Nomenclature

$A\;$

area

${C_p}$

pressure coefficient

$\overline {{C_p}} $

time averaged pressure coefficient

${C^{\prime}_p}$

turbulent pressure coefficient

${C_{pT}}$

total pressure coefficient

$\langle {C_p}\rangle $

area-weighted averaged pressure coefficient

${C_\mu }$

blowing coefficient

$D$

diameter

$DMD$

dynamic mode decomposition

$f$

frequency

$Re$

Reynolds number

$LES$

large eddy simulation

${L_s}$

slot length

$St$

Strouhal number

$t$

time

$\hat u,\hat v,\hat w$

velocity components

$U,V,W$

time averaged velocity components

$u,v,w$

turbulent velocity components

Greek symbol

$\phi $

phase

$\rho $

density

$\theta $

boundary layer momentum thickness

2.0 Numerical methodology

2.1 Geometry definition and numerical scheme

Figure 1 shows the computational domain and mesh. It includes the pulsed jet actuator and a large downstream area in which the pulsed jet develops. The diameter of the actuator is $D = 0.1965$ m. The computational domain is also cylindrical with a diameter of $5D$ and a length of $11D$ . The width and length of the slot is ${W_s} = 2$ mm and ${L_s} = 10$ mm, respectively. In this study, the loudspeaker cone is simplified as a diaphragm by keeping the cavity volume approximately unchanged: the diaphragm position does not change the volume of the cavity significantly [Reference Utturkar, Mittal, Rampunggoon and Cattafesta25]. This simplification, therefore, provides a close approximation to the experimental conditions. A detailed description of the actuator geometry can be found in Oxlade [Reference Oxlade26]. The origin of the coordinate system is defined as the centre of the base. The pressure coefficient ${C_p}$ , Equation (1), is defined as

(1) \begin{equation}{C_p} = \frac{{p - {p_\infty }}}{{\frac{1}{2}\rho U_\infty ^2}},\end{equation}

where ${p_\infty }$ is the pressure in the far field, ${U_\infty }$ is the reference velocity. Following Oxlade et al. [Reference Oxlade, Morrison, Qubain and Rigas3], ${U_\infty }$ is defined as 15m/s in this study. The area-weighted averaged pressure coefficient on the base surface, Equation (2), is

(2) \begin{equation}\langle{C_p}\rangle = \frac{1}{A_b}\int_{\phi}\int_r C_p (r,\phi)r\ dr\ d\phi,\end{equation}

where ${A_b}$ is the area of the base. The total pressure coefficient, Equation (3), is defined as

(3) \begin{equation}{C_{pT}} = \frac{{{p_T} - {p_\infty }}}{{\frac{1}{2}\rho U_\infty ^2}},\end{equation}

where ${p_T}$ is the total pressure. The pressure coefficient ${C_p}$ and the velocity components $\left( {\hat u,\hat v,\hat w} \right)$ are further decomposed into the time averaged component $\overline {{C_p}} $ , $\left( {U,V,W} \right)$ and the turbulent component ${C^{\prime}_p}$ , $\left( {u,v,w} \right)$ with Equation (4):

(4) \begin{align} {C_p} & = \overline {{C_p}} + {{\tilde C}_p} + {C^{\prime}_p},\nonumber\\\left( {\hat u,\hat v,\hat w} \right) & = \left( {U,V,W} \right) + \left( {u,v,w} \right).\end{align}

The pulsed jet velocity is expressed as a blowing coefficient, Equation (5), which is given by

(5) \begin{equation}{C_\mu } = \frac{{u_f^2{A_j}}}{{U_\infty ^2{A_b}}},\end{equation}

where ${u_f}$ is the amplitude of the pulsed jet, defined as the amplitude of the Fourier component of the jet centre line velocity, and ${A_j}$ is the area of the slot.

