Two-fluid Flow Model Research Articles

A two-fluid multiphase flow model coupled with the population balance equations for the flow of a multidispersed particle system

SummaryWe present a two-fluid multiphase flow model that can be applied to the flow of polydisperse particle systems. In the model, the flow of polydisperse particle systems with different N particle sizes is considered only as two phases of fluid and particle phases, without considering the individual N phases, and the motion of particles with different diameters is represented by the motion characteristics of fluid and granular phases. The equations for the granular phase are obtained by averaging the Euler equations for the particles of different particle sizes. And the velocity of particles of different particle sizes is calculated by an explicit approximation, similar to determining the slip velocity in a mixed model of multiphase flow without solving the Eulerian momentum equation. Also, the conservation equation of particles of different particle sizes was obtained by adding the source term to the population balance equation, which is caused by the difference in the particle velocity of the particles with different particle sizes. The proposed method was compared with the results of experimental studies presented in the previous study to verify the validity of the method. The method has the advantage of not only visualizing the particle size fraction distribution but also of being stable in calculation and not increasing the computational effort significantly in studying the flow of polydisperse particle systems with particle size distribution.

Spectral stability of travelling waves in a thin-layer two-fluid Couette flow

We consider linear stability of travelling waves in a thin-film model for two-fluid Couette flow when a thin layer of the more viscous fluid resides next to the stationary wall. We prove that in a neighbourhood of a bifurcation point, characterized by a positive integer k b , the principal branch ( k b = 1 ) is spectrally stable while all other branches ( k b > 1 ) are spectrally unstable. For larger amplitude travelling waves, we establish a number of conditional theorems where the conditions were checked with help of computer assist for a set of parameter values. Using these theorems, we rigorously confirm earlier numerical evidence (D. Papageorgiou & S. Tanveer, Proc. R. Soc. A , (doi: 10.1098/rspa.2019.0367 )) on stability and instability of travelling waves over a range of parameters.

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.

Diffuse-Interface Blended Method for Imposing Physical Boundaries in Two-Fluid Flows.

Multiphase flows are commonly found in chemical engineering processes such as distillation columns, bubble columns, fluidized beds and heat exchangers. The physical boundaries of domains in numerical simulations of multiphase flows are generally defined by a conformal unstructured mesh which, depending on the complexity of the physical system, results in time-consuming mesh generation which frequently requires user-intervention. Furthermore, the resulting conformal unstructured mesh could potentially contain a large number of skewed elements, which is undesirable for numerical stability and accuracy. The diffuse-interface approach allows for the use of a simple structured meshes to be used while still capturing the desired physical (e.g., solid-fluid) boundaries. In this work, a novel diffuse-interface method for the imposition of physical boundaries is developed for the incompressible two-fluid multiphase flow model. This model is appropriate for dispersed multiphase flows which are pervasive in chemical engineering processes, in that this flow regime results in high levels of mass and energy transfer between phases. A diffuse interface is used to define the physical boundaries and boundary conditions are imposed by blending the conservation equations from the two-fluid model with that of the nondeformable solid. The results from the diffuse-interface method are compared with results from a conformal unstructured mesh for different interface functions and widths. For small interface widths, the accuracy of the flow profile is unaffected by the choice of interface function and the phase fraction distribution and flow behavior are within 3% compared to those from a conformal mesh. As the interface width increases, the diffuse-interface solution deviates from the conformal mesh solution in both the localized gas fraction and the overall gas hold-up, resulting in a difference up to 30%. In the case of flow past a cylinder, where the solid interacts with the flow, the presence of the diffuse interface extends the thickness of the solid boundary and results in a deviation from the conformal mesh solution as time increases.

HEAT TRANSFER IN COUPLE STRESS TWO-FLUID FLOW MODEL: EFFECTS OF MODIFIED HEAT FLUX, ELECTROMAGNETIC FORCE, AND UNEVEN HEAT SOURCE/SINK

The aim of the present investigation is to study the variations due to uneven heat flux and an electromagnetic field on the couple stress fluid flow of a binary mixture traveling through a stretched cylindrical surface with developed heat flux. The partial differential equations governing the fluid transport system are derived into a set of ordinary differential equations using suitable nondimensional parameters and similarity transformations, which could then be computationally solved using the Runge-Kutta Fehlberg method. The main aspect of the current problem is that it discusses the entire scenario in terms of the liquid and dust phases. Using the MATLAB code, a graphical representation of the relevant parameters is obtained. The results were carried out for our system, and it reveals that the velocity and thermal gradient of both the phases of fluid and dust behave in an opposite trend for rising mass concentration, where the curvature parameter makes the rise in the same manner. Furthermore, the thermal transport profiles for both phases decline for the rising thermal relaxation parameters.

Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

We show that the one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the model. This result extends upon earlier work on the shallow water equations (SWE), with the important difference that we include non-conservative pressure terms in the analysis, and that we propose a formulation that holds for ducts with an arbitrary cross-sectional shape, with the 2D channel and circular pipe geometries as special cases.The second novel result of this work is the formulation of a finite volume scheme for the TFM that satisfies a discrete form of the continuous energy equation. This discretization is derived in a manner that runs parallel to the continuous analysis. Due to the non-conservative pressure terms it is essential to employ a staggered grid, which requires careful consideration in defining the discrete energy and energy fluxes, and the relations between them and the discrete model. Numerical simulations confirm that the discrete energy is conserved.

Kinematic stability and simulations of the variational two-fluid model for slug flow

The two-fluid short-wave theory (TF-SWT) mode of the one-dimensional two-fluid model (TFM) [A. Clausse and M. Lopez de Bertodano, “Natural modes of the two-fluid model of two-phase flow,” Phys. Fluids 33, 033324 (2021)] showed that the incompressible kinematic and Kelvin–Helmholtz instabilities are the source of the long-standing ill-posed question. Here, the stability of the short wave mode is analyzed to obtain an unstable incompressible well-posed TFM for vertical slug flow, where inertial coupling and drag play the key role. Then, a computational method is implemented to perform non-linear simulations of slug waves. Linear stability analyses, i.e., characteristics and dispersion, of a variational TF-SWT for vertical slug flows are presented. The current TFM is constituted with a lumped-parameter model of inertial coupling between the Taylor bubble and the liquid. A characteristic analysis shows that this conservative model is parabolic, and it provides a base upon which other models can be constructed, including short-wave damping mechanisms, like vortex dynamics. The dispersion analysis shows that depending on the interfacial drag, the problem can be kinematic unstable. A new kinematic condition in terms of the inertial coupling and the interfacial drag is derived that is consistent with previous theoretical and experimental results. The material waves, which are predicted by linear stability theory, then develop into nonlinear slug waveforms that are captured by the numerical simulations. These and the horizontal stratified flow waves of previous research illustrate the TFM capability to model interfacial structures that behave like waves. Otherwise, when the physics of the TF-SWT waves is ignored, the model is ill-posed.

NUMERICAL SIMULATION OF SUPERCRITICAL AND SUBCRITICAL INJECTION OF CRYOGENIC NITROGEN BASED ON THE HOMOGENEOUS EQUILIBRIUM MODEL OF TWO-PHASE FLOW

Multidimensional numerical simulations of the outflow of the dense turbulent jet of cryogenic nitrogen into a chamber filled with nitrogen at normal temperature are performed using the homogeneous equilibrium model of the two-phase flow and the new accurate analytical equation of state (EOS) of real gas for nitrogen. The applicability of the new EOS is verified in a wide range of density (from 0 to the liquid density in the triple point) and temperature (from 100 K to 5000 K). The results of calculations are compared with available experimental data on nitrogen density variation in the jets. The satisfactory agreement of the results is obtained. The developed EOS allows the identification of two-phase flow regions and the mass fractions of liquid and gas based on the local instantaneous values of density and temperature in the jet without using the two-fluid flow model.

