Two-fluid Flow Research Articles

On the interaction of Taylor length scale size droplets and isotropic turbulence

Droplets in turbulent flows behave differently from solid particles, e.g. droplets deform, break up, coalesce and have internal fluid circulation. Our objective is to gain a fundamental understanding of the physical mechanisms of droplet–turbulence interaction. We performed direct numerical simulations (DNS) of 3130 finite-size, non-evaporating droplets of diameter approximately equal to the Taylor length scale and with 5 % droplet volume fraction in decaying isotropic turbulence at initial Taylor-scale Reynolds number $\mathit{Re}_{\unicode[STIX]{x1D706}}=83$. In the droplet-laden cases, we varied one of the following three parameters: the droplet Weber number based on the r.m.s. velocity of turbulence ($0.1\leqslant \mathit{We}_{rms}\leqslant 5$), the droplet- to carrier-fluid density ratio ($1\leqslant \unicode[STIX]{x1D70C}_{d}/\unicode[STIX]{x1D70C}_{c}\leqslant 100$) or the droplet- to carrier-fluid viscosity ratio ($1\leqslant \unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x1D707}_{c}\leqslant 100$). In this work, we derive the turbulence kinetic energy (TKE) equations for the two-fluid, carrier-fluid and droplet-fluid flow. These equations allow us to explain the pathways for TKE exchange between the carrier turbulent flow and the flow inside the droplet. We also explain the role of the interfacial surface energy in the two-fluid TKE equation through the power of the surface tension. Furthermore, we derive the relationship between the power of surface tension and the rate of change of total droplet surface area. This link allows us to explain how droplet deformation, breakup and coalescence play roles in the temporal evolution of TKE. Our DNS results show that increasing $\mathit{We}_{rms}$, $\unicode[STIX]{x1D70C}_{d}/\unicode[STIX]{x1D70C}_{c}$ and $\unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x1D707}_{c}$ increases the decay rate of the two-fluid TKE. The droplets enhance the dissipation rate of TKE by enhancing the local velocity gradients near the droplet interface. The power of the surface tension is a source or sink of the two-fluid TKE depending on the sign of the rate of change of the total droplet surface area. Thus, we show that, through the power of the surface tension, droplet coalescence is a source of TKE and breakup is a sink of TKE. For short times, the power of the surface tension is less than $\pm 5\,\%$ of the dissipation rate. For later times, the power of the surface tension is always a source of TKE, and its magnitude can be up to 50 % of the dissipation rate.

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

A nonlinear flow-transition criterion for the onset of slugging in horizontal channels and pipes

In this work, the interfacial instability and transition of a two-fluid flow from a stratified state to large amplitude waves or slugs is considered. By combining an asymptotic approximation of the linear Orr-Sommerfeld analysis with nonlinear resonant wave interaction theory, a novel nonlinear slug-transition criterion is derived. This criterion corresponds to a bounding condition on the upper fluid’s velocity in order to limit the amount of energy (provided by the linear instability) which is transferred to long waves through resonant wave interactions. It is proposed that such a condition can predict the formation of large-amplitude long waves and/or slugs. Quantitative comparisons of the onset of slugging are made between the prediction by the nonlinear transition criterion and the experimental measurements carried out in a horizontal square channel. Good agreement is observed. An additional heuristic model is developed which generalizes the transition criterion to flow through horizontal pipes. Comparisons are made for flows through different pipe diameters and over a wide range of fluid properties. Good agreement between the present theoretical predictions and the experimental measurements is also observed.

A SIMPLE APPROACH TO ANALYZE THE FULLY DEVELOPED TWO-PHASE MAGNETOCONVECTION TYPE FLOWS IN INCLINED PARALLEL-PLATE CHANNELS

A numerical approach is presented herein to analyze steady-state, fully developed MHDtype stratified two-fluid flow with heat transfer in inclined parallel-plate channels. The one dimensional model is extended to include several effects, such as buoyancy, moving plates, porous layers, inclined applied magnetic field, Joule and viscous heating and volumetric heat generation/absorption. Evaluation includes axial induced magnetic field, although its effect on the flow is negligible. The plates are maintained at constant temperatures, and the thermophysical properties are considered to be constant. The governing non-linear and coupled equations are solved through a well-established numerical routine to solve stiff two-point boundary value problems. The results are presented and validated for a large range of governing parameters and situations. Keywords   , ,, ,

