Shock Mach Number Research Articles

Numerical Simulations of Shock-driven Heavy Fluid Layer

The present study presents the numerical simulations for a shocked-heavy fluid layer with a stratified N2/SF6/N2 configuration. Simulations were conducted using a third-order modal discontinuous Galerkin method to solve the compressible two-component Euler equations. The results were validated against experimental data, confirming the accuracy of the computational approach. Dynamics of the heavy fluid layer were found to be strongly influenced by the shock Mach numbers Ms = 1.15, 1.25, 1.5. At a lower Mach number Ms = 1.15, the interface deformations remained smooth and relatively symmetric, with limited vorticity generation and weak perturbations. Baroclinic effects at this stage were minimal, and the instability growth remained linear. As the Mach number increased to Ms = 1.25, the interaction became nonlinear, leading to the formation of small-scaled vortex structures driven by moderate baroclinic effects. Interface mixing intensified as rotational motion increased. At the highest Mach number Ms = 1.5, the interface rapidly evolved into chaotic structures, characterized by significant vorticity amplification, vortex roll-up, and the onset of turbulence. The baroclinic vorticity, resulting from the misalignment of pressure and density gradients, dominated the vorticity production mechanism, particularly at higher Mach numbers. Quantitative analysis demonstrated that average vorticity, baroclinic vorticity, and enstrophy grew rapidly with increasing Mach numbers. Enstrophy, which quantifies turbulence intensity, exhibited pronounced growth at Ms = 1.5, marking the transition to turbulent mixing.

The Effect of Contraction–Expansion Nozzle on High-Temperature Shock Tube Flow

To achieve higher enthalpy and pressure, the technique of variable cross-section drive is effectively combined with the heating of light gas to enhance the intensity of the incident shock wave. A study was conducted to predict the impact of variable cross-sections on the performance of high-temperature shock tube flow using a shock tube with a 2.6:1 diameter ratio between the driver and driven sections. The driver section was filled with a helium–argon gas mixture (mass ratio of 1:9), while the driven section contained dry air. Under total pressure conditions of 14.5 MPa and total temperature of 3404 K, as well as total pressure of 45 MPa and total temperature of 4845 K in the driver section, corresponding to driven section pressures of 10 kPa and 80 kPa, the results of chemical non-equilibrium numerical simulations were compared to experimental measurements of the incident shock Mach number and total pressure. The results indicated the following: First, after adding the contraction–expansion nozzle, the incident shock accelerated through the contraction section and reflected within the contraction section. Strong oscillations occurred during the flow, with increasing intensity as the throat size decreased. Second, without the nozzle, the shock velocity increased and then decreased. However, with the nozzle, the Mach number was highest near the nozzle exit and gradually decreased thereafter. Third, the presence of the nozzle led to the formation of a distinct fan-shaped wavefront, accompanied by significant variations in flow variables such as pressure, temperature, and Mach number in the region. This phenomenon was attributed to the interaction between the shock wave and the nozzle geometry, which altered the flow dynamics. Finally, as the throat size decreased, the intensity of the incident shock also decreased. After reflecting at the end of the shock tube, the total pressure in the driven section also decreased. The numerical simulations employed a multi-component, multi-temperature chemical non-equilibrium model, validated against experimental data, to accurately capture the complex flow behavior and wave interactions within the shock tube.

