Theory Of Shock Dynamics Research Articles

Realization of a shock-tube facility to study the Richtmyer-Meshkov instability driven by a strong shock wave.

A shock-tube facility capable of generating a planar shock with the Mach number higher than 3.0 is developed for studying Richtmyer-Meshkov instability induced by a strong shock wave (referred to as strong-shock RMI). Shock enhancement is realized through the convergence of shock within a channel with the profile determined by using shock dynamics theory. The facility is designed considering the repeatability of shock generation, transition of shock profile, and effects of viscosity and flow choking. By measuring the dynamic pressure of the tube flow using pressure sensors and capturing the shock movement through the high-speed shadowing technique, the reliability and repeatability of the shock tube for generating a strong planar shock are first verified. Particular emphasis is then placed on the ability of the facility to study strong-shock RMI, for which a thin polyester film is adopted to form the initial interface separating gases of different densities. The results indicate that the shock tube is reliable for conducting strong-shock RMI experiments.

Freeze-out of perturbation growth for shocked heavy fluid layers by eliminating reverberating waves

The shock wave accelerating a heavy fluid layer can induce reverberating waves that continuously interact with the first and second interfaces. In order to manipulate the perturbation growths at fluid-layer interfaces, we present a theoretical framework to eliminate the reverberating waves. A model is established to predict the individual freeze-out (i.e. stagnation of perturbation growth) for the first and second interfaces under specific flow conditions determined based on the shock dynamics theory. The theoretical model quantifies the controllable parameters required for freeze-out, including the initial amplitudes of the first and second interfaces, the interface coupling strength and the maximum initial layer thickness preventing the second interface's phase reversal. The effectiveness of the model in predicting individual freeze-out for the first and second interfaces is validated numerically over a wide range of initial conditions. The upper and lower limits of initial amplitudes for the freeze-out of the whole fluid-layer width growth are further predicted. Within this amplitude range, a slightly higher initial amplitude for the second interface is specified, effectively arresting the growth of the entire fluid-layer width before the phase reversal of the second interface.

On divergent Richtmyer–Meshkov instability of a light/heavy interface

Abstract

Flowfield characteristics in sidewall compression inlets

In this work, the flow fields in three-dimensional (3D) sidewall compression inlets are investigated theoretically and numerically. Combining spatial dimension reduction theory with the shock dynamics theory, a detailed analysis of the flowfield in the compression part of an inlet is proposed. Using this analytical approach, wave configurations are determined and flowfield parameters are solved; numerical simulations are conducted to validate these theoretical results. The effects of the attack angle and yaw angle on the pressure ratio, wave structures, and total pressure recovery performance are discussed in detail. Findings offer an effective way to study shock/shock interaction (SSI) in sidewall inlets, which is essential to the designing of such inlets.

Note: A contraction channel design for planar shock wave enhancement.

A two-dimensional contraction channel with a theoretically designed concave-oblique-convex wall profile is proposed to obtain a smooth planar-to-planar shock transition with shock intensity amplification that can easily overcome the limitations of a conventional shock tube. The concave segment of the wall profile, which is carefully determined based on shock dynamics theory, transforms the shock shape from an initial plane into a cylindrical arc. Then the level of shock enhancement is mainly contributed by the cylindrical shock convergence within the following oblique segment, after which the cylindrical shock is again "bent" back into a planar shape through the third section of the shock dynamically designed convex segment. A typical example is presented with a combination of experimental and numerical methods, where the shape of transmitted shock is almost planar and the post-shock flow has no obvious reflected waves. A quantitative investigation shows that the difference between the designed and experimental transmitted shock intensities is merely 1.4%. Thanks to its advantage that the wall profile design is insensitive to initial shock strength variations and high-temperature gas effects, this method exhibits attractive potential as an efficient approach to a certain, controllable, extreme condition of a strong shock wave with relatively uniform flow behind.

