Stationary Shock Wave Research Articles

Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux

We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac [Formula: see text]-source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac [Formula: see text]-source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux’s connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax’s conditions — even for arbitrarily small shocks — and may also have no viscous profile. The Rankine–Hugoniot condition — embedding the Dirac [Formula: see text]-source — and the entropy condition are geometrically represented by “Flux Projections” that support the analytical method proposed in this paper. We then obtain a [Formula: see text] analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian–Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit.

Adaptive moving window technique for SPH simulation of stationary shock waves

A novel adaptive moving window (AMW) technique is developed for simulating stationary shock waves in a moving coordinate system associated with a simulation box. The velocity of this system is adjusted by an iterative feedback algorithm with the purpose of establishing a desirable position of shock front. Galilean transformations are iteratively used to maintain the shape of the flow profile. The moving coordinate system has advantages in simulation over a conventional coordinate system associated with the intact material at rest. In particular, a truly steady flow regime can be established in this moving window (MW). Comparative simulation using the smoothed particle hydrodynamics method to study shock propagation in copper confirms that the AMW converges to a steady state faster than a MW technique with a fixed velocity. We demonstrate that AMW can be applied for very weak shocks. Advantage of AMW is revealed in SPH simulation of the front structure for weak shocks in porous copper that result in a partial pore compaction. Specifically, we clarify the mechanism of the partial pore collapse for shock velocities below a knee on a calculated shock Hugoniot, where this knee corresponds to the complete collapse of pores.

An experimental and kinetic modeling study on the autoignition characteristics of ammonia-based ternary fuels under reflected shock wave conditions

Ammonia has received considerable attention as a renewable and carbon-free fuel in the context of global carbon neutrality. The high-temperature autoignition characteristics of NH3-H2-DME binary and ternary blended fuels were investigated using a shock tube experiment under conditions of T = 1300–2000 K, p = 0.14, 1.0 MPa, and ϕ = 0.5, 1.0, 2.0. The experiment focused on the effect of the proportion of H2 + DME as the binary active fuel on the ignition delay time reduction rate of pure NH3 blended fuel under various operating conditions. The experimental results showed that the composition of binary active fuels affects the ignition-promoting effect of the active fuel. H2 + DME (1:1) could result in a greater reduction in activation energy for the blended fuels, and the effectiveness of different of active fuels varies considerably depending on the operating conditions. Furthermore, a model for NH3-H2-DME fuels was developed, and the mechanism of action of H2 + DME (1:1) and H2 as active fuels on the ignition characteristics of the blended fuel was thoroughly investigated. Chemical kinetic analysis showed that one of the most important factors for promoting ignition by H2 + DME (1:1) and H2 is the alteration of the pathways for generating reactive H radicals in the initial stage of the reaction, which accelerates the chain reactions to facilitate ignition.

Structure, morphology, and magnetic properties of Fe microparticles as impact on shock waves

To analyze the stability of the physical and chemical properties of Fe, samples were investigated by subjecting them to shockwaves. The magnetic response of Fe to the external magnetic field mainly depends on the degree of magnetic ordering and the temperature of the sample. Powder X-ray diffraction (PXRD) characterization techniques were employed to analyze the crystallinity and cell volume of the structure obtained from Rietveld refinement. The sample was exposed to a shockwave ∼ 2.2 Mach with a different set of repetitions (such as 50, 100, 150, and 200). Refined data from PXRD shows no structural change in the crystallinity of the system. Williamson-Hall analysis demonstrates that the stress and strain between Fe change during the application of shocks. Thus, a noticeable change in the volume of the unit cell is observed. By subjecting the particles to repetitive shockwaves, their size was reduced. This was confirmed through imaging with a scanning electron microscope (SEM), which compared the particle images before and after the application of shockwaves from different sets. The magnetic field and temperature dependencies were applied to measure the ambient and shocked conditions. The magnetization results indicate an increase in magnetic moment due to the influence of the shockwave conditions. Also, 100 shocks show a transition behavior from ferromagnetic to paramagnetic. Based on its convincing reversible magnetic phase transition behavior, Fe is a potential material for applications in magnetic sensors and spintronics.

