Circulation production model and unified formation number of compressible vortex rings generated by a shock tube

To investigate the formation number of compressible vortex rings (CVRs), a shock-tube apparatus with an open end is designed to generate CVRs and the flow structures are measured by using particle image velocity (PIV) and time-resolved schlieren techniques. A series of experiments were designed by varying the two governing factors: Mach numbers of the incident shock (Ms= 1.28, 1.48, and 1.59) and driven section length (DL = 100, 200, and 300 mm). By theoretically analyzing the shock diffraction problem, a slipstream model is proposed to predict the circulation generation of CVRs. Comparing with the PIV results, this model well predicts the circulation for Ms=1.28 but slightly underestimates the circulation for Ms= 1.48 and 1.59. Then, an alternative model based on the variation of Ms is proposed and well predicts the circulation generation. Based on the general definition of the vortex formation time and the circulation production model, we newly define the physical formation time of CVRs and then determine the formation number (denoted by F*) when CVRs pinch off. The formation number of CVRs (F*≈3.5) is found to coincide with the optimal vortex formation number originated from incompressible vortex rings (ICVRs). This consistency generalizes the principle of optimal vortex formation into compressible flows. However, both the PIV and schlieren results demonstrate that the CVRs for different Ms pinch off in different modes. With the aim of modulating F* of CVRs, a converging nozzle is designed, and we found that F*≈3.5 is remained for Ms = 1.28 but F* = 5.5 and 6.0 is obtained for Ms= 1.48 and 1.59. Furthermore, an extension of the Kelvin–Benjamin variational principle is explored to explain the unified formation number of CVRs and ICVRs.