Initial Shock Mach Number Research Articles

Experimental investigation of breakup times of droplet under the action of shock wave

In this paper, the morphological characteristics and breakup time of droplet under the action of shock wave were investigated experimentally. Morphological characteristics of droplet were captured by high-speed photography. By varying the diameter of droplets and taking the influence of the shock wave Mach number into consideration, eight tests were performed in the experiments. The experimental results showed that under the action of the shock wave, the droplet underwent two different stages: the deformation stage and the atomization stage. In the deformation stage, the droplet evolved from spherical shape to pineal shape under the action of accompanying airflow induced by shock wave. In the atomization stage, the growth of unstable wave on the surface of the droplet caused by the Rayleigh–Taylor instability was the main reason for the droplet to be atomized. The effects of initial diameter and shock Mach number on atomization time were analysed. The decrease of initial diameter and the increase of shock Mach number reduced the droplet atomization time. The Mach number played a dominant role in the breakup process of the droplet.

On the interactions between a propagating shock wave and evaporating water droplets

One-dimensional numerical simulations based on the hybrid Eulerian–Lagrangian approach are performed to investigate the interactions between propagating shock waves and dispersed evaporating water droplets in two-phase gas–droplet flows. Two-way coupling for interphase exchanges of mass, momentum, and energy is adopted. A parametric study on shock attenuation, droplet evaporation, motion, and heating is conducted, through considering various initial droplet diameters (5 µm–20 µm), number densities (2.5 × 1011/m3–2 × 1012/m3), and incident shock Mach numbers (1.17–1.9). It is found that the leading shock may be attenuated to the sonic wave and even to the subsonic wave when the droplet volume fraction is large and/or the incident shock Mach number is low. Attenuation in both strength and propagation speed of the leading shock is mainly caused by momentum transfer to the droplets that interact at the shock front. Total pressure recovery is observed in the evaporation region, whereas pressure loss results from shock compression, droplet drag, and pressure gradient force behind the shock front. Recompression of the region between the leading shock and the two-phase contact surface is observed when the following compression wave is supersonic. After a critical point, this region gets stable in width and interphase exchanges in mass, momentum, and energy. However, the recompression phenomenon is sensitive to the droplet volume fraction and may vanish with high droplet loading. For an incident shock Mach number of 1.6, recompression only occurs when the initial droplet volume fraction is below 3.28 × 10−5.

The transition to turbulence in shock-driven mixing: effects of Mach number and initial conditions

The effects of incident shock strength on the mixing transition in the Richtmyer–Meshkov instability (RMI) are experimentally investigated using simultaneous density–velocity measurements. This effort uses a shock with an incident Mach number of 1.9, in concert with previous work at Mach 1.55 (Mohaghar et al., J. Fluid Mech., vol. 831, 2017 pp. 779–825) where each case is followed by a reshock wave. Single- and multi-mode interfaces are used to quantify the effect of initial conditions on the evolution of the RMI. The interface between light and heavy gases ($\text{N}_{2}/\text{CO}_{2}$, Atwood number, $A\approx 0.22$; amplitude to wavelength ratio of 0.088) is created in an inclined shock tube at $80^{\circ }$ relative to the horizontal, resulting in a predominantly single-mode perturbation. To investigate the effects of initial perturbations on the mixing transition, a multi-mode inclined interface is also created via shear and buoyancy superposed on the dominant inclined perturbation. The evolution of mixing is investigated via the density fields by computing mixed mass and mixed-mass thickness, along with mixing width, mixedness and the density self-correlation (DSC). It is shown that the amount of mixing is dependent on both initial conditions and incident shock Mach number. Evolution of the density self-correlation is discussed and the relative importance of different DSC terms is shown through fields and spanwise-averaged profiles. The localized distribution of vorticity and the development of roll-up features in the flow are studied through the evolution of interface wrinkling and length of the interface edge, which indicate that the vorticity concentration shows a strong dependence on the Mach number. The contribution of different terms in the Favre-averaged Reynolds stress is shown, and while the mean density-velocity fluctuation correlation term, $\langle \unicode[STIX]{x1D70C}\rangle \langle u_{i}^{\prime }u_{j}^{\prime }\rangle$, is dominant, a high dependency on the initial condition and reshock is observed for the turbulent mass-flux term. Mixing transition is analysed through two criteria: the Reynolds number (Dimotakis, J. Fluid Mech., vol. 409, 2000, pp. 69–98) for mixing transition and Zhou (Phys. Plasmas, vol. 14 (8), 2007, 082701 for minimum state) and the time-dependent length scales (Robey et al., Phys. Plasmas, vol. 10 (3), 2003, 614622; Zhou et al., Phys. Rev. E, vol. 67 (5), 2003, 056305). The Reynolds number threshold is surpassed in all cases after reshock. In addition, the Reynolds number is around the threshold range for the multi-mode, high Mach number case ($M\sim 1.9$) before reshock. However, the time-dependent length-scale threshold is surpassed by all cases only at the latest time after reshock, while all cases at early times after reshock and the high Mach number case at the latest time before reshock fall around the threshold. The scaling analysis of the turbulent kinetic energy spectra after reshock at the latest time, at which mixing transition analysis suggests that an inertial range has formed, indicates power scaling of $-1.8\pm 0.05$ for the low Mach number case and $-2.1\pm 0.1$ for the higher Mach number case. This could possibly be related to the high anisotropy observed in this flow resulting from strong, large-scale streamwise fluctuations produced by large-scale shear.

