In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, $P(s)$, is continuous and shows a power-law decay. For systems with short-range attractions, however, $P(s)$ appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing “discontinuity” does not mean that the transition is discontinuous. In fact, it signifies the coexistence of two distinct length scales, associated with the largest cluster and smaller ones, respectively. The comparison between the largest and second largest clusters indicates that their growth rates with system size are rather different. However, both cluster sizes tend to diverge in the large system size limit, suggesting that the jamming transition of systems with short-range attractions is still continuous. In the framework of the two-scale scenario, we also derive a generalized hyperscaling relation. With robust evidence, our work challenges the former single-scale view of the rigidity percolation.

The oil-free compressor is a key component in fabricating polyethylene terephthalate (PET) bottles for beverages and water. At present, the main compressor type used for blowing PET bottles is the reciprocating compressor. However, compared to screw compressors, reciprocating compressors have shortcomings of high energy consumption and too many consumable parts. Many manufacturers of PET bottles in Asia are seeking to replace reciprocating compressors with screw compressors, as we know. Screw compressors can be classified as single-screw compressors (SSC) and twin-screw compressors. Since the load in a twin-screw compressor is far larger than that in an SSC, SSCs are more suitable for being developed for high-pressure applications such as PET bottle blowing. This paper presents a performance study on an oil-free single-screw compressor as the first stage of the PET compressor unit. A 5.4 m³·min⁻¹ prototype and its test rig were developed. The thermophysical process of the moist air is theoretically analyzed. The pressure loss on the flow path and the influence of the important parameters are experimentally investigated. It is found that water vapor cannot be separated during the adiabatic compression process. The results also show that the pressure loss from the discharging duct to the check valve accounts for the largest percentage of the total pressure loss. The experimental results further show that the discharge capacity and shaft power increase almost linearly with the motor speed. The efficiency declines with increasing injected water temperature. The discharge capacity and shaft power all increase with the injected water flowrate, and an optimum flowrate is found to ensure a highest isentropic efficiency. With the increase in discharge pressure, the discharge capacity decreases, and the shaft power increases. The isentropic efficiency is found to have its maximum value at a certain discharge pressure.

Large eddy simulation (LES) approach was used to investigate jumps of primary frequency of shear layer flow over a cavity. Comparisons between computational results and experimental data show that LES is an appropriate approach to accurately capturing the critical values of velocity and cavity length of a frequency jump, as well as characteristics of the separated shear layer. The drive force of the self-sustained oscillation of impinging shear layer is fluid injection and reinjection. Flow patterns in the shear layer and cavity before and after the frequency jump demonstrate that the frequency jump is associated with vortex–corner interaction. Before frequency jump, a mature vortex structure is observed in shear layer. The vortex is clipped by impinging corner at approximately half of its size, which induces strong vortex–corner interaction. After frequency jump, successive vortices almost escape from impinging corner without the generation of a mature vortex, thereby indicating weaker vortex–corner interaction. Two wave peaks are observed in the shear layer after the frequency jump because of: (1) vortex–corner interaction and (2) centrifugal instability in cavity. Pressure fluctuations inside the cavity are well regulated with respect to time. Peak values of correlation coefficients close to zero time lags indicate the existence of standing waves inside the cavity. Transitions from a linear to a nonlinear process occurs at the same position (i.e., x/H = 0.7) for both velocity and cavity length variations. Slopes of linear region are solely the function of cavity length, thereby showing increased steepness with increased cavity length.