavenumber of $k_h$. The red, magenta and cyan symbols respectively indicate the wavenumbers of $k_h\pm k_{s1}$, $k_h\pm k_{s2}$ and $k_h\pm k_{s3}$. The green symbols indicate the wavenumber of the upstream-propagating acoustic wave $k_a=-\omega _s/a_\infty$.

Shock cell spacings (Powell Reference Powell1953) and standing wave wavelengths (Panda Reference Panda1999) are both important length scales for predicting screech frequencies. The relationship between these two length scales was discussed in Mercier et al. (Reference Mercier, Castelain and Bailly2017) and Li et al. (Reference Li, Liu, Hao, Zhang and He2021). Here, the connection between these two length scales is analysed via the wavenumber spectra of the normalized amplitude fields of the chosen SPOD modes. As shown in figures 9 and 10, the upstream-propagating guided jet modes with the wavenumbers of $k_h-k_{s1}$, $k_h-k_{s2}$ and $k_h-k_{s3}$, respectively, close the screech feedback loops of the A1, A2 and A3 modes. Based on the wavenumber relation of the standing wave, the standing wave patterns with the wavenumbers of $k_{s1}$, $k_{s2}$ and $k_{s3}$ should respectively exist in the flow fields of the jets that produce the screech tones in the A1, A2 and A3 modes. Figure 11 shows that the wavenumbers of high-amplitude regions are around $k_{s1}$, $k_{s2}$ and $k_{s3}$ in the radial range $0 \leqslant Y/D \leqslant 0.5$. In the study of Panda (Reference Panda1999), the wavenumber of standing wave $k_{sw}$ is considered as $k_h-k_a$. For the A1, A2 and A3 modes, during the process of getting away from the jet axis, the high-amplitude regions corresponding to $k_{s1}$, $k_{s2}$ and $k_{s3}$ approach $k_h-k_a$, respectively. In different screech modes, the axial wavenumber of the standing wave pattern corresponds to the different wavenumber peaks of averaged shock structures. As shown in figure 8, the corresponding relationships between the shock cell spacings and standing wave wavelengths are different in the A1, A2 and A3 modes. In the axial range of the first three shock cells, there are nearly three anti-nodes of the standing wave pattern for the A1 mode, four anti-nodes for the A2 mode, and five anti-nodes for the A3 mode. The above results will be used to predict the screech frequencies in the following subsections.

Figure 11. Normalized wavenumber spectra for the amplitude fields of the chosen SPOD modes for the jets at (a) $M_j=1.12$, (b) $M_j=1.28$, and (c) $M_j=1.40$. The spectrum of the averaged schlieren image is also shown at the bottom of each panel as a reference. The dashed vertical green lines indicate the wavenumber of $k_h-k_a$.

3.5. Mode staging process of axisymmetric screech modes

The weakest link theory (Tam & Tanna Reference Tam and Tanna1982; Tam, Seiner & Yu Reference Tam, Seiner and Yu1986) suggested that the interaction between the K–H wavepacket and shock cells leads to a redistribution of energy. The waves with the wavenumbers of the sum and difference of $k_h$ and $k_s$ will be energized. At screech frequencies, the wavenumbers of upstream-propagating guided jet modes coincide with these interaction wavenumbers, i.e. $k_h-k_{s}$ (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021a). The weakest link theory can be viewed as a first approximation of the condition for the occurrence of the absolute instability (Nogueira et al. Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a,Reference Nogueira, Jordan, Jaunet, Cavalieri, Towne and Edgington-Mitchellb). Here, the mode staging process of the axisymmetric screech modes for the impinging jets is analysed by combining the weakest link theory and the dispersion relations of upstream-propagating guided jet modes.