Effects of a depletion layer on flow in a rotating channel

An analytical model is presented for pressure-driven two-fluid flow between two parallel flat plates rotating about an axis perpendicular to the plates. The aim is to determine the lubricating effects due to a near-wall low-viscosity depletion layer on the axial as well as transverse flows in a rotating channel. It is found that the primary and secondary flow fields may be affected differently by the system rotation, the thickness of the depletion layer, and the viscosity ratio. For a very thin depletion layer, the wall lubrication reduces to a Navier slip condition, where the slip length is found to be the same as the one for a non-rotating channel at the leading order, while the higher-order slip length is a function of a complex rotation parameter. The Fåhræus–Lindqvist effect in a rotating environment is also examined.

Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels

Data generated by direct numerical simulations (DNS) of bubbly up-flow in a periodic vertical channel is used to generate closure relationships for a simplified two-fluid model for the average flow. Nearly spherical bubbles, initially placed in a fully developed parabolic flow, are driven relatively quickly to the walls, where they increase drag and slowly reduce the flow rate. Once the flow rate has been decreased enough, some of the bubbles move back into the channel interior and the void fraction there approaches the value needed to balance the weight of the mixture and the imposed pressure gradient. A database is generated by averaging the DNS results over planes parallel to the walls, and a Model Averaging Neural Network (MANN) is used to find the relationships between unknown closure terms in a simple model equations for the average flow and the resolved variables. The closure relations are then tested, by following the evolution of different initial conditions, and it is found that the model predictions are in reasonably good agreement with DNS results.

New analytical solutions to the two-phase water faucet problem

The one-dimensional water faucet problem is one of the classical benchmark problems originally proposed by Ransom to study the two-fluid two-phase flow model. With simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. The water faucet problem and its analytical solutions have been widely used for code assessment, benchmark, and numerical verification. In this work, we present a new set of analytical solutions to the water faucet problem at the steady-state condition, with the gas-phase density’s effect on pressure distribution considered. This new set of analytical solutions is used in a rigorous numerical verification process from which the anticipated second-order spatial accuracy is achieved for a second-order spatial discretization scheme. On the contrary, the same anticipated order of accuracy could not be obtained using the Ransom solutions as the reference. In addition, extended Ransom transient solutions for the gas-phase velocity and pressure are derived with the assumption of decoupled liquid and gas pressures. Numerical benchmark on the extended Ransom solutions is also presented.

Linear stability analysis and direct numerical simulation of two-layer channel flow

We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.

Inviscid instability of two-fluid free surface flow down an incline

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The ...

CFD modeling of the gas–solid two-fluid flow in polyethylene FBRs: From traditional operation to super-condensed mode

A computational fluid dynamics (CFD) model based on the Eulerian–Eulerian approach, coupled with polymerization kinetics, has been implemented to describe the gas–solid flow behaviors in a gas phase polyethylene fluidized bed reactor (FBR). The model is firstly validated by the classic equations and reference data, and then extended to simulate the gas-phase polyethylene FBR under super-condensed mode for the first time. The influence of three important operating conditions (i.e., inlet gas velocity, inlet gas temperature, and condensable component isopentane concentration) on gas–solid flow hydrodynamics is investigated to give qualitative and quantitative insights on how these factors affecting FBR performances (like bed expansion, bed temperature and productivity) under super-condensed mode through CFD approach. The simulation results demonstrate that with the increase of inlet gas velocity, the bed expansion increases rapidly and the number of particles at the lower portion of the bed reduces while raises at the higher position. The rise of the bed temperature is mainly contributed by the increased exothermic reaction rate as the inlet gas temperature increasing, which is also influenced by the concentration of condensable component isopentane. The simulation results show that bed temperature decreases with the increase of condensable isopentane concentration in gas stream. It is found that the method of heat removal through the latent heat of vaporization is extremely significant and can effectively increases reactor productivity without enlarging the size of the reactor under super-condensed mode.

