3.1. Hydrodynamic resonance in an acoustically compact configuration

Vortical–acoustic resonance in the compact configuration sketched in figure 1(a), that is, the cavities are small compared with the acoustic wavelength (Nakiboglu et al. Reference Nakiboglu, Belfroid, Golliard and Hirschberg2011; Nakiboglu, Manders & Hirschberg Reference Nakiboglu, Manders and Hirschberg2012; Dai Reference Dai2020), is calculated with the compressible and incompressible models in this subsection. The geometrical parameters are $H^*=50$ mm, $D^*=25$ mm and $L^*=25$ mm. The Mach number averaged over the cross-section of the flow duct varies over the range 0.03 to 0.15, and the velocity profile is prescribed by $f= (1-(2y)^m)(m+1)/m$, where the parameter $m=8$ is used. The mean flow is assumed unchanged along the entire duct–cavity configuration, which means that a jet velocity profile is formed and the mean velocity is zero at $|y|>H/2$ in the cavity segment. The discontinuity in $\mathrm {d}f/\mathrm {d} y$ at $y=\pm H/2$ in the cavity segment causes a convergence problem in solving the travelling modes, and inflexion points are not defined at $y=\pm H/2$. By empirically introducing a resistive sheet with a small resistance $R=0.2M_0$ at $y=\pm H/2$ (Dai & Aurégan Reference Dai and Aurégan2018; Dai Reference Dai2020), the convergence problem is mitigated and the instability characteristics of the shear flow are similar to those of a hyperbolic–tangent velocity profile (Michalke Reference Michalke1965; Schmid & Henningson Reference Schmid and Henningson2001). The resistive sheets result in a pressure jump $\Delta p=Rv$ at $y=\pm H/2$ in the cavity segment. It is noted that this choice of the mean flow model leads to a piecewise homogeneous problem, so that the travelling mode analysis is rather simple, and to the continuity of the mean velocity between the cavity and duct segments, so that the disturbances are continuous at the interfaces between segments when mass and momentum conservation are enforced.

The spatial distributions of the antisymmetric and symmetric global modes for $M_0=0.09$ are presented in figures 2(a–d) and 3(a–d), where the fields have been normalized by $v$ at $x=L/4$ and $y=-H/2$. As expected, for an acoustically compact problem, the spatial distributions calculated from the compressible and incompressible models are almost exactly the same. A compact global mode here is mainly formed by a downstream-travelling unstable vortical mode (the antisymmetric or symmetric KH mode) and multiple upstream-travelling acoustic modes. The primary analysis approach is to first separate the travelling modes into three groups, namely vortical and $\pm x$ acoustic, then seeing the spatial variation of disturbances of each group. An analysis of the multiple feedback-loop channels is given in Appendix B. The modal shape of the global modes indicates that strong oscillations are concentrated in the opening area of the cavities, thus an approximate way to examine the propagation link ($\boldsymbol{\mathsf{P}}_{u}$ and $\boldsymbol{\mathsf{P}}_{d}$) of the feedback loop is to see the variation of $v$ along the line between the two edges of the upper or lower cavity, as shown in figures 2(e,f) and 3(e,f). Around the feedback loop, the phase change associated with the RPR upstream feedback is small and the phase change of the KH modes is close to $2{\rm \pi}$, which agree with the previous understanding of the Rossiter modes in compact cases.

Figure 2.

Spatial distribution of the antisymmetric global mode for $M_0=0.09$ calculated from the compressible ((a,c), where $\omega _G=0.4054 + 0.0036\mathrm {i}$) and incompressible ((b,d), where $\omega _G=0.4055 + 0.0046\mathrm {i}$) models. Amplitude (e) and phase (f) of $v$ along the opening of the low cavity (at the point just below $y=-H/2$). Solid lines, compressible model; dashed lines, incompressible model.

Figure 3.

Spatial distribution of the symmetric global mode for $M_0=0.09$ calculated from the compressible ((a,c), where $\omega _G=0.4126 + 0.0051\mathrm {i}$) and incompressible ((b,d), where $\omega _G=0.4114 + 0.0089\mathrm {i}$) models. For the descriptions of (e,f), see figure 2.

The global mode frequency $\omega _G$ as a function of $M_0$ is presented in figure 4. It is shown that $\mathrm {Re}(\omega _G)$ follows the Strouhal law ($Sr=\mathrm {Re}(\omega )L/ 2{\rm \pi} M_0$), which is in line with the previous experiments of incompressible or acoustically compact cavity flow oscillations (Knisely & Rockwell Reference Knisely and Rockwell1982; Gharib & Roshko Reference Gharib and Roshko1987; Nakiboglu et al. Reference Nakiboglu, Belfroid, Golliard and Hirschberg2011, Reference Nakiboglu, Manders and Hirschberg2012). Note that the spatial distributions at the other Mach numbers in figure 4 are all similar to those shown in figures 2 and 3 for $M_0=0.09$. The phase difference of the KH modes between the upstream and downstream edges is almost constant and close to $2{\rm \pi}$ for all these Mach numbers, which explains the Strouhal law and also the coincidence between the compressible and incompressible results of $Sr$.