A uGMRT and MeerKAT Study of Radio Relics in the Low-mass Merging Cluster PSZ2 G200.95−28.16

Abstract Diffuse radio sources known as radio relics are direct tracers of shocks in the outskirts of merging galaxy clusters. PSZ2 G200.95–28.16, a low-mass merging cluster (M 500 = (2.7 ± 0.2) × 1014 M ⊙), features a prominent radio relic, first identified by R. Kale et al. We name this relic as the Seahorse. The MeerKAT Galaxy Cluster Legacy Survey has confirmed two additional radio relics, R2 and R3, in this cluster. We present new observations of this cluster with the upgraded Giant Metrewave Radio Telescope (uGMRT) at 400 and 650 MHz, paired with Chandra X-ray data. The largest linear sizes for the three relics are 1.53 Mpc, 1.12 kpc, and 340 kpc. All three radio relics are polarized at 1283 MHz. Assuming the diffusive shock acceleration model, the spectral indices of the relics imply shock Mach Numbers of 3.1 ± 0.8 and 2.8 ± 0.9 for the Seahorse and R2, respectively. The Chandra X-ray surface brightness map shows two prominent subclusters, but the relics are not perpendicular to the likely merger axis, as typically observed; no shocks are detected at the locations of the relics. We discuss possible merger scenarios in light of the low mass of the cluster and the radio and X-ray properties of the relics. The relic R2 follows the correlation known in the radio relic power and cluster-mass plane, but the Seahorse and R3 relics are outliers. We have also discovered a radio ring in our 650 MHz uGMRT image that could be an Odd Radio Circle candidate.

Predicting non-ideal effects from the diaphragm opening process in shock tubes

Shock tubes are instrumental in studying high-temperature kinetics and simulating high-speed flows. They rapidly increase the thermodynamic conditions of test gases, making them ideal for examining chemical reactions and generating high-enthalpy flows for aerodynamic research. However, non-ideal effects, stemming from factors like diaphragm opening processes and viscous effects, can significantly influence thermodynamic conditions behind the shock wave. This study investigates the impact of various diaphragm opening patterns on the shock parameters near the driven section's end wall. Experiments were conducted using helium and argon as driver and driven gases, respectively, at pressures ranging from 1.32 to 2.09 bar and temperatures from 1073 to 2126 K behind the reflected shock. High-speed imaging captured different diaphragm rupture profiles, classified into four distinct types based on their dynamics. Results indicate that the initial stages of diaphragm opening, including the rate and profile of opening, play crucial roles in the resulting incident shock Mach number and test time. A sigmoid function was employed to fit the diaphragm opening profiles, allowing for accurate categorization and analysis. New correlations were developed to predict the incident shock attenuation rate and post-shock pressure rise, incorporating parameters such as diaphragm opening time, rupture profile constants, and normalized experimental Mach number. The results emphasize the importance of considering diaphragm rupture dynamics in shock tube experiments to achieve accurate predictions of shock parameters.

Computational investigation of moving shock interaction with a granular particle curtain using a coupled Eulerian–Lagrangian approach

The objective of this research is to investigate the influence of a moving shock wave on a dense particle curtain using a coupled supersonic Eulerian–Lagrangian solver. The numerical investigation seeks to gain a deeper understanding of the interactions between particles and gas flow through a four-way coupling approach that incorporates interparticle collisions. Upon interaction, the particle curtain undergoes rapid expansion, generating transmitted and reflected shocks. This work focuses on how different particle arrangements affect the fluid flow dynamics and vice versa. By systematically varying the shock Mach number, we elucidate the effects of these interactions across a range of shock strengths, providing a comprehensive picture of the phenomenon. Our findings reveal the crucial role of particle properties and interparticle forces in shaping the dynamics of shock–granular interactions, paving the way for a deeper understanding of multiphase shock flows in various applications.

