Novel methods for evaluation of the Reynolds number of synthetic jets

Abstract

The paper proposes four novel methods of evaluation of synthetic jet velocity amplitude (Reynolds number in dimensionless form). The methods are based on the measurement of synthetic jet actuator electrical input (alternating current and voltage) and are applicable for loudspeaker-based actuators with air as the working fluid. Experimental validations are performed by means of hot-wire anemometry and laser Doppler vibrometry. Uncertainty and limitation of the methods are discussed, including a proposal of an adequate incompressibility criterion. Ranges of applicability are specified. Additionally, the results are compared with available literature, namely with another method based on cavity pressure measurements, a good consistency is found.

Appendix: derivation of \(\Updelta I\) for method D

Momentum Eq. (22) of a SJA with incompressible fluid contains driving force, $F,$ which is proportional to an unknown input electrical current, $I.$ If the current is established, the displacement amplitude for steady state can be evaluated from the analysis of Eq. (22). To do so, an additional equation for the current must be assembled.

We are assuming a typical arrangement of a modern function generator to which the SJA is wired up directly, without any amplifier. The function generator (such as Agilent 33220A, used in the present experiments, see part Sect. 4.1) has a voltage source connected in series to an embedded internal resistance, $R_{\mathrm{G}},$ (typically $R_{\mathrm{G}} = 50\,\varOmega$). The voltage source generates a voltage, $E_{\mathrm{G}},$ that is usually not the same as the voltage set-up by a user, $E_{\mathrm{set}},$ in a function generator menu. The voltage drop, $E_{\mathrm{set}},$ appears only on the output resistance, $R_{\mathrm{Z}},$ whose value is preset by users via a menu (default values typically are $R_{\mathrm{Z}} = 50\,\varOmega$). The relation between $E_{\mathrm{G}}$ and $E_{\mathrm{set}}$ follows from this arrangement and is given by a constant $k_{\mathrm{G}} = 1 + \frac{R_{\mathrm{G}}}{R_{\mathrm{Z}}}$:

$$E_{\mathrm{G}} = k_{\mathrm{G}} E_{\mathrm{set}}$$

(28)

The electrical circuit with the SJA contains an in-series connected voltage source, $E_{\mathrm{G}},$ internal resistance, $R_{\mathrm{G}},$ cable resistance, $R_{\mathrm{W}},$ and a voltage drop across the SJA loudspeaker voice-coil, $E$ (see Fig. 6). The equation for the current follows from this arrangement:

$$k_{\mathrm{G}} E_{\mathrm{set}} - \left( R_{\mathrm{G}} + R_{\mathrm{W}}\right) I - \left( \underbrace{ R_{\mathrm{L}} I + \overbrace{Bl \dot{x}_{\mathrm{D}}}^{E_{\mathrm{emf}}} }_{E}\right) = 0.$$

(29)

Because we assume only cases with very low frequencies, the influence of loudspeaker inductance, $L_{\mathrm{L}},$ in voltage $E,$ is neglected [compare with Eq. (2)]. Assuming harmonic waveforms for quantities $I,\,E_{\mathrm{set}},\,\dot{x}_{\mathrm{D}}$ [with amplitude given by Eq. (25)], and taking $\delta ,\,\varOmega,$ and $Bl$ as constant, we get the following steady-state current amplitude:

$$\Updelta I= \frac{k_{\mathrm{G}} \Updelta E_{\mathrm{set}}}{\sqrt{ \left( R_{\mathrm{emfa}}+R_{\mathrm{sum}}\right) ^2 + R_{\mathrm{emfb}}^2 }},$$

(30)

where:

$$R_{\mathrm{sum}}= R_{\mathrm{G}}+R_{\mathrm{W}}+R_{\mathrm{L}},$$

(31)

$$R_{\mathrm{emfa}}= \frac{2 \delta \left( Bl \omega \right) ^2}{ m^* \left[ \left( \varOmega ^2-\omega ^2\right) ^2+\left( 2 \delta \omega \right) ^2 \right] },$$

(32)

$$R_{\mathrm{emfb}}= \frac{Bl^2 \omega \left( \varOmega ^2-\omega ^2\right) }{ m^* \left[ \left( \varOmega ^2-\omega ^2\right) ^2+\left( 2 \delta \omega \right) ^2 \right] }.$$

(33)

Fig. 6

Electrical circuit of the synthetic jet actuator