The dispersion relation of the vortex sheet model (Lessen, Fox & Zien Reference Lessen, Fox and Zien1965; Tam & Hu Reference Tam and Hu1989) can be written as

(3.1) \begin{equation} \frac{1}{\left(1-\dfrac{k M_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{equation}

and

(3.2)$$\begin{gather} \gamma_i = \sqrt{k^2 - \frac{1}{T} (\omega - M_a k)^2}, \end{gather}$$

(3.3)$$\begin{gather}\gamma_o = \sqrt{k^2 - \omega^2}. \end{gather}$$

Here, $I_m$ and $K_m$ are modified Bessel functions of the first and second kinds, respectively. All quantities have been normalized by the ideally expanded jet diameter $D_j$ and the ambient sound velocity $a_ \infty$. Also, $M_a = U_j / a_\infty$ is the acoustic Mach number, and ${T = T_j /T_\infty}$ is the temperature ratio. The relation between the ideally expanded jet Mach number $M_j$ and the acoustic Mach number $M_a$ is $M_j = M_a /\sqrt {T}$. For given values of wavenumber $k$ and azimuthal wavenumber $m$, the dispersion relation (3.1) has many real roots or eigenvalues. The entire set of eigenvalues can be characterized by $m$ and a radial wavenumber $n$ (Tam & Ahuja Reference Tam and Ahuja1990). For convenience, the jet mode that corresponds to the $m$th azimuthal mode and the $n$th radial mode will be designated by $(m,n)$, where $m = 0, 1, 2, \ldots$ and $n = 1, 2, 3, \ldots$. The pressure eigenfunctions of guided jet modes are given by

(3.4a,b)\begin{equation} p_i = I_0(\gamma_i r) (0\leqslant r/D_j \leqslant 0.5) , \quad p_o = A\,K_0(\gamma_o r) (0\leqslant r/D_j \leqslant 0.5). \end{equation}

These two functions are matched at $r/D_j=0.5$ by the coefficient $A$.

The dispersion relations of the guided jet modes $(0,2)$ for the cases associated with the staging process from the A1 mode to the A2 mode are shown in figure 12. The wavenumbers energized by the interaction between the K–H wavepacket and shock cells, i.e. $k_h-k_{s1}$ (red dashed lines) and $k_h-k_{s2}$ (magenta dashed lines), are also displayed. Here, the K–H waves are assumed to be non-dispersive waves with the phase velocities that are listed in table 2 for different screech modes. As shown in figure 12(a), the red dashed line intersects the dispersion relation of the $(0,2)$ mode, and the frequency of this intersection ($St_j=0.73$) is just above the screech frequency of the $M_j=1.09$ jet ($St_j=0.69$). The guided jet mode that is energized by the interaction between the K–H wavepacket and the dominant shock wavenumber of $k_{s1}$ closes the screech feedback loop of the A1 mode (Nogueira et al. Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a). For this $M_j$, no intersection is found for the magenta dashed line with the upstream-propagating part of the dispersion relation. This implies that at this condition, $k_h-k_{s2}$ does not correspond to an upstream-propagating mode that is supported by the jet. The values of $k_{s1}$ and $k_{s2}$ decrease with the increase of $M_j$. As shown in figure 12(b), the red dashed line intersects the upstream branch of the dispersion relation near the cut-on frequency of the guided jet mode, and the intersection point of the magenta dashed line and the dispersion relation gradually moves to the upstream branch. When $M_j=1.16$, as shown in figure 12(c), there is no intersection between the red dashed line and the dispersion relation of the $(0,2)$ mode, which means that $k_h-k_{s1}$ does not correspond to an upstream-propagating guided jet mode. The dominant screech frequency of the $M_j=1.16$ jet is in the A2 mode and below the frequency of the intersection of the magenta dashed line and the dispersion relation. This result agrees with that of Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a) and indicates that the A2 mode is closed by the guided jet mode that is stimulated by the interaction between the K–H wavepacket and $k_{s2}$ (Nogueira et al. Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a).

Figure 12. Dispersion relations of the upstream-propagating guided jet modes $(0,2)$ (green lines) for the jets at (a) $M_j=1.09$, (b) $M_j=1.12$, and (c) $M_j=1.16$. The red and magenta dashed lines respectively indicate the wavenumbers of $k_h-k_{s1}$ and $k_h-k_{s2}$. The wavenumber of acoustic waves is represented by the grey dashed line. The yellow and orange squares respectively indicate the screech frequencies of the A1 and A2 modes.

