2.1. Computational set-up

We perform a large eddy simulation of the turbulent cavity flow using the compressible flow solver CharLES, which uses a second-order finite volume spatial discretisation scheme and a third-order accurate Runge–Kutta scheme for time stepping (Ham & Iaccarino Reference Ham and Iaccarino2004; Ham et al. Reference Ham, Mattsson and Iaccarino2006). The Vreman model is used for sub-grid-scale modelling (Vreman Reference Vreman2004), and the Harten–Lax–van Leer scheme (Toro et al. Reference Toro, Spruce and Speares1994) is used as the shock capturing scheme. The computational domain (not to scale) is shown in figure 1. The incoming streamwise velocity profile in the boundary layer uses a one-seventh power law that smoothly joins with the free stream. The inlet profile is superimposed with random Fourier modes to simulate unsteady fluctuations at the inlet (Bechara et al. Reference Bechara, Bailly, Lafon and Candel1994). The initial boundary layer thickness at the leading edge is set to be $\delta _0/D=0.167.$ The far-field and outlet boundary conditions are specified as sponge zones to dampen the exiting waves and prevent any numerical reflections (Freund Reference Freund1997). No-slip and adiabatic boundary conditions are specified along the cavity walls. Periodicity is enforced in the spanwise direction.

The origin of the Cartesian coordinate system is set at one of the spanwise boundaries, as shown in figure 1. The $x,\,y$ and $z$ coordinates denote the streamwise, transverse and spanwise directions. The spanwise-periodic extent is two times the depth of the cavity. The inlet is located at three depths from the leading edge of the cavity, and the outlet is specified at seven depths from the aft cavity wall. The far-field boundary condition is specified at nine depths in the transverse direction from the cavity leading edge. We perform local refinement near the actuator at the leading-edge location in all three directions. The computational set-up is the same as the one used in our earlier numerical and experimental studies and has been verified extensively (Sun et al. Reference Sun, Liu, Cattafesta, Ukeiley and Taira2019; Zhang et al. Reference Zhang, Sun, Arora, Cattafesta, Taira and Ukeiley2019; Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021).

2.2. Actuator set-up

We compute impulse response and perform flow control of cavity flow using a wall-normal unsteady blowing jet near the leading edge of the cavity. The actuator set-up is chosen to match our previous numerical and experimental studies (Zhang et al. Reference Zhang, Sun, Arora, Cattafesta, Taira and Ukeiley2019; Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021; Singh et al. Reference Singh, Ukeiley, Zhang, Cattafesta and Taira2022). The streamwise location of the actuator is set at $x_{\!f}/D=-0.0698$ , and the streamwise width of the actuator slot is $\varLambda _{\!f}/D=0.0175$ . The unsteady blowing at a non-dimensional spanwise actuation wavenumber, $\beta$ , is prescribed by modifying the wall-normal velocity boundary condition as

(2.1) \begin{equation} u_{\textit{jet}}(x,z,t)=\eta (t)h(x,x_{\!f},\varLambda _{\!f})\cos (\beta z), \end{equation}

where $\eta (t)$ is a temporal waveform of the unsteady actuation, $h(x,x_{\!f},\varLambda _{\!f})$ is the streamwise shape function of the actuation and the last term accounts for the spanwise sinusoidal behaviour. The exact form of $\eta (t)$ will be examined in this study later. The streamwise spatial profile of the actuator is specified using a hyperbolic tangent function to prevent numerical discontinuity near the actuator slot edges as

(2.2) \begin{equation} h(x,x_{\!f},\varLambda _{\!f})=\frac {1}{4}\left [1+\tanh (\kappa _1(x-x_{\!f}+\varLambda _{\!f}/\kappa _2))\right ]\left [1-\tanh (\kappa _1(x-x_{\!f}-\varLambda _{\!f}/\kappa _2))\right ]\!, \end{equation}

where $\kappa _1=2000$ and $\kappa _2=2.6$ (Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021). The spatial profile of the actuator for an instance of $\beta =\pi$ is shown in figure 1.

To determine the phase-sensitivity function, (2.1) can be used. In this case, $\eta (t)$ takes a temporal waveform of a regularised impulse

(2.3) \begin{equation} \eta _u(t) = u_{\textit{impulse}}(t)= \frac {\epsilon }{\sqrt {2\pi }\sigma } \exp \left [-\frac {1}{2}\frac {(t-t_0)^2}{\sigma ^2}\right ]\!, \end{equation}

where $\epsilon$ is the magnitude of impulse perturbation, $t_0$ is the time-instant of the impulse perturbation and $\sigma$ is the pulse width of the impulse. For the current study, we consider an impulse perturbation of magnitude $\epsilon =0.05$ , and $\sigma$ is chosen to be 1 % of the dominant oscillation period.

We consider open-loop flow control using two periodic waveforms with an actuation frequency $\omega _{\!f}=2\pi f_c$ and a Strouhal number ${\textit{St}}_L=f_cL/U_\infty$ . One is a non-sinusoidal phase-sensitivity-based actuation waveform (discussed later in § 4.3). The other is a sinusoidal actuation waveform at the same frequency $\eta _{\textit{sine}}(t)=A\sin (\omega _{\!f} t)$ , where $A$ is the sinusoidal actuation amplitude.

The actuation cost is measured using the unsteady momentum coefficient, $C_\mu$ . The unsteady momentum coefficient for active flow control cases is defined as

(2.4) \begin{equation} C_\mu = \frac {1}{1/2 U_\infty ^2 w\delta _0}\left [\frac {\omega _{\!f}}{2\pi }\int _{T_{\!f}}\int _{l_x}\int _{l_z} u_{\textit{jet}}(x,z,\omega _{\!f} t)^2 \,{\rm d}z\,{\rm d}x\,{\rm d}t\right ]\!, \end{equation}

where $T_{\!f}=2\pi /\omega _{\!f}$ is the actuation time period, $l_x$ and $l_z$ are the spanwise extents of actuation in the $x$ and $z$ directions, respectively, and $w=2D$ is the spanwise extent of the cavity. In this study, we consider $ C_\mu =0.02$ in line with previous studies (Williams et al. Reference Williams, Cornelius and Rowley2007; Zhang et al. Reference Zhang, Sun, Arora, Cattafesta, Taira and Ukeiley2019; Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021) to assess the control performance.

Since the control objective in this study is to reduce the pressure fluctuations within the cavity, the control performance is measured using relative change of overall root mean squared (r.m.s.) values of pressure distribution between the controlled $(\tilde {p}_{\textit{rms}}^{\textrm {c}})$ and uncontrolled flows $(\tilde {p}_{\textit{rms}}^{\textit{unc}})$ in the cavity as

(2.5) \begin{equation} \Delta \tilde {p}_{\textit{rms}}=\frac {\tilde {p}_{\textit{rms}}^{\textit{unc}}- \tilde {p}_{\textit{rms}}^{\textrm {c}}}{\tilde {p}_{\textit{rms}}^{\textit{unc}}}\quad \text{and} \quad \tilde {p}_{\textit{rms}}=\int _{{V}}\frac {p_{\textit{rms}}}{\frac {1}{2}\rho U_\infty ^2} {\rm d}V, \end{equation}

where $\tilde {p}_{\textit{rms}}$ is volume-integrated r.m.s. of the pressure fluctuations and the spatial domain of integration, $V=\left \{(x,\,y,\,z)/D \in ([0,6],[-1,1.5],[0,2])\right \}$ .

3.1. Flow physics

Let us consider supersonic turbulent flow over an open rectangular cavity at $M_\infty =1.4$ and $\textit{Re}=10\,000$ . The representative instantaneous flow field visualised using an iso-surface of $Q$ -criterion and coloured by the coefficient of pressure is shown in figure 2(a) and $Q$ -criterion is defined as $Q={1/2}\left(|\Omega|^2-|S|^2\right)$ , where $S$ and $\Omega$ are the symmetric strain-rate and antisymmetric rotation-rate tensors, respectively (Hunt et al. Reference Hunt, Wray and Moin1988). Here, the coefficient of pressure, $C_p$ , is defined as $(p-p_\infty )/(({1}/{2})\rho U_\infty ^2)$ . The numerical schlieren visualised using $||\boldsymbol{\nabla }\rho ||$ for a spanwise plane $z=0$ is shown in the background. The shear layer emanating from the leading edge rolls up, forming large-scale spanwise vortical structures at one third of the cavity length from the leading edge. Since the flow is supersonic, we also observe compression waves in the numerical schlieren. These compression waves are generated due to these vortical structures obstructing the free-stream flow. These can also be seen as the large density gradients coincide with the low-pressure vortical structures. Past mid-cavity, we observe the emergence of small-scale structures around the large-scale structures. These large structures convect downstream and impinge on the aft wall, generating strong acoustic waves that can travel upstream through the subsonic regime within the cavity and perturb the leading-edge shear layer, creating a feedback loop. The vortical structures and the compression waves produce violent fluctuations inside the cavity.

Figure 2.

(a) Instantaneous flow field visualised using iso-surface of $Q(D/U_\infty )^2=14$ and coloured by the coefficient of pressure, $C_p$ . The background shows non-dimensionalised numerical schlieren at a spanwise plane $z=0$ in grey. (b) The normalised r.m.s. pressure along the cavity walls and a slice at a spanwise plane $z=0$ . The pressure probes along the shear layer are shown in circles. (c) Time-averaged and spanwise-averaged streamwise velocity $\tilde {u}/U_\infty$ along with streamlines and (d) Reynolds stress $\tau _{xy}/U_\infty ^2$ distribution. (e–g) Frequency spectra $|P(f)| (\textrm {Pa}^2\,\textrm {Hz}^{-1})$ of the pressure probes in the shear layer located at $(x,\,y,\,z)/D=(0.75,0,1),\,(3,0,1)$ and $(5.25,0,1)$ , respectively, with superposed Rossiter mode frequencies denoted by the vertical dotted lines.

