Stiffened Gas Equation Research Articles

Pseudosteady shock refractions over an air–water interface

Shock refraction in a gas–liquid interface is ubiquitous in nature and engineering. This study investigates the shock refraction phenomena in air–water interfaces for various inclination angles. The interface inclination angles are achieved using a tiltable vertical shock tube. The time-resolved schlieren images are compared with numerical simulations performed using the BlastFoam solver in the OpenFOAM software. The stiffened gas equation of state is used to model water in the simulations. The shock polar analysis using modified shock relations for a stiffened gas is used to elucidate the refraction patterns. A regular refraction pattern with a reflected shock wave and a bound precursor refraction with a regular reflection are observed experimentally for the first time in an air–water system. Further, a new free precursor refraction pattern with a Mach reflection is observed. The transition criteria and the corresponding boundaries for each refraction pattern are demarcated in the ( $M_S, \theta _w^c$ )-plane. The refraction sequence and the range for various incident shock strength regimes are also identified.

Modelling and simulation on compressible multi-component gas-liquid flows with chemical reaction and phase transition effects

In this paper, mathematical models and numerical algorithm are conducted for simulating compressible two-phase multi-component reactive flows, coupling with the finite-rate-based chemistry and the instantaneous-equilibrium phase transition model. Distinctive from previous multi-component two-phase governing equations, all volume fraction transport equations, with non-conservative characteristics, of gaseous components are substituted with a thermal equilibrium assumption inside the gaseous mixture for closure. Moreover, based on the temperature polynomial (NASA-7) fitted gas properties and the stiffened gas equation of state (SG-EOS) described liquid properties, the instantaneous-equilibrium phase transition model is re-derived, and the solving strategy of the transient phase transition model is detailly provided using an algebraical approach rather than the iterative process. Having included these elements, the proposed model is validated on three canonical tests, first for pure reactive flows, second for pure two-phase flows, and third for complex compressible multi-component two-phase reactive flows with phase transition effects: hydrogen-oxygen detonation interacting with a water droplet. The cases have well demonstrated the proposed model capability and the expected fluid physics in both the multiphase and reactive flows are reproduced.

On a doubly reduced model for dynamics of heterogeneous mixtures of stiffened gases, its regularizations and their implementations.

We deal with the reduced four-equation model for the dynamics of heterogeneous compressible binary mixtures with the stiffened gas equations of state. We study its further reduced form, with the excluded volume concentrations, and with a quadratic equation for the common pressure of the components; this form can be called a quasi-homogeneous form. We prove new properties of the equation, derive simple formulas for the squared speed of sound, and present an alternative proof for a formula that relates it to the squared Wood speed of sound; also, a short derivation of the pressure balance equation is given. For the first time, we introduce regularizations of the heterogeneous model (in the quasi-homogeneous form). Previously, regularizations of such types were developed only for the homogeneous mixtures of perfect polytropic gases, and it was unclear how to cover the case considered here. In the 1D case, based on these regularizations, we construct new explicit two-level in time and symmetric three-point in space finite-difference schemes without limiters and provide numerical results for various flows with shock waves.

REGULARIZED EQUATIONS FOR DYNAMICS OF THE HETEROGENEOUS BINARY MIXTURES OF THE NOBLE-ABEL STIFFENED-GASES AND THEIR APPLICATION

We consider the so-called four-equation model for dynamics of the heterogeneous compressible binary mixtures with the Noble-Abel stiffened-gas equations of state. We exploit its quasi-homogeneous form arising after excluding the volume concentrations from the sought functions and based on a quadratic equation for the common pressure of the components. We present new properties of this equation and a simple formula for the squared speed of sound, suggest an alternative derivation for a formula relating it to the squared Wood speed of sound and state the pressure balance equation. For the first time, we give quasi-gasdynamic-type regularization of the heterogeneous model (in the quasi-homogeneous form), construct explicit two-level in time and symmetric three point in space finite-difference scheme without limiters to implement it in the 1D case and present numerical results.

