A compressible two-layer model for transient gas–liquid flows in pipes

This work is dedicated to the modeling of gas–liquid flows in pipes. As a first step, a new two-layer model is proposed to deal with the stratified regime. The starting point is the isentropic Euler set of equations for each phase where the classical hydrostatic assumption is made for the liquid. The main difference with the models issued from the classical literature is that the liquid as well as the gas is assumed compressible. In that framework, an averaging process results in a five-equation system where the hydrostatic constraint has been used to define the interfacial pressure. Closure laws for the interfacial velocity and source terms such as mass and momentum transfer are provided following an entropy inequality. The resulting model is hyperbolic with non-conservative terms. Therefore, regarding the homogeneous part of the system, the definition and uniqueness of jump conditions is studied carefully and acquired. The nature of characteristic fields and the corresponding Riemann invariants are also detailed. Thus, one may build analytical solutions for the Riemann problem. In addition, positivity is obtained for heights and densities. The overall derivation deals with gas–liquid flows through rectangular channels, circular pipes with variable cross section and includes vapor–liquid flows.     

A two-fluid model for reactive dilute solid–liquid mixtures with phase changes

Based on the Eulerian spatial averaging theory and the Müller–Liu entropy principle, a two-fluid model for reactive dilute solid–liquid mixtures is presented. Initially, some averaging theorems and properties of average quantities are discussed and, then, averaged balance equations including interfacial source terms are postulated. Moreover, constitutive equations are proposed for a reactive dilute solid–liquid mixture, where the formation of the solid phase is due to a precipitation chemical reaction that involves ions dissolved in the liquid phase. To this end, principles of constitutive theory are used to propose linearized constitutive equations that account for diffusion, heat conduction, viscous and drag effects, and interfacial deformations. A particularity of the model is that the mass interfacial source term is regarded as an independent constitutive variable. The obtained results show that the inclusion of the mass interfacial source term into the set of independent constitutive variables permits to easily describe the phase changes associated with precipitation chemical reactions.     

A first order relaxation model for the prediction of the local interfacial area density in two-phase flows

Many energy production and chemical processes involve vapor/liquid two-phase flows. Mass and energy are often exchanged between the vapor and the liquid phases, and the fluid mechanics of the two-phase system is strongly influenced by the exchange of momentum between each phase. Significantly, the transport of mass, energy and momentum between the phases takes place across interfaces. Therefore the interfacial area density (i.e. the interfacial area per unit volume) has to be accurately known in order to make reliable predictions of the interfacial transfers. Indeed, the interfacial area density must be known for both steady and transient two-phase flows. It is the purpose of this paper to present a first order relaxation model which is derived from the Boltzmann transport equation, and which accurately describes the evolution of interfacial area density for bubbly flows. In particular, the local, instantaneous interfacial area densities and volume fractions are predicted for vertical flow of a vapor/liquid bubbly flow involving both bubble clusters and individual bubbles.     

Forced capillary-gravity wave motion of two-layer fluid in three-dimensions

A class of problems associated with forced capillary-gravity wave motion in a channel are analyzed in the presence of surface and interfacial tensions in a two-layer fluid in both the cases of finite and infinite water depths. The two and three-dimensional Green functions associated with the capillary-gravity wave problems in the presence of surface and interfacial tensions are derived using the fundamental source potentials. Using the two-dimensional Green function along with Green’s second identity, the expansion formulae for the velocity potentials associated with the capillary-gravity wavemaker problems in two-dimensions are obtained. The two-dimensional results are generalized to derive the expansion formulae for the velocity potentials associated with the forced capillary-gravity wave motion in the presence of surface and interfacial tensions in three-dimensions. Certain characteristics of the eigen-system associated with the expansion formulae are derived. The velocity potentials associated with the free oscillation of capillary-gravity waves in a closed basin and semi-infinite open channel in the presence of surface and interfacial tensions are obtained. The utility of the forced motion in a channel is demonstrated by analyzing the capillary-gravity wave reflection by a wall in a channel in the presence of surface and interfacial tensions. Long wave equations associated with capillary-gravity wave motion in the presence of surface and interfacial tensions are derived under shallow water approximation and the associated dispersion relation are obtained. Various expansion formulae and Green functions derived in the present study will be useful for analyzing a large class of physical problems in ocean engineering and mathematical physics.     

