Shock Mach Number Research Articles

Deep Chandra observations of merging galaxy cluster ZwCl 2341+0000

Context. Knowledge of X-ray shock and radio relic connection in merging galaxy clusters has been greatly extended in terms of both observation and theory over the last decade. ZwCl 2341+0000 is a double-relic merging galaxy cluster; previous studies have shown that half of the southern relic is associated with an X-ray surface brightness discontinuity, while the other half not. The discontinuity was believed to be a shock front. Therefore, it is a mysterious case of an only partial shock-relic connection. Aims. By using the 206.5 ks deep Chandra observations, we aim to investigate the nature of the southern surface brightness discontinuity. Meanwhile, we aim to explore new morphological and thermodynamical features. Methods. We perform both imaging and spectroscopic analyses to investigate the morphological and thermodynamical properties of the cluster. In addition to the X-ray data, we utilize the GMRT 325 MHz image and JVLA 1.5 GHz and 3.0 GHz images to compute radio spectral index maps. Results. Surface brightness profile fitting and the temperature profile suggest that the previously reported southern surface brightness discontinuity is better described as a sharp change in slope or as a kink. This kink is likely contributed by the disrupted core of the southern subcluster. The radio spectral index maps show spectral flattening at the south-eastern edge of the southern relic, suggesting that the location of the shock front is 640 kpc away from the kink, where the X-ray emission is too faint to detect a surface brightness discontinuity. We update the radio shock Mach number to be ℳradio, S = 2.2 ± 0.1 and ℳradio, N = 2.4 ± 0.4 for the southern and northern radio relics based on the injection spectral indices. We also put a 3σ lower limit on the X-ray Mach number of the southern shock to be ℳX-ray, S > 1.6. Meanwhile, the deep observations reveal that the northern subcluster is in a perfect cone shape, with a ∼400 kpc linear cold front on each side. This type of conic subcluster has been predicted by simulations but is observed here for the first time. It represents a transition stage between a blunt-body cold front and a slingshot cold front. Strikingly, we found a 400 kpc long gas trail attached to the apex of the cone, which could be due to the gas stripping. In addition, an over-pressured hot region is found in the south-western flank of the cluster.

A 3.5 Mpc long radio relic in the galaxy cluster ClG 0217+70

Context. Megaparsec-scale radio sources in the form of halos and relics are often detected in dynamically disturbed galaxy clusters. Although they are associated with merger-induced turbulence and shocks, respectively, their formation is not fully understood. Aims. We aim to identify the mechanisms responsible for particle acceleration and magnetic field amplification in the halo and relics of the galaxy cluster ClG 0217+70. Methods. We observed ClG 0217+70 with LOFAR at 141 MHz and with VLA at 1.5 GHz, and we combined these observations with VLA 1.4 GHz archival data to study the morphological and spectral properties of the diffuse sources. We added Chandra archival data to examine the thermal and non-thermal properties of the halo. Results. Our LOFAR and VLA data confirm the presence of a giant radio halo in the cluster centre and multiple relics in the outskirts. The radio and X-ray emission from the halo are correlated, implying a tight relation between the thermal and non-thermal components. The diffuse radio structure in the south-east, with a projected size of 3.5 Mpc, is the most extended radio relic detected to date. The spectral index across the relic width steepens towards the cluster centre, suggesting electron ageing in the post-shock regions. The shock Mach numbers for the relics derived from the spectral index map range between 2.0 and 3.2. However, the integrated spectral indices lead to increasingly high Mach numbers for the relics farther from the cluster centre. This discrepancy could be because the relation between injection and integrated spectra does not hold for distant shocks, suggesting that the cooling time for the radio-emitting electrons is longer than the crossing time of the shocks. The variations in the surface brightness of the relics and the low Mach numbers imply that the radio-emitting electrons are re-accelerated from fossil plasma that could originate in active galactic nuclei.