Figure 4.

Global mode frequency as a function of $M_0$ in the compact case, calculated from the compressible and incompressible models. The legend in (b) also applies to (a).

The temporal growth rate of the global modes, $-\mathrm {Im}(\omega _G)$ according to the $\exp (\mathrm {i} \omega t)$ convention, in figure 4(b) shows the increasing discrepancy between the compressible and incompressible models as $M_0$ is increased. The value of $\mathrm {Im}(\omega _G)$ calculated from the incompressible model shows a linear increase with $M_0$. This is because the incompressible global modes calculated with different $M_0$ actually describe the same hydrodynamic feedback loop, which is indicated by the constant $Sr$ in figure 4(a) and the wavenumbers shown in figure 5(b). The wavenumbers calculated from the incompressible model overlap for the seven $M_0$ and the corresponding $\mathrm {Re}(\omega _G)$. For the seven $M_0$ and the corresponding $\omega _G$, the wavenumbers shift from those calculated with real-valued frequencies, but they still overlap. The latter overlap requires $\mathrm {Im}(\omega _G)$ being proportional to $\mathrm {Re}(\omega _G)$, and being proportional to $M_0$. On the other hand, figure 5(a) shows that the wavenumbers from the compressible model vary with $M_0$ and the KH modes show the stabilization effect of the Mach number on the KH convective instability (Miles Reference Miles1958). Although the wavenumbers only represent the propagation link ($\boldsymbol{\mathsf{P}}_{u}$ and $\boldsymbol{\mathsf{P}}_{d}$) of the feedback loop, the deviations of the compressible results of the wavenumbers and $\mathrm {Im}(\omega _G)$ from their incompressible counterparts show the same trend with increasing $M_0$. It is noted that the compressible and incompressible wavenumbers in a lined flow duct have also been shown to be close to each other at a low Mach number but are not exactly the same (Marx & Aurégan Reference Marx and Aurégan2013). It is also noted that, for the present parameters, figure 4(b) shows a stable $\textrm {KH}+\textrm {RPR}$ feedback loop, which can be destabilized by reducing the shear layer thickness (Yamouni et al. Reference Yamouni, Sipp and Jacquin2013) or reducing $R$ (Dai Reference Dai2020), as discussed below.

Figure 5.

(a) Wavenumbers in the cavity segment calculated from the compressible (crosses) and incompressible (squares) models with $M_0$ and $\omega _G$ of the antisymmetric modes in figure 4; (b) wavenumbers in the cavity segment calculated from the incompressible model with $M_0$, $\omega _G$ (squares) and $\mathrm {Re}(\omega _G)$ (diamonds) of the antisymmetric modes in figure 4.

