Rankine Hugoniot Research Articles

A physics-based study of the stagnation enthalpy rise of moving normal shocks

Normal shocks are generally treated by defining a stationary adiabatic shock discontinuity with a supersonic upstream velocity. A fundamental feature of this approach is that the problem is steady, and the stagnation enthalpy is constant across the shock, greatly simplifying the analysis. However, shocks are generally not stationary, but rather are unsteady flow features often translating into a quiescent medium. Moreover, the stagnation enthalpy of a fluid will rise in a moving shock, an often unexpected result for students studying compressible flow. In this paper, stationary normal shocks are analyzed using the classic Rankine–Hugoniot equations. The analysis is then transformed to consider a moving shock propagating into a quiescent medium. This analysis produces a simple expression relating the stagnation enthalpy rise across a moving normal shock as a function of the shock strength as measured by either the shock pressure ratio or differential pressure across the shock. An unsteady integral control volume analysis is then used to show that the cause for the stagnation enthalpy rise is the work done by the fluid downstream of the shock to compress the upstream fluid upstream.

Advective Accretion onto a Nonspherical Accretor in White Dwarf and Neutron Star Binaries: A New Scenario of Shock Formation

Abstract Numerous studies on hydrodynamics of the Keplerian as well as the sub-Keplerian accretion disk around a compact object (e.g., white dwarf (WD), neutron star (NS), or a black hole) have attempted to explain the observed UV, soft, and hard X-ray spectra. Although, when the compact object (e.g., a WD or an NS) has a finite surface, its rapid rotation, the stellar magnetic field could cause deformation of the spherical symmetry. Earlier studies on the Keplerian disk showed that a deviation from the spherical symmetry of the compact object could affect the observed light curve and spectra at high frequencies. Here, we have explored the effect of the nonspherical nature of a compact object on the hydrodynamics of an optically thin, geometrically thick sub-Keplerian advective flow. We find that due to the nonspherical shape of the central accretor, there is a possibility to trigger Rankine–Hugoniot shock in the sub-Keplerian advective flow close to the accretor without considering any general relativistic effect or presence of the hard surface of the star. Our results are more relevant for accretion onto a WD as hardly any general relativistic effect will come into the picture. We propose that some observational features, e.g., high significance of fitting the spectra with multi-temperature plasma models rather than single-temperature models, and variable efficiency of X-ray emission (X-ray luminosity in comparison with the optical and UV luminosity of the disk) in nonmagnetic cataclysmic variables can be explained by the presence of a shock in the sub-Keplerian advective flow.

Wedge-Induced Oblique Detonations with Small Heat Release

The present work exploits simplifications arising in weakly exothermic detonations when the postshock conditions are supersonic to investigate the structure of wedge-induced oblique detonations. These simplifications enable the linearized Euler equations (employed here in characteristic form) to be efficiently solved numerically, subject to the linearized Rankine–Hugoniot jump conditions across the leading oblique shock. A first set of computations employs one-step first-order Arrhenius chemistry appropriate for describing detonations when the postshock chemistry exhibits a thermal-explosion character. In that case, the relevant chemical-kinetic parameter of order unity is the product of the heat release and the activation energy divided by the square of the postshock thermal enthalpy. The transition from the shock to the detonation wave is continuous at small , begins to develop spatially decaying oscillations as increases, and develops a singularity at the shock at a critical value of ; above which, the transition must become discontinuous and involve a triple point. Parametric results are presented in a plane of the wedge angle and the incident-flow Mach number: the two important controlling parameters. The triple point is found to develop when the incident-flow Mach number falls below a critical value that exhibits a U-shaped dependence on the wedge angle, becoming large at both high and low wedge angles and reflecting large differences between shock angles with and without heat release in those two extremes. Additional computations are performed for a three-step branched-chain scheme with the heat-release step having zero activation energy and for very fuel-lean hydrogen–air detonations with postshock temperatures above crossover. These cases, for which ignition develops as a chain-branching explosion, do not develop a singularity at the shock; although they display many of the features identified with the Arrhenius chemistry, including oscillations and appearance of a precursor point indicative of criticality. The results suggest a strong potential influence of the chemistry on the transition.