The open-source code OpenFOAM is used for the present numerical study. A three-dimensional transient solver based on the PIMPLE algorithm is used to solve compressible Navier-Stokes equations, Equation (6), given by

(6) \begin{equation}\frac{{\partial \left( {\rho {\textbf{u}}} \right)}}{{\partial t}} + \nabla \cdot \left( {\rho {\textbf{uu}}} \right) = - \nabla p + \nabla \cdot \tau ,\end{equation}

where ${\textbf{u}}$ is the velocity vector, $\tau $ is the stress tensor. The PIMPLE algorithm combines the PISO (Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), with the advantage of reduced Courant number restriction. LES is performed with the standard Smagorinsky subgrid model, which has been extensively verified by Krajnović and Fernandes [Reference Krajnović and Fernandes12] and Parkin et al. [Reference Parkin, Thompson and Sheridan13]. The numerical scheme is second-order in time discretion. A first/second-order linear-upwind discretisation is used for space terms. The velocity boundary of the computational domain is defined as a zero-gradient boundary, and a non-reflecting pressure boundary is used to minimise the pressure reflection. A no-slip boundary condition is used for the wall boundary of the actuator. A universal wall function based on Spalding’s law is used to reduce the mesh requirement in the near-wall region, reducing the computational cost. A sparse mesh is used for the far field. In the vicinity of the body and in the wake region, the mesh is refined: specifically, the mesh near the slot is further refined to simulate the pulsed jet. A mesh with a total number of 4.5m nodes is used for the pulsed jet computational domain. Since the velocity varies with phase, an adjustable time interval (keeping the Courant number less than 0.7) is used to save computational time while satisfying the CFL condition. Approximately 1,500 time steps are resolved in every pulsed-jet period.

2.2 Actuator boundary condition

In the current study, the pulsed jet actuator is included in the computational domain. The diaphragm periodically ingests and expels the working fluid through a slot to generate the pulsed jet. The diaphragm boundary condition is prescribed as sinusoidal motion in time and parabolic distribution in space. Mane et al. [Reference Mane, Mossi, Rostami, Bryant and Castro27] demonstrate that for piezoelectric diaphragm a parabolic surface approximation is more accurate than a logarithmic one. Hence, taking the diaphragm centre as the coordinate origin moving in the $x$ direction, the time-dependent diaphragm profile, Equation (7), is described by

(7) \begin{equation}x\left( {y,{\rm{\;}}z,t} \right) = \left( {a{y^2} + by + c{z^2} + dz + e} \right){\rm{sin}}\left( {2\pi ft} \right),\end{equation}

where $a$ , $b$ , $c$ , $d$ , $e$ are constants. For an axisymmetric surface with the maximum displacement in the centre and zero displacement at the edge, the constants $c = a$ , $b = d = 0$ , and $e = - \frac{1}{4}aD_d^2$ , where ${D_d}$ is the diameter of the diaphragm. Thus, Equation (7) becomes Equation (8)

(8) \begin{equation}x\left( {y,z,t} \right) = \left( {a{y^2} + a{z^2} - \frac{1}{4}aD_d^2} \right){\rm{sin}}\left( {2\pi ft} \right).\end{equation}

The velocity profile, Equation (9), is achieved by taking the time derivative of Equation (8):

(9) \begin{equation}{\hat u_d}\left( {y,z,t} \right) = \left( {a{y^2} + a{z^2} - \frac{1}{4}aD_d^2} \right)2\pi f{\rm{cos}}\left( {2\pi ft} \right).\end{equation}

The remaining unknown constant, $a$ , is estimated assuming mass conservation with the incompressible flow. Then, since the solver is compressible, an iterative calculation is conducted with a coarse mesh to modify this number to match the pressure or velocity in the corresponding experiment.

The boundary conditions for the computational domain (Fig. 1(a)) can be summarised in Table 1. For the far-field, a zero-gradient boundary is used for the velocity and a non-reflectional boundary condition is used for the pressure. For the diaphragm boundary, as described above, a time-dependent parabolic velocity profile or moving mesh is used. The non-slip wall boundary condition is applied to the boundaries except the diaphragm and the far-field, including the actuator wall, the base and the stationary geometry inside the actuator, etc.