Diffuse interface model for high speed cavitating underwater systems

High speed underwater systems involve many modelling and simulation difficulties related to shocks, expansion waves and evaporation fronts. Modern propulsion systems like underwater missiles also involve extra difficulties related to non-condensable high speed gas flows. Such flows involve many continuous and discontinuous waves or fronts and the difficulty is to model and compute correctly jump conditions across them, particularly in unsteady regime and in multi-dimensions. To this end a new theory has been built that considers the various transformation fronts as ‘diffuse interfaces’. Inside these diffuse interfaces relaxation effects are solved in order to reproduce the correct jump conditions. For example, an interface separating a compressible non-condensable gas and compressible water is solved as a multiphase mixture where stiff mechanical relaxation effects are solved in order to match the jump conditions of equal pressure and equal normal velocities. When an interface separates a metastable liquid and its vapor, the situation becomes more complex as jump conditions involve pressure, velocity, temperature and entropy jumps. However, the same type of multiphase mixture can be considered in the diffuse interface and stiff velocity, pressure, temperature and Gibbs free energy relaxation are used to reproduce the dynamics of such fronts and corresponding jump conditions. A general model, based on multiphase flow theory is thus built. It involves mixture energy and mixture momentum equations together with mass and volume fraction equations for each phase or constituent. For example, in high velocity flows around underwater missiles, three phases (or constituents) have to be considered: liquid, vapor and propulsion gas products. It results in a flow model with 8 partial differential equations. The model is strictly hyperbolic and involves waves speeds that vary under the degree of metastability. When none of the phase is metastable, the non-monotonic sound speed is recovered. When phase transition occurs, the sound speed decreases and phase transition fronts become expansion waves of the equilibrium system. The model is built on the basis of asymptotic analysis of a hyperbolic total non-equilibrium multiphase flow model, in the limit of stiff mechanical relaxation. Closure relations regarding heat and mass transfer are built under the examination of entropy production. The mixture equation of state (EOS) is based on energy conservation and mechanical equilibrium of the mixture. Pure phases EOS are used in the mixture EOS instead of cubic one in order to prevent loss of hyperbolicity in the spinodal zone of the phase diagram. The corresponding model is able to deal with metastable states without using Van der Waals representation.     

Pure steam condensation model with laminar film in a vertical tube

A new physical model for calculating the liquid film thickness and condensation heat transfer coefficient in a vertical condenser tube is proposed by considering the effects of gravity, liquid viscosity, and vapor flow in the core region of the flow. To estimate the velocity profile in the liquid film, the liquid film was assumed to be in Couette flow forced by the interfacial velocity at the liquid–vapor interface. For simplifying the calculation procedures, the interfacial velocity was estimated by introducing an empirical power-law velocity profile. The resulting film thickness and heat transfer coefficient from the model were compared with the experimental data and the results obtained from the other condensation models. The results demonstrated that the proposed model described the liquid film thinning effect by the vapor shear flow and predicted the condensation heat transfer coefficient from experiments reasonably well.     