Thermochemical effects on hypersonic shock waves interacting with weak turbulence

The interaction between a weakly turbulent free stream and a hypersonic shock wave is investigated theoretically by using linear interaction analysis (LIA). The formulation is developed in the limit in which the thickness of the thermochemical nonequilibrium region downstream of the shock, where relaxation toward vibrational and chemical equilibrium occurs, is assumed to be much smaller than the characteristic size of the shock wrinkles caused by turbulence. Modified Rankine–Hugoniot jump conditions that account for dissociation and vibrational excitation are derived and employed in a Fourier analysis of a shock interacting with three-dimensional isotropic vortical disturbances. This provides the modal structure of the post-shock gas arising from the interaction, along with integral formulas for the amplification of enstrophy, concentration variance, turbulent kinetic energy (TKE), and turbulence intensity across the shock. In addition to confirming known endothermic effects of dissociation and vibrational excitation in decreasing the mean post-shock temperature and velocity, these LIA results indicate that the enstrophy, anisotropy, intensity, and TKE of the fluctuations are much more amplified through the shock than in the thermochemically frozen case. In addition, the turbulent Reynolds number is amplified across the shock at hypersonic Mach numbers in the presence of dissociation and vibrational excitation, as opposed to the attenuation observed in the thermochemically frozen case. These results suggest that turbulence may persist and get augmented across hypersonic shock waves despite the high post-shock temperatures.

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.

Cumulative energy effect in the shock-discontinuity interaction under real-gas conditions

A recently proposed technique for improving ignition timing is investigated for real-gas effects. A correction to the model describing the cumulative energy effect in the presence of the shock-discontinuity interaction was done by re-considering the Rankine–Hugoniot relations and the shock refraction equations. Non-dissociating gas in thermal equilibrium state, with excited translational, rotational, vibrational, and electronic degrees of freedom is considered. When applied to N2 gas at T = 3000 K, 11% density increase and 17% temperature and 7% pressure jump decrease were found, compared to that in ideal gas. The derived relations are largely built on experimental data thus avoiding the complexity of careful accounting for all real gas effects and still offering satisfactory precision levels. The main result is that in real gases the effect in the interaction can be stronger than that in an ideal gas. The findings can be of interest in design of efficient combustors for hypersonic systems, detonation, space and craft design, shock-flame interaction, in astrophysics, and fusion research.

Shock-induced melting of two-dimensional Yukawa systems from TH−PH Hugoniot curves

The TH−PH Hugoniot curves of compressional shocks in 2D Yukawa systems are derived from the combination of the Rankine–Hugoniot relation around the shock front and the universal relationship for the temperature in the postshock region. From the equation of state of 2D Yukawa liquids, the equilibrium melting curve for 2D Yukawa systems is derived using the two variables of the temperature T and the pressure P. It is found that the obtained TH−PH Hugoniot curves are intercepted by the equilibrium melting curve, indicating the existence of shock-induced phase transition at these crossing points. To confirm this prediction, molecular dynamical simulations of 2D Yukawa systems of κ=0.75 for the conditions around the crossing point are performed. In the postshock region, the calculated various diagnostics of static structural measures, like the Voronoi diagram, the defect ratio, the probability distribution of the shape factors ξ, the pair correlation function g(r), and the static structure factor S(q), suggest that, for our studied system, the shock-induced melting happens when the compressional speed of the boundary is 0.212a0ωpd<vleft<0.283a0ωpd, the same as the prediction from the crossing point.

On the stability of the improved Aw–Rascle–Zhang model with Chaplygin pressure

In this paper, we mainly study the stability of Riemann problem for the improved Aw–Rascle–Zhang model which describes the formation and dynamics of traffic jams. First of all, we construct the classical Riemann solutions by elementary waves with the method of characteristic analysis. With the generalized Rankine–Hugoniot and entropy conditions, we prove the existence and uniqueness of δ-shock wave for arbitrary convex F(u) in this model. Then through a small perturbation, we analyze the wave interactions of different kinds of waves. As a result, we get the stability for the Riemann problem by letting the perturbed parameter ε→0.

Riemann problem for a two‐dimensional steady pressureless relativistic Euler equations

AbstractWe consider the Riemann problem for a two‐dimensional steady pressureless relativistic Euler equations. The delta shock wave is discovered in the Riemann solutions. It is shown that Dirac delta function develops in the state variable describing the number density of particles. By virtue of a suitable generalized Rankine–Hugoniot relation and entropy condition, we establish the existence and uniqueness for delta‐shock solution. Furthermore, we analyze in detail the interactions of delta shock waves and vacuum states.

