# Helmholtz Resonance Frequency of the Synthetic Jet Actuator
## 1 Introduction  
A typical synthetic jet actuator (SJA) is a zero-net mass-flux device. A synthetic jet (SJ) is directly formed by the fluid system in which the synthetic jet actuator is located. The SJ is produced by the interaction of a train of vortices that are formed by successive suction and ejection of the fluid across an orifice. The working fluid is alternately sucked into and ejected from the cavity by the motion of a diaphragm that bounds the cavity. Cavity volume oscillation may be caused by a loudspeaker or piezoelectric actuator. The case in which a loudspeaker is used is the subject of the present work. A synthetic jet is a novel flow technique that may be useful for cooling applications, among others, in electronic systems. Recently, synthetic jets have been extensively studied. A number of positions can be found in the literature regarding properties and applications of SJs. The latest references include, among others, the review articles [1, 2], in which the authors present a detailed overview of SJ characteristics and parameters that can influence the enhancement of fluid flow and heat transfer. In addition, research on the acoustic and flow properties of an SJ actuator driven by a loudspeaker was recently conducted in [3, 4]. It should be emphasized that the literature cited above represents only a small portion of the total volume of SJ research, which began at the end of the last century [5, 6, 7]. The frequency characteristics are among the SJ topics studied. A synthetic jet actuator with a single cavity and one orifice is characterized by two resonance frequencies: the firstresulting from the properties of the actuator and the second resulting from the Helmholtz resonance system. These frequencies were the subject of the research of [PERSON] et al. [8; 9], in which the authors showed that the output of the actuator is maximized at the two resonance frequencies considered. Consequently, the SJ actuator energetic efficiency is also maximized at the resonance frequency, as reported by [PERSON] and [PERSON] in [10]. Investigations on flow characteristics of SJ in connection with resonance frequencies have been performed by, among others, [PERSON] et al. [11], [PERSON] and [PERSON] [12], [PERSON] [13], [PERSON] et al. [14; 15; 16], and [PERSON] et al. [17; 18]. It has been concluded that the characteristic SJ velocities are also maximized at the resonance frequencies. Due to the application of SJs in cooling systems, the heat transfer intensity is an important issue in SJ investigations. The results of the conducted research show that excitation frequencies have an impact on heat transfer coefficients [19; 20]. It has been noted that the SJ heat transfer characteristics depend on the dimensionless stroke length, which is the parameter that nondimensionalizes the frequency. Thus, knowledge about resonance frequencies, including the Helmholtz resonant frequency, can also be of significance in SJ heat transfer issues. The main goal of the present research was to determine the mathematical formula of a model to calculate the Helmholtz resonance frequency in the case of synthetic jet actuators. A problem associated with analytical modeling of SJ actuator characteristics was raised by [PERSON] et al. [14; 15; 16]. In this study, the researchers proposed a lumped element model based on an analogy with electrical circuitry. In this model, the components of a piezo-electrically driven SJ actuator were modeled as analogous actuators of an electrical circuit. The modification of the lumped element model was proposed by [PERSON] and [PERSON] [21]. In their model, the mechanical movement of the diaphragm was decoupled from the fluid phenomenon in the cavity, which allowed the modeling of the SJ actuator to be separately investigated and validated. A general lumped parameter model for an SJ actuator with an electromagnetic or piezoelectric driver was presented by [PERSON] [22].