X-ray cavities in TNG-Cluster: AGN phenomena in the full cosmological context

ABSTRACT Active galactic nuclei (AGNs) feedback from supermassive black holes (SMBHs) at the centres of galaxy clusters plays a key role in regulating star formation and shaping the intracluster medium, often manifesting through prominent X-ray cavities embedded in the cluster’s hot atmosphere. Here we show that X-ray cavities arise naturally due to AGN feedback in TNG-Cluster. This is a new suite of magnetohydrodynamic cosmological simulations of galaxy formation and evolution, and hence of galaxy clusters, whereby cold dark matter, baryon dynamics, galactic astrophysics, and magnetic fields are evolved together consistently. We construct mock Chandra X-ray observations of the central regions of the 352 simulated clusters at z = 0 and find that $\sim$39 per cent contain X-ray cavities. Identified X-ray cavities vary in configuration with some still attached to their SMBH, while others have buoyantly risen. Their size ranges from a few to several tens of kpc. TNG-Cluster X-ray cavities are underdense compared to the surrounding halo and filled with hot gas ($\sim 10^8$ K); 25 per cent of them are surrounded by an X-ray bright and compressed rim associated with a weak shock (Mach number $\sim$1.5). Clusters exhibiting X-ray cavities are preferentially strong or weak cool-cores, are dynamically relaxed, and host SMBHs accreting at low Eddington rates. We show that TNG-Cluster X-ray cavities originate from episodic, wind-like energy injections from central AGN. Our results illustrate the existence and diversity of X-ray cavities simulated in state-of-the-art models within realistic cosmological environments and show that these can form without necessarily invoking bipolar, collimated, or relativistic jets.

Influence of flow nonuniformities and real gas effects on cylindrical shock wave convergence

Convergence of cylindrical shock in argon is studied both experimentally and numerically. Shock tube experiments are conducted, where a planar shock is first transformed to a cylindrical shape and then converged to its focal axis. Numerical simulations of the converging shock using equations of state for an ideal gas and a real gas (SESAME 5173 model) are conducted and compared. High temporal resolution data of cylindrical shock convergence is presented. When comparing the trajectories of the converging shock of initial shock Mach number (MS) of 4.63, the convergence exponent (α) in experiments is found to be 0.833. This α value in experiments is higher than the value obtained from computations with argon treated as an ideal gas but agrees well with the real gas computations. It is revealed that the form of convergence varies with different MS. An asymptotic approach of α toward the self-similar solution for high MS is attributed to an earlier transition of shock motion to self-similarity, while a significantly higher α observed at lower MS is attributed to the negative influence of upstream nonuniformities and weaker initiation of the shock. It is found that even before the shock reflection, real gas effects are significant enough to affect the convergence of the shock and limit the extreme conditions predicted by the ideal gas computations. For an MS of 4.63, the maximum temperature reached is 9250 K before reflection, leading to 0.12% of the argon gas undergoing the first stage of ionization.

Numerical Study of Bifurcation Structures in Reflected Shock-Wave/Laminar-Boundary-Layer Interaction Within an End-Wall Tube

Abstract The primary aim of this study is to analyze the unsteady characteristics of the interaction between a reflected shock wave and a laminar boundary layer in an end-wall shock tube. Our direct numerical simulations at shock Mach numbers of Ms = 1.9, 2.5, and 3.5 using a fifth-order WENO scheme and three-step Runge–Kutta time integration method revealed inhomogeneity and anisotropy in the shock bifurcation. Surprisingly, the upper and lower bifurcated structures maintain a notably asymmetric flow during the forward propagation of the reflected shock bifurcation. The inverse flow in the bifurcation resembles a crooked earthworm structure, exhibiting high-frequency oscillations indicative of instability. However, at higher shock intensities, the earthworm transforms into a stable strip-like configuration, facilitating the entrapment of inverse flow and leading to rapid bifurcation height growth and early convergence. Additionally, isolated islands with high density, temperature, and pressure emerge in the transitional region behind the bifurcated shocks, due to variations in wave propagation speed.

Weak collisionless shocks mediated by ion gyroviscosity.

Collisionless shocks are ubiquitous in space and astrophysical plasmas, and they are essential dynamical features of these systems. Lacking Coulomb collisions, these shocks are mediated by the anomalous dissipation provided by nonlinear plasma instabilities. By numerically resolving the structure of a steady-state, ion gyroviscous shock, we show that ion gyroviscosity, alone, can produce weak (M≲1.1, where M is the sonic Mach number) shocks in a collisionless, magnetized plasma. We emphasize that this effect does not require an appeal to plasma microturbulence. Moreover, while most collisionless systems may be unsuitable to support purely gyroviscous shocks, we argue that gyro-viscous heating may be an overlooked mechanism, generally; and it may be a key driver within magnetohydrodynamic shocks at large. Representative examples include the plasma environments produced on the plasma liner experiment and the magnetized liner inertial fusion platforms.

