Stiffened Gas Equation Of State Research Articles

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.

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.

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.

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.

Two-Phase Flow of CO2 with Phase Transfer

A model for two-phase pipeline flow of CO2 with phase transfer is presented. Two different relaxation models for phase transfer are developed. The system of equations is solved by splitting it into a hyperbolic conservation law and a relaxation ODE, solved by a multi-stage (MUSTA) finite volume scheme and the backward Euler method, respectively. The stiffened-gas equation of state is used for calculating thermodynamic properties. Simulation results from a depressurisation case of a CO2 pipe, causing phase change and cooling, are presented, showing that statistical rate theory predicts solutions close to those of an equilibrium model with instant phase transfer.

Depressurization of carbon dioxide in pipelines–Models and methods

A model for two-phase carbon dioxide transport with phase transfer in pipelines is presented. The stiffened-gas equation of state is used for calculating thermodynamical properties. The convergence and accuracy of two numerical methods–a centred multi-stage (MUSTA) scheme and an upwind Roe scheme–are compared. Results for the depressurization of a pipe, causing phase change and cooling, are presented, showing the importance of an accurate numerical method. The effect of a discontinuous sound velocity on the depressurization is discussed.

On Solutions to Equilibrium Problems for Systems of Stiffened Gases

We consider an isolated system of N immiscible fluids, each following a stiffened-gas equation of state. We consider the problem of calculating equilibrium states from the conserved fluid-mechanical properties, i.e., the partial densities and internal energies. We consider two cases; in each case mechanical equilibrium is assumed, but the fluids may or may not be in thermal equilibrium. For both cases, we address the issues of existence, uniqueness, and physical validity of equilibrium solutions. We derive necessary and sufficient conditions for physically valid solutions to exist, and prove that such solutions are unique. We show that for both cases, physically valid solutions can be expressed as the root of a monotonic function in one variable. We then formulate efficient algorithms which unconditionally guarantee global and quadratic convergence toward the physically valid solution.

A Pressure Relaxation Closure Model for One-dimensional, Two-material Lagrangian Hydrodynamics Based on the Riemann Problem

Despite decades of development, Lagrangian hydrodynamics of strengthfree materials presents numerous open issues, even in one dimension. We focus on the problem of closing a system of equations for a two-material cell under the assumption of a single velocity model. There are several existing models and approaches, each possessing different levels of fidelity to the underlying physics and each exhibiting unique features in the computed solutions. We consider the case in which the change in heat in the constituent materials in the mixed cell is assumed equal. An instantaneous pressure equilibration model for a mixed cell can be cast as four equations in four unknowns, comprised of the updated values of the specific internal energy and the specific volume for each of the two materials in the mixed cell. The unique contribution of our approach is a physics-inspired, geometry-based model in which the updated values of the sub-cell, relaxing-toward-equilibrium constituent pressures are related to a local Riemann problem through an optimization principle. This approach couples the modeling problem of assigning sub-cell pressures to the physics associated with the local, dynamic evolution. We package our approach in the framework of a standard predictor-corrector time integration scheme. We evaluate our model usingmore » idealized, two material problems using either ideal-gas or stiffened-gas equations of state and compare these results to those computed with the method of Tipton and with corresponding pure-material calculations.« less

Two-Dimensional Two-Fluid Two-Phase Flow Simulation Using an Approximate Jacobian Matrix for HLL Scheme

In this article, a numerical method to solve the two-set, eight-equation, compressible, two-fluid, two-phase flow model is developed in two dimensions as an extension of the earlier one-dimensional version. The multidimensional two-fluid model can be effectively solved by a finite-volume method in a rotated reference frame. In order to estimate the fastest wave speeds in the hyperbolic equation system for the Harten–Lax–van Leer (HLL) scheme, we first regard the liquid phase as compressible by taking the stiffened-gas equation of state. Then we derive the two-dimensional approximate Jacobian matrix and obtain the associated eight analytic eigenvalues. Using the HLL scheme, we solve a few two-phase flow problems including shape cavitation and underwater explosion, demonstrating application of the present numerical method to meaningful problems.

A new pressure relaxation closure model for one-dimensional two-material Lagrangian hydrodynamics

We present a new model for closing a system of Lagrangian hydrodynamics equations for a two-material cell with a single velocity model. We describe a new approach that is motivated by earlier work of Delov and Sadchikov and of Goncharov and Yanilkin. Using a linearized Riemann problem to initialize volume fraction changes, we require that each material satisfy its own p dV equation, which breaks the overall energy balance in the mixed cell. To enforce this balance, we redistribute the energy discrepancy by assuming that the corresponding pressure change in each material is equal. This multiple-material model is packaged as part of a two-step time integration scheme. We compare results of our approach with other models and with corresponding pure-material calculations, on two-material test problems with ideal-gas or stiffened-gas equations of state.