The distribution of normalised r.m.s. pressure $p_{\textit{rms}}/(({1}/{2})\rho _\infty U_\infty ^2)$ along the cavity walls and on a plane $z=0$ is visualised in figure 2(b). We observe large fluctuations on the aft wall of the cavity due to the impingement of the vortical structures. We also observe a secondary region with high r.m.s. of pressure fluctuations in the mid-cavity due to the convection of the large-scale spanwise vortical structures.

The time-averaged and spanwise-averaged statistics are shown in figure 2(c–d) using streamwise velocity with streamlines and Reynolds stress, respectively. We observe the presence of a large recirculation zone encompassing the length of the cavity and secondary recirculation zone near the leading edge of the cavity. From the Reynolds stress distribution, we see mixing in the shear-layer region past $x/D\approx 2$ . This region coincides with the formation of large-scale spanwise structures and their breakdown past mid-cavity.

This complex nonlinear flow physics comprises various time scales. The resonant acoustic tones in the cavity that occur due to the acoustic feedback are referred to as the Rossiter modes. These resonant frequencies in the cavity can be determined based on the incoming free-stream Mach number using Rossiter’s semi-empirical formula (Heller et al. Reference Heller, Holmes and Covert1971)

(3.1) \begin{equation} {\textit{St}}_L^n = \frac {n-\alpha }{1/k+M_\infty /\sqrt {1+(\gamma -1)M_\infty ^2/2}}, \end{equation}

where ${\textit{St}}_L^n=fL/U_\infty$ is the length-based Strouhal number for $n$ th Rossiter mode, $\gamma =1.4$ is the specific heat ratio, $k=0.65$ is the averaged convection speed of disturbance and $\alpha =0.38$ is the phase-delay Rossiter mode (Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021).

The frequency spectra $|P(f)|$ of three pressure probes in the shear layer at the locations $(x,\,y,\,z)/D=(0.75,0,1),\, (3.0,0,1),\,(5.25,0,1)$ are shown in figure 2(e–g). The frequency spectra are computed using 50 convective time units based on the length of the cavity for the uncontrolled case. The black dotted lines indicate the predicted first eight Rossiter modes. The dominant time scales match well with the predicted frequencies and earlier studies (Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021). In the shear layer closer to the leading edge, the second Rossiter mode ${\textit{St}}_L^{II}=0.6$ is dominant, and the first and third Rossiter modes, ${\textit{St}}_L^I=0.23$ and ${\textit{St}}_L^{III}=0.96$ , are sub-dominant. Moving towards the mid-cavity, we observe a strong peak at the fourth Rossiter mode ${\textit{St}}_L^{IV}=1.32$ . The dominant frequency shifts to the lower second Rossiter mode near the aft wall. We observe that the maximum oscillation energy is observed in the mid-cavity region due to the formation and convection of the large-scale spanwise structures.

Figure 3.

Phase description for turbulent cavity flow. (a) Instantaneous spanwise-averaged pressure fluctuation $p^\prime _z /(({1}/{2})\rho _\infty U_\infty ^2)$ . (b) The real component of the DMD mode $\varPhi _r/(({1}/{2})\rho _\infty U_\infty ^2)$ of spanwise-averaged pressure at ${\textit{St}}_L\approx 1.32$ . (c) Phase space $a$ – $a^H$ obtained using the inner product of the pressure fluctuations with the real component of the DMD mode.

3.2. Definition of phase

The dominant features of the cavity flows are characterised by the various phases of formation, convection and the impingement of the large vortical structures in the cavity. We aim to identify the low-dimensional phase dynamics that can effectively capture these various phases of vortex convection in the cavity. To this end, we need to identify the dominant time scale and spatial structures associated with the convection process. From the frequency spectra of the pressure probes shown in figure 2(e–g), we observe a broadband spectrum with various dominant frequencies depending on the spatial location. Since the large-vortex roll-up occurs about the mid-cavity, the dominant Rossiter mode associated with the convection is ${\textit{St}}_L^{IV}=1.32$ based on the mid-cavity pressure probe (see figure 2 f).

To support this analysis, we leverage modal analysis (Taira et al. Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017) to identify a low-dimensional space that can capture the dominant physics. More specifically, we apply dynamic mode decomposition (DMD) (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010; Tu Reference Tu2013) to identify the spatial structure corresponding to the vortex convection at the corresponding dominant frequency. Since we are mainly interested in capturing the large vortical structures, the DMD is performed on the spanwise-averaged pressure field. The DMD analyses 30 cycles of flow fields associated with the dominant frequency, ${\textit{St}}_L^{IV}\approx 1.32$ , using approximately 80 snapshots per cycle. The instantaneous spanwise-averaged pressure fluctuations $p^\prime _z (x,y,t)$ and the real component of the DMD mode $\varPhi _r(x,y)$ corresponding to ${\textit{St}}_L\approx 1.32$ are shown in figures 3(a) and 3(b), respectively. The DMD mode highlights the shear-layer region around the mid-cavity.

We can capture the relative phases of the convective process of these large-scale structures by projecting the instantaneous pressure field onto the DMD mode. Since the DMD mode is complex and the real and imaginary parts are shifted by a phase, we consider the inner product $\langle \boldsymbol{\cdot }, \boldsymbol{\cdot }\rangle$ of the instantaneous pressure fields with the real component of the DMD mode given by

(3.2) \begin{equation} a(t) \equiv \langle p^\prime _z(x,y,t), \varPhi _r(x,y) \rangle . \end{equation}

We now establish a two-dimensional phase space that can capture the dominant flow physics by defining the phase $\theta$ and amplitude $r$ based on the time signal $a(t)$ . While several ways exist to compute the phase and amplitude from a temporal signal, we consider the widely used Hilbert transform (Boashash Reference Boashash1992; Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2001) that also results in removing high-frequency fluctuations. Using the Hilbert transform of $a(t)$ denoted by $a^H(t)$ , the phase and amplitude can be defined in the normalised $a-a^H$ phase space as

(3.3) \begin{align} \theta (t)\equiv \tan ^{-1}\left (\frac {a^H(t)}{a(t)}\right )\!,\quad r(t)\equiv \sqrt {a(t)^2+{a^H}(t)^2}. \end{align}

Here, we note that, since the DMD modes are unique up to an arbitrary phase shift, the phase variable $\theta$ discussed in this study could be shifted, and absolute values are not unique, but the relative phase dynamics is of importance. The absolute value of phase is used to compute the phase-sensitivity function and to design the actuation waveform, as discussed later. These quantities are computed from offline analysis of the uncontrolled flow; the actuation waveform is prescribed a priori and executed without real-time feedback. Thus, the absolute value of phase is used only for waveform design and, where applicable, for initial phase alignment. We maintain consistency of $\theta$ using the same DMD mode for all the analysis. We also note that the phase obtained depends on the method used to compute the instantaneous phase. In addition to the Hilbert transform, one may also consider $a(t)$ and the temporal derivative, $\dot {a}(t)$ , for obtaining a phase space which results in a phase shift of $\theta (t)$ relative to that obtained using the Hilbert transform. The comparison between these methods is provided in Appendix A.

Figure 4.

Instantaneous spanwise-averaged pressure fluctuation fields corresponding to various phases $\theta =[0,\pi /2,\pi ,3\pi /2]$ and radii, $r_1\approx 0.51$ (a,b, f,g) and $r_2\approx 0.8$ (c,d,h,i) respectively in the (e) $a-a^H$ phase space.

Figure 4. Long description

Panel A: A heat map showing pressure fluctuations at theta equals 0 and radius r1. The horizontal axis is labeled x/D, and the vertical axis is labeled r1. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel B: A heat map showing pressure fluctuations at theta equals pi/2 and radius r1/2. The horizontal axis is labeled x/D, and the vertical axis is labeled r1/2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel C: A heat map showing pressure fluctuations at theta equals 0 and radius r2. The horizontal axis is labeled x/D, and the vertical axis is labeled r2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel D: A heat map showing pressure fluctuations at theta equals pi/2 and radius r2/2. The horizontal axis is labeled x/D, and the vertical axis is labeled r2/2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel E: A scatter plot showing the phase space with various radii labeled. The horizontal axis is labeled a, and the vertical axis is labeled aH. Different points are marked with different colors and radii. Panel F: A heat map showing pressure fluctuations at theta equals pi and radius r1. The horizontal axis is labeled x/D, and the vertical axis is labeled r1. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel G: A heat map showing pressure fluctuations at theta equals 3pi/2 and radius r1/2. The horizontal axis is labeled x/D, and the vertical axis is labeled r1/2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel H: A heat map showing pressure fluctuations at theta equals pi and radius r2. The horizontal axis is labeled x/D, and the vertical axis is labeled r2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations. Panel I: A heat map showing pressure fluctuations at theta equals 3pi/2 and radius r2/2. The horizontal axis is labeled x/D, and the vertical axis is labeled r2/2. The color scale ranges from blue to red, indicating different levels of pressure fluctuations.

Figure 5.

Instantaneous flow fields (a–h) corresponding to various phases $\theta =[0,\pi /2,\pi ,3\pi /2]$ and radii, $r_1\approx 0.51$ (a,b,e, f) and $r_2\approx 0.8$ (c,d,g,h) are visualised using iso-surface of $Q(D/U_\infty )^2=14$ and coloured by the coefficient of pressure. The background shows numerical schlieren at a spanwise plane $z=0$ in grey.