Spontaneous singularity formation in converging cylindrical shock waves

We develop a nonlinear, Fourier-based analysis of the evolution of a perturbed, converging cylindrical strong shock using the approximate method of geometrical shock dynamics (GSD). This predicts that a singularity in the shock-shape geometry, corresponding to a change in Fourier-coefficient decay from exponential to algebraic, is guaranteed to form prior to the time of shock impact at the origin, for arbitrarily small, finite initial perturbation amplitude. Specifically for an azimuthally periodic Mach-number perturbation on an initially circular shock with integer mode number q and amplitude proportional to epsilon > 1. The constant of proportionality is obtained as a function of gamma and is independent of the initial shock Mach number M-0. Singularity formation corresponds to the transition from a smooth perturbation to a faceted polygonal form. Results are qualitatively verified by a numerical GSD comparison.

The reflection of cylindrical shock wave segments on cylindrical concave wall segments

The reflection of a two-dimensional cylindrical shock wave segment on a concave-cylindrical wall segment was investigated from an experimental and numerical perspective. The images obtained from experiment show no qualitative difference between cylindrical shock behavior and how a plane shock would behave in terms of the features developed. The length of the shock’s Mach stem was plotted against subtending angle. Two limits are highlighted, one where the shock’s radius is much larger than the wall’s radius and another where the wall has the larger radius: the former being akin to a plane shock interacting with a cylindrical wall segment. The increase in initial shock Mach number was observed to affect the type of Mach reflection that is formed as well as the transition point to a transitioned Mach reflection.

On the propagation and multiple reflections of a blast wave travelling through a dusty gas in a closed box

This paper concerns the propagation of shock waves in an enclosure filled with dusty gas. The main motivation for this problem is to probe the effect on such dynamics of solid particles dispersed in the fluid medium. This subject, which has attracted so much attention over recent years given its important implications in the study of the structural stability of systems exposed to high-energy internal detonations, is approached here in the framework of a hybrid numerical two-way coupled Eulerian-Lagrangian methodology. In particular, insights are sought by considering a relatively simple archetypal setting corresponding to a shock wave originating from a small spherical region initialized on the basis of available analytic solutions. The response of the system is explored numerically with respect to several parameters, including the blast intensity (via the related value of the initial shock Mach number), the solid mass fraction (mass load), and the particle size (Stokes number). Results are presented in terms of pressure-load diagrams. Beyond practical applications, it is shown that a kaleidoscope of fascinating patterns is produced by the “triadic” relationships among multiple shock reflection events and particle-fluid and particle-wall interaction dynamics. These would be of great interest to researchers and scientists interested in fundamental problems relating to the general theory of pattern formation in complex nonlinear multiphase systems.

