Effect of synthetic jet modulation schemes on the reduction of a laminar separation bubble

The response of a laminar separation bubble to synthetic jet forcing with various modulation schemes is investigated via direct numerical simulations. A simple sinusoidal waveform is considered as a reference case, and various amplitude modulation schemes, including the square-wave “burst” modulation, are employed in the simulations. The results indicate that burst modulation is less effective at reducing the length of the flow separation than the sinusoidal forcing primarily because burst modulation is associated with a broad spectrum of input frequencies that are higher than the target frequency for the flow control. It is found that such high-frequency forcing delays vortex roll-up and promotes vortex pairing and merging, which have an adverse effect on reducing the separation bubble length. A commonly used amplitude modulation scheme is also found to have reduced effectiveness due to its spectral content. A new amplitude modulation scheme which is tailored to impart more energy at the target frequency is proposed and shown to be more effective than the other modulation schemes. Experimental measurements confirm that modulation schemes can be preserved through the actuator and used to enhance the energy content at the target modulation frequency. The present study therefore suggests that the effectiveness of synthetic jet-based flow control could be improved by carefully designing the spectral content of the modulation scheme.

Figure 1

Synthetic jet profiles. SIN: sinusoidal actuation [Eq. (2)]; BM: burst modulation with $f_m=1,f_c=10,N=10$, and $\mathrm{DC}=50\%$ [Eq. (3)]; AM: sinusoidal amplitude modulation with $f_m=1$ and $f_c=10$ [Eq. (4)].

Figure 2

Schematic of a simulation setup.

Figure 3

(a) Instantaneous vortical structure visualized by the isosurface of the second invariant of velocity gradients $\left(Q_2\right)$ for the baseline (no control) case. The color contour represents the streamwise velocity. (b) Time averaged skin friction $(C_f$, solid line) and pressure $(C_p$, dashed line) coefficient profiles. The separation location is determined by the first zero-crossing of $C_f$, and the reattachment point is identified by the peak $C_p$ location. $L_{\mathrm{sep}}$ represents the mean separation bubble length. (c) Mean displacement thickness along the streamwise direction. The separation bubble height $\left(h_{\mathrm{sep}}\right)$ is defined by the maximum displacement thickness.

Figure 4

Velocity spectra at $x/\delta =40$ and 50, located 2δ above from the wall for the baseline case. A discrete peak associated with the vortex shedding in the bubble is observed and is identified as $f_0$.

Figure 5

Time averaged fields for the control cases with the ZNMF synthetic jet. Streamwise velocity contours for the sinusoidal jet case with $F_m^{+}=1$ (a), and the burst modulation jet with $F_c^{+}=10,F_m^{+}=1$, and $\mathrm{DC}=50\%$ (b). Skin friction (solid line) and pressure (dashed line) coefficient profiles for the sinusoidal jet (c) and the burst modulation jet (d). Mean displacement thickness along the streamwise direction for the sinusoidal jet (e) and the burst modulation jet (f).

Figure 6

Instantaneous flow fields for the control cases with the ZNMF synthetic jet. 3D vortical structure visualized by the isosurface of the second invariant of the velocity gradient $\left(Q_2\right)$ for the sinusoidal (SIN) jet case with $F_m^{+}=1$ (a), and the burst modulation (BM) jet with $F_c^{+}=10,F_m^{+}=1$, and $\mathrm{DC}=50\%$ (b). Spanwise vorticity $\left({\omega }_{\mathrm{z}}\right)$ contours for the SIN jet (c) and the BM jet (d).

Figure 7

Synthetic jet profiles for the burst modulation cases listed in Table 1 (a)–(e).

Figure 8

Mean separation bubble length $\left(L_{\mathrm{sep}}\right)$ versus the momentum ratio coefficient $\left(C_{\mu }\right)$. (a)–(e) BM jet controls listed in Table 1. SIN: sinusoidal jet control case shown for the comparison. The solid line represents the quadratic best fit for the BM cases.

Figure 9

(a) Frequency spectra of the synthetic jet velocity input signal. (b) Power spectra of vertical velocity monitored on the synthetic jet outlet port in the flow field. The spectral energy is normalized by the RMS of synthetic jet input velocity. SIN: sinusoidal jet at $F_m^{+}=1$; BM: burst modulated jet with $F_c^{+}=10,F_m^{+}=1$, and $\mathrm{DC}=50\%$.