Interaction of strong converging shock wave with SF6 gas bubble

Interaction of a strong converging shock wave with an SF6 gas bubble is studied, focusing on the effects of shock intensity and shock shape on interface evolution. Experimentally, the converging shock wave is generated by shock dynamics theory and the gas bubble is created by soap film technique. The post-shock flow field is captured by a schlieren photography combined with a high-speed video camera. Besides, a three-dimensional program is adopted to provide more details of flow field. After the strong converging shock wave impact, a wide and pronged outward jet, which differs from that in planar shock or weak converging shock condition, is derived from the downstream interface pole. This specific phenomenon is considered to be closely associated with shock intensity and shock curvature. Disturbed by the gas bubble, the converging shocks approaching the convergence center have polygonal shapes, and the relationship between shock intensity and shock radius verifies the applicability of polygonal converging shock theory. Subsequently, the motion of upstream point is discussed, and a modified nonlinear theory considering rarefaction wave and high amplitude effects is proposed. In addition, the effects of shock shape on interface morphology and interface scales are elucidated. These results indicate that the shape as well as shock strength plays an important role in interface evolution.

Calibration and verification of the reaction rate for an insensitive PBX

To investigate the detonation characteristics of an insensitive Plastic bonded explosive (PBX), the reaction rate was obtained and used in the numerical simulation of diverse case problems in the one-dimensional and two dimensional space. To calibrate the reaction rate, Detonation shock dynamics (DSD) theory was adopted, and two-dimensional steady-state experimental data were used to determine the shock front curvature function D n (κ). The reaction rate was calibrated by numerically integrating the master equation of the DSD theory. For the calibrated reaction rate, the simulation of the Large scale gap test (LSGT) and EIDS CAP test with regard to the shock sensitivity problem were carried out. Two numerical example were investigated using the reaction rate. First, an underwater explosion, which is basically a one-dimensional problem, was solved. Second, to also validate the reaction rate detonation wave progress, a corner turning problem was solved. Considering the results of the simulations and the experimental test, it was concluded that the calibrated reaction rate well describes the detonation characteristics of insensitive PBX with a reasonable degree of accuracy. The present study is the first case to obtain the reaction rate of an insensitive castable PBX using the DSD theory. Moreover the reaction rate obtained was verified using different detonation tests such as the gap sensitivity test, the underwater explosion test and the detonation wave propagation test.

Three-dimensional shock wave configurations induced by two asymmetrical intersecting wedges in supersonic flow

This study explores the three-dimensional (3D) wave configurations induced by 3D asymmetrical intersecting compression wedges in supersonic and hypersonic inviscid flows. By using the “spatial dimension reduction” approach, the problem of 3D steady shock/shock interaction is converted to that of the interaction of two moving shock waves in the characteristic two-dimensional (2D) plane. Shock polar theory is used to analyze the shock configurations in asymmetrical situations. The results show that various shock configurations exist in 3D asymmetrical shock wave interactions, including regular interaction, transitioned regular interaction, single Mach interaction, inverse single Mach interaction, transitional double Mach interaction, weak shock interaction, and weak single Mach interaction. All of the above 3D steady shock/shock interactions have their corresponding 2D moving shock/shock interaction configurations. Numerical simulations are performed by solving the 3D inviscid Euler equations with the non-oscillatory, non-free parameters, dissipative (NND) numerical scheme, and good agreement with the theoretical analysis is obtained. Furthermore, the comparison of results show that the concept of the “virtual wall” in shock dynamics theory is helpful for understanding the mechanism of two-dimensional shock/shock interactions.

Geometrical shock dynamics applied to condensed phase materials

Taylor blast wave (TBW) theory and geometrical shock dynamics (GSD) theory describe a radially expanding shock wave front through an inert material, typically an ideal gas, in the strong blast wave limit and weak acoustic limit respectively. We simulate a radially expanding blast shock in air using a hydrodynamic simulation code and numerically describe the intermediate region between these two limits. We test our description of the intermediate shock phase through a two-dimensional simulation of the Bryson and Gross experiment. We then apply the principles of GSD to materials that follow the Mie–Gruneisen equation of state, such as plastics and metals, and derive an equation that accurately relates the acceleration, velocity and curvature of the shock through these materials. Along with detonation shock dynamics (DSD), which describes detonation shock propagation through high explosive fluids, we develop a hybrid DSD/GSD model for the simulation of heterogeneous explosives. This model enables computationally efficient simulation of the shock front in high explosive/inert mixtures consisting of simple or complex geometric configurations. We simulate an infinite two-dimensional slab consisting of one half explosive, PBXN-9, and one half aluminium and model the boundary angle conditions using shock polar analysis. We also simulate a series of high explosive unit cells embedded with aluminium spherical particles, and we compare the propagation of the detonation shock front with a direct numerical simulation performed with the ALE3D code.

