Shock Mach Number Research Articles

Diffusive Shock Acceleration by Multiple Weak Shocks

The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, M ≲ 3, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diffusive shock acceleration (DSA). We examine the effects of DSA by multiple shocks on the spectrum of accelerated CRs by including in situ injection/acceleration at each shock, followed by repeated re-acceleration at successive shocks in the test-particle regime. For simplicity, the accelerated particles are assumed to undergo adiabatic decompression without energy loss and escape from the system, before they encounter subsequent shocks. We show that in general the CR spectrum is flattened by multiple shock passages, compared to a single episode of DSA, and that the acceleration efficiency increases with successive shock passages. However, the decompression due to the expansion of shocks into the cluster outskirts may reduce the amplification and flattening of the CR spectrum by multiple shock passages. The final CR spectrum behind the last shock is determined by the accumulated effects of repeated re-acceleration by all previous shocks, but it is relatively insensitive to the ordering of the shock Mach numbers. Thus multiple passages of shocks may cause the slope of the CR spectrum to deviate from the canonical DSA power-law slope of the current shock.

Propagation of Shock Waves of Varying Curvature

When a shock wave having variable concave curvature propagates, it can develop a kink followed by the development of a reflected shock. A typical example is a plane incident shock encountering a surface with concave curvature, the part of the shock adjacent to the surface curves forward and subsequently develops into a Mach reflection with a Mach stem, shear layer and reflected shock. The physical mechanisms associated with the evolution of the shock profile was evaluated for shock waves with initial profiles comprising a cylindrical arc, placed in-between two straight segments, propagating in a converging channel. The temporal variation of the pressure distribution immediately behind the shock wave was studied using CFD. This revealed a pressure imbalance in the region where the curved (which was initially cylindrical) and straight shock segments meet. This imbalance occurs due to the difference in the propagation behaviour of curved and planar shock waves, and results in the development of reflected shocks on the shock front. The angle at which the channel walls converge, the initial curvature radius, and the shock Mach number, was varied between 40 and 60 degrees, 130 and 190 mm and 1.1 to 1.4, respectively. The variation with time of the pressure-gradient distribution and the maximum pressure gradient behind theshock wave was evaluated. From this, the trajectory angle of the triple points, and the rate at which the reflected shocks develop, was deduced. It was found that when shock waves with larger curvature radii propagate in channels with lower wall angles, the reflected shocks develop at a slower rate, and the triple points follow a steeper trajectory. Consequently, the likelihood of reflected shocks emerging on the shock front, within the duration of the shock propagation, is reduced. This is due to the triple points intersecting the walls, before reflected shocks can fully develop. Similarly, when the shock Mach number is higher, the trajectory angle of the triple points is greater, and they intersect the walls before the reflected shocks can emerge.

Oblique High Mach Number Heliospheric Shocks: The Role of α Particles

AbstractSpacecraft observations of heliospheric shocks often find oscillations in the magnetic field and density both, upstream and downstream. The downstream magnetic oscillations of oblique collisionless shocks were detected by Wind with 10.9 samples s−1 and DSCOVR spacecraft with high temporal resolution of 50 samples s−1. The density oscillations associated with the shocks are also evident in proton and α particle density by Wind (with much lower temporal resolution). Recently, we have investigated low Mach number low‐β oblique shock oscillations using satellite data and 2.5D hybrid modeling with electrons modeled as fluid and ions modeled as particles and found that α particles—an important component of heliospheric plasma—may affect considerably the downstream oscillations and the shock structure. The objective of the present study is to investigate the effects of α particles on high Mach number heliospheric shocks dynamics, oscillations, nonstationarity, and shock‐front rippling. We extend the study to high Mach number shocks (M > 3), investigate several α particle typical densities, and compare the results for the various shock parameters. We model the effects of α particles on the shock ramp, wake, and downstream oscillation structure and the kinetic properties of proton and α particle velocity distributions at various locations downstream of the shocks. Using the 2.5D hybrid model we found that the modeled high Mach number quasi‐perpendicular shock magnetic and density structures are significantly affected by α particles with typical solar wind relative abundances, suggesting that the observed high Mach number shocks are similarly affected by α particles.

Pulse-burst spontaneous Raman thermometry of unsteady wave phenomena in a shock tube.

