# Numerical Simulation of Vacuum Leak Jet and Jet Noise
## 3 Discussion
### Effect of Leak Hole Size
#### 3.2.2 Aeroacoustic Characteristics  
The essence of sound waves is the transmission phenomenon of pressure disturbances in the medium. Whether a sound wave can propagate from the disturbance source to the far field depends on the existence of a medium. As such, mechanisms of the generation and propagation of supersonic jet noise in atmospheric environment are quite different from those in vacuum environment. In atmospheric environment, the instability waves travel downstream extracting energy from the potential core and  
Figure 8: Velocity contour of different leak hole diameter. (a) 1 mm; (b) 0.5 mm. Figure 9: Density contour with pressure contour map of different leak hole diameter. (a) 1 mm; (b) 0.5 mm. making the amplitude grow rapidly. When it interacts with the shock cell structure, strong reflected waves will be generated and travel upstream. The consistent characteristic impedance makes it possible for most energy of the reflected waves to penetrate the shear layer and reach the leak hole lip which is susceptible to external disturbance due to the very thin shear layer. Therefore, new instability wave is excited, travelling downstream and a complete feedback loop is formed, leading to the excitation of screech tones [20]. However, in vacuum environment, the extremely low molecular number density makes the characteristic impedance inside and outside the shear layer very different. So, the instability waves cannot penetrate the shear layer to reach far field, but be reflected upstream along the shear layer and new instability wave is excited at the leak hole lip, thereby forming the feedback loop. The reflection path of pressure fluctuation forms an approximate cavity resonance structure, which is different from the conventional cavity resonance. On the other hand, obvious density, pressure, and temperature gradients existing in vacuum leak jet make the mechanism of sound generation very complicated. Furthermore, the acoustic impedance change also complicates the pressure fluctuation reflection path. Figure 10 shows the pressure fluctuation contour of different leak hole diameter. It can be seen obviously that for a vacuum leak jet shown in Figures 7a and 10, as the leak hole diameter decreases, both forward waves and backward waves get weaker and weaker. However, there is a certain boundary where a clear reflection phenomenon occurs and the reflected wave feeds back along this boundary. Knudsen number, \(Kn\), which is a dimensionless parameter representing the flow rarefaction, can be used to analyze the vacuum jet. Knudsen number is defined by \(Kn=\lambda/L\), where \(\lambda\) is mean free path of air molecules and \(L\) is the characteristic length. Theoretically, \(\lambda\) is calculated by:  
\[\lambda=\frac{\text{k}_{0}T}{\sqrt{2}\pi d^{2}p} \tag{5}\]  
where k\({}_{0}\) is Boltzmann constant (k\({}_{0}=1.38\times 10^{-23}\)) and \(d\) is the average diameter of air molecules (\(d=3.74\times 10^{-10}\)\(m\)). According to the value of \(Kn\), the flow can be divided into four types, namely, \(Kn\leq 0.01\) for continuous flow, \(0.01<Kn\leq 0.1\) for slip flow, \(0.1<Kn\leq 10\) for transition flow and \(Kn>10\) for molecular flow. Continuous flow and slip flow still satisfy the continuity hypothesis and can still be calculated using the Navier-Stokes (N-S) equation, while the rarefied effect must be considered when computing transition flow and molecular flow. Choosing leak hole diameter as the characteristic length, \(Kn\) contour is shown in Figure 11.  
Figure 10: Pressure fluctuation contour of different leak hole diameter (The arrow marked in the figure represents the direction of fluctuation wave propagation.). (**a**) 1 mm; (**b**) 0.5 mm. It can be seen obviously that there is a good agreement between the reflection boundary and \(Kn\), where \(Kn=0.3\) for 2 mm leak hole, \(Kn=0.6\) for 1 mm leak hole and \(Kn=1.2\) for 0.5 mm leak hole.