Development of realistic liquid equations of state for multi-phase CFD modelling

Several equations of state (EOS) have been incorporated into a novel algorithm to solve a system of multi-phase equations in which all phases are assumed to be compressible to varying degrees. The EOSs are used to both supply functional relationships to couple the conservative variables to the primitive variables and to calculate accurately thermodynamic quantities of interest, such as the speed of sound. Each EOS has a defined balance of accuracy, robustness and computational speed; selection of an appropriate EOS is generally problem-dependent. This work employs an AUSM+-up method for accurate discretisation of the convective flux terms with modified low-Mach number dissipation for added robustness of the solver. In this paper we show a newly-developed time-marching formulation for temporal discretisation of the governing equations with incorporated time-dependent source terms, as well as considering the system of eigenvalues that render the governing equations hyperbolic.

Development of accurate, robust liquid equations of state for multi-phase CFD simulations with a modified AUSM+-up scheme

Eight equations of state (EOS) have been evaluated for the simulation of compressible liquid water properties, based on empirical correlations, the principle of corresponding states and thermodynamic relations. The IAPWS-IF97 EOS for water was employed as the reference case. These EOSs were coupled to a modified AUSM+-up convective flux solver to determine flow profiles for three test cases of differing flow conditions. The impact of the non-viscous interaction term discretisation scheme, interfacial pressure method and selection of low-Mach number diffusion were also compared. It was shown that a consistent discretisation scheme using the AUSM+-up solver for both the convective flux and the non-viscous interfacial term demonstrated both robustness and accuracy whilst facilitating a computationally cheaper solution than discretisation of the interfacial term independently by a central scheme. The simple empirical correlations gave excellent results in comparison to the reference IAPWS-IF97 EOS and were recommended for developmental work involving water as a cheaper and more accurate EOS than the more commonly used stiffened-gas model. The correlations based on the principles of corresponding-states and the modified Peng-Robinson cubic EOS also demonstrated a high degree of accuracy, which is promising for future work with generic fluids. Further work will encompass extension of the solver to multiple dimensions and to account for other source terms such as surface tension, along with the incorporation of phase changes. © 2013.

3D relativistic hydrodynamic computations using lattice-QCD-inspired equations of state

In this communication, we report results of three-dimensional hydrodynamic computations, by using equations of state with a critical end Point as suggested by the lattice QCD. Some of the results are an increase of the multiplicity in the mid-rapidity region and a larger elliptic-flow parameter nu(2). We discuss also the effcts of the initial-condition fluctuations and the continuous emission.

Realistic equations of state for the primeval universe

Equations of state for the early universe including realistic interactions between constituents are formulated. Under certain hypotheses, these equations are able to generate an inflationary regime prior to the period of the nucleosynthesis. The resulting accelerated expansion is intense enough to solve the flatness and horizon problems. In the cases of a curvature parameter. equal to 0 or + 1, the model is able to avoid the initial singularity and offers a natural explanation for why the universe is in expansion. All the results are valid only for a matter-antimatter symmetric universe.

Liquid silicate equation of state : using shock waves to understand the properties of the deep Earth

The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe₂SiO₄ (fayalite), Mg₂SiO₄ (forsterite), CaFeSi₂O₆ (hedenbergite), an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆ (anorthite-hedenbergite), and an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆-CaMgSi₂O₆(anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi₂O₆ (diopside), CaAl₂Si₂O₈ (anorthite), and MgSiO₃ (enstatite), which were previously determined by shock wave methods, are also presented.

The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaO-MgO-Al₂O₃-SiO₂-FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean.

The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe2+ coordination, there is a threshold of Fe concentration (molar X_Fe ≤ 0.06) permitted in a liquid for which its density can still be approximated by linear mixing of end-member volumes.

High pressure states in condensed matter: I. High pressure behavior of the iron-sulphur system with applications to the earth's core. II. Empirical equation of state for organic compounds at high pressures

Part I:

The earth's core is generally accepted to be composed primarily of iron, with an admixture of other elements. Because the outer core is observed not to transmit shear waves at seismic frequencies, it is known to be liquid or primarily liquid. A new equation of state is presented for liquid iron, in the form of parameters for the 4th order Birch-Murnaghan and Mie-Grüneisen equations of state. The parameters were constrained by a set of values for numerous properties compiled from the literature. A detailed theoretical model is used to constrain the P-T behavior of the heat capacity, based on recent advances in the understanding of the interatomic potentials for transition metals. At the reference pressure of 10⁵ Pa and temperature of 1811 K (the normal melting point of Fe), the parameters are: ρ = 7037 kg/m³, K_S0 = 110 GPa, K_S' = 4.53, K_S" = -.0337 GPa-1, and γ = 2.8, with γ α ρ^-1.17. Comparison of the properties predicted by this model with the earth model PREM indicates that the outer core is 8 to 10 % less dense than pure liquid Fe at the same conditions. The inner core is also found to be 3 to 5% less dense than pure liquid Fe, supporting the idea of a partially molten inner core. The density deficit of the outer core implies that the elements dissolved in the liquid Fe are predominantly of lower atomic weight than Fe. Of the candidate light elements favored by researchers, only sulfur readily dissolves into Fe at low pressure, which means that this element was almost certainly concentrated in the core at early times. New melting data are presented for FeS and FeS₂ which indicate that the FeS₂ is the S-hearing liquidus solid phase at inner core pressures. Consideration of the requirement that the inner core boundary be observable by seismological means and the freezing behavior of solutions leads to the possibility that the outer core may contain a significant fraction of solid material. It is found that convection in the outer core is not hindered if the solid particles are entrained in the fluid flow. This model for a core of Fe and S admits temperatures in the range 3450K to 4200K at the top of the core. An all liquid Fe-S outer core would require a temperature of about 4900 K at the top of the core.

Part II.

The abundance of uses for organic compounds in the modern world results in many applications in which these materials are subjected to high pressures. This leads to the desire to be able to describe the behavior of these materials under such conditions. Unfortunately, the number of compounds is much greater than the number of experimental data available for many of the important properties. In the past, one approach that has worked well is the calculation of appropriate properties by summing the contributions from the organic functional groups making up molecules of the compounds in question. A new set of group contributions for the molar volume, volume thermal expansivity, heat capacity, and the Rao function is presented for functional groups containing C, H, and O. This set is, in most cases, limited in application to low molecular liquids. A new technique for the calculation of the pressure derivative of the bulk modulus is also presented. Comparison with data indicates that the presented technique works very well for most low molecular hydrocarbon liquids and somewhat less well for oxygen-bearing compounds. A similar comparison of previous results for polymers indicates that the existing tabulations of group contributions for this class of materials is in need of revision. There is also evidence that the Rao function contributions for polymers and low molecular compounds are somewhat different.

EQUATION OF STATE FOR POLYMERS

In the theoretical study on equation of state for polymers, much attention has been paid to the polymer in liquid state, but less to that in solid state. Therefore, some empirical and semi-empirical equations of state have been used to describe its pressure-volume-temperature (P-V-T) relations.

Conceptual problem for noncommutative inflation and the new approach for nonrelativistic inflationary equation of state

In a previous paper, we connected the phenomenological noncommutative inflation of Alexander, Brandenberger and Magueijo [ Phys. Rev. D 67 081301 (2003)] and Koh and Brandenberger [ J. Cosmol. Astropart Phys. 2007 21 ()] with the formal representation theory of groups and algebras and analyzed minimal conditions that the deformed dispersion relation should satisfy in order to lead to a successful inflation. In that paper, we showed that elementary tools of algebra allow a group-like procedure in which even Hopf algebras (roughly the symmetries of noncommutative spaces) could lead to the equation of state of inflationary radiation. Nevertheless, in this paper, we show that there exists a conceptual problem with the kind of representation that leads to the fundamental equations of the model. The problem comes from an incompatibility between one of the minimal conditions for successful inflation (the momentum of individual photons being bounded from above) and the Fock-space structure of the representation which leads to the fundamental inflationary equations of state. We show that the Fock structure, although mathematically allowed, would lead to problems with the overall consistency of physics, like leading to a problematic scattering theory, for example. We suggest replacing the Fock space by one of two possible structures that we propose. One of them relates to the general theory of Hopf algebras (here explained at an elementary level) while the other is based on a representation theorem of von Neumann algebras (a generalization of the Clebsch-Gordan coefficients), a proposal already suggested by us to take into account interactions in the inflationary equation of state.

Free energy functional expansion as the generalized approach to the equation of state of dense fluids

A version of the thermodynamic perturbation theory based on a scaling transformation of the partition function has been applied to the statistical derivation of the equation of state in a highpressure region. Two modifications of the equations of state have been obtained on the basis of the free energy functional perturbation series. The comparative analysis of the experimental PV T- data on the isothermal compression for the supercritical fluids of inert gases has been carried out. © 2012.

A squeezed state holomorphic phase space representation of equations of motion

A mapping scheme is presented which takes quantum operators associated to bosonic degrees of freedom into complex phase space integral kernel representatives. The procedure consists of using the Schrödinger squeezed state as the starting point for the construction of the integral mapping kernel which, due to its inherent structure, is suited for the description of second quantized operators. Products and commutators of operators have their representatives explicitly written which reveal new details when compared to the usual q-p phase space description. The classical limit of the equations of motion for the canonical pair q-p is discussed in connection with the effect of squeezing the quantum phase space cellular structure. © 1993.

Impulsive Partial Hyperbolic Functional Differential Equations of Fractional Order with State-Dependent Delay

MSC 2010: 26A33, 34A37, 34K37, 34K40, 35R11