Refraction of cylindrical converging shock wave at an air/helium gaseous interface

Refraction of a cylindrical converging shock wave at an inclined air/helium interface is investigated. Experimentally, based on the shock dynamics theory, a special wall profile is designed to generate a perfectly cylindrical converging shock wave. A soap film technique is developed to form an inclined discontinuous air/helium interface, and high-speed schlieren photography is adopted to capture the flow. Numerical simulations are also performed to compare with the experimental counterparts and to show details of refraction. In this work, two initial incident angles (45° and 60°) are considered. As the incident shock converges inward, the shock intensity increases while the incident angle decreases, causing possible transitions among the wave patterns. For the case of 45°, an irregular refraction of free precursor refraction (FPR) first occurs and gradually transits into regular refraction, while for the case of 60°, various irregular refractions of twin von Neumann refraction (TNR), twin regular refraction (TRR), free precursor von Neumann refraction (FNR), and FPR occur in sequence. The transition sequences do not belong to any groups described in the planar counterpart, indicating that the classification of the refraction phenomenon in the planar case is not exhaustive or cannot be applied to the converging case. It is also the first time to observe the transition from FNR to FPR, providing an experimental evidence for the previous numerical results. It is deemed that the difference between the velocities of the incident and transmitted shocks propagating along the interface is the primary factor that induces the transitions among wave patterns.

Theoretical approach to one-dimensional detonation instability

Detonation instability is a fundamental problem for understanding the micro-behavior of a detonation front. With the theoretical approach of shock dynamics, detonation instability can be mathematically described as a second-order ordinary difference equation. A one-dimensional detonation wave can be modelled as a type of oscillators. There are two different physical mechanisms controlling the behaviors of a detonation. If the shock Mach number is smaller than the equilibrium Mach number, the fluid will reach the sonic speed before the end of the chemical reaction. Then, thermal chock occurs, and the leading shock becomes stronger. If the shock Mach number is larger than the equilibrium Mach number, the fluid will be subsonic at the end of the chemical reaction. Then, the downstream rarefaction waves propagate upstream, and weaken the leading shock. The above two mechanisms are the basic recovery forces toward the equilibrium state for detonation sustenance and propagation. The detonation oscillator concept is helpful for understanding the oscillating and periodic behaviors of detonation waves. The shock dynamics theory of detonation instability gives a description of the feedback regime of the chemical reaction, which causes variations of the leading shock of the detonation.

Parallel implementation of geometrical shock dynamics for two dimensional converging shock waves

Geometrical shock dynamics (GSD) theory is an appealing method to predict the shock motion in the sense that it is more computationally efficient than solving the traditional Euler equations, especially for converging shock waves. However, to solve and optimize large scale configurations, the main bottleneck is the computational cost. Among the existing numerical GSD schemes, there is only one that has been implemented on parallel computers, with the purpose to analyze detonation waves. To extend the computational advantage of the GSD theory to more general applications such as converging shock waves, a numerical implementation using a spatial decomposition method has been coupled with a front tracking approach on parallel computers. In addition, an efficient tridiagonal system solver for massively parallel computers has been applied to resolve the most expensive function in this implementation, resulting in an efficiency of 0.93 while using 32 HPCC cores. Moreover, symmetric boundary conditions have been developed to further reduce the computational cost, achieving a speedup of 19.26 for a 12-sided polygonal converging shock.

Embedded particle size distribution and its effect on detonation in composite explosives

This paper discusses the effects of stochastically varying inert particle parameters on the long-term behaviour of detonation front propagation. The simulation model involves a series of cylindrical high explosive unit cells, each embedded with an inert spherical particle. Detonation shock dynamics theory postulates that the velocity of the shock front in the explosive fluid is related to its curvature. In our previous work, we derived a series of partial differential equations that govern the propagation of the shock front passing over the inert particles and developed a computationally efficient simulation environment to study the model over extremely long timescales. We expand upon that project by randomising several properties of the inert particles to represent experimental designs better. First, we randomise the particle diameters according to the Weibull distribution. Then we discuss stochastic particle spacing methods and their effects on the predictability of the shock wave speed. Finally, we discuss mixtures of plastic and metal particles and material inconsistency among the particles.