A high-speed temperature diagnostic based on spontaneous Raman scattering (SRS) was demonstrated using a pulse-burst laser. The technique was first benchmarked in near-adiabatic ${{\rm H}_2} \text{-} {\rm air}$ flames at a data-acquisition rate of 5 kHz using an integrated pulse energy of 1.0 J per realization. Both the measurement precision and accuracy in the flame were within 3% of adiabatic predictions. This technique was then evaluated in a challenging free-piston shock tube environment operated at a shock Mach number of 3.5. SRS thermometry resolved the temperature in post-incident and post-reflected shock flows at a repetition rate of 3 kHz and clearly showed cooling associated with driver expansion waves. Collectively, this Letter represents a major advancement for SRS in impulsive facilities, which had previously been limited to steady state regions or single-shot acquisition.

The TeV Cosmic-Ray Bump: A Message from the Epsilon Indi or Epsilon Eridani Star?

A recently observed bump in the cosmic-ray (CR) spectrum from 0.3 to 30 TV is likely caused by a stellar bow shock that reaccelerates preexisting CRs, which further propagate to the Sun along the magnetic field lines. Along their way, these particles generate an Iroshnikov-Kraichnan (I-K) turbulence that controls their propagation and sustains the bump. Ad hoc fitting of the bump shape requires six adjustable parameters. Our model requires none, merely depending on three physical unknowns that we constrain using the fit. These are the shock Mach number, M, its size, l ⊥, and the distance to it, ζ obs. Altogether, they define the bump rigidity R 0. With M ≈ 1.5-1.6 and R 0 ≈ 4.4 TV, the model fits the data with ≈0.08% accuracy. The fit critically requires the I-K spectrum predicted by the model and rules out the alternatives. These attributes of the fit make an accidental agreement highly unlikely. In turn, the R 0 and M derived from the fit impose the distance-size relation on the shock: . For sufficiently large bow shocks, l ⊥ = 10-3-10-2 pc, we find the distance of ζ obs = 3-10 pc. Three promising stars in this range are the Scholz's Star at 6.8 pc, Epsilon Indi at 3.6 pc, and Epsilon Eridani at 3.2 pc. Based on their current positions and velocities, we propose that Epsilon Indi and Epsilon Eridani can produce the observed spectral bump. Moreover, Epsilon Eridani's position is only ~6°.7 off of the magnetic field direction in the solar neighborhood, which also changes the CR arrival direction distribution. Given the proximity of these stars, the bump appearance may change in a relatively short time.

Losing the shock wave front profile due to interaction with turbulence

In this study, losing the shock wave profiles under interactions with grid turbulence was investigated experimentally and theoretically. We demonstrated that the shock wave contrast on side-view schlieren images gradually decreased to an undetectable level in the experiment. This shock wave ‘vanishment’ occurred at a low shock Mach number with a high turbulent Mach number. With a relatively strong shock wave, the contrast of the shock wave remained detectable although the shock wave profile region was expanded. To understand the shock wave vanishment phenomenon during interaction with turbulence, we established a shock wave vanishment model based on the solution of a one-dimensional interaction between a shock wave and forward induced flow. The criterion of the occurrence of local vanishment of the shock wave was derived as , where is the turbulent Mach number and is the shock Mach number. The proposed shock vanishment model involves the effect of the interaction length and the shock wave recovery characteristics. The derived model explains the vanishment of weak shock wave profile during turbulence with an interaction length of ten times the order of the integral scale of the turbulence, as observed in the experiment.

Impacts of grid turbulence on the side projection of planar shock waves

In this study, the characteristics of side-projected planar shock waves propagating through grid turbulence were investigated using a counter-driver shock tube. At the location of visualization of the shock–turbulence interactions, the range of the shock Mach numbers was $$M_{\mathrm {s}} = 1.01{-}1.15$$ and the representative values of the turbulent Mach numbers $$\widetilde{M_{\mathrm {t}}}$$ were $$ 0.005{-}0.014$$ . The interaction length between the shock wave and grid turbulence was in the range approximately from $$-50$$ to $$300$$ times the integral length scale of the turbulence. For the interaction involving the strongest turbulence with $$\widetilde{M_{\mathrm {t}}} = 0.014$$ , the weakest planar shock wave with $$M_{\mathrm {s}} = 1.01$$ was largely deformed and could not be detected on the projected shadowgraph and schlieren images. The density changes on the grayscale in the projected images of the shock waves weakened and expanded multidimensionally. This undetectable profile of the shock wave could indicate that the shock wave locally lost discontinuous property change profile. The shock waves with $$M_{\mathrm {s}} \ge 1.05$$ did not show the undetectable profile in the projected image. In these cases, the projected thickness of the shock wave increased with an increase in the interaction length. The increase in the projected thickness became larger as the turbulent Mach number increased, and the shock Mach number decreased.

