Compressible Two-fluid Model Research Articles

The Cauchy Problem for a Non-conservative Compressible Two-Fluid Model with Far Field Vacuum in Three Dimensions

The Cauchy Problem for a Non-conservative Compressible Two-Fluid Model with Far Field Vacuum in Three Dimensions

Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model

Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model

Thermodynamically Compatible Discretization of a Compressible Two-Fluid Model with Two Entropy Inequalities

Thermodynamically Compatible Discretization of a Compressible Two-Fluid Model with Two Entropy Inequalities

An Integrated Modeling Approach for Vertical Gas Migration Along Leaking Wells Using a Compressible Two-Fluid Flow Model

Gas migration behind casings can occur in wells where the annular cement barrier fails to provide adequate zonal isolation. A direct consequence of gas migration is annular pressure build-up at wellhead, referred to as sustained casing pressure (SCP). Current mathematical models for analyzing SCP normally assume gas migration along the cemented interval to be single-phase steady-state Darcy flow in the absence of gravity and use a drift-flux model for two-phase transport through the mud column above the cement. By design, such models do not account for the possible simultaneous flow of gas and liquid along the annulus cement or the impact of liquid saturation within the cemented intervals on the surface pressure build-up. We introduce a general compressible two-fluid model which is solved over the entire well using a newly developed numerical scheme. The model is first validated against field observations and used for a parametric study. Next, detailed studies are performed, and the results demonstrate that the surface pressure build-up depends on the location of cement intervals with low permeability, and the significance of two-phase co-current or counter-current flow of liquid and gas occurs along cement barriers that have an initial liquid saturation. As the magnitude of the frictional pressure gradient associated with counter-current of liquid and gas can be comparable to the relevant hydrostatic pressure gradient, two-phase flow effects can significantly impact the interpretation of the wellhead pressure build-up.

On instability of a generic compressible two-fluid model in

This paper concerns the study of instability of a generic compressible two-fluid model in the whole space , where the capillary pressure is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, , Evje et al (2016 Arch. Ration. Mech. Anal. 221 1285–316) established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu et al (2022 arXiv:2204.10706) proved global stability of the constant equilibrium state for the case (corresponding to ). We investigate instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, . By employing the Hodge decomposition technique and making a detailed analysis of the Green’s function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space Hk , thus leading to a global instability result for the linearized problem. Moreover, with the help of this global linear instability result together with a local existence theorem of classical solutions to the original nonlinear system, we can show instability of the nonlinear problem in the sense of Hadamard by carefully analysing the properties of the semigroup. Therefore, our results show that for the case , the constant equilibrium state of the two-fluid model is globally linearly unstable and locally nonlinearly unstable in the sense of Hadamard, which is different from the cases (Evje et al 2016 Arch. Ration. Mech. Anal. 221 1285–316) and (corresponding to ) (Wu et al 2022 arXiv:2204.10706) where the constant equilibrium state of the two-fluid model was proved to be globally nonlinearly stable.

The Global Solvability of the Non-conservative Viscous Compressible Two-Fluid Model with Capillarity Effects for Some Large Initial Data

The Global Solvability of the Non-conservative Viscous Compressible Two-Fluid Model with Capillarity Effects for Some Large Initial Data

A Numerical Approach for the Existence of Dissipative Weak Solutions to a Compressible Two-fluid Model

A Numerical Approach for the Existence of Dissipative Weak Solutions to a Compressible Two-fluid Model

Asymptotic limits of dissipative turbulent solutions to a compressible two-fluid model

In this paper we study the asymptotic limits of a compressible two-fluid model in an unbounded domain Ω with general initial data. By applying refined related entropy method and carrying out detailed analysis on the oscillations of velocity, we derive rigorously that the dissipative turbulent solutions (velocity) of the compressible two-fluid model converge to the strong solution of the quasi-geostropic equation when there is a Coriolis force and Ω=R2×T1, and while the two-fluid model has no Coriolis force term and Ω=R3, its solutions converge to the strong solution of incompressible Euler equations.