Converging cylindrical shocks in ideal magnetohydrodynamics

We consider a cylindrically symmetrical shock converging onto an axis within the framework of ideal, compressible-gas non-dissipative magnetohydrodynamics (MHD). In cylindrical polar co-ordinates we restrict attention to either constant axial magnetic field or to the azimuthal but singular magnetic field produced by a line current on the axis. Under the constraint of zero normal magnetic field and zero tangential fluid speed at the shock, a set of restricted shock-jump conditions are obtained as functions of the shock Mach number, defined as the ratio of the local shock speed to the unique magnetohydrodynamic wave speed ahead of the shock, and also of a parameter measuring the local strength of the magnetic field. For the line current case, two approaches are explored and the results compared in detail. The first is geometrical shock-dynamics where the restricted shock-jump conditions are applied directly to the equation on the characteristic entering the shock from behind. This gives an ordinary-differential equation for the shock Mach number as a function of radius which is integrated numerically to provide profiles of the shock implosion. Also, analytic, asymptotic results are obtained for the shock trajectory at small radius. The second approach is direct numerical solution of the radially symmetric MHD equations using a shock-capturing method. For the axial magnetic field case the shock implosion is of the Guderley power-law type with exponent that is not affected by the presence of a finite magnetic field. For the axial current case, however, the presence of a tangential magnetic field ahead of the shock with strength inversely proportional to radius introduces a length scale \documentclass[12pt]{minimal}\begin{document}$R=\sqrt{\mu _0/p_0}\,I/(2\,\pi )$\end{document}R=μ0/p0I/(2π) where I is the current, μ0 is the permeability, and p0 is the pressure ahead of the shock. For shocks initiated at r ≫ R, shock convergence is first accompanied by shock strengthening as for the strictly gas-dynamic implosion. The diverging magnetic field then slows the shock Mach number growth producing a maximum followed by monotonic reduction towards magnetosonic conditions, even as the shock accelerates toward the axis. A parameter space of initial shock Mach number at a given radius is explored and the implications of the present results for inertial confinement fusion are discussed.

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

AbstractWe present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number ${\approx }1.2$, the density ratio (precisely $|A_0|\approx 0.67$) and the perturbation shape (dominant spherical wavenumber $\ell _0=40$ and amplitude-to-initial radius of $3\, \%$): the incident shock travels from the lighter fluid to the heavy fluid or, inversely, from the heavy to the light fluid. After describing the computational problem we present results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. Turbulent statistics are developed in Part 2 (Lombardini, M., Pullin, D. I. & Meiron, D. I. J. Fluid Mech., vol. 748, 2014, pp. 113–142). A wave-diagram analysis of the radially symmetric flow highlights that the light–heavy mixing layer is processed by consecutive reshocks, and not by reverberating rarefaction waves as is usually observed in planar geometry. Less surprisingly, reshocks process the heavy–light mixing layer as in the planar case. In both configurations, the incident imploding shock and the reshocks induce Richtmyer–Meshkov (RM) instabilities at the density layer. However, we observe differences in the mixing-layer growth because the RM instability occurrences, Rayleigh–Taylor (RT) unstable scenarios (due to the radially accelerated motion of the layer) and phase inversion events are different. A small-amplitude stability analysis along the lines of Bell (Los Alamos Scientific Laboratory Report, LA-1321, 1951) and Plesset (J. Appl. Phys., vol. 25, 1954, pp. 96–98) helps quantify the effects of the mean flow on the mixing-layer growth by decoupling the effects of RT/RM instabilities from Bell–Plesset effects associated with geometric convergence and compressibility for arbitrary convergence ratios. The analysis indicates that baroclinic instabilities are the dominant effect, considering the low convergence ratio (${\approx } 2$) and rather high ($\ell >10$) mode numbers considered.

