Visualization of synthetic jet formation in air

Abstract

This experimental study deals with a round synthetic jet (SJ) issuing into quiescent surroundings. Flow visualization in air is used to identify different flow field regimes. Hot-wire anemometry and theoretical evaluations are used to quantify parameters in terms of the Reynolds (Re) and Stokes (S) numbers. To verify the theoretical evaluation, additional experiments were performed using the laser Doppler vibrometry. Four regimes of oscillatory suction and extrusion are distinguished and presented by means of a Re–S parameter map: (a) creeping flow without SJ formation, (b) SJ formation and propagation without vortex rollup, (c) SJ with vortex rollup, and (d) vortex structure breakdown, instability and transition to turbulence. Differences in the SJ regimes at low, moderate and high Stokes numbers are found. While all four (a–d) regimes are identified for lower (S < 10) and moderate (S = 10–30) Stokes numbers, the SJ formation process for higher Re and S is coupled with the laminar–turbulent transition. The results are reasonably consistent with those in the available literature.

Graphical Abstract

Appendices

Appendix A: Relative uncertainties of the Reynolds and Stokes numbers

Uncertainty analysis was performed according to the guidelines of BIMP et al. (2008).

Uncertainties of the Reynolds number

For the Reynolds number based on the hot-wire experiment, errors in nozzle diameter, temperature, barometric pressure were estimated at 0.04 mm, 0.7, and 1.5 %, respectively. The error resulting from the temperature loading effect was 6.1 %. The calibration errors were estimated at 1.0–10.8 %, depending on the velocity. Finally, the typical relative uncertainties of the Reynolds numbers were within 17.2 and 10.7 % for experiments with nozzle diameters of D = 1.5 mm and D = 5 mm, respectively. The evaluation was made for cases Figs. 4b and 3d from Table 1, respectively. The confidence level of the uncertainties was 95 %.

For the Reynolds number based on the theoretical evaluation, errors in nozzle diameter, the effective diameter of the diaphragm, temperature, and barometric pressure were estimated at 0.04 mm, 3.8, 0.7, and 1.5 %, respectively. The diaphragm velocity errors were estimated at 16.1 % and 6.0 % for evaluations with nozzle diameters of D = 1.5 mm and D = 5 mm, respectively. Finally, the relative uncertainties of the Reynolds numbers were within 18.3 and 10.1 % for evaluations with nozzle diameters of D = 1.5 mm and D = 5 mm, respectively. The confidence level of the uncertainties was 95 %.

For the Reynolds number based on the LDV experiment, errors in nozzle diameter, the effective diameter of the diaphragm, temperature, and barometric pressure were estimated at 0.04 mm, 3.8, 0.7, and 1.5 %, respectively. Finally, the relative uncertainties of the Reynolds numbers were within 8.7 and 7.8 % for experiments with nozzle diameters of D = 1.5 mm and D = 5 mm, respectively. The confidence level of the uncertainties was 95 %.

Uncertainties of the Stokes number

For the Stokes number evaluation, errors in nozzle diameter, temperature, and barometric pressure were estimated at 0.04 mm, 0.7, and 1.5 %, respectively. Finally, the Stokes number relative uncertainties were within 4.0 and 1.6 % for experiments with nozzle diameters of D = 1.5 mm and D = 5 mm, respectively. The confidence level of the uncertainties was 95 %.

Appendix B: Sinusoidal character of the diaphragm and SJ flow oscillations

The diaphragm surface velocity was measured using laser Doppler vibrometry (LDV). Assuming the continuity equation, incompressibility, a rigid (piston-like) diaphragm, and the slug flow model, the velocity at the orifice (u ₀) was evaluated during the driven cycle—see the curve marked as “u ₀, LDV evaluation” in Fig. 10. This velocity was found to be very close to the ideal sine waveform, as Fig. 10 illustrates. To quantify the very small differences, the crest factors of both curves were calculated. For the u ₀ curve and for the ideal sine curve, the crest factors were 1.44 and 2^0.5, respectively—i.e., the difference in both was only 1.8 %. It can be concluded that direct LDV measurement of the diaphragm velocity confirms the sinusoidal character of the diaphragm oscillations.

Fig. 10

Phase-locked velocity component at the SJ actuator orifice (x = 0, r = 0) for parameters of the 6b case (D = 1. 5 mm, f = 1.5 mm—see parameters in Table 1)

Unlike the sinusoidal character of the diaphragm oscillations, the velocity cycle generated by the SJ actuator can exhibit significant deviations from a sinusoidal character, resulting from the fluid dynamics during SJ formation. Namely, the flow field patterns of the extrusion and suction strokes are basically different. While the extrusion stroke generates a streamwise velocity component (and a radial entrainment component is of lesser importance), the flow field during the suction stroke assumes the characteristics of a three-dimensional (centripetal) sink. Therefore, the velocity vectors are inclined towards the axis in the off-axis regions during the suction stroke. To demonstrate these effects for the present SJ actuator, a hot-wire velocity measurement was performed at the SJ actuator orifice (x = 0, r = 0). Figure 10 shows the results as the U + U _P velocity—see “U + U _P, CTA measurement” curve. During the extrusion stroke, the U + U _P curve corresponds reasonably well with the LDV evaluation. It is worth noting here that this agreement is important for evaluation of the Reynolds number. Namely, the Re is based on the time-mean orifice velocity U ₀, which is defined solely from the extrusion stroke—see Eq. (1).

On the other hand, there were evident differences between the curves, “U + U _P, CTA measurement” and “u ₀, LDV evaluation”, during the suction stroke. This effect demonstrates the above mentioned three-dimensional character of the suction stroke.

Another meaningful effect was an occurrence of around-zero gaps on the U + U _P curve between the extrusion and suction strokes. These gaps are typical for all hot-wire measurements of SJs. They indicate that the velocity magnitude is below the low end of the hot-wire calibration range—see Smith and Glezer (1998). However, a link-up between the extrusion and suction strokes is usually a rather smooth line, occurring near the ideal sine curve. Surprisingly, Fig. 10 shows a deflecting tendency of the U + U _P curve near these gaps. Apparently, this effect is linked with the present low Reynolds numbers because no similar effect was observed at a higher Re range—cf. SJ experiments by Trávníček et al. (2006) at Re = 8,200. An estimation of a more probable link-up over these gaps is shown by the dotted (near-sine) lines in Fig. 10. Note that this effect is negligible for evaluation of the time-mean orifice velocity U ₀.