Wide-range Equation Of State Research Articles

A numerical model for the electrical and shock wave characteristics of underwater pulsed spark discharge

Underwater pulsed spark discharge has been widely used in industrial fields as a source of shock waves or acoustic waves, and numerical modeling of the discharge and pressure wave characteristics is necessary to improve the application performance. In this paper, a numerical model is proposed that couples the circuit equation, the mass and energy conservation equations, and a momentum conservation equation based on the Rankine–Hugoniot conditions. A tabulated wide range equation of state and conductivity data of water are used, and various physical processes during the plasma channel expansion are considered, including the energy flow and mass exchange between the channel and the surrounding water due to thermal radiation, evaporation, and condensation. The model self-consistently solves the circuit current and voltage, the plasma channel parameters including composition, temperature, conductivity, pressure, etc., and the pressure profile at a certain distance from the discharge channel. The calculated results show good consistency with the experimental measurements, and three sets of experimental results from other literature are tested to further verify the applicability and effectiveness of the model.

Influence of the phase interface mass transfer characteristics on the cavitation bubble collapse in water

The dependence of cavitation bubble collapse on such characteristics of the mass transfer across the bubble surface as the phase interface temperature jump, the ratio of the interface displacement speed due to the mass exchange to the speed of the thermal motion of the vapor molecules, the difference between the evaporation and condensation coefficients on the cavitation bubble collapse in water is studied. The evaporation and condensation coefficients range from 0.002 to 0.2. In the numerical technique, the dynamics of the vapor in the bubble and the surrounding liquid is governed by the gas dynamics equations closed with wide-range equations of state. Evaporation and condensation are described by the Hertz–Knudsen–Langmuir expressions. It has been found that the influence of the mass transfer characteristics considered is insignificant at small values of the evaporation and condensation coefficients, and monotonically grows with their increasing. In particular, the vapor mass in the bubble at the end of collapse is underestimated by 10% if the temperature jump is not taken into account, is overestimated by 20% if non-zero ratio of the interface displacement speed due to mass exchange to the thermal motion speed of vapor molecules is ignored and is overestimated by 80% if the difference between the evaporation and condensation coefficients is not allowed for. Similar errors in the maximum pressure in the bubble during its collapse can reach 10%, 15% and 35%, respectively.

MARPLE: программное обеспечение для мультифизического моделирования в задачах сплошных сред

The research code MARPLE was originally created to model high-speed dynamic processes caused by the action of high-intensity energy fluxes on matter. At present, it is a universal tool able to solve various continuum mechanics problems. The implemented physical models are the following: single-fluid two-temperature MHD model of plasma dynamics, including electron-ion energy exchange and generalized Ohm's law; model of electrical and thermal conductivity taking into account the anisotropy in the magnetic field; radiative heat transfer: models pertinent to optically thin as well as optically thick media: techniques for taking into account radiative cooling losses, spectral multigroup diffusion transfer, laser radiation propagation etc.; model of multicomponent flow. Calculations are performed using wide-range equations of state, transport and optical data. The MARPLE code utilises modern computational technologies based on block structured and unstructured meshes consisting of tetrahedral, hexahedral, prismatic elements and their combinations. The solvers implement conservation laws using high-resolution techniques. We apply the physical splitting to solve the governing system. The object-oriented approach to software design is used, as well as methods of object and generic programming (C++ implementation language). Design of computational domains is provided by means of integrated SALOME open source CAD-CAE system. Marple works as MPI application for modern HPC systems. The paper presents examples of problems in plasma dynamics, magnetohydrodynamics, astrophysics, and solid thermomechanics solved by means of the MARPLE code.

Helmholtz Free Energy Equation of State for 3He–4He Mixtures at Temperatures Above 2.17 K

The work presents the first wide-range equation of state (EOS) for 3He–4He mixtures based on the reduced Helmholtz free energy multi-fluid approximation model. It covers the temperature range from 2.17 to 300 K and the pressure from the vapor pressure up to 3 MPa for any given mixture 3He mole fraction. In this model, the 4He and 3He reduced Helmholtz free energy equations and departure functions from the literature are employed and only five unknown mixture parameters are needed for each given departure function. The parameters and the best model for the concerned binary mixture were determined by the Levenberg–Marquardt optimization method. With the best developed model, the liquid, gaseous, and saturated thermophysical properties of the mixture can be mostly described with an accuracy better than 5%. Furthermore, a database for the thermophysical properties of 3He–4He mixtures is generated and provided for interpolation in temperature, pressure, and 3He mole fraction. The current EOS and database can be applied to the design and optimization of ultra-low temperature refrigerators.