Generalization of the Stefan model to allow for both velocity and temperature jumps

We obtain an expression for the energy dissipation due to an evolving nonmaterial interface across which the mass density, velocity, stress, energy density, heat flux, entropy density, and temperature may be discontinuous. This expression is a sum of three terms: the product of the interfacial mass flux with the interfacial energy release; the scalar product of the interfacial velocity slip with the interfacial friction; and, the product of the interfacial temperature jump, scaled by the interfacial temperature average, with the interfacial heating. When the surface in question is a phase interface, we propose, on the basis of the interfacial dissipation inequality, supplemental relations that determine the interfacial energy release, the interfacial friction, and the interfacial heating constitutively as functions of the interfacial mass flux, the interfacial velocity slip, and the scaled interfacial temperature jump. As a step toward an understanding of the role that such interfacial relations may serve in theories for phase transitions, we investigate a problem involving the solidification of a pure substance in the absence of flow. Received February 17, 1999     

Dynamics and rheology of the morphology of immiscible polymer blends – on modeling and simulation

 The material properties of heterogeneous polymer blends are crucially influenced by their morphology, i.e., by the spatial structure of the blend components and by the specific configuration of the interfaces separating the phases. Hence, in order to understand the behavior of experimentally obtained morphologies, one is interested in modeling the relevant dynamics of the morphology subject to external flow. Thus one can study, e.g., through the interfacial stress tensor the rheological properties due to the interfaces. The balance equations used for that purpose are based on a Cahn-Hilliard equation for the local concentration, the continuity equation, and a modified Navier-Stokes equation for the local velocity. The essential material and processing parameters such as surface tension, viscosity and volume fraction of both polymers, and imposed shear rate are taken into consideration as model coefficients. By regarding hydrodynamic interaction, which is proved to be important in case of immiscible blends, the interfacial relaxation is described properly. Simulations in both three and two dimensions agree at least qualitatively with experimental results concerning droplet deformation, droplet coalescence, and interfacial rheological properties of the blend. Received: 25 September 2000 Accepted: 24 April 2001     

Experimental study of non-uniform inlet conditions and three-dimensional effects of vertical air–water two-phase flow in a narrow rectangular duct

To study the three-dimensional interfacial structure development in vertical two-phase flow, air–water upflow experiments were performed in a rectangular duct. Various non-uniform two-phase profiles were created by injecting air from individually controlled spargers at the duct inlet into uniformly injected water flow. A four-sensor conductivity probe was used to measure local void fraction, interfacial area concentration, bubble velocity and Sauter mean diameter at three axial locations to record the development of two-phase parameters. Experimental results showed that the lateral development across the wider dimension of the duct was significant with a non-uniform inlet profile when compared to a uniform inlet profile. It is postulated that lift, wall and turbulent forces are the major contributors to the lateral distribution of the two-phase interfacial structures making this an useful experiment for benchmarking three-dimensional two-fluid models. In examining the interfacial area, the shearing-off of group 1 bubbles (defined as the smaller spherical and distorted bubbles) from the skirt region of group 2 bubbles (defined as the bigger cap and churn bubbles), the coalescence of group 2 bubbles due to wake entrainment, and random collision are the major source and sink mechanisms of interfacial area concentration.     

Forced convective film condensation inside vertical tubes

A theoretical study of forced convective film condensation inside vertical tubes is presented. We propose a unified procedure for predicting the pressure gradient and condensation heat transfer coefficient of a vapor flowing turbulently in the core and associated with laminar or turbulent film on the tube wall. The analysis for the vapor flows is performed under the condition that the velocity profiles are locally self-similar. The laminar and turbulent film models equate the gravity, pressure and viscous forces, and consider the effect of interfacial shear. The transition from laminar to turbulent film depends not only on the liquid Reynolds number but also on the interfacial shear stress. In this work we also proposed a new eddy viscosity model which is divided into three regions: the inner region in liquid condensate near the wall; the interface region including both liquid and vapor; and the outer region for the vapor core. Comparisons of the theory with some published experimental data showed good agreement.     

