## 2 Theoretical Background
### Jet-Flow Characteristics Outside the Nozzle  
Since the fluid medium is generally accelerated from a low speed to a relatively high speed in rock erosion application, only the convergent nozzle and the Laval nozzle are discussed here. The flow characteristics of a jet emitted from the nozzle exit are related to the nozzle configuration and the pressure ratio of the ambient pressure to the inlet pressure, \(P_{a}/P_{i}\), which affects the acceleration process of fluid inside the nozzle and determines the interaction of the jet fluid with the environment fluid outside the nozzle [26; 27; 28; 29]. When a jet is formed using a convergent nozzle, the jet will be in three different flow patterns depending on the pressure ratio. For the convergent nozzle, there is a critical pressure ratio to distinguish the flow states of the jet. The critical pressure ratio can be calculated by Equation (2) [27; 29]. To be specific, when \(P_{a}/P_{i}>R_{1}\), the flow velocity at the nozzle exit is subsonic and the jet is a subsonic jet. Moreover, when \(P_{a}/P_{i}=R_{1}\), the flow velocity at the nozzle exit is just the speed of sound and the jet is a sonic jet. In addition, when \(P_{a}/P_{i}<R_{1}\), the flow velocity at the nozzle exit does not continue to increase and is still equal to the speed of sound, which is attributed to the chocking of the convergent nozzle. Then, the sonic flow at the nozzle exit will inflate into a supersonic jet and shock waves will occur in the jet beam. This is because the jet fluid is in an under-expanded state relative to the ambient fluid under this pressure ratio [26; 36]. \[R_{1}=\frac{P_{c1}}{P_{i}}=\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}} \tag{2}\]  
where \(R_{1}\) is critical pressure ratio of convergent nozzle; \(P_{c1}\) is critical ambient pressure of convergent nozzle (MPa); \(P_{i}\) is inlet pressure (MPa); and \(k\) is adiabatic exponent. For the case of the Laval nozzle, there are five different flow states for the jet formed by the nozzle under different pressure ratios. Correspondingly, there are three critical pressure ratios, \(R_{2}\), \(R_{41}\), and \(R_{42}\), which can be calculated by the Equations (3)-(5), respectively. The first case is that, when \(P_{a}/P_{i}>R_{42}\), the flow velocity at all locations inside the nozzle is subsonic and the jet is a subsonic jet. The second case is, when \(R_{a1}<P_{a}/P_{i}\leq R_{42}\), the fluid can reach supersonic velocity at the divergent section near the throat of the nozzle and a normal shock wave is formed at the divergent section. The supersonic flow passes through the normal shock wave and then decelerates to subsonic flow, and finally forms a subsonic jet. The third case is, when \(R_{2}<P_{a}/P_{i}<R_{41}\), the normal shock wave moves outward and is located outside the nozzle. A supersonic jet can be achieved under this pressure condition. It should be noted that the formation of normal shock wave is due to the over-inflated state of the jet fluid relative to the ambient fluid. The fourth case is, when \(P_{a}/P_{i}=R_{2}\), a supercritical jet can be obtained through this nozzle and no shock wave will occur in the jet. In addition, the last case is, when \(P_{a}/P_{i}<R_{2}\), a supercritical flow can be achieved at the nozzle exit, and the supercritical jet will be further accelerated through expansion outside nozzle exit and expansion waves will occur in the jet, which is attributed to the under-expanded state of jet fluid relative to the ambient fluid [27; 29; 37]. \[R_{2}=\frac{P_{c2}}{P_{i}}=\pi(\lambda_{e}) \tag{3}\]  
\[R_{a1}=\pi(\lambda_{e})\frac{\frac{k+1}{k-1}\lambda_{e}^{2}-1}{\frac{k+1}{k-1 }-\lambda_{e}^{2}} \tag{4}\]  
\[R_{a2}=\pi(\lambda_{e}^{\prime}) \tag{5}\]where \(R_{2}\), \(R_{a1}\), and \(R_{a2}\) are critical pressure ratios of Laval nozzle; \(P_{c2}\) is critical ambient pressure of Laval nozzle, MPa; \(\lambda_{e}\) and \(\lambda_{e}^{{}^{\prime}}\) are supersonic and subsonic velocity coefficients, respectively; and \(\pi(\lambda_{e})\) is gas dynamics function.