Table 1. A summary of the boundary conditions for the main variables

3.0 Boundary condition validation

In this section, the moving boundary condition and the velocity boundary condition are compared based on the round pulsed jet actuator studied experimentally by Feero et al. [Reference Feero, Lavoie and Sullivan24]. The configuration investigated is $f/{f_H} = 1.07$ ( ${f_H}$ is the Helmholtz resonance frequency), $\overline {{p_c}} = $ 160Pa ( $\overline {{p_c}} $ is the cavity pressure). The boundary condition of the diaphragm is iterated to match the cavity pressure. The jet profile on the orifice outlet plane is chosen as the criterion for validation.

Initially, 30 periods are calculated to achieve a stable result, after which another 30 periods are calculated to provide a phase-locked average. The numerical and experimental velocity magnitude distribution near the orifice outlet plane at $x/{D_o} = 0.075$ ( ${D_o}$ is the diameter of the orifice) is normalised by the maximum centre-line velocity, illustrated in Fig. 2. As noted by Feero et al. [Reference Feero, Lavoie and Sullivan24], and Mu et al. [Reference Mu, Yan, Wei and Sullivan28], the hot-wire measurements in the experiment do not resolve reverse flow well, leading to uncertainty at the ends of the nozzle ( $y/{D_o} = \pm 0.5$ ). Therefore, the comparison between the numerical results and the experimental ones is focused on $ - 0.45 \leq y/{D_o} \leq 0.45$ . This figure shows that the phase-locked average of the velocity distributions is identical for the two boundary conditions. Both results show a qualitatively good agreement with the experimental ones by Feero et al. [Reference Feero, Lavoie and Sullivan24], especially in the suction phase. There is only a small difference near the orifice edge in the blowing phase.

Figure 2. Spatial distribution of normalised velocity magnitude near the orifice outlet plane over six phases. —, Moving boundary. – –, Velocity boundary. -o-, Experimental result by Feero et al. [Reference Feero, Lavoie and Sullivan24].

Figure 3. $Q$ criterion iso-surface contoured by normalised streamwise velocity $\hat u/{\hat u_{max}}$ (contour range: $ - 0.6 \le \hat u/{\hat u_{max}} \le 0.6$ ) at $\phi = 0,1/2\pi ,\pi ,3/2\pi $ . (a–d) Moving boundary condition. (e–h) Velocity boundary condition. The transparent grey area shows the wall boundary of the actuator. For more details of the actuator geometry, see Feero et al. [Reference Feero, Lavoie and Sullivan24].

The flow structures identified by the $Q$ -criterion ( $Q \gt $ 0) from both the moving boundary and the velocity boundary are shown in Fig. 3. The streamwise velocity is normalised by its maximum and imposed on the iso-surface of $Q$ -criterion. These figures illustrate that the vorticity structure produced by these two methods is almost identical in the downstream region outside the actuator. The coherent structure inside the cavity, however, shows significant differences. By comparing Fig. 3(a)–(d) with Fig. 3(e)–(h), it is clear that with the velocity boundary, the flow structure inside the cavity is weaker and smaller than that achieved by the moving boundary condition. In the moving boundary condition, the diaphragm boundary is a moving wall boundary, which is the same as the piezoelectric diaphragm motion, while the velocity boundary condition uses a velocity inlet/outlet boundary to approximate such motion. Therefore, with the velocity boundary condition, a considerable portion of the flow is sucked into the diaphragm instead of being reflected. Hence, inside the cavity, the velocity boundary result is probably less accurate than the moving boundary result.

The computational cost of the moving boundary condition is compared with the velocity boundary condition by assessing the computational time with the same computational resource and numerical settings. Since mesh deformation is performed at every iteration in the moving boundary method, to complete the same computation the computational time with the moving boundary condition ${t_m}$ compared to that with the velocity boundary condition ${t_v}$ is about ${t_m}/{t_v} \approx 250{\rm{\% }}$ . In conclusion, when the downstream interaction between the pulsed jet with the outer fluid is of interest, the velocity boundary condition can achieve very similar results compared to the moving boundary condition but with a lower computational cost. Conversely, if the fluid motion inside the cavity is important, the moving boundary condition should be used.