On mixing enhancement by secondary baroclinic vorticity in a shock–bubble interaction

To investigate the intrinsic mechanism for mixing enhancement by variable-density (VD) behaviour, a canonical VD mixing extracted from a supersonic streamwise vortex protocol, a shock–bubble interaction (SBI), is numerically studied and compared with a counterpart of passive-scalar (PS) mixing. It is meaningful to observe that the maximum concentration decays much faster in a VD SBI than in a PS SBI regardless of the shock Mach number ( $Ma=1.22 - 4$ ). The quasi-Lamb–Oseen-type velocity distribution in the PS SBI is found by analysing the azimuthal velocity that stretches the bubble. Meanwhile, for the VD SBI, an additional stretching enhanced by the secondary baroclinic vorticity (SBV) production contributes to the faster-mixing decay. The underlying mechanism of the SBV-enhanced stretching is further revealed through the density and velocity difference between the light shocked bubble and the heavy ambient air. By combining the SBV-accelerated stretching model and the initial shock compression, a novel mixing time estimation for VD SBI is theoretically proposed by solving the advection–diffusion equation under a deformation field of an axisymmetric vortex with the additional SBV-induced azimuthal velocity. Based on the mixing time model, a mixing enhancement number, defined by the ratio of VD and PS mixing time further, reveals the contribution from the VD effect, which implies a better control of the density distribution for mixing enhancement in a supersonic streamwise vortex.

High-Speed X-Ray Stereo Digital Image Correlation in a Shock Tube

X-ray stereo digital image correlation (DIC) measurements were performed at 10 kHz on the internal surface of a jointed structure in a shock tube at a shock Mach number of 1.42 and compared with optical stereo DIC measurements on the outer, visible surface of the structure. The shock tube environment introduces temperature and density gradients in the gas through which the structure was imaged, resulting in spatial and temporal index of refraction variations. These variations cause bias errors in optical DIC measurements due to beam-steering but have minimal influence on x-ray DIC measurements. These results demonstrate the utility of time-resolved x-ray DIC measurements in complicated environments where optical measurements suffer severe errors and/or are precluded by lack of optical access.

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Ji et al. (2020) [40]), in this paper a higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averaged flow variables and their cell-averaged derivatives, the nonlinear high-order compact reconstruction can be obtained for these variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination of non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in GKS is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function provides the Navier-Stokes (NS) flux function directly without separating the inviscid and viscous terms, which simplifies the numerical discretization of the NS equations directly on unstructured mesh. Based on the time-accurate flux solver, the two-stage fourth-order time discretization can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same order Runge-Kutta time stepping method. The current high-order GKS can uniformly capture acoustic and shock waves without identifying troubled cells and implementing additional limiting procedure. In addition, the fourth- up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme in most flow problems. The fourth-order GKS on unstructured mesh will be used in the computations from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems have advantages in the development of high-order schemes.

Nonthermal Particle Acceleration at Highly Oblique Nonrelativistic Shocks

Abstract The nonthermal acceleration of electrons and ions at an oblique, nonrelativistic shock is studied using large-scale particle-in-cell simulations in one spatial dimension. Physical parameters are selected to highlight the role of electron preheating in injection and subsequent acceleration of particles at high Mach number shocks. Simulation results show evidence for the early onset of the diffusive shock acceleration process for both electrons and ions at a highly oblique subluminal shock. Ion acceleration efficiencies of ≲5% are measured at late times, though this is not the saturated value.

Flow Nonuniformities Behind Accelerating and Decelerating Shock Waves in Shock Tubes

Shock tubes are used to investigate the chemical kinetics and radiative properties of gasses. Analyses of the flow conditions produced in shock tubes typically assume a single, constant shock speed even though the shock speed varies due to driver effects, boundary-layer growth, and diaphragm rupture effects. This paper will investigate the dependence of flow properties upon the entire shock speed history for a 100-mm-diam, 8-m-long shock tube using the FROSST axisymmetric shock tube Navier–Stokes code. A perfect gas is applied to remove the complexity of thermochemistry and allow a focus upon the flow details. Shock speed profiles ranging from accelerating to decelerating are simulated to qualitatively represent different experimental conditions, and the resulting test slugs compared. All shocks reach the test section at the same final shock Mach number of 6.53. Pressure variations caused by the accelerating shock are found to combine with entropy gradients introduced by the changing shock strength, yielding differences in maximum temperature of the test slug by as much as 56%, and differences in maximum pressure as high as 22% despite the shared tube-end Mach number. Local temperature and pressure around gas packets in the test slug are found to relax significantly through time in response to changing postshock conditions. Test slug variations due to boundary-layer effects are found to be present, but overwhelmed by the relaxation processes under many conditions.

