Real-time feedback control of three-dimensional Tollmien-Schlichting waves using a dual-slot actuator geometry

The growth of Tollmien-Schlichting (TS) waves is experimentally attenuated using a single-input and single-output (SISO) feedback system, where the TS wave packet is generated by a surface point source in a flat-plate boundary layer. The SISO system consists of a single wall-mounted hot wire as the sensor and a miniature speaker as the actuator. The actuation is achieved through a dual-slot geometry to minimize the cavity near-field effects on the sensor. The experimental setup to generate TS waves or wave packets is very similar to that used by Li and Gaster [J. Fluid Mech. 550, 185 (2006)]. The aim is to investigate the performance of the SISO control system in attenuating single-frequency, two-dimensional disturbances generated by these configurations. The necessary plant models are obtained using system identification, and the controllers are then designed based on the models and implemented in real-time to test their performance. Cancellation of the rms streamwise velocity fluctuation of TS waves is evident over a significant domain.

Figure 1

Schematic of experiment for generating TS waves from a surface point source.

Figure 2

Schematic of the control components mounted on a removable disk: (a) the dual actuator slots and (b) the wall-mounted hot-wire sensor. All dimensions shown in the close-up view are in mm. The hot wire protrudes above the plate surface by about 0.5 mm.

Figure 3

Picture showing a close-up view of the dual actuator slot geometry and the wall hot wire used in control experiments. The traverse hot wire is aligned with the wall hot wire before performing boundary-layer measurements downstream of the control system.

Figure 4

Feedback control block diagram: $G_d$ is transfer function between the point source and the wall sensor; $G_a$ is the transfer function between the actuator and the wall sensor; the controller is $K$ and the filter $F$.

Figure 5

Experimental boundary layer profiles scaled and compared to the Blasius boundary layer at $X=445$ to 635 mm. The black dashed line shows the Blasius profile.

Figure 6

Impulse input for generating TS wave packets in the boundary layer.

Figure 7

Time history of the normalized streamwise velocity fluctuations through the entire boundary layer, at $X=500\mathrm{mm}$ for an impulse excitation of the point source.

Figure 8

Bode plot of the FRD model $G_a$.

Figure 9

Bode plot showing performance of the controller in experiment with point source. The break points in phase are circled in black.

Figure 10

Nyquist plot of the loop gain in the feedback control of TS waves generated from a point source. The first and the last frequencies are shown to indicate the sense of direction.

Figure 11

Cancellation percentage map: $A_{TS,\mathrm{int}}$. Color bar shows cancellation due to control (%). Figure also shows the location of the actuator dual slots and the wall hot wire. The flow direction is from left to right.

Figure 12

Spanwise distribution of the integrated ${u^{\prime }}_{rms}$ at 385 mm downstream of the wall hot wire ($X=950\mathrm{mm}$), with and without control.

Figure 13

A surface plot of the spanwise distribution of the integrated ${u^{\prime }}_{rms}$ at all the streamwise locations in the measurement grid, with and without control.

Figure 14

Growth of the integrated ${u^{\prime }}_{rms}$ profiles plotted at various streamwise locations along the centerline, inside the control wedge shown in the cancellation percentage map: (a) linear plot and (b) semilog plot. The data are fitted with best-fit straight lines on the semilog plot.