Four-way coupled second-order method of two-fluid model for gas-particle flow and the numerical simulation in horizontal channels

In this study, a four-way coupled second-order method of Eulerian-Eulerian two-fluid model is proposed to numerically investigate the gas-particle flow, where exist particle-particle collisions, however, the particle turbulence is far from equilibrium to satisfy the Boussinesq approximation. The second-order two-fluid model is developed by coupling the gas phase model of Reynolds-averaged Navier-Stokes method closed by the second-order Reynolds stress turbulence model with the particle simulation equations of second-order moment method of kinetic theory of granular flow. The four-way coupling method is established considering the particle-particle collisions and fluid-particle interactions, which are represented by the momentum exchange between two phases, fluid turbulence generation and/or dissipation due to the presence of particles, and the particle fluctuation due to drag fluctuations. Furthermore, the model is applied to the pneumatic conveying in a horizontal channel to demonstrate the Eulerian-Eulerian two-fluid modeling of the gas-particle flow in the median concentration of the particles. The model is found superior to the two-way coupled method and the classic isotropic method of kinetic theory of granular flow. The simulation results are in good agreement with the experimental results.

Modeling Injector Fuel Spray Dynamics for Soot Reduction in Heavy Duty Diesel Engine

Efficient, reliable and adaptable high-torque diesel engines are environmental polluters due to high soot emissions. Inefficient combustion due to inadequacy of injected fuel into the chamber is the major contributor of high soot emission. This work models the internal and the external dynamics of fuel spray inside the combustion chamber. The two-Fluid nozzle flow model is developed to investigate internal dynamics and cavitation inception. A realizable k- ε turbulence model is developed to capture the initiation of the external fuel flow and air turbulence negotiation inside the chamber. KH-RT fuel droplet breakup model is developed for the stated conditions. The trajectory of the evaporating droplet is predicted by the integration on the forces balance on the droplet in the Lagrangian frame of reference. The obtained results show that different sets of injection and backpressures modify the velocity and the pressure profiles, discharge and area coefficients and critical cavitation conditions. The effects of injection pressure and ambient conditions on spray characteristics are assessed in external spray phase. These results facilitate the development of better understanding of fuel spray dynamics for improved combustion process and subsequent reduction in soot formation for high-torque diesel engine.

Mathematical study on two-fluid model for flow of K\u2013L fluid in a stenosed artery with porous wall

The present communication presents a theoretical study of blood flow through a stenotic artery with a porous wall comprising Brinkman and Darcy layers. The governing equations describing the flow subjected to the boundary conditions have been solved analytically under the low Reynolds number and mild stenosis assumptions. Some special cases of the problem are also presented mathematically. The significant effects of the rheology of blood and porous wall of the artery on physiological flow quantities have been investigated. The results reveal that the wall shear stress at the stenotic throat increases dramatically for the thinner porous wall (i.e. smaller values of the Brinkman and Darcy regions) and the rate of increase is found to be 18.46% while it decreases for the thicker porous wall (i.e. higher values of the Brinkman and Darcy regions) and the rate of decrease is found to be 10.21%. Further, the streamline pattern in the stenotic region has been plotted and discussed.

Dynamic control method of particle distribution uniformity in the rolling circulating fluidized bed (RCFB)

This paper mainly proposed a dynamic control method of particle distribution uniformity in a rolling circulating fluidized bed (RCFB) by using an air distribution plate. An Euler–Euler numerical simulation approach which is called two-fluid flow model is used to simulate the behaviors multiphase flow within the RCFB. Six different tilt angles θ which are θ = 2°, 5°, 8°, 10°, 12° and 15° are used for the fixed inclination state of the RCFB, meanwhile, eight different inclined angles φ which are φ = −2°, −5°, 0°, 2°, 5°,8°, 10° and 12° are used for the location of the air distribution plate. After performing the simulation, the following three different conclusions are obtained. Firstly, with the increase of θ, serious uneven distribution behaviors for particles are founded at the lower position of the riser which is a part of the RCFB. Secondly, by adjusting φ, the uneven phenomenon of particles distribution behaviors is able to be alleviated no matter how θ changes for the RCFB. Thirdly, a function relation between θ and φ is obtained, which is verified by using the dynamic control method based on the dynamic mesh model in the RCFB.