X-ray bubbles in the circumgalactic medium of TNG50 Milky Way- and M31-like galaxies: signposts of supermassive black hole activity

ABSTRACT The TNG50 cosmological simulation produces X-ray emitting bubbles, shells, and cavities in the circumgalactic gas above and below the stellar discs of Milky Way- and Andromeda-like galaxies with morphological features reminiscent of the eROSITA and Fermi bubbles in the Galaxy. Two-thirds of the 198 MW/M31 analogues inspected in TNG50 at z = 0 show one or more large-scale, coherent features of overpressurized gas that impinge into the gaseous halo. Some of the galaxies include a succession of bubbles or shells of increasing size, ranging from a few to many tens of kpc. These are prominent in gas pressure, X-ray emission, and gas temperature, and often exhibit sharp boundaries with typical shock Mach numbers of 2–4. The gas in the bubbles outflows with maximum (95th pctl) radial velocities of ∼100–1500 km s−1. TNG50 bubbles expand with speeds as high as 1000–2000 km s−1 (about 1–2 kpc Myr−1), but with a great diversity and with larger bubbles expanding at slower speeds. The bubble gas is at 106.4−7.2 K temperatures and is enriched to metallicities of $0.5-2~\rm Z_{\odot }$. In TNG50, the bubbles are a manifestation of episodic, kinetic, wind-like energy injections from the supermassive black holes at the galaxy centres that accrete at low Eddington ratios. According to TNG50, X-ray, and possibly γ-ray, bubbles similar to those observed in the Milky Way should be a frequent feature of disc-like galaxies prior to, or on the verge of, being quenched. They should be within the grasp of eROSITA in the local Universe.

Contribution of Mach number to the evolution of the Richtmyer-Meshkov instability induced by a shock-accelerated square light bubble

The Richtmyer-Meshkov (RM) instability has long been an interesting subject due to its fundamental significance in scientific research. In the current study, the contribution of shock Mach number on the evolution of the RM instability induced by a shock-accelerated square light bubble is investigated numerically. The shock Mach number causes significant changes in flow morphology, resulting in complex wave patterns, vorticity generation, vortex formation, and bubble deformation. The Mach number effects are explored in detail through various physical phenomena such as vorticity production, kinetic energy, dissipation rate, and enstrophy.

Characteristics of shock tube generated compressible vortex rings at very high shock Mach numbers

Compressible vortex rings are usually formed at the open end of a shock tube. They show exciting flow phenomena during their formation, evolution, and propagation depending on the shock Mach number (Ms) and exit flow conditions. This study considers high shock tube pressure ratio (PR) cases showing hitherto unknown, spectacular flow structures. With hydrogen as a driver section gas at high PR, a supersonic compressible vortex ring having vortex ring Mach number (Mv) >1 is obtained for the first time. The formation of multiple triple points and the corresponding slipstream shear layers and, thus, multiple counter-rotating vortex rings (CRVRs) behind the primary vortex ring at different radial locations, in addition to the usual CRVRs, appears to be a unique characteristic for high Mach number vortex rings. During the formation stage, a vortex layer of reverse circulation than that of the primary vortex ring gets generated from the outer wall of the shock tube. The instability of such a vortex layer creates another series of opposite circulation vortices, which later interfere with the primary vortex core considerably. Also, a near stationary slipstream vortex and multiple fast-moving tiny vortices of opposite circulation to the slipstream vortex are observed near the central zone. Mechanisms for the formation of these complex vortical structures are identified. The implications of these phenomena on the vortex ring's geometric and kinematic characteristics, such as ring diameter, core diameter, circulation, and translational velocity, are discussed in detail, illustrating their differences with low vortex ring Mach number cases considering 0.31 < Mv < 1.08.

