Theory Of Shock Dynamics Research Articles

Two shock interaction using new theory of shock dynamics

This paper deals with the use of NTSD (the new theory of shock dynamics) to study the propagation of discontinuities, when more than one discontinuity is present in the initial data for a model equation. Even when the initial data is a C N 0 function ( N large) with a leading discontinuity, the paper demonstrates how NTSD can be effectively used by replacing the initial data by piecewise analytic data in suitable intervals. A comparative study of use of NTSD with other methods of solution, including the exact one, is undertaken here.

Long time behaviour of the solution of a system of equations from new theory of shock dynamics

The new theory of shock dynamics gives a system of n + 2 equations, the solution of which determines the propagation of a discontinuity in the initial condition. For the model equation u t + u u x = 0, the long time behaviour of the solution is investigated for various confined initial data. The suitability of using Padé approximants for the solution is examined. Special exact solutions are obtained.

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 case study with the new theory of shock dynamics

A new theory of shock dynamics (NTSD) with analytic considerations and a numerical solution has already been discussed in Part I and Part II. This paper deals with the numerical simulation of the system of ordinary differential equations of the new theory of shock dynamics for the equation u t + ( u 2 2 ) x = 0 when u x ( x, 0) = φ′( x) < 0 in the region behind the shock.

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 Note on the New Theory of Shock Dynamics

The new theory of shock dynamics proposed by Ravindran and Prasad has been considered with initial values taken from the similarity solution for a single conservation equation and the results have been compared with those obtained by Lax-Wendroff finite difference method.

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.

On converging shock waves

An approximate theory of shock dynamics is used to study the behaviour of converging cylindrical shocks. For cylindrical shocks with regular polygonal-shaped cross sections, exact solutions are found, showing that an original polygonal shape repeats at successive intervals with successive contractions in scale. In this sense, these shapes are stable, and the successive Mach numbers increase according to exactly the same formula as for a circular cylindrical shock. The behaviour for initial shock shapes close to these and the general tendency of perturbed circular shapes to become polygonal, not necessarily regular, is explored numerically. Further analytical results are provided for rectangular shapes. Comments are made on the interpretation of regular reflection in this theory and on converging shocks in three dimensions.

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.

Shock dynamics in non-uniform media

The theory of shock dynamics in two dimensions is reformulated to treat shock propagation in a non-uniform medium. The analysis yields a system of hyperbolic equations with source terms representing the generation of disturbances on the shock wave as it propagates into the fluid non-uniformities. The theory is applied to problems involving the refraction of a plane shock wave at a free plane gaseous interface. The ‘slow–fast’ interface is investigated in detail, while the ‘fast–slow’ interface is treated only briefly. Intrinsic to the theory is a relationship analogous to Snell's law of refraction at an interface. The theory predicts both regular and irregular (Mach) refraction, and a criterion is developed for the transition from one to the other. Quantitative results for several different shock strengths, angles of incidence and sound-speed ratios are presented. An analogy between shock refraction and the motion of a force field in unsteady one-dimensional gasdynamics is pointed out. Also discussed is the limiting case for a shock front to be continuous at the interface. Comparison of results is made with existing experimental data, with transition calculations based on three-shock theory, and with the simple case of normal interaction.

The focusing of weak shock waves

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.

Distortion of near-sonic shocks by layers with weak thermal fluctuations

The focusing effects of turbulence in the propagation medium upon near-sonic shocks is studied by extending the theory of shock dynamics to a medium with random temperature and pressure variations. A perturbation procedure is employed to solve the resulting non-linear partial differential equation. The results are applied in a discussion of sonic boom distortion in the terrestrial boundary layer.