Realistic Equation Of State Research Articles

The Influx-Management Envelope Considering Real-Fluid Behavior

Summary The influx-management envelope (IME), Culen et al. (2016), is a decision-making tool for how to deal with an influx during managed-pressure-drilling (MPD) operations that offers a substantial improvement over the traditional MPD well-control matrix (WCM). However, in the case study by Gabaldon et al. (2017), it has been found that the simplified analytical solution introduced by Culen et al. (2016) makes the IME inaccurate for many real-world applications. This paper extends the original IME that considers the gas migration in the annulus as a single bubble, without making the simplifying assumptions that are required to make the equations explicit and analytically solvable. The underlying equations that are required to develop the IME are derived from first principles, and it is shown that using realistic equations of state (EOSs) for gas, and a single-bubble-type model, the equations for the IME can be numerically solved yielding less conservative limits. The proposed approach is significantly faster than constructing the IME through high-fidelity simulations. The proposed method allows for the calculation of a peak circulating pressure at the surface and the maximum weak-point pressure for a given kick size and initial shut-in pressure, as well as a kick envelope with respect to formation limits. This makes the contributions of this paper also relevant for traditional well-control situations. The effect of considering various simplifying assumptions on the resulting IME is studied, and different scenarios that compare the results of the proposed approach with that of the original single-bubble equation for the IME (Culen et al. 2016) are presented.

Computational modelling of multi-material energetic materials and systems

ABSTRACT In this paper, we present a systematic roadmap for developing a robust and parallel multi-material reactive hydrodynamic solver that integrates historically stable algorithms with new and current modern methods to solve explosive system design problems. The Ghost Fluid Method and Riemann solvers were used to enforce appropriate interface boundary conditions. Improved performance in terms of computational work and convergence properties was achieved by modifying a local node sorting strategy that decouples ghost nodes, allowing us to set material boundary conditions via an explicit procedure, removing the need to solve a coupled system of equations numerically. The locality and explicit nature of the node sorting concept allows for greater levels of parallelism and lower computational cost when populating ghost nodes. Non-linear numerical issues endemic to the use of real Equations of State in hydro-codes were resolved by using more thermodynamically consistent forms allowing us to accurately resolve large density gradients associated with high energy detonation problems at material interfaces. Pre-computed volume tables were implemented adding to the robustness of the solver base.

Possibility of rapid neutron star cooling with a realistic equation of state

Abstract Whether fast cooling processes occur or not is crucial for the thermal evolution of neutron stars. In particular, the threshold of the direct Urca process, which is one of the fast cooling processes, is determined by the interior proton fraction $Y_p$, or the nuclear symmetry energy. Since recent observations indicate the small radius of neutron stars, a low value is preferred for the symmetry energy. In this study, simulations of neutron star cooling are performed adopting three models for the equation of state (EoS): Togashi, Shen, and LS220 EoSs. The Togashi EoS has been recently constructed with realistic nuclear potentials under finite temperature, and found to account for the small radius of neutron stars. As a result, we find that, since the direct Urca process is forbidden, the neutron star cooling is slow with use of the Togashi EoS. This is because the symmetry energy of Togashi EoS is lower than those of other EoSs. Hence, in order to account for observed age and surface temperature of isolated neutron stars with the use of the Togashi EoS, other fast cooling processes are needed regardless of the surface composition.

Assessment of numerical schemes for complex two-phase flows with real equations of state

Some accidental scenarii studied in the framework of the nuclear safety analysis involve liquids undergoing strong pressure drops at high temperature. In order to perform realistic simulations of such situations, a code based on a model that can handle both the thermodynamical disequilibrium between liquid and vapor and complex equations of state is required. We propose herein to test a homogeneous model built on the basis of the Euler system of equations and complemented by a mixture pressure law. The latter is defined in accordance with the Gibbs relation on the basis of the phasic pressures which are defined through a look-up table based on the IAPWS-97 formulation. A wide range of verification problems (Riemann problems) is then studied to assess the behavior of the numerical schemes for this complex equation of state. The tested relaxation scheme is the best compromise between accuracy and stability. At last, a simple test case of vaporization near a wall is investigated in order to test some return to thermodynamical-equilibrium time-scale based on the nucleation theory.

