Beyond the limitation of geometrical shock dynamics for diffraction over wedges

Abstract

Geometrical Shock Dynamics (GSD) is a simplified model for nonlinear shock wave propagation for which the front evolution is governed by a local relation between the geometry of the shock and its velocity, so-called A − M rule. Numerous studies have proven the ability of the GSD model to estimate correctly the leading shock front in interaction with obstacles. Nevertheless, a solution for the problem of diffraction over a convex corner does not always exist, especially for weak shocks. To overcome this limitation, we propose an ad-hoc modification of the A−M relation for two-dimensional configurations: an extra term based on the transverse variation of the Mach number is added. This new closure is fitted against experimental observations, which ensures, by construction, a correct behaviour for expansive shocks. A Lagrangian numerical solver is developed, for which this new term is activated only on specific parts of the front. Results of this new model are compared with the original GSD model, experiments, and Eulerian simulations for several cases of increasing complexity. A noticeable improvement of the solution is observed.

An immersed boundary method for geometrical shock dynamics

“…Slight discrepancies, such as underestimation of the triple point trajectories, may be imputed to the non conservativity of the model formulation. In the future we plan to extend our method to weak shocks following the Lagrangian idea developed in [1], and to develop a conservative high order formulation to improve the computational efficiency.…”

“…The need to develop a method for rapidly calculating the mechanical effects of shocks is, however, of prime importance in many applications such as the prevention of industrial accidents and the computation of structure loading, or the definition of source models for acoustic propagation codes. Recent researches have shown that a promising solution [1,2] could be envisaged thanks to the Geometrical Shock Dynamics model (GSD).…”

“…However, studies have shown that relatively good results can be obtained even for weaker shocks (see e.g. [5,1]). In comparison to the equations of fluid dynamics, the main advantage of GSD is the significant reduction in the cost of calculation since the computation of the flow behind the shock is not necessary.…”

“…The GSD model is therefore composed of Eqs. ( 4)-( 5), together with the A − M relation (1) in which the term F is neglected. This model is hyperbolic, meaning 50 that discontinuities can develop on the front.…”

GSD models for blast propagation]{Two-dimensional geometrical shock dynamics for blast wave propagation and post-shock flow effects

“…Since only a limited number of shock dynamics problems can be solved analytically using GSD, many algorithms were developed to numerically implement GSD models including front tracking methods [6,[10][11][12][16][17][18][19][20], finite difference [5] and finite volume schemes [21,22] based on the conservation form of GSD, and a recent level-set fast marching approach [23]. Among these schemes, the front tracking-based Lagrangian schemes appear to be the most popular ones that have been used for a wide range of shock dynamics problems since accuracy and speed can be well balanced.…”