The feedback-loop closure principle ($k_{fl}=1$) implies two conditions of a system resonant mode: the phase condition ($\mathrm {arg}(k_{fl})=2j{\rm \pi}$, where $j$ is an integer) and the gain or amplitude condition ($|k_{fl}|=1$). The real part of the global mode frequency $\mathrm {Re}(\omega _G)$ is mainly decided by the phase condition, whereas the temporal growth rate $\mathrm {Im}(\omega _G)$ is associated with the amplitude condition. For each global mode, no matter whether the mode is globally stable, neutral or unstable, both $\mathrm {arg}(k_{fl})=2j{\rm \pi}$ and $|k_{fl}|=1$ should be satisfied at the global mode frequency $\omega _G$. For stable, neutral and unstable global modes, the loop gain at $\mathrm {Re}(\omega _G)$ is respectively $|k_{fl}|<1$, $|k_{fl}|=1$ and $|k_{fl}|>1$. Therefore, a loop-gain criterion of global instability can be obtained: $|k_{fl}|$ being larger or smaller than unity at $\mathrm {Re}(\omega _G)$ determines whether the global mode is unstable or stable (Alvarez & Kerschen Reference Alvarez and Kerschen2005; Rowley et al. Reference Rowley, Williams, Colonius, Murray and Macmynowski2006; Dai Reference Dai2020). Note that the above loop-gain criterion of global instability is slightly different from Powell's gain condition, i.e. a unit gain at a real frequency, which was used to describe the saturated and stable state of self-sustained oscillations (Powell Reference Powell1961). To better understand the above discussions, the eigenvalues of $\boldsymbol{\mathsf{M}}_{fl}$ and the wavenumbers of the travelling modes in the cavity segment are plotted in figure 6, as the complex frequency is perturbed from $\omega _G$. It is shown that, at $\omega _G$ for the antisymmetric global mode, one eigenvalue of $\boldsymbol{\mathsf{M}}_{fl}$ is unity. Another eigenvalue of $\boldsymbol{\mathsf{M}}_{fl}$, close to unity, is associated with the symmetric global mode, whose feedback loop is not closed yet. Note that the global modes with multiple KH wavelengths in the cavity opening region (Nakiboglu et al. Reference Nakiboglu, Belfroid, Golliard and Hirschberg2011, Reference Nakiboglu, Manders and Hirschberg2012; Yamouni et al. Reference Yamouni, Sipp and Jacquin2013) are damped here, because the shear layers are rather thick. First, the variation of $\mathrm {Re}(\omega )$ leads to $k_{fl}$ rotating around zero in the complex plane (the trajectory of a $k_{fl}$ is not exactly a circle, however) as shown in figure 6(a), since the wavelengths of the travelling modes change with $\mathrm {Re}(\omega )$ as shown in figure 6(b). This demonstrates how the multiple Rossiter frequencies are selected by the loop phase condition: $\mathrm {arg}(k_{fl})=2j{\rm \pi}$, where $j$ is an integer. Second, the variation of $\mathrm {Im}(\omega )$ leads to the change of $|k_{fl}|$, since the spatial growth rate ($\pm \mathrm {Im}(k)$ respectively for the waves travelling in the $\pm x$ directions) of the travelling vortical and acoustic modes, being positive or negative, varies with $\mathrm {Im}(\omega )$. It has been revealed that, in general, a thinner shear layer potentially results in stronger self-sustained oscillations in cavity flows (Knisely & Rockwell Reference Knisely and Rockwell1982; Gharib & Roshko Reference Gharib and Roshko1987; Nakiboglu et al. Reference Nakiboglu, Manders and Hirschberg2012). The stronger self-sustained oscillations of a fully nonlinear flow system are understood to be associated with the stronger global instabilities, i.e. larger $-\mathrm {Im}(\omega _G)$ (Yamouni et al. Reference Yamouni, Sipp and Jacquin2013). It was also known that reducing the shear layer thickness leads to a larger spatial growth rate of the KH instability waves (Schmid & Henningson Reference Schmid and Henningson2001). Figure 6 casts light on the link between the increase in the spatial growth rate of travelling waves and flow destabilization. The former would cause an increase of $|k_{fl}|$ if $\mathrm {Im}(\omega )$ remains unchanged. To maintain the $k_{fl}=1$ condition for a resonant mode, $\mathrm {Im}(\omega )$ should decrease, that is, a global mode destabilization. It is noted that, in the above local–global analysis of the feedback loop, both frequency and wavenumber can be complex and a real-valued frequency is not more special than a complex frequency (Gallaire & Chomaz Reference Gallaire and Chomaz2004; Doaré & de Langre Reference Doaré and de Langre2006; Stewart et al. Reference Stewart, Waters and Jensen2009; Jordan et al. Reference Jordan, Jaunet, Towne, Cavalieri, Colonius, Schmidt and Agarwal2018), thus $\mathrm {Im}(\omega _G)$ decreasing or increasing, rather than $\omega _G$ crossing the real axis, is referred to as global mode destabilization or stabilization in this article. Such a local–global relation of the feedback loop applies to both acoustic and vortical–acoustic resonances.

Figure 6.

Variations of (a) the eigenvalues of $\boldsymbol{\mathsf{M}}_{fl}$ and (b) the wavenumbers in the cavity segment as the frequency is perturbed from $\omega _G=0.4054 + 0.0036\mathrm {i}$ (the antisymmetric global mode frequency for $M_0=0.09$ calculated with the compressible model). Red, $\mathrm {Re}(\omega )$ varying from $\mathrm {Re}(\omega _G)-0.01$ (circles) to $\mathrm {Re}(\omega _G)+0.01$ (triangles); blue, $\mathrm {Im}(\omega )$ varying from $\mathrm {Im}(\omega _G)-0.01$ (circles) to $\mathrm {Im}(\omega _G)+0.01$ (triangles).

3.2. Acoustic resonator

To study the influence of an AR on the $\textrm {KH}+\textrm {RPR}$ feedback loop, the depth of the cavities will be increased, $D^*=200$ mm ($D/L=8$), whereas all the other geometrical and flow parameters remain the same as those in § 3.1. Before discussing the vortical–acoustic resonance in an AR in § 3.3, we first examine the resonator acoustics without flow. Acoustic resonant modes in cavities have been studied by Tam (Reference Tam1976), Koch (Reference Koch2005), Hein et al. (Reference Hein, Koch and Nannen2012) and others, here, only the modes needed for the later discussions are calculated. The first antisymmetric and symmetric acoustic resonant modes are respectively presented in figures 7 and 8, where the modes are normalized so that $|p|=1$ at the cavity bottoms.

Figure 7.

Spatial distribution of (a) $\mathrm {Re}(\,p)$ and (b) $\mathrm {Re}(v)$ of the antisymmetric acoustic resonant mode at a real-valued frequency $\omega _A=0.3673$ (a trapped mode). (c) Variation of $p$ (red) and $v$ (blue) along $y$ at $x=L/2$. Dashed lines, $\mathrm {Re}(\,p)$ and $\mathrm {Re}(\textit {v})$; solid lines, $|p|$ and $|v|$. (d) Wavenumbers of the travelling waves. (e) Mode coefficients of the travelling waves in the $+x$ direction, where the modes are sorted according to their orders. The legend in (d) also applies to (e).