On the frequency correlations of low-frequency QPOs with kilohertz QPOs in accreting millisecond X-ray pulsars

ABSTRACT We investigate frequency correlations of low frequency (LF, <80 Hz) and kHz quasi-periodic oscillations (QPOs) using the complete RXTE data sets on six accreting millisecond X-ray pulsars (AMXPs) and compare them to those of non-pulsating neutron star (NS) low-mass X-ray binaries with known spin. For the AMXPs SAX J1808.4−3658 and XTE J1807−294, we find frequency-correlation power-law indices that, surprisingly, are significantly lower than in the non-pulsars, and consistent with the relativistic precession model (RPM) prediction of 2.0 appropriate to test-particle orbital and Lense–Thirring precession frequencies. As previously reported, power-law normalizations are significantly higher in these AMXPs than in the non-pulsating sources, leading to requirements on the NS specific moment of inertia in this model that cannot be satisfied with realistic equations of state. At least two other AMXPs show frequency correlations inconsistent with those of SAX J1808.4−3658 and XTE J1807−294, and possibly similar to those of the non-pulsating sources; for two AMXPs no conclusions could be drawn. We discuss these results in the context of a model that has had success in black hole (BH) systems involving a torus-like hot inner flow precessing due to (prograde) frame dragging, and a scenario in which additional (retrograde) magnetic and classical precession torques not present in BH systems are also considered. We show that a combination of these interpretations may accommodate our results.

Role of crustal physics in the tidal deformation of a neutron star

In the late inspiral phase, gravitational waves from binary neutron star mergers carry the imprint of the equation of state due to the tidally deformed structure of the components. If the stars contain solid crusts, then their shear modulus can affect the deformability of the star and, thereby, modify the emitted signal. Here, we investigate the effect of realistic equations of state (EOSs) of the crustal matter, with a realistic model for the shear modulus of the stellar crust in a fully general relativistic framework. This allows us to systematically study the deviations that are expected from fluid models. In particular, we use unified EOSs, both relativistic and non-relativistic, in our calculations. We find that realistic EOSs of crusts cause a small correction, of $\sim 1\%$, in the second Love number. This correction will likely be subdominant to the statistical error expected in LIGO-Virgo observations at their respective advanced design sensitivities, but rival that error in third generation detectors. For completeness, we also study the effect of crustal shear on the magnetic-type Love number and find it to be much smaller.

Improved analytic modeling of neutron star interiors

Studies of neutron stars are extremely timely given the recent detection of gravitational waves from a binary neutron star merger GW170817, and an International Space Station payload NICER currently in operation that aims to determine radii of neutron stars to a precision better than 5%. In many cases, neutron star solutions are constructed numerically due to the complexity of the field equations with realistic equations of state. However, in order to relate observables like the neutron star mass and radius to interior quantities like central density and pressure, it would be useful to provide an accurate, analytic modeling of a neutron star interior. One such solution for static and isolated neutron stars is the Tolman VII solution characterized only by two parameters (e.g. mass and radius), though its agreement with numerical solutions is not perfect. We here introduce an improved analytic model based on the Tolman VII solution by introducing an additional parameter to make the analytic density profile agree better with the numerically obtained one. This additional parameter can be fitted in terms of the stellar mass, radius and central density in an equation-of-state-insensitive way. In most cases, we find that the new model more accurately describes realistic profiles than the original Tolman VII solution by a factor of 2-5. Our results are first-step calculations towards constructing analytic interior solutions for more realistic neutron stars under rotation or tidal deformation.