Figure 4. A whole orthographic view of the Q criterion iso-surface contoured by normalised streamwise velocity (contour range: $ - 0.1 \le \hat u/{\hat u_{max}} \le 0.3$ ).

4.0 Characteristics of the pulsed jet actuator

In the following sections, the pulsed jet actuator for control of the bluff body wake is studied with the velocity boundary condition, since it requires less computational cost and the analysis is focused on the area away from the diaphragm. The pulsed jet frequency is $f = 200$ Hz, the Strouhal number based on the boundary layer momentum thickness $\theta = 2.14$ mm [Reference Oxlade, Morrison, Qubain and Rigas3] is $S{t_\theta } = 0.029$ , and the pulsed jet velocity coefficient at the slot outlet is ${C_\mu } = 0.034$ .

The pulsed jet structures generated by the actuator are identified with the Q-criterion and illustrated in Fig. 4. It shows that the actuator generates a series of counter-rotating vorticity rings moving downstream, which confirms that the pulsed jet structures observed experimentally in the 2D PIV plane by Oxlade et al. [Reference Oxlade, Morrison, Qubain and Rigas3] are in the form of rings. Tracing the pulsed jet generated near the slot edge (Fig. 5), in the initial stage ( $\phi = \frac{1}{2}\pi $ ), the pulsed jet remains coherent as a pair of rings in the vicinity of the inner and outer edges of the slot. The spanwise variation of the jet structure is weak, thus the jet is quasi-two-dimensional. As these vorticity rings propagate downstream ( $\phi = \pi \sim 2\pi $ ), the perturbation in the azimuthal direction grows, which leads the vorticity ring to gradually break up into three-dimensional structures. After $\phi = 2\pi $ , the separated vorticity rings rapidly decay, breaking down into small-scale turbulence and eventually dissipating. From their generation near the slot edge, the pulsed jet rings lose coherence after about two periods.

Figure 5. Amplified view of the Q criterion iso-surface contoured by the normalised streamwise velocity at $\phi = \frac{1}{2}\pi ,\pi ,\frac{3}{2}\pi ,2\pi $ (contour range: $ - 0.1 \le \hat u/{\hat u_{max}} \le 0.3$ ).

A typical history of the normalised cavity pressure ${C_p}/{C_{pmax}}$ , the diaphragm velocity ${\hat u_d}/{\hat u_{dmax}}$ , and the streamwise velocity on the slot outlet plane $\hat u/{\hat u_{max}}$ is in Fig. 6. It shows that the phase of the cavity pressure fluctuation is about $\frac{2}{5}\pi $ ahead of the pulsed jet and the diaphragm velocity. This is a physical phenomenon because the pressure gradient is a prerequisite for the formation of the pulsed jet [Reference Glezer and Amitay1]. The diaphragm velocity indicates the cavity size: positive ${\hat u_d}/{\hat u_{dmax}}$ means that the diaphragm is compressing the cavity, thus when the positive ${\hat u_d}/{\hat u_{dmax}}$ comes to an end in the time series, the cavity size is minimum; in turn, negative ${\hat u_d}/{\hat u_{dmax}}$ indicates that the cavity is expanding. By correlating the pressure fluctuation to the diaphragm motion, it is evident that the maximum pressure appears when the cavity size is close to the maximum, while the minimum pressure appears when the cavity size is near a minimum. A theoretical analysis is conducted to analyse the cavity pressure phase.

Figure 6. A typical history of the normalised cavity pressure ${C_p}/{C_{pmax}}$ , the normalised diaphragm motion velocity ${\hat u_d}/{\hat u_{dmax}}$ and the normalised streamwise velocity $\hat u/{\hat u_{max}}$ on the slot outlet plane.

