Gasdynamic wave interaction in two spatial dimensions

Abstract

We examine the interaction of shock waves by studying solutions of the twodimensional Euler equations about a point. The problem is reduced to linear form by considering local solutions that are constant along each ray and thereby exhibit no length scale at the intersection point. Closed-form solutions are obtained in a unified manner for standard gasdynamics problems including oblique shock waves, PrandtlMeyer flow and Mach reflection. These canonical gas dynamical problems are shown to reduce to a series of geometrical transformations involving anisotropic coordinate stretching and rotation operations. An entropy condition and a requirement for geometric regularity of the intersection of the incident waves are used to eliminate spurious solutions. Consideration of the downstream boundary conditions leads to a formal determination of the allowable downstream matching criteria. By retaining the time-dependent terms, an approach is suggested for future investigation of the open problem of the stability of shock wave interactions. IntroductionRiemann problems in two spatial dimensions are germane to many problems in experimental, analytical and computational gasdynamics. Familiar examples include the interaction of oblique shock waves in steady supersonic flow, the self-similar problem of transition to Mach reflection from a wedge and the treatment of discontinuous shock fronts in numerical shock fitting algorithms.The well-known approach to the interaction of shock waves has its basis in the work of Rankine (1870) and Hugoniot (1889) regarding the propagation of shock waves. The solution for oblique shock waves was then obtained by Meyer (1908) through a Galilean transformation that imposed a velocity component parallel to the shock front. Also due to Meyer is the development of the Prandtl-Meyer function for expansion through an expansion fan. The development of the theory of shock wave interactions by Courant, Friedrichs, von Neumann and independently by Weise is described by Courant & Friedrichs (1948). By hypothesizing the existence of a vortex sheet at the intersection point, solutions for several interacting shock waves and/or expansion fans may be constructed that are consistent with the experimental observations. A similar approach has been developed for shock-shear layer interactions. The interaction solutions are represented graphically by the intersection in the pressure-flow deflection angle, p-δ, plane of the loci of possible downstream states for the interacting waves. Because of the nonlinear interaction of the waves, problems of non-uniqueness

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

“…Taking into account the physical meaning of the wave interaction, we shall limit our considerations to configurations containing at most two incoming rays. Other configurations are indeed geometrically irregular (Sanderson 2004), in the sense that a slight geometrical perturbation of one of the incoming rays would break the single-node interaction pattern into multiple nodes. This requirement also excludes the case of incoming wave fans centred on the node.…”

Reactant Jetting in Unstable Detonation

“…It is remarkable that all local features of the flow are thereby completely determined, up to an unknown length scale. Sanderson et al [13,12] demonstrate that beyond smooth regions of the main shock front, a strictly limited set of possible solutions are admissible at discontinuities on the shock front, even for vanishingly weak disturbances. The global flow field therefore adapts to accommodate the existence of the jet flow structure at some length scale determined by interaction of the local and global scales.…”

“…Satisfaction of the conservation laws at any point along the main detonation front requires that it remain contiguous, with potentially finite curvature, or that it be punctuated by the shock interaction structure described above [12,13], at some indeterminate length scale, and that the surrounding shock layer flow distort to accommodate the necessity of its existence. The undetermined length scales are derived from the interaction between the flow length scales and chemical length scales,…”

Shock/expansion fan interaction with real gas effects for Earth and Mars atmospheric conditions

“…They found that the expansion waves generated from the train trailing edge propagated upstream and reduced the overpressure region created by the compression waves inside the tunnel. Sanderson 16 investigated the interaction between an oblique shock undergoing Mach reflection and expansion fan considering two‐dimensional (2D) Euler equations. Hillier 5 performed simulations for the interaction of expansion fan and an incident oblique shock of opposite family.…”

“…From the literature reported findings, it is noticed that the presence of an expansion fan significantly affects various flow phenomena, such as its presence in the vicinity of an SWBLI can lead to lower flow separation. 3,4,[6][7][8][9][10][11] Furthermore, the presence of an expansion fan can delay the occurrence of Mach reflection, 5,16,17 can diminish the shock strength passing through ducts, 18,19 and decrease the overpressure region created by a high-speed train entering a tunnel, 15 and can lead to high drag in SCB. 22,23 It is observed that the majority of the studies are limited to airflow medium in the lower enthalpy range where perfect gas assumptions are true.…”