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The recent discovery of shear shock wave generation and propagation in the porcine brain suggests that this new shock phenomenology may be responsible for a broad range of traumatic injuries. Blast-induced head movement can indirectly lead to shear wave generation in the brain, which could be a primary mechanism for injury. Shear shock waves amplify the local acceleration deep in the brain by up to a factor of 8.5, which may tear and damage neurons. Currently, there are numerical methods that can model compressional shock waves, such as comparatively well-studied blast waves, but there are no numerical full-wave solvers that can simulate nonlinear shear shock waves in soft solids. Unlike simplified representations, e.g., retarded time, full-wave representations describe fundamental physical behavior such as reflection and heterogeneities. Here we present a piecewise parabolic method-based solver for one-dimensional linearly polarized nonlinear shear wave in a homogeneous medium and with empirical frequency-dependent attenuation. This method has the advantage of being higher order and more directly extendable to multiple dimensions and heterogeneous media. The proposed numerical scheme is validated analytically and experimentally and compared to other shock capturing methods. A Riemann step-shock problem is used to characterize the numerical dissipation. This dissipation is then tuned to be negligible with respect to the physical attenuation by choosing an appropriate grid spacing. The numerical results are compared to ultrasound-based experiments that measure planar polarized shear shock wave propagation in a tissue-mimicking gelatin phantom. Good agreement is found between numerical results and experiment across a 40 mm propagation distance. We anticipate that the proposed method will be a starting point for the development of a two- and three-dimensional full-wave code for the propagation of nonlinear shear waves in heterogeneous media.

In the present study, a particle method of characteristics is put forward to simulate the evolution of one-dimensional shock waves in barotropic gaseous, closed-conduit, open-channel, and two-phase flows. All these flow phenomena can be described with the same set of governing equations. The proposed scheme is established based on the characteristic equations and formulated by assigning the computational particles to move along the characteristic curves. Both the right- and left-running characteristics are traced and represented by their associated computational particles. It inherits the computational merits from the conventional method of characteristics (MOC) and moving particle method, but without their individual deficiencies. In addition, special particles with dual states deduced to the enforcement of the Rankine–Hugoniot relation are deliberately imposed to emulate the shock structure. Numerical tests are carried out by solving some benchmark problems, and the computational results are compared with available analytical solutions. From the derivation procedure and obtained computational results, it is concluded that the proposed PMOC will be a useful tool to replicate one-dimensional shock waves.

General conservation laws are considered, and the concept of a weak solution is extended to the case of an equation involving three space variables and time. Four-dimensional vector calculus is used to develop general jump conditions at a shock wave in the material. To illustrate the use of this result, jump conditions at a shock in unsteady three-dimensional compressible gas flow are presented. It is then proved rigorously that these reduce to the commonly assumed conditions in coordinates normal and tangential to the shock face. A similar calculation is also outlined for an unsteady three-dimensional shock in magnetohydrodynamics, and in a chemically reactive fluid. The technique is available for determining shock–jump conditions in quite general continuous media.

A one-dimensional model for the numerical simulation of transport effects in small-scale, i.e., low Reynolds number, shock tubes is presented. The conservation equations have been integrated in the lateral directions and three-dimensional effects have been introduced as carefully controlled sources of mass, momentum and energy, into the axial conservation equations. The unsteady flow of gas behind the shock wave is reduced to a quasi-steady flow by choosing a coordinate system attached to the shock. The boundary layer problem is thereby reduced to a laminar solution, similar to the Blasius solution, with the exception that the wall velocity can be nonzero. The resulting one-dimensional equations are then solved numerically using a two-step Lax-Wendroff/ MacCormack scheme with flux correction transport. For validation purposes, comparisons are performed against previously published shock structure and low Reynolds number shock tube experiments; good agreement is observed. The model has been used to predict the performance of a 10µm shock tube and the result of this simulation shows the possibility of shock wave disappearance at lower pressure ratios for a micro-scale shock tube.

In many shock wave problems, the assumption of cylinder symmetry is useful, and allows 3D situations to be simulated with a 2D numerical code, thereby reducing computational cost, and increasing computational speed. In this paper we use the regularized smoothed particle hydrodynamics (RSPH) description for cylinder symmetry (Omang in “A numerical study of shock reflection phenomena. Exploring the capabilities of regularized smoothed particle hydrodynamics to handle shock waves”, 2005; Omang et al. in Shock Waves 14(4):293–298, 2005; Omang et al. in J Comput Phys 213(1):391–412, 2006) to study 3D shock reflection and shock interaction phenomena. In the first example we study the shock reflection pattern resulting from a torus-shaped shock reflecting both from the symmetry axis and from an object in the flow field. At the symmetry axis a characteristic Mach reflection is formed, with a triple point, a curved Mach stem, and a weak slip layer. The shock also interacts with the object, producing additional reflected shocks. The interactions between the different shock waves, resulting for instance in strong local pressure and density peaks, are studied in closer detail.