Structural Properties of Stress Relaxation and Convergence from Viscoelasticity to Polyconvex Elastodynamics

We consider a model of stress relaxation approximating the equations of elastodynamics. Necessary and sufficient conditions are derived for the model to be equipped with a global free energy and to have positive entropy production. The resulting class allows for both convex and non-convex equilibrium potentials. For convex equilibrium potentials, we prove a strong dissipation estimate and two relative energy estimates for: the relative entropy of the relaxation process and the modulated relative energy. Both give convergence results to smooth solutions. For polyconvex equilibrium potentials, an augmenting of the system of polyconvex elastodynamics and the null-Lagrangian structure suggest an appropriate notion of relative energy. We prove convergence of viscosity approximations to polyconvex elastodynamics in the regime where the limit solution remains smooth. A modulated relative energy is also obtained for the polyconvex case which shows stability of relaxation approximations.     

A thermodynamic and dynamic subgrid closure model for two‐material cells

A general and robust subgrid closure model for two‐material cells is proposed. The conservative quantities of the entire cell are apportioned between two materials, and then, pressure and velocity are fully or partially equilibrated by modeling subgrid wave interactions. An unconditionally stable and entropy‐satisfying solution of the processes has been successfully found. The solution is valid for arbitrary level of relaxation. The model is numerically designed with care for general materials and is computationally efficient without recourse to subgrid iterations or subcycling in time. The model is implemented and tested in the Lagrange‐remap framework. Two interesting results are observed in 1D tests. First, on the basis of the closure model without any pressure and velocity relaxation, a material interface can be resolved without creating numerical oscillations and/or large nonphysical jumps in the problem of the modified Sod shock tube. Second, the overheating problem seen near the wall surface can be solved by the present entropy‐satisfying closure model. The generality, robustness, and efficiency of the model make it useful in principle in algorithms, such as ALE methods, volume of fluid methods, and even some mixture models, for compressible two‐phase flow computations. Copyright © 2013 John Wiley & Sons, Ltd.     

Heat transfer and interfacial drag in countercurrent steam-water stratified flow

Local condensation heat transfer coefficients and interfacial shear stresses have been measured for countercurrent stratified flow of steam and subcooled water in rectangular channels over a wide range of inclination angles (4–87°) at two aspect ratios. Dimensionless correlations for the interfacial friction factor have been developed that show that it is a function of the liquid Reynolds number only. Empirical correlations of the heat transfer coefficient, based upon the bulk flow properties, have also been set up for the whole body of data encompassing the different inclination angles and aspect ratios. These indicate that the Froude number as a dimensionless gas velocity is a better correlating parameter than the gas Reynolds number. As an alternative approach, a simple dimensionless relationship for the beat transfer coefficient was obtained by analogy between heat and momentum transfer through the interface. Finally, a turbulence-centered model has been modified by using measured interfacial parameters for the turbulent velocity and length scales, resulting in good agreement with the data.     

Experimental study on local characteristics of oil–water dispersed flow in a vertical pipe

The local flow characteristics of oil–water dispersed flow in a vertical upward pipe were studied experimentally. The inner diameter and length of the test section are 40 mm and 3800 mm, respectively. A double-sensor conductivity probe was used to measure the local interfacial parameters, including interfacial area concentration, oil phase fraction, interfacial velocity, and oil drops Sauter mean diameter. The water flow rates varied from 0.12 m/s to 0.89 m/s, while the oil flow rates ranged from 0.024 m/s to 0.198 m/s. Typical radial profiles of interfacial area concentration, oil phase fraction, interfacial velocity, and oil drops Sauter mean diameter are presented. An interesting phenomenon is that the local and cross-section-averaged interfacial area concentrations display concave change with water flow rate under constant oil flow rate. The physical mechanism of such a variation is discussed in details.     