The phase space $a-a^H$ for the spanwise-averaged cavity flow is shown in figure 3(c). Since turbulent cavity flow exhibits a dynamics with broadband spectra, we observe a noisy limit cycle due to the cycle-to-cycle variation in the amplitude and the time period with an averaged frequency of ${\textit{St}}_L=1.32$ . The radius $r$ characterises the pressure fluctuations within the cavity at this frequency, as it is based on the magnitude of the inner product with the DMD mode. The phase $\theta$ captures the various stages of vortex convection within the cavity. The correlation between the phase variable and the flow physics is depicted in figure 4. To identify the dependence on radius $r$ , we compare the flow physics at two arbitrary radii as a demonstration. The instantaneous spanwise-averaged pressure fluctuation fields corresponding to two average radii, $r_1\approx 0.51$ and $r_2\approx 0.8$ (corresponding cycles are highlighted in the phase space figure 4 e) at four different phases are shown in figure 4. The associated $xy$ -view of instantaneous flow fields are shown in figure 5.

For all the phases, the flow fields corresponding to the smaller radius, $r_1\approx 0.51$ , exhibit weaker vortical structures with lower fluctuations within the cavity, whereas the cycle corresponding to the larger radius shows the presence of stronger spanwise vortical structures within the cavity. The highlighted arrows in figure 4(a,c) show a smaller low-pressure region for $r_1\approx 0.51$ , and larger and stronger low-pressure region for $r_2\approx 0.8$ at $x=2D$ . The instantaneous flow fields (figure 5 a,c) show the corresponding spanwise roll-up formation in the shear layer at $x/D\approx 2$ . The increased fluctuations can also be observed by the strong compression waves seen in the background schlieren for instantaneous flow fields corresponding to $r_2\approx 0.8$ (figure 5 c).

Now, let us consider the evolution of flow fields relative to the phase $\theta$ . For both radii $r$ , when $\theta =0$ (figures 4 a,c, and 5 a,c), we observe the presence of a strong spanwise vortex closer to $x=2D$ . We then observe the convection of this strong vortex to mid-cavity at $\theta =\pi /2$ (figures 4 b,d, and 5 b,d). As we move further along $\theta$ , at phase $\theta =\pi$ (figures 4 f,h, and 5 e,g), we observe that the strong large-scale structure diffuses, and we see the emergence of several small-scale structures around this vortex, about the mid-cavity $x=3D$ . This large-scale structure then loses coherence and we observe several small-scale structures from $x/D=3.5$ to 4 at the phase $\theta =3\pi /2$ as seen in figures 4(g,i) and 5(f,h). The relative position of the low-pressure region associated with spanwise roll-up is captured well by the phase variable defined by the frequency of ${\textit{St}}_L^{IV}=1.32$ . Thus, by projecting the instantaneous flow field onto the DMD mode, we can capture the cycle of vortex convection using the phase variable.

4.1. Phase-reduction approach for periodic flows

Let us consider compressible periodic fluid flows governed by the Navier–Stokes equations

(4.1) \begin{equation} \dot {\boldsymbol{q}}=\mathcal{N}(\boldsymbol{q}(\boldsymbol{x},t)), \end{equation}

where $\boldsymbol{q}=[\rho ,\,\rho u_x,\,\rho u_y,\,\rho u_z,\,e]^T$ , $\boldsymbol{x}=[x,y,z]^T$ and $\mathcal{N}$ is the nonlinear Navier–Stokes operator. Here, $\rho , u_x,\, u_y,\, u_z$ and $e$ are the density, velocities in streamwise, transverse and spanwise directions and energy, respectively.

The periodic solution, $\boldsymbol{q}_0(\boldsymbol{x},t)$ represents a limit cycle and satisfies $\boldsymbol{q}_0(\boldsymbol{x},t)={}\boldsymbol{q}_0(\boldsymbol{x},t+T)$ , where $T$ is the time period of the oscillation with oscillation frequency $\omega _n=2\pi /T$ . For such a flow, the phase $\theta \in [0,2\pi )$ defined on the limit cycle satisfies

(4.2) \begin{equation} \dot {\theta }(t) = \omega _n. \end{equation}

Given this perspective, we can map the flow state on the limit cycle $\boldsymbol{q}_0(\boldsymbol{x},\theta )$ to the single scalar phase variable $\theta$ . For periodic flows, the phase variable $\theta$ can be defined on the phase space formed by characteristic observable of the fluid system. For instance, the lift coefficient has been used to define $\theta$ for periodic wake flows (Taira & Nakao Reference Taira and Nakao2018). In this work, we consider the phase space defined based on the global DMD mode for turbulent oscillatory flows as shown in figure 3 and the corresponding phase description is discussed in § 4.2.

If we have a stable limit cycle solution, this concept of phase can be extended to the vicinity of the limit cycle solution through a generalised phase variable $\varTheta (\boldsymbol{q}(\boldsymbol{x},t))$ such that $\varTheta (\boldsymbol{q})=\theta$ . Thus combining (4.1) and (4.2) and following chain rule, the generalised phase variable satisfies

(4.3) \begin{equation} \dot {\theta }=\dot {\varTheta }(\boldsymbol{q}) = \int _{\mathcal{D}}\boldsymbol{\nabla} _{\boldsymbol{q}} \varTheta (\boldsymbol{q}) \boldsymbol{\cdot }\dot {\boldsymbol{q}} {\rm d} \boldsymbol{x} = \int _{\mathcal{D}}\boldsymbol{\nabla} _{\boldsymbol{q}}\varTheta (\boldsymbol{q}) \boldsymbol{\cdot }\mathcal{N}(\boldsymbol{q}) {\rm d} \boldsymbol{x} = \omega _n, \end{equation}

where $\mathcal{D}$ corresponds to the fluid region.

We can now obtain the phase response to sufficiently small perturbations. We consider the perturbation of the form $\epsilon \boldsymbol{p}(\boldsymbol{x},t)$ added to the governing equations as,

(4.4) \begin{equation} \dot {\boldsymbol{q}}=\mathcal{N}(\boldsymbol{q}(\boldsymbol{x},t))+\epsilon \boldsymbol{p}(\boldsymbol{x},t). \end{equation}

The phase dynamics can be derived as

(4.5) \begin{align} \dot {\theta }(t)&=\dot {\varTheta }(\boldsymbol{q}) = \int _{\mathcal{D}} \boldsymbol{\nabla} _{\boldsymbol{q}}\varTheta (\boldsymbol{q}) \boldsymbol{\cdot }\dot {\boldsymbol{q}} {\rm d} \boldsymbol{x} = \int _{\mathcal{D}} \boldsymbol{\nabla} _{\boldsymbol{q}} \varTheta (\boldsymbol{q}) \boldsymbol{\cdot }\left [\mathcal{N}(\boldsymbol{q}(\boldsymbol{x},t))+\epsilon \boldsymbol{p}(\boldsymbol{x},t)\right ] {\rm d}\boldsymbol{x} \nonumber \\ &\approx \omega _n + \epsilon \int _{\mathcal{D}}\boldsymbol{Y}(\boldsymbol{x},\theta )\boldsymbol{\cdot }\boldsymbol{p}(\boldsymbol{x},t) {\rm d} \boldsymbol{x}, \end{align}

where $\boldsymbol{Y}(\boldsymbol{x},\theta )\equiv \boldsymbol{\nabla} _{\boldsymbol{q}} \varTheta (\boldsymbol{q})|_{\boldsymbol{q}=\boldsymbol{q}_0(\boldsymbol{x},\theta )}$ is the first-order spatial phase-sensitivity function. It quantifies the linear phase response, i.e. phase advancement or delay, for any arbitrary small perturbation. We can further simplify the (4.5), by considering a perturbation of the form $\boldsymbol{p}(\boldsymbol{x},t)=\boldsymbol{h}(\boldsymbol{x})\eta (t)$ such that

(4.6) \begin{align} \dot {\theta }(t)=\omega _n + \epsilon \int _{\mathcal{D}}\boldsymbol{Y}(\boldsymbol{x},\theta )\boldsymbol{\cdot }\boldsymbol{h}(\boldsymbol{x})\eta (t) {\rm d} \boldsymbol{x} = \omega _n + \epsilon Z(\theta )\eta (t), \end{align}

where $Z(\theta )=\int _{\mathcal{D}}\boldsymbol{Y}(\boldsymbol{x},\theta )\boldsymbol{\cdot }\boldsymbol{h}(\boldsymbol{x}) {\rm d} \boldsymbol{x}$ is the effective phase-sensitivity function and $\boldsymbol{h}(\boldsymbol{x})$ is the spatial profile of perturbation.

There are various ways to compute the phase-sensitivity function for periodic fluid flows: the adjoint method (Kawamura & Nakao Reference Kawamura and Nakao2013, Reference Kawamura and Nakao2015; Kawamura et al. Reference Kawamura, Godavarthi and Taira2022), the direct method (Taira & Nakao Reference Taira and Nakao2018; Khodkar & Taira Reference Khodkar and Taira2020; Loe et al. Reference Loe, Nakao, Jimbo and Kotani2021; Nair et al. Reference Nair, Taira, Brunton and Brunton2021) and the Jacobian-free approach (Iima Reference Iima2021). In this study, we use an extension of the direct impulse-based method to compute phase sensitivity, as this method can be used in both simulations and experiments. The direct impulse-based method for periodic flows is based on evaluating the phase shift at various phases in response to added impulse perturbations. Once the phase space and phase variables are determined, at a phase, $\theta _0=\theta (t_0)$ , a small impulse perturbation of the form $\epsilon \eta (t)=\epsilon \delta (t-t_0)$ , where $\delta (t)$ is the Dirac-delta distribution, is added to the periodic flow. The phase of the perturbed flow, $\theta _{\textit{pert}}$ , is then obtained from the phase space of the perturbed flow. For periodic flow, this small impulse creates a phase shift once the transient effects of the perturbation decay. This shift can be measured using the phase difference between the perturbed and the unperturbed flows as, $g(\theta _0;\epsilon )=\lim _{t\rightarrow \infty }\theta _{\textit{pert}}(t)-\theta _{\textit{unpert}}(t)$ (Taira & Nakao Reference Taira and Nakao2018). Thus the first-order phase response is given by $Z(\theta _0)=g(\theta _0; \epsilon )/\epsilon$ .

