Theory Of Geometrical Shock Dynamics Research Articles

On dynamics of imploding shock waves in a mixture of gas and dust particles

In this paper, the generalized analytical solutions for one-dimensional adiabatic flow behind the imploding shock waves propagating in a dusty gas are obtained using the geometrical shock dynamics theory. The dusty gas is assumed to be a mixture of a perfect gas and spherically small solid particles, in which solid particles are continuously distributed. Anand׳s shock jump relations for a dusty gas are taken into consideration to explore the effects due to an increase in (i) the propagation distance from the centre of convergence, (ii) the mass fraction of solid particles in the mixture and (iii) the ratio of the density of solid particles to the initial density of the gas, on the shock velocity, pressure, temperature, density, velocity of mixture, speed of sound, adiabatic compressibility of mixture and the change-in-entropy across the shock front. The results provided a clear picture of whether and how the presence of solid particles influences the flow field behind the imploding shock front.

Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects

In this paper, the generalized analytical solution for one dimensional adiabatic flow behind the strong imploding shock waves propagating in a non-ideal gas is obtained by using Whitham’s geometrical shock dynamics theory. Landau and Lifshitz’s equation of state for non-ideal gas and Anand’s generalized shock jump relations are taken into consideration to explore the effects due to an increase in (i) the propagation distance from the centre of convergence, (ii) the non-idealness parameter and, (iii) the adiabatic index, on the shock velocity, pressure, density, particle velocity, sound speed, adiabatic compressibility and the change in entropy across the shock front. The findings provided a clear picture of whether and how the non-idealness parameter and the adiabatic index affect the flow field behind the strong imploding shock front.

Focusing of strong shocks in an elliptic cavity

Focusing of strong shock waves in a gas-filled thin chamber with an elliptic boundary is investigated theoretically and numerically. The process of reflection and convergence of cylindrical shocks generated at one of the focal points of the chamber is studied by means of Whitham's theory of geometrical shock dynamics (1957, 1959). The current calculations are based on the modified version of this theory, which takes into account the non-homogeneous flow ahead of the advancing shock, Apazidis and Lesser (1996). There are two main results of the present study indicating a qualitatively different behavior of strong shocks as compared to the linear acoustic case. (1) - the form of the reflected shock is not cylindrical and (2) - the reflected, converging shock does not focus at the geometrical second focal point of the chamber. The dependence of the form of the reflected shock and the location of its focusing point on the strength of the shock and the eccentricity of the chamber is investigated.

Oblique shock reflection from an axis of symmetry: shock dynamics and relation to the Guderley singularity

Oblique shock reflection from an axis of symmetry is studied using Whitham's theory of geometrical shock dynamics, and the results are compared with previous numerical simulations of the phenomenon by Hornung (2000). The shock shapes (for strong and weak shocks), and the location of the shock-shock (for strong shocks), are in good agreement with the numerical results, though the detail of the shock reflection structure is, of course, not resolved by shock dynamics. A guess at a mathematical form of the shock shape based on an analogy with the Guderley singularity in cylindrical shock implosion, in the form of a generalized hyperbola, fits the shock shape very well. The smooth variation of the exponent in this equation with initial shock angle from the Guderley value at zero to 0.5 at 90° supports the analogy. Finally, steady-flow shock reflection from a symmetry axis is related to the self-similar flow.

A higher-order Godunov method for the hyperbolic equations modelling shock dynamics

A numerical method of solution is developed for steady–state hyperbolic equations in conservation form. An upwind direction is identified and a smooth mapping is introduced to align the equations with that direction. A flux F ( u ) in the upwind direction is regarded as the ‘state’ variable and advanced numerically on a grid according to a second–order extension of Godunov' method. The solution u is determined from the computed flux, and a constraint on the mapping is that the Jacobian, det( ∂F ∂u ), is of one sign. The method is applied to the equations modelling shock–wave propagation according to Whitham' theory of geometrical shock dynamics. In this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times. Shock propagation in both uniform and non–uniform gases is considered and several example calculations are presented. A further application of the numerical method is inviscid supersonic flow, as governed by the steady Euler equations. The steady supersonic flow in a converging–diverging duct is computed to illustrate the numerical method for this latter application.