Outgoing shock waves at collapse of a cavitation bubble in water

Formation and propagation of a shock pulse arising at collapse of a cavitation bubble in water are numerically studied. The initial bubble radius R0 is varied in the range 1–4 mm, the liquid pressure p∞ in the range 1–20 bar, and the liquid temperature T∞ in the range 20–80 °C. The dynamics of the vapor in the bubble and the surrounding liquid is governed by the equations of gas dynamics with allowing for the heat conductivity, heat and mass exchange on the bubble surface. Wide-range equations of state are applied. The numerical technique is based on the Godunov method on moving grids thickened to the bubble surface. A discontinuous approximation of the numerical shock pulse in the vicinity of its front part is used. The case of R0=1 mm, p∞=1 bar, T∞=20 °C is considered as a reference one. It is revealed that the influence of R0 in terms of the dimensionless distance from the collapse center r/R0 is not large and is mainly realized during the shock pulse formation. The variation in p∞ mostly leads to quantitative changes of the pulse characteristics. In particular, the maximum rate of decrease in the pulse pressure peak grows with r (the distance to the collapse center) from that of a regression proportional to r−3/2 to that of a regression according to r−4/3. With rising T∞, the pulse gets weaker and becomes pronounced at larger distance from the collapse center. With increasing T∞ to about 45 °C at R0=1.92 mm and p∞=1 bar, the pulse turns out to be shockless. With the increase of T∞ to approximately 40 °C, the maximum rate of decay in the pulse pressure amplitude is reduced to that of a regression proportional to r−1. It is shown that the obtained results are in good agreement with known experimental data.

Progress on wide-range equation of state for hydrogen and deuterium

Hydrogen is the most abundant element in nature and an important object of astrophysics and ICF research. This paper briefly presents an overview of the research progress in wide-range equation of state and especially comments assessment of the most recent shock compression experiments on Omega laser facility and the theoretical models. Based on the previous work, the wide-range equation of state of hydrogen is constructed by using the improved chemical free energy model, the first-principle numerical simulation results and the multi-parameter equation of state model, which is applicable in the temperature range of 20−108 K and the density range of 10−7−2000 g/cm3. Compared with experimental results, such as those of shock compression experiment, static high pressure isotherm experiment and sound velocity experiment, the newly-constructed wide-range equation of state for hydrogen has high confidence and provides high precision data for astrophysics, inertial confinement fusion, international thermonuclear experimental reactor and other engineering physics designs. The construction and validation method of the hydrogen wide-range equation of state can also be applied to its isotope deuterium. In comparison with current models published abroad, the deuterium wide-range equation of state constructed by this method is in better agreement with the experimental data of principal and secondary Hugoniot published in 2019. This paper also points out the temperature-density regimesthat need attention in future research.

Small non-sphericity of a convergent shock wave arising in a cavitation bubble in acetone during its collapse

A numerical study of the features of the growth of small deformations of a radially-convergent shock wave in a collapsing cavitation bubble in acetone has been performed. The liquid temperature and pressure are 273.15 K and 15 bar, respectively. At the beginning of collapse, the bubble radius is 500 μm, the bubble is filled with the saturated vapor of acetone. The initial bubble non-sphericity in the form of the second, fourth and sixth spherical harmonics are considered. The dynamics of the vapor in the bubble and the surrounding liquid are governed by the axisymmetric gas dynamics equations closed by wide-range equations of state. The bubble surface is explicitly traced. It is shown that the small non-sphericity of the shock wave during its convergence, similarly to that of the surface of a bubble during its collapse, increases with decreasing its radius in the form of oscillations with growing amplitude. Moreover, the dependence of the increase in the amplitude of those oscillations on the radius of the shock wave in the interval from its formation to its reaching the “hot” bubble nucleus with a radius r < 0.25 μm, can be described by a power law with an index of -1.18.