A Stability Criterion for Two-Fluid Interfaces and Applications

We derive a new stability criterion for two-fluid interfaces that ensures the existence of “stable” local solutions that do not break down too fast due to Kelvin–Helmholtz instabilities. It can be seen both as a two-fluid generalization of the Rayleigh–Taylor criterion and as a nonlinear version of the Kelvin stability condition. We show that gravity can control the inertial effects of the shear up to frequencies that are high enough for the surface tension to play a relevant role. This explains why surface tension is a necessary condition for well-posedness while the (low frequency) main dynamics of interfacial waves are unaffected by it. In order to derive a practical version of this criterion, we work with a nondimensionalized version of the equations and allow for the possibility of various asymptotic regimes, such as the shallow water limit. This limit being singular, we have to derive a new symbolic analysis of the Dirichlet–Neumann operator that includes an infinitely smoothing “tail” accounting for the contribution of the bottom. We then validate our criterion by comparison with experimental data in two important settings: air–water interfaces and internal waves. The good agreement we observe allows us to discuss the scenario of wave breaking and the behavior of water-brine interfaces, and to propose a formula for the maximal amplitude of interfacial waves. We also show how to rigorously justify two-fluid asymptotic models used for applications and how to relate some of their properties to Kelvin–Helmholtz instabilities.     

Direct numerical simulation of water droplet coalescence in the oil

Coalescence of two water droplets in the oil was simulated using Computational Fluid Dynamics (CFD) techniques. The finite volume numerical method was applied to solve the Navier–Stokes equations in conjunction with the Volume of Fluid (VOF) approach for interface tracking. The effects of some parameters consisting of the collision velocity, off-center collision parameter, oil viscosity and water–oil interfacial tension on the coalescence time were investigated. The simulation results were validated against the experimental data available in the literature. The results revealed that quicker coalescence could be achieved if the head-on collisions occur or the droplets approach each other with a high velocity. In addition, low oil viscosities or large water–oil interfacial tensions cause less coalescence time. Moreover, a correlation was developed to predict coalescence efficiency as a function of the mentioned parameters.     

A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments

We propose a Bhatnagar–Gross–Krook (BGK) kinetic model in which the collision frequency is a linear combination of polynomials in the velocity variable. The coefficients of the linear combination are determined so as to enforce proper relaxation rates for a selected group of moments. The relaxation rates are obtained by a direct numerical evaluation of the full Boltzmann collision operator. The model is conservative by construction. Simulations of the problem of spatially homogeneous relaxation of hard spheres gas show improvement in accuracy of controlled moments as compared to solutions obtained by the classical BGK, ellipsoidal-statistical BGK and the Shakhov models in cases of strong deviations from continuum.     

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.     

Statistical modeling and comparison with experimental data of tire–soil interaction for combined longitudinal and lateral slip

The interaction of a tire with a soft terrain has multiple sources of uncertainties such as the mechanical properties of the terrain, and the interfacial properties between the tire and the terrain. These uncertainties are best characterized using statistical methods such as the development of stochastic models of tire–soil interaction. The quality of the models can be assessed via statistical validation measures or metrics. Although validation of stochastic tire–soil interaction models has recently been reported with good results, it involves longitudinal slip only without considering lateral slip which can occur simultaneously with longitudinal motion. This paper presents results of the validation of a simple stochastic tire–soil interaction model for the more complicated case of combined slip. The statistical methods used for validation include the development of a Gaussian process metamodel, the calibration of model parameters using the approach of the maximum likelihood estimate in conjunction with new test data. The validation of the calibrated model, when compared with test data, is obtained using four validation metrics with good results.     

Energy relaxation method for chemical non‐equilibrium flow computations

In this paper, an algorithm for chemical non‐equilibrium hypersonic flow is developed based on the concept of energy relaxation method (ERM). The new system of equations obtained are studied using finite volume method with Harten–Lax–van Leer scheme for contact (HLLC). The original HLLC method is modified here to account for additional species and split energy equations. Higher order spatial accuracy is achieved using MUSCL reconstruction of the flow variables with van Albada limiter. The thermal equilibrium is considered for the analysis and the species data are generated using polynomial correlations. The single temperature model of Dunn and Kang is used for chemical relaxation. The computed results for a flow field over a hemispherical cylinder at Mach number of 16.34 obtained using the present solver are found to be promising and computationally (25%) more efficient. The present solver captures physically correct solution as the entropy conditions are satisfied automatically during the computations. Copyright © 2007 John Wiley & Sons, Ltd.