Solenoidal linear forcing for compressible, statistically steady, homogeneous isotropic turbulence with reduced turbulent Mach number oscillation

This study investigates a solenoidal linear forcing scheme with reduced oscillation of a turbulent Mach number MT for direct numerical simulations (DNS) of statistically steady, homogeneous isotropic turbulence. A conventional linear forcing scheme results in a large temporal oscillation of MT, where the maximum MT reaches about 1.1 times the time-averaged MT. Therefore, strong shocklets are generated when MT becomes large although such strong shocklets hardly appear when MT is close to the time-averaged value. DNS with the proposed forcing scheme confirms that the temporal oscillation of MT is effectively reduced by adjusting a forcing coefficient with a ratio between velocity variance and its steady state value prescribed as a parameter. The time-dependent forcing coefficient results in the variation of the power input to kinetic energy. Therefore, the temporal oscillation of the Reynolds number for this forcing scheme is as large as that for the conventional linear forcing. The ratio between the solenoidal and dilatational kinetic energy dissipation rates increases with MT, and the MT dependence is consistent between the present solenoidal linear forcing and the low-wavenumber solenoidal forcing in wavenumber space. The skewness and flatness of the velocity derivative become large compared with incompressible turbulence when MT exceeds 0.6. Both average and root-mean-squared fluctuation of the shock Mach number of shocklets increase with MT. The most typical thickness of shocklets decreases with MT and asymptotically approaches about 1.5 times the Kolmogorov scale. The shocklet thickness normalized by the Kolmogorov scale hardly depends on the Reynolds number.

Exploring the role of cosmological shock waves in the Dianoga simulations of galaxy clusters

ABSTRACT Cosmological shock waves are ubiquitous to cosmic structure formation and evolution. As a consequence, they play a major role in the energy distribution and thermalization of the intergalactic medium (IGM). We analyze the Mach number distribution in the Dianoga simulations of galaxy clusters performed with the SPH code gadget-3. The simulations include the effects of radiative cooling, star formation, metal enrichment, supernova, and active galactic nuclei feedback. A grid-based shock-finding algorithm is applied in post-processing to the outputs of the simulations. This procedure allows us to explore in detail the distribution of shocked cells and their strengths as a function of cluster mass, redshift, and baryonic physics. We also pay special attention to the connection between shock waves and the cool-core/non-cool-core (CC/NCC) state and the global dynamical status of the simulated clusters. In terms of general shock statistics, we obtain a broad agreement with previous works, with weak (low-Mach number) shocks filling most of the volume and processing most of the total thermal energy flux. As a function of cluster mass, we find that massive clusters seem more efficient in thermalizing the IGM and tend to show larger external accretion shocks than less massive systems. We do not find any relevant difference between CC and NCC clusters. However, we find a mild dependence of the radial distribution of the shock Mach number on the cluster dynamical state, with disturbed systems showing stronger shocks than regular ones throughout the cluster volume.

Numerical investigation of condensation shock and re-entrant jet dynamics around a cavitating hydrofoil using a dynamic cubic nonlinear subgrid-scale model

Cavity shedding mechanisms, such as those due to re-entrant jets (RJ) and condensation shocks (CS), are important but challenging topics in cavitating flows. This study investigates unsteady cavity shedding around a NACA66 hydrofoil using a self-defined compressible cavitation solver based on OpenFOAM with a dynamic cubic nonlinear subgrid-scale model. The predictions give satisfactory agreement with experimental data for the quantitative pressure evolution and the cavity shedding behavior induced by the re-entrant jet and the pressure wave. Moreover, the numerical data provides deeper insight into the pressure wave characteristics (e.g. propagation speed, wave intensity, shock Mach number, etc.) during cavity contraction. Multi-perspective analyses demonstrate that the pressure wave is a condensation shock which is responsible for the attached cavity abruptly disappearing. Additionally, the different influences of the re-entrant jet and the condensation shock on the local flow patterns are compared in detail in terms of the duration, average motion velocity, cavity behavior, formation mechanism and pressure intensity. The results show the condensation shock is characterized by fast propagation speed, short duration and high pressure pulse, which differs from the re-entrant jet features. This research provides an improved understanding of the cavity shedding mechanism around a cavitating hydrofoil and demonstrates the effectiveness of the current compressible cavitation solver.

Numerical Study of Air Flow Induced by Shock Impact on an Array of Perforated Plates.