Figure 7. (a) A schematic of the CV bounded by the CS (dashed line). (b) Comparison of the numerical result with the theoretical one calculated using Equation (13).

Taking the cavity as a control volume (CV) bounded by the control surfaces (CS), which includes the diaphragm, the slot inlet, and the cavity wall (Fig. 7(a)), the conservation of mass, Equation (10), can be written as

(10) \begin{equation}\mathop \int \nolimits_{CV} \frac{{\partial \rho }}{{\partial t}}d{\mathcal{V}} + \mathop \int \nolimits_{CS} \rho \left( {{\textbf{U}} \cdot {\textbf{n}}} \right)dA = 0,\end{equation}

where ${\mathcal{V}}$ is the volume of the control volume, $A$ is the area of the control surface. Since the pressure fluctuation propagates with the speed of sound, we assume that in the CV, the pressure and the density are spatially uniform but change in time, thus Equation (10) becomes Equation (11),

(11) \begin{equation}\frac{{\partial \rho }}{{\partial t}}{\mathcal{V}} + \rho \mathop \int \nolimits_{C{S_s}} {\hat u_s}dA - \rho \mathop \int \nolimits_{C{S_d}} {\hat u_d}dA = 0,\end{equation}

where ${\hat u_s}$ is the streamwise velocity of the pulsed jet, ${\hat u_d}$ is the streamwise velocity of the diaphragm, $d$ denotes diaphragm, and $s$ denotes the slot. Rearranging this equation to get Equation (12),

(12) \begin{equation}\frac{1}{\rho }\frac{{\partial \rho }}{{\partial t}}{\mathcal{V}} = \frac{{\partial ln\rho }}{{\partial t}}{\mathcal{V}} = \mathop \int \nolimits_{C{S_d}} {\hat u_d}dA - \mathop \int \nolimits_{C{S_s}} {\hat u_s}dA.\end{equation}

With the assumption of the ideal gas law, $\rho = p/RT$ , and constant temperature, we obtain Equation (13),

(13) \begin{equation}\frac{{\partial lnp}}{{\partial t}} \propto \frac{1}{{\mathcal{V}}}\left( {\mathop \int \nolimits_{C{S_d}} {{\hat u}_d}dA - \mathop \int \nolimits_{C{S_s}} {{\hat u}_s}dA} \right).\end{equation}

To validate the model given by Equation (13), the theoretical result is compared to the numerical one with phase-locked averages. The theoretical estimate of $\partial lnp/\partial t$ is computed by calculating the right-hand side of the equation with the streamwise diaphragm velocity ${u_d}$ , the streamwise velocity at the inlet of the slot ${u_s}$ and the cavity volume ${\mathcal{V}}$ . The numerical estimate of $\partial lnp/\partial t$ is obtained directly. Figure 7(b) shows that the model is able to represent the phase change of the pressure well. The normalised magnitude of the pressure gradient $\partial lnp/\partial t$ also shows good agreement.

The model takes the difference between axial mass flux at the diaphragm and that at the slot inlet, so it clarifies whether the flow can be adequately described by temporal changes with a spatially averaged velocity alone. Practically, then one could use a harmonically varying, parabolic-shaped diaphragm at the inlet, and measured velocity at the slot to compare to simulation data. Equation (13) shows that the phase of the cavity pressure is determined by time integration of the pulsed jet and the diaphragm motion. The domain from the diaphragm face to the external flow supports a spatially evolving pressure wave that predominantly propagates downstream, and ignores any pressure reflections inside the cavity. Pressure changes downstream of the slot can also propagate upstream to the diaphragm since the flow inside the slot is subsonic. The pressure propagating downstream manifests as a pressure wave which causes a large pressure fluctuation on the base.