Self-organization of collisionless shocks: from a laminar profile to a rippled time-dependent structure

A collisionless shock structure results from the nonlinear interaction between charged particles and electromagnetic fields. Yet, a collisionless shock is globally governed by the mass, momentum and energy conservation requirements. A stable shock structure must ensure that the fluxes of the conserved quantities are constant on average, and, therefore, is determined by this necessity. Here, we study an observed high upstream temperature high Mach number shock and show that the conservation laws cannot be fulfilled unless the shock is spatially inhomogeneous along the shock front and time-dependent.

Electron Acceleration at Quasi-parallel Nonrelativistic Shocks: A 1D Kinetic Survey

We present a survey of 1D kinetic particle-in-cell simulations of quasi-parallel nonrelativistic shocks to identify the environments favorable for electron acceleration. We explore an unprecedented range of shock speeds v sh ≈ 0.067–0.267c, Alfvén Mach numbers MA=5–40 , sonic Mach numbers Ms=5–160 , as well as the proton-to-electron mass ratios m i/m e = 16–1836. We find that high Alfvén Mach number shocks can channel a large fraction of their kinetic energy into nonthermal particles, self-sustaining magnetic turbulence and acceleration to larger and larger energies. The fraction of injected particles is ≲0.5% for electrons and ≈1% for protons, and the corresponding energy efficiencies are ≲2% and ≈10%, respectively. The extent of the nonthermal tail is sensitive to the Alfvén Mach number; when MA≲10 , the nonthermal electron distribution exhibits minimal growth beyond the average momentum of the downstream thermal protons, independently of the proton-to-electron mass ratio. Acceleration is slow for shocks with low sonic Mach numbers, yet nonthermal electrons still achieve momenta exceeding the downstream thermal proton momentum when the shock Alfvén Mach number is large enough. We provide simulation-based parameterizations of the transition from thermal to nonthermal distribution in the downstream (found at a momentum around pi,e/mivsh≈3mi,e/mi ), as well as the ratio of nonthermal electron to proton number density. The results are applicable to many different environments and are important for modeling shock-powered nonthermal radiation.

Computational Study of Shocked V-Shaped N2/SF6 Interface across Varying Mach Numbers

The Mach number effect on the Richtmyer–Meshkov instability (RMI) evolution of the shocked V-shaped N2/SF6 interface is numerically studied in this research. Four distinct Mach numbers are taken into consideration for this purpose: Ms=1.12,1.22,1.42, and 1.62. A two-dimensional space of compressible two-component Euler equations is simulated using a high-order modal discontinuous Galerkin approach to computational simulations. The numerical results show good consistency when compared to the available experimental data. The computational results show that the RMI evolution in the shocked V-shaped N2/SF6 interface is critically dependent on the Mach number. The flow field, interface deformation, intricate wave patterns, inward jet development, and vorticity generation are all strongly impacted by the shock Mach number. As the Mach number increases, the V-shaped interface deforms differently, and the distance between the Mach stem and the triple points varies depending on the Mach number. Compared to lower Mach numbers, higher ones produce larger rolled-up vortex chains. A thorough analysis of the Mach number effect identifies the factors that propel the creation of vorticity during the interaction phase. Moreover, kinetic energy and enstrophy both dramatically rise with increasing Mach number. Lastly, a detailed analysis is carried out to determine how the Mach number affects the temporal variations in the V-shaped interface’s features.