On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions

We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes (Mellet and Vasseur, Commun Math Phys 281(3):573–596, 2008) (see also the compressible Oldroyd-B model with stress diffusion (Barrett et al., Commun Math Sci 15:1265–1323, 2017). Another interesting connection is that it is formally the equations of compressible magnetohydrodynamic (MHD) flows without resistivity in two dimensions under the action of vertical magnetic field (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019). Under weak assumptions on the initial data which can be discontinuous, unbounded and large as well as involve transition to pure single-phase points or regions, we show existence of global weak solutions with finite energy. The essential novelty of this work, compared with previous works on the same model, is that transition to each single-phase flow is allowed without any constraints between adiabatic constants or two densities. It means that one of the phases can vanish in a point while the other can persist. The lack of enough regularity for each densities brings up essential difficulties in the two-component pressure compared with the single-phase model, i.e., compressible Navier-Stokes equations. The key points to achieve the main result rely on the variables reduction technique for the pressure function, domain separation, and some new estimates. As a byproduct, we obtain the existence of global weak solutions to the compressible MHD system without resistivity in two dimensions under the action of non-negatively vertical magnetic field, which represents a step forward to the study of the global large solution to the compressible MHD system without resistivity.

On a Non-conservative Compressible Two-Fluid Model in a Bounded Domain: Global Existence and Uniqueness

In this work, we consider the Dirichlet problem for a non-conservative compressible two-fluid model in three dimensions. In particular, capillary pressure is taken into account in the sense that $$P^+ - P^-=f\ne 0$$ where f is assumed to be a strictly decreasing function near the equilibrium. This work aims to prove that this assumption has an essential stabilization effect on the model in bounded domains.

A hyperbolic two-fluid model for compressible flows with arbitrary material-density ratios

Abstract

Large time behavior for a compressible two-fluid model with algebraic pressure closure and large initial data

In this paper, we consider a compressible two-fluid system with a common velocity field and algebraic pressure closure in dimension one. Existence, uniqueness and stability of global weak solutions to this system are obtained with arbitrarily large initial data. Making use of the uniform-in-time bounds for the densities from above and below, exponential decay of weak solution to the unique steady state is obtained without any smallness restriction to the size of the initial data. In particular, our results show that degeneration to single-fluid motion will not occur as long as in the initial distribution both components are present at every point.

Compressible multiphase particle-in-cell method (CMP-PIC) for full pattern flows of gas-particle system

The compressible multiphase flows in a gas-particle system often arise in many engineering applications and nature phenomena. This study focuses on the development of a novel method, i.e. the compressible multiphase particle-in-cell (CMP-PIC), which can be capable of simulating all pattern flows of gas-particle system from dilute to dense and granular flows. The equivalent relation of the momentum equations of the particle phase between the Baer and Nunziato (B-N) model and the MP-PIC model is the key support for the modeling of gas phase, particle phase and coupling effects. For the gas phase, the governing equations are constructed in the Eulerian frame and the B-N model is taken as reference to derive the equations of gas phase and two-way coupling terms. A transport five-equation model for multi-material compressible flows is adopted in gas equation. Additionally, the disperse particle is tracked in the Lagrangian coordinate based on the particle-in-cell (PIC) method. The collisions among particles are simulated with coarse-grained discrete element method (DEM) model which provides more physical manner rather than empirical collision stress model. Moreover, a two-way coupling model is derived based on a comparative study with B-N model. Thereafter a set of unified HLL/HLLC solvers is developed for the discretization of the convective and nozzling terms of gas phase equations. A high order interpolation operator with certain enlarged smooth length is applied for the robust computation of coupling effects. Overall, the advantages of traditional compressible two-fluid model, DEM and PIC model are integrated into the current CMP-PIC model, and it can simulate the transition flows from dilute to dense or vice versa, which is difficult for traditional methods. This Eulerian-Lagrangian approach is implemented in an in-house parallel code and validated against experimental observations involving different shock induced multiple phase flows. The qualitative and quantitative comparison shows that numerical results perform good consistence with experimental and theoretical results.

Low Mach number limit of compressible two-fluid model

In this paper, we consider the low Mach number limit of the weak solutions to the viscous compressible two-fluid model with one velocity and a pressure of two components in three spatial dimensions. For well-prepared initial data, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Navier–Stokes equations as the Mach number tends to zero. Here, we allow that the two densities converge to different limits. Furthermore, the rates of convergence are also obtained.

Numerical analysis of jet breakup based on a modified compressible two-fluid-LES model

Jet breakup with a conical nozzle was studied including in-nozzle flow based on a modified compressible two-fluid model coupled with the large eddy simulation model in the OpenFOAM framework. In order to capture the interface, the phase equation and the pressure equation were modified by adding the quasi-VOF model and surface tension respectively. The effects of temperature and pressure on the air thermal properties were taken into account and the density of diesel was fitted linearly with respect to the pressure according to experimental data. In addition, the Ranz-Marshall model was applied to consider the interphase heat transfer. The modified compressible two-fluid model was verified by comparing the calculated mass flow rates, discharge coefficients, spray momentum fluxes and effective jet velocities under different injection conditions with the available experimental results. It was demonstrated that the numerical simulation results agreed well with the experimental measurements with minor discrepancies which were caused by the numerical assumptions and the experimental uncertainty. Then the mechanism of jet breakup and the effects of the ambient temperature on jet breakup were investigated. It is found that jet breakup is divided into two regions, i.e. the intact region and the jet breakup region. The intact region is dominated by the spatial instability, and the jet breakup region is dominated by the combination of the temporal instability and the spatial instability. It's worth mentioning that the surface tension plays a dual role in inhibiting atomization of droplets and promoting the pinch-off of diesel column and ligament.

