Shock jump relations for multiphase mixtures with stiff mechanical relaxation

Examples of multiphase mixtures for which velocity and pressure relaxation are a stiff phenomenon are involved in many practical applications dealing with condensed phase mixtures, solid alloys, propellants and solid explosives, specific composite materials, micro- and nano-structured mixtures, etc. Shock relations for the mixture necessitate the determination of the volume fraction jump or any other thermodynamic variable jump. Examination of the shock dispersion mechanism suggests such jump relations. These relations are the phase Hugoniots which are compatible with the mixture energy equation. The corresponding model is conservative and symmetric. It fulfils the single-phase limit and guarantees volume fraction positivity. The shock relations are validated over a large set of experimental data and provide a remarkable agreement. R. Saurel, O. Le Métayer, J. Massoni and S. Gavrilyuk also belong to INRIA SMASH Project.     

Investigation of the evolutionarity of discontinuities in binary mixture flows through a porous medium

The discontinuity surfaces (shock waves) that arise in nonisothermal carbon dioxide-water binary mixture flows through a porous medium are considered. In the plane of determining parameters the discontinuity adiabats are investigated and their evolutionarity diagrams are plotted. It is shown that one of the adiabat branches corresponds to the displacement fronts at which there are no temperature jumps and phase transitions and the other branch to temperature jumps and phase transition fronts. The adiabat branches may intersect at a point that corresponds to the Jouguet point for the parameters both ahead of and behind the finite-amplitude jump. It is shown that in the neighborhood of this double Jouguet point the adiabat behavior differs from the classical adiabat behavior at single Jouguet points.     

The Structure of Precipitation Fronts for Finite Relaxation Time

When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper’s simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax’s stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.     

Numerical study of unsteady turbulent cavitating flows

The simulation of cavitating flows is a challenging problem both in terms of modelling the physics and developing robust numerical methodologies. Such flows are characterized by important variations of the local Mach number, compressibility effects on turbulence and involve thermodynamic phase transition. To simulate these flows by applying homogeneous models and Reynolds averaged codes, the turbulence modelling plays a major role in the capture of unsteady behaviours. This paper presents a one-fluid compressible Reynolds-Averaged Navier–Stokes (RANS) solver with a simple equation of state (EOS) for the mixture. A special focus is devoted to the turbulence model influence. Unsteady numerical results are given for Venturi geometries and comparisons are made with experimental data.     

Non-conservative pressure-based compressible formulation for multiphase flows with heat and mass transfer

A pressure-based compressible multiphase flow solver has been developed based on non-conservative discretization of the mixture continuity equation. The formulation is an extension of the single phase incompressible pressure-correction approach, such that it can be applied to both two-phase flows using interface resolving methods and general n-phase ensemble-averaged mixture flows. The formulation is currently presented with the single pressure and single temperature assumption, but extension to multiple temperatures is straightforward. A robust treatment of phase change allows the method to model conditions with rapid phase change such as expansion through nozzles and valves. The method has been validated thoroughly using canonical single phase problems such as the shock tube, tank filling and sudden valve closure problems. Multiphase flow validation has been carried out for sound propagation in mixtures using the ensemble-averaged model and pressure wave transmission and reflection across an air-water interface, using the level set interface tracking method. The method has been used to study sound propagation in saturated steam-water systems under thermodynamic non-equilibrium, where the expected drastic reduction in the speed of sound is reproduced. Finally the method is applied to the problem of critical (choked) flow in a nozzle for a saturated steam-water system.     

Numerical simulation of multiphase cavitating flows around an underwater projectile

The present simulation investigates the multiphase cavitating flow around an underwater projectile. Based on the Homogeneous Equilibrium Flow assumption, a mixture model is applied to simulate the multiphase cavitating flow including ventilated cavitation caused by air injection as well as natural cavitation that forms in a region where the pressure of liquid falls below its vapor pressure. The transport equation cavitating model is applied. The calculations are executed based on a suite of CFD code. The hydrodynamics characteristics of flow field under the interaction of natural cavitation and ventilated cavitation is analyzed. The results indicate that the ventilated cavitation number is under a combined effect of the natural cavitation number and gas flow rate in the multiphase cavitating flows.     