Shock-driven three-fluid mixing with various chevron interface configurations

When a shock wave crosses a density interface, the Richtmyer–Meshkov instability causes perturbations to grow. Richtmyer–Meshkov instabilities arise from the deposition of vorticity from the misaligned density and pressure gradients at the shock front. In many engineering applications, microscopic surface roughness will grow into multi-mode perturbations, inducing mixing between the fluid on either side of an initial interface. Applications often have multiple interfaces, some of which are close enough to interact in the later stages of instability growth. In this study, we numerically investigate the mixing of a three-layer system with periodic zigzag (or chevron) interfaces, calculating the dependence of the width and mass of mixed material on properties such as the shock timing, chevron amplitude, multi-mode perturbation spectrum, density ratio, and shock mach number. The multi-mode case is also compared with a single-mode perturbation. The Flash hydrodynamic code is used to solve the Euler equations in three dimensions with adaptive grid refinement. Key results include a significant increase in mixed mass when changing from a single-mode to a multi-mode perturbation on one of the interfaces. The mixed width is mainly sensitive to the density ratio and chevron amplitude, whereas the mixed mass also depends on the multi-mode spectrum. Steeper initial perturbation spectra have lower mixed mass at early times but a greater mixed mass after the reflected shock transits back across the layer.

Comprehensive analysis of normal shock wave propagation in turbulent non-ideal gas flows with analytical and neural network methods

Shock wave propagation in gases through turbulent flow has wide-reaching implications for both theoretical research and practical applications, including aerospace engineering, propulsion systems, and industrial gas processes. The study of normal shock propagation in turbulent flow over non-ideal gas investigates the changes in pressure, density, and flow velocity across the shock wave. The Mach number is derived for the system and explored across various gas molecule quantities and turbulence intensities. This study analytically investigated the normal shock wave propagation in turbulent flow of adiabatic gases with modified Rankine–Hugoniot conditions. Artificial neural network (ANN) techniques are used to estimate the solutions for shock strength and Mach number training validation phases of back-propagated neural networks with the Levenberg–Marquardt algorithm. The results reveal that pressure ratio with density ratio increase for higher values of increase in the turbulence level as well as intermolecular forces. A reverse trend is observed in velocity coefficient after shock in the presence of adiabatic gas. The regression coefficient values obtained using the network model ranged from 0.999 99 to 1, indicating an almost perfect correlation. These findings demonstrate that the ANN can predict the Mach number with high accuracy.

Numerical investigation of the reflection of unsteady decelerating incident shock waves

The incident shock attenuation phenomenon in shock tube has received widespread interest because of its inevitable influence on experimental gas properties. However, few studies have investigated the cascading effects of the resulting nonuniformity on shock reflection in shock tunnels before the nozzle. This paper describes a numerical study on the unsteady reflection of a decelerating incident shock driven by a decelerating piston, as a simplification of the complex nonideal factors. The initial decay and uniform parameter distributions are generated based on the non-inertial frame. The results indicate that the existence of the nonuniform area forces the reflected shock wave region to undergo a transition process of attenuation and then stability. The final values of the gas parameters in zone 5 will, therefore, deviate from those given by the traditional relation for an ideal shock tube, which are only dependent on the terminal shock Mach number. This affects the determination of the total temperature T5, which is difficult to measure directly. We discuss the reconstruction of the nonuniform region with the attenuation trajectory of the incident shock wave and find that there is no one-to-one correspondence between this trajectory and the resulting nonuniformity, which introduces additional uncertainties to the predictions. Thus, an analogy method is developed through the multilevel block building algorithm, allowing the total temperature T5 to be determined using the total pressure p5 considering the effect of nonuniformity. Further assessments verify the applicability of this method in cases with multidimensional and viscous interactions.