Micro shock tube flows were simulated using unsteady 2D Navier–Stokes equations combined with boundary slip velocities and temperature jumps conditions. These simulations were performed using the parallel version of a multi-block finite-volume home code. Different initial low pressures and shock tube diameters allow to have the scaling ratio ReD/4L vary. The numerical results show a strong attenuation of the shock wave strength with a decrease of the hot flow values along the tube. When the scaling ratio decreases the shock waves can transform into compression waves. Comparison to the existing 1D models also shows the limit of these models.

The interaction of two oblique axisymmetrical shock waves in a uniform supersonic flow has been investigated both theoretically and experimentally. The first shock is produced by a conical ring with a sharp leading edge located at the exit of a supersonic nozzle. The second shock is generated by a centre-body placed in the test chamber and along the symmetry axis of the nozzle. This flow field is absolutely free of three-dimensional edge effects. The displacement of this mobile centre-body leads to an interaction of an oblique shock wave with a diverging shock where both shock strengths are changing. Experimental results have shown three different structures of the two shock wave interaction and hysteresis effects between them. A description of these shock wave interactions is obtained by numerically solving the Euler and Navier-Stokes equations. Comparison with experiment shows the importance of viscous effects through shock/boundary-layer interaction for a correct description and understanding of these complex flows.

A spatially and temporally local analysis is provided for unsteady, oblique shock waves, in which the flow is assumed to be two-dimensional or axisymmetric. Three unsteady parameters, in a laboratory frame, are viewed as the known independent variables. These are the upstream Mach number, the shock Mach number, and the angle of the shock relative to the instantaneous upstream velocity. Other steady and unsteady parameters, such as the velocity turn angles and downstream Mach numbers, are evaluated in closed form, in terms of these three quantities. Trends are assessed, and a sensitivity analysis is provided. It is suggested that the theory may find application in converting a shock capturing algorithm, at an early time during the computational process, into a shock fitting algorithm.

A contact discontinuity tracking method with a specially designed moving grid is developed to eliminate the interface smearing completely. In order to precisely locate the contact surface, an updated Riemann solver for unsteady one-dimensional inviscid flows is also developed to allow consideration of the specific heat ratio change across the shock wave. These two new computational techniques are illustrated in a high Mach number shock tube flow field computation.

The problem of a plane shock wave that propagates in an air media and then is reflected from a parabolic concave reflector and focuses at some region is considered. The shock focusing can greatly magnify the pressure and the temperature. The purpose of this study is to numerically simulate the shock focusing process of the reflection of shock waves from the parabolic reflectors with different depths and to analyze their associated flow fields in detail. The present solver developed is to solve the Euler equations using an improved, implicit, upwind Total Variation Diminishing scheme in a finite-volume approach. The effects of reflectors with different depths and of the incident shock Mach numbers on shock focusing are investigated. The real-gas effect is taken into account through a proper correction of the specific heat ratio of air, when high temperature occurs due to shock focusing.

The study of the propagation of a shock down a tube of slowly varying cross sectional area has proved to be most valuable in understanding the dynamics of shocks. A particular culmination of this work has been the theory of geometrical shock dynamics due to Whitham (1957, 1959). In this theory the motion of a shock may be approximately computed independently of a determination of the flow field behind the shock. In this paper the propagation of a shock down such a tube is reconsidered. It is found that the motion of the shock is described by an infinite sequence of ordinary differential equations. Each equation is coupled to all of its predecessors but only to its immediate successor, a feature which allows the system to be closed by truncation. Of particular relevance is the demonstration that truncation at the first equation in the sequence yields the A-M relation that is the basis for Whitham's highly successful theory. Truncation at the second equation yields the next level of approximation. The equations so obtained are investigated with analytic solutions being found in the strong shock limit for the propagation of cylindrical and spherical shock waves. Implementation of the theory in the numerical scheme of geometrical shock dynamics allows the computation of shock motion in more general geometries. In particular, investigation of shock diffraction by convex corners of large angular deviation successfully yields the observed inflection point in the shock shape near the wall. The theory developed allows account to be taken of non-uniform flow conditions behind the shock. This feature is of particular interest in consideration of underwater blast waves in which case the flow behind the shock decays approximately exponentially. Application of the ideas developed here provides an excellent description of this phenomenon.