Mixture Momentum Equations Research Articles

Two phase discharge flow prediction in safety valves

Safety relief valves (SRV) are necessary elements in the protection of any pressurised system and the prediction of the expected discharge flows is an important consideration for the valve sizing to ensure that rupture pressures do not occur. The high speed flows that occur inside the SRV are complex particularly when a two-phase flow is involved and lead to a less capable protection device which result in larger valves compared to single phase flows. In this paper the ability of a CFD based two phase mixture model to predict the critical flows of air and water through a safety valve is examined. An industrial refrigeration safety relief valve of ¼″ inlet bore size has been tested experimentally over a pressure range of 6–15 barg and air mass qualities from 0.1 to 1 when discharging to near atmospheric conditions for a fully open condition. A two-dimensional mixture model consisting of mixture mass, momentum, and energy equations, combined with a liquid mass equation and the standard k–ε turbulence model for mixture turbulent transport has been used to predict the two phase flows through the valve. The mixture model results have been compared with the Homogenous Equilibrium Model (HEM) commonly used for in valve sizing in non flashing two phase flow conditions. The accuracy of the models over the two phase flow range are quantified and discussed.

Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations

In this work we study a compressible gas-liquid models highly relevant for wellbore operations like drilling. The model is a drift-flux model and is composed of two continuity equations together with a mixture momentum equation. The model allows unequal gas and liquid velocities, dictated by a so-called slip law, which is important for modeling of flow scenarios involving for example counter-current flow. The model is considered in Lagrangian coordinates. The difference in fluid velocities gives rise to new terms in the mixture momentum equation that are challenging to deal with. First, a local (in time) existence result is obtained under suitable assumptions on initial data for a general slip relation. Second, a global in time existence result is proved for small initial data subject to a more specialized slip relation.

Prediction of two-phase flow through a safety relief valve

Safety relief valves are necessary elements in any pressurised system. The flow inside the safety relief valve shows a number of interesting, yet complicated, features especially when a two-phase flow is involved. Consequently, developing an efficient and accurate means for predicting the safety relief valve performance and understanding the flow physics is a demanding objective. In this article, the ability of a two-phase mixture model to predict the critical flows of air and water through a safety valve is examined. An industrial refrigeration safety relief valve of ¼” inlet bore size has been tested experimentally over a pressure range of 6–15 barg and air mass qualities from 0.23 to 1 when discharging to near atmospheric conditions for a range of valve lift positions. A two-dimensional mixture model consisting of mixture mass, momentum and energy equations, combined with a liquid mass equation and the standard k-e turbulence model for mixture turbulent transport has been used to predict the two-phase flows though the valve. The mixture model results have been compared with the homogenous equilibrium model and the homogenous non-equilibrium model adopted by the ISO standard. It has been shown that the mixture model can be used satisfactorily to predict the mass flows for the above conditions. Overall, the accuracy of the two-phase air mass flow for given inlet liquid flow rates can be predicted to within 15%.

A Full Eulerian Fluid-Membrane Coupling Method with a Smoothed Volume-of-Fluid Approach

AbstractA novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach. The present method is based on a full Eulerian fluid-(bulk) structure coupling solver (Sugiyama et al., J. Comput. Phys., 230 (2011) 596-627), with the bulk structure replaced by elastic membranes. In this study, a closed membrane is consid-ered, and it is described by a volume-of-fluid or volume-fraction information generally called VOF function. A smoothed indicator (or characteristic) function is introduced as a phase indicator which results in a smoothed VOF function. This smoothed VOF function uses a smoothed delta function, and it enables a membrane singular force to be incorporated into a mixture momentum equation. In order to deal with a membrane deformation on the Eulerian mesh, a deformation tensor is introduced and updated within a compactly supported region near the interface. Both the neo-Hookean and the Skalak models are employed in the numerical simulations. A smoothed (and less dissipative) interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane. The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach. The present method is validated by using the spherical membrane deformation problems, and is applied to a pressure-driven flow with the biconcave membrane capsules (red blood cells).

