UC San Diego

Shock wave focusing can lead to extreme thermodynamic conditions, and applications have been extended to a variety of areas such as civil engineering and medical treatment. Among all numerical approaches, geometrical shock dynamics (GSD) is a model capable of efficiently predicting the position, shape and strength of a shock. Compared to the traditional Euler method that solves the inviscid Euler equations, GSD is a reduced-order model derived from the method of characteristics that is more computationally efficient since it only considers the motion of the shock front instead of the entire flow field.

Whitham’s original theory of GSD successfully relates the change of area upon the shock front to the shock motion with an assumption of a uniform state behind the shock, so it is able to accurately predict the behavior of a shock wave with constant properties behind it. However, the truncation of the post-shock flow term discredits its application to shock waves with decaying properties behind (e.g., in the case of blast waves). In this study three two-dimensional GSD models were first reviewed with a focus on how the post-shock flow effect is accounted for. It turned out that the completeness of the post-shock flow term determines the accuracy of GSD for blast waves, but prior knowledge of the particular blast is required to achieve full completeness. The point-source GSD (PGSD) model thus stands out since it encodes the analytical solution to blast propagation and is independent of the initial energy content of the point-explosion. Then a general framework based on PGSD was proposed aiming at efficiently solving the irregular reflection phase of blast focusing problems. Lagrangian simulations were thus performed for the symmetric interaction between two cylindrical blasts and compared to the experimental results. An agreement in attenuation of the maximum pressure at the Mach stem was observed but an overestimation of the Mach stem growth at its early stage by PGSD was also seen. To address this issue, an alternative model called PGSDSS was developed that combines PGSD and the shock-shock approximate theory for cylindrical shock reflection off a straight surface.

Another advantage of PGSD is its capability to be extended to three dimensions. Unlike the traditional three-dimensional GSD studies using triangulated meshes, in this study the shock surface is represented by a point cloud arranged in an octree data structure, such that a fast k-nearest neighbor search is possible without the need of connectivity information. Differential geometric properties required by PGSD are obtained by computing the moving least squares (MLS) surface that approximates the underlying shock surface. The resulting MLS-PGSD model was utilized to investigate first the propagation of a single spherical micro-blast in air and then its reflection off a solid wall. A good agreement of the blast front contour at different time instants with the experimental results was reached.