Stiffened Gas Equation Research Articles

Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

The Noble–Abel Stiffened-Gas (NASG) equation of state (Le Métayer, O. and Saurel, R. proposed in 2016) is extended to variable attractive and repulsive effects to improve the liquid phase accuracy when large temperature and pressure variation ranges are under consideration. The transition from pure phase to supercritical state is of interest as well. The gas phase is considered through the ideal gas assumption with variable specific heat rendering the formulation valid for high temperatures. The liquid equation-of-state constants are determined through the saturation curves making the formulation suitable for two-phase mixtures at thermodynamic equilibrium. The overall formulation is compared to experimental characteristic curves of the phase diagram showing good agreement for various fluids (water, oxygen). Compared to existing cubic equations of state, the present one is convex, a key feature for computations with hyperbolic flow models.

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas–liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

Liquid-vapor flows in heated ducts: Reference solutions

Analytical one-dimensional steady state solutions in a heated duct of constant area are determined for two-phase liquid-gas and liquid-vapor mixtures. First, the exact solution for single-phase flow is recalled in the different flow regimes (subsonic and supersonic) for given inflow conditions. These solutions are extended in the context of duct connected to an upstream tank and to a downstream pressure outflow condition. Then, reference solutions for various two-phase flow models are developed for compressible two-phase flows. They are applied in the context of ideal gas and stiffened gas equation of state for liquid-gas and liquid-vapor flows. These solutions corresponds to limit situations of partial equilibrium among the phases. The first limit situation corresponds to a two-phase flow model where both phases evolve in mechanical equilibrium only. The second one corresponds to a two-phase model where the phases evolve in both mechanical and thermal equilibria. The last one corresponds to a model describing a liquid-vapor mixture in thermodynamical equilibrium. The three limit situations are compared to show the impact given the choice of one or another equilibrium on fluid state and in particular on mass flow rate in the duct.

Effects of Different Equations of State on the Oblique Shock Wave Reflection in Solids

The equations of state of solids under high pressure are more complicated than that of gases in a variety of forms. While the existing studies on the oblique shock wave reflection usually take just one of the equations of state, lacking of the comparisons among them. This paper aims at the investigation of oblique shock wave reflection in solids through shock polar methodology. Four different forms of equations of state are considered, principal shock taking with vN particle velocity relationship and second shock taking with Grüneisen equation of state(1Du2Grüneisen), principal and second shock both taking with vN particle velocity relationship (1Du2Du), principal shock taking with vN particle velocity relationship and second shock taking with stiffened gas equation of state (1Du2Gamma), and principal and second shock both taking with stiffened gas equation of state (1Gamma2Gamma). The effects of different equations of state on the pressure behind the reflected shock show: different combination of EOS may lead to diverse results. In our opinion, the 1Du2Grüneisen and 1Du2Du shows almost the same trends in most conditions and is suitable for the theoretical analysis of oblique reflection, and 1Du2Gamma and 1Gamma2Gamma should be carefully employed in studies to avoid the incorrect result.

Mathematical modeling of impact of two metal plates using two-fluid approach

The paper is devoted to the development of the two-fluid mathematical model and the computational algorithm for the modeling of two metal plates impact. In one-dimensional case the governing system of equations comprises seven equations: three conservation laws for each fluid and transfer equation for the volume fraction of one of the fluids. Both fluids are considered to be compressible and equilibrium on velocities. Pressures equilibrium is used as fluids interface condition. The system has hyperbolic type but could not be written in the conservative form because of nozzling terms in the right-hand side of the equations. The algorithm is based on the Harten–Lax–van Leer numerical flux function. The robust computation in the presence of the interface boundary is carried out due to the special pressure relaxation procedure. The problem is solved using stiffened gas equations of state for each fluid. The parameters in the equations of state are calibrated using the results of computations using wide-range equations of state for the metals. In simulations of metal plates impact we get two shocks after the initial impact that propagate to the free surfaces of the samples. The characteristics of shock waves are close (maximum relative error in characteristics of shocks is not greater than 7%) to the data from the wide-range equations of states computations.

Improvement of three-dimensional mathematical model for the simulation of impact of high-speed metallic plates

The work is devoted to the development of the authors software package Turbulence Problem Solver originally created for the study of three-dimensional problems of hydrodynamic instabilities. Mathematical model is based on the three-dimensional two-component Euler equations, the numerical algorithm is based on the grid-characteristics scheme of the second approximation order in space. The functionality of the mathematical model and the numerical algorithm of the package is extended to consider the problem of high-speed metallic plates impact. The detailed description of the proposed numerical approach is presented. The problem of high-speed metallic plates impact is solved using stiffened gas equation of state. The parameters of the equation of state are calibrated on the basis of computational results obtained with the use of wide-range equations of states for the metals.

A REGULARIZED STIFFENED-GAS EQUATION OF STATE

An alternative to the stiffened-gas equation of state is proposed that corrects for unphysical behavior, namely the returning of negative pressure values for densities lower than the reference density. The question of generalizing a given isentropic equation of state is first addressed. Then, a regularized stiffened-gas equation of state is derived, which is the main goal of this work. Finally, the use of the new equation of state is briefly illustrated for a classical computational fluid dynamics benchmark concerning the evaluation of liquid impact on a wall.