4.2. Phase response for turbulent oscillatory flows

Traditionally, the phase-reduction analysis is applied to periodic systems. Since many high-speed fluid flows are turbulent, exhibiting a dominant oscillatory behaviour, we extend the phase-reduction approach for such complex oscillatory flows. The computation of phase response of such flows is challenging due to a couple of factors. Firstly, due to the chaotic dynamics (resulting in the absence of strict limit cycles in phase space), there is no asymptotic phase shift as measured using the direct impulse-based method. Further, due to the broadband nature of turbulent flows superimposed on the dominant frequency, there is cycle-to-cycle variation in frequency and time period. The overview of phase reduction for turbulent oscillatory flows is shown in figure 6.

Given these challenges, there have been a limited number of studies that provide heuristic approaches to compute phase-sensitivity functions for low-dimensional chaotic oscillators (Josić & Mar Reference Josić and Mar2001; Kurebayashi et al. Reference Kurebayashi, Fujiwara, Nakao and Ikeguchi2012; Schwabedal et al. Reference Schwabedal, Pikovsky, Kralemann and Rosenblum2012; Imai et al. Reference Imai, Suetani and Aoyagi2022; Tönjes & Kori Reference Tönjes and Kori2022). Among these methods, the approach proposed by Tönjes & Kori (Reference Tönjes and Kori2022) facilitates a direct extension of the impulse-based method to compute the phase-sensitivity function. Their approach measures the averaged phase response behaviour of the chaotic system relative to the average oscillatory period of the unperturbed system. We implement their approach to compute the averaged phase-sensitivity function for turbulent cavity flow.

Figure 6.

Overview of continuous pulse method to compute phase sensitivity of chaotic flows. (a) Definition of phase space using DMD mode of the dominant frequency ${\textit{St}}_L=1.32$ . (b) First instance of impulse perturbation added at the leading edge with a spanwise wavenumber $\beta =\pi$ , when $\theta _0=3\pi /2$ and the corresponding perturbed trajectory in the $a-a^H$ phase space and as $a(t)$ vs $t$ . (c) Pulse-train perturbation $u_{{impulse}(t)}$ and phase-sensitivity function $Z(\theta )$ measured using averaged time shift between perturbed and unperturbed trajectories. The shaded areas represent the standard deviation of $Z(\theta )$ .

Figure 6. Long description

Panel A: A 3D phase space visualization shows chaotic flow patterns with red and blue streamlines. A line graph plots a(t) against time (t), displaying oscillatory behavior. A heatmap shows the spatial distribution of the DMD mode of the dominant frequency. Panel B: A phase space plot compares unperturbed and perturbed trajectories, with the perturbed trajectory shown in red. An inset highlights the impulse perturbation at the leading edge. Panel C: A line graph depicts a pulse-train perturbation over time (t/Td), with vertical lines representing individual pulses. Another line graph shows the phase-sensitivity function Z(θ) against θ, with shaded areas indicating standard deviation.

Following the phase dynamics for periodic flows in (4.6), the phase dynamics of perturbed chaotic flows can be described as

(4.7) \begin{align} \dot {\theta } = \omega _d+D_\omega (t) + \epsilon Z(\theta )\eta (t), \end{align}

where $\omega _d$ is the dominant frequency in the phase space and $D_\omega (t)$ denotes the fluctuations about the averaged dominant frequency due to the chaotic flow dynamics. Note that, $\int _0^\infty D_\omega (t)\,{\rm d}t =0$ .

Extending the direct impulse-based method, we consider a continuous pulse-train perturbation whenever a trajectory crosses a phase $\theta _0$ (Tönjes & Kori Reference Tönjes and Kori2022). We consider the perturbation of the form, $\eta (t)=\sum _i\delta (t-t_i)$ (approximated using Gaussian impulse profile in (2.3)), where $t_i$ is the time instant corresponding to phase $\theta (t_i)=\theta _0$ of the perturbed trajectory. The phase dynamics of continuous $N$ -pulse perturbed chaotic flows is given by

(4.8) \begin{align} \dot {\theta } = \omega _d+D_\omega (t) + \epsilon \sum \limits _{i=1}^N Z(\theta )\delta (t-t_i). \end{align}

For sufficiently large $N$ , the averaged phase dynamics about the unperturbed system is given as

(4.9) \begin{align} \langle \dot {\theta } \rangle _{t} = \omega _d + \epsilon \sum \limits _{i=1}^N \langle Z(\theta )\delta (t-t_i) \rangle _{t}, \end{align}

where $\langle \boldsymbol{\cdot }\rangle _t$ is temporal average. Thus, the averaged phase sensitivity can be computed based on the shift in the average time period and can be written in terms of the average periods of perturbed, $T_\epsilon$ and unperturbed, $T_d$ as

(4.10) \begin{align} Z(\theta ) = \frac {2\pi }{\epsilon }\frac {T_d - T_\epsilon }{T_d}. \end{align}

The overall procedure of computing the phase-sensitivity function for turbulent oscillatory flows is as follows and is shown in figure 6:

1. (i) Defining phase variable $\theta$ : we define the phase variable $\theta$ using the inner product between spanwise-averaged pressure field and the DMD mode associated with ${\textit{St}}_L=1.32$ as $a(t)=\langle p^\prime (x,y,t),\varPhi _r(x,y)\rangle$ (see figure 6 a). The phase and radius variables are defined using the $a-a^H$ phase space (unperturbed trajectory in figure 6 b) using (3.3).

2. (ii) Addition of impulse perturbation (figure 6 b): to measure the response of the system at each phase, we introduce an impulse perturbation at the leading edge at a spanwise wavenumber $\beta$ . An impulse perturbation is added to the unperturbed flow at a time instant $t_1$ corresponding to a phase $\theta _0$ . An instance of this process for $\beta =\pi$ and $\theta _0=3\pi /2$ is shown in figure 6(b). The profile of this perturbation following (2.3) with a zoomed-in view at the actuator location can be seen for $\beta =\pi$ . The simulation is then evolved with the added impulse and the perturbed trajectory is then obtained as $a_{ {pert}}(t) =\langle p_{ {pert}}^\prime (x,y,t),\varPhi _r(x,y)\rangle$ , where $p_{ {pert}}^\prime$ is the spanwise-averaged pressure field for the perturbed cavity flow. The phase variable $\theta _{ {pert}}(t)$ is measured using $a_{ {pert}}-a^H_{ {pert}}$ phase space and an instance of this perturbed trajectory is shown in red in figure 6(b). The second impulse is introduced at the next time instant $(t_2)$ when the perturbed trajectory crosses $\theta _{ {pert}}(t_2)=\theta _0$ .

3. (iii) Computing phase sensitivity through pulse-train method (figure 6 c): the impulse addition is repeated for $N$ cycles for each $\theta _0$ at $t_i,i=1,2,\boldsymbol{\cdot}s ,N$ such that $\theta _{ {pert}}(t_i)=\theta _0$ . An example of a pulse-train perturbation, $u_{\textit{impulse}}(t)$ is shown in (figure 6 c). The time instances of the impulses are plotted relative to the dominant averaged period, $T_d=2\pi /\omega _d$ , to visualise the cycle-to-cycle variations of the trajectory. Note that, while the pulses are varying in time, the time of the pulse correspond to same phase on the limit cycle. The phase dynamics can be seen in figure 15(a). The averaged phase sensitivity $Z(\theta )$ is then measured as the cumulative shift in the time period of the perturbed trajectory relative to the unperturbed trajectory (equation (4.10)). For practical applications of this method, we use a finite number of pulses, $N$ , and the converged phase sensitivity relative to $N$ is considered for further analyses. The phase-sensitivity function $Z(\theta )$ after 13 cycles is shown in figure 6(c).

4.3. Optimal actuation waveform for fast synchronisation

While the flow control objective for cavity flows is aimed at suppressing the overall pressure fluctuations, it is also desirable for the control to take effect and suppress the fluctuations as quickly as possible to minimise structural fatigue. Based on the averaged phase-sensitivity function, we now consider an optimal actuation waveform that quickly modifies the dominant frequency of the vortex-convection dynamics in the shear layer (Zlotnik et al. Reference Zlotnik, Chen, Kiss, Tanaka and Li2013; Godavarthi et al. Reference Godavarthi, Kawamura and Taira2023). We achieve this by considering a waveform that can quickly synchronise to the forcing frequency. While lock-on/synchronisation is undesirable for cavity flows, the phase-based analysis identifies the optimal waveform in a linear sense. Since the underlying flow physics is multi-modal with nonlinear interactions among these frequencies, we hypothesise that the obtained waveform with sufficiently small amplitude at a detuned frequency will suppress the dominant cavity tone. The large vortical structures in the cavity responsible for the violent fluctuations arise due to the acoustic feedback. We aim to quickly modify the frequency of the vortex convection to disrupt this feedback and break down the large-scale structures, thereby reducing the overall pressure fluctuations within the cavity. To this end, we obtain an actuation waveform for fast frequency modification through the lens of synchronisation.

