Comparison of geometrical shock dynamics and kinematic models for shock-wave propagation

Abstract

Geometrical Shock Dynamics (GSD) is a simplified model for nonlinear shock wave propagation. It is based on the decomposition of the shock front into elementary ray tubes with a simple relation linking its local curvature and velocity. This relation is obtained under the assumption of strong shock in order to neglect the effect of the post-shock flow on the front. More recently, a new simplified model, referenced as the Kinematic model, was proposed. This model is obtained by combining the threedimensional Euler equations and the Rankine-Hugoniot relations at the front, which leads to an equation for the normal variation of the shock Mach number at the wave front. In the same way as GSD, the Kinematic model is closed by neglecting the post-shock flow effects. Although each model's approach is different, we prove here their structural equivalence: the Kinematic model can be rewritten under the form of GSD with a specific A − M relation. Both models are thus compared through a wide variety of examples including experimental data or Eulerian simulations results when available. Attention is drawn to the simple cases of compression ramps and convex corners' diffraction. The analysis is completed by the more complex cases of the diffraction over a cylinder, a sphere, a mound and a trough.

Beyond the limitation of geometrical shock dynamics for diffraction over wedges

“…In the case of diffraction over a convex corner, θ − θ 0 < 0. Then, from (18), the Mach number decreases along the shock from M 0 to M w , the value at the wall. Since θ w − θ 0 and M 0 are known, the wall Mach number is calculated with the expression holding on the last characteristic:…”

“…appears to work remarkably well in a large number of configurations [34,7,18]. Finally, the GSD model is composed of the geometrical system (2) and the A − M relation (4).…”

“…The GSD model has no solution in the area between the wall and the red dotted line. as the Kinematic model [18], or theoretical modifications of the A − M relation [15,14], do not seem much more efficient to remove this restriction.…”

“…Since the GSD model works well in compressive zones [18], the additional term has to be active only in expansive regions of the shock front, where the curvature, κ, is positive. In this way, the function H is introduced:…”

“…Some modifications of GSD, such as its extension to post-shock flow [4], or another treatment of the wall condition [1], are able to recover the inflection point experimentally observed for strong shocks, but do not remove the limitation. The more recent Kinematic model [26,27,16], is no more efficient to remove this restriction [18]. Oshima et al [15] studied several extensions of Whitham's A − M rule to improve the GSD model.…”