Optimum Configuration of an Expanding-Contracting-Nozzle for Thrust Enhancement by Bubble Injection

This paper addresses the concept of thrust augmentation through bubble injection into an expanding-contracting nozzle. Two-phase models for bubbly flow in an expanding-contracting nozzle are developed, in parallel with laboratory experiments and used to ascertain the geometry configuration for the nozzle that would lead to maximum thrust enhancement upon bubble injection. For preliminary optimization of experimental setup’s design, a quasi 1-D approach is used. Averaged flow quantities (such as velocities, pressures, and void fractions) in a cross section are used for the analysis. The mixture continuity and momentum equations are numerically solved simultaneously along with equations for bubble dynamics, bubble motion, and an equation for conservation of the total bubble number. Various geometric parameters such as the exit and inlet areas, the area of the bubble injection section, the presence of a throat and its location, the length of the diffuser section and the length of the contraction section are varied, and their effects on thrust enhancement are studied. Investigation on the effect of the injected void fraction is also carried out. The key objective function of the optimization is the normalized thrust parameter, which is the thrust with bubble injection minus the thrust with liquid only divided by the inlet liquid momentum. An approximate analytical expression for the normalized thrust parameter was also derived starting from the mixture continuity and momentum equations. This analytical expression involved flow variables only at three locations; inlet section, injection section, and outlet section, and the expression is simple enough to produce a quick concept design of the diffuser-nozzle thruster. The numerical and analytical approaches are verified against each other and the limitations of the analytical approach are discussed.

Analytical solution for two-phase flow in a wellbore using the drift-flux model

This paper presents analytical solutions for steady-state, compressible two-phase flow through a wellbore under isothermal conditions using the drift flux conceptual model. Although only applicable to highly idealized systems, the analytical solutions are useful for verifying numerical simulation capabilities that can handle much more complicated systems, and can be used in their own right for gaining insight about two-phase flow processes in wells. The analytical solutions are obtained by solving the mixture momentum equation of steady-state, two-phase flow with an assumption that the two phases are immiscible. These analytical solutions describe the steady-state behavior of two-phase flow in the wellbore, including profiles of phase saturation, phase velocities, and pressure gradients, as affected by the total mass flow rate, phase mass fraction, and drift velocity (i.e., the slip between two phases). Close matching between the analytical solutions and numerical solutions for a hypothetical CO 2 leakage problem as well as to field data from a CO 2 production well indicates that the analytical solution is capable of capturing the major features of steady-state two-phase flow through an open wellbore, and that the related assumptions and simplifications are justified for many actual systems. In addition, we demonstrate the utility of the analytical solution to evaluate how the bottomhole pressure in a well in which CO 2 is leaking upward responds to the mass flow rate of CO 2–water mixture.

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. The model’s predictions are validated in multi-dimensions against experiments of high velocity projectile impact onto a liquid tank. Simulations are compared to experiments and reveal excellent quantitative agreement regarding shock and cavitation pocket dynamics as well as projectile deceleration versus time. Then model’s capabilities are illustrated for flow computations around underwater missiles.

On the Wave Structure of Two‐Phase Flow Models

We explore the relationship between two common two‐phase flow models, usually denoted as the two‐fluid and drift‐flux models. They differ in their mathematical description of momentum transfer between the phases. In this paper we provide a framework in which these two model formulations are unified. The drift‐flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation, which gives the relative velocity between the phases as a function of the flow parameters. This closure law is in general highly complex, which makes it difficult to analyze the model algebraically. To facilitate the analysis, we express the quasi‐linear formulation of the drift‐flux model as a function of three parameters: the derivatives of the slip with respect to the vector of unknown variables. The wave structure of this model is investigated using a perturbation technique. Then we rewrite the drift‐flux model with a general slip relation such that it is expressed in terms of the canonical two‐fluid form. That is, we replace the mixture momentum equation and the slip relation with equivalent evolution equations for the momentums of each phase. We obtain a mathematically equivalent formulation in terms of a two‐fluid model and by this bridge some of the gap between the drift‐flux model and the two‐fluid model. Finally, the effect of the various exchange terms on the wave structure of the two‐fluid model is investigated.

Development of a commercial code-based two-fluid model for bubble plumes

Bubble plume models are applied to study the de-stratification of lake water, aeration of reservoirs, wastewater treatment, and gas injection into liquid metals. Several existing models exemplify numerical modeling as problem-specific art, solve a mixture momentum equation, and have limitations in the availability of documentation, definition of boundary conditions, and post- and pre-processing capabilities. The transfer of problem-specific models to a client or to a multidisciplinary research and development team is a difficult process. The questions addressed in this study were as follows: (a) can one use a commercial code as a basis to develop a user-friendly, efficient model that simulates two-phase flow in bubble plumes? (b) what are the capabilities and limitations of such a model? The two-fluid model developed has flexibility in the definition of the multiphase and viscous models, easily understood definition of boundary conditions, simple definition of spatial dimensionality and time dependency, efficient numerical solution, clear documentation and user-friendly pre- and post-processing capabilities. Water and air phase velocities, water turbulent kinetic energy, and air volume fraction are predicted with accuracy similar to that of existing problem-specific models. A strategy to overcome some under-dispersion and include air–water mass transfer effects through user-defined functions is discussed.