Turbulent mixing driven by spherical implosions. Part 2. Turbulence statistics

AbstractWe present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number $\approx $1.2, the density ratio (precisely $|A_0|\approx 0.67$) and the perturbation shape (dominant spherical wavenumber $\ell _0=40$ and amplitude-to-initial radius of 3 %): the incident shock travels from the lighter fluid to the heavy one, or inversely, from the heavy to the light fluid. In Part 1 (Lombardini, M., Pullin, D. I. & Meiron, D. I., J. Fluid Mech., vol. 748, 2014, pp. 85–112), we described the computational problem and presented results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. In particular, it was shown that both configurations reach similar convergence ratios $\approx $2. Here, turbulent mixing is studied through various turbulence statistics. The mixing activity is first measured through two mixing parameters, the mixing fraction parameter $\varTheta $ and the effective Atwood ratio $A_e$, which reach similar late time values in both light–heavy and heavy–light configurations. The Taylor-scale Reynolds numbers attained at late times are estimated at approximately 2000 in the light–heavy case and 1000 in the heavy–light case. An analysis of the density self-correlation $b$, a fundamental quantity in the study of variable-density turbulence, shows asymmetries in the mixing layer and non-Boussinesq effects generally observed in high-Reynolds-number Rayleigh–Taylor (RT) turbulence. These traits are more pronounced in the light–heavy mixing layer, as a result of its flow history, in particular because of RT-unstable phases (see Part 1). Another measure distinguishing light–heavy from heavy–light mixing is the velocity-to-scalar Taylor microscales ratio. In particular, at late times, larger values of this ratio are reported in the heavy–light case. The late-time mixing displays the traits some of the traits of the decaying turbulence observed in planar Richtmyer–Meshkov (RM) flows. Only partial isotropization of the flow (in the sense of turbulent kinetic energy (TKE) and dissipation) is observed at late times, the Reynolds normal stresses (and, thus, the directional Taylor microscales) being anisotropic while the directional Kolmogorov microscales approach isotropy. A spectral analysis is developed for the general study of statistically isotropic turbulent fields on a spherical surface, and applied to the present flow. The resulting angular power spectra show the development of an inertial subrange approaching a Kolmogorov-like $-5/3$ power law at high wavenumbers, similarly to the scaling obtained in planar geometry. It confirms the findings of Thomas & Kares (Phys. Rev. Lett., vol. 109, 2012, 075004) at higher convergence ratios and indicates that the turbulent scales do not seem to feel the effect of the spherical mixing-layer curvature.

Shock-induced ignition of thermally sensitive explosives

The process of planar detonation ignition, induced by a constant‐velocity piston or equivalently by a shock reflected from a stationary wall, is investigated using high‐resolution one‐dimensional numerical simulations. The standard one‐step model with Arrhenius kinetics, which models thermally sensitive explosives, is employed. Emphasis is on comparing and contrasting the results of the finite activation temperature simulations with high activation temperature asymptotic predictions and previous simulations. During the induction phase, it is shown that the asymptotic results give qualitatively good predictions. However, for parameters representative of gaseous explosives, subsequent to thermal runaway at the piston and the formation of a reaction wave, the high activation temperature asymptotic theory is qualitatively incorrect for moderately high activation temperatures. It is shown that the results are very sensitive to the value of the activation temperature, especially the distance from the piston at which a secondary shock forms and the degree of unsteadiness in the reaction wave which moves away from the piston. The dependence of the ignition evolution on the other parameters (initial shock Mach number, heat of reaction and polytropic index) is also investigated. It is shown that qualitative predictions regarding the dependence of the ignition evolution on each of the parameters can be elucidated from finite activation temperature homogeneous explosion calculations together with the high activation temperature asymptotic shock ignition results. It is found that for sufficiently strong initiating shocks the ignition evolution is qualitatively different from cases studied previously in that no secondary shock forms. For a high polytropic index, corresponding to a simple equation of state model for condensed phase explosives, the results are in much better qualitative agreement with the asymptotic theory.