We examine the characteristics of the guided jet modes that are energized by the interactions between the K–H wavepacket and $k_{s2}$ during the mode staging process from the A1 mode to the A2 mode. Here, $k_h-k_{s2}$ intersects the dispersion relation of the $(0,2)$ mode at $(St_j, kD_j)=(0.59, -9.77)$ in figure 12(a). As shown in figure 13(a), the pressure eigenfunction of this guided jet mode is mainly trapped in the jet core. Therefore, for the $M_j=1.09$ jet, a duct-like mode with a positive group velocity but negative phase velocity is energized by the interaction between the K–H wavepacket and $k_{s2}$, and because of the positive group velocity, this jet mode is unable to close the resonance. As shown in figure 12(b), $k_h-k_{s2}$ intersects the dispersion relation near the saddle point of the guided jet mode at $M_j=1.12$, and the eigenfunction shows some amplitudes outside the shear layer in figure 13(b). In figure 12(c), $k_h-k_{s2}$ intersects the upstream branch of the dispersion relation at $M_j=1.16$. Furthermore, as shown in figure 13(c), the eigenfunction of this jet mode has significant support outside the shear layer. With the increase of $M_j$, the jet mode energized by the interaction between the K–H wavepacket and $k_{s2}$ gradually changes from a downstream-propagating duct-like mode to an upstream-propagating mode that has support outside the shear layer, which agrees with the results of the wavenumber spectra shown in figures 9 and 10.

Figure 13. Pressure eigenfunction distributions of guided jet modes at the intersections of the dispersion relations of the $(0,2)$ mode and $k_h-k_{s2}$: (a) $St_j=0.59$, $kD_j=-9.77$ for the $M_j=1.09$ jet; (b) $St_j=0.75$, $kD_j=-5.02$ for the $M_j=1.12$ jet; and (c) $St_j=0.69$, $kD_j=-4.53$ for the $M_j=1.16$ jet.

The mode staging process between the A2 and A3 modes is shown in figure 14 in a similar manner to figure 12. With the increase of $M_j$, the line of $k_h-k_{s2}$ gradually has no intersection with the dispersion relation of the upstream-propagating guided jet mode. The screech frequency of the A3 mode is just below the frequency of the intersection of ${k_h-k_{s3}}$ and the upstream branch of the dispersion relation. Meanwhile, as shown in figure 15, the eigenfunctions of the jet modes energized by the interactions of the K–H wavepacket and $k_{s3}$ gradually have more amplitudes outside the jet shear layer. These results imply further that the screech feedback loop of the A3 mode is closed by the guided jet mode that is energized by the interaction between the K–H wavepacket and $k_{s3}$.

Figure 14. Wavenumbers of the upstream-propagating guided modes of the second radial order (green lines) for the jets at (a) $M_j=1.22$, (b) $M_j=1.28$, and (c) $M_j=1.33$. The red, magenta and cyan dashed lines respectively indicate the wavenumbers of $k_h-k_{s1}$, $k_h-k_{s2}$ and $k_h-k_{s3}$. The wavenumber of acoustic waves is represented by the grey dashed line. The orange and red squares respectively indicate the screech frequencies of the A2 and A3 modes.

Figure 15. Pressure eigenfunction distributions of guided jet modes at the intersections of the dispersion relations of the $(0,2)$ mode and $k_h-k_{s3}$: (a) $St_j=0.66$, $kD_j=-4.70$ for the $M_j=1.22$ jet; (b) $St_j=0.59$, $kD_j=-4.16$ for the $M_j=1.28$ jet; and (c) $St_j=0.53$, $kD_j=-3.82$ for the $M_j=1.33$ jet.

As shown in figure 2, the screech frequencies of the A1 and A2 modes terminate near the cut-on frequency of the upstream-propagating guided jet mode $(0,2)$. However, this is not the case for the A3 mode. The acoustic results in figure 2(b) show that the impinging jet at $M_j=1.47$ does not generate a strong tone of the A3 screech mode. As shown in figure 16, the line of $k_h-k_{s3}$ still intersects the upstream branch of the dispersion relation at $M_j=1.47$. This result cannot be explained by the theory of Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a). The termination of the A3 mode will be discussed in the following subsections.

Figure 16. Wavenumbers of the upstream-propagating guided modes of the second radial order (green lines) for the $M_j=1.47$ jet. The red, magenta and cyan dashed lines respectively indicate the wavenumbers of ${k_h-k_{s1}}$, $k_h-k_{s2}$ and $k_h-k_{s3}$. The wavenumber of acoustic waves is represented by the grey dashed line.