Hybrid flux-splitting schemes for a common two-fluid model

The aim of this paper is to construct hybrid flux vector splitting (FVS) and flux difference splitting (FDS) schemes for a commonly used two-fluid model consisting of two separate momentum equations. This is done by refining ideas previously applied to develop hybrid FVS/FDS schemes for a simpler two-phase model consisting of a mixture momentum equation [J. Comput. Phys. 175 (2002) 674]. More specifically, we seek to construct upwind type of schemes which are not based on calculations of the full eigenstructure of Jacobi matrices as needed by approximate Riemann solvers like the Roe scheme. Based on a crude approximation of the eigenstructure of the model, we derive schemes of the van Leer and FVS type. We demonstrate that these schemes possess desirable stability properties, but are excessively diffusive. By adapting ideas originally suggested by Wada and Liou [SIAM J. Sci. Comput. 18 (1997) 633] for the Euler equations, we suggest a mechanism for removing numerical dissipation. We present numerical simulations where we compare the performance of the resulting schemes with that of the Roe scheme, and by that shed light on the issues of accuracy, efficiency, and robustness of the proposed schemes. Particularly, we consider the classical water faucet problem as well as a stiff separation problem which locally involves transition from two-phase to single-phase flow. Results from these test cases show that we are able to construct hybrid FVS/FDS schemes which properly combine the accuracy of FDS in the resolution of sharp mass fronts and the robustness of FVS which ensures stability under stiff conditions.

Performance Analysis of Cavitating Flow in Centrifugal Pumps Using Multiphase CFD

A multi-phase CFD method is used to analyze centrifugal pump performance under developed cavitating conditions. The differential model employed is the homogeneous two-phase Reynolds-Averaged-Navier-Stokes equations, wherein mixture momentum and volume continuity equations are solved along with vapor volume fraction continuity. Mass transfer modeling is provided for the phase change associated with sheet cavitation. Using quasi-3D (Q3D) analysis, steady and time-dependent analyses were performed across a wide range of flow coefficients and cavitation numbers. Characteristic performance trends associated with off-design flow and blade cavitation are observed. The rapid drop in head coefficient at low cavitation numbers (breakdown) is captured for all flow coefficients. Local flow field solution plots elucidate the principal physical mechanisms associated with the onset of breakdown.

Numerical modeling and experimental investigation on the characteristics of molten pool during laser processing

The article illustrates melting processes in fixed meshes, based on mixture continuity equations, momentum equations and energy equations for both liquid and solid phases by introducing scalar of liquid fraction and applying source-based method. The model was applied to laser processing of AISI 304 stainless steel. The effects of heat conduction, Marangoni flow and thermal buoyancy on melting process and shape of molten pool were analyzed and physical fundament of laser processing were evidenced. Compared to the experimental outcome, the proposed model and numerical techniques are validated.

A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction

An implicit algorithm for the computation of viscous two-phase flows is presented in this paper. The baseline differential equation system is the multi-phase Navier–Stokes equations, comprised of the mixture volume, mixture momentum and constituent volume fraction equations. Though further generalization is straightforward, a three-species formulation is pursued here, which separately accounts for the liquid and vapor (which exchange mass) as well as a non-condensable gas field. The implicit method developed here employs a dual-time, preconditioned, three-dimensional algorithm, with multi-block and parallel execution capabilities. Time-derivative preconditioning is employed to ensure well-conditioned eigenvalues, which is important for the computational efficiency of the method. Special care is taken to ensure that the resulting eigensystem is independent of the density ratio and the local volume fraction, which renders the scheme well-suited to high density ratio, phase-separated two-fluid flows characteristic of many cavitating and boiling systems. To demonstrate the capabilities of the scheme, several two- and three-dimensional examples are presented.

The effect of free-floating dendrites and convection on macrosegregation in direct chill cast aluminum alloys: Part I: model development

To obtain a better understanding of the evolution of macrosegregation during the direct chill casting of aluminum alloys, a binary mixture model has been developed that accounts for the redistribution of alloying elements through the transport of free-floating dendrites and fluid flow in the melt and mushy zones of a solidifying ingot. Separate and distinct mixture momentum equations are employed to account for momentum transfer in two-phase regions which include a slurry of free-floating dendrites and a rigid solid matrix. A scaling analysis revealed that the flow of liquid and free-floating dendrites is controlled by a balance between buoyancy forces arising from mixture density variations and the development of an adverse pressure gradient. In a companion paper, the model is used to predict macrosegregation in Al–4.5 wt% Cu and Al–6.0 wt% Mg billets.