Numerical Studies of Cosmic‐Ray Injection and Acceleration

A numerical scheme that incorporates a injection model into a combined gasdynamics and cosmic-ray (CR) diffusion-convection code has been developed. The hydro/CR code can follow in a very cost-effective way the evolution of CR-modified planar quasi-parallel shocks by adopting subzone shock tracking and multilevel adaptive mesh refinement techniques. An additional conservative quantity, S = Pg/ργg−1, is introduced to follow the adiabatic compression accurately in the precursor region, especially in front of strong, highly modified shocks. The thermal leakage injection model is based on the nonlinear interactions of the suprathermal particles with self-generated MHD waves in quasi-parallel shocks. The particle injection is followed numerically by filtering the diffusive flux of suprathermal particles across the shock to the upstream region according to a velocity-dependent transparency function that controls the fraction of leaking particles. This function is determined by a single parameter, , which should depend on the strength of postshock wave turbulence but is modeled as a constant parameter in our simulations. We have studied CR injection and acceleration efficiencies during the evolution of CR-modified planar shocks for a wide range of initial shock Mach numbers, M0, assuming a Bohm-like diffusion coefficient. For expected values of the injection process is very efficient when the subshock is strong, leading to fast and significant modification of the shock structure. As the CR pressure increases, the subshock weakens and the injection rate decreases accordingly so that the subshock does not disappear. Although some fraction of the particles injected early in the evolution continue to be accelerated to ever higher energies, the postshock CR pressure reaches an approximate time-asymptotic value because of a balance between fresh injection/acceleration and advection/diffusion of the CR particles away from the shock. In the strong shock limit of M0 30, the injection and acceleration processes are largely independent of the initial shock Mach number for a given , while they are sensitively dependent on M0 for M0 < 30. We conclude that the injection rates in strong parallel shocks are sufficient to lead to rapid nonlinear modifications to the shock structures and that self-consistent injection and time-dependent simulations are crucial to understanding the nonlinear evolution of CR-modified shocks.

Saturation of Richtmyer’s impulsive model

In this Brief Communication, we use results from two-dimensional simulations to calculate for what product of perturbation wave number (k) and initial amplitude (a0) the Richtmyer impulsive model will fail in its prediction of shock wave driven instability growth as a function of initial shock Mach number. For “large” ka0, we suggest a formula which bounds the maximum value of the ratio of the amplitude growth rate over the interface speed by one minus the ratio of the interface speed and the transmitted shock speed.

Reflection and focusing of toroidal shock waves from coaxial annular shock tubes

Focusing of a toroidal shock wave emitted from a coaxial annular shock tube and the resulting shock wave reflection were studied numerically with a dispersion-controlled scheme. The numerical code was first applied to prediction of a shock wave discharged from the open-end of a circular shock tube into ambient air. The numerical interferogram was constructed from the numerical result, taking three-dimensional density distribution into account, to directly compare with an experimental interferogram. Obtaining good agreement between these results, the numerical method was applied to shock waves discharged from a coaxial annular shock tube with various initial shock Mach numbers and in different configurations. It was found that intense shock focusing appears when the toroidal shock waves merge at the axis of symmetry. A somewhat unusual shock wave reflection was also observed in the case of higher shock Mach numbers.

Focusing of Toroidal Shock Waves from Co-Axial Annular Shock Tubes.

Focusing of a toroidal shock wave emitted from a co-axial annular shock tube and the resulting shock wave reflection were studied numerically with a dispersion-controlled scheme. A shock wave discharged from an open-end of a circular shock tube into ambient air was calculated first for comparison with an experimental interferogram. After obtaining good agreement, further three cases were calculated with various initial shock Mach numbers. It was found that intense shock focusing appears when toroidal shock waves merge at the axis of symmetry. A somewhat unusual shock wave reflection was also observed in the case of higher shock Mach numbers.

