Theoretical Prediction for the Mach-Disk Height in Two-Dimensional Supersonic Underexpanded Jets

Abstract

Supersonic jets are in common use in aerospace, where the Mach-disk height is an important parameter to the aerothermodynamic and acoustic effects of the related interaction. To understand the Mach disk and predict its height, two-dimensional supersonic underexpanded jets are analyzed and a theoretical model is proposed in this work. First, through analyses of the viscous term in the Navier–Stokes equation, the dominance of an inviscid mechanism is verified upstream of the Mach disk. The pressure downstream of the Mach disk is a primary parameter to the prediction of the Mach-disk height. Then, the region dominated by the inviscid mechanism is divided into the following three regions: homogeneous, simple, and nonsimple by contour lines of the Riemann invariants. Analytical description of the simple region is achieved through analyses of the characteristic curves. Fan-shaped approximation to the nonsimple region is deducted based on analyses of expansion fans. Finally, a theoretical prediction of the Mach-disk height is developed based on the pressure downstream of the Mach disk. For either the freejet or the lateral jet interaction, it is feasible to obtain prediction with errors below 10%, which is much better than existing empirical models, especially for supersonic underexpanded jets.