Following (4.9), synchronisation to an external periodic force with a forcing frequency, $\omega _{\!f}$ can be analysed using the relative phase dynamics, $\psi (t)=\theta (t)-\omega _{\!f} t$ as

(4.11) \begin{equation} \dot {\psi }(t) = \Delta \omega + \epsilon Z(\psi +\omega _{\!f} t)\eta (\omega _{\!f} t), \end{equation}

where $\Delta \omega = \omega _d-\omega _{\!f}$ . Synchronisation is achieved when $\dot {\psi }(t) \rightarrow 0$ . Since this equation has explicit temporal dependence, we can transform it into an autonomous form by averaging over a period of forcing (Kuramoto Reference Kuramoto1984; Ermentrout & Kopell Reference Ermentrout and Kopell1991). Considering the relative phase $\psi (t)$ as slowly varying over the averaging period we obtain

(4.12) \begin{equation} \dot {\varPsi }(t) = \Delta \omega + \epsilon \varGamma (\varPsi ),\quad \varGamma (\varPsi )=\frac {1}{2\pi }\int \limits _{0}^{2\pi } Z(\varPsi +\phi )\eta (\phi )\, {\rm d}\phi , \end{equation}

where $\varPsi$ is the averaged relative phase which can be held fixed during the averaging over the fast phase $\phi$ , and $\varGamma (\varPsi )$ is the phase-coupling function. Therefore, the synchronisation condition to an external forcing can be theoretically identified as $\epsilon \min \varGamma (\psi ) \leqslant -\Delta \omega \leqslant \epsilon \max \varGamma (\psi )$ (Nakao Reference Nakao2016).

Now, we seek an optimal actuation waveform to achieve quick frequency modification by maximising $|\dot {\psi }_*|$ , where $\psi _*$ is the stable fixed point of the relative phase dynamics. The cost function for maximisation can then be expressed as

(4.13) \begin{equation} \mathcal{J}(\eta ) = -\varGamma ^\prime (\psi _*) - \lambda \big ( \langle \eta ^2\rangle - 1 \big) - \mu \left [\Delta \omega +\epsilon \varGamma (\psi _*)\right ], \end{equation}

where $\lambda$ and $\mu$ are Lagrangian multipliers and $\langle \boldsymbol{\cdot }\rangle = ({1}/{2\pi }) \int _{0}^{2\pi } ( \boldsymbol{\cdot }) {\rm d}\theta$ . The first term of the cost function $-\varGamma ^\prime (\psi _*)$ is the speed of synchronisation, the second term constrains the actuation energy and the third term enforces the synchronisation condition.

Once the phase sensitivity is obtained, this optimisation problem can be solved analytically using the calculus of variations (Zlotnik et al. Reference Zlotnik, Chen, Kiss, Tanaka and Li2013). Here, without loss of generality, we consider the fixed point, $\psi _*=0$ since the long-term synchronisation dynamics is independent of the choice of initial phase. The phase-sensitivity-based actuation waveform is obtained analytically as

(4.14) \begin{equation} \eta _{Z}(\theta ;\Delta \omega /\epsilon ) = -\frac {Z^\prime (\theta )}{2\lambda } - \frac {(\Delta \omega /\epsilon ) Z(\theta )}{\langle Z^2 \rangle },\quad \lambda = \frac {1}{2}\sqrt {\frac {\langle (Z^\prime )^2\rangle }{1-\frac {(\Delta \omega /\epsilon )^2}{ \langle Z^2 \rangle }}}, \end{equation}

where $Z^\prime (\theta )$ is the derivative of the effective phase-sensitivity function with respect to $\theta$ . Using (4.14), we can determine the actuation waveform that can quickly synchronise to a frequency $\omega _{\!f}$ and the associated magnitude of perturbation, $\epsilon$ . While this phase-sensitivity-based waveform is optimised for fast frequency modification, it is to be noted that this is theoretically valid within the linear regime in the vicinity of a perfect limit cycle. Since we are concerned with turbulent oscillatory flows, we investigate the efficacy of this phase-sensitivity-based actuation waveform by performing numerical simulations.

5.1. Phase-sensitivity functions

We assess the first-order phase response of the cavity flows about the dominant oscillation cycle corresponding to the convection of the large-scale spanwise structures in the cavity flow using the pulse-train method discussed in § 4.2. While the phase variable is defined using spanwise-averaged quantities, phase-reduction theory governs the timing of perturbations rather than the dimensionality of actuation. Three-dimensional forcing is therefore compatible with the present framework, with its influence entering through projection associated with the dominant oscillation. We introduced impulsive perturbations from the actuation at the leading edge with spanwise actuation wavenumbers $\beta =\pi$ and $2\pi$ for each phase at every time the trajectory reaches $\theta =\theta _0$ defined based on the spanwise DMD modes to compute the averaged phase response for each phase $\theta _0$ . We considered such simulations for various $\theta _0$ for both the spanwise wavenumbers.

The phase-sensitivity functions obtained after perturbing for 13 cycles are shown in figure 7(a–b). The trend of the phase-sensitivity function is confirmed to converge after 8 perturbation cycles. The plotted phase-sensitivity functions are computed using the average of the last five impulse perturbations, and the shaded area represents the standard deviation of the last five cycles of impulse perturbation. We note that, since the phase sensitivity is computed based on the spanwise-averaged flow field, the phase sensitivity based on three-dimensional perturbation characterises the most sensitive phase where the three-dimensional perturbation can cause delay/advancement of the two-dimensional large-scale spanwise vortex convection. We observe that, for both $\beta =\pi$ and $2\pi$ , the phase-sensitivity function shows a similar trend. The magnitude of phase sensitivity is also similar, albeit slightly higher at the sensitive phase for $\beta =\pi$ . For both the wavenumbers, the most sensitive phase is indicated as $\theta =3\pi /2$ , where the impulse response is observed to cause phase delay. We note that, while the absolute value of $\theta$ is dependent on the chosen phase space, the flow state associated with a phase and the waveform of the phase-sensitivity function encodes the effect of the perturbation on the respective flow state. From figure 5(d,h), $\theta =3\pi /2$ is mapped to the flow state when the large-scale spanwise vortex is convected past the mid-cavity. This phase precedes the phase, $\theta =0$ , where the strong spanwise vortex is observed at $x/D=2$ . This indicates that introducing a small three-dimensional impulse perturbation at both $\beta =\pi$ and $2\pi$ at the leading edge at $\theta =3\pi /2$ is convected through the shear layer and significantly modifies the formation of the strong spanwise vortex and causes a delay of the vortex-convection process.

Figure 7.

(a–b) Phase-sensitivity functions computed using the pulse-train method for $\beta =\pi$ and $2\pi$ , respectively. The shaded areas represent the standard deviation of $Z(\theta )$ . Phase-sensitivity-based actuation waveforms, $\eta _Z$ , as a function of (c) $\theta$ and (d–e) time, for actuation frequencies of $\omega _{\!f}=0.8\omega _d$ and $1.2\omega _d$ and spanwise wavenumbers of $\beta =\pi$ and $2\pi$ .

Using the phase-sensitivity functions, we then compute the actuation waveform $(\eta _Z)$ for fast frequency modification using (4.14) for actuation frequencies within $20\,\%$ range of the dominant frequency for the magnitude $\epsilon$ corresponding to $C_\mu =0.02$ . The waveforms for both $\beta =\pi$ and $2\pi$ for $\omega _{\!f}=0.8\omega _d$ and $1.2\omega _d$ are shown in figure 7(c). The corresponding temporal variations are shown in figure 7(d,e). While we show the waveforms for the two frequencies, the waveforms associated with $\omega _{\!f}=0.9\omega _d$ and $1.1\omega _d$ lie between them. Since the phase-sensitivity functions are similar, the fast modification waveforms are also similar for $\beta =\pi$ and $2\pi$ . These waveforms resemble a stretched pulse with a significant amount of actuation concentrated between $\theta =\pi$ and $2\pi$ , targeting the sensitive phase of $\theta =3\pi /2$ with minimal actuation input for the other phases.

Figure 8.

Distribution of $p_{\textit{rms}}$ along the cavity walls and a spanwise slice of $z=0$ for controlled flows using phase-sensitivity-based and sinusoidal actuation waveforms at $C_\mu =0.02$ for actuation frequencies $\omega _{\!f}=0.8\omega _d,0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ and spanwise wavenumbers $\beta =\pi$ and $2\pi$ .

Figure 8. Long description

Panel A: A heat map showing the distribution of values along the cavity walls and a spanwise slice for controlled flows using phase-sensitivity-based actuation waveforms at an actuation frequency of 0.8 times the reference frequency. The color scale ranges from blue to red, indicating lower to higher values respectively. The heat map shows a gradient of values with higher values concentrated near the right end of the cavity. Panel B: A heat map similar to Panel A but with an actuation frequency of 0.9 times the reference frequency. The distribution pattern is similar but with slight variations in the intensity and location of higher values. Panel C: A heat map with an actuation frequency of 1.1 times the reference frequency. The pattern shows a more uniform distribution of values compared to Panels A and B. Panel D: A heat map with an actuation frequency of 1.2 times the reference frequency. The higher values are more dispersed and less concentrated near the right end compared to the previous panels. Panel E: A heat map showing the distribution of values for sinusoidal actuation waveforms at an actuation frequency of 0.8 times the reference frequency. The pattern is similar to Panel A but with noticeable differences in the intensity and distribution of higher values. Panel F: A heat map with an actuation frequency of 0.9 times the reference frequency for sinusoidal actuation waveforms. The distribution shows a gradient similar to Panel B but with variations in the intensity of higher values. Panel G: A heat map with an actuation frequency of 1.1 times the reference frequency for sinusoidal actuation waveforms. The pattern is more uniform compared to Panels E and F. Panel H: A heat map with an actuation frequency of 1.2 times the reference frequency for sinusoidal actuation waveforms. The higher values are more dispersed and less concentrated near the right end compared to the previous panels.

