A study of the flow structure for Mach reflection in steady supersonic flow

Abstract

In this paper we study the waves generated over the slipline and their interactions with other waves for Mach reflection in steady two-dimensional supersonic flow. We find that a series of expansion and compression waves exist over the slip line, even in the region immediately behind the leading part of the reflected shock wave, previously regarded as a uniform flow. These waves make the leading part of the slipline, previously regarded as straight, deviate nonlinearly towards the reflecting surface. When the transmitted expansion waves from the upper corner first intersect the slipline, an inflexion point is produced. Downstream of this inflexion point, compression waves are produced over the slipline. By considering the interaction between the various expansion or compression waves, we obtain a Mach stem height, the shape and position of the slipline and reflected shock wave, compared well to computational fluid dynamics (CFD) results. We also briefly consider the case with a subsonic portion behind the reflected shock wave. The global flow pattern is obtained through CFD and the starting point of the sonic line is identified through a simple analysis. The sonic line appears to coincide with the first Mach wave from the upper corner expansion fan after transmitted from the reflected shock wave.

Time history of regular to Mach reflection transition in steady supersonic flow

“…Notably, due to the pressure decreases along the stream tube below the slipline, a series of expansion waves (R s ) are created above the slipline (s) to balance this pressure decrease. These waves and their interactions with other flow structures have been first studied in Gao & Wu (2010) for steady MR. In the next section, we will show that including these waves leads to a better prediction of the Mach stem height during this stage.…”

“…In the next section, we will show that including these waves leads to a better prediction of the Mach stem height during this stage. Now let us describe in short the possible interactions, similarly as in the steady case (Gao & Wu 2010). The interaction of (R c ) with (r), for instance at point G, produces transmitted Mach waves (R t ) and slipline (c t ).…”

“…model of Gao & Wu (2010). An additional expression is needed to determine the velocity of the triple point.…”

“…Inspired by the numerical simulation of Kudryavtsev et al (2002), we will perform more refined numerical simulations to identify a multiple shock interaction structure preceding the pure unsteady MR structure. For the pure unsteady MR stage, which occupies the last but main part of the transition process, we include, in the analysis, compression and expansion waves generated over the slipline and these simple waves have been first considered by Gao & Wu (2010) for steady MR and have been shown to have important influence on the Mach stem height in the steady flow case. By considering these simple waves, we expect to have a more accurate prediction of the Mach stem height before and after it stabilizes.…”

Study on Mach stems induced by interaction of planar shock waves on two intersecting wedges

“…By using this analytical model, they obtained a very simple algebraic expression for the shape of the 2D Mach stem, and it was consistent with Dewey's hypothesis. Gao and Wu [16] considered the interaction between the various expansion or compression waves. They obtained a Mach stem height, the shape and position of the slip line, and reflected shock wave.…”