3.6. Characteristics of upstream- and downstream-propagating waves

The wavenumber spectra of the chosen SPOD modes are shown in figure 9. The sign of the phase velocity ($\omega _s/k$) is determined by the sign of the axial wavenumber $k$. In the current analyses, in order to reconstruct the upstream- and downstream-propagating waves approximately, bandpass filters in the wavenumber domain are applied to isolate the upstream- and downstream-propagating waves in the screech feedback loop. The negative and positive wavenumber domains are used respectively to reconstruct the upstream-propagating waves and the downstream-propagating waves. The approximatively reconstructed upstream- and downstream-propagating waves at screech frequencies are presented in figure 17. Amplitude fields have been normalized by the maximum amplitude in each case for clarity. The axial locations of the rear edges of shock cells are extracted from averaged schlieren images and shown by the red arrows in figure 17.

Figure 17. Approximately reconstructed upstream (a–c) and downstream (d–f) propagating waves at the screech frequencies for (a,d) the $M_j=1.12$ jet (A1 mode), (b,e) the $M_j=1.28$ jet (A2 mode), and (c, f) the $M_j=1.40$ jet (A3 mode). The top image of each panel is the normalized amplitude field of waves. The colour scales range from 0 (blue) to 1 (yellow). The bottom image of each panel is the real part of the waves. Blue contours indicate negative values, and yellow contours indicate positive values. Red arrows indicate the axial positions of the shock reflection points at the jet shear layer.

As shown in figure 17, there are high-amplitude regions of the upstream-propagating waves at multiple shock tips for all these three screech modes. These high-amplitude regions should be related to the upstream motions of shock tips that lead to the generation of upstream-propagating acoustic waves (Edgington-Mitchell et al. Reference Edgington-Mitchell, Weightman, Lock, Kirby, Nair, Soria and Honnery2021b). As discussed in the previous subsections, the upstream-propagating guided jet mode that is energized by the interaction of the K–H wavepacket and $k_{s1}$ closes the screech feedback loop of the A1 mode. As shown in figure 17(a), some high-amplitude regions are near the impinging plate, where the shock cells are with shorter spacings. These are the energy associated mainly with the $k_h - k_{s2}$ interaction, which is also shown in the wavenumber spectrum of figure 9(a). In the $M_j$ range 1.12–1.16, the screech mode transforms from the A1 mode to the A2 mode. Before the mode switch occurs, the energy associated with the interaction of the K–H wavepacket and $k_{s2}$ becomes significant (Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022). As shown in figures 17(b) and 17(c), the higher amplitudes of upstream-propagating waves in the A2 and A3 modes are concentrated mainly at the axial regions downstream of the third shock cell, where the spacings of shock cells exhibit obvious changes in the axial direction. The upstream-propagating guided jet modes that are energized by the interactions of the K–H wavepacket with $k_{s2}$ and $k_{s3}$ close the screech feedback loops of the A2 and A3 modes, respectively. These two shock wavenumber peaks represent the axial variations of shock spacings that occur further downstream. Thus the approximate reconstructions of upstream-propagating waves can be viewed as the spatial representations of the corresponding wavenumber spectra. The screech tones of different modes might be associated with the shock cells at different distances downstream of the jets (Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022). For the A1 mode, the K–H wavepacket interacts with the first several shock cells to energize the upstream-propagating guided jet modes that close the screech feedback loop, and the acoustic resonances of the A2 and A3 modes are associated mainly with the axial regions in which the spacings of shock cells exhibit obvious axial variations. As shown in the amplitude fields of downstream-propagating waves in figure 17, the standing wave patterns along the jet shear layer are removed. However, periodic modulations within the jet core are exhibited for all these jets. The spatial scales of these modulations match the shock cell spacings closely. These spatial modulations may be attributed to the interference between the K–H wavepacket ($k_h$) and the duct-like mode with the wavenumber of $k_h+k_s$ (Edgington-Mitchell et al. Reference Edgington-Mitchell, Wang, Nogueira, Schmidt, Jaunet, Duke, Jordan and Towne2021a), or the direct modulations of shock cells on the K–H wavepacket (Nogueira et al. Reference Nogueira, Self, Towne and Edgington-Mitchell2022c).