Numerical simulation of the impact of high-speed metallic plates using two approaches

The paper is devoted to the mathematical modeling of the problem of two metal plates impact using two approaches. In the first approach the problem is solved using three-dimensional Euler equations and the stiffened-gas equation of state for the media. The parameters of the equation of state are calibrated using wide-range equations of state computations of the parameters of shock waves which form after the impact. The second approach is based on one-dimensional two-fluid seven-equation model. In simulations of metal plates impact we get two shocks after the initial impact that propagate to the free surfaces of the samples. The characteristics of shock waves are close (maximum relative error in characteristics of shocks is not greater than 7%) to the data from the wide-range equations of states computations.

Numerical and experimental investigation of oblique shock wave reflection off a water wedge

Shock wave interaction with solid wedges has been an area of much research in past decades, but so far very few results have been obtained for shock wave reflection off liquid wedges. In this study, numerical simulations are performed using the inviscid Euler equations and the stiffened gas equation of state to study the transition angles, reflection patterns and triple point trajectory angles of shock reflection off solid and water wedges. Experiments using an inclined shock tube are also performed and schlieren photography results are compared to simulations. Results show that the transition angles for the water wedge cases are within 5.3 % and 9.2 %, for simulations and experiments respectively, compared to results obtained with the theoretical detachment criterion for solid surfaces. Triple point trajectory angles are measured and compared with analytic solutions, agreement within $1.3^{\circ }$ is shown for the water wedge cases. The transmitted wave in the water observed in the simulation is quantitatively studied, and two different scenarios are found. For low incident shock Mach numbers, $M_{s}=1.2$ and 2, no shock wave is formed in the water but a precursor wave is induced ahead of the incident shock wave and passes the information from the water back into the air. For high incident shock Mach numbers, $M_{s}=3$ and 4, precursor waves no longer appear but instead a shock wave is formed in the water and attached to the Mach stem at every instant. The temperature field in the water is measured in the simulation. For strong incident shock waves, e.g. $M_{s}=4$, the temperature increment in the water is up to 7.3 K.

Effects of air chemistry and stiffened EOS of air in numerical simulations of bubble collapse in water

Numerical simulations of bubble collapse show that the combined effects of air chemistry at high temperature and a stiffened gas equation of state for air at high pressures lead to significantly lower peak temperature and pressure during the evolution.

High-order spectral volume scheme for multi-component flows using non-oscillatory kinetic flux

Abstract In this paper, an arbitrary high-order compact method is developed for compressible multi-component flows with a stiffened gas equations of state (EOS). The main contribution is combining the high-order, conservative, compact spectral volume scheme (SV) with the non-oscillatory kinetic scheme (NOK) to solve the quasi-conservative extended Euler equations of compressible multi-component flows. The new scheme consists of two parts: the conservative part and the non-conservative part. The original high order compact SV scheme is used to discretize the conservative part directly. In order to treat the equation of state of the stiffened gas, the NOK scheme is utilized to compute the numerical flux. Then, careful analysis is made to satisfy the necessary condition to avoid unphysical oscillation near the material interfaces. After that, a high-order compact scheme for the non-conservative part is obtained. This new scheme has the following advantages for numerical simulations of compressible multi-component stiffened gas: high order accuracy with compact stencil and oscillation-free near the material interfaces. Numerical tests demonstrate the good performance and the efficiency of the new scheme for multi-component flow simulations.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

Transient simulation of a two-phase loop thermosyphon with a model out of thermodynamic equilibrium

Numerical investigation of two-phase loop thermosyphon (2PLT) in steady and transient states is addressed. A one-dimensional two-phase flow model describing a liquid-gas mixture in both mechanical and thermal equilibrium but out of thermodynamic equilibrium is developed. The model considers subcooled liquid and over heated vapor as well as phase transition (evaporation and condensation). The flow model is solved with a specific hyperbolic solver based on Godunov method and Harten-Lax-van Leer-Contact (HLLC) Riemann solver. A parametric study on the thermal power at the evaporator is performed in steady and transient states, the aim being to determine the effects of thermal power increase at the evaporator on the loop behavior. The comparison between Goodwin and Stiffened Gas (SG) equation of state (EOS) models shows fair agreement for latent heat of vaporization, specific volume and enthalpy for both liquid and vapor phases. Simulation of four test cases, corresponding to different evaporator thermal loads, is also carried out in transient state showing that loop response is correctly reproduced by this numerical approach, novel in the context of thermosyphon loops.

A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer–Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer–Nunziato model, namely Schwendeman–Wahle–Kapila's Godunov-type scheme [39] and Tokareva–Toro's HLLC scheme [44]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.