5.2. Open-loop flow control

We can now perform open-loop flow control at the actuation frequencies of $\omega _{\!f}=0.8\omega _d,0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ at $\beta =\pi$ and $2\pi$ using both $\eta _Z$ and a sinusoidal waveform, $\eta _{\textit{sine}}$ at the same actuation frequency by introducing unsteady blowing at the leading-edge actuator.

The distribution of r.m.s. fluctuations on the cavity walls and a spanwise slice at plane $z=0$ for all the cases of actuation frequencies using both $\beta =\pi$ and $2\pi$ and waveforms, $\eta _Z$ and $\eta _{\textit{sine}}$ , are shown in figure 8. Overall, we observe a significant reduction in the pressure fluctuations for the considered controlled cases compared with the uncontrolled flow field (seen in figure 2 b). In contrast to the uncontrolled flow field wherein large fluctuations are observed on the aft wall and the shear-layer region, the controlled flows show a reduction in $p_{\textit{rms}}$ along both the shear layer and the cavity walls.

Let us consider the controlled flow scenarios with spanwise actuation wavenumber of $\beta =\pi$ and compare the effect of actuation waveform on the $p_{\textit{rms}}$ distribution. The main difference is observed within the cavity along the shear layer for all the actuation frequencies. We observe lower r.m.s. fluctuations along the shear layer across the cavity length when actuated using phase-sensitivity-based waveform $\eta _Z$ . The difference between the control performance of the waveforms is more significant for higher actuation frequencies, $\omega _{\!f}=1.1\omega _d$ and $1.2\omega _d$ . The difference can be inferred from the narrower transverse spatial extent of large fluctuations over the shear layer when actuated using $\eta _Z$ , whereas the actuation using sinusoidal waveform shows a broader transverse spatial extent of large fluctuations. This narrower spatial region at higher actuation frequencies could indicate the weakening of the large-scale vortical structures resulting in reduction of pressure fluctuations. Overall, for actuation using $\beta =\pi$ , the phase-sensitivity-based actuation waveform achieves more reduction of $p_{\textit{rms}}$ with actuation frequencies higher than the dominant frequency of the uncontrolled flow.

Figure 9.

Instantaneous $xy$ - and $xz$ - views of controlled flow fields visualised using isosurface of $Q(D/U_\infty )^2=14$ coloured using the contour of pressure coefficient with numerical schlieren of spanwise slice $z=0$ in the background. Control using phase-sensitivity-based $\eta _Z$ and sinusoidal waveforms $\eta _{\textit{sine}}$ using (top) $\beta =\pi$ and $\omega _{\!f}=1.1\omega _d$ and using (bottom) $\beta =2\pi$ and $\omega _{\!f}=1.2\omega _d$ .

When the spanwise actuation wavenumber is $\beta =2\pi$ , the $p_{\textit{rms}}$ distribution of the controlled flows shows a similar distribution for both the actuation waveforms, $\eta _Z$ and $\eta _{\textit{sine}}$ , when actuated using $\omega _{\!f}=0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ . In contrast to $\omega _{\!f}=0.8\omega _d$ , the $p_{\textit{rms}}$ distribution shows larger fluctuations over the shear layer and the cavity walls when actuated using the phase-sensitivity-based waveform $\eta _Z$ compared with the sinusoidal actuation waveform $\eta _{\textit{sine}}$ . Even when using sinusoidal actuation waveform, $\omega _{\!f}=0.8\omega _d$ shows larger fluctuations over the shear layer than the other actuation frequencies. Additionally, high-frequency actuation $\omega _{\!f}=1.2\omega _d$ shows a narrower transverse extent of large fluctuations when compared with lower actuation frequencies along the shear layer, especially until the mid-cavity from the leading edge. Further, when comparing the control performance of the two actuation wavenumbers $\beta =\pi$ and $2\pi$ , we observe that $\beta =2\pi$ achieves more attenuation of pressure fluctuations within the cavity when actuated using both the waveforms.

To further investigate the flow physics of the controlled flows, we visualise the instantaneous flow fields in figure 9 for a couple of actuator parameter combinations using both waveforms: (a) $\beta =\pi$ and $\omega _{\!f}=1.1\omega _d$ , (b) $\beta =2\pi$ and $\omega _{\!f}=1.2\omega _d$ . These instantaneous visualisations are provided to give physical context to the flow evolution, while quantitative assessment of control effectiveness is established through integrated pressure metrics and the modal analysis presented later. When compared with the uncontrolled flow fields shown in figures 2(a) and 4(a–h), the controlled flows exhibit weakened the large spanwise structures. We observe the presence of more small-scale structures close to the leading edge and shear-layer thickening due to the introduction of leading-edge actuation. For all the shown controlled flows, we observe a suppression of fluctuations in the shear layer, which is also seen as weaker compression waves in the background schlieren than the uncontrolled flow. The wall-normal leading-edge actuation introduces streamwise vortices into the shear layer as seen from the $xz$ -view of the controlled flow fields.

When actuated with $\beta =\pi$ (figure 9 a), the controlled flows using both the waveforms show stronger vortices in the shear layer seen from the envelope of compression waves ( $xy$ -view) when compared with cavity flows actuated with $\beta =2\pi$ (figure 9 b). This was observed as a wider transverse region with large fluctuations in the $p_{\textit{rms}}$ distribution in figure 8. For $\beta =\pi$ case, actuation using $\eta _Z$ results in the emergence of a larger number of small-scale structures until $x/D=2.5$ when compared with $\eta _{\textit{sine}}$ actuation (see $xz$ -view). When actuated using sinusoidal waveform at the same actuation frequency of $\omega _{\!f}=1.1\omega _d$ , we observe longer streamwise elongated structures about $x/D=2$ whereas we observe shorter streamwise vortices and their mixing around $x/D=2$ resulting in a more stable shear layer when actuated using $\eta _Z$ . Thus, this also weakens the effect of impingement of these vortices on the aft wall as can be observed from the lower $p_{\textit{rms}}$ fluctuations near the aft wall in figure 8 for $\beta =\pi$ .

Let us consider the controlled flow scenario with $\beta =2\pi$ . The pressure distributions show similar variation for $\omega _{\!f}=0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ . Here, we visualise the controlled flow for $\omega _{\!f}=1.2\omega _d$ for $\beta =2\pi$ as a representative case. Since the actuation is at a higher spanwise wavenumber and frequency, we observe more number of smaller streamwise vortices for $\beta =2\pi$ (see $xz$ -view for $\beta =2\pi$ in figure 9). Similar to the actuation cases with $\beta =\pi$ , we observe a larger number of small-scale structures and the interaction among them from $x/D=0$ to $2$ when actuated using $\eta _Z$ waveform when compared with $\eta _{\textit{sine}}$ waveform. Both sinusoidal and phase-sensitivity-based waveforms, when actuated at $\beta =2\pi$ , have a more stabilising effect on the shear layer, resulting in small-scale weaker vortical structures, thereby attenuating pressure fluctuations within the cavity.

Figure 10.

Premultiplied pressure spectra at a probe placed mid-cavity along the shear layer $(x,y,z)/D=(3,0,1)$ for uncontrolled flow shown in black dotted line and controlled flows using $\beta =\pi$ (blue) and $2\pi$ (purple). Controlled cases correspond to the actuation frequencies, $0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ (shown as green dashed lines) actuated using $\eta _Z$ (a–c) and $\eta _{\textit{sine}}$ waveforms (d–f), respectively.

Figure 10. Long description

Panel A: A line graph shows the premultiplied pressure spectra for uncontrolled flow in black dotted lines and controlled flows in blue and purple lines. The x-axis is labeled |p|StL and ranges from 0 to 0.10. The y-axis is labeled StL and ranges from 0.5 to 3.5. A green dashed line indicates an actuation frequency of 0.9ωd. Panel B: A line graph similar to Panel A but with a green dashed line indicating an actuation frequency of 1.1ωd. Panel C: A line graph similar to Panel A but with a green dashed line indicating an actuation frequency of 1.2ωd. Panel D: A line graph similar to Panel A but with a green dashed line indicating an actuation frequency of 0.9ωd. Panel E: A line graph similar to Panel A but with a green dashed line indicating an actuation frequency of 1.1ωd. Panel F: A line graph similar to Panel A but with a green dashed line indicating an actuation frequency of 1.2ωd.

Figure 11.

Temporal variation of integrated $p_{\textit{rms}}$ ratio between the controlled and uncontrolled flows when actuated using spanwise wavenumbers of $\beta =\pi$ (a–d) and $\beta =2\pi$ (e–h) and actuation frequencies of $\omega _{\!f}=0.8\omega _d,0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ , respectively, for both the actuation waveforms, $\eta _Z$ and $\eta _{\textit{sine}}$ .

Figure 12.

Suppression of pressure fluctuations ( $\Delta p_{\textit{rms}}$ ) and the corresponding speed of reduction for the considered control scenarios (SR, speed of reduction).

Figure 12. Long description

A line graph with the horizontal axis labeled as omega f over omega d and the vertical axis labeled as delta p r m s. The graph includes multiple data series represented by different symbols and colors. The legend indicates two different eta values: eta Z and eta sine, each with different beta values (beta equals pi and beta equals 2 pi). The data points show trends for these different scenarios, with lines connecting the points to illustrate the changes in delta p r m s as omega f over omega d varies. The color bar on the right indicates the speed of reduction (SR) values, ranging from 0.05 to 0.09.