Boundary of Metastable Region within Equations of State by Nigmatulin–Bolotnova

The present paper considers the determination of the two-phase region boundary within a simplified version of the wide-range equations of state of water and steam by Nigmatulin–Bolotnova (EOS NB) in the case the determination is based on the temperature-dependence of the saturated vapor pressure. This version of EOS NB can be used when the temperature and pressure ranges under investigation do not include the area of abnormal features of water at temperatures close to freezing ones (lower than $$10^{o}C$$ ) and when some errors in adiabatic speed of sound in a nearly-critical zone are acceptable. In particular, it is applied to numerically simulating dynamics of cavitation bubbles under their strong collapse. It is shown that in determining the two-phase region boundary within the simplified version of EOS NB on the basis of the temperature-dependencies of the saturated vapor pressure, a number of problems can occur. A simple modification of one of the known temperature-dependencies of the saturated vapor pressure, free of such problems, is presented.

Numerical Simulation of Convergence of Nonspherical Shock Waves in a Cavitation Bubble

An efficient numerical technique is developed for calculation of radially convergent shock waves arising inside an axially symmetric cavitation bubble during its collapse in a quiescent liquid. It allows one to consider the case in which the bubble remains nearly spherical at the end of its collapse and the shock wave significantly deviate from the spherical one only in a small central region. Motion of liquid and vapor is governed by the equations of gas dynamics. Heat conductivity of liquid and vapor as well as evaporation and codensation on the bubble surface are taken into account, wide-range equations of state are applied. The technique is based on a TVD-modification of the Godunov method, developed by the authors. Numerical convergence of the technique with refining the computational grid along the radial and angular coordinates is illustrated using a problem of collapse of a cavitation bubble in tetradecane. Resolution of non-sphericity of the convergent shock waves in the bubble is taken as a criterion of the numerical convergence. A case of collapse of a bubble is considered in which the radius of the bubble is initially 0.5 mm and the bubble is slightly prolate along the axis of symmetry. The liquid pressure and temperature are 60 bar and 663.15 K. The vapor in the bubble at the beginning of collapse is at the state of saturation at a temperature of the surrounding liquid with a pressure of 10.4 bar.

Influence of the Liquid Pressure on Collapse of a Vapor Bubble in Hot Acetone

Compression of the content of a vapor cavitation bubble during its collapse in hot acetone is studied. The temperature of liquid is 419 K, its pressure is varied in the range from 15 to 150 bar. A realistic model is used, in which the dynamics of vapor in the bubble and of the surrounding liquid is governed by the gas dynamics equations. The nonstationary heat conductivity of vapor and liquid, and the non-equilibrium heat and mass transfer across the bubble surface are taken into account. Wide-range equations of state are applied. It is shown that as the liquid pressure increases, three scenarios of the bubble content compression are successively realized. In the first scenario, the compression is nearly uniform. In the second and third scenarios, the bubble content is compressed by radially convergent isentropic and shock waves, respectively. The shock wave scenario, which arises starting with a pressure of about 90 bar, is realized in three versions.

Ab initio based equation of state of dense water for planetary and exoplanetary modeling

Context. The modeling of planetary interiors requires accurate equations of state (EOSs) for the basic constituents with proven validity in the difficult pressure–temperature regime extending up to 50 000 K and hundreds of megabars. While EOSs based on first-principles simulations are now available for the two most abundant elements, hydrogen and helium, the situation is less satisfactory for water where no wide-range EOS is available despite its requirement for interior modeling of planets ranging from super-Earths to planets several times the size of Jupiter. Aims. As a first step toward a multi-phase EOS for dense water, we develop a temperature-dependent EOS for dense water covering the liquid and plasma regimes and extending to the super-ionic and gas regimes. This equation of state covers the complete range of conditions encountered in planetary modeling. Methods. We use first-principles quantum molecular dynamics simulations and the Thomas-Fermi extension to reach the highest pressures encountered in giant planets several times the size of Jupiter. Using these results, as well as the data available at lower pressures, we obtain a parametrization of the Helmholtz free energy adjusted over this extended temperature and pressure domain. The parametrization ignores the entropy and density jumps at phase boundaries but we show that it is sufficiently accurate to model interior properties of most planets and exoplanets. Results. We produce an EOS given in analytical form that is readily usable in planetary modeling codes and dynamical simulations (a fortran implementation is provided). The EOS produced is valid for the entire density range relevant to planetary modeling, for densities where quantum effects for the ions can be neglected, and for temperatures below 50 000K. We use this EOS to calculate the mass-radius relationship of exoplanets up to 5000 MEarth, explore temperature effects in the wet Earth-like, ocean planets and pure water planets, and quantify the influence of the water EOS for the core on the gravitational moments of Jupiter.