Effects of Nuclei Size Distribution on the Dynamics of a Spherical Cloud of Cavitation Bubbles

The nonlinear dynamics of a spherical bubble cloud with nuclei size distribution are studied numerically. The spectrum of nuclei is assumed uniform initially. The simulations employ a nonlinear continuum bubbly mixture model with consideration of the presence of bubbles of different sizes. This model is then coupled with the Rayleigh-Plesset equation for the dynamics of bubbles. A numerical method based on the integral representation of the mixture continuity and momentum equations in the Lagrangian coordinates is developed to solve this set of integro-differential equations. Computational results show that the nuclei size distribution has significant effects on the cloud dynamics in comparison to the results for a single bubble size. One important effect is that the bubble collapse is always initiated near the surface of the cloud, even if the cloud has a very small initial void fraction. This effect has an important consequence, namely that the geometric focusing of the bubbly shock wave is always a part of the nonlinear dynamics associated with the collapse of a spherical cloud with nuclei size distribution. The strength of the shock and the oscillation structure behind the shock front are suppressed due to the effects of multiple bubble sizes. Far-field acoustic pressures radiated by two bubble clouds, one of equal-size bubbles and the other with bubble size distribution, are also compared. It is found that the cloud containing bubbles of different sizes emits a larger noise than the cloud of identical bubbles. Explanations for this effect are also presented.

Properties and Averaged Equations for Flows of Bubbly Liquids

Averaged properties of bubbly liquids in the limit of large Reynolds and small Weber numbers are determined as functions of the volume fraction, mean relative velocity, and velocity variance of the bubbles using numerical simulations and a pair interaction theory. The results of simulations are combined with those obtained recently for sheared bubbly liquids [19] and the mixture momentum and continuity equations to propose a complete set of averaged equations and closure relations for the flows of bubbly liquids at large Reynolds and small Weber numbers.

Extension of the continuum model for transport phenomena occurring during metal alloy solidification—I. The conservation equations

In this study, models for simulating transport phenomena occurring during solidification of a binary metal alloy are reviewed, with emphasis placed on the benefits and shortcomings of existing continuum and two-phase approaches. Linkages between the two approaches are discussed, and volume-averaging procedures inherent in the two-phase model are used to develop an extension of the continuum model which retains its computational convenience, while eliminating its inability to treat important features such as solutal undercooling, nucleation, stereological characteristics of the solid/liquid interface, solid movement in the form of floating or settling crystals, and shrinkage. Two approaches to development of a mixture momentum equation are considered, one involving evolution from the liquid momentum conservation equation and the other involving summation of the liquid and solid momentum equations. Special features of both approaches are discussed. In a companion paper, additional models are developed to account for the transport (floating and settling) of solid crystals in the melt, solutal undercooling, and nucleation.

Prediction of ammonia releases using ADREA‐AF code

AbstractThe ADREA‐HF code is a 3‐D time dependent finite volume code for vapour cloud dispersion under development at NCSR “DEMOKRITOS” and JRC ISPRA. ADREA‐HF is especially suitable for complex terrain and is also designed to treat liquid phase change in the vapour cloud (two phase releases). The fluid dynamics and thermodynamics are described by the mixture mass, momentum and internal energy equations whereas mass conservation of the heavy fluid component is fulfilled through a separate mass transport equation. The system is assumed to be in thermodynamic equilibrium, but the liquid and gas phases can have different velocitites. The present work falls within the framework of the code validation for two phase relases. The experiment selected for comparision was the DT1 Run from the Desert Tortoise ammonia spill series of experiments. The results are given in terms of ammonia concentrations as well as temperatures. The comparison with the experimental data is discussed.

A Three-Field Diffusion Model of Three-Phase, Three-Component Flow for the Transient Three-Dimensional Computer Code IVA2/001

IVA 2/001 is a computer code for simulating transient, three-dimensional, three-phase, three-component nonhomogeneous (three velocity fields), nonequilibrium flow in a cylindrical porous body (including nuclear reactor cores if desired). Each velocity field consists of an inert and a noninert component. A separated equation of field mass, inert mass concentration in each of the fields, and entropy of the fields together with mixture momentum equations for the flow are solved by a semi-implicit numerical method with an analytical reduction to the pressure or pressure velocity problem and backward substitution. The flexibility of the method in describing three velocity fields of arbitrary direction is demonstrated. The solution procedure of the hydrodynamic problem is described. Finally, a numerical example and a comparison with experimental data demonstrate that the IVA2 method is a powerful tool for numerical multiphase flow simulation.

Brownian motion in a flowing fluid revisited

It is shown how the phenomenon of osmosis may be treated using the phenomenological theory of Brownian motion in a flowing fluid. The theory is also generalized to include viscous stresses in the particle and mixture momentum equations.