Numerical simulations of underwater explosions using a compressible multi-fluid model

We present a novel solver for simulating compressible multi-fluid multiphase flow in underwater explosions (UNDEXs). The developed solver uses a modified version of Saurel's six-equation model, which includes an additional total mixture energy equation to resolve discrepancies in the thermodynamic states predicted under shock conditions. Additionally, we integrate a more precise stiffened gas equation of state (SG-EOS) that is determined using a novel method to enhance the accuracy of predicting experimental data based on a shock Hugoniot curve. We also propose a solution procedure using the modified Saurel's six-equation model on a three-dimensional (3D) structured Cartesian grid system. This involves discretizing the equation system using a Godunov scheme with a two-fluid Harten-Lax-van Leer-Contact approximate Riemann solver and a MUSCL-Hancock primitive scheme with total-variation-diminishing limiters, achieving a second-order extension. Both the dimensional splitting and fractional-step methods are utilized to model one-dimensional (1D) operators, splitting them into sequential operators. The modified model is validated for 1D and 3D problems, including the water–air shock tube, cavitation, shock–bubble interaction, and UNDEX problems in a free field, near a free surface, and near a rigid dam. Our simulations accurately predict the shockwave propagation, shock and free-surface interactions, cavitation evolution, and water jetting impact characteristics, exhibiting satisfactory agreement with those of previous studies. The proposed solver provides insight into the effects of UNDEXs on rigid structures, with potential applications in engineering and defense. The proposed method for determining the SG-EOS parameters can be applied to other areas of research involving high-pressure multi-phase flows.

Analysis on physical-constraint-preserving high-order discontinuous Galerkin method for solving Kapila's five-equation model

This work focuses on the analysis and development of a high-order physical-constraint-preserving discontinuous Galerkin (DG) method for compressible two-fluid flows by solving Kapila's five-equation model with the ideal gas equation of state. The proposed method is proved to possess the following two attractive advantages: first, the high-order DG discretization intrinsically satisfies the mechanical equilibrium criterion around an isolated material interface, and the numerical oscillations due to the use of high-order approximations are well suppressed by using a cell-based characteristic-wise weighted essentially non-oscillatory (WENO) limiter without destroying the mechanical equilibrium criterion; second, it maintains the physical-constraint-preserving property, including the bound-preserving property of volume fractions and positivity-preserving property of partial densities and internal energy. A thorough analysis on the sufficient condition is given to achieve the physical-constraint-preserving property. Several benchmark test cases are examined to validate the accuracy and robustness of the proposed method. Further discussion on the application of the physical-constraint-preserving DG method for solving Kapila's five-equation model with the stiffened gas equation of state is also presented.

A positivity‐preserving Lagrangian discontinuous Galerkin scheme with exact Riemann solver for gas‐water compressible flows

AbstractIn this study, a new cell‐centered Lagrangian discontinuous Galerkin (DG) scheme is presented to simulate gas‐water compressible flows. The two‐phase flows are governed by the compressible Euler equation with ideal and stiffened gas equations of state. We integrate the Lagrangian DG scheme with the exact gas‐water Riemann solver for calculating the numerical fluxes so that the inherent stiff features of gas‐water compressible flows are well addressed. Furthermore, in order to guarantee the positivity of the density and internal energy during the high Mach number calculation, the positivity‐preserving limiter and strong stability preserving temporal integral are incorporated to ensure the numerical stability. Six numerical examples are tested to show the accuracy, robustness and positivity‐preserving property of the present scheme. It can be found that the present scheme can handle challenging numerical cases involving large density ratio (up to 1000) and strong shock. The results obtained with the typical HLLC flux are also given for comparison and the present scheme with the exact Riemann solver shows higher accuracy, particularly in the presence of large density ratio at the gas‐water interface.

A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent flows with shocks and material interfaces. We consider the gamma-based model by Shyue (1998) [58] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac (2019) [53] and Coquel et al. (2021) [15] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner (2013) [27] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-finding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform profiles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the specific entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM, while we consider two kinds of fluctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP fluctuations. Under a given condition on the time step, both EC and CP fluctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results have allowed us to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the proposed DGSEM with either CP, or EC fluctuations, while the scheme with CP fluctuations is shown to offer better resolution capabilities.