The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

Comprehensive study of autoignition characteristics of propane

Ignition delay times (IDT) for stoichiometric propane (C3H8) diluted with nitrogen were measured in a shock tube facility under reflected shock wave conditions at pressures ranging from 1 to 10 atm and temperatures between 850 and 1500 K. The experiments were limited to a maximum pressure of 10 atm due to the facility’s constraints. In addition, numerical simulations were conducted using several detailed kinetic mechanisms at pressures from 1 to 30 atm and three equivalence ratios (φ = 0.5, 1, and 2) to provide comparative insights. The results indicated that IDT decreases as pressure increases, with a more significant reduction observed between 1 and 10 atm compared to 10 to 30 atm. While most models exhibited similar trends and minimal discrepancies, the GRI Mech 3.0 mechanism demonstrated a slower prediction of ignition delay times at temperatures below 1250 K. In contrast, the POLIMI model exhibited a relatively faster prediction at temperatures above 1250 K, with the deviation between the two models becoming more pronounced as pressure increased. A comparative analysis revealed that the experimental predictions of propane autoignition behavior were in good agreement with the results obtained using the ARAMCO 3.0 mechanism. To further understand the chemistry governing the autoignition process of C3H8, a sensitivity analysis was performed for a stoichiometric mixture at three distinct temperatures (850 K, 1200 K, and 1550 K).

Linear interaction approximation for shock/disturbance interaction in a Noble–Abel stiffened gas

When departure from the ideal gas equation of state is considered, the Noble–Abel stiffened gas model is an appealing and versatile candidate due to its simple form. The linear interaction approximation formalism is extended to consider non-ideal gas effects introduced by this equation of state. Kovásznay decomposition and adequate definition of the energy of disturbances are provided in the context of this equation of state. Changes with respect to ideal gas are investigated on transfer functions, critical angle, and compression factor. Those differences yield concrete effects on the damping and transfer of fluctuations across shock waves. Those changes are further illustrated by considering the interaction of an entropy spot with a Mach 3 stationary shock wave.

Structure of shock wave in tungsten and titanium metals by using navier-stokes equation

Using the Navier–Stokes equation, we present the structure of a one-dimensional stationary shock wave formed in condensed alloy materials such as tungsten and titanium. In this study, the shock wave structure in the mentioned materials has been analysed by using several important parameters such as material viscosity, the Mie-Gruneisen parameter, and Mach number (M). One of our derived results is shock thickness, which is found to be of the order of 10−6 meters. In addition, we address a number of key insights into the shock wave interaction with tungsten and titanium. It should be noted that the validity of the model discussed in this study is limited to high M values, i.e., M ≥ 2.0.

ПОЛУАНАЛИТИЧЕСКИЙ МЕТОД РЕШЕНИЯ УРАВНЕНИЙ ГАЗОВОЙ ДИНАМИКИ В ПЕРЕМЕННЫХ ЭЙЛЕРА

This paper presents a semi-analytical method for solving a system of equations of gas dynamics in Eulerian coordinates. Since only spatial derivatives are replaced by finite differences, the system of gas dynamic equations is reduced to a system of ordinary differential equations on a spatial grid. An approximate analytical solution of this system of differential equations for a small time-interval is used to describe the dynamics of a gas in the entire required time interval. Verification was carried out on one-dimensional test problems on the decay of an arbitrary discontinuity and the propagation of stationary shock waves of various intensities. To compare one-dimensional problems, the solution of test problems is given by the simple-to-implement basic particle-in-cell method. It is shown that the semianalytical method has high accuracy of calculations, and is also the most universal method for calculating applied problems.

Analysis of shock waves in a mixture theory of a thermoelastic solid and fluid with distinct temperatures

The continuum theory of mixtures is used to describe a medium consisting of a solid phase and fluid phase, both compressible and attributed distinct deformation, temperature, entropy, and internal energy fields. Continuum balance laws are presented, and constitutive equations consistent with thermodynamic restrictions are postulated. Jump conditions for shock waves are established. Evolution equations for transient, i.e., unsteady, shock wave propagation are newly derived for an ideal mixture consisting of an arbitrary number of nonlinear thermoelastic solids, whereby inviscid thermoelastic fluid behavior is attained for a given constituent under more restrictive constitutive assumptions. Detailed calculations of Hugoniot response and shock decay are specialized to the case of a two-phase mixture of a single solid and a single fluid, where physical properties pertinent to lung parenchymal tissue containing air are assigned. Local constitutive behaviors for this lung mixture model correspond to a compressible neo-Hookean solid phase and an ideal gas fluid phase. Predictions of the model offer new information on origins of soft tissue trauma, in the context of prior experimental observations, for dynamic impact or shock loading of the lung.

Implementation of exactly well‐balanced numerical schemes in the event of shockwaves: A 1D approach for the shallow water equations

AbstractThis article presents a numerical technique that ensures the exact solution of stationary shockwaves, known as the hydraulic jump, for the one‐dimensional shallow water equations with the geometric source term. The mathematical description of the hydraulic jump is given by the Rankine–Hugoniot condition, at the interface of a control volume, where one variable in the system, the discharge, is supposed to be constant while the remaining variables are discontinuous. However, at the discrete level, the solution could be, for instance, a 3‐state jump, consisting of extra (intermediate) states resulting in the so‐called numerical anomaly. The anomaly is due to the nonlinearity of the Hugoniot locus and the fact that, at these points, a naive formulation is unlikely to avoid the inaccurate representation. The concept used here is simple, limit all discontinuous variables, that is, the surface flow data inside the cell that contains a shock with the data of the nearby cells where the shock is traveling to, allowing the solution to be linear. When used with a number of well‐known numerical solvers, namely Lax–Friedrichs, HLL, and Roe schemes, this approach is confirmed to accurately capture the moving and stationary shockwaves in the trans‐critical flow over nonflat beds.

