Constant Adiabatic Index Research Articles

Relativistic hydrodynamic approach to the classical ideal gas

The necessary and sufficient condition for a conservative perfect fluid energy tensor to be the energetic evolution of a classical ideal gas is obtained. This condition forces the square of the speed of sound to have the form ${c}_{s}^{2}=\frac{\ensuremath{\gamma}p}{\ensuremath{\rho}+p}$ in terms of the hydrodynamic quantities, energy density $\ensuremath{\rho}$ and pressure $p$, $\ensuremath{\gamma}$ being the (constant) adiabatic index. The inverse problem for this case is also solved, that is, the determination of all the fluids whose evolutions are represented by a conservative energy tensor endowed with the above expression of ${c}_{s}^{2}$, and it shows that these fluids are, and only are, those fulfilling a Poisson law. The relativistic compressibility conditions for the classical ideal gases and the Poisson gases are analyzed in depth and the values for the adiabatic index $\ensuremath{\gamma}$ for which the compressibility conditions hold in physically relevant ranges of the hydrodynamic quantities $\ensuremath{\rho}$, $p$ are obtained. Some scenarios that model isothermal or isentropic evolutions of a classical ideal gas are revisited, and preliminary results are presented in applying our hydrodynamic approach to looking for perfect fluid solutions that model the evolution of a classical ideal gas or of a Poisson gas.

PHYSICAL-CONSTRAINT-PRESERVING CENTRAL DISCONTINUOUS GALERKIN METHODS FOR SPECIAL RELATIVISTIC HYDRODYNAMICS WITH A GENERAL EQUATION OF STATE

ABSTRACT The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax–Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L 1-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc.

Simulating cosmic ray physics on a moving mesh

We discuss new methods to integrate the cosmic ray (CR) evolution equations coupled to magneto-hydrodynamics (MHD) on an unstructured moving mesh, as realised in the massively parallel AREPO code for cosmological simulations. We account for diffusive shock acceleration of CRs at resolved shocks and at supernova remnants in the interstellar medium (ISM), and follow the advective CR transport within the magnetised plasma, as well as anisotropic diffusive transport of CRs along the local magnetic field. CR losses are included in terms of Coulomb and hadronic interactions with the thermal plasma. We demonstrate the accuracy of our formalism for CR acceleration at shocks through simulations of plane-parallel shock tubes that are compared to newly derived exact solutions of the Riemann shock tube problem with CR acceleration. We find that the increased compressibility of the post-shock plasma due to the produced CRs decreases the shock speed. However, CR acceleration at spherically expanding blast waves does not significantly break the self-similarity of the Sedov-Taylor solution; the resulting modifications can be approximated by a suitably adjusted, but constant adiabatic index. In first applications of the new CR formalism to simulations of isolated galaxies and cosmic structure formation, we find that CRs add an important pressure component to the ISM that increases the vertical scale height of disk galaxies, and thus reduces the star formation rate. Strong external structure formation shocks inject CRs into the gas, but the relative pressure of this component decreases towards halo centres as adiabatic compression favours the thermal over the CR pressure.

Isothermal Gaseous Detonation Model

Introduction. Approximate models, as compared to exact ones, usually use additional physical assumptions that permit "screening" unimportant details of the investigated phenomenon and simplifying the system of equations to be solved. Therefore, approximate models enhance the graphicness of obtained results, facilitate the theoretical analysis of the phenomenon, and shorten the time of computer calculations. If, moreover, the above models feature also a high accuracy, then their practical importance increases, since it enables them to describe the phenomenon not only qualitatively but quantitatively as well. In modeling the propagation of the detonation wave (DW) in a chemically active gas, its front is often considered as a compression shock with instantaneous heat release, at which the mass, momentum, and energy conservation laws should be observed [1–3]. Such a situation takes place when the thickness of the zone of major energy release due to the chemical reaction (for a gaseous detonation of the order of the size of the detonation cell [4]) is small compared to the characteristic linear scale of the whole fl ow, for example, the diameter of the tube in which the detonation propagates. Such a tube can be considered to be "wide" [5], and the difference of the gas-dynamic parameters of the DW front from the ideal detonation (without friction and heat transfer losses) is negligibly small. There exist well-tested methods (for example, the method described in [6] for calculating the parameters of the steady ideal detonation, often called the Chapman–Jouguet detonation) [3, 7 that are based on the assumption of chemical equilibrium of the composition of gaseous detonation products. The condition of chemical equilibrium of detonation products means that the chemical reaction rate is much higher than the rate of change in the thermodymamic parameters of the substance, i.e., the chemical reaction is instantaneous. Estimates of the characteristic times of the processes in which the chemical equilibrium is reestablished are given in [8]. Data of equilibrium calculations agree fairly well with experimental data; in the fi rst place it concerns the detonation front velocity DCJ measured with a high accuracy in experiments [9]. In spite of its apparent simplicity, the equilibrium detonation model is rather complicated for theoretical analysis, since in the computational iterative process one has to solve an expanded system of equations containing, apart from the gas dynamics equations, chemical equilibrium formulas, thermochemical relations, and atomic balance equations for all possible components of the detonation reaction. Therefore, other approaches are needed for modeling the detonation in gases. For instance, from the viewpoint of the classical one-dimensional theory for the analytical description of the gas dynamics of the detonation process [1–3], one usually uses an approximate model with highly idealized notions on detonation products, which is justifi ed only by the possibility of simple algebraic transformations with the use of conservation laws. In particular, it is assumed that detonation products represent an ideal inert gas with a constant adiabatic index �� and the chemical energy is released in the form of heat Q = const instantaneously at the DW front. The reasoning behind what values of �� and Q should be used in the given model is usually omitted thereby. The accuracy of the description of the thermodynamic part of internal