Cavitating flow behind a backward facing step

Flow topology and unsteady cavitation dynamics in the wake of a backward facing step was investigated using high speed videography and time-resolved X-ray densitometry, along with static and dynamic pressure measurements. The measurements are used to inform the understanding of underlying mechanisms of observed flow dynamics as they are related to shock wave induced instabilities. The differences in cavity topology and behavior at different cavitation numbers are examined to highlight flow features such as the pair of cavitating spanwise vortices in the shear layer and propagating bubbly shock front. The shock speeds at different cavitation conditions are estimated based on void-fraction measurements using the X-T diagram and compared to those computed using the one-dimensional Rankine Hugoniot jump condition. The two speeds(computed and measured) are found to be comparable. The effect of compressibility of the bubbly mixture is determined by estimating the local speed of sound using homogeneous ‘frozen model’ and homogeneous ‘equilibrium model’. The current estimation of sound speed suggests that expression used to determine the local speed of sound is closer to the classic ‘frozen model’ than the ‘equilibrium model’ of speed of sound.

Shock wave propagation through air: a reactive molecular dynamics study

Shock compression of air is observed in numerous situations ranging from explosions to hypersonic vehicle entry into atmosphere. In an effort to develop continuum-based equation of state for air subjected to shock compression, it is necessary to understand the dynamics of the shock compression process with regards to formation of new chemical species in air at the molecular level. With this in perspective, three different models of air (consisting a mixture of O 2 , N 2 and CO 2 gas, with or without H 2 O based on presence of humidity) are subjected to shock compression ranging from 0.5 km s −1 to 5.0 km s −1 particle velocities. Thermodynamic quantities are evaluated to plot Rankine Hugoniot planes for the three different air models: dry air at mean sea level (MSL), humid air at MSL and dry air at high altitude level of 36 000 ft above MSL. It is observed that high shock velocities eventually results in dissociation of gaseous molecules and formation of new gaseous species which has been quantified in the manuscript.

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.

The multiplication of distributions in the study of delta shock waves for zero-pressure gasdynamics system with energy conservation laws

In this article, we study the delta shock wave for zero-pressure gasdynamics system with energy conservation laws in the frame of $$\alpha $$ -solutions defined in the setting of distributional products. By reformulating the system, we construct within a convenient space of distributions, all solutions which include discontinuous solutions and Dirac delta measures. We also establish the generalized Rankine–Hugoniot jump conditions for delta shock waves. The $$\alpha $$ -solutions which we constructed coincide with the solution obtained through different methods.

Shock wave hydrodynamics of nano-carbons

Dynamic deformation of nano-carbons by shock waves is an important object in technological applications as well as in basic sciences. We report, on hydrodynamic response of two types of nano-carbon systems: graphene nano-flakes (GNF) and carbon nano-spheres (CNS) by subjecting them to the Klosky bar shock test (at strain rate 102–104/s). Data of stress (σ), strain (ε), and strain rate (ἑ) were obtained with time to analyse the behaviour of constitutive parameter (σ–ε). In elastic region GNF showed superior stress sensitivity at least by fivefold over CNS, whereas, stress accumulation ability of CNS was found to be ten times better than GNF. In plastic region both the systems were behaved quiet complexly. They comprised of various stages of deformation like inter–particle separation, micro–structure gliding, fracture, and perforation. To obtain hydrodynamic variables a few thermodynamic assumptions like matrix homogeneity, linear volume deformity, negligible temperature rise were made to set up the Lagrange–Rankine–Hugoniot model. Interplay of built–in pressure (P), particle velocity (UP), shock velocity (US), specific volume (V/VO), density (ρ), shock energy (E), behind and ahead of the shock wavefront led to the establishment of the equation of state for the system. Theoretical shock profile was vis-à-vis compared with the experimentally obtained shock results. Distribution of impulse pressure over the topology of the nano-carbons was examined that exhibited non-uniform shock energy dissipation pattern with peak pressure ~250 N/m2. Our calculations revealed that, almost ~65% shock energy was damped within GNF and ~89% in CNS. Details of the analysis are presented.

Discrete-time modeling and output regulation of gas pipeline networks

In this work, a discrete-time output regulator design is proposed for a class of gas pipeline networks to meet various operating requirements in energy scheduling. Based on the isothermal Euler equations, linearized continuous-time gas pipeline network models with boundary actuation and sensing in the infinite-dimensional space are established, with consideration of Rankine–Hugoniot conditions at junction joints. Cayley–Tustin bilinear transformation is applied for model time discretization without any spatial approximation, by which the continuous-time model with unbounded operators is transformed into an infinite-dimensional discrete-time system with all bounded operators and essential continuous-time properties are invariant under this transformation. Based on the internal model principle, the discrete output regulator is constructed and its solvability conditions are provided. Considering the unavailability of the full state information, observer design methods for state estimation of exogenous and pipeline systems are proposed in order to construct an output feedback regulator. Additionally, a stability analysis of the considered pipeline system is provided. Finally, two simulation examples representing a single gas pipeline and a star-shaped pipeline network are given to demonstrate the applicability of the proposed method.

