Nonstationary loads in shock diffraction

The purpose of the paper is to analyze the nonlinear process of plane shock reflection from a cylinder or sphere. In contrast to [1], where the S. K. Godunov method of first-order accuracy is used, the solution is carried out by using an explicit difference scheme of second-order accuracy [2, 3]. Curves of the pressure distribution over the surface are presented and a comparison is given with the results of experiments and the theory of regular (I) and Mach (II) reflections [4], which assume rectilinearity of the shocks and perpendicularity of the Mach wave to the reflecting surface. A formula describing the maximum reflection pressure on the body surface is presented. The results of computations supplement the known information about the singularities of the reflection process obtained experimentally and theoretically.     

Vorticity produced by shock wave diffraction

Numerical simulations with a monotonicity preserving flow solver have been performed to study shock diffraction phenomena and shock wave generated vorticity. The computations were performed using the conservative Finite Element Method-Flux Corrected Transport (FEM-FCT) scheme, which has been shown to have an excellent predictive capability for various compressible flows with both strong and weak shocks. An adaptive unstructured methodology based on adapting to high density and entropy gradients was used in conjunction with a conservative shock-capturing scheme to adequately resolve strong and weak flowfield gradients. The chief interest was the formation of vorticity arising from shock wave propagation over a sharp corner and the high accuracy and resolution of the interacting compressible wave features. Numerical simulations were compared with previous experimental results and exhibited remarkably good agreement in terms of compressible wave propagation, as well as vorticity development and transport. The computations also allowed insight into the fundamental fluid dynamics, specifically shock diffraction, vortex convection and shock-vortex interactions.     

Three-dimensional diffraction of a shock wave

The diffraction of a shock wave (M₀=4.7) at an angle close to 180 has been experimentally investigated for the three-dimensional case. Interferograms of the flow and the pressure distribution on the back wall in the course of its interaction with the diffracted wave were obtained. Rotation of the flow structure behind the shock wave relative to the axis of symmetry was observed as the flow pattern develops in time and space.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 200–201 March–April, 1993.     

Theory of diffraction of weak shock waves

A numerical solution is considered to the universal nonlinear boundary-value diffraction problem which occurs in various problems of weak interaction [1, 2] in the asymptotic analysis of the flow in a region with large gradients of the parameters near the point of intersection of the incident, diffracted, and reflected waves. The analytical solutions to this type of problem usually approximately satisfy the conditions on the diffracted front, the position of which is not known beforehand, but is found along with the solution. In the present paper, the problem is solved by the numerical method of [3], which reduces the initial boundary-value problem for the system of short-wave equations with an unknown boundary to the solution of a series of boundary-value problems with a fixed boundary. The problem of the diffraction of a weak shock wave on a wedge with a finite apex angle is considered as an application of the solution. The data calculated by the asymptotic theory agree significantly better with the experimental data [5] than the theoretical data of [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 176–178, November–December, 1984.     

Shear layer behavior resulting from shock wave diffraction

All previous studies on shock wave diffraction in shock tubes have spatial and temporal limitations due to the size of the test sections. These limitations result from either the reflection of the expansion wave, generated at the corner, from the top wall and/or of the reflection of the incident diffracted shock from the bottom wall of the test section passing back through the region of interest. This has limited the study of the evolution of the shear layer and its associated vortex, which forms a relatively small region of the flow behind the shock with an extent of only a few centimeters, and yet is a region of significant interest. A special shock tube is used in the current tests which allow evolution of the flow to be examined at a scale about an order of magnitude larger than in previously published results, with shear layer lengths of up to 250 mm being achieved without interference from adjacent walls. Tests are presented for incident shock wave Mach numbers of nominally 1.3–1.5. Studies have been undertaken with wall angles of 10, 20, 30 and 90°. Significant changes are noted as the spatial and temporal scale of the experiment increases. For a given wall angle, the flow behind the incident shock is not self-similar as is usually assumed. Both shear layer instability and the development of turbulent patches become evident, neither of which have been noted in previous tests.     

