Bow-shock Instability with Edged Body in Low-? Flow

Abstract

In order to find out mechanism of bow-shock instability, which was experimentally observed in a low flow, three-dimensional CFD analysis has been conducted with discontinuous Galerkin finite element method on unstructured grids. We examined dependency of the instability on body shapes and found that a shock wave around a circular cone tends to be unstable even if using grids that hardly causes carbuncle phenomenon. Grid convergence analysis also suggested that this instability seems to be physical one, and the bow-shock instability must be induced in a flow around an edged blunt body that affects vortex structures in a shock layer. Moreover, we found that an critical condition about the ratio of specific heat exists for this instability. The greatly affects the shock stand-off distance, so the shock thickness may have an important role on the interaction between the shock and the vortices, which is a main factor of this instability. A supersonic vehicle, with which flight speed is more than two times as that of current aircrafts, is demanded in order to shorten traveling time of a long-distance transportation. However, some critical issues caused by shock waves, such as wave drag, heat load, and sonic boom, prevent form realizing affordable supersonic transports. In particular, the problems of noise must be resolved for practical use of it. Although many attempts have been made to conquer them, existing methods can reduce sonic boom only by trade-off with the vehicle weight or the heat load. In order to propose a new shock wave application that is different from existing methods based on steady shock waves, we have focused on an instability of bow shock wave. Actively using this instability, the shock wave itself may be weakened; therefore, wave drag, heat load, and sonic boom are also reduced all together. In the experiments using a ballistic range, the bow-shock instability was observed in a low- gas as shown in Fig. 1. 1,2 Under the experiments with various conditions, it was concluded that the instability occurs depending on Mach number, ambient gas pressure, and curvature of the blunt body. They also suggested that a cause of the instability is chemical reaction in the shock layer. However, since it is difficult to analyze the flowfield behind the shock wave in the experiments, the mechanism of the instability has not been revealed yet. So, it is expected that computational fluid dynamics (CFD) enables a detailed analysis of the instability and clarifying the cause of it. On the other hand, in CFD with a strong shock wave, shock-capturing schemes often become unstable when the shock wave is parallel to computational grids; this is so-called carbuncle phenomenon. 3 The carbuncle phenomenon also occurs depending on various factors such as flow conditions, computational grids, and flux function schemes. Therefore, an appropriate computational condition is required to extract the physical instability by keeping off the carbuncle phenomenon. In this study, in order to find out the mechanism of the bow-shock instability observed ahead of a blunt body, three-dimensional CFD analysis has been conducted with the discontinuous Galerkin (DG) method 4 on unstructured grids. As mentioned above, we should carefully distinguish the physical instability from numerical one in the obtained results. We have thus examined robustness of Riemann solvers against the carbuncle phenomenon in the condition we want 5 and employed a AUSM family scheme in the present paper. From the past studies, 6 we found that it is difficult to observe the bow-shock instability with semi-ellipsoids even when the aspect ratios are relatively large. In this paper, we first examine dependency of the instability on body shapes with an edged body such as circular cone and circular