Numerical study on the mitigation effect of glass particles filling a partially confined space on a blast wave

The interaction of a shock wave and particles filling the inside of a straight tube was numerically investigated to understand the mitigation mechanism on a blast wave outside the tube. A shock wave propagating along the particle layer inside the straight tube induced differences in the velocity and temperature between the particle layer and shocked air, inducing energy transfer through a drag effect, heat transfer, and nozzling term. Because the particle layer hardly moved inside the straight tube, the drag effect and nozzling term were too small to mitigate the blast wave. On the other hand, the heat transfer from the air to the particle layer absorbed several tens of percent of the energy released by the high explosive and was the dominant factor in mitigating the blast wave. To estimate the effect of the heat transfer between the particle layer and shocked air, we proposed a novel index estimated by the shocked-air properties, using the Rankine–Hugoniot relation. The numerical data for the heat transfer rate were simply and adequately estimated by the novel index, which indicated that the shocked-air properties were responsible for the heat transfer rate used to determine the mitigation of the blast wave.

Scaling of Hugoniot curves for shock-compressed liquids

In previous studies of shock compression, pressure P, specific volume v, specific internal energy e, shock velocity Us, and particle velocity up have typically been presented in a dimensional form. For different materials, the plots of P−v, P−up, or Us−up, often called Hugoniot curves, are different. Here, we predict the behavior of shock-compressed liquids through proper scaling of the Rankine–Hugoniot (RH) equations and dimensionless Hugoniot curves. The characteristic density and velocity scales are the density ρ0 and bulk speed of sound cb0 of the undisturbed liquid, respectively. Two dimensionless numbers arise from the scaled RH equations, one for the initial condition of pressure and the other for the initial condition of internal energy. Under normal conditions, these two numbers do not affect the solutions of the dimensionless RH equations. The dimensionless Hugoniot curves P/(ρ0cb02) vs vρ0, Us/cb0 vs up/cb0, and P/(ρ0cb02) vs up/cb0 of different liquids merge reasonably well onto a single curve. The dimensionless Hugoniot curve vρ0 vs Us/cb0 or vρ0 vs up/cb0, often omitted in the previous work, is thus found to be useful in the understanding of shock compression. The vρ0 vs Us/cb0 curve clearly shows that the dependence of the specific volume ratio vρ0 on Us/cb0 is different for moderate and strong shocks. For a moderate strength shock (Us/cb0≲10), a new approximation relation is proposed for shock velocity Us and particle velocity up as (Us−up)/cb0≈(Us/cb0)n, where the exponent is determined empirically as n=0.55−0.6. This new approximation relation is different from the commonly used linear relation between Us and up and better predicts the behavior of shock-compressed liquids. Using the new approximation relation, the ratio vρ0 under moderate strength shocks can be approximated by a power law vρ0≈(Us∗)n−1. For stronger shocks, the decrease in the specific volume ratio is slower and is bounded.

Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics

The motivation of the present study is to derive the solution of the Riemann problem for modified Chaplygin gas equations in the presence of constant external force. The analysis leads to the fact that in some special circumstances delta shock appears in the solution of the Riemann problem. Also, the Rankine–Hugoniot relations for delta shock wave which are utilized to determine the strength, position and propagation speed of the delta shocks have been derived. Delta shock wave solution to the Riemann problem for the modified Chaplygin gas equation is obtained. It is found that the external force term, appearing in the governing equations, influences the Riemann solution for the modified Chaplygin gas equation.

Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy

In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of $$1{/}\sqrt{3}$$ times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).

Kinetic Simulation of Nonrelativistic Perpendicular Shocks of Young Supernova Remnants. IV. Electron Heating

Abstract High-Mach-number collisionless shocks are found in planetary systems and supernova remnants (SNRs). Electrons are heated at these shocks to temperatures well above the Rankine–Hugoniot prediction. However, the processes responsible for causing the electron heating are still not well understood. We use a set of large-scale particle-in-cell simulations of nonrelativistic shocks in the high-Mach-number regime to clarify the electron heating processes. The physical behavior of these shocks is defined by ion reflection at the shock ramp. Further interactions between the reflected ions and the upstream plasma excites electrostatic Buneman and two-stream ion–ion Weibel instabilities. Electrons are heated via shock surfing acceleration, the shock potential, magnetic reconnection, stochastic Fermi scattering, and shock compression. The main contributor is the shock potential. The magnetic field lines become tangled due to the Weibel instability, which allows for parallel electron heating by the shock potential. The constrained model of electron heating predicts an ion-to-electron temperature ratio within observed values at SNR shocks and in Saturn’s bow shock.