Validation Strategy of Reduced-Order Two-Fluid Flow Models Based on a Hierarchy of Direct Numerical Simulations

Whereas direct numerical simulation (DNS) have reached a high level of description in the field of atomization processes, they are not yet able to cope with industrial needs since they lack resolution and are too costly. Predictive simulations relying on reduced order modeling have become mandatory for applications ranging from cryotechnic to aeronautic combustion chamber liquid injection. Two-fluid models provide a good basis in order to conduct such simulations, even if recent advances allow to refine subscale modeling using geometrical variables in order to reach a unified model including separate phases and disperse phase descriptions based on high order moment methods. The simulation of such models has to rely on dedicated numerical methods and still lacks assessment of its predictive capabilities. The present paper constitutes a building block of the investigation of a hierarchy of test-cases designed to be amenable to DNS while close enough to industrial configurations, for which we propose a comparison of two-fluid compressible simulations with DNS data-bases. We focus in the present contribution on an air-assisted water atomization using a planar liquid sheet injector. Qualitative and quantitative comparisons with incompressible DNS allow us to identify and analyze strength and weaknesses of the reduced-order modeling and numerical approach in this specific configuration and set a framework for more refined models since they already provide a very interesting level of comparison on averaged quantities.

Effect of eccentricity on radial force and cavitation characteristics in the reactor coolant pump

Nuclear reactor coolant pump as one of the most critical equipment is the only one rotating equipment in first loop system of nuclear power plant. Due to the asymmetric structure of the pump body, especially the existence of outlet segment lead to a certain of radial force, the magnitude of radial force directly affects the work stability of the reactor coolant pump. The nuclear reactor coolant pump could stability work under those transient complex conditions is an important index of its performance. To study the cavitation characteristics and radial force of reactor coolant pump on transient cavitation, a prototype pump and those exhibiting different gravity center offsets are analyzed numerically with CFD software ANSYS CFX by employing RNG k-? model and two-fluid two-phase flow model. Through the experiment-combined simulation, the variations of cavitation characteristics and radial force of the reactor coolant pump under different eccentricities are characterized. As revealed from the results, the flow characteristics of the internal flow field of the nuclear main pump change after the axis is offset by different distance. The influence of eccentricity on the cavitation of the nuclear main pump is mainly manifested at the impeller inlet from cavitation inception severe cavitation. When the eccentricity is 5 mm, the cavitation performance is improved. The effect of eccentricity on the radial force of impeller is reflected in the variation of force direction. Compared with other plans, the radial force is superior in transient cavitation under the eccentricity of 5 mm.

A novel pressure-free two-fluid model for one-dimensional incompressible multiphase flow

A novel pressure-free two-fluid model formulation is proposed for the simulation of one-dimensional incompressible multiphase flow in pipelines and channels. The model is obtained by simultaneously eliminating the volume constraint and the pressure from the widely used two-fluid model (TFM). The resulting ‘pressure-free two-fluid model’ (PF-TFM) has a number of attractive features: (i) it features four evolution equations (without additional constraints) that can be solved very quickly with explicit time integration methods; (ii) it keeps the conservation properties of the original two-fluid model, and therefore the correct shock relations in case of discontinuities; (iii) its solutions satisfy the two TFM constraints exactly: the volume constraint and the volumetric flow constraint; (iv) it offers a convenient form to analytically analyse the equation system, since the constraint has been removed.A staggered-grid spatial discretization and an explicit Runge-Kutta time integration method are proposed, which keep the constraints exactly satisfied when numerically solving the PF-TFM. Furthermore, for the case of strongly imposed boundary conditions, a novel adapted Runge-Kutta formulation is proposed that keeps the volumetric flow exact in time while retaining high order accuracy. Several test cases confirm the theoretical properties and show the efficiency of the new pressure-free model.

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

Abstract

Solute dispersion in two-fluid flowing through tubes with a porous layer near the absorbing wall: Model for dispersion phenomenon in microvessels