Modelling detonation waves in condensed energetic materials: multiphase CJ conditions and multidimensional computations

A hyperbolic multiphase flow model with a single pressure and a single velocity but several temperatures is proposed to deal with the detonation dynamics of condensed energetic materials. Temperature non-equilibrium effects are mandatory in order to deal with wave propagation (shocks, detonations) in heterogeneous mixtures. The model is obtained as the asymptotic limit of a total non-equilibrium multiphase flow model in the limit of stiff mechanical relaxation only (Kapila et al. in Phys Fluids 13:3002–3024, 2001). Special attention is given to mass transfer modelling, that is obtained on the basis of entropy production analysis in each phase and in the system (Saurel et al. in J Fluid Mech 607:313–350, 2008). With the help of the shock relations given in Saurel et al. (Shock Waves 16:209–232, 2007) the model is closed and provides a generalized ZND formulation for condensed energetic materials. In particular, generalized CJ conditions are obtained. They are based on a balance between the chemical reaction energy release and internal heat exchanges among phases. Moreover, the sound speed that appears at sonic surface corresponds to the one of Wood (A textbook of sound, G. Bell and Sons LTD, London, 1930) that presents a non-monotonic behaviour versus volume fraction. Therefore, non-conventional reaction zone structure is observed. When heat exchanges are absent, the conventional ZND model with conventional CJ conditions is recovered. When heat exchanges are involved interesting features are observed. The flow behaviour presents similarities with non ideal detonations (Wood and Kirkwood in J Chem Phys 22:1920–1924, 1950) and pathological detonations (Von Neuman in Theory of detonation waves, 1942; Guenoche et al. in AIAA Prog Astron Aeronaut 75: 387–407, 1981). It also present non-conventional behaviour with detonation velocity eventually greater than the CJ one. Multidimensional resolution of the corresponding model is then addressed. This poses serious difficulties related to the presence of material interfaces and shock propagation in multiphase mixtures. The first issue is solved by an extension of the method derived in Saurel et al. (J Comput Phys 228(5):1678–1712, 2009) in the presence of heat and mass transfers. The second issue poses the difficult mathematical question of numerical approximation of non-conservative systems in the presence of shocks associated to the physical question of energy partition among phases for a multiphase shock. A novel approach is used, based on extra evolution equations used to retain the information of the material initial state. This method insures convergence in the post-shock state. Thanks to these various theoretical and numerical ingredients, one-dimensional and multidimensional unsteady detonation waves computations are done, eventually in the presence of material interfaces. Convergence of the numerical hyperbolic solver against ZND multiphase solution is reached. Material interfaces, shocks, detonations are solved with a unified formulation where the same equations are solved everywhere with the same numerical scheme.     

A numerical method to simulate turbulent cavitating flows

The objective of this paper is to develop a numerical method for simulating multiphase cavitating flows on unstructured grids. The multiphase medium is represented using a homogeneous mixture model that assumes thermal equilibrium between the liquid and vapor phases. We develop a predictor–corrector approach to solve the governing Navier–Stokes equations for the liquid/vapor mixture, together with the transport equation for the vapor mass fraction. While a non-dissipative and symmetric scheme is used in the predictor step, a novel characteristic-based filtering scheme with a second order TVD filter is developed for the corrector step to handle shocks and material discontinuities in non-ideal gases and mixtures. Additionally, a sensor based on vapor volume fraction is proposed to localize dissipation to the vicinity of discontinuities. The scheme is first validated for simple one dimensional canonical problems to verify its accuracy in predicting jump conditions across material discontinuities and shocks. It is then applied to two turbulent cavitating flow problems – over a hydrofoil using RANS and over a wedge using LES. Our results show that the simulations are in good agreement with experimental data for the above tested cases, and that the scheme can be successfully applied to both RANS and LES methodologies.     

Multiphase flow model developed for simulating gas hydrate transport in horizontal pipe

The hydrate formation or dissociation in deep subsea flow lines is a challenging problem in oil and gas transport systems. The study of multiphase flows is complex while necessary due to the phase changes (i.e., liquid, solid, and gas) that occur with increasing the temperature and decreasing the pressure. A one-dimensional multiphase flow model coupled with a transient hydrate kinetic model is developed to study the characteristics of the multiphase flows for the hydrates formed by the phase changes in the pipes. The multiphase flow model is derived from a multi-fluid model, while has been widely used in modelling multiphase flows. The heat convection between the fluid and the ambient through the pipe wall is considered in the energy balance equation. The developed multiphase flow model is used to simulate the procedure of the hydrate transport. The results show that the formation of the hydrates can cause hold-up oscillations of water and gas.     