Figure 8.

Spatial distribution of (a) $\mathrm {Re}(\,p)$ and (b) $\mathrm {Re}(v)$ of the symmetric acoustic resonant mode at a complex frequency $\omega _A=0.3885 + 0.1350\mathrm {i}$. For the descriptions of (c–e), see figure 7.

The antisymmetric acoustic resonant mode displayed in figure 7(a–c) is a trapped mode. The coefficients of the $+x$ travelling modes in the cavity segment (at $x=0$) and in the right duct (at $x=L$) are shown in figure 7(e), where the travelling modes have been normalized so that $|P_n(y)|_{{max}} = 1$. Note that the coefficient of the $n$th $-x$ travelling mode in the cavity segment (at $x=L$) or in the left duct (at $x=0$), which is not shown, is exactly the same as that of the $n$th $+x$ travelling mode. The split between symmetric and antisymmetric travelling modes in such a symmetric system can be seen: compared with the antisymmetric counterparts, the symmetric travelling modes have vanishingly small amplitudes. The zero amplitude of the outgoing propagative (cut-on) mode in each duct, as indicated numerically in figure 7(e) by the vanishingly small mode coefficient of the $n=1$ mode in the duct, means that acoustic energy of the trapped mode cannot escape, and consequently the trapped mode occurs at a real-valued frequency. Such a wave trapping mechanism in a symmetric system, i.e. because of symmetric mismatch, the antisymmetric trapped mode is decoupled from the outgoing plane waves in the two semi-infinite ducts which are the only cut-on modes in the ducts at the trapped mode frequency, has been extensively studied (Evans & Linton Reference Evans and Linton1991; Evans, Levitin & Vassiliev Reference Evans, Levitin and Vassiliev1994; Evans & Porter Reference Evans and Porter1997; Pagneux Reference Pagneux2013). The symmetric acoustic resonant mode, shown in figure 8(a–c), is assembled by symmetric travelling modes. The non-vanishing amplitude of the outgoing propagative mode in each duct, as shown in figure 8(e), means radiation of acoustic energy to infinity, thus the resonant mode is damped. Owing to the outgoing propagative modes in the ducts, $|k_{fl}|<1$ if the frequency remains real valued. To satisfy the feedback-loop closure principle for a resonant mode ($k_{fl}=1$), $\mathrm {Im}(\omega )$ increases. As shown in figure 8(d), for the two $+x$ propagative modes in the cavity segment $\mathrm {Im}(k)>0$, since $\mathrm {Im}(\omega _A)>0$.

The acoustic resonance frequency of the configuration can be approximately estimated with a one-dimensional model, assuming a resonance happens when $\lambda /2=H+2D$ or $\lambda /4=D$, where $\lambda$ is the wavelength. However, the two resonant modes shown in figures 7 and 8 are close to but not exactly the same as the $\lambda /2$-resonance between the two cavity bottoms (see figure 7c) or the $\lambda /4$-resonance in each cavity (see figures 7c and 8c). The $\lambda /2$- and $\lambda /4$-resonance frequencies are $\omega _{\lambda /2}=0.3491$ and $\omega _{\lambda /4}=0.3927$. In the present case, $\omega _{\lambda /2}<\omega _{A,anti}<\mathrm {Re}(\omega _{A,sym})<\omega _{\lambda /4}$. The discrepancies in frequency depend on the geometrical parameters, and in general they reduce as the cavity depth is increased (Koch Reference Koch2005; Hein et al. Reference Hein, Koch and Nannen2012). It is noted here that for $D/L=1$ in § 3.1, the lowest acoustic resonance frequency is $\omega _{A,anti}=1.9387$ when $R=0$, which is much higher than the frequency range considered there.

For a comparison later in § 3.3, the acoustic resonant modes for the configuration $D/L=8$ without flow but with a resistive sheet $R=0.018$ added at the entrance of each cavity are also calculated here. The complex frequencies of the first antisymmetric and symmetric resonant modes are respectively $\omega _A=0.3761 + 0.0044\mathrm {i}$ and $\omega _A=0.3881 + 0.1415\mathrm {i}$, which indicate that the influence of the small $R$ on $\mathrm {Re}(\omega _A)$ is extremely small and $\mathrm {Im}(\omega _A)$ is only slightly increased owing to the damping effect of $R$. Note that since the symmetry holds after the resistive sheets are introduced, the antisymmetric resonant mode, a lightly damped mode, however, is still a trapped mode (the outgoing propagative mode in each duct still has a zero amplitude).

