# Frequency Modulation and Erosion Performance of a Self-Resonating Jet
## 1 Introduction  
Water jet technology has been developed for a variety of commercial cleaning applications as chemical stripping procedures used today are costly, dangerous to personnel, and environmentally unsafe without substantial controls. The increasing demands of modern industry promote the rapid development of water jet technology, and advanced water jet technologies, such as cavitating water jets and pulsed water jets, have been introduced continually. In contrast to non-cavitating jets, cutting is achieved by the energy from collapsing cavitation bubbles. The pressure from these imploding bubbles is extremely high, and is focused at many small areas on the eroding surface [1]. As for the pulsed water jet, it takes advantage of the water hammer pressures produced by the slugs' impact (which are much higher than the stagnation pressures generated by a continuous jet) and cycling of loading [2]. The use of a fluidic oscillator to achieve the pulsed performance was investigated [3]. Besides, several self-resonating nozzle design concepts, which integrate cavitating jets and pulsed jets, have been developed [4; 5; 6]. The oscillations are obtained passively without any moving parts and no need for any additional external source of power, which makes these nozzles superior to conventional nozzles. However, there are also some limitations for these self-resonating nozzles since strong oscillations are only achieved in a certain range of both geometrical and operating parameters [7]. Considerable promise has been assigned to the Helmholtz nozzle due to its high efficiency and simplicity. [PERSON] [8] concluded that jet instabilities coupled with the Helmholtz resonance can generate very powerful pressure oscillations with the jet frequency slightly higher than the fundamental frequency of the resonator. [PERSON] and [PERSON] [9] optimized the shape of the impinging edge based on equations of the disturbance wave and two-dimensional vortex, which are the two predominant sources of the generation of pulsation and cavitation. They found that a conical impinging edge with an angle of 120\({}^{\circ}\) (Figure 1) outperformed other nozzles with different impinging edges. The three elements (jet shear-layer instability, Helmholtz resonance, and a coupling (feedback) mechanism) of the loop accounted for the oscillating mechanism [10]. [PERSON] and [PERSON] [11] also concluded that the vapor around the bulk flow acts as an accumulator due to its compressibility. This theory, termed the Gas-Spring Theory, inspired scholars to design nozzles in which the jet is modulated by the vapor phase rather than acoustic modulation. This LPHF nozzle (whose working condition is low pressure and high flow rate) has been proven to enhance cleaning efficiency. Although having similar configurations, the LPHF nozzle and conventional Helmholtz nozzles have different optimal ranges of structural and operating parameters due to different oscillating mechanisms. For the conventional Helmholtz nozzle, the high frequencies (order of kHz) involved, corresponding to sound wave lengths in water of less than 0.3 m, suggest that acoustic oscillator or resonator concepts should be of particular interest [7]. As for the LPHF nozzle, the oscillating period is 3-5 s [12], which means the time of phase transition is more than the relaxation time of the jet passing through the cavity, and the vapor phase is predominant in the oscillating process. To have a better understanding of the vapor phase effect on the oscillating mechanism, numerous experiments have been conducted to study cavitation structures. The various limitations of the measurement techniques have resulted in notable efforts to use numerical simulations for cavitating flows in recent years. Based on the assumption of a homogenous equilibrium medium proposed by [PERSON] et al. [13], many cavitation models have been proposed which require the mixture density to be defined. One approach to this kind of model was based on the state equation [14; 15].