基于相变松弛法则的水下爆炸空化模型研究

水下爆炸过程中存在着大量的空化现象,空化的产生、演化及其溃灭过程对于水下冲击波传播、爆炸气泡运动以及水下结构物冲击损伤都会产生重要影响。本文基于多相可压缩流体理论模型,考虑空化发生过程中汽-液两相流体亚平衡状态下两相之间发生的热力学-化学平衡机制,分析汽-液两相介质之间的质量和热量交换,从而实现对相变过程的自动捕捉。该系统的控制方程采用分步法处理,首先利用二阶MUSCL-Hancock格式和HLLC黎曼求解器来求解齐次双曲型方程,再采用牛顿迭代法求解相变方程。数值测试结果表明,本文的计算模型对于空化相变过程具有较好的捕捉能力。最后将该模型应用到水下近水面爆炸空化的数值模拟当中,研究发现空泡的溃灭压力峰值约为冲击波压力峰值的15%,有效作用时间是冲击波载荷有效作用时间的2倍以上。本文的空化相变模型能够为水下爆炸空化现象的机理研究提供重要支撑。     

A Bayesian model selection analysis of equilibrium and nonequilibrium models for multiphase flow in porous media

The classical constitutive relations for multiphase flows in porous media assume instantaneous and local phase-equilibrium. Several alternative nonequilibrium/dynamic constitutive relations have been proposed in the literature including the works of Barenblatt, and Hassanizadeh and Gray. This work applies a Bayesian model selection framework in order to examine the relative efficacy of these three models to represent experimental observations. Experimental observations of multiphase displacement processes in natural porous media are often sparse and indirect, leading to considerable uncertainty in control conditions. Data from three core-scale drainage experiments are considered. Gaussian prior probability models are assumed for key multiphase flow parameters and measurements. Accurate numerical simulation approximations using the three constitutive relation models are implemented. The model selection analysis comprises a data-assimilation stage that calibrates the assumed model to the data while quantifying uncertainty. The second stage is the computation of the maximum likelihood estimate and its application to compute the Bayesian Information Criterion. It is observed that Barenblatt’s nonequilibrium model is more likely to match data from unstable displacements that involve higher viscosity ratios of the invading phase to the resident fluid. At the lowest viscosity ratio, there is no delineation between the goodness of fit obtained using the classical model and the model proposed by Hassanizadeh and Gray, and both outperform Barenblatt’s nonequilibrium model.     

Phase discontinuities in water flows through a porous medium

Flow of a fluid through a porous medium is considered with allowance for heat conduction processes and phase transitions. Discontinuities in flows between both single-phase zones saturated with water and steam and single-and two-phase zones saturated with an equilibrium steam-water mixture are studied. It is shown that only the evaporation fronts are evolutionary for a convex-downward shock adiabat of the discontinuity inside the steam-water mixture. The structure of these fronts is considered and a condition supplementary to the conservation laws and necessary for the well-posed formulation of problems whose solution contains this front is found from the condition of existence of a discontinuity structure between the water (steam) and the steam-water mixture.     

Theory of multiphase mixtures—A thermomechanical formulation

The paper presents a theory of mixtures with the nonzero interfacial area between the constituents of the mixture. The conservation laws are physically motivated by utilizing a volume averaging procedure and by the definition of a mapping transformation. It is shown that the theory constructed in this manner is consistent with the theory of mixtures with a vanishingly small interfacial area and that a second law of thermodynamics can be assigned for each phase of the mixture. The conservation laws are examined for invariance properties with the principle of the material frame indifference, and a particular constitutive assumption is discussed. Also presented in the paper are the conservation laws in the integral form and the jump conditions for the singular surfaces in the multiphase mixture.     

Longitudinal force on an embedded filament

Summary A thin elastic filament embedded in an elastic medium is subjected to a concentrated longitudinal load. For two-dimensional geometry and a relatively stiff filament the application of the load gives rise to a system of wedge-like and cylindrical waves. The dynamic shear stresses at the interface of the filament and the matrix, and in the region of the cylindrical waves are determined by means of Fourier transform techniques and Cagniard's method [2]. At the wave fronts of the wedge-like waves the jumps in the shear stresses are computed. Along the filament, the magnitudes of propagating discontinuities decrease exponentially. Along rays normal to the wave fronts of the wedge-like waves, the magnitudes of propagating discontinuities remain unchanged.     

Pressure calculations in disperse and continuous multiphase flows

In many models for disperse two-phase flows, the pressure of the disperse phase is often assumed to be the same as that of the continuous phase, or differ only by an amount caused by the surface tension. This type of model is referred to as an equilibrium pressure model. Recent research indicates that the stress difference between the phases caused by dynamics of the motion can be significantly important in the modeling of disperse two-phase flows. Although this difference is still ignored in most calculations of disperse multiphase flows for various reasons, when an equilibrium pressure model is applied to continuous multiphase flows, a conceptual difficulty arises. For instance, the equilibrium pressure model cannot be used to study the tensile break of a sponge with interconnected pores, because the air in the pores can never go into tension while the sponge material does not break without tension.     