Since the motivation of the control strategy is to weaken the dominant frequency by introducing a slightly different time scale, we visualise the premultiplied pressure spectra at a mid-cavity probe placed along the shear layer for a variety of actuation cases in figure 10. The pressure spectra are computed using 40 oscillation cycles with respect to ${\textit{St}}_L=1.32$ (25 convective time units based on the cavity length). All the controlled cases effectively suppress the dominant tone of ${\textit{St}}_L=1.32$ and the overall pressure fluctuations. The actuation frequency for each control scenario is shown using the green dashed line. A secondary peak with much lower energy is observed at ${\textit{St}}_L=1.32$ , indicating a successful frequency detuning. For the same $\beta =\pi$ , actuation using $\eta _Z$ results in a secondary peak with lower energy content at ${\textit{St}}_L=1.32$ similar to the case of $\eta _Z$ . However, we also observe a weaker peak in the spectra at the actuation frequency when compared with the sinusoidal actuation waveform, thereby overall reduction in fluctuations in the cavity when actuated with phase-sensitivity-based waveform at $\beta =\pi$ . This can also be attributed to the presence of the smaller streamwise vortices in the shear layer, which results in the breakdown of the large-scale spanwise vortices being more effective than those introduced with sinusoidal actuation, weakening the feedback loop. Like $\beta =\pi$ , actuation at $\beta =2\pi$ can effectively suppress the dominant tone at ${\textit{St}}_L=1.32$ . Further, a more apparent reduction in overall fluctuations when compared with $\beta =\pi$ is seen for sinusoidal waveform over a frequency band of $1.5\lt {\textit{St}}_L\lt 2.5$ . Thus, introducing actuation at a frequency slightly different from the dominant frequency can suppress the overall fluctuations by disrupting the feedback loop.

The phase-sensitivity-based actuation waveform is designed to quickly modify the flow frequency. While the previous analysis so far considered the asymptotic statistics of controlled flows, we now investigate how quickly we can suppress the pressure fluctuations within the cavity. We visualise the ratio of the integrated $p_{\textit{rms}}$ between the controlled and uncontrolled flows as, $\tilde {p}^{\textrm {c}}_{\textit{rms}}/\tilde {p}^{\textit{unc}}_{\textit{rms}}$ within the cavity using the domain of integration as $(x,\,y,\,z)/D \in ([0,6],[-1,1.5],[0,2])$ for all the actuation scenarios. The speed of reduction (SR) is quantified using

(5.1) \begin{equation} \textrm {SR} = \frac {\frac {\tilde {p}^{\textrm {c}}_{\textit{rms}}}{\tilde {p}^{\textit{unc}}_{\textit{rms}}}\bigg |_{t=0}-\frac {\tilde {p}^{\textrm {c}}_{\textit{rms}}}{\tilde {p}^{\textit{unc}}_{\textit{rms}}}\bigg |_{t=t_0}}{t_0}, \end{equation}

where $t_0$ is the time instant of first inflection point for $p^{\textrm {c}}_{\textit{rms}}/p^{\textit{unc}}_{\textit{rms}}$ . An instance for computing SR is shown in figure 11(h).

When using the actuation wavenumber $\beta =\pi$ ((a)–(d)), the phase-sensitivity-based actuation waveform achieves quicker suppression of pressure fluctuations compared with that of sinusoidal actuation, as predicted by the phase-based analysis. Further, the control performance is better at higher frequencies (c)–(d) than the dominant frequency, as the $p_{\textit{rms}}$ ratio achieves a lower value much quicker than at lower actuation frequencies. For these high-frequency actuation scenarios, the inflection point is observed at approximately six convective time units based on the cavity length for the phase-sensitivity-based waveform. When $\beta =2\pi$ , for $\omega _{\!f}=0.9\omega _d,1.1\omega _d$ and $1.2\omega _d$ the initial variation of the integrated $p_{\textit{rms}}$ ratio is similar for both phase-sensitivity-based and sinusoidal waveforms, and we observe similar reduction and inflection points for $\omega _{\!f}=0.9,1.2\omega _d$ . However, for $\omega _{\!f}=1.1\omega _d$ , we observe that the sinusoidal waveform achieves quicker suppression than the phase-sensitivity-based waveform. In addition, $\omega _{\!f}=0.8\omega _d$ shows an oscillating $p_{\textit{rms}}$ ratio for phase-sensitivity-based waveform, suggesting that the actuation energy might not be enough to achieve frequency modification to a lower frequency for $\beta =2\pi$ .

The summary of control performance combining the overall reduction of pressure fluctuations and the SR for all the considered actuation scenarios is provided in table 1 and is visualised in figure 12. The data points in figure 12 are coloured using the corresponding SR. We observe that the phase-sensitivity-based actuation waveform outperforms the sinusoidal waveform both in terms of the magnitude of reduction and the SR for $\beta =\pi$ . The maximum reduction observed for the phase-sensitivity-based waveform is approximately 41 % for $(\beta ,\omega _{\!f})=(\pi ,1.1\omega _d)$ with a similar reduction of 38 % for $(\beta ,\omega _{\!f})=(\pi ,1.2\omega _d)$ . The controlled case with a sinusoidal waveform achieves a maximum reduction of 32 % for $(\beta ,\omega _{\!f})=(\pi ,0.9\omega _d)$ and other similar reductions of 29 % for $(\beta ,\omega _{\!f})=(\pi ,1.1\omega _d)$ and $(\pi ,1.2\omega _d)$ . Thus, the phase-sensitivity-based actuation waveform achieves 10 % higher attenuation of pressure fluctuations compared with the sinusoidal waveform at $\beta =\pi$ . We observe an interesting trend for $\beta =2\pi$ . The sinusoidal actuation waveform achieves a maximum reduction of 46 % for $(\beta ,\omega _{\!f})=(2\pi ,0.9\omega _d)$ followed by similar reductions of 44 %, 44 % and 43 % for $(\beta ,\omega _{\!f})=(2\pi ,0.8\omega _d),(2\pi ,1.1\omega _d),(2\pi ,1.2\omega _d)$ , respectively. The phase-sensitivity-based actuation waveform suppresses the pressure fluctuations by 46 % for $(\beta ,\omega _{\!f})=(2\pi ,1.2\omega _d)$ with similar reductions of 42 % and 43 % for $(\beta ,\omega _{\!f})=(2\pi ,0.9\omega _d),(2\pi ,1.1\omega _d)$ , respectively. However, for $(\beta ,\omega _{\!f})=(2\pi ,0.8\omega _d)$ , the phase-sensitivity-based waveform suppresses the pressure fluctuations by 30 %. This indicates that an overall higher-frequency actuation achieves better reduction when using a phase-sensitivity-based actuation waveform. Further, for $\beta =2\pi$ , both the magnitude and speed of pressure fluctuation reduction are similar for both sinusoidal and phase-sensitivity-based actuation waveforms for $(\beta ,\omega _{\!f})=(2\pi ,0.9\omega _d)$ and $(2\pi ,1.2\omega _d)$ . However, for $(\beta ,\omega _{\!f})=(2\pi ,1.1\omega _d)$ , the sinusoidal waveform achieves a quicker reduction of pressure fluctuations within the cavity.

Table 1.

Summary of unsteady cavity flow control scenarios.

Table 1. Long description

The table presents a comparison of various control scenarios for unsteady cavity flow, focusing on the reduction of pressure fluctuations and the SR. It has 12 rows and 7 columns. The columns are labeled as Cases, β, ωf/ωd, Stc, prms, Δprms x 100 percent, and SR. The rows are grouped under three main categories: Uncontrolled, ηz, and ηsine. Each row provides specific values for the parameters β, ωf/ωd, Stc, prms, Δprms x 100 percent, and SR. Notable trends include the phase-sensitivity-based actuation waveform outperforming the sinusoidal waveform in terms of the magnitude of reduction and the SR for certain values of β and ωf/ωd. The table highlights the maximum reduction observed for different waveforms and values, indicating the effectiveness of each control scenario.

Figure 13.

Instantaneous $xy$ - and $xz$ - views of controlled flow fields visualised using isosurface of $Q(D/U_\infty )^2=14$ coloured using the contour of pressure coefficient with numerical schlieren of spanwise slice $z=0$ in the background at $tL/U_{\infty }=6$ . Control using phase-sensitivity-based $\eta _Z$ and sinusoidal waveforms $\eta _{\textit{sine}}$ using (top) $(\beta ,\omega _{\!f})=(\pi ,1.1\omega _d)$ and (bottom) $(2\pi ,1.1\omega _d)$ .

Figure 13. Long description

Panel A: Two line graphs compare the perturbation evolution about a periodically varying base state. The x-axis represents the dimensionless time (tL/U∞) and the y-axis represents the normalized perturbation amplitude (Prms/Prms^unc). The graphs show data for two different phase angles, β = π and β = 2π, with two different waveforms, ηZ(ωft) and ηsine(ωft), indicated by different colors. Panel B: Two sets of visualizations show the controlled flow fields using isosurfaces colored by the pressure coefficient and numerical schlieren of spanwise slices in the background. The visualizations are for two different phase angles, β = π and β = 2π, and two different waveforms, ηZ(ωft) and ηsine(ωft). The x-axis represents the dimensionless streamwise distance (x/D).