A thermal equilibrium approach to modelling multiple solid–fluid interactions with phase transitions, with application to cavitation

This paper presents a new approach to model phase transitions in multi-material problems based on the assumption of thermal equilibrium. The method is used to construct an equation of state for water which captures the liquid–vapour phase transition and is suitable for simulating cavitating flows. A key advantage of this approach is that the physics of the phase transition is entirely encoded within the equation of state. Complex multi-physics models can therefore be augmented to include the physics of phase transitions without further complicating the partial evolution equations. This is substantiated by integrating the model within a diffuse interface scheme for coupled solid–fluid dynamics, and implemented using a structured adaptive mesh refinement (SAMR) strategy. The combined model facilitates the fully-coupled simulation of complex multi-material, multi-physics problems featuring elastoplastic solids, gases, cavitating liquids and reactive materials. Evaluation of the model is shown to add little overhead – for parts of the domain without cavitation the equations reduce to those of the underpinning model with simple stiffened gas equation of state for liquid components; for parts of the domain where cavitation does occur, it requires a non-linear relaxation with only one degree of freedom. The method is applied to several strenuous test cases for cavitating flows and shown to be in good agreement with previously published works.

A Three-dimensional Multi-species Flow Solver for the Euler Equations Combined with a Stiffened Gas Equation of State

Although numerical simulation in fluid mechanics is undergoing a significant development due to the dazzling evolution of computing means, complex physical phenomena, such as multidimensional viscous effects in turbomachinery and cavitation, remain mysterious and attract the curiosity of several researchers. Highresolution shock captures are often obtained by the WENO family of schemes, except that in problems that depend on discontinuities and shocks, an appearance of numerical oscillations weakens its ability to provide adequate captures. The use of the characteristic construction methods prevents this type of oscillation. The present paper contributes to the numerical resolution of multi-species flows of viscous, compressible, or incompressible fluids with shocks and discontinuities. The proposed numerical model can handle various configurations with a unique method based on a conservative and consistent threedimensional finite volume scheme with an aligned mesh. The system of equations is a set of Euler equations coupled with a two-parameters generalized state equation of state in three-dimensional Cartesian coordinates. This system is solved using a Roe type approximate Riemann solver, and second-order precision is obtained using limiters. The obtained numerical results maintain a nonoscillatory flow near the discontinuities, which makes the method satisfactory and shows its accuracy and robustness in different cases.

Simulations of water-vapor two-phase flows with non-condensable gas using a Noble-Able-Chemkin stiffened gas equation of state

In order to simulate some accidental scenarii which might affect a pressurized water reactor, a homogeneous two-phase water-vapor flow model taking into account non-condensable gas is considered. The liquid phase and the gaseous phase are assumed to be immiscible, but the gaseous phase is composed of two miscible components. Due to these hybrid hypotheses of miscibility, the configurations with at least one missing field are carefully examined. A semi-analytical equation of state is chosen for the liquid water, which is an extension of the Noble-Able stiffened gas equation of state. Its accuracy is assessed with respect to the reference equation of state IAPWS. The homogeneous model is first verified thanks to Riemann problems. Then, some simulations aiming at reproducing SUPERCANON experiment are presented. The amount of air dissolved in liquid water is shown to have a strong influence on numerical results, in particular on the sound speed. Some out of thermodynamical equilibrium simulations are also presented, using two time scales describing the return towards equilibrium.

An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato (1986) [5]) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [47]). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al. (2014) [10]; Gassner et al. (2016) [32]). Here, we build upon the framework provided in (Renac (2019) [55]) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (2013) [12], while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

Numerical simulation of in-nozzle flow characteristics under flash boiling conditions

Flash boiling spray has been extensively investigated because it has the potential in improving spray atomization. However, some phenomena, such as flow rate reduction, exit velocity choke and drastic bubble generation near the exit, etc., cannot be explained by existing theory of subcooled liquid jet and spray atomization. The phase change process occurring inside the nozzle still needs further exploration and investigation. However, due to the small dimensions of the nozzle, it is challenging to measure the flow characteristics experimentally. In this research, a one-dimensional two-phase flow model incorporating modified Stiffened Gas Equation of State is used to simulate the in-nozzle flow characteristics under flash boiling conditions. The model was validated against previous experimental data. The simulation results of the phase change process inside the nozzle was compared qualitatively with experimental data taken from a 2D optical nozzle. Different from subcooled conditions, superheated working liquid inside the nozzle evaporates more drastically as the result of local pressure distribution change. The flow velocity and mass flow rate are affected accordingly as well. In addition, the effects of the injection pressure, ambient pressure, and liquid temperature on these phenomena are also discussed. It was found that the ambient pressure and liquid temperature can affect the bubble generation by altering the evaporation rate of the fuel, while the injection pressure only influence the flow velocity and change the evaporation time of the liquid.