Thermal Characteristics of Air in the Problem of Hypersonic Motion of Bodies in the Earth’s Atmosphere

Introduction. The air fl ow around bodies moving in the atmosphere with hypersonic velocities is accompanied by a sharp increase in the air temperature due to the high energy of the fl ow and its dissipation in strong shock waves (SW) characteristic of such fl ows [1]. As the velocity of motion increases, there is an increase in the air temperature behind the shock wave, which leads to the initiation of a number of high-temperature processes such as vibrational excitation of molecules and their dissociation and the formation of molecular and atomic ions. These physicochemical processes have their own relaxation times. When these times are much longer than the characteristic time of the change in the fl ow parameter (in the stationary fl ow the above situation is equivalent to a relaxation length larger than the linear size), the thermodynamical properties of air can be described in the ideal gas approximation with a constant adiabatic index. In the opposite case of a short relaxation time, the thermal parameters of the medium can be assumed to be in equilibrium. To describe the intermediate region where the times are comparable, it is necessary to use physical and chemical kinetics equations [2]. It is known that the above high-temperature processes can not only infl uence locally the fl ow parameters, but also determine its character in general. Nevertheless, considering the hypersonic fl ow without account for the nonequilibrium processes is justifi ed [3], since it permits forming an approximate hydrodynamic fl ow pattern. The present paper gives the results of calculations of the equilibrium properties of a high-temperature air and describes a model of the processes of physicochemical kinetics, which, along with the gas dynamics equations, forms the basis for simulating the hypersonic motion of bodies in the Earth’s atmosphere. Equilibrium Characteristics of Air. In the ideal gas approximation, the system of gas-dynamic equations is closed by the equation of state

Modeling of Explosion Gas Dynamics with Account of Detonation

The physical and hydrodynamic processes in the initial phase of explosion of condensed explosives in the air have been considered. The role of the processes of energy release connected with the explosive detonation has been analyzed. The equations of formal kinetics for modeling the processes of transformation of the original substance into detonation products have been described. The results obtained with the use of the equation of state of an ideal gas with a constant adiabatic index have been compared with calculations, where for the equation of state wide-range tables of properties of the air and explosion products were used. The stage of detonation of an explosive is included in the self-consistent hydrodynamic model used for describing the explosion processes from the moment of initiation of the detonation wave to the moment the air shock wave is formed, as well as in describing its propagation and attenuation.

Crisis of a vorticity source during heat supply with regard to the real properties of a diatomic gas

On the example of a diatomic gas consisting of a mixture of nitrogen and oxygen (air), the characteristics of energy-consuming vorticity source flow were investigated, with the dependence of heat capacity on temperature being taken into account. The statement of this problem undergoes a change if account is taken of the dependence of heat capacity and adiabatic index on temperature in the range from several hundred to several thousand degrees Kelvin (a calorically nonideal gas). The phenomenon of a thermal crisis, that is, choking of a stationary flow on attainment of unity by the radial component of the Mach number, was studied. The characteristic dimensions of the heating region and the characteristic time of heating have been determined. Estimations of the error incurred in a model of an ideal gas with constant heat capacity and adiabatic index have been made. The dependences of the characteristics of the vorticity source on the gas circulation and flow rate, as well as on the position of the energy supply region and its extension were investigated.