Solving the Riemann problem for realistic astrophysical fluids

We present new methods to solve the Riemann problem both exactly and approximately for general equations of state (EoS) to facilitate realistic modeling and understanding of astrophysical flows. The existence and uniqueness of the new exact general EoS Riemann solution can be guaranteed if the EoS is monotone regardless of the physical validity of the EoS. We confirm that: (1) the solution of the new exact general EoS Riemann solver and the solution of the original exact Riemann solver match when calculating perfect gas Euler equations; (2) the solution of the new Harten-Lax-van Leer-Contact (HLLC) general EoS Riemann solver and the solution of the original HLLC Riemann solver match when working with perfect gas EoS; and (3) the solution of the new HLLC general EoS Riemann solver approaches the new exact solution. We solve the EoS with two methods, one is to interpolate 2D EoS tables by the bi-linear interpolation method, and the other is to analytically calculate thermodynamic variables at run-time. The interpolation method is more general as it can work with other monotone and realistic EoS while the analytic EoS solver introduced here works with a relatively idealized EoS. Numerical results confirm that the accuracy of the two EoS solvers is similar. We study the efficiency of these two methods with the HLLC general EoS Riemann solver and find that analytic EoS solver is faster in the test problems. However, we point out that a combination of the two EoS solvers may become favorable in some specific problems. Throughout this research, we assume local thermal equilibrium.

Tidal Love numbers of neutron stars in f(R) gravity

The recent detection of gravitational waves from a neutron star merger was a significant step towards constraining the nuclear matter equation of state by using the tidal Love numbers (TLNs) of the merging neutron stars. Measuring or constraining the neutron star TLNs allows us in principle to exclude or constraint many equations of state. This approach, however, has the drawback that many modified theories of gravity could produce deviations from General Relativity similar to the deviations coming from the uncertainties in the equation of state. The first and the most natural step in resolving the mentioned problem is to quantify the effects on the TLNs from the modifications of General Relativity. With this motivation in mind, in the present paper we calculate the TLNs of (non-rotating) neutron stars in R^2-gravity. More precisely, by solving numerically the perturbation equations, we calculate explicitly the polar and the axial l=2 TLNs for three characteristic realistic equations of state and compare the results to General Relativity. Our results show that while the polar TLNs are slightly influenced by the R^2 modification of General Relativity, the axial TLNs can be several times larger (in terms of the absolute value) compared to the general relativistic case.

Entropy stable modeling of non-isothermal multi-component diffuse-interface two-phase flows with realistic equations of state

In this paper, we consider mathematical modeling and numerical simulation of non-isothermal compressible multi-component diffuse-interface two-phase flows with realistic equations of state. A general model with the general reference velocity is derived rigorously through thermodynamical laws and Onsager’s reciprocal principle, and it is capable of characterizing compressibility and partial miscibility between multiple fluids. We prove a novel relation between the pressure, temperature and chemical potentials, which results in a new formulation of the momentum conservation equation indicating that the gradients of chemical potentials and temperature become the primary driving force of the fluid motion except for the external forces. A key challenge in numerical simulation is to develop entropy stable numerical schemes preserving the laws of thermodynamics. Based on the convex–concave splitting of Helmholtz free energy density with respect to molar densities and temperature, we propose an entropy stable numerical method, which solves the total energy balance equation directly, and thus, naturally satisfies the first law of thermodynamics. Unconditional entropy stability (the second law of thermodynamics) of the proposed method is proved by estimating the variations of Helmholtz free energy and kinetic energy with time steps. Numerical results validate the proposed method.

Constraint on energy-momentum squared gravity from neutron stars and its cosmological implications

Deviations from the predictions of general relativity due to energy-momentum squared gravity (EMSG) are expected to become pronounced in the high density cores of neutron stars. We derive the hydrostatic equilibrium equations in EMSG and solve them numerically to obtain the neutron star mass-radius relations for four different realistic equations of state. We use the existing observational measurements of the masses and radii of neutron stars to constrain the free parameter, $\alpha ,$ that characterizes the coupling between matter and spacetime in EMSG. We show that $-10^{-38}\,\mathrm{cm^{3}/erg}<\alpha <+10^{-37}\,\mathrm{cm^{3}/erg}$. Under this constraint, we discuss what contributions EMSG can provide to the physics of neutron stars, in particular, their relevance to the so called \textit{hyperon puzzle} in neutron stars. We also discuss how EMSG alters the dynamics of the early universe from the predictions of the standard cosmological model. We show that EMSG leaves the standard cosmology safely unaltered back to $t\sim 10^{-4}$ seconds at which the energy density of the universe is $\sim 10^{34}\,\mathrm{erg\,cm^{-3}}$.