One-dimensional magneto-hydrodynamics simulation of magnetically driven solid liner implosions on FP-1 facility

As an important cylindrical-convergent drive technology, magnetically driven solid liner implosion has been widely used in the high energy density physics (HEDP) experiments for different researches, such as the properties of condensed matter at an extreme pressure, the hydrodynamic behaviors of imploding systems, and the properties and behaviors of dense plasmas. On the 2.2 MA FP-1 facility (with a rise time of 7 s), implosions of aluminum liners and their impact on target liners are studied experimentally for exploring the applications of instability and ejecta mixing. A one-dimensional Lagrangian codeMADE1D is developed to study liner implosions numerically, which is based on magneto-hydrodynamics model with material strength, wide-range equation of state, Lee-More conductivity, and SCG (Steinberg, Cochran and Guinan) constitutive model. The code is based on the finite difference method. The finite difference equations are written in the covariant form for both Cartesian and cylindrical coordinates which enables the accurate simulation of different load geometries. Numerical results, such as the simulated velocity and radius at inner surface of the liner and target, agree well with the measurements. It shows that FP-1 has the ability to accelerate a 0.5 mm thick aluminum liner with an initial radius of 1.5 mm to a speed of more than 1.1 km/s, and the corresponding velocity of inner surface is more than 1.5 km/s due to the cylindrical convergence effect. In our calculation, most of the liner keeps solid throughout the implosion, though its outer surface is melted due to the Ohmic heating. A cylindrical converging shock about 8-10 GPa can be obtained by setting a target with an initial radius of 8-11 mm inside the liner coaxially. The numerical results show that since the imploding liner is fully magnetized when it impacts the target, the shock and the corresponding reflect release wave run faster than in the unmagnetized target. This means that the target will spall near the liner-target interface, though they are impedance-matched acoustically. The movement of the shocked target can be affected by the pre-filled gas inside. Increasing the gas pressure makes the target lose its velocity quickly, and the rebound radius increases as well. By adjusting the load design and gas pressure appropriately, we can obtain the right implosion process to meet the study requirement.

Liquid temperature dependence of the shock formation in a cavitation bubble

The dependence of the radially convergent shock wave formation in a cavitation bubble on the surrounding liquid temperature TL in the range from 273.15 to 400 K is investigated at the liquid pressure equal to 50 bar. Realistic mathematical model is applied, in which the effects of the liquid compressibility, the heat conductivity of the vapor and liquid, the evaporation and condensation on the bubble surface are taken into account, wide-range equations of state are utilized. The governing equations of the vapor and liquid dynamics are solved numerically using a modification of the Godunov method of the second order of accuracy. It has been found that a radially convergent shock wave arises in the bubble in 273.15≤T_L≤375 К. In this interval, the distance between the shock wave formation position and the bubble surface decreases with decreasing the liquid temperature. The possibility of using a known simplified criterion of the formation of a shock wave inside a bubble to estimate its formation position under the studied conditions is considered. It is shown that with applying that criterion the shock wave formation position turns out to be correctly predicted at T_L≈325 К, while at T_L&gt;325 К and T_L&lt;325 К it is predicted closer to and more distant from the bubble surface, respectively.

Calculation of thermodynamic functions of aluminum plasma for high-energy-density systems

The results of calculating the degree of ionization, the pressure, and the specific internal energy of aluminum plasma in a wide temperature range are presented. The TERMAG computational code based on the Thomas–Fermi model was used at temperatures Т > 105 K, and the ionization equilibrium model (Saha model) was applied at lower temperatures. Quantitatively similar results were obtained in the temperature range where both models are applicable. This suggests that the obtained data may be joined to produce a wide-range equation of state.