基于VPM-THINC/QQ模型的波浪高保真模拟

为实现波浪传播的高保真数值模拟,采用包含单元均值和点值(volume-average/point-value method,VPM)的有限体积法求解纳维-斯托克斯方程和具有二次曲面性质和高斯积分的双曲正切函数(THINC method with quadratic surface representation and Gaussian quadrature,THINC/QQ)方法来重构自由面,建立以开源求解库OpenFOAM底层函数库为基础的VPM-THINC/QQ模型. 在本模型中添加推板造波法实现波浪的产生功能,采用松弛法实现消波功能,构建高精度黏性流数值波浪水槽. 分别采用VPM-THINC/QQ模型和InterFoam求解器(OpenFOAM软件包中广泛使用的多相流求解器)开展规则波的数值模拟,重点探究网格大小和时间步长等因素对波浪传播过程的影响,定量地分析波高衰减程度;为验证本模型的适应性,对长短波进行模拟. 结果表明,在相同网格大小或时间步长条件下,VPM-THINC/QQ模型的预测结果与参考值吻合较好,波高衰减较少,且无相位差,在波浪传播过程的模拟中呈现出良好的保真性. 本文工作 为波浪传播的模拟研究提供了一种高精度的黏性数值波浪水槽模型.      

Modelling interactions between waves and diffused interfaces

When simulating multiphase compressible flows using the diffuse-interface methods, the test cases presented in the literature to validate the modellings with regard to interface problems are always textbook cases: interfaces are sharp and the simulations therefore easily converge to the exact solutions. In real problems, it is rather different because the waves encounter moving interfaces which consequently have already undergone the effects of numerical diffusion. Numerical solutions resulting from the interactions of waves with diffused interfaces have never been precisely investigated and for good reasons, the results obtained are extremely dependent on the model used. Precisely, well-posed models present similar and important issues when such an interaction occurs, coming from the appearance of a wave-trapping phenomenon. To circumvent those issues, we propose to use a thermodynamically-consistent pressure-disequilibrium model with finite, instead of infinite, pressure-relaxation rate to overcome the difficulties inherent in the computation of these interactions. Because the original method to solve this model only enables infinite relaxation, we propose a new numerical method allowing infinite as well as finite relaxation rates. Solutions of the new modeling are examined and compared to literature, in particular we propose the study of a shock on a water–air interface, but also for problems of helium–air and water–air shock tubes, spherical and nonspherical bubble collapses.     

Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

We consider dynamics of chains of rigid masses connected by links described by irreversible, piecewise linear constitutive relation: the force-elongation diagram consists of two stable branches with a jump discontinuity at the transition point. The transition from one stable state to the other propagates along the chain and excites a complex system of waves. In the first part of the paper (Cherkaev et al., 2004, Transition waves in bistable structures. I. Delocalization of damage), the branches could be separated by a gap where the tensile force is zero, the transition wave was treated as a wave of partial damage. Here we assume that there is no zero-force gap between the branches. This allows us to obtain steady-state analytical solutions for a general piecewise linear trimeric diagram with parallel and nonparallel branches and an arbitrary jump at the transition. We derive necessary conditions for the existence of the transition waves and compute the speed of the wave. We also determine the energy of dissipation which can be significantly increased in a structure characterized by a nonlinear discontinuous constitutive relation. The considered chain model reveals some phenomena typical for waves of failure or crushing in constructions and materials under collision, waves in a structure specially designed as a dynamic energy absorber and waves of phase transitions in artificial and natural passive and active systems.     

A hybrid phase field multiple relaxation time lattice Boltzmann method for the incompressible multiphase flow with large density contrast

A hybrid phase field multiple relaxation time lattice Boltzmann method (LBM) is presented in this paper for simulation of multiphase flows with large density contrast. In the present method, the flow field is solved by a lattice Boltzmann equation. Concurrently, the interface of two fluids is captured by solving the macroscopic Cahn‐Hilliard equation using the upwind scheme. To be specific, for simulation of the flow field, an lattice Boltzmann equation (LBE) model developed in Shao et al. (Physical Review E, 89 (2014), 033309) for consideration of density contrast in the momentum equation is used. Moreover, in the present work, the multiple relaxation time collision operator is applied to this LBE to enable simulation of problems with large viscosity contrast or high Reynolds number. For the interface capturing, instead of solving another set of LBE as in many phase field LBMs, the macroscopic Cahn‐Hilliard equation is directly solved by using a weighted essentially non‐oscillatory scheme. In this way, the present hybrid phase field LBM shares full advantages of the phase field LBM while enhancing numerical stability. The ability of the present method to simulate multiphase flow problems with large density contrast is demonstrated by several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.