Evolution of heat transfer at the stagnation point during the detached bow shock establishment

The diffraction of a shock wave over a stationary body is a problem of interest associated with the starting of shock tubes and expansion tubes which are well suited to studies of hypersonic flows. However, these facilities are characterized by very short test times. The transient parameters during the establishment of the detached bow shock in such impulsive facilities are important for both data processing and experimental design. In the present study, numerical simulations are conducted to investigate the diffraction of a normal shock wave over a sphere and the subsequent transient phenomena in a viscous perfect-gas flow field. The incident shock Mach number ranges from 3 to 5 with a specific heat ratio of 1.4. Based on the theoretical description of the reflected shock position during bow shock formation, approximate solutions for the time histories of the stagnation-point heat flux are also derived. The analytical and numerical results agree well. The results show that the stagnation-point pressure and heat flux approach their steady-state values much more rapidly than the shock detachment distance does.

Analysis of the blast waves from the explosions of stoichiometric, rich, and lean propane/oxygen mixtures

The blast waves from a series of explosions of stoichiometric, rich, and lean propane/oxygen mixtures have been analysed. The explosive mixtures were contained in hemispherical soap bubbles, 0.05 m in radius, with total masses in the order of 1 g. The blast waves were measured with a series of piezoelectric transducers, flush mounted in the horizontal surface supporting the charges, at various distances from the centres of the explosions. The measured time history of hydrostatic overpressure from each transducer was least-squares-fitted to the modified Friedlander equation to provide the best estimates of the peak hydrostatic overpressure immediately behind the primary shock and the positive phase duration. The times-of-arrival (TOA) of the primary shocks at each gauge location were used to determine the shock Mach numbers as functions of distance, and these values were used in a Rankine–Hugoniot relationship to calculate the peak hydrostatic overpressures, also as functions of distance. For all explosive mixtures, there was excellent agreement between the direct gauge measurements and the values from the TOA analyses. In order to compare the relative strengths of the blast waves from the three mixtures, the measured results were scaled to those for a 1-kg charge using the masses of propane and applying Hopkinson’s cube root scaling. This analysis showed that the blast wave from the lean mixture, for which there was an excess of oxygen, was stronger than that from a stoichiometric mixture, indicating a more efficient detonation. The blast wave from the rich mixture, for which there was a deficiency of oxygen, was weaker than that from the stoichiometric mixture, but gradually strengthened relatively, probably due to the afterburning of the undetonated propane in the presence of atmospheric oxygen. The peak overpressures as functions of distance and the overpressure time histories from the three types of explosive were compared with predictions by a Propane Blast interface and in all cases showed excellent agreement. The interface predictions were based on measurements from a nominal 20-ton propane/oxygen explosion, and the agreement with results from charges with masses of less than 1 g indicates the validity of Hopkinson’s cube root scaling applied to propane/oxygen explosions over a range of charge masses in excess of six orders of magnitude.

Surface conductivity effects during shock wave reflection off curved surfaces

Previous work has shown that heat transfer into a plane conducting surface affects shock wave reflection properties. In the more complex reflection case of a curved surface the situation would be different because of the increased number of reflection patterns, with a rapid change in reflection geometry as the wave propagates up the surface. Test pieces of different thermal conductivities (0.19 W/mK and 401 W/mK) and hydraulically smooth surfaces are used on both convex and concave cylindrical surfaces. The test pieces are placed in identical positions on either side of a plane of symmetry and at the same incident angles in a shock tube. Tests are performed at incident shock Mach numbers of 1.22< M <1.5 and imaged using high-speed shadowgraph photography. The images are analyzed both quantitatively through reflection angle measurements and qualitatively by searching for asymmetry. Both the qualitative and quantitative measurements clearly indicate that the thermal conductivity of the surface affects the reflection patterns due to the complex transient thermal environment. Asymmetry in the reflection patterns is found at all Mach numbers. In contrast to the plane wall case the lack of theory limits comparison, where in that case it was shown that the lower the thermal conductivity the closer the experimental results approached the ideal adiabatic case. Similar comparisons in the relationship of reflection types between different materials are found in the current case. The findings could be refined by using higher imaging frame rates and spatial resolution, and sophisticated numerical modeling.