This study is focused on the propagation behavior and attenuation characteristics of a planar incident shock wave when propagating through an array of perforated plates. Based on a density-based coupled explicit algorithm, combined with a third-order MUSCL scheme and the Roe averaged flux difference splitting method, the Navier–Stokes equations and the realizable k-ε turbulence model equations describing the air flow are numerically solved. The evolution of the dynamic wave and ring vortex systems is effectively captured and analyzed. The influence of incident shock Mach number, perforated-plate porosity, and plate number on the propagation and attenuation of the shock wave was studied by using pressure- and entropy-based attenuation rates. The results indicate that the reflection, diffraction, transmission, and interference behaviors of the leading shock wave and the superimposed effects due to the trailing secondary shock wave are the main reasons that cause the intensity of the leading shock wave to experience a complex process consisting of attenuation, local enhancement, attenuation, enhancement, and attenuation. The reflected shock interactions with transmitted shock induced ring vortices and jets lead to the deformation and local intensification of the shock wave. The formation of nearly steady jets following the array of perforated plates is attributed to the generation of an oscillation chamber for the inside dynamic wave system between two perforated plates. The vorticity diffusion, merging and splitting of vortex cores dissipate the wave energy. Furthermore, the leading transmitted shock wave attenuates more significantly whereas the reflected shock wave from the first plate of the array attenuates less significantly as the shock Mach number increases. The increase in the porosity weakens the suppression effects on the leading shock wave while increases the attenuation rate of the reflected shock wave. The first perforated plate in the array plays a major role in the attenuation of the shock wave.

Study on Size Distribution and Flow Characteristics of Condensed Products in Solid Rocket Motor

The size distribution of condensed products during the combustion of aluminized propellants and flow characteristics of the gas-solid two-phase flow in solid rocket motor were studied in this paper. Firstly, based on the laser scattering technology, an online detection system for condensed products in plume was established, and the size detection of condensed products in the plume of solid rocket motor is carried out. Secondly, a numerical model of two-phase flow in solid rocket motor is established by combining the real size distribution of products in the plume with discrete phase model through the Rosin-Rammler distribution function. Besides, numerical simulation research is carried out under the same experimental conditions, focusing on the influence of condensed products with real size on the characteristics of solid rocket motor. The results show that the innovation measurement system can be used to obtain the size distribution characteristic of condensed products in the plume. At the particle size of stable stage, the mean size, D v 50 , is 104 μm, which is the smallest among all stages. It is also suggested that condensed products at the end stage have the most impact on the flow behavior in solid rocket motor, in that the shock structure, Mach number, and temperature distribution in the near field of plume are significantly changed.

A multiplicity of regimes of transonic flow in channels with branching

The flow in 2D channel with a flat plate located along the plane of symmetry of the channel at a certain distance from the inlet was numerically studied. Straight segments formed the channel walls. At the inlet and the outlet of the channel, a supersonic flow was set. The first and third parts of the channel diverged. The middle part converged. Solutions to the Euler equations were obtained using the Ansys CFX finite volume solver. Four solutions were found that differed in the position of the shock waves. The hysteresis in the shock wave position versus Mach number given at the inlet was found. In a certain range of the inlet Mach number, there were asymmetric solutions of the equations.

Mach number dependence of ion-scale kinetic instability at collisionless perpendicular shock: Condition for Weibel-dominated shock

We investigate ion-scale kinetic plasma instabilities at the collisionless shock using linear theory and nonlinear particle-in-cell (PIC) simulations. We focus on the Alfvén ion cyclotron (AIC), mirror, and Weibel instabilities, which are all driven unstable by the effective temperature anisotropy induced by the shock-reflected ions within the transition layer of a strictly perpendicular shock. We conduct linear dispersion analysis with a homogeneous plasma model to mimic the shock transition layer by adopting a ring distribution with finite thermal spread to represent the velocity distribution of the reflected ions. We find that, for wave propagation parallel to the ambient magnetic field, the AIC instability at lower Alfvén Mach numbers tends to transition to the Weibel instability at higher Alfvén Mach numbers. The instability property is, however, also strongly affected by the sound Mach number. We conclude that the instability at a strong shock with Alfvén and sound Mach numbers both in excess of ∼20–40 may be considered as Weibel-like in the sense that the reflected ions behave essentially unmagnetized. Two-dimensional PIC simulations confirm the linear theory and find that, with typical parameters of young supernova remnant shocks, the ring distribution model produces magnetic fluctuations of the order of the background magnetic field, which is smaller than those observed in previous PIC simulations for Weibel-dominated shocks. This indicates that the assumption of the gyrotropic reflected ion distribution may not be adequate to quantitatively predict nonlinear behaviors of the dynamics in high Mach number shocks.