The resonant modes presented in figures 7 and 8 are eigensolutions without forcing. As an acoustic forcing is introduced from the left duct, the wave scattering of the resonator is shown in figure 9, which is used to further discuss the purely acoustic resonances in figures 7 and 8 where $R=0$, and to provide the forced acoustic fields inside the deep cavities at and far from the acoustic resonance frequencies that will be compared with the global modes in § 3.3 where $R \neq 0$. The calculation of wave scattering based on the scattering matrix method has been described in Dai & Aurégan (Reference Dai and Aurégan2018) and Dai (Reference Dai2020). The plane-wave transmission and reflection coefficients of the resonator are plotted in figure 9(a), which shows a zero transmission caused by the symmetric resonant mode. As the difference between the incident frequency and the resonant mode frequency is increased, the amplitude of the excited oscillations in the resonator decreases, as shown in figure 9(d–f). The antisymmetric resonant mode cannot be excited by an incoming plane wave owing to symmetric mismatch, thus its effect is not seen in figure 9(a). In figure 9(b,c), the resonator is excited by the first $+x$ cross-mode in the left duct. At the frequency of the antisymmetric resonant mode, the forcing excites a stronger oscillation in the resonator compared with figure 9(d–f), since the antisymmetric resonance has a larger quality factor. Note that, when $R=0$, the quality factor of such a perfect resonance is infinity and the amplitude of response is also infinity. The present numerics indicate the pressure amplitude at the cavity bottoms shown in figure 9(b) to be $|p|=1.15 \times 10^8$ if $R=0$. In figure 9 and § 3.3, the pressure is shown in $|p|$ rather than $\mathrm {Re}(\,p)$ for a better visual comparison (see the acoustic pressure nodes inside the deep cavities in figures 9(e,f) and 14(d,e)). At $y=0$, $|p|=0$ and $|p|\neq 0$ respectively indicate antisymmetric and symmetric resonant modes.

Figure 9.

(a) Transmission and reflection coefficients for a plane-wave excitation (thin lines: $R=0$, thick lines: $R=0.018$). (b,c) Value of $|p|$ excited by the first $+x$ cross-mode in the left duct of amplitude $|p|_{{max}}=1$ at $x=0$, at frequencies denoted by red vertical lines in (a): $\omega =\mathrm {Re}(\omega _A)$ where $\omega _A=0.3761 + 0.0044\mathrm {i}$ (antisymmetric acoustic resonance) and $\omega =0.3949$ (frequency in figure 12e). (d–f) Value of $|p|$ excited by the $+x$ plane wave in the left duct of amplitude $|p|=1$, at frequencies denoted by blue vertical lines in (a): $\omega =\mathrm {Re}(\omega _A)$ where $\omega _A=0.3881 + 0.1415\mathrm {i}$ (symmetric acoustic resonance), $\omega =0.5476$ (frequency in figure 14d) and $\omega =0.7708$ (acoustic anti-resonance). In (b–f), $R=0.018$.

3.3. Vortical–acoustic resonance in an AR

The problem sketched in figure 1(b) is discussed in this subsection. At low Mach numbers, the $\textrm {KH}+\textrm {RPR}$ feedback loop occurring at the junction between the flow duct and the deep cavities is still acoustically compact, but this compact region is influenced by the acoustics of the resonator (Peters Reference Peters1993; Kriesels et al. Reference Kriesels, Peters, Hirschberg, Wijnands, Iafrati, Riccardi, Piva and Bruggeman1995).

The global mode frequency as a function of the Mach number is plotted in figure 10. At $M_0=0.03$, the global mode frequency for $D/L=8$, being very close to those for $D/L=1$, approximately agrees with the Strouhal law. As $M_0$ increases, $\mathrm {Re}(\omega _G)$ of the antisymmetric mode progressively deviates from the Strouhal law, and the deviation becomes more noticeable as $\mathrm {Re}(\omega _G)$ gets close to the frequency of the antisymmetric acoustic resonant mode, $\mathrm {Re}(\omega _A)$. The temporal growth rate, $-\mathrm {Im}(\omega _G)$, shows a global mode destabilization when $\mathrm {Re}(\omega _G)$ is around $\mathrm {Re}(\omega _A)$, and a maximum temporal growth rate occurs when $\mathrm {Re}(\omega _G)= \mathrm {Re}(\omega _A)$ (Yamouni et al. Reference Yamouni, Sipp and Jacquin2013), which in this case happens at $M_0 \approx 0.09$. Note that the acoustic resonance frequency might be slightly shifted owing to the convection of the acoustic waves by the flow. This shift, however, should be extremely small since the flow speed is low, thus the maximum temporal growth rate happens when $\mathrm {Re}(\omega _G)$ equals to the $\mathrm {Re}(\omega _A)$ calculated without flow. For the symmetric modes, the influence of the heavily damped acoustic resonant mode on the global modes is rather weak near the frequency crossing point. The deviation of $\mathrm {Re}(\omega _G)$ from the Strouhal law is small. However, $\mathrm {Im}(\omega _G)$ shows a stabilization phenomenon over a $M_0$ range where $\mathrm {Re}(\omega _G)$ is much larger than $\mathrm {Re}(\omega _A)$. It is noted that, in the grey region in figure 10(b) (also in figure 15), $\mathrm {Im}(\omega _G)$ is high. For example, $\omega _G = 0.5213 + 0.0843 \mathrm {i}$ when $M_0=0.11$. We do not try to find the $M_0$ at which a peak $\mathrm {Im}(\omega _G)$ occurs, but mark that region in grey.