3.7. Phase criteria

In this subsection, the frequencies of acoustic resonances are predicted by combining the phase criterion of an acoustic feedback loop with the results of current experiments. The phase criterion can be written as (Edgington-Mitchell Reference Edgington-Mitchell2019)

(3.5)\begin{equation} \frac{N}{f_s} = \frac{h}{U_c} + \frac{h}{U_{-}} + \psi,\end{equation}

where $U_c$ and $U_{-}$ are respectively the convection velocities of the downstream- and upstream-propagating waves that constitute the feedback loop, $f_s$ is the screech frequency, $h$ is the distance between the nozzle exit plane and the downstream reflection point, $\psi$ accounts for any additional delay associated with different components of the feedback loop, and $N$ is an integer value that can be considered as the number of concurrent disturbances in the acoustic feedback loop at any moment (Edgington-Mitchell Reference Edgington-Mitchell2019). By combining the relation between the wavenumber and the phase velocity of a wave, (3.5) can be written further as

(3.6)\begin{equation} (k_h-k_{-})\times h + 2 {\rm \pi}f_s \psi = \Delta k \times h + 2 {\rm \pi}f_s \psi = 2 {\rm \pi}N, \end{equation}

where $\Delta k$ is the wavenumber difference between the downstream-propagating K–H wavepacket and the upstream-propagating guided jet mode. Here, the downstream-propagating K–H wavepacket is considered to be neutrally stable and $k_h$ is calculated by using $U_c$ listed in table 2 for different screech modes. The wavenumber of the upstream-propagating guided jet mode $k_{-}$ is obtained from the vortex-sheet model (3.1), and $\psi$ is assumed to be $0$, i.e. the in-phase resonance criterion is considered. Using the dispersion relations of the downstream- and upstream-propagating waves, the frequencies at which (3.6) is satisfied are selected as the predicted frequencies.

For impinging jets, the impinging plate is usually viewed as the downstream reflection point, and $h$ in (3.6) is equal to the impinging distance (Ho & Nosseir Reference Ho and Nosseir1981; Bogey & Gojon Reference Bogey and Gojon2017; Gojon & Bogey Reference Gojon and Bogey2017). In the present study, not only the impinging distance at the jet axis ($5D$) but also the impinging distances along the upper and lower lip lines ($5.9D$ and $4.1D$) are used to predict tonal frequencies. As shown in figure 18, based on these three impinging distances, the frequencies and mode staging behaviours of acoustic resonances cannot be predicted. These results imply that the current acoustic resonances of underexpanded jets that impinge on an inclined plate are not due to the feedback loop between the nozzle exit and the plate.

Figure 18. Comparison of acoustic resonance frequencies obtained from the acoustic measurements and the phase criterion (3.6) between the nozzle exit and the impinging plate: (a) $h=5.9D$, the impinging distance at the upper lip line; (b) $h=5.0D$, the impinging distance at the jet axis; and (c) $h=4.1D$, the impinging distance at the lower lip line. The blue crosses indicate the results of acoustic measurements. The dashed grey lines indicate the frequency ranges in which the upstream-propagating guided jet mode $(0,2)$ is propagable.

The upstream- and downstream-propagating waves in the screech feedback loop are assumed to exchange energy at the nozzle exit and at a shock cell location, and the location of the fourth shock cell is used to predict screech frequencies in axisymmetric A1 and A2 modes (Mancinelli et al. Reference Mancinelli, Jaunet, Jordan and Towne2019, Reference Mancinelli, Jaunet, Jordan and Towne2021). In this subsection, in order to predict tonal frequencies, the location of the downstream reflection point is determined by analysing the movements of upstream- and downstream-propagating waves relative to the averaged shock structures. The upstream- and downstream-propagating waves of the screech feedback loop are reconstructed approximately in § 3.6, and the movements of waves during half of the screech period are displayed in figure 19. In different screech modes, the parts with the same phases of the upstream- and downstream-propagating waves are encountered at the rear edge of the third shock cell. Therefore, the location of the third shock cell is considered as the downstream reflection point for different screech modes. It should be noted that as discussed in previous subsections, the acoustic resonance in different screech modes is associated with the interaction of the K–H wavepacket and shock cells in various axial regions. Thus identifying a specific location of interaction is inconstant and case-related. The location of this downstream reflection point is just used to predict resonance frequencies.