Verification Studies for the Noh Problem Using Nonideal Equations of State and Finite Strength Shocks

The Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan–Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which, in combination with the Rankine–Hugoniot jump conditions, provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case, the solution can be applied to shocks of arbitrary strength, but for the cylindrical and spherical geometries, it is required that the analysis be restricted to strong shocks. The exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode free Lagrangian (FLAG).

Analysis of Calculation of the Critical Temperature of Metals

General considerations on the form of the real equation of state make it possible to assume that the ratio of the melting temperature to the critical temperature is constant for each group of elements. This assumption is verified for two groups of elements, the critical temperatures of which are measured. It is shown that, for certain groups of elements, the critical temperatures of which are calculated instead of being measured, this assumption is also valid. The critical temperatures of certain elements are calculated or refined.

Extended I-Love relations for slowly rotating neutron stars

Observations of gravitational waves from inspiralling neutron star binaries---such as GW170817---can be used to constrain the nuclear equation of state by placing bounds on stellar tidal deformability. For slowly rotating neutron stars, the response to a weak quadrupolar tidal field is characterized by four internal-structure-dependent constants called "Love numbers." The tidal Love numbers $k_2^\text{el}$ and $k_2^\text{mag}$ measure the tides raised by the gravitoelectric and gravitomagnetic components of the applied field, and the rotational-tidal Love numbers $\mathfrak{f}^\text{o}$ and $\mathfrak{k}^\text{o}$ measure those raised by couplings between the applied field and the neutron star spin. In this work we compute these four Love numbers for perfect fluid neutron stars with realistic equations of state. We discover (nearly) equation-of-state independent relations between the rotational-tidal Love numbers and the moment of inertia, thereby extending the scope of I-Love-Q universality. We find that similar relations hold among the tidal and rotational-tidal Love numbers. These relations extend the applications of I-Love universality in gravitational-wave astronomy. As our findings differ from those reported in the literature, we derive general formulas for the rotational-tidal Love numbers in post-Newtonian theory and confirm numerically that they agree with our general-relativistic computations in the weak-field limit.

Numerical Analysis of Vapor-Liquid Phase Change Using Real Equation of State

Numerical analysis of vapor-liquid phase change was performed. Real equation of state was used to consider the compressibility of liquid and to reproduce phenomena affected by the speed of sound. An example of such phenomenon is choked flow. The CFD method was inspected by comparing with existing experiments.

A complete study of the precision of the concentric MacLaurin spheroid method to calculate Jupiter’s gravitational moments

A few years ago, Hubbard (2012, ApJ, 756, L15; 2013, ApJ, 768, 43) presented an elegant, non-perturbative method, called concentric MacLaurin spheroid (CMS), to calculate with very high accuracy the gravitational moments of a rotating fluid body following a barotropic pressure-density relationship. Having such an accurate method is of great importance for taking full advantage of the Juno mission, and its extremely precise determination of Jupiter gravitational moments, to better constrain the internal structure of the planet. Recently, several authors have applied this method to the Juno mission with 512 spheroids linearly spaced in altitude. We demonstrate in this paper that such calculations lead to errors larger than Juno’s error bars, invalidating the aforederived Jupiter models at the level required by Juno’s precision. We show that, in order to fulfill Juno’s observational constraints, at least 1500 spheroids must be used with a cubic, square or exponential repartition, the most reliable solutions. When using a realistic equation of state instead of a polytrope, we highlight the necessity to properly describe the outermost layers to derive an accurate boundary condition, excluding in particular a zero pressure outer condition. Providing all these constraints are fulfilled, the CMS method can indeed be used to derive Jupiter models within Juno’s present observational constraints. However, we show that the treatment of the outermost layers leads to irreducible errors in the calculation of the gravitational moments and thus on the inferred physical quantities for the planet. We have quantified these errors and evaluated the maximum precision that can be reached with the CMS method in the present and future exploitation of Juno’s data.