A two-fluid four-equation model with instantaneous thermodynamical equilibrium

We consider an equilibrium version of a common two-fluid model for pipe flow, containing one mixture mass equation and one mixture energy equation. This model can be derived from a five-equation model with instantaneous thermal equilibrium, to which additional phase relaxation terms are added. An original contribution of this paper is a quasilinear formulation of the instantaneous phase relaxation limit. From this, the mixture sound speed intrinsic to the model can be extracted. This allows us to directly prove some subcharacteristic conditions with respect to a previously established model hierarchy of different relaxation processes. These subcharacteristic conditions reveal the fundamental insight of this paper; in the hierarchy, thermodynamic versus velocity relaxation both reduce the mixture sound velocity with a factor that is independent of whether the other type of relaxation has been performed.

The Noble-Abel Stiffened-Gas equation of state

Hyperbolic two-phase flow models have shown excellent ability for the resolution of a wide range of applications ranging from interfacial flows to fluid mixtures with several velocities. These models account for waves propagation (acoustic and convective) and consist in hyperbolic systems of partial differential equations. In this context, each phase is compressible and needs an appropriate convex equation of state (EOS). The EOS must be simple enough for intensive computations as well as boundary conditions treatment. It must also be accurate, this being challenging with respect to simplicity. In the present approach, each fluid is governed by a novel EOS named “Noble Abel stiffened gas,” this formulation being a significant improvement of the popular “Stiffened Gas (SG)” EOS. It is a combination of the so-called “Noble-Abel” and “stiffened gas” equations of state that adds repulsive effects to the SG formulation. The determination of the various thermodynamic functions and associated coefficients is the aim of this article. We first use thermodynamic considerations to determine the different state functions such as the specific internal energy, enthalpy, and entropy. Then we propose to determine the associated coefficients for a liquid in the presence of its vapor. The EOS parameters are determined from experimental saturation curves. Some examples of liquid-vapor fluids are examined and associated parameters are computed with the help of the present method. Comparisons between analytical and experimental saturation curves show very good agreement for wide ranges of temperature for both liquid and vapor.

Development of Numerical Simulation Method for Compressible Gas-Liquid Two-Phase Flows

A numerical simulation method of compressible gas-liquid two-phase flow is developed for analyses of a cavitation bubble. Thermodynamic state of both phases is described with stiffened gas equation of state. Interface of two phases is captured by Level-Set method. As internal energy jump between two phases is critical for the stability of computation, total energy equation is modified so that inviscid flux of energy is smoothly connected across the interface. Detail of governing equations as well as their discretization is described followed by the result of one-dimensional simple example computation.

Investigation on the laser-induced shock pressure with condensed matter model

In this paper, based on the basic shock wave relations and the stiffened gas equation of state, the laser-induced shock pressures, which are generated with homogenous rectangular laser pulse at power density of several GW/cm2, are proposed theoretically and experimentally. The laser-induced shock pressures are affected by the laser power density as well as the shock velocities, initial densities and adiabatic exponents of the target material and confined water which are considered to be condensed matter other than “solid” or “perfect gas” in the previous work. Performed on Al-2024 alloys, the laser-induced shock wave was measured with the poly(vinylidene fluoride) transducer and recorded by the oscilloscope. The laser-induced shock pressure derived from the proposed method agrees much better with the experimental results than that using the “solid” or “perfect gas” theory.

Numerical simulations of compressible flows using multi-fluid models

Numerical simulations of two-fluid flow models based on the full Navier–Stokes equations are presented. The models include six and seven partial differential equations, namely, six- and seven-equation models. The seven-equation model consists of a non-conservative equation for volume fraction evolution of one of the fluids and two sets of balance equations. Each set describes the motion of the corresponding fluid, which has its own pressure, velocity, and temperature. The closure is achieved by two stiffened gas equations of state. Instantaneous relaxation towards equilibrium is achieved by velocity and pressure relaxation terms. The six-equation model is deduced from the seven-equation model by assuming an infinite rate of velocity relaxation. In this model, a single velocity is used for both fluids. The numerical solutions are obtained by applying the Strang splitting technique. The numerical solutions are examined in a set of one, two, and three dimensions for both the six- and seven-equation models. The results indicate very good agreement with the experimental results. There is an insignificant difference between the results of the two models, but the six-equation model is much more economical compared to the seven-equation model.

Simulating cavitating liquid jets using a compressible and equilibrium two-phase flow solver

In internal combustion engines the injection of high-pressure liquid fuel into a low-pressure gas through a nozzle passage is an important process to atomize the liquid and achieve optimal fuel–air mixing. A Computational Fluid Dynamics (CFD) model is developed in the present work to simulate the internal- and external-nozzle flow fields in an integrated way. The model assumes that the flow within and near the nozzle is continuous, and an Eulerian flow solver is developed using the general conservation laws of fluid dynamics. Differences in the thermodynamic states of the liquid and gas phases are modeled with a Stiffened Gas Equation of State (EOS). A practical phase equilibrium solver is developed, and is implemented into the Eulerian flow solver to predict phase changes in the flows – in particular, cavitation of the liquid within the injector nozzle passage. The combined equilibrium solver is applied to single-component and two component flows with one component being non-condensable air. A number of test problems are simulated to verify the numerical methods and validate the proposed models. These include two-phase shock tube problems, a converging–diverging nozzle flow problem, a submerged liquid jet problem, and a cavitating liquid jet problem.