# Effects of Nozzle Exit Angle on the Pressure Characteristics of SRWJs Used for Deep-Hole Drilling
## 2 Generation of SRWJs  
When a turbulent air jet is excited with periodic acoustic signals by transducers or loudspeakers located at the upstream of the nozzle, the jet structure in the shear layer will organize into discrete ring vortices if the excitation frequency is equal to the predominant frequency of the non-excited jet [11]. The potential of this phenomenon for submerged waterjets was recognized and used to produce SRWJs under a critical Strouhal number of 0.3 [10]. The Strouhal number, \(S_{d}\), is defined as [11]:  
\[S_{d}=\frac{f_{v}D_{j}}{U_{j}} \tag{1}\]  
where \(f_{v}\) is the structuring frequency of the jet, \(D_{j}\) is the jet diameter, and \(U_{j}\) is the jet velocity.  
The passive excitation needed for successful generation of a SRWJ can be obtained by shaping the nozzle with a suitable acoustic chamber, so as to feed back the pressure waves which occur at the nozzle exit. The possible types of nozzle are, \"pulser\", \"organ-pipe\", \"pulser-fed\", and \"laid-back pulser\", as mentioned above [10]. Since the organ-pipe nozzle is the most promising self-resonating nozzle and has wide applications, it is used here to give a detailed description of the SRWJ generation process. The geometrical structure of the organ-pipe nozzle and the principles of the operation of SRWJs are schematically shown in Figure 1.  
An organ-pipe nozzle consists of two abrupt area contractions (the upstream one and the downstream one) and a resonant chamber. The upstream area contraction is characterized by \((D_{i}/D_{c})^{2}\), and the downstream one is characterized by \((D_{c}/D_{c})^{2}\)). Disturbances are triggered at the downstream contraction when the pressurized water is flowing through the nozzle, because of the abrupt change in the flow field. The disturbances propagate in the form of waves and the ones that propagate upwards are reflected at the upstream area contraction, due to the impedance change caused by the variation of the flow area. The reflected waves superpose the incident ones, and a standing wave appears if \(L_{c}\) satisfies the following equation:  
\[L_{c}=\left\{\begin{array}{cc}\frac{2n-1}{4}\lambda&\text{for}\,\left(\frac{ D_{i}}{D_{c}}\right)^{2}\gg 1,\,\left(\frac{D_{c}}{D_{c}}  
Furthermore, peak resonance caused by the standing wave will excite the shear layer of the jet to organize into large-scale structures if \(f_{0}\) is equal to, or is near, \(f_{v}\)[12]. The large-scale structures then develop into vortex rings, whose volume fluctuations are another predominant source accounting for the generation of a SRWJ. Therefore, by equaling \(f_{0}\) and \(f_{v}\), the following relation between \(L_{c}\) and \(D_{e}\) can be obtained using Equations (1)-(3):  
\[f_{0}=f_{v}\Rightarrow\frac{L_{c}}{D_{e}}=\frac{K_{n}c}{U_{j}S_{d}}=\frac{K_{ n}}{M_{a}S_{d}} \tag{4}\]  
where \(M\)a is the Mach number, and \(K_{n}\) is the mode number and is described as:  
\[K_{n}=func\left(n,\frac{D_{c}}{D_{e}},M_{a}\right)=\left\{\begin{array}{ll} \frac{2n-1}{4}\lambda&\text{for }\left(\frac{D_{i}}{D_{e}}\right)^{2}\gg 1,\ \left(\frac{D_{i}}{D_{e}} \right)^{2}\gg 1\\ \frac{n}{2}\lambda&\text{for }\left(\frac{D_{i}}{D_{e}}\right)^{2}\gg 1,\ \left(\frac{D_{i}}{D_{e}} \right)^{2}\approx 1\end{array}\right. \tag{5}\]  
It should be noted that the jet diameter, \(D_{j}\), has been approximated to the exit diameter of the nozzle, \(D_{e}\), in Equation (4). As a result, a SRWJ can be successfully produced by an organ-pipe nozzle which meets the geometrical parameters in Equation (4). More detailed information can be found in the previous research [10; 12; 13; 14].