Analyzing Richtmyer–Meshkov Phenomena Triggered by Forward-Triangular Light Gas Bubbles: A Numerical Perspective

In this paper, we present a numerical investigation into elucidating the complex dynamics of Richtmyer–Meshkov (RM) phenomena initiated by the interaction of shock waves with forward-triangular light gas bubbles. The triangular bubble is filled with neon, helium, or hydrogen gas, and is surrounded by nitrogen gas. Three different shock Mach numbers are considered: Ms=1.12,1.21, and 1.41. For the numerical simulations, a two-dimensional system of compressible Euler equations for two-component gas flows is solved by utilizing the high-fidelity explicit modal discontinuous Galerkin technique. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The numerical model explores the impact of the Atwood number on the underlying mechanisms of the shock-induced forward-triangle bubble, encompassing aspects such as flow evolution, wave characteristics, jet formation, generation of vorticity, interface features, and integral diagnostics. Furthermore, the impacts of shock strengths and positive Atwood numbers on the flow evolution are also analyzed. Insights gained from this numerical perspective enhance our understanding of RM phenomena triggered by forward-triangular light gas bubbles, with implications for diverse applications in engineering, astrophysics, and fusion research.

Hypersonic inlet flow field reconstruction dominated by shock wave and boundary layer based on small sample physics-informed neural networks

High Mach number inlets face complex challenges such as shock wave/boundary layer interference, adversely impacting the aerodynamic characteristics and stable operating boundaries. Traditional computational fluid dynamics (CFD) simulations for performance and flow field analysis are slow, and data-driven deep learning technologies struggle with predicting flow fields in complex aerodynamic phenomena like shock wave/boundary layer interactions, boundary layer separation, and Mach disks. This study introduces a rapid, robust method for reconstructing hypersonic viscous inlet flow fields using inlet geometry design parameters. This method is called shock wave and boundary layer dominated two-dimensional multiphysical field reconstruction model based on multi-scale sensory field fusion residual physical information neural network (PIMSRF_ResNet), ensuring high-precision training data. Using the optimal Pareto solution set from multi-objective optimization with a small sample intelligence method, 150 sets of multi-physics fields under Ma10 conditions with varying geometric design parameters are calculated. Compared to traditional pure data-driven models, PIMSRF_ResNet's structural similarity in the test set reaches 0.9773, with a peak signal-to-noise ratio close to 30 dB and a correlation coefficient exceeding 98%. These experimental results demonstrated that the proposed method is a promising tool for real-time prediction of complex internal flow fields dominated by high Mach number shock waves and boundary layers.

Quasi-perpendicular shocks of galaxy clusters in hybrid kinetic simulations

Context. Cosmic ray acceleration in galaxy clusters is still an ongoing puzzle, with relativistic electrons forming radio relics at merger shocks and emitting synchrotron radiation. These shocks are also potential sources of ultra-high-energy cosmic rays, gamma rays, and neutrinos. Our recent work focuses on electron acceleration at low Mach number merger shocks in the hot intracluster medium which is characterized by high plasma beta. Using particle-in-cell (PIC) simulations, we previously showed that electrons are energized through the stochastic shock-drift acceleration process, which is facilitated by multi-scale turbulence, including ion-scale shock surface rippling. For the present work, we performed hybrid-kinetic simulations in a range of various quasi-perpendicular foreshock conditions, including plasma beta, magnetic obliquity, and the shock Mach number. Aims. We study the ion kinetic physics, which is responsible for the shock structure and wave turbulence, that in turn affects the particle acceleration processes. We cover the spatial and temporal scales, which allow the development of large-scale ion turbulence modes in the system. Methods. We applied a recently developed generalized fluid-particle hybrid numerical code that can combine fluid modeling for both electrons and ions with an arbitrary number of kinetic species. We limited this model to a standard hybrid simulation configuration with kinetic ions and fluid electrons. The model utilizes the exact form of the generalized Ohm’s law, allowing for an arbitrary choice of mass and energy densities, as well as the charge-to-mass ratio of the kinetic species. Results. We show that the properties of ion-driven multi-scale magnetic turbulence in merger shocks are in agreement with the ion structures observed in PIC simulations. In typical shocks with the sonic Mach number Ms = 3, the magnetic structures and shock front density ripples grow and saturate at wavelengths reaching approximately four ion Larmor radii. Only shocks with Ms ≳ 2.3 develop ripples. At very weak shocks with Ms ≲ 2.3, weak turbulence is formed downstream of the shock. We observed a moderate dependence of the strength of magnetic field fluctuations on the quasi-perpendicular magnetic field obliquity. However, as the field obliquity decreases, the shock front ripples exhibit longer wavelengths. Finally, we note that the steady-state structure of Ms = 3 shocks in high-beta plasmas shows evidence that there is little difference between 2D and 3D simulations. The turbulence near the shock front seems to be a 2D-like structure in 3D simulations.