Figure 10.

Global mode frequency $\omega _G$ as a function of $M_0$: comparison between the deep-cavity case ($D/L=8$) and the shallow-cavity case ($D/L=1$). In (a), $\omega _A$ of acoustic resonant modes ($0.3671 + 0.0044\mathrm {i}$ and $0.3881 + 0.1415\mathrm {i}$) are calculated with $R=0.018$ without flow. The legend in (a) also applies to (b).

The different $\mathrm {Im}(\omega _G)$ trajectories of the antisymmetric and symmetric global modes explain the previous experiments, from a linear point of view, that vortex structures are not shed simultaneously from the upper and lower cavity leading edges but with a shift of half a period (Peters Reference Peters1993; Kriesels et al. Reference Kriesels, Peters, Hirschberg, Wijnands, Iafrati, Riccardi, Piva and Bruggeman1995; Oshkai & Yan Reference Oshkai and Yan2008; Wang, Deng & Liu Reference Wang, Deng and Liu2018). Note that both antisymmetric and symmetric global modes could be unstable near the crossing point of $\mathrm {Re}(\omega _G)$ and $\mathrm {Re}(\omega _A)$ if the shear layer thickness is reduced, but the antisymmetric one should win in the competition owing to its larger temporal growth rate.

To explain the different $\omega _G$ trajectories of the antisymmetric and symmetric global modes in figure 10, we adopt the view sketched in figure 11 that the vortical–acoustic resonance in an acoustic resonator and the associated system instability are governed by the KH instability waves and dual feedbacks: the RPR and AR feedbacks (Powell Reference Powell1990; Fabre & Hirschberg Reference Fabre and Hirschberg2000; Rienstra & Hirschberg Reference Rienstra and Hirschberg2018).

Figure 11.

Vortical–acoustic resonance in an AR: KH instability and dual feedbacks. Adapted from Rienstra & Hirschberg (Reference Rienstra and Hirschberg2018).

The spatial distributions of the antisymmetric (figures 12 and 13) and symmetric (figure 14) global modes are plotted for different $M_0$, where the fields have been normalized by $v$ at $x=L/4$ and $y=-H/2$. Figures 12(a,f) and 14(a,f) show that, when $\mathrm {Re}(\omega _G)$ is much lower than $\mathrm {Re}(\omega _A)$, at a very low Mach number $M_0=0.03$, the acoustic effect of the resonator barely shows in the spatial distributions, i.e. the whole resonator is compact. As $\mathrm {Re}(\omega _G)$ increases with the increasing $M_0$, the acoustic effect of the resonator on the $|p|$ distribution progressively appears. Note in figure 10(a) that, for global and acoustic resonant modes of the same symmetry, $\mathrm {Re}(\omega _G)$ only coincides with $\mathrm {Re}(\omega _A)$ at one particular value of $M_0$ as $\mathrm {Re}(\omega _G)$ varies continuously with $M_0$, since the antisymmetric or symmetric acoustic resonant mode only occurs at one discrete frequency in the frequency range considered. However, the influence of the resonator on the global modes appears over continuous Mach number and frequency ranges. It is also interesting to compare the $|p|$ distribution here with those shown in figure 9. The excitation frequencies in figures 9(c) and 9(e) respectively equal to $\mathrm {Re}(\omega _G)$ in figures 12(e) and 14(d). The resemblance in the $|p|$ distribution inside the cavities can be observed when the frequencies coincide, even though one is an externally excited field whereas the other is an eigenfunction describing the two-way excitation between vortical and acoustic motions in an AR.

Figure 12.

Spatial distributions of the antisymmetric global modes for $M_0=0.03$, 0.06, 0.09, 0.12 and 0.15 (rows 1 to 5). Column 1, $|p|$; column 2, $\mathrm {Re}(v)$; columns 3 and 4, amplitude and phase of $v$ along the opening of the low cavity (at the point just below $y=-H/2$). Blue lines, vortical; red lines, $-x$ acoustic; cyan lines, $+x$ acoustic; black lines, total (all travelling modes).

Figure 13.

Spatial distributions of the antisymmetric global modes for $M_0=0.0985$ (row 1, $\omega _G=0.3787 - 0.0175\mathrm {i}$) and $0.0990$ (row 2, $\omega _G=0.3792 - 0.0172\mathrm {i}$), between which the mode switching occurs. See figure 12 for the descriptions of the columns.

Figure 14.