Figure 19. Movements of approximately reconstructed upstream- and downstream-propagating waves at the jet shear layer ($Y/D=0.5$): (a) $M_j=1.12$ jet (${\rm NPR}=2.20$, A1 mode); (b) $M_j=1.28$ jet (${\rm NPR}=2.70$, A2 mode); and (c) $M_j=1.40$ jet (${\rm NPR}=2.70$, A2 mode). The approximately reconstructed upstream-propagating waves are indicated by the dashed black lines. The approximately reconstructed downstream-propagating waves are indicated by the solid blue lines. The red arrows indicate the axial positions of the shock reflection points at the jet shear layer.

As shown in figure 4, the impinging plate has only slight effects on the first three shock cells. Thus $h$ in (3.6) is calculated as $3 \times \lambda _s$, and the axial spacing of a shock cell $\lambda _s$ is obtained by the formula of Pack (Reference Pack1950):

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

Gojon et al. (Reference Gojon, Bogey and Marsden2016) demonstrated that the number of cells in the standing wave pattern between the nozzle exit plane and the downstream reflection point is the $N$ of (3.6). As shown in the normalized amplitude fields of figure 8, $N$ is equal to 3, 4 and 5 for the A1, A2 and A3 modes, respectively. The reflection coefficients at the nozzle exit and the downstream reflection point are not accounted for in the present model. Thus this is a simplified version of the model in Mancinelli et al. (Reference Mancinelli, Jaunet, Jordan and Towne2021).

As shown in figure 20, based on the current experimental results, the screech frequencies calculated by (3.6) have good agreement with the results of acoustic measurements. By combining the phase criterion and the right parameters, screech frequencies of different screech modes can be predicted precisely (Nogueira et al. Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a). As shown in figure 16, the vanishing of the A3 mode cannot be explained by combining the weakest link theory and the dispersion relation of the upstream-propagating guided jet mode. Thus the phase criterion for the A3 screech mode is examined further. At several $M_j$, the averaged schlieren images and the first SPOD modes at the screech frequencies of the A3 mode are shown in figure 21. There are always five cells in the standing wave pattern between the nozzle exit and the third shock reflection point along the upper lip line of jets. For the axisymmetric mode A3, the same phase criterion needs to be met at all the azimuthal positions. With the increase of $M_j$, the inclined plate gradually invades the third shock cell. In figures 21(a–c), the rear edge of the third shock cell is nearly vertical to the jet axis. The amplitudes of the standing wave patterns along the upper and lower lip lines are similar. As shown in figures 21(d–f), when $M_j$ is larger than 1.40, the rear edge of the third shock cell is obviously inclined. The asymmetry of averaged shock structures and the differences between the $h$ values along the upper and lower lip lines become distinct with the increase of $M_j$. The phase criterion is not satisfied at the lower side of the jet gradually. Thus the intensity of the standing wave pattern at the lower side becomes weaker relative to the upper side of the jets. When $M_j$ is larger than 1.45, the jets would not generate the screech tone of the A3 mode.

Figure 20. Comparison of acoustic resonance frequencies obtained from the acoustic measurements and the phase criterion (3.6) between the nozzle exit and the rear edge of the third shock cell. The blue crosses indicate the results of acoustic measurements. The dashed grey lines indicate the frequency ranges in which the upstream-propagating guided jet mode $(0,2)$ is propagable. The black solid, dashed and dotted lines indicate the results of the phase criteria of $N=3$, 4 and 5, respectively.

Figure 21. The first SPOD modes at the screech frequencies in A3 mode for the jets at (a) $M_j=1.33$ ($St_j=0.51$), (b) $M_j=1.35$ ($St_j=0.50$), (c) $M_j=1.38$ ($St_j=0.48$), (d) $M_j=1.40$ ($St_j=0.47$), (e) $M_j=1.42$ ($St_j=0.46$), and ( f) $M_j=1.44$ ($St_j=0.45$). The top image of each panel is the averaged schlieren image. The bottom image of each panel is the normalized amplitude field of the SPOD mode. The colour scales range from 0 (blue) to 1 (yellow). The red arrows indicate the axial positions of the shock reflection points at the jet shear layer.