The Dynamics of a High Mach Number Quasi-perpendicular Shock: MMS Observations

Abstract Shock parameters at Earth’s bow shock in rare instances can approach the Mach numbers predicted at supernova remnants. We present our analysis of a high Alfvén Mach number (M A = 27) shock utilizing multipoint measurements from the Magnetospheric Multiscale spacecraft during a crossing of Earth’s quasi-perpendicular bow shock. We find that the shock dynamics are mostly driven by reflected ions, perturbations that they generate, and nonlinear amplification of the perturbations. Our analyses show that reflected ions create modest magnetic enhancements upstream of the shock, which evolve in a nonlinear manner as they traverse the shock foot. They can transform into proto-shocks that propagate at small angles to the magnetic field and toward the bow shock. The nonstationary bow shock shows signatures of both reformation and surface ripples. Our observations indicate that although shock reformation occurs, the main shock layer never disappears. These observations are at high plasma β, a parameter regime that has not been well explored by numerical models.

Understanding the radio relic emission in the galaxy cluster MACS J0717.5+3745: Spectral analysis

Radio relics are diffuse, extended synchrotron sources that originate from shock fronts generated during cluster mergers. The massive merging galaxy cluster MACS J0717.5+3745 hosts one of the more complex relics known to date. We present upgraded Giant Metrewave Radio Telescope band 3 (300−500 MHz) and band 4 (550−850 MHz) observations. These new observations, combined with published VLA and the new LOFAR HBA data, allow us to carry out a detailed, high spatial resolution spectral analysis of the relic over a broad range of frequencies. The integrated spectrum of the relic closely follows a power law between 144 MHz and 5.5 GHz with a mean spectral slope α = −1.16 ± 0.03. Despite the complex morphology of this relic, its subregions and the other isolated filaments also follow power-law behaviors, and show similar spectral slopes. Assuming diffusive shock acceleration, we estimated a dominant Mach number of ∼3.7 for the shocks that make up the relic. A comparison with recent numerical simulations suggests that in the case of radio relics, the slopes of the integrated radio spectra are determined by the Mach number of the accelerating shock, with α nearly constant, namely between −1.13 and −1.17, for Mach numbers 3.5 − 4.0. The spectral shapes inferred from spatially resolved regions show curvature, we speculate that the relic is inclined along the line of sight. The locus of points in the simulated color-color plots changes significantly with the relic viewing angle. We conclude that projection effects and inhomogeneities in the shock Mach number dominate the observed spectral properties of the relic in this complex system. Based on the new observations we raise the possibility that the relic and a narrow-angle-tailed radio galaxy are two different structures projected along the same line of sight.

2D single-mode Richtmyer–Meshkov instability

We study the evolution of the single-mode Richtmyer–Meshkov instability for a wide range of Atwood numbers, shock strengths and perturbation amplitudes using Youngs’ hydrodynamical simulation code TURMOIL. We compare our results to previously published analytic models for the impulsively-driven growth rate, and propose a modification to them to treat the reduction of growth found at high initial perturbation amplitudes and high Mach numbers. It is known that the overall asymmetry between bubbles and spikes and their eventual deceleration can be interpreted as the result of nonlinear coupling to higher harmonic modes. However, we find that for light-to-heavy interfaces at moderate to high impinging shock Mach numbers, the shape of the growing bubbles varies in time, with the initial curved bubble surfaces flattening and inverting to generate a second low velocity jet. For high shock Mach numbers and low initial surface amplitudes, the process of inversion can recur on numerous occasions. We interpret this as being the result of vorticity deposited by the transmitted shock in the bulk of the heavy material, away from the initial interface.

Insights into the shockwave attenuation in miniature shock tubes

Miniature shock tubes are finding growing importance in a variety of interdisciplinary applications. There is a lack of experimental data to validate the existing shock tube flow models that explain the shockwave attenuation in pressure-driven miniature shock tubes. This paper gives insights into the shock formation and shock propagation phenomena in miniature shock tubes of 2, 6 and 10 mm square cross-sections operated at diaphragm rupture pressure ratios in the range 5–25 and driven section initially at ambient conditions. Pressure measurements and visualization studies are carried out in a new miniature table-top shock tube system using nitrogen and helium as driver gases. The experimental findings are validated using a shock tube model explained in terms of two regions: (i) the shock formation region, dominated by wave interactions due to the diaphragm's finite rupture time; and (ii) the shock propagation region, where the shockwave attenuation occurs mainly due to wall effects and boundary layer growth. Correlations to predict the variation of shock Mach number in the shock formation region and shock propagation region work well for the present findings and experimental data reported in the literature. Similar flow features are observed in the shock tubes at the same dimensionless time stamps. The formation of the planar shock front scales proportionally with the diameter of the shock tube. The peak Mach number attained by the shockwave is higher as the shock tube diameter increases.