Study on Mach stems induced by interaction of planar shock waves on two intersecting wedges

The properties of Mach stems in hypersonic corner flow induced by Mach interaction over 3D intersecting wedges were studied theoretically and numerically. A new method called “spatial dimension reduction” was used to analyze theoretically the location and Mach number behind Mach stems. By using this approach, the problem of 3D steady shock/shock interaction over 3D intersecting wedges was transformed into a 2D moving one on cross sections, which can be solved by shock-polar theory and shock dynamics theory. The properties of Mach interaction over 3D intersecting wedges can be analyzed with the new method, including pressure, temperature, density in the vicinity of triple points, location, and Mach number behind Mach stems. Theoretical results were compared with numerical results, and good agreement was obtained. Also, the influence of Mach number and wedge angle on the properties of a 3D Mach stem was studied.

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.

A cylindrical converging shock tube for shock-interface studies

A shock tube facility for generating a cylindrical converging shock wave is developed in this work. Based on the shock dynamics theory, a specific wall profile is designed for the test section of the shock tube to transfer a planar shock into a cylindrical one. The shock front in the converging part obtained from experiment presents a perfect circular shape, which proves the feasibility and reliability of the method. The time variations of the shock strength obtained from numerical simulation, experiment, and theoretical estimation show the desired converging effect in the shock tube test section. Particular emphasis is then placed on the problem of shock-interface interaction induced by cylindrical converging shock waves. For this purpose, membrane-less gas cylinder is adopted to form the interface between two different fluids while the laser sheet technique to visualize the flow field. The result shows that it is convenient to perform such experiments in this facility.

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.

Application of the Level Set Method for Propagation of Small Curvature Bending Detonation Wave

Based on the detonation shock dynamics (DSD) theory, Level Set method for propagation of detonation wave was researched. In order to compute the propagation of small curvature bending detonation wave accurately, and according to characteristic of the Hamilton-Jacobi equation, the unstructured quadrilateral and Galerkin finite element difference scheme for solving Level Set function equation was studied.The finite-element code LS-DYNA was used to compute the experimental models about the propagation of detonation wave and the motion of flyers driven by explosive. The computation results are in good agreement with the experimental data. The results indicate that the difference scheme is of high accuracy and universal application.

Parametric study of cylindrical converging shock waves generated based on shock dynamics theory

In our previous work, the technique of generating cylindrical converging shock waves based on shock dynamics theory was proposed. In the present work, a further study is carried out to assess the influence of several parameters including the converging angle θ0, the incident planar shock Mach number M0, and the shock tube height h on the wall profile and the converging shock wave. Combining the high-speed schlieren photography and the numerical simulation with the shock dynamics theory, the characteristics of wall profiles, cylindrical converging shock waves, and thermodynamic properties for different controllable parameters are analyzed. It is found that these parameters have great effects on shapes of the wall profile and experimental investigation favors large values of M0 and h and moderate θ0. The experimental sequences of schlieren images indicate that the shocks moving in the converging part are of circular shapes, which further verifies the method in our previous work. In addition, the changes of the shock Mach number, pressure, temperature, and density are obtained quantitatively. The results show that higher pressure and temperature can be reached in the converging part at the same distance to the center of convergence for larger incident shock Mach numbers, larger shock tube heights, or smaller converging angles. All the database will be useful for understanding the shock focusing and further investigating the Richtmyer-Meshkov instability induced by the converging shock waves.

Converging shocks in elastic-plastic solids

We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy e=e(s)(I(1))+e(h)(ρ,ς), where e(s) accounts for shear through the first invariant of the Cauchy-Green tensor, and e(h) represents the hydrostatic contribution as a function of the density ρ and entropy ς. In the strong-shock limit, reached as the shock approaches the axis or origin r=0, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., e(h)=e(h)(ρ), with a power-law dependence e(h) is proportional to ρ(α), shock dynamics predicts that for a converging shock located at r=R(t) at time t, the Mach number increases as M is proportional to [log(1/R)](α), independently of the space index s, where s=2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(ρ) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M is proportional to R(-(s-1)) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part e(h) is that of an ideal gas, is also tested, recovering the strong-shock limit M is proportional to R(-(s-1)/n(γ)) originally derived by Whitham for perfect gases, where γ is inherently related to the maximum compression ratio that the material can reach, (γ+1)/(γ-1). From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the hydrostatic part of the energy essentially commands the strong-shock behavior, the shear modulus and yield stress modify the compression ratio and velocity of the shock far from the axis or origin. A characterization of the elastic-plastic transition in converging shocks, which involves an elastic precursor and a plastic compression region, is finally exposed.