Figure 10

Instantaneous flow fields for the sinusoidal jet control at $F^{+}=2$. (a) Vortex structures visualized by the isosurface of the second invariant of velocity gradients $\left(Q_2\right)$. (b) Spanwise vorticity $\left({\omega }_{\mathrm{z}}\right)$ contours.

Figure 11

(a) Waveform of synthetic jet velocity for AM1: amplitude modulation given by Eq. (4), $F_c^{+}=10,F_m^{+}=1$, and $t^{+}=t{f}_0$. (b) waveform for AM2: tailored amplitude modulation scheme given by Eq. (9). (c) Spectra of the synthetic jet input velocities for AM1 and AM2. (d) Power spectra of vertical velocity monitored on the synthetic jet outlet port in the flow field.

Figure 12

Time averaged fields for the control cases with the amplitude modulated synthetic jet. Streamwise velocity contours for the AM1 jet (a) and AM2 jet (b) with $F_c^{+}=10,F_m^{+}=1$. Skin friction (solid line) and pressure (dashed line) coefficient profiles for AM1 (c) and AM2 (d). Mean displacement thickness along the streamwise direction for AM1 (e) and AM2 (f).

Figure 13

Instantaneous flow fields for the control cases with the amplitude modulated synthetic jet. 3D vortical structure visualized by the isosurface of the second invariant of velocity gradients $\left(Q_2\right)$ for the AM1 jet (a) and AM2 jet (b) with $F_c^{+}=10,F_m^{+}=1$. Spanwise vorticity $\left({\omega }_{\mathrm{z}}\right)$ contours for the AM1 jet (c) and AM2 jet (d).

Figure 14

Mean separation bubble length $\left(L_{\mathrm{sep}}\right)$ versus the momentum ratio coefficient $\left(C_{\mu }\right)$. The solid line represents the best quadratic fit for the BM cases. $F_m^{+}=1$ unless otherwise noted.

Figure 15

Decrease of separation bubble length $\left(\mathrm{\Delta }{L}_{\mathrm{sep}}\right)$ normalized by the uncontrolled length $\left(L_{\mathrm{sep},0}\right)$ versus the spectral momentum ratio coefficient at $F^{+}=1\left[C_{\mu }\left(F^{+}=1\right)\right]$. The solid line represents the best power-law fit.

Figure 16

Picture and schematic of the zero-net mass-flux actuator employed in the present study. All dimensions are in mm.

Figure 17

Normalized energy spectra of the output jet velocities from the actuator measured in the experiments for the various input energy modulation schemes; BM: burst modulation [Eq. (3)] with $\mathrm{DC}=50\%$; AM1: amplitude modulation by Eq. (4); and AM2: amplitude modulation by Eq. (9). The modulation frequency is $f_m$ and the actuator driving frequency is $f_c=10f_m$.

Figure 18

Time and spanwise averaged, streamwise velocity contours with different grid resolutions. (a) coarse (384 × 96 × 32), (b) medium (512 × 128 × 32), and (c) fine (768 × 256 × 32) resolutions.

Figure 19

Comparisons of the time and spanwise averaged $\left(\overline{u},\overline{v}\right)$ and root-mean-squared fluctuation $\left(u_{\mathrm{rms}}^{\prime },v_{\mathrm{rms}}^{\prime }\right)$ velocity profiles extracted at a number of streamwise locations for the different grid resolutions. The arrow indicates the scale of the inflow freestream velocity.

Figure 20

Spanwise correlation coefficients for the streamwise $\left(C_{u^{\prime }{u}^{\prime }}\right)$ and vertical $\left(C_{v^{\prime }{v}^{\prime }}\right)$ velocity fluctuations at $y/\delta =1.5$ and various streamwise locations for the baseline case with $L_z=8\delta$ and $N_z=32(L_z$: spanwise domain size; $N_z$: number of grid points in the spanwise direction).

Figure 21

Comparisons of the time and spanwise averaged $\left(\overline{u},\overline{v}\right)$ and root-mean-squared fluctuation $\left(u_{\mathrm{rms}}^{\prime },v_{\mathrm{rms}}^{\prime }\right)$ velocity profiles extracted at a number of streamwise locations for the different spanwise domain sizes and resolutions. The arrow indicates the scale of the inflow freestream velocity. $L_z$: spanwise domain size; $N_z$: number of grid points in the spanwise direction.

Figure 22

A schematic of the synthetic jet outlet slot. The dashed box indicates the control volume for the analysis of pressure drop through the slot.

Figure 23

A schematic for the control volume (dashed line) near the wall location where the synthetic jet is modeled as the velocity boundary condition. The control volume is taken along the local streamlines.