Numerical model for electrical explosion of copper wires in water

This paper presents a simple but quite accurate numerical model for analyzing electrical explosion of copper wires in water. The numerical model solves a circuit equation coupled with one-dimensional magneto-hydrodynamic (MHD) equations with the help of appropriate wide-range equation of state (EOS) and electrical conductivity for copper. The MHD equations are formulated in a Lagrangian form to identify the interface between the wire and surrounding water clearly. A quotidian EOS (QEOS) that is known as the simplest form of EOS is utilized to build wide-range EOS for copper. In the QEOS, we consider the liquid-vapor phase transition, which is critical in analyzing the wire explosion system. For the electrical conductivity of copper, a semi-empirical set of equations covering from solid state to partially ionized plasma state are employed. Experimental validation has been performed with copper wires of various diameters, which are exploded by a microsecond timescale pulsed capacitive discharge. The simulation results show excellent agreements with the experimental results in terms of temporal motions of a plasma channel boundary and a shock front as well as current and voltage waveforms. It is found that the wire explodes (vaporizes) along the liquid branch of a binodal curve irrespective of wire dimension and operating voltage. After the explosion, the wire becomes a plasma state right away or after the current pause (dwell), depending on the operating conditions. It is worth noting that such a peculiar characteristic of wire explosion, i.e., current pause and restrike, is well simulated with the present numerical model. In particular, it is shown that the wire cools down along the vapor branch of the binodal curve during the current dwell, due to a significant difference of thermodynamic characteristics across the binodal curve. The influence of radiation for studying nonideal plasmas with a wire explosion technique and a physical process for shock wave formation by an exploding wire is discussed as well.

The formation of a crater on the surface of the cathode in the explosion of micro tip

Present the results of numerical simulation of the formation of a crater on the surface of the cathode during the explosion of micro tip. The simulation was performed using the two-dimensional magnetohydrodynamic program, which used wide-range equation of state and the table of conductivity of the metal, based on experimental data. It is shown that the electric explosion of micro tip with parameters typical unit ecton may form on the cathode surface of the crater with a radius of a few microns.

Simulation of femtosecond pulsed laser ablation of metals

In this paper a mathematical model for femtosecond laser ablation of metals is proposed, based on standard two-temperature model connected with 1D hydrodynamic equations. Wide-range equation of state has been developed. The simulation results are compared with experimental data for aluminium and copper. A good agreement for both metals with numerical results and experiment shows that this model can be employed for choosing laser parameters to better accuracy in nanoparticles production by ablation of metals.

Simulation of laser ablation of metals for nanoparticles production

In this paper a mathematical model for femtosecond laser ablation of metals is proposed, based on standard two-temperature model connected with 1D hydrodynamic equations. Wide-range equation of state has been developed. The simulation results are compared with experimental data for aluminium and copper. A good agreement for both metals with numerical results and experiment shows that this model can be employed for choosing laser parameters to better accuracy in nanoparticles production by ablation of metals.

Wide-range equation of state for gas and liquid plasma

A fundamentally new correction to the interaction of charged particles in plasma is proposed and substantiated. The substitution of this correction into the generalized Saha equation makes it possible to extrapolate these equations to zero temperature. This yields physically reasonable values of the ionization rate and thermodynamic functions and can significantly extend the applicability range of the Saha model to extremely high densities and moderate temperatures, i.e., the liquid plasma state. In passing, deficiencies are corrected inherent in the classical Debye model and the inadequacy of the model of simple harmonic oscillators is demonstrated. Liquid plasma is more accurately described by the Thomas–Fermi model with quantum and exchange corrections. An ultrafast algorithm has been developed for the solution of equations of this model. The calculations based on this model are joined with the calculations by the generalized Saha model by means of a special interpolation yielding a single wide-range equation of state for gas and liquid plasma. This equation of state is used to calculate shock adiabats with allowance for radiation contribution. The role of shell effects has been analyzed.

Equation of state of compounds of lithium isotopes with hydrogen isotopes

A semi-empirical wide-range equation of state of compounds of lithium isotopes with hydrogen isotopes is proposed. This equation allows thermodynamic properties to be calculated both in the range of comparatively small densities, pressures, and energies available for experimental studies and in the range of superhigh densities, pressures, and energies where the states can only be estimated at the moment by calculations in accordance with theoretical models. The equation of state contains empirical functions, which allow the composition of the isotopes and the influence of the hydroxide admixture on the compound properties to be taken into account. The capabilities of the equation of state are demonstrated by an example of the description of experimental and numerical data characterizing thermodynamic and thermophysical properties of several compounds of lithium and hydrogen isotopes.