Self-induced oscillations in the shock wave flow pattern formed in a stationary supersonic flow over a double wedge

Abstract

Numerical simulations of a two-dimensional supersonic flow of an inviscid perfect gas over a double wedge in the Mach numbers range 5рM р9, revealed the existence of self-induced oscillations in the shock wave flow pattern in a narrow range of geometrical parameters.We consider a supersonic flow of an inviscid perfect gas over a double wedge ͑Fig. 1͒ in the range of high Mach numbers for which the shock waves are attached to the leading edges of both the first and the second wedges. The interaction of these two waves results in the formation of a complex shock wave flow pattern. Such flow fields can occur during flight of supersonic/hypersonic aircrafts or in the course of the re-entry of space shuttles. A numerical solution of this problem, in a stationary formulation, was conducted elsewhere. 1 Our recent success in discovering the existence of hysteresis processes in numerous cases of the interaction of supersonic flows with various geometries 2 motivated us to thoroughly complement these studies and check whether a hysteresis phenomenon exists in the case of a supersonic flow over a double wedge.Not only did our study reveal that there is a hysteresis, we also found out that there are self-induced oscillations in the shock wave flow pattern for various angles of inclination of the second wedge.The flow is described by the nonstationary Euler equations for an inviscid perfect diatomic gas (␥ϭ1.4). The parameters in the problem are the free stream flow Mach number, M , the ratio of the lengths of the surfaces of the double wedge, L 1 /L 2 , the angle of the first wedge, 1 , and the angle of the second wedge, 2 , or alternatively the difference between the two wedge angles, ⌬ϭ 2 Ϫ 1 .A W-modification of Godunov's scheme 3 that has second-order accuracies both in space and time was used in the calculations. The stationary solution was determined by settling of the nonstationary solution with time. Since the inclination of the first shock wave and the induced flow field behind it could be determined analytically, the size of the computational domain was reduced to include only the region of the interaction of the two shock waves ͑dashed line in Fig. 1͒. In order to damp the numerical oscillations that occur behind strong shock waves in stationary flows the following technique was employed. At each time step two independent calculations were performed using standard and diagonal stencils ͑Fig. 2͒, and their average was used at the following time step. This extension of the stencil enabled us to damp the numerical fluctuations and to avoid the need to use artificial viscosity. The investigation was performed with the following parameters: L 1 /L 2 ϭ2, M ϭ9, 1 ϭ15°, and ⌬ was changed in the course 20°→35°→20°. The goal was to check whether there is a hysteresis phenomenon. This was done by changing the value of ⌬ while keeping all the other parameters fixed. ⌬ was continuously changed during one dimensionless unit of time by 0.2°, and then five units of dimensionless time were spent to enable the solution to settle. ...

Computational confirmation of an abnormal Mach reflection wave configuration

“…Numerical studies on the similar problem were conducted elsewhere. 2,3,6,9,10 However, the existence of an abnormal oMR͑InMR+ InMR͒ wave configuration, to the best of our knowledge, has never been reported until the present work. Figure 2 schematically shows the MR configuration for two shock waves of opposite families.…”

“…Related research can still be found in the recent literature. [2][3][4][5][6] In the present work, an abnormal MR configuration of asymmetric shock waves is computationally confirmed over a double-wedge geometry ͑Fig. …”

Geometric criterion for RR↔MR transition in hypersonic double-wedge flows

“…high frequency over the wedge surface which can cause vital damages to the high-speed vehicles. 21,27 However, the oscillation phenomenon seems to be dependent on the numerical techniques and grid resolution. Wave configurations of oMR and oRR can alternate during an oscillation period in a previous computational study 27 where a W-modification of Godnuv's scheme and relatively coarse mesh were used.…”

“…It was found that hysteresis and self-induced oscillations in the shock flow pattern can result in extremely high and unsteady loads on the wedge surfaces. 21,27 Thermally nonequilibrium effects on the transition and the oscillation were reported by Hu et al 22 where the thermal properties of the components of air are temperature dependent. Advanced RR→ MR transition due to downstream influence has been first reported in a theoretical dual-solution regime.…”

“…24 Figures 2͑a͒ and 2͑b͒ show the fundamental flow features for shock interactions of SW3 and LSW2 in which an overall RR ͑oRR͒ and an oMR, respectively, emerge. Six-and seven-shock patterns 22,27 were used to distinguish the above two different wave configurations. However, such a taxonomy is incomplete.…”