On the Noble-Abel stiffened-gas equation of state

The inviscid hydrodynamics of inert compressible media governed by the Euler equations of motion only require knowledge of a caloric equation of state e(p, v) for the material relating the internal energy e to the fluid pressure p and specific volume v (or density). For departures from the ideal gas behavior, simple equations of state such as the stiffened gas, Noble-Abel, or a hybrid recently generalized by Le Métayer and Saurel [“The Noble-Abel stiffened-gas equation of state,” Phys. Fluids 28, 046102 (2016)] can correctly model compressible flows in gases, liquids, and solids. However, reactive and multicomponent descriptions require a formal definition of temperature. In the present note, we formulate a general thermodynamically based method to determine the thermal equation of state T(p, v) compatible with a generic e(p, v) relation. We apply our method to the Noble-Abel Stiffened Gas equation of state and recover the closed form solution of Le Métayer and Saurel. We also show that variations of the model taking its exponent different from the ratio of specific heats do not permit to define a thermodynamic temperature.

Efficient exact solution procedure for quasi-one-dimensional nozzle flows with stiffened-gas equation of state

We propose an efficient and accurate exact solution procedure for the converging–diverging nozzle flows of compressible liquids governed by a stiffened-gas equation of state (SG-EOS). First, we elaborate on how to formulate a complete SG-EOS and suggest a new method to determine the parameters of SG-EOS. Next, we derive the relations for the quasi-one-dimensional nozzle flow of the SG-EOS liquids and propose an efficient solution procedure to obtain the exact solution at various boundary conditions. The proposed solution procedure can accurately calculate the position of the shock wave regardless of the number of computational grid points. We then verify the solution procedure against the previous results for the air and water nozzle flows. We also investigate the influence of the SG-EOS parameters on the nozzle flow of liquid water containing a shock wave. The proposed solution procedure can be used for the basic design of a compressible liquid nozzle and the validation of compressible two-phase flow model.

A thermodynamic closure for the simulation of multiphase reactive flows

A simple thermodynamic closure for the simulation of multiphase reactive flows is presented. It combines a fully explicit thermodynamic closure appropriate for weakly thermal multiphase flow simulations, with the classical variable heat capacity ideal gas thermodynamic closure, commonly used for reactive flows simulations. Each liquid and gas component is assumed to follow the recent Noble-Abel Stiffened Gas equation of state, fully described by a set of five parameters. A new method for setting these parameters is presented and validated through comparisons with NIST references. Comparisons with a well-known cubic equation of state, Soave-Redlich-Kwong, are also included. The Noble-Abel Stiffened-Gas equation of state is then extended as to cope with variable heat capacity, to make the mixture thermodynamic closure appropriate for multiphase reactive flows.

A modified equation of state for water for a wide range of pressure and the concept of water shock tube

An equation of state (EOS) for compressible water is presented for the application over a wide range of pressure. The proposed EOS is a modified form of the Noble Abel Stiffened Gas equation (NASG) which is originally developed for the prediction of saturation properties of water. Three well-known EOS for water viz., the Tait EOS, the Stiffened Gas EOS, and the NASG EOS are used in the comparative study of the proposed EOS. Accuracy of the density estimates of the modified EOS is quantified by comparing with the published NIST data. The evaluative study of the EOS is performed over the pressure range of 1×105 Pa to 1×109 Pa and temperature range of 280 K–370 K. The analysis reveals the supremacy of the modified version of the NASG EOS over the original version for its improved accuracy and suggests its applicability over a wide range of pressure. The problem of water shock tube is selected in the study as an application of the EOS and to demonstrate the robustness of the modified EOS. The superiority of the modified NASG EOS over the Tait EOS in modelling non-isothermal flow problems with high accuracy is also revealed in the study.

