Shock waves in supercritical granular flow impacting a pyramidal wedge

Supercritical granular flows, such as debris flow and snow avalanches, interacting with obstacles can generate interesting flow phenomena like shock, vacuum, and dead zone, which is crucial for improving hazard prediction and mitigation. This study employs a high-resolution depth-integrated continuum model, in which the topography-gradient approach is used to account for the influence of solid obstacles, enabling efficient and accurate simulation of flow-obstacle interactions. The steady-state behavior of supercritical gravity-driven, free-surface flow impacting pyramidal wedge obstacles is investigated. Three distinct shock wave patterns are identified in the interaction between supercritical granular flow and a pyramidal wedge: oblique shock, attached bow shock, and detached bow shock. At small wedge angles, a classical oblique shock forms, attaching to both sides of the wedge. As the wedge angle increases, we observe an attached bow shock structure—a transitional regime bridging the classical oblique shock and the detached bow shock. This attached bow shock retains connection to the obstacle but exhibits a curved shock front markedly different from the linear oblique case. At even larger wedge angles, the shock detaches entirely, forming a stationary bow shock upstream of the wedge, and the flow structure in this regime closely resembles experimental observations for supercritical flows past blunt bodies. The interaction modes are jointly regulated by the wedge angle and the upstream Froude number, where a phase diagram for their transition is proposed based on a series of numerical experiments. Interestingly, we found that the tangent of the attached bow shock satisfies the classical oblique shock theory, suggesting that the attached bow shock results from a bow shock broken up by the wedge obstacle due to the terrain effects, which does not exist in gas dynamics or hydraulics.