Spatial distributions of the symmetric global modes for $M_0=0.03$, 0.06, 0.09, 0.12 and 0.15 (rows 1 to 5). See figure 12 for the descriptions of the columns.

Note that the different variation in $|p|$ at the cavity bottoms with frequency shown in figures 12–14 from those in figure 9 might be due to the compressibility effect changing with $M_0$ and also the variation in global mode distribution (the global modes are normalized by $v$ at a constant position).

We now analyse the phase relation and the Strouhal number. The variations of $v$ along the line between the upstream and downstream edges of the lower cavity are plotted in the third and fourth columns in figures 12–14. By observing those variations, one can analyse the effect of the AR on the $\textrm {KH}+\textrm {RPR}$ loop that mainly happens at the cavity openings. For the antisymmetric modes, figure 12(p,q,r,s,t) shows that the phase difference of the vortical disturbances between the edges reduces as $M_0$ is increased. Nevertheless, the phase difference of the $-x$ acoustic disturbances between the edges also changes with $M_0$, and the total phase variation around the feedback loop, which is approximately represented by the phase variation of the vortical disturbances from the upstream to the downstream edge plus the phase variation of the $-x$ acoustic disturbances from the downstream to the upstream edge, remains close to an integral multiple of $2{\rm \pi}$. For the symmetric modes, figure 14(p,q,r,s,t) shows that the variation of the phase difference of the vortical disturbances between the edges with the increasing $M_0$ is small. The phase variations are further displayed in figure 15(a,b), which shows that the variation of the phase difference of the KH waves between cavity edges is directly linked to the $-x$ acoustic disturbances, required by loop closure.

Figure 15.

Phase difference (a,b) and amplitude ratio (c,d) of $v$ at the upstream and downstream cavity edges (blue and red crosses, $\pm x$ disturbances; solid squares, the KH waves; circles, the phase change of $-x$ disturbances plus the phase change of $+x$ disturbances). (e,f) Wavenumbers of the KH waves (solid squares and diamonds, $\mathrm {Re}(k_{{KH}})$ and $\mathrm {Im}(k_{{KH}})$ calculated with $\omega _G$; hollow squares and diamonds, $\mathrm {Re}(k_{{KH}})$ and $\mathrm {Im}(k_{{KH}})$ calculated with $\mathrm {Re}(\omega _G)$). Rows 1 and 2: antisymmetric and symmetric global modes.

An interpretation of the variation of the KH phase difference between the upstream and downstream edges in the antisymmetric and symmetric global modes, from the dual-feedback view, is as follows. In the antisymmetric global modes in figure 12, the acoustic resonant mode is a nearly perfect resonant mode with a high quality factor, as discussed in § 3.2. At frequencies near $\mathrm {Re}(\omega _A)$, the AR feedback, having an amplitude much higher than that of the RPR feedback at the cavity leading edges, dominates in the total feedback. Thus, the quick variation of the KH phase difference near $\mathrm {Re}(\omega _A)$ is due to the quick phase change with frequency in the response function of the AR (Tonon et al. Reference Tonon, Golliard, Hirschberg and Ziada2011). On the other hand, the heavily damped symmetric acoustic resonant mode has a minor influence on the loop phase relation in figure 14, since the corresponding response function has a low magnitude and a slowly varying phase.

The Strouhal number defined as $Sr=\mathrm {Re}(\omega _G)L/2{\rm \pi} M_0$ is reformed,

(3.1)\begin{equation} Sr=\frac{\mathrm{Re}(\omega_G)L}{2{\rm \pi} M_c} \frac{M_c}{M_0}, \end{equation}

where $M_c$ is the convection velocity of the KH waves. The first term on the right-hand side of (3.1) equals the ratio between the cavity length and the KH wavelength, and equals the phase difference of the KH waves between cavity edges, i.e. $\Delta \phi _{KH}/ 2{\rm \pi} =L/\lambda _{KH}=\mathrm {Re}(\omega _G)L/2{\rm \pi} M_c$. Thus, the essential mechanism of the frequency deviation from the Strouhal law shown in figure 10(a), which reflects the deviation of $\Delta \phi _{KH}$ from $2{\rm \pi}$, is the upstream feedback being affected by the AR, or the coexistence of RPR and AR feedbacks. Note that for a parallel velocity profile, $M_c/M_0$ also depends on $Sr$ (Michalke Reference Michalke1965; Schmid & Henningson Reference Schmid and Henningson2001), and this relation also influences the finally selected frequency of a global mode.