On generation and convergence of polygonal-shaped shock waves

A process of generation and convergence of shock waves of arbitrary form and strength in a confined chamber is investigated theoretically. The chamber is a cylinder with a specifically chosen form of boundary. Numerical calculations of reflection and convergence of cylindrical shock waves in such a chamber filled with fluid are performed. The numerical scheme is similar to the numerical procedure introduced by Henshaw et al. (1986) and is based on a modified form of Whitham's theory of geometrical shock dynamics (1957, 1959). The technique used in Whitham (1968) for treating a shock advancing into a uniform flow is modified to account for non-uniform conditions ahead of the advancing wave front. A new result, that shocks of arbitrary polygonal shapes may be generated by reflection of cylindrical shocks off a suitably chosen reflecting boundary, is shown. A study is performed showing the evolution of the shock front's shape and Mach number distribution. Comparisons are made with a theory which does not account for the non-uniform conditions in front of the shock. The calculations provide details of both the reflection process and the shock focusing.

Accounting for transverse flow in the theory of geometrical shock dynamics

The theory of geometrical shock dynamics is deduced as a formal approximation to the equations of gas dynamics in two dimensions, thus overcoming the approximation of quasi-one-dimensional flow utilized in the original derivation. Successive approximations may be obtained by using the truncation scheme developed by Best (1991) for considering shock propagation down a tube of slowly varying cross section. The zero-order approximation yields the theory as developed by Whitham (1957). The first-order approximation includes a term which is additional to those appearing in the equations derived by Best (1991). The characteristic speeds for disturbances propagating on the shock are computed and a hierarchical wave structure is evident. In view of this structure, comment is made upon the problem of blast diffraction by a convex corner.

A new numerical method for shock wave propagation based on geometrical shock dynamics

In this paper, a new numerical method for calculating the motion of shock waves in two and three dimensions is presented. The numerical method is based on Whitham’s theory of geometrical shock dynamics, which is an approximate theory that determines the motion of the leading shockfront explicitly. The numerical method uses a conservative finite difference discretization of the equations of geometrical shock dynamics. These equations are similar to those for steady supersonic potential flow, and thus the numerical method developed here is similar to ones developed for that context. Numerical results are presented for shock propagation in channels and for converging cylindrical and spherical shocks. The channel problem is used in part to compare this new numerical method with ones developed earlier. Converging cylindrical and spherical shocks are calculated to analyse their stability.

Focusing of Reflected Shock Waves Analyzedby Means of Geometrical Shock Dynamics.

In the present work, a numerical scheme for the unsteady motion of shock waves in gases is developed based on Whitham's theory of geometrical shock dynamics (GSD) . To investigate the effect of the initial shock strength on the shock reflection and diffraction on a wall, the two-dimensional focusing of cylindrical blast waves reflected from a concave elliptic wall is analyzed under various initial conditions. The results are compared with finite difference simulations based on the piecewise linear method (PLM) as well as with previous experimental results. It is found that geometrical shock dynamics is a practical and economical method to analyze the shock focusing process of the blast waves produced by exploding wires. Pressure changes along the major axis of the ellipse show good agreement when the present computations are compared with experimental results as well as with PLM simulations.

A generalisation of the theory of geometrical shock dynamics

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.

A numerical scheme for shock propagation in three dimensions

A numerical scheme for shock propagation in three space dimensions is presented. The motion of the leading shock surface is calculated by using Whitham’s theory of geometrical shock dynamics. The numerical scheme is used to examine the focusing of initially curved shock surfaces and the diffraction of shocks in a pipe with a 90° bend. Numerical and experimental results for the corresponding two-dimensional or axi-symmetrical cases are used to compare with the new and more complicated three-dimensional results.

Numerical shock propagation in non-uniform media

A general numerical scheme is developed to calculate the motion of shock waves in gases with non-uniform properties. The numerical scheme is based on the approximate theory of geometrical shock dynamics. The refracted shockfronts at both planar and curved gas interfaces are calculated. Both regular and irregular refraction patterns are obtained, and in particular, precursor-irregular refraction systems are found using the approximate theory. The numerical results are compared with recent theoretical and experimental investigations. It is shown that the shockfronts determined using geometrical shock dynamics are in good agreement with the actual shock waves.

Numerical shock propagation using geometrical shock dynamics

A simple numerical scheme for the calculation of the motion of shock waves in gases based on Whitham's theory of geometrical shock dynamics is developed. This scheme is used to study the propagation of shock waves along walls and in channels and the self-focusing of initially curved shockfronts. The numerical results are compared with exact and numerical solutions of the geometrical-shock-dynamics equations and with recent experimental investigations.