Linearly Decoupled Energy-Stable Numerical Methods for Multicomponent Two-Phase Compressible Flow

In this paper, for the first time we propose two linear, decoupled, energy-stable numerical schemes for multi-component two-phase compressible flow with a realistic equation of state (e.g. Peng-Robinson equation of state). The methods are constructed based on the scalar auxiliary variable (SAV) approaches for Helmholtz free energy and the intermediate velocities that are designed to decouple the tight relationship between velocity and molar densities. The intermediate velocities are also involved in the discrete momentum equation to ensure the consistency with the mass balance equations. Moreover, we propose a component-wise SAV approach for a multi-component fluid, which requires solving a sequence of linear, separate mass balance equations. We prove that the methods preserve the unconditional energy-dissipation feature. Numerical results are presented to verify the effectiveness of the proposed methods.

Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility

A general diffuse interface model with a realistic equation of state (e.g. Peng–Robinson equation of state) is proposed to describe the multi-component two-phase fluid flow based on the principles of the NVT-based framework which is an attractive alternative recently over the NPT-based framework to model the realistic fluids. The proposed model uses the Helmholtz free energy rather than Gibbs free energy in the NPT-based framework. Different from the classical routines, we combine the first law of thermodynamics and related thermodynamical relations to derive the entropy balance equation, and then we derive a transport equation of the Helmholtz free energy density. Furthermore, by using the second law of thermodynamics, we derive a set of unified equations for both interfaces and bulk phases that can describe the partial miscibility of multiple fluids. A relation between the pressure gradient and chemical potential gradients is established, and this relation leads to a new formulation of the momentum balance equation, which demonstrates that chemical potential gradients become the primary driving force of fluid motion. Moreover, we prove that the proposed model satisfies the total (free) energy dissipation with time. For numerical simulation of the proposed model, the key difficulties result from the strong nonlinearity of Helmholtz free energy density and tight coupling relations between molar densities and velocity. To resolve these problems, we propose a novel convex–concave splitting of Helmholtz free energy density and deal well with the coupling relations between molar densities and velocity through very careful physical observations with a mathematical rigor. We prove that the proposed numerical scheme can preserve the discrete (free) energy dissipation. Numerical tests are carried out to verify the effectiveness of the proposed method.

A roe-type numerical solver for the two-phase two-fluid six-equation model with realistic equation of state

In the nuclear industry, a method based on a staggered grid is used in two-phase flow system codes such as RELAP, TRAC, and CATHARE. Solving the two-phase two-fluid model with this method is complicated. The objective of this article is to develop a new solver, which is mathematically consistent and algebraically simpler than existing codes. The extension of existing shock-capturing upwind schemes for single-phase flows is our way. A numerical solver with a Roe-type numerical flux is formulated based on a very well-structured Jacobian matrix. We formulate the Jacobian matrix with arbitrary equation of state and simplify the Jacobian matrix to a simple and structured form with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the characteristic polynomial of the Jacobian matrix is simple and suitable for analytical analysis. Results from the characteristic analysis of the two-phase system are consistent with well-known facts, such as the ill-posedness of the basic two-phase two-fluid model which assumes all pressure terms are equal. An explicit numerical solver, with a Roe-type numerical flux, is constructed based on the characteristic analysis. A critical feature of the method is that the formulation does not depend on the form of equation of state and the method is applicable to realistic two-phase problems. We demonstrate solver performance based on three two-phase benchmark problems: two-phase shock-tube problem, faucet flow problem, and Christensen boiling pipe problem. The solutions are in excellent agreement with analytical solutions and numerical solutions from a system code. The new solver provides essential framework for developing a more accurate and robust solver for realistic reactor safety analysis. However, improvements on the new solver is necessary for achieving a high-order accuracy and increasing the robustness.

Dark matter effect on realistic equation of state in neutron stars

In this work we apply relativistic mean-field theory in neutron stars assuming that fermionic dark matter is trapped inside the star and interacts directly with neutrons by exchanging Standard Model Higgs bosons. For realistic values of the parameters of the model we compute numerically the equation of state, and we compare it to the standard one. Furthermore, the mass-to-radius relation for both equations of state (pure neutron matter as well as admixed DM-neutron star) is shown, and the highest star mass for both cases is reported.