Scaling behavior of density gradient accelerated mixing rate in shock bubble interaction

Variable-density mixing in shock bubble interaction, a canonical flow of Richtermyer-Meshkov instability, is studied by the high-resolution simulation. While the dissipation mainly controls the passive scalar mixing rate, an objective definition of variable-density mixing rate characterizing the macroscopic mixing formation is still lacking, and the fundamental behavior of mixing rate evolution is not yet well understood. Here, we first show that the variable-density mixing of shock bubble interaction is distinctly different from the previous observations in the passive scalar mixing. The widely-accepted hyperbolic conservation of the first moment of concentration in the scalar mixing, i.e., the conservation of the mean concentration, is violated in variable-density flows. We further combine the compositional transport equation and the divergence relation for the miscible flows to provide the evidence that the existence of density gradient accelerated mixing rate, decomposed by the accelerated dissipation term and redistributed diffusion term, contributes to the anomalous decrease or increase of the mean concentration depending on Atwood number. Further analyzing a number of simulations for the cylindrical or spherical bubbles under a broad range of shock Mach numbers, Reynolds numbers, and Peclet numbers, the density gradient accelerated mixing rate exhibits weak dependent on Peclet numbers, and identifies an Atwood number range with high mixing rate, which can be theoretically predicted based on the mode of hyperbolic conservation violation behavior.

Ignition behavior and the onset of quasi-detonation in methane-oxygen using different end wall reflectors

The shock wave focusing technique is beneficial to the development of detonation-based engines, which can shorten the long run-up distance that is usually required for a detonation, reducing the weight of the detonation engine. In this work, we performed experiments using different end wall reflectors to facilitate the formation and intensity of the combustion wave in a stoichiometric methane-oxygen mixture diluted by argon. By changing shock wave intensity, the effects of the incident shock Mach number (Ma) on the ignition delay times in two reflectors are systematically investigated. The experimental results show that the conical reflector creates an abrupt pressure rise in the apex, resulting in a 64.5% rise in the acceleration of the reflected shock velocity compared with the planar reflector. The introduction of a conical reflector significantly shortens the ignition delay time by an order of magnitude and facilitates the amalgamation of the leading reflected shock wave and the subsequent combustion wave. The experimental results also illustrate that conical reflectors promote the formation of the second ignition. However, only a weak deflagration wave is formed in a planar reflector, and a strong deflagration wave and quasi-detonation wave are observed by using a conical reflector. Our observation sheds light on the mechanism of ignition modes induced by the shape of reflectors.

Ion Acceleration Efficiency at the Earth’s Bow Shock: Observations and Simulation Results

Abstract Collisionless shocks are some of the most efficient particle accelerators in heliospheric and astrophysical plasmas. Here we study and quantify ion acceleration at Earth’s bow shock with observations from NASA’s Magnetospheric Multiscale (MMS) satellites and in a global hybrid-Vlasov simulation. From the MMS observations, we find that quasiparallel shocks are more efficient at accelerating ions. There, up to 15% of the available energy goes into accelerating ions above 10 times their initial energy. Above a shock-normal angle of ∼50°, essentially no energetic ions are observed downstream of the shock. We find that ion acceleration efficiency is significantly lower when the shock has a low Mach number (M A < 6) while there is little Mach number dependence for higher values. We also find that ion acceleration is lower on the flanks of the bow shock than at the subsolar point regardless of the Mach number. The observations show that a higher connection time of an upstream field line leads to somewhat higher acceleration efficiency. To complement the observations, we perform a global hybrid-Vlasov simulation with realistic solar-wind parameters with the shape and size of the bow shock. We find that the ion acceleration efficiency in the simulation shows good quantitative agreement with the MMS observations. With the combined approach of direct spacecraft observations, we quantify ion acceleration in a wide range of shock angles and Mach numbers.