Incompressible inviscid limit of the viscous two-fluid model with general initial data

In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space \({\mathbb {R}}^3\) with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.

Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ≥1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-ε estimates to the regular solutions. At the second level, we show global existence of weak solutions to the system with artificial pressure by sending ε to 0 and deriving uniform-in-δ estimates. Then global weak solution to the original system is constructed by vanishing δ. The key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field. This is accomplished by following the technique developed in [15,26] and using the new technique of variable reduction developed by Vasseur et al. [33] in the context of compressible two-fluid model so as to handle the cross terms.

Time periodic solutions for a three-dimensional non-conservative compressible two-fluid model

Time periodic solutions for a three-dimensional non-conservative compressible two-fluid model

A dual grid method for the compressible two-fluid model which combines robust flux splitting methodology with high-resolution capturing of incompressible dynamics

The speed of sound in two-phase pipe flow systems is often several orders of magnitude greater than the travelling speed of hydraulic information (volume fractions). Dynamically simulating such flows requires resolution of acoustic and hydraulic waves existing at vastly different spatial and temporal scales. If simulated on the same numerical grid, the need for accuracy in hydraulic waves will necessitate an exaggerated resolution of acoustic waves. Likewise, time steps restricted by the speed of sound are small compared to the time scales active in hydraulic waves. This constitutes a waste of computational potential. The method proposed herein decouples the hydraulic and acoustic scales, greatly improving computational efficiency.The proposed dual grid method solves a four-equation compressible two-fluid model on a principal grid which robustly accounts for the pressure evolution and conserves mass and momentum. An incompressible two-equation model is at the same time solved on a finer grid, resolving the details of the hydraulic evolution. Information from both model formulations is coupled through the terms of the governing transport equations, providing consistency between the grids. Accurate and finely resolved schemes can then be employed for the incompressible two-fluid model without suffering from the time and stability restrictions otherwise enforced by acoustic waves. At the same time, the Hybrid Central-Upwind flux splitting scheme of Evje and Flåtten (2005a) allows for an explicit and numerically robust treatment of the acoustic waves without losing hydraulic accuracy.The dual grid method is tested against four dissimilar problems: A shock tube problem, the water faucet problem, a surge wave and pressure wave problem and a roll-wave case. In all problems, the proposed scheme provided significant increases in computational efficiency and accuracy as compared with a single grid arrangement.

Analysis of time integration methods for the compressible two-fluid model for pipe flow simulations

In this paper we analyse different time integration methods for the two-fluid model and propose the BDF2 method as the preferred choice to simulate transient compressible multiphase flow in pipelines. Compared to the prevailing Backward Euler method, the BDF2 scheme has a significantly better accuracy (second order) while retaining the important property of unconditional linear stability (A-stability). In addition, it is capable of damping unresolved frequencies such as acoustic waves present in the compressible model (L-stability), opposite to the commonly used Crank–Nicolson method. The stability properties of the two-fluid model and of several discretizations in space and time have been investigated by eigenvalue analysis of the continuous equations, of the semi-discrete equations, and of the fully discrete equations. A method for performing an automatic von Neumann stability analysis is proposed that obtains the growth rate of the discretization methods without requiring symbolic manipulations and that can be applied without detailed knowledge of the source code.The strong performance of BDF2 is illustrated via several test cases related to the Kelvin–Helmholtz instability. A novel concept called Discrete Flow Pattern Map (DFPM) is introduced which describes the effective well-posed unstable flow regime as determined by the discretization method. Backward Euler introduces so much numerical diffusion that the theoretically well-posed unstable regime becomes numerically stable (at practical grid and timestep resolution). BDF2 accurately identifies the stability boundary, and reveals that in the nonlinear regime ill-posedness can occur when starting from well-posed unstable solutions. The well-posed unstable regime obtained in nonlinear simulations is therefore in practice much smaller than the theoretical one, which might severely limit the application of the two-fluid model for simulating the transition from stratified flow to slug flow. This should be taken very seriously into account when interpreting results from any slug-capturing simulations.