It is shown in figure 10(a) that the reduction of the Strouhal number of the antisymmetric global modes continues even though $M_0$ is no longer close to $M_0=0.09$. This might be linked to the global mode switching, that is, the total phase change around the feedback loop jumping from $2j{\rm \pi}$ to $2(\kern1.5pt j-1){\rm \pi}$, where $j$ is an integer. The mode switching is shown in figures 12(r,s) and 15(a). Here, the switching happens slightly later than the maximum temporal growth rate as $M_0$ is increased, which is in line with the result of Yamouni et al. (Reference Yamouni, Sipp and Jacquin2013). The switching is found to occur suddenly during the increase of $M_0$. Figure 13 elaborates the mode switching and shows that the switching is linked to the $-x$ acoustic disturbances. The $\pm {\rm \pi}$ phase jump and the zero amplitude suggest that $v$ of the $-x$ acoustic disturbances changes sign between the downstream and upstream edges. Such a phenomenon, not happening in the compact case, indicates the interaction between the RPR and AR feedbacks here. Note that the phase jump can only be observed near the entrance of the cavities ($y=\pm H/2$) where both the $\textrm {KH}+\textrm {RPR}$ loop and the additional AR feedback are at play. At positions inside the deep cavities, the disturbance field, almost completely governed by the AR, is uniform in the $x$ direction. The phase jump happening in the $-x$ acoustic disturbances rather than in the KH wave also suggests that the mode switching is not a vortical but an acoustic effect.

The present linear global mode analysis for the compact and deep-cavity cases might also provide some insight into the frequency lock-in phenomenon in experiments. In the compact case, since the phase difference of the KH waves between the edges is almost constant and approximately equals to $2{\rm \pi}$, owing to the fact that the solo RPR feedback is nearly hydrodynamic, the oscillation frequency roughly follows the Strouhal law. When an AR is involved, the total feedback is affected by the additional AR feedback, thus $\Delta \phi _{KH}/ 2{\rm \pi}$ accordingly changes to satisfy the loop phase relation. Moreover, $\Delta \phi _{KH}/ 2{\rm \pi}$ varies quickly with frequency when $\mathrm {Re}(\omega _G)$ is near the $\mathrm {Re}(\omega _A)$ of an acoustic resonant mode which has a small $\mathrm {Im}(\omega _A)$. Such a linear mechanism of frequency deviation from the Strouhal law is described in this work. It is noted that a linear mode-coupling mechanism has also been discussed for the frequency lock-in in vortex-induced vibrations (de Langre Reference de Langre2006). The measured frequency of self-sustained cavity flow oscillations also depends on the oscillation amplitude which affects the convection velocity of the vortical motions ($M_c/M_0$). At high amplitudes, the shear layer, strongly perturbed by the intense feedback acoustic waves at the separation point, rolls up into concentrated vortex structures (Ho & Huerre Reference Ho and Huerre1984). A minimum value of $M_c/M_0$ happens at the Mach number of maximum oscillation amplitude (Ma et al. Reference Ma, Slaboch and Morris2009). In some cases, clear vortex structures may not be observed, but the shear layer can be thickened by medium amplitude oscillations, which also leads to a decrease of the convection velocity (Boujo, Bauerheim & Noiray Reference Boujo, Bauerheim and Noiray2018).

The coexistence of the RPR and AR feedbacks also explains the stabilization and destabilization shown in figure 10(b). As shown in the compact case in § 3.1, the amplitude of the RPR feedback disturbances is very small at the upstream separation points. For the antisymmetric modes in the deep-cavity case, the AR feedback is strong near the frequency of the acoustic resonant mode. Thus, compared with the solo RPR feedback, the amplitude of $-x$ acoustic disturbances at the leading edges is increased in the deep-cavity case, which decreases the ratio between the amplitudes of the $-x$ feedback disturbances at $x=L$ and at $x=0$. For the symmetric modes, the weak AR feedback at the upstream edge may have a comparable amplitude as the RPR feedback but an opposite phase. Those two feedbacks then cancel each other at the upstream edge, which leads to an extremely large ratio between the amplitudes of the $-x$ feedback disturbances at $x=L$ and at $x=0$. To satisfy the amplitude relation of the feedback loop for a global mode, the imaginary part of the frequency should be adjusted (see discussion in figure 6). The above analysis is clearly confirmed in figure 15(d) for the symmetric global modes, even though the plot only approximately represents $\boldsymbol{\mathsf{P}}_{u}$ and $\boldsymbol{\mathsf{P}}_{d}$ of the feedback loop. It demonstrates the large ratio between $|v|$ at the downstream edge and $|v|$ at the upstream edge, for the $-x$ feedback disturbances owing to the mutual cancellation between RPR and AR feedbacks at the upstream edge and also for the symmetric KH mode owing to the amplitude requirement for loop closure. The large $\mathrm {Im}(k_{\mathrm {KH}})$ indicated in figure 15(d) is linked to the increase of $\mathrm {Im}(\omega _G)$ (stabilization), as shown in figure 15(f). For the antisymmetric global modes, $|v|$ ratios for the $-x$ feedback disturbances and for the KH wave do not show the same trend, as shown in figure 15(c), which seems due to the global mode switching. A simple and intuitive demonstration of the feedback-loop destabilization is not obtained at this moment, although figure 15(e) shows the reduction of $\mathrm {Im}(k_{\mathrm {KH}})$ over the $M_0$ range of destabilization.