Моделирование ударно-волновых эффектов в металлах на основе релаксационной модели с дислокационной кинетикой пластических сдвигов

In the range of weak shock waves (before absorption of elastic precursors by the shock wave) the effects of shock wave deformation of aluminum and beryllium alloys are numerically studied by the relaxation model with dislocation kinetics of plastic shear. Both the structure of non-stationary and stationary shock wave profiles are modeled. The plastic flow behind the front of elastic precursors is studied.

Investigation of structure and mechanical properties under quasi-static and planar impact loading of aluminum composite reinforced with Al2O3 nanoparticles of different shape

We report on an investigation of the structure and mechanical properties under quasi-static and planar impact loading of Al-based nanocomposites reinforced with alumina nanoparticles of ball, plate and sheet shape fabricated through accumulative roll bonding (ARB) for 4 and 10 cycles. The distribution of the nanoparticles and structural characteristics of the matrix were revealed using transmission electron microscopy. The microhardness, ultimate and yield strengths and ductility under tension at the strain rate of 1 × 10−3 s−1 were measured as mechanical characteristics under the quasi-static loading. The maximum pressure of the impact compression, Hugoniot elastic limit and spall strength were calculated using the free surface velocity data recorded during the impact by aluminum flyer-plates with the impact velocity of 630 ± 30 m/s. Our results show that nanoparticles tend to agglomeration inside the composite. The size of agglomerates depends on the nanoparticle shape and a number of ARB cycles. Only the part of the ball-shaped nanoparticles can be distributed uniformly as separate nanoparticles after 10 ARB cycles. The introduction of the nanoparticles into aluminum assists in the structural refinement in the nanocomposites relative to alumina-free aluminum. This effect is mostly exhibited in the composite reinforced with the ball-shaped nanoparticles, some of which are distributed separately. The nanoparticle of different shape affects the mechanical properties ambivalently under both quasi-static and shock-wave conditions due to their various distribution in the matrix and different properties of the agglomerates. The spall strength decreases with the nanoparticle introduction because the particles and their agglomerates are the stress concentrators and crack origins.

Quantitative and Sensitive Mid-Infrared Frequency Modulation Detection of HCN behind Shock Waves

Despite its key role for the study and modeling of nitrogen chemistry and NOx formation in combustion processes, HCN has only rarely been detected under high-temperature conditions. Here, we demonstrate quantitative detection of HCN behind incident and reflected shock waves using a novel sensitive single-tone mid-infrared frequency modulation (mid-IR-FM) detection scheme. The temperature-dependent pressure broadening of the P(26) line in the fundamental CH stretch vibration band was investigated in the temperature range 670K≤T≤1460K, yielding a pressure broadening coefficient for argon of 2γAr296K=(0.093±0.007)cm−1atm−1 and a temperature exponent of nAr=0.67±0.07. The sensitivity of the detection scheme was characterized by means of an Allan analysis, showing that HCN detection on the ppm mixing ratio level is possible at typical shock wave conditions. In order to demonstrate the capability of mid-IR-FM spectroscopy for future high-temperature reaction kinetic studies, we also report the first successful measurement of a reactive HCN decay profile induced by its reaction with oxygen atoms.

An unmagnetized strongly coupled plasma: heavy ion acoustic shock wave excitations

The propagation of heavy ion-acoustic shock waves (HIASWs) are studied in an unmagnetized strongly coupled plasma having inertial positively charged heavy ions (HIs) fluid and inertia-less (α, q)-distributed electrons and Maxwellian light ions (MLIs). To examine such phenomena, the Burgers equation (BE), modified Burgers equation (mBE), and mixed mBE (mmBE) are derived without considering the stretching of viscosity coefficient of HIs via the reductive perturbation technique (RPT). In addition, the appropriate stationary shock wave solutions of the nonlinear equations, particularly, mBE and mmBE is formulated for the first time. With the changes of plasma parameters, the basic features of HIASWs are examined in both of strongly coupled regime (SCR) and weakly coupled regime (WCR). It is described that the basic features (viz., polarity, amplitude, width, etc.) of the HIASWs are remarkably modified, and for that the adiabatic HIs, (α, q)-distributed electrons, MLIs and viscosity coefficient of HIs played an important role. The presented investigation might be beneficial to comprehend and predict the behavior of HIASWs structures in space and astrophysical environments (SAEs) and the future experimental studies in plasma laboratory (PL).

Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism

By rigorous analysis, it is proven that from discontinuous Lagrangians, which are invariant with respect to the Galilean group, Rankine–Hugoniot conditions for propagating discontinuities can be derived via a straight forward procedure that can be considered an extension of Noether’s theorem. The use of this general procedure is demonstrated in particular for a Lagrangian for viscous flow, reproducing the well known Rankine–Hugoniot conditions for shock waves.

Propagation of a planar shock wave along a convex–concave ramp

The propagation of a normal shock wave along a coupled convex–concave surface of equal radii has been analysed experimentally and numerically in this study. The experimental and numerical studies were conducted using a similar geometry as of that used by Ram et al. (J. Fluid Mech., vol. 768, 2015, pp. 219–239) for studying the shock wave transition from regular reflection to Mach reflection. Many interesting flow features such as shock wave transitions over the ramp, characteristics of the induced flow behind the shock wave and the development of a stationary separation shock wave have been observed in the study. The numerical results are validated with experimental data. While the shock wave transitions over the ramp are found to depend mainly on the ramp geometry, the characteristics of the stationary shock wave and the flow separation in the concave region of the ramp surface have been found to vary with the shock wave Mach numbers.

Delta-shock waves and Riemann solutions to the generalized pressureless Euler equations with a composite source term

In this paper, we are concerned with the Riemann problem for the generalized pressureless Euler equations with a composite source term, which covers the important Eulerian droplet model associated with aircrafts and airways. The Riemann solutions exhibit exactly two types of structures: delta-shocks and vacuum solutions. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. It is shown that the composite source term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore, which is a new and interesting phenomenon different from the homogeneous generalized pressureless Euler equations. On the other hand, compared with previous results on the nonhomogeneous generalized pressureless Euler equations, different from the nonhomogeneous case with friction where the state variable u changes linearly with respect to t, here u changes exponentially with respect to t under the influence of composite source term, but with a velocity quite different from the nonhomogeneous case with dissipation. It is also shown that, as the composite source term vanishes wholly or partly, the Riemann solutions converge to the corresponding ones of the homogeneous system or the nonhomogeneous system. These results will give us valuable insights into later research on the (generalized) pressureless Euler equations and other conservation laws with more complicated source terms, such as discontinuous source terms or singular source terms. Finally, two typical examples are given to show the application of our results on the Eulerian droplet model and the nonlinear geometric optics system with a source term.

Full analytical ultrarelativistic 1D solutions of a planar working surface

ABSTRACT We show that the 1D planar ultrarelativistic shock tube problem with an ultrarelativistic polytropic equation of state can be solved analytically for the case of a working surface, i.e. for the case when an initial discontinuity on the hydrodynamical quantities of the problem form two shock waves separating from a contact discontinuity. The procedure is based on the extensive use of the Taub jump conditions for relativistic shock waves, the Taub adiabatic, and performing Lorentz transformations to present the solution in a system of reference adequate for an external observer at rest. The solutions are found using a set of very useful theorems related to the Lorentz factors when transforming between systems of reference. The energy dissipated inside the working surface is relevant for studies of light curves observed in relativistic astrophysical jets and so, we provide a full analytical solution for this phenomenon assuming an ultrarelativistic periodic velocity injected at the base of the jet.

Homogeneous relaxation and shock wave problems: Assessment of the simplified and generalized Bernoulli trial collision schemes

The impetus of this paper is to assess the newly suggested direct simulation Monte Carlo (DSMC) collision schemes, that is, the “Simplified Bernoulli Trails (SBT)” and “Generalized Bernoulli Trials (GBT)” schemes in the prediction of the higher-order moments of the velocity distribution function for both confined and non-confined gas flows. Two fundamental rarefied gas dynamics problems are considered: spatially homogeneous relaxation process of a gas flow from a non-Maxwellian condition given by Bobylev–Krook–Wu exact (analytical) solution of the Boltzmann equation and the stationary shock wave problem. To perform the relaxation test, SBT and GBT schemes were implemented in the DSMC0F program. For the shock wave test, changes were made in the DSMC1 code to include the SBT and GBT schemes. A detailed comparison of the SBT and GBT collision schemes in treating the higher-order moments of the velocity distribution function and comparison with theory and the solution of the standard No-Time-Counter (NTC) method and its new variant, nearest neighbor scheme, using the DS1 code, is reported. Some higher moments beyond the usual moments were computed. The results of the fourth moment of the velocity distribution function in the homogeneous relaxation problem show that while both collision schemes produce identical results at an ample time, the initial relaxation process indicates the difference between the schemes. Even though the NTC schemes required a large number of particles per cell to produce the same results as the theory, the SBT scheme successfully simulates the solution using a low number of particles per cell.