Shock wave interaction with a fast convection-reaction driven flame

Premixed flames are highly sensitive to changes in pressure. It is found that following the passage of a shock wave, a conventional diffusion-reaction deflagration can accelerate to become a fast convection-reaction driven flame that itself can further markedly accelerate. An induction zone is formed between the shock and the combustion wave. This is a relatively long region in which small changes in the dependent variables occur. Treating these as asymptotic perturbations, an intermediate zone that connects the rear of the induction zone to the front of the fast flame must also be considered. Matching conditions between the zones and boundary conditions at the shock and piston are calculated and the evolution of the shock wave, induction zone, and fast flame system is investigated. Across the induction zone there is strong acoustic coupling such that the resulting behavior of the combined system leads to thermal runaway. The time for this blow up to occur is investigated as a function of initial mass flux through the fast flame and initial shock Mach number. It is also shown that there is the possibility of hot spots occurring in the induction zone which then leads to secondary ignition ahead of the fast flame. We have also investigated the effect on primary and secondary ignition of the activation energy of the simple one-step chemistry used in this work.

Unstructured adaptive remeshing finite element method for dusty shock flow

The passage of planar shocks in a dusty gas was investigated to note effects due to particle loading and initial shock Mach number. Two-phase flow equations have been added to a conservative, monotonic flow solver to allow study of compressible particle and droplet flows, which are of importance for shock propagation in two-phase flows and spray propulsion systems. The formulation developed herein employed a conservative Eulerian treatment for the gas and particle phases. The computations were performed using the finite element method-flux corrected transport (FEM-FCT) scheme, which has shown excellent predictive capability of various compressible flows which include both strong and weak shocks. The flux limiting technique was modified to provide monotonic particle velocity fields to increase the scheme's computational stability. Adaptive unstructured methodology based on adapting to high gradients of both the fluid and particle densities was used in conjunction with the conservative shock-capturing scheme to adequately resolve strong flowfield gradients. The shock attenuation of this scheme was then compared with previous experimental and numerical results and was found to yield robust predictions. Various interphase coupling terms were also considered to note their effect on the shock attenuation.

Shock wave attenuation in partially confined channels

An approximate analytical solution is presented for the attenuation of planar shock waves in channels with perforated walls. The problem is considered as quasi-one-dimensional. Good agreement is found between the theoretical results and available experimental data regarding the rate of shock wave attenuation within the range of initial shock Mach numbers between 1.1 and 4 and perforation ratios between 4.5 × 10−3 and 0.53. A correlation for the discharge coefficient of a single hole perforation is presented which gives quantitatively good agreement with particular experimental observations.

Ionizing argon boundary layers. Part 2. Shock-tube side-wall boundary-layer flows

A combined experimental and theoretical investigation was conducted on the shock-tube side-wall ionizing boundary-layer induced by a shock wave moving into argon. The dual-wavelength interferometric boundary-layer data were obtained by using a 23 cm diameter Mach—Zehnder interferometer with the 10 × 18 cm Hypervelocity Shock Tube at initial shock Mach numbers of 13 and 16, an initial pressure of 5 torr and a temperature of 300° K. The plasma density and electron number density in the boundary layer were measured and compared with numerical profiles obtained by using an implicit finite-difference scheme for a two-temperature, chemical non-equilibrium, laminar boundary-layer flow in ionizing argon. The analysis included the variations of transport properties based on elastic-scattering cross-sections, effects of chemical reactions, radiation-energy losses and electron-sheath wall boundary conditions. Considering the difficulties involved in such complex plasma flows, satisfactory agreement was obtained between the analyses and experiments. A comparison was made with the flat-plate case and despite the very different velocity boundary conditions at the wall for the two flows the experimental data appear to be quite similar. The experimental bump in the profile of electron number density which was found in the flat-plate case was not found in the side-wall case. Comparisons and discussions of the results for the different types of boundary layer are presented, including a comparison between experimentally derived and analytical plasma-temperature profiles.

Corner-Expansion Flow of Ionized Argon

The recombination rate constant for ionized argon as a function of temperature was determined by using dual-frequency laser interferometry to measure total density and electron number density in a 15° corner-expansion flow. The flow was induced by a normal shock wave in a 4 in. × 7 in. (10.2 cm × 17.8 cm) hypervelocity shock tube over a range of initial shock Mach numbers [Formula: see text] and pressures [Formula: see text] Torr at room temperature T1 ~ 300 °K, with impurity levels less than 150 parts per million.