# Helmholtz Resonance Frequency of the Synthetic Jet Actuator
## 4 Discussion  
### Helmholtz Resonance Frequency Model  
The Helmholtz resonance frequency is usually recognized to be [23]:  
\[f_{\rm H}=\frac{1}{2\pi}\cdot\sqrt{\frac{\kappa\cdot A^{2}\cdot p_{a}}{m\cdot V}} \tag{2}\]  
where: \(\kappa\)--specific heat ratio, \(A\)--orifice surface area, \(p_{a}\)--ambient pressure and:  
\[m=\rho\cdot A\cdot l_{e} \tag{3}\]  
where \(l_{e}\) represents the effective orifice length. In turn, assuming that air is an ideal gas, the isentropic speed of sound is given by:  
\[v=\sqrt{\frac{\kappa\cdot p_{a}}{\rho}} \tag{4}\]  
where \(\rho\) is air density. Considering the above, it can be written that:  
\[f_{\rm H}=\frac{v}{2\pi}\cdot\sqrt{\frac{A}{V\cdot l_{e}}} \tag{5}\]  
The effective (modified) length of the orifice \(l_{e}\) resulting from the application of the Bernoulli equation in the modeling of synthetic jet actuation [18] is evaluated according to:  
\[l_{\rm e}=l+\Delta l \tag{6}\]  
Typically, the additive length \(\Delta l\) can be adopted [23]:  
\[\Delta l=0.62\cdot d \tag{7}\]  
Figure 9: Effect of cavity volume and orifice length on the Helmholtz resonance frequency—the case of constant orifice diameter. Values of the constant factor greater than 0.62 in the above equation have also been used in some considerations. For example, in acoustics investigations it is equal to 0.8 [27]. The Helmholtz resonance frequency is also obtained basing on the lumped element model of a synthetic jet actuator. In this case, the components of a synthetic jet actuator are modeled as elements of an analogous electrical circuit. According to the model [15], Helmholtz resonance frequency is given by:  
\[f_{\mathrm{H}}=\frac{1}{2\pi}\cdot[(M_{\mathrm{aN}}+M_{\mathrm{arad}})(C_{ \mathrm{aC}})]^{-0.5} \tag{8}\]  
where: \(M_{\mathrm{aN}}\) and \(M_{\mathrm{arad}}\) are the effective  
Taking into account the impact of the SJ actuator geometry on the effective orifice length \(l_{e}\) in the model M2, the experimental results were compared with those obtained from the M2 model. Figure 11 presents the first comparison as a ratio of the frequencies modeled by the M2 model to the Helmholtz resonance frequencies obtained from the experimental value of \(f_{\text{H-M2}}/f_{\text{H-exp}}\) as a function of the \(l/d\) ratio. All tested cases were included. As can be seen, the convergence of the model with the experimental values decreases with the increase in the \(l/d\) ratio. To merge the data series from Figure 11, an additional dimensionless variable--surfaces ratio \(\sigma\)--was introduced:  
\[\sigma=\frac{\frac{\pi}{4}d^{2}}{\frac{\pi}{4}D^{2}}=\frac{d^{2}}{D^{2}} \tag{14}\]  
Figure 12 presents the second comparison of the\(f_{\text{H-M2}}/f_{\text{H-exp}}\) ratio as a function of \(\sigma\) at a constant value of the orifice length. This applies to the SJ actuator geometries from d1-1/V-1--dl-1/V-9 to d1-6/V-1--dl-6/V-9 (54 cases). As can be seen,\(f_{\text{H-M2}}/f_{\text{H-exp}}\) increases  
Figure 11: The ratio of frequencies obtained from the M2 model to the experimental results as a function of \(l/d\). \[m=\rho\cdot A\cdot l_{e} \tag{3}\]0 The comparison of Helmholtz resonance frequencies obtained from experiments and models: Equations (5) and (11). with the increase in the ratio of the orifice cross-sectional area to the cross-sectional area of the cavity. The theoretical model and the experiment converge at \(\sigma\) = 0.04. The above discussion shows that the compliance of the model M2 with the obtained experimental results was achieved for a small range of the cases considered. Using the regression method, the analytical relationship between the effective orifice length and the geometrical parameters \(l\)/\(d\) and \(\sigma\) was obtained for the range of the cases investigated. This is given by:  
\[m=\rho\cdot A\cdot l_{e} \tag{3}\]1  
In this manner, the M3 model was created.