Three-dimensional simulations of X-ray cavities inflated by radio galaxies

ABSTRACT Vast cavities in the intergalactic medium are excavated by radio galaxies. The cavities appear as such in X-ray images because the external medium has been swept-up, leaving a hot but low-density bubble surrounding the radio lobes. We explore here the predicted thermal X-ray emission from a large set of high-resolution three-dimensional simulations of radio galaxies driven by supersonic jets. We assume adiabatic non-relativistic hydrodynamics with injected straight and precessing jets of supersonic gas emitted from nozzles. Images of X-ray bremsstrahlung emission tend to generate oval cavities in the soft keV bands and leading arcuate structures in hard X-rays. However, the cavity shape is sensitive to the jet-ambient density contrast, varying from concave-shaped at η = 0.1 to convex for η = 0.0001 where η is the jet/ambient density ratio. We find lateral ribs in the soft X-rays in certain cases and propose this as an explanation for those detected in the vicinity of Cygnus A. In bi-lobed or X-shaped sources and in curved or deflected jets, the strongest X-ray emission is not associated with the hotspot but with the relic lobe or deflection location. This is because the hot high-pressure and dense high-compression regions do not coincide. Directed towards the observer, the cavity becomes a deep round hole surrounded by circular ripples. With short radio-mode outbursts with a duty cycle of 10 per cent, the intracluster medium simmers with low Mach number shocks widely dissipating the jet energy in between active jet episodes.

Детонация горючей газовой смеси при взаимодействии ударной волны с эллиптической областью тяжелого инертного газа

On the basis of the Euler equations, the interaction of a shock wave in a combustible gas with an elliptical bubble of an inert gas of increased density is numerically simulated within a plane two-dimensional formulation. The finite-volume Godunov-type method of the second order of approximation is applied. Gas combustion is modeled using the Korobeinikov-Levin two-stage kinetics. Various values of the Mach number of the incident wave and the elongation of the inert bubble are considered, and the refraction and focusing of the incident shock are described. Qualitatively different regimes of gas detonation initiation have been found, including direct initiation by a strong wave, ignition upon reflection of an average-intensity wave from the gas interface, and upon focusing of secondary shock waves at lower shock Mach numbers. The dependence of the ignition mode on the shock intensity and the shape of the bubble is determined.

Strong shock as a stringent test for Onsager-Burnett equations.

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)2470-004510.1103/PhysRevE.96.013106] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

Effect of wavy wall strut fuel injector on shock wave development and mixing enhancement of fuel and air for a scramjet combustor

Abstract The shear mixing and streamline vortices are the notable parameters to influence the air–fuel mixing in hypersonic flows. The shock wave development and Mach number significantly influence the shear mixing phenomenon. Hence, this research introduced an unconventional strut and tested its performance for the generation of shock waves at different flow conditions (M = 2,4,6). The Reynolds-averaged Navier–Stokes equations are solved to evaluate the performance of the new strut. Both the DLR scramjet strut injector and wavy wall strut injector are assessed for the shear mixing development. Turbulence for the association of shock waves, mixing layer, and the boundary layer has been modeled with the SST k-ω model. The variation in shock development and its interactions are investigated further with an increase in Mach number. The scramjet flow structure differentiation found the increased number of oblique shock waves with the wavy wall strut fuel injector. It increases the turbulence level with increased streamline vortices, turbulent intensity, and turbulent kinetic energy. The shock wave generation analysis at different Mach numbers (M = 2,4,6) found fewer interactions between the shock wave and shear layer with increased Mach number. From the examination of shock wave generation and its interaction with the shear layer and analysis of turbulent parameters, it is found that the wavy wall strut has an appreciable effect on shock-induced blend augmentation of fuel and air.

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.

Experimental Analysis of Acoustic Performance of Porous Ducts using a Small- scale Shock tube

A Shock tube is a device often used to recreate the shock phenomena by several methods. The present work is a preliminary study to analyse the efficiency of baffles in suppressing the noise generated from the shock associated jet ejected from the duct exit. The experiment was performed at a shock Mach number of 1.27 using a diaphragm based, small scale, open ended shock tube. Parametric study conducted by varying the baffle dimensions such as outer diameter, hole diameter and porosity reveals that the baffles with hole diameter less than half of the baffle diameter and with a greater porosity is a cost effective and efficient noise suppressor for shock associated jets.