Figure 8(a) shows the time-averaged total pressure distribution on a plane near the slot. This figure shows that the total pressure loss is concentrated in the vicinity of the sharp slot edge which leads to flow separation. The total pressure loss inside the cavity is relatively small since the flow velocity is low there. Loss in total pressure from diaphragm to slot outlet is due to static pressure drop not recovered by an increase in velocity. A time series of the total pressure coefficient at the slot inlet $\langle {C_{pTi}}\rangle $ at $x/{L_s} = - 1$ , the slot outlet $\langle {C_{pTo}}\rangle $ at $x/{L_s} = 0$ and the difference between the inlet and the outlet $\langle {C_{pTio}}\rangle $ is shown on Fig. 8(c). In the constant blowing jet, the total pressure near the slot inlet is higher than that near the slot outlet, while in the pulsed jet the periodic change of the cavity pressure and jet direction results in total pressure variation. Thus, depending on the phase of the pulsed jet, the jet direction can be from the slot inlet to the outlet or reverses, and the total pressure in the slot inlet can be either higher or lower than the outlet.

Figure 8. The mean total pressure $\overline {{C_{pT}}} $ distribution in the vicinity of the slot for (a) the sharp edge slot and (b) the round edge slot. (c) A time series of the area-weighted averaged total pressure coefficient at the slot inlet $\langle {C_{pTi}}\rangle $ , the slot outlet $\langle {C_{pTo}}\rangle $ , and the difference between the inlet and the outlet $\langle {C_{pTio}}\rangle $ for the slot with a sharp edge. (d) Area-weighted averaged mean total pressure coefficient $\langle \overline {{C_{pT}}} \rangle $ along the slot. $x/{L_s} = 0$ is the outlet of the slot, and $x/{L_s} = - 1$ is the entry of the slot.

The mean total pressure coefficient variation inside the slot is plotted in Fig. 8(d). Since the pulsed jet is characterised by periodic changes in the jet direction and cavity pressure, the mean total pressure coefficient is not constantly decreasing towards the outside. However, it is still clear that the total pressure coefficient dramatically decreases near the slot edge, i,e., at near $x/{L_s} = - 1$ and $x/{L_s} = 0$ . To reduce the total pressure loss through the slot, a round edge is applied to the slot. Figure 8(b) and (d) show that the total pressure loss in the round edge slot is lower than that in the sharp edge slot. By using a round edge, the total pressure coefficient in the middle of the slot ( $x/{L_s} = - 0.5$ ) is increased from $\langle \overline {{C_{pT}}} \rangle = 0.02$ to $\langle \overline {{C_{pT}}} \rangle = 0.05$ . However, since the vortex rings generated by a round edge is weaker than that generated by the sharp edge, it is likely to provide a reduced influence on the wake.

5.0 Dynamic mode decomposition of pressure field

In this section, DMD [Reference Schmid29] is applied to study the pressure field downstream of the actuator and on the actuator base. DMD is a data-based equation-free decomposition method to extract the dominating mode of a dynamic system. It is able to extract low-rank physically interpretable flow features of a high-dimensional field according to their frequency. In the present work, following the exact DMD algorithm described by Kutz et al. [Reference Kutz, Fu and Brunton30], the DMD modes of the pressure field in the range $0 \le x/D \le 10$ , $ - 2.5 \le y/D \le 2.5$ have been decomposed without excluding the long-time mean. The number of sampling points in the spatial coordinate is $400 \times 200$ . The temporal coordinate, each period ( $T{U_\infty }/D = 0.38$ ) is sampled with 100 snapshots. A total number of five periods are sampled to conduct the decomposition. With the DMD modes $\phi $ , the approximate state of the dynamic system $x\left( t \right)$ can be reconstructed by Equation (14),

