Onset conditions for Mach disk formation in underexpanded jet flows

In this study, the formation conditions of the Mach disk in an underexpanded jet flow were numerically and theoretically investigated under sonic injection conditions and the assumption of an axisymmetric flow. The numerical results demonstrated that the threshold nozzle-pressure ratio (NPR) at which the Mach disk occurred was situated between 3.03 and 3.12, which is lower than those reported in the previous studies. Since the oscillation frequency of the Mach disk was approximately constant over a wide range of NPRs and the amplitude was weak, it was regarded as a steady shock. In addition, the Mach disk was found to form when the total pressure loss reached approximately 40%. To predict the appearance of the Mach disk, we proposed a theoretical model based on a quasi-one-dimensional flow by considering the Mach disk as a normal shock wave on the axis. Based only on injection and ambient conditions, the total-pressure loss ratio derived from the model was in good agreement with that obtained from the numerical simulations, indicating that the proposed model provides useful knowledge for Mach disk occurrence prediction.