Numerical simulation of the interaction between a planar shock wave and a backward-facing triangular bubble containing gases with different Atwood numbers

The interaction between a shock wave and an interface delineating two gases engenders intricate flow physics, with particular attention drawn to the hydrodynamic instability due to its practical significance. Previous studies have primarily focused on elucidating different wave patterns and instabilities evolution at the interface during the initial phase of shock interaction with cylindrical or spherical bubbles. However, scant literature has shifted its focus toward exploring the long-term morphology of bubbles, especially those characterized by polygonal interfaces. Notably, the detailed examination of shock interaction with a polygonal interface, such as a triangular one with a constant incident angle, remains largely unexplored in existing literature. Recently, the longtime evolution of detailed flow structures across the interface of shock-forward-facing triangular bubbles was captured by Kundu et al. [“A study on dynamics of shock-accelerated forward-facing triangular bubbles at different Atwood numbers,” Phys. Fluids 36, 016110 (2024)] through numerical simulation. In this study, the dynamics of a shock-accelerated backward-facing triangular interface containing various gases, namely, Sulfur Hexafluoride, Refrigerant-22, Argon, Neon, and Helium, is studied for a shock Mach number of 1.21. Simulations were performed by solving the two-dimensional Euler equation using low-dissipative advection upwind splitting methods (AUSMD), in conjunction with a derived ninth-order upwind scheme and a four-stage third-order Runge–Kutta scheme for temporal integration. The development of Richtmyer–Meshkov (RM) and Kelvin–Helmholtz (K–H) instabilities at the interface, mixing, and normalized movements of backward-facing triangular bubbles is captured at different Atwood (At) numbers.

Formation and propagation characteristics of a weak shock wave in maglev tube

The propagation of the weak shock wave (WSW) to the tunnel exits and their radiation as micro-pressure waves (MPWs) may cause sonic booms or structural resonance of buildings, posing potential hazards to humans, animals, and buildings in the exit's environment. The characteristics of the WSW and sonic booms of a maglev train/tube coupling model were studied based on the two-dimensional axisymmetric unsteady Reynolds average Navier–Stokes turbulence model. In the later stage of a MPW, the formation mechanism, geometry, and kinematic characteristics of compressible vortex rings (CVRs) were systematically analyzed. The inertial effect causes the initial wavefront to gradually transition from a Gaussian-shape waveform to a triangular waveform during its propagation, eventually coalescing into a WSW. The overpressure, density jump, and shock Mach number at the WSW location all increase with the increasing train speed, while the WSW thickness decreases accordingly. The formation distance of the WSW is inversely proportional to the amplitude of the initial wavefront gradient, and the WSW directly causes the occurrence of the exit sonic boom. The MPW amplitude has significant directionality with a largest value in the axial direction. Within the speed range of 450–700 km/h, the sound pressure level of the MPW exceeds the hearing threshold and even reaches the feeling threshold. The evolution of CVRs includes primary CVR, secondary CVR, and Kelvin–Helmholtz vortices. Primary CVR has the greatest impact on the axial MPW among them. The occurrence of CVRs will cause a second small noise level other than the sonic boom.