THE COUPLED EVOLUTION OF ELECTRONS AND IONS IN CORONAL MASS EJECTION-DRIVEN SHOCKS

We present simulations of coronal mass ejections (CMEs) performed with a new two-temperature coronal model developed at the University of Michigan, which is able to address the coupled thermodynamics of the electron and proton populations in the context of a single fluid. This model employs heat conduction for electrons, constant adiabatic index (γ = 5/3), and includes Alfven wave pressure to accelerate the solar wind. The Wang-Sheeley-Arge empirical model is used to determine the Alfven wave pressure necessary to produce the observed bimodal solar wind speed. The Alfven waves are dissipated as they propagate from the Sun and heat protons on open magnetic field lines to temperatures above 2 MK. The model is driven by empirical boundary conditions that includes GONG magnetogram data to calculate the coronal field, and STEREO/EUVI observations to specify the density and temperature at the coronal boundary by the Differential Emission Measure Tomography method. With this model, we simulate the propagation of fast CMEs and study the thermodynamics of CME-driven shocks. Since the thermal speed of the electrons greatly exceeds the speed of the CME, only protons are directly heated by the shock. Coulomb collisions low in the corona couple the protons and electrons allowing heat exchange between the two species. However, the coupling is so brief that the electrons never achieve more than 10% of the maximum temperature of the protons. We find that heat is able to conduct on open magnetic field lines and rapidly propagates ahead of the CME to form a shock precursor of hot electrons.

A SEMI-ANALYTIC FORMULATION FOR RELATIVISTIC BLAST WAVES WITH A LONG-LIVED REVERSE SHOCK

This paper performs a semi-analytic study of relativistic blast waves in the context of gamma-ray bursts (GRBs). Although commonly used in a wide range of analytical and numerical studies, the equation of state (EOS) with a constant adiabatic index is a poor approximation for relativistic hydrodynamics. Adopting a more realistic EOS with a variable adiabatic index, we present a simple form of jump conditions for relativistic hydrodynamical shocks. Then we describe in detail our technique of modeling a very general class of GRB blast waves with a long-lived reverse shock. Our technique admits an arbitrary radial stratification of the ejecta and ambient medium. We use two different methods to find dynamics of the blast wave: (1) customary pressure balance across the blast wave and (2) the "mechanical model". Using a simple example model, we demonstrate that the two methods yield significantly different dynamical evolutions of the blast wave. We show that the pressure balance does not satisfy the energy conservation for an adiabatic blast wave while the mechanical model does. We also compare two sets of afterglow light curves obtained with the two different methods.

A MULTIDIMENSIONAL RELATIVISTIC HYDRODYNAMIC CODE WITH A GENERAL EQUATION OF STATE

The ideal gas equation of state with a constant adiabatic index, although commonly used in relativistic hydrodynamics, is a poor approximation for most relativistic astrophysical flows. Here we propose a new general equation of state for a multi-component relativistic gas which is consistent with the Synge equation of state for a relativistic perfect gas and is suitable for numerical (special) relativistic hydrodynamics. We also present a multidimensional relativistic hydrodynamics code incorporating the proposed general equation of state, based on the HLL scheme, which does not make use of a full characteristic decomposition of the relativistic hydrodynamic equations. The accuracy and robustness of this code is demonstrated in multidimensional calculations through several highly relativistic test problems taking into account nonvanishing tangential velocities. Results from three-dimensional simulations of relativistic jets show that the morphology and dynamics of the relativistic jets are significantly influenced by the different equation of state and by different compositions of relativistic perfect gases. Our new numerical code, combined with our proposed equation of state is very efficient and robust, and unlike previous codes, it gives very accurate results for thermodynamic variables in relativistic astrophysical flows.

Corotational instability of inertial-acoustic modes in black hole accretion discs: non-barotropic flows

