NUMERICAL INVESTIGATION ON THE VARIATION CHARACTERISTICS OF MACH DISK IN SUPERSONIC UNDEREXPANDED JETS

Underexpanded jets constitute a fundamental and widely observed flow regime in supersonic flows and are of significant practical importance. While sonic underexpanded jets have been studied extensively and a relatively complete theoretical framework exists, underexpanded jets that originate from a supersonic nozzle exit have received comparatively little attention. In the present study, numerical simulations are carried out for supersonic underexpanded jets generated by a Laval nozzle. The investigation focuses on the structural features of the Mach disk, specifically the first shock cell length and the Mach disk diameter, which serve as the main geometric descriptors. To identify the key parameters governing these quantities, a dimensional analysis is first performed. Based on the identified dimensionless groups, a series of comparative test cases are designed by systematically varying the exit Mach number and the exit-to-ambient pressure ratio. The steady, compressible Navier–Stokes equations are solved to obtain the corresponding flow fields. The numerical results reveal that the Reynolds number has only a minor impact on the formation and variation of the shock-wave pattern within the parameter range considered. In contrast, both the first shock cell length and the Mach disk diameter are predominantly controlled by the combined effect of the exit-to-ambient pressure ratio and the exit Mach number. In particular, the first shock cell length is found to scale with the exit Mach number raised to the power 2.5 and the pressure ratio raised to the power 0.5. The Mach disk diameter scales with the exit Mach number and with the excess of the pressure ratio over unity raised to the power 0.66. These scaling laws are observed to hold consistently for all simulated conditions. Finally, empirical correlations for the first shock cell length and the Mach disk diameter are derived by fitting the numerical data. Comparisons between the correlation predictions and the simulation results confirm their accuracy and reliability. The proposed correlations thus offer a practical and validated reference for predicting the shock structure in supersonic underexpanded jets, facilitating both fundamental research and design.