Let us investigate the flow physics responsible for quick attenuation of the pressure fluctuations within the cavity based on the instantaneous flow fields in figure 13. The flow fields of the actuation scenarios correspond to $tL/U_{\infty }=6$ , where the actuation is turned on at $tL/U_{\infty }=0$ . We consider $\omega _{\!f}=1.1\omega _d$ as we observe a different trend of SR for phase-sensitivity-based and sinusoidal waveforms for $\beta =2\pi$ . Since the control performance is similar at $\beta =\pi$ for both $\omega _{\!f}=1.1\omega _d$ and $1.2\omega _d$ , the same observations also hold for $\omega _{\!f}=1.2\omega _d$ . The initial condition for introducing actuation is the same for all the control scenarios. For $\beta =\pi$ , from the instantaneous flow fields, the phase-sensitivity-based actuation waveform achieves a breakdown of the large-scale spanwise vortex structure within just six convective time units, whereas the sinusoidal waveform shows a more gradual change in weakening the spanwise vortex structure near $x/D=3$ . Actuation using $\eta _Z$ shows a more compact and larger number of streamwise vortices at $1\lt x/D\lt 2$ when compared with $\eta _{\textit{sine}}$ actuation, which comprises elongated and fewer streamwise vortices. For $\eta _Z$ , these smaller streamwise vortices convect along the shear layer and quickly modify the spanwise vortex roll-up through spanwise mixing at $x/D\gt 1.75$ , preceding the streamwise location of the large-scale spanwise vortex, as identified by the phase-sensitivity function in figure 4. Since the sinusoidal actuation introduced elongated streamwise vortices, we observe the emergence of small-scale structures and a gradual loss of coherence and mixing past $x/D\gt 2$ .

For $\beta =2\pi$ , as shown in the $xz$ -view, the phase-sensitivity-based waveform actuation similarly introduces a more compact and larger number of streamwise vortices at $0\lt x/D\lt 1$ when compared with sinusoidal actuation waveform. While the sinusoidal actuation waveform breaks the large-scale spanwise vortices, we observe strong fluctuations and the emergence of coherence for the phase-sensitivity-based actuation waveform at streamwise location $1.5\lt x/D\lt 2.5$ . The streamwise vortices are observed to align in a closely packed manner in the spanwise direction at $x/D=1.5$ . These closely packed vortices appear to cancel out their influence, reducing the actuation effect. This is also seen as a local thickening of the shear layer at $x/D=1.5$ in the $xy$ -view of the flow field. Hence, for a few specific $(\beta ,\omega _d)=(2\pi ,1.1\omega _d)$ , it is possible that sinusoidal actuation achieves quicker suppression than the phase-sensitivity-based actuation waveform. However, we observe similar control performance for the higher actuation frequency of $1.2\omega _d$ .

Figure 14.

The $xz$ - and $xy$ -views of DMD modes of the spanwise velocity components of controlled cases using phase-sensitivity-based $\eta _Z$ and sinusoidal waveforms $\eta _{\textit{sine}}$ with (a) $\beta =\pi$ and $\omega _{\!f}=1.1\omega _d$ , (b) $\beta =2\pi$ and $\omega _{\!f}=1.1\omega _d$ and (c) $\beta =2\pi$ and $\omega _{\!f}=1.2\omega _d$ . The DMD modes are visualised using isosurface of $\pm 0.05$ (red/blue).

We further investigate the control effect using DMD of the three-dimensional flow fields. The spatial DMD modes will reveal the effect of three-dimensional unsteady actuation and the persistence of flow structures introduced by the actuation in the cavity. We extract the DMD modes of the controlled flow fields using $\eta _Z$ and $\eta _{\textit{sine}}$ actuation corresponding to the actuation parameters, $(\beta ,\omega _{\!f})=(\pi ,1.1\omega _d),(2\pi ,1.1\omega _d)$ and $(2\pi ,1.2\omega _d)$ . The Strouhal numbers of these DMD modes correspond to 1.45 and 1.58 for $\omega _{\!f}=1.1\omega _d$ and $1.2\omega _d$ , respectively. These modes are computed using controlled flow fields for 25 actuation cycles after transients, with 58 snapshots for each cycle. The real component of the DMD modes of spanwise velocity are visualised in figure 14. We observe that, for $\beta =\pi$ (figure 14 a), for both $\eta _Z$ and $\eta _{\textit{sine}}$ actuations, the dominant spanwise wavenumber remains the same as the actuation wavenumber $\beta =\pi$ for the entirety of the cavity length. Here, the persistence of $\beta =\pi$ structures falls in line with the assumption and hypothesis for the phase-reduction analysis. Hence, the phase-sensitivity-based waveform works well, as hypothesised, and outperforms the sinusoidal waveform at all actuation frequencies indicated in figures 8, 9, 11 and 12.

For the actuation with a wavenumber of $\beta =2\pi$ (figure 14 b,c), the dominant spanwise wavenumber matches with $\beta =2\pi$ for $\eta _{\textit{sine}}$ for both the forcing frequencies. This indicates that the three-dimensional actuation structures persists throughout the cavity length. However, for $\eta _Z$ actuation at $\omega _{\!f}=1.1\omega _d$ , we observe several smaller-scale spanwise structures over $0\lt x/D\leqslant 1.5$ . This indicates the presence of nonlinear effects introduced by the $\eta _Z$ actuation. These structures coincide with the small-scale streamwise vortices introduced due to actuation, as discussed in figure 13. These structures then start to merge and the dominant spanwise wavenumber shifts to $\beta =\pi$ past $x/D\geqslant 1.5$ . We also observed that $\eta _Z$ -based actuation at $\beta =2\pi$ at a higher frequency of $\omega _{\!f}=1.2\omega _d$ achieves a faster fluctuation reduction speed from figures 11(h) and 12. The DMD mode corresponding to this control case is in figure 14( c). Similar to $\omega _{\!f}=1.1\omega _d$ , the actuation results in several small-scale spanwise structures over $0\lt x/D\leqslant 1.5$ . Unlike the lower-frequency case, for the $\omega _{\!f}=1.2\omega _d$ case, $\beta =2\pi$ is maintained over $1.5\lt x/D\lt 5$ figure 14( c). Since the formation of the large-scale structures in the baseline cavity flow occurs past $x/D=2$ , the higher-frequency forcing results in a breakdown of this structure. Hence for the control parameter combination $\beta =2\pi$ and $\omega _{\!f}=1.2\omega _d$ , $\eta _Z$ waveform shows a better control performance.

For $\beta =2\pi$ , $\eta _Z$ -based actuation results in smaller-scale spanwise structures near the cavity leading edge, over $0\lt x/D\lt 1.5$ , irrespective of the frequency. The three-dimensional actuation effects result in earlier onset of nonlinear flow response, moving the controlled flow dynamics beyond the assumptions of phase-reduction theory. These nonlinear effects become beneficial to the breakdown of the large-scale structures at higher forcing frequencies and show a reduced performance for certain lower forcing frequencies. This also explains the transient control effect where the sinusoidal waveform outperforms the phase-sensitivity-based waveform in SR for $\beta =2\pi$ for certain actuation frequencies. We also note that the asymptotic fluctuation reduction is similar for both $\eta _Z$ and $\eta _{\textit{sine}}$ (figure 12) for $\beta =2\pi$ . This indicates that the three-dimensional actuation and the nonlinear effects at $\beta =2\pi$ at high frequencies result in breakdown of large-scale vortical structures and achieve fluctuation reduction for both the waveforms in the asymptotic sense. This invites opportunities for further studies into characterising the effect of nonlinearity on fast synchronisation control.

5.3. Outlook

The current phase-based analysis is based on spanwise-averaged flow physics and can capture/modify the large-scale spanwise vortex-convection dynamics within the cavity. The phase-based control objective is framed to modify this dominant frequency, disrupting the feedback loop and, hence, indirectly reducing the pressure fluctuations within the cavity. For the current cavity flow control, both the sinusoidal and phase-sensitivity-based actuation waveform achieve a similar reduction of $46\,\%$ with an actuation wavenumber of $\beta =2\pi$ . While the phase-based control waveform does not further enhance the reduction speed, we attributed this to the three-dimensional alignment and the nonlinear interaction of actuation at this wavenumber. Further, this observation is specific to the current problem set-up, and the effect of the variation of the cavity geometry, inflow conditions and leading-edge boundary layer thickness on the phase sensitivity and phase-based control is yet to be investigated. In addition, the performance of the phase-sensitivity-based actuation waveform is significantly better than the sinusoidal waveform for $\beta =\pi$ . A combination of actuation wavenumbers and frequencies is yet to be investigated as a possibility to achieve an even higher attenuation of pressure fluctuations.

Phase-based analysis for turbulent flows can serve as a guideline for formulating timing-based flow control strategies for complex, unsteady flows. Exploring connections between the resolvent analysis and phase-based analysis could be advantageous for designing efficient flow control strategies, as resolvent analysis identifies the most amplified response to perturbations while phase-based analysis identifies the most sensitive time to introduce actuation. Liu et al. (Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021) identified that $(\beta =2\pi ,{\textit{St}}_L=2.49)$ achieves a pressure fluctuation reduction of 52 % on the cavity walls guided by resolvent analysis. There also have been previous studies to identify secondary actuator location on the bottom cavity walls supplementing the leading-edge actuation (Godavarthi et al. Reference Godavarthi, Cattafesta, Ukeiley and Taira2024). Thus, a combination of resolvent-based actuation at the leading edge and phase-based optimal control at a secondary actuation location could achieve a quicker and higher attenuation of pressure fluctuations within the cavity and is to be studied.

Further, with recent developments in data-driven techniques, machine learning has been used to identify and model the dynamics in low-order coordinates (Lusch et al. Reference Lusch, Kutz and Brunton2018; Champion et al. Reference Champion, Lusch, Kutz and Brunton2019; Linot & Graham Reference Linot and Graham2020). Recently, phase-based analysis has been used with machine learning methods to obtain the phase response and a description of periodic and transient flows (Fukami et al. Reference Fukami, Nakao and Taira2024; Smith et al. Reference Smith, Fukami, Sedky, Jones and Taira2024; Yawata et al. Reference Yawata, Fukami, Taira and Nakao2024). With these promising advancements, phase-based analysis can potentially develop unsteady flow control strategies for unsteady flows.