An efficient algorithm for the multicomponent compressible Navier–Stokes equations in low- and high-Mach number regimes

The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows, including nearly incompressible low-Mach number flows, flows with strong shocks, multicomponent flows with high density ratios and interfacial physics, inviscid and viscous flows, as well as flows featuring combinations of these phenomena and various interactions between them. Although many highly efficient algorithms have been proposed for simulating each type of the flows mentioned above, the construction of a universal solver is known to be challenging. Extreme cases, such as incompressible and highly compressible flows, or inviscid and highly viscous flows, require different numerical treatments in order to maintain the efficiency, stability, and accuracy of the method.Linearized block implicit (LBI) factored schemes (see e.g. Briley and McDonald, 1980 [1], Beam and Warming, 1978 [2]) are known to provide an efficient way of solving the compressible Navier–Stokes equations (compressible NSEs) implicitly, allowing us to avoid stability restrictions at low Mach number and high viscosity. However, the methods’ splitting error has been shown to grow and dominate physical fluxes as the Mach number approaches zero (see Choi and Merkle, 1985 [3]). In this paper, a splitting error reduction technique is proposed to solve the issue. A shock-capturing algorithm from [4] is reformulated in terms of finite differences, extended to the stiffened gas equation of state (SG EOS) and combined with the LBI factored scheme to stabilize the method around flow discontinuities at high Mach numbers. A novel stabilization term is proposed for low-Mach number applications. The resulting algorithm is shown to be efficient in both low- and high-Mach number regimes. Next, the algorithm is extended to a multicomponent case using an interface capturing strategy (see e.g. Saurel and Abgrall, 1999 [5]) with surface tension as a continuous surface force (see Perigaud and Saurel, 2005 [6]). Special care is taken to avoid spurious oscillations of pressure and generation of artificial acoustic waves in the numerical mixture layer. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.

A localized artificial diffusivity method to simulate compressible multiphase flows using the stiffened gas equation of state

SummaryThe development of numerical approaches to perform direct numerical simulations of compressible multiphase flows has been an active field of research for several years. Proper treatment of fluid interfaces is crucial as important physics occur in this infinitesimally small region. Furthermore, the compressibility of the fluid requires proper treatment of discontinuities. Artificial diffusivity is among a number of methods widely used for compressible flows. This study develops a general form of consistent artificial diffusion fluxes and extends the localized artificial diffusivity method for high‐order central schemes to solve multiphase flows with an interface‐capturing method. These fluxes ensure an oscillation‐free interface for pressure, velocity, and temperature without employing a sharpening technique. Moreover, the high‐order representation of all scales in the flow helps capture the wide range of instabilities inherent in these flows. The goal is to develop an approach capable of performing high‐fidelity simulations supported by physics‐driven validation. This is achieved by solving the five‐equation model with the stiffened‐gas equation of state using the proposed method for multicomponent and multiphase flows on a variety of 1D and 2D problems.

Damping effect on impact pressure from liquid droplet impingement on wet wall

Damping of the impact pressure from liquid droplet impingement (LDI) on a wet wall was studied by numerical simulation and experiment. The numerical simulation was carried out for the impact of an axisymmetric spherical droplet on a wet wall using a compressible form of the Euler equations combined with the stiffened gas equation. The impact pressures on the wall were highly damped by the influence of the liquid film prevailing over the wet wall, and the damping effect was formulated as a function of the liquid-film thickness to droplet diameter. The physical mechanism of the liquid-film damping effect is due to the two-stage compression during LDI and its weakening by the diffraction of the shock wave propagated in the liquid film. In order to understand the liquid-film damping effect obtained from the numerical simulation, experiments on LDI erosion on a wet wall were carried out for various liquid temperatures, which generated a thinner liquid film on the wall at higher temperatures by the viscous effect. The experimental results indicated that the LDI erosion rate increased with rising liquid temperatures, which corresponds to the erosion-rate growth at thinner liquid-film thicknesses. This result is consistent with the liquid-film damping effect obtained from the numerical simulation.