(14) \begin{equation}x\left( t \right) \approx \mathop \sum \limits_{k = 1}^N {\phi _k}exp\left( {{\omega _k}t} \right){a_k},\end{equation}

where ${\omega _k} = log\left( {{\lambda _k}} \right)/{\rm{\Delta }}t$ , $\lambda $ is the DMD eigenvalue. The growth rate $\sigma $ is defined as $\sigma = Re\left( \omega \right)$ , indicating whether the corresponding modes are growing, decaying or stable. The oscillation frequency of the DMD modes $f$ is defined as $f = \left| {Im\left( \omega \right)/2\pi } \right|$ . Following Schmid et al. [Reference Schmid, Violato and Scarano31], the amplitude of each DMD mode $\left| {{a_k}} \right|$ is achieved by projecting the data sequence on the identified dynamic modes. Assembling the multiple data sequences into matrix form $X$ , the Equation (14) can be rewritten as Equation (15),

(15) \begin{equation}\underbrace {\left[ {\begin{array}{c@{\quad}c@{\quad}c}| & | & \cdots \\[4pt]{{x_1}} & {{x_2}} & \cdots \\[4pt]| & | & \cdots \end{array}} \right]}_{\textbf{X}} \approx \underbrace {\left[ {\begin{array}{c@{\quad}c@{\quad}c}| & | & \cdots \\[4pt]{{\phi _1}} & {{\phi _2}} & \cdots \\[4pt]| & | & \cdots \end{array}} \right]}_{\rm{\Phi }}\underbrace {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{a_1}}&0 & \cdots \\[4pt]0 & {{a_2}} & \cdots \\[4pt] \vdots & \vdots & \ddots \end{array}} \right]}_a\underbrace {\left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 &{{\lambda _1}} & \cdots & {\lambda _1^{m - 2}}\\[4pt]1 &{{\lambda _2}} & \cdots & {\lambda _2^{m - 2}}\\[4pt] \vdots & \vdots & \ddots & \vdots \end{array}} \right]}_{{V_{and}}}.\end{equation}

Figure 9(a) shows the amplitude of the DMD modes against the corresponding frequency. The mean mode is omitted. Four DMD modes dominate the pressure field, each one corresponding to $f = 200$ Hz, 400Hz, 600Hz and 800Hz. Detailed information can be found in Table 2. Since negative frequency is unphysical, $Im\left( \omega \right)/2\pi $ is used as $x$ coordinate to illustrate the full map of the growth rate $\sigma $ (Fig. 9(b)). The first four dominating DMD modes are located in the close vicinity of the $\sigma = 0$ axis. The detailed growth rate (see Table 2) indicates that these DMD modes are stabilised without growing or decaying. Since the pressure wave is generated periodically, the peak magnitude between two successive waves is neither growing or decaying.

Table 2. Summary of the first four dominant DMD modes

Figure 9. DMD spectrum of the actuator pressure field in the downstream. (a) The amplitude $\left| a \right|$ of each DMD mode against frequency. (b) The growth rate $\sigma $ of each DMD mode, the size of the cycle indicating the amplitude of the mode.

The first three modes are illustrated in Fig. 10. The first is characterised by a pair of high-amplitude regions in the vicinity of the slot and a large wave downstream. The other two DMD modes show that downstream a series of high-frequency small-scaled harmonic structures exist. The oscillation frequency of these structures is several times higher than the first mode.

Figure 10. Visualisation of the first three DMD modes in the downstream region.

The time-resolved pressure data on the base is collected from 64 equispaced probe locations described by Rigas et al. [Reference Rigas, Oxlade, Morgans and Morrison32] to analyse the structure of the pressure wave on the base. Since the spatial resolution is significantly lower than the temporal resolution, an augmented matrix obtained by stacking the state multiple times is used to perform DMD [Reference Tu, Rowley, Luchtenburg, Brunton and Kutz33, Reference Brunton, Johnson, Ojemann and Kutz34]. The amplitude map and the growth map are identical to that of the downstream DMD (Table 2) and therefore is not displayed. The first three base DMD modes, which oscillate at frequency $f = 200$ Hz, 400Hz, and 600Hz respectively, are displayed in Fig. 11. These modes are axisymmetric since the pressure wave generated by the axisymmetric actuator is axisymmetric.

Figure 11. Visualisation of the first three DMD modes on the actuator base.