Numerical study of shock diffraction in dusty gases

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Unsteady three-dimensional compressible vortex flows generated during shock wave diffraction

The results of an experimental and numerical investigation into the behaviour of the spiral vortex generated by shock wave diffraction over edges yawed to the incident shock wave are presented. Three-dimensional numerical simulations reveal significant distortion and bending of the free vortex in regions near the boundary of the flow domain, so as to meet it at a right angle. The results of numerical simulations were found to mimic the experimentally obtained photographs very well. The numerical results are used to explain the various features of the resultant flow fields, with particular emphasis placed on the behaviour and properties of the spiral vortex, as it evolves with time. The effects of bending on the structure of the vortex are examined. The rate of circulation production for the three-dimensional shock diffraction cases was calculated, and the trends observed correlated with those for the much published two-dimensional diffraction case.     

A note on numerical simulation of vortical structures in shock diffraction

In numerical simulation of the Euler equations, the slipstream or shear layer that appears behind a diffracted shock wave may develop small discrete vortices using fine computational meshes. Similar phenomena were also observed in the simulation of a Mach reflection that is accompanied by a shear layer. However, these small vortices have never been observed in any shock-tube experiment, although the wave pattern and the shape of the main vortex agree very well with visualization results. Numerical solutions obtained with coarse grids may agree better with experimental photos than those with very fine grids because of the pollution of the small vortices. This note tries to investigate the effect of viscosity on the small vortices by comparing the solutions of the laminar Navier-Stokes equations and the turbulence model. It is found that the small vortices are still observed in the solution of the laminar Navier-Stokes equations, although they can be suppressed by using the turbulence model. Numerical and experimental factors that are responsible for the deviation of the laminar solutions from experimental results are discussed. The secondary vortex in shock diffraction is successfully simulated by solving the Navier-Stokes equations.Received: 28 March 2003, Accepted: 6 May 2003, Published online: 11 June 2003     

Computation of shock wave diffraction at a ninety degrees convex edge

This paper presents a numerical study of shock wave diffraction at a sharp ninety degrees edge, using an explicit second-order Godunov-type Euler scheme based upon the solution of a generalized Riemann problem (GRP). The Euler computations produce flow separation very close to the diffraction edge, leading to a realistic development of the separated shear layer and subsequent vortex roll-up. The diffracted shock wave, and the secondary shock wave, are both reproduced well. In addition a pair of vortex shocks are shown to form, extending well into the vortex core.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Weak shock waves and shear bands in compressible,inextensible thermoelastic solids

A study of weak shock waves propagating into a solid, which is compressible but temperature-dependent extensible in a specified direction is presented. The inextensible solid is also considered. The constitutive equations of constrained thermoelastic material are written as the summation of constrained and unconstrained counterparts of the relevant quantities. The equation of motion of weak shock waves, which is recovered by the theory of singular surfaces, reduces to an eigenvalue problem. The solution of this eigenvalue problem yields the speeds of propagation of weak shock waves. In the case of an undeformed solid, the speeds of these waves are explicitly expressed. Additionally, a discussion on the ductility limits of constrained thermoelastic material subjected to the uniaxial and biaxial extensions is presented.     

Analysis of shock wave diffraction by a right-angle corner in a real gas

In addition to physical experiments, numerical experiments are widely used for the investigation of complex gasdynamic and magnetogasdynamic flows with various internal surfaces of discontinuity and are found to yield the most complete quantitative results. In the investigation of two-or three-dimensional flows by numerical methods, however, the interpretation of the results poses more than a trivial problem, because the internal discontinuity surfaces are diffuse in the majority of situations. Some surfaces vanish altogether (such as the contact surfaces in application of the particle-in-cell method), while others, such as second shocks, can be identified by investigation of the flow parameters in the suspicious region. The kind of experience needed for the latter is most readily acquired in the calculation and analysis of complex flows with known qualitative singularities. A suitable candidate for this standard flow is the flow associated with shock diffraction, the qualitative singularities of which have been investigated in sharp detail in experimental work [1, 2]. The particle-in-cell (PIC) method [3, 4] is used in the present work to investigate the flow associated with diffraction of a shock wave by a 90 corner. The scheme of the described flow is borrowed from Skews [1] and has been used in our earlier work [5], in which we showed that PIC analysis yields a correct expression for the diffracted shock and exhibits the presence of flow separation in the resulting second shock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 125–129, September–October, 1976.     