Present study concerns the solute dispersion analysis in a two-fluid model of blood flow through tubes with a thin porous layer near the absorbing wall. A thin porous layer near the wall represents an endothelial glycocalyx layer the composition of which may be attributed to the accumulation of carbohydrates, absorbed plasma proteins and macromolecules in a layer at the wall that affects the plasma flow (Secomb et al., 1998. A model for red blood cell motion in glycocalyx-lined capillaries. Am. J. Physiol. 274 (Heart Circ. Physiol. 43), H1016-H1022; Pries et al., 2000. The endothelial surface layer. Eur. J. Physiol. 440, 653–666). Two-fluid model for blood flow is considered in which the central region of the blood vessel is occupied by Herschel-Bulkley fluid with variable viscosity and a peripheral region of plasma surrounded over the central region is occupied by a Newtonian fluid with constant viscosity. Governing equations are solved analytically for all the regions (central, intermediate and porous regions) and the approach of Sankarasubramanian and Gill (Sankarasubramanian, R, Gill, W N, 1973. Unsteady convective diffusion with interphase mass transfer. Proc. R. Soc. London Ser. A 333, 115–132) is followed to solve the diffusion equation by series expansion method. The impact of porous layer (and hence the porous layer parameters), varying viscosity, plasma layer and wall absorption parameters on the diffusion process like convective, dispersion and mean concentration have been analyzed. A comparative investigation of solute dispersion between Newtonian and non-Newtonian fluids has also been discussed. It is perceived that the presence of such a layer reduces the convective and axial dispersion for a highly reactive wall. A remarkable observation is that a reduced porosity near the wall contributes towards reduced average concentration of the solvent. The analysis of the porous layer inside the arterial wall may be used to understand the diffusion process for flow through arteries with a deposition of a porous layer near the absorbing wall.

Influence of Flow Field's Radial Dimension on Ventilated Supercavitating Flow

To investigate the influence of flow field's radial dimension on the flow of the portion gas-leakage supercavity, based on the two-fluid multiphase flow model and SST turbulence model, considering the compressibility of ventilated gas, a 3D simulation model of ventilated supercavity was established to simulate the flow of the supercavitation, which was consistent with water tunnel experiment. The effect of flow field's radial dimension on ventilated supercavity's dimension and pressure distribution was studied. The results show that flow field's radial dimension has a significant effect on the ventilated supercavity's dimension and pressure distribution. When flow field's radial dimension ratio is 6.5 times lower than the maximum diameter of supercavity, the supercavity cannot be formed to completely enclose the underwater vehicle. With the increase of flow field's radial dimension, the pressure inside and outside the supercavity decreases, and there is a pronounced increase in supercavity dimension. When flow field's radial dimension ratio is 54.0 times greater than the maximum diameter of supercavity, the dimension and pressure distribution of ventilated supercavity remain unchanged, which coincides with the theoretical results. In addition, the calculation results provide a criterion for simulating the shape of ventilated supercavity in the open environment, which can be used to guide engineering practice.

Development of a thermal-hydraulic analysis code for helically coiled once-through steam generator

Helically coiled once-through steam generators being compact in size and most efficient for heat transfer have been widely used in many SMRs (Small and Medium-size Reactor). It is very imperative to study the hydraulic characteristics of the helically coiled once-through steam generator prior to their integration with SMRs. In this study a thermal hydraulics code that uses a two-fluid two-phase flow model with the consideration of thermal non-equilibrium is developed for thermal-hydraulic characteristics analyses of helically coiled steam generators. A pressure-velocity coupling algorithm is applied to solve equations and a set of heat transfer correlations and frictional pressure drop correlations are embedded into the code.Validation of the code is done using experimental data of pressure drop and heat transfer for helical coil tubes. The code is also tested for simulation of the transient as well as steady state thermal-hydraulic characteristics of OTSG (Once-Through Steam Generator) of IRIS (International Reactor Innovative and Secure) and results are compared with the design data and the RELAP5’s result, the simulation results are found in close agreement with these available results.

Prediction Method of Cavitation Jet Wave Attenuation Based on Five-Equation Two-Fluid Model

In this paper, the seven-equation two-fluid flow model is streamlined to a five-equation numerical calculation method. This method is applied to predict the wave attenuation of the cavitation jet. Compared with 9 different experimental schemes in two separate laboratories, it is found that the velocity flow characteristics are different from normal transient flow when the cavitation jet is formed in the pipeline. Such a difference in velocity flow characteristics will cause changes in the prediction deviation of pressure response time and amplitude. A numerical method for predicting the attenuation of pressure fluctuation with the cavitation jet is presented taken into account these flow characteristics. In this method, the unsteady friction model is opened after the jet wave transmits 3–5 cycles, and the cavitation jet wave attenuation is well simulated. This calculation method could provide a research basis for pipeline leakage, vibration, noise, and other fields that need accurate pressure signals.