We study the effect of corotation resonance on the inertial-acoustic oscillations (p-modes) of black-hole accretion discs. Previous works have shown that for barotropic flows (where the pressure depends only on the density), wave absorption at the corotation resonance can lead to mode growth when the disc vortensity, $\zeta=\kappa^2/(2\Omega\Sigma)$ (where $\Omega, \kappa, \Sigma$ are the rotation rate, radial epicyclic frequency and surface density of the disc, respectively), has a positive gradient at the corotation radius. Here we generalize the analysis of the corotation resonance effect to non-barotropic fluids. We show that the mode instability criterion is modified by the finite radial Brunt-V\"as\"al\"a frequency of the disc. We derive an analytic expression for the reflectivity when a density wave impinges upon the corotation barrier, and calculate the frequencies and growth rates of global p-modes for disc models with various $\alpha$-viscosity parameterizations. We find that for disc fluids with constant adiabatic index $\Gamma$, super-reflection and mode growth depend on the gradient of the effective vortensity, $\zeta_{\rm eff} = \zeta/S^{2/\Gamma}$ (where $S \equiv P/\Sigma^{\Gamma}$ measures the entropy): when $d\zeta_{\rm eff}/dr > 0$ at the corotation radius, wave absorption leads to amplification of the p-mode. Our calculations show that the lowest-order p-modes with azimuthal wave number $m=2, 3, 4,...$ have the largest growth rates, with the frequencies approximately in (but distinct from) the $2:3:4...$ commensurate ratios. We discuss the implications of our results for the high-frequency quasi-periodic oscillations observed in accreting black-hole systems.

Numerical study of the viscous hypersonic flow over a blunt delta wing

A previously developed two-dimensional Navier-Stokes solver is extended to three dimensions. The hypersonic steady flow of an perfect gas (with a constant adiabatic index) over a delta wing with a spherical nose and a cylindrical leading edge is computed as an example. The characteristic features of the flow are analyzed. Pressure, heat flux, and friction coefficient distributions are computed in the shock layer and on the wing surface. The results are compared with available numerical and experimental data.

On the efficacy of imploding plasma liners for magnetized fusion target compression

A new theoretical model is formulated to study the idea of merging a spherical array of converging plasma jets to form a “plasma liner” that further converges to compress a magnetized plasma target to fusion conditions [Y. C. F. Thio et al., “Magnetized target fusion in a spheroidal geometry with standoff drivers,” Current Trends in International Fusion Research II, edited by E. Panarella (National Research Council Canada, Ottawa, Canada, 1999)]. For a spherically imploding plasma liner shell with high initial Mach number (M=liner speed/sound speed) the rise in liner density with decreasing radius r goes as ρ∼1∕r2, for any constant adiabatic index γ=dlnp∕dlnρ. Accordingly, spherical convergence amplifies the ram pressure of the liner on target by the factor A∼C2, indicating strong coupling to its radial convergence C=rm∕R, where rm(R)=jet merging radius (compressed target radius), and A=compressed target pressure/initial liner ram pressure. Deuterium-tritium (DT) plasma liners with initial velocity ∼100km∕s and γ=5∕3, need to be hypersonic M∼60 and thus cold in order to realize values of A∼104 necessary for target ignition. For optically thick DT liners, T<2eV, n>1019–1020cm−3, blackbody radiative cooling is appreciable and may counteract compressional heating during the later stages of the implosion. The fluid then behaves as if the adiabatic index were depressed below 5∕3, which in turn means that the same amplification A=1.6×104 can be accomplished with a reduced initial Mach number M≈12.7(γ−0.3)4.86, valid in the range (10<M<60). Analytical calculations indicate that the hydrodynamic efficiency for plasma liners assembled by current and anticipated plasma jets is <4%. A new similarity model for fusion α-particle heating of the collapsed liner indicates that “spark” ignition of the DT liner fuel does not appear to be possible for magnetized fusion targets with typical threshold values of areal density ρR<0.02gcm−2.

Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index

The role of the equation of state (EoS) for a perfectly conducting, relativistic magnetized fluid is the main subject of this work. The ideal constant Γ-law EoS, commonly adopted in a wide range of astrophysical applications, is compared with a more realistic EoS that better approximates the single-specie relativistic gas. The paper focuses on three different topics. First, the influence of a more realistic EoS on the propagation of fast magnetosonic shocks is investigated. This calls into question the validity of the constant Γ-law EoS in problems where the temperature of the gas substantially changes across hydromagnetic waves. Secondly, we present a new inversion scheme to recover primitive variables (such as rest-mass density and pressure) from conservative ones that allows for a general EoS and avoids catastrophic numerical cancellations in the non-relativistic and ultrarelativistic limits. Finally, selected numerical tests of astrophysical relevance (including magnetized accretion flows around Kerr black holes) are compared using different equations of state. Our main conclusion is that the choice of a realistic EoS can considerably bear upon the solution when transitions from cold to hot gas (or vice versa) are present. Under these circumstances, a polytropic EoS can significantly endanger the solution.

Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics

We explore the role of the Lagrangian map for Lie symmetries in magnetohydrodynamics (MHD) and gas dynamics. By converting the Eulerian Lie point symmetries of the Galilei group to Lagrange label space, in which the Eulerian position coordinate x is regarded as a function of the Lagrange fluid labels x0 and time t, one finds that there is an infinite class of symmetries in Lagrange label space that map onto each Eulerian Lie point symmetry of the Galilei group. The allowed transformation of the Lagrangian fluid labels x0 corresponds to a fluid relabelling symmetry, including the case where there is no change in the fluid labels. We also consider a class of three, well-known, scaling symmetries for a gas with a constant adiabatic index γ. These symmetries map onto a modified form of the fluid relabelling symmetry determining equations, with non-zero source terms. We determine under which conditions these symmetries are variational or divergence symmetries of the action, and determine the corresponding Lagrangian and Eulerian conservation laws by use of Noether's theorem. These conservation laws depend on the initial entropy, density and magnetic field of the fluid. We derive the conservation law corresponding to the projective symmetry in gas dynamics, for the case γ = (n + 2)/n, where n is the number of Cartesian space coordinates, and the corresponding result for two-dimensional (2D) MHD, for the case γ = 2. Lie algebraic structures in Lagrange label space corresponding to the symmetries are investigated. The Lie algebraic symmetry relations between the fluid relabelling symmetries in Lagrange label space, and their commutators with a linear combination of the three symmetries with a constant adiabatic index are delineated.

DARBOUX CLASS OF COSMOLOGICAL FLUIDS WITH TIME-DEPENDENT ADIABATIC INDICES

A one-parameter family of time-dependent adiabatic indices is introduced for any given type of cosmological fluid of constant adiabatic index by a mathematical method belonging to the class of Darboux transformations. The procedure works for zero cosmological constant at the price of introducing a new constant parameter related to the time dependence of the adiabatic index. These fluids can be the real cosmological fluids that are encountered at cosmological scales and they could be used as a simple and efficient explanation for the recent experimental findings regarding the present day accelerating universe. In addition, new types of cosmological scale factors, corresponding to these fluids, are presented.

Homogeneous spheres with constant adriatic index

Compressible homogeneous spheres with constant adiabatic index γ were studied for their dynamical stability by Chandrasekhar and he found that for each value of u (≡ mass to size ratio), there is a value of γ = γc, such that for γ (1/3), the physically viable models of homogeneous density distribution are not possible. If the configuration is considered to be isentropic, then both γ and the speed of sound become infinite throughout the configuration.

The spectrum of adiabatic stellar oscillations

The stability analysis for spherically symmetric stellar equilibrium models with respect to ‘‘small’’ adiabatic Lagrangian perturbations leads to the consideration of a class of densely defined, linear symmetric operators in Hilbert space, which are induced by certain singular vector–integro–partial differential operator. The extension properties of these operators as well as the spectral properties of the linear self-adjoint extensions which are chosen by physical boundary conditions are investigated. For this, the equilibrium models are assumed to be polytropic, with a constant adiabatic index only near the center and near the boundary of the star. Among others it is shown that the operators of the class having a polytropic index near the boundary which is ≥1 are in particular essentially self-adjoint and have a closure with a pure point spectrum.

The spectrum of radial adiabatic stellar oscillations

The stability analysis with respect to ‘‘small’’ radial adiabatic perturbations of spherically symmetric stellar equilibrium models which are polytropic with a constant adiabatic index only near the center and the boundary of the star leads to the consideration of a class of singular minimal Sturm–Liouville operators. It is shown that the physical boundary conditions choose in a unique way the corresponding Friedrichs extensions. Moreover, all linear self-adjoint extensions of the members of the class are determined and are shown to have a purely discrete spectrum.

Relativistic oblique magnetohydrodynamic shocks

Special relativistic magnetohydrodynamic shock waves in a perfect gas of infinite conductivity and constant adiabatic index are analysed. It is shown that the Rankine-Hugoniot equations for such shocks may be reduced to a seventh degree polynomial for the downstream dynamical volume ω, with the polynomial coefficients depending on the upstream state (ω equals the specific volume times the ratio of the energy density of the fluid (omitting electromagnetic terms) to the fluid rest mass energy density). In the non-relativistic limit, the polynomial equation reduces to a relation between the upstream and downstream Alfvénic Mach numbers, previously obtained by Cabannes. The equation for ω classifies in a natural way both shocks in which the electric field may be eliminated by transforming to the de Hoffman–Teller frame, and shocks for which this is not possible. The equation is used to determine the downstream state of relativistic shocks for a given upstream state as specified by the plasma beta, magnetic field obliquity, and flow speed.