Weak shock waves in isotropic solids at finite temperatures up to the melting point

Propagation speeds and Rankine–Hugoniot relations for weak shock waves in isotropic solids are derived analytically in order to elucidate mechanical and thermal properties of the waves. In the analysis, we adopt a new continuum model for the solids, which takes into account explicitly microscopic thermal vibration of the constituent atoms. As the model is valid in a wide temperature range up to the melting point, we can discuss the relations at high temperatures even near the melting point. Typical numerical results are also shown and discussed as illustrations.      

The Mach reflection triple-point locus for internal and external conical diffraction of a moving shock wave

This paper examines the different behavior that occurs for the Mach reflection triple-point loci between the two fundamental axisymmetric cases, these being the external diffraction by a cone and the internal diffraction within a conically contracting channel. From equations derived in this paper using a shock dynamics approach, it has been shown that, for external diffraction over a cone, a possible solution is that the triple-point locus is a straight line which corresponds to the experimental results available, while for internal diffraction along a conically converging channel, it cannot be straight and is, in fact, a convex curve. In the latter case, a transition point is noted on the triple-point locus before which the locus is nearly straight but after which the curvature becomes marked. The second region diminishes as a proportion of the total locus with decreasing half cone angle.For the external case, a set of simple, axisymmetric equations are derived which allow a rapid estimation of the triple point locus angle and the Mach stem strength for any incident shock Mach number and cone angle combination. The equations for internal diffraction are similar and allow a quick computation of both the curved triple-point locus and the strength of the diffracting front of the shock wave. A comparison with experiment has been carried out and agreement is good.     

X-Ray diffraction evidence for the role of stacking faults in plastic deformation of solids under shock loading

A mechanism responsible for the high speed shear relaxation immediately behind shock fronts is suggested. The shear stress generated by the shock front causes the growth of two-dimensional defects in the crystal lattice, known as stacking faults (SF). Increasing the SF concentration and area leads to the absorption of impact energy. A breach of the lattice symmetry due to the SF presence causes an additional shift in peaks of the x-ray diffraction pattern obtained from the shock compressed material. Thus pulse x-ray diffraction is the only method that experimentally measures both the dilatational and deviatoric components of the deformation, which takes place during shock wave passage.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Type of reflection of a near-wall shock wave in the process of diffraction from a square channel

The results of an experimental and numerical investigation of the process of diffraction of shock waves from a square channel at a ninety-degree convex corner are presented for various incident shock wave Mach numbers M₀ (1.4<M₀<7). The type of reflection of the near-wall fragment of the diffracting shock wave from the wall and the wave velocity are determined as functions of M₀, direction, and time. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 145–151, January–February, 2000. The work was carried out with partial support from the Russian Foundation for Basic Research (project No. 96-02-16170a).     

Experimental studies of ignition and transition to detonation induced by the reflection and diffraction of shock waves

This paper presents results from a program of experimental studies of ignition induced by the interaction of an initially planar shock wave with an obstacle in its path. With the aid of pressure measurements, spark schlieren photography and smoked foil techniques it is shown how, given favourable initial conditions, the two-dimensional multiple shock reflection and diffraction can promote ignition and transition to detonation in reactive gaseous mixtures. Comparison of the results with those of a non-reactive gas distinguishes the gas dynamic and chemical processes involved, and experimentally determined detonation cell sizes are compared with values predicted using chemical kinetic rate data. The systems investigated were argon, air, propane-air, propane-oxygen-argon and ethylene-oxygen-argon. Received: 3 December 1998 / Accepted: 27 October 1999     

Weak Singularities of Planar Vector Fields Under Weak Perturbation

We characterize the elements of the set H _n of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in H _n , both on the plane and on the Poincaré sphere. We use this information to characterize elements of the set W _n of smooth vector fields on ² beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in W _n . We also determine the set of elements of H _n that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.