Tracking interface and common curve dynamics for two-fluid flow in porous media

The movements of fluid–fluid interfaces and the common curve are an important aspect of two-fluid-phase flow through porous media. The focus of this work is to develop, apply and evaluate methods to simulate two-fluid-phase flow in porous medium systems at the microscale and to demonstrate how these results can be used to support evolving macroscale models. Of particular concern is the problem of spurious velocities that confound the accurate representation of interfacial dynamics in such systems. To circumvent this problem, a combined level-set and lattice-Boltzmann method is advanced to simulate and track the dynamics of the fluid–fluid interface and of the common curve during simulations of two-fluid-phase flow in porous media. We demonstrate that the interface and common curve velocities can be determined accurately, even when spurious currents are generated in the vicinity of interfaces. Static and dynamic contact angles are computed and shown to agree with existing slip models. A resolution study is presented for dynamic drainage and imbibition in a sphere pack, demonstrating the sensitivity of averaged quantities to resolution.

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

We propose in this work an adaptive variational multiscale method for two-fluid flows with surface tension. A level set function is used to provide a precise position of the interfaces. The implementation of the surface tension in the context of the Continuum Surface Force is proposed to circumvent the capillary time step restriction. The obtained system is then solved using a new derived Variational Multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios. Combined with an a posteriori error estimator, we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two-phase incompressible isothermal flows. Stable and accurate results are obtained for all two- and three-dimensional numerical examples. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation.

Magneto Hydro Dynamic two fluid flow of blood through stenosed artery

When blood flow through artery, the two-phase nature of blood as a suspension becomes important as the diameter of the red blood cell (RBC) becomes comparable to the tube diameter. The aim of the present study is to analyzed the effect of magnetic field on the plug flow region, shear stress in the core and plasma layer in two-fluid flow of blood through stenosed artery. Besides magnetic field, the effect of Womersley parameter, thickness of stenosis and width of plasma layer are also discussed. Generated data are analyzed and discussed through graphs.

Computational free-surface fluid–structure interaction with application to floating offshore wind turbines

Abstract In this paper, we propose a computational fluid–structure interaction (FSI) framework for the simulations of the interaction between free-surface flow and floating structures, such as offshore wind turbines. The framework is based on a suitable combination of the finite element method (FEM) and isogeometric analysis (IGA), and has good efficiency, accuracy and robustness characteristics. The free-surface phenomena are modeled using the Navier–Stokes equations of incompressible two-fluid flow in conjunction with the level set method. The FEM-based moving-domain ALE-VMS technique is employed to discretize the fluid mechanics equations, while the IGA-based rotation-free shell and beam/cable formulation is employed to model the mechanics of floating structures. The kinematic and traction compatibility at the fluid–structure interface is handled by means of a recently-developed augmented Lagrangian FSI formulation with formal elimination of the Lagrange multiplier variable. A quasi-direct coupling strategy is adopted to handle the large added mass, and implemented by means of a matrix-free technique. The mathematical formulation of the FSI problem and its algorithmic implementation are described in detail, and two numerical test cases are presented. The first case is a free-surface-flow benchmark example of a solitary wave impacting a fixed, rigid platform. The second case is a set of full-scale free-surface-FSI simulations of the OC3-Hywind floating wind turbine design subjected to wave action. The computational results are compared with experimental and simulation data, with good agreement achieved in all cases where such data was available. Wind-turbine computations in the regime of high-amplitude waves are also presented.

Application of Jacobian-free Newton–Krylov method in implicitly solving two-fluid six-equation two-phase flow problems: Implementation, validation and benchmark

This work represents a first-of-its-kind successful application to employ advanced numerical methods in solving realistic two-phase flow problems with two-fluid six-equation two-phase flow model. These advanced numerical methods include high-resolution spatial discretization scheme with staggered grids (high-order) fully implicit time integration schemes, and Jacobian-free Newton–Krylov (JFNK) method as the nonlinear solver. The computer code developed in this work has been extensively validated with existing experimental flow boiling data in vertical pipes and rod bundles, which cover wide ranges of experimental conditions, such as pressure, inlet mass flux, wall heat flux and exit void fraction. Additional code-to-code benchmark with the RELAP5-3D code further verifies the correct code implementation. The combined methods employed in this work exhibit strong robustness in solving two-phase flow problems even when phase appearance (boiling) and realistic discrete flow regimes are considered. Transitional flow regimes used in existing system analysis codes, normally introduced to overcome numerical difficulty, were completely removed in this work. This in turn provides the possibility to utilize more sophisticated flow regime maps in the future to further improve simulation accuracy.

CFD-aided mathematical modeling of thermal vapor compressors in multiple effects distillation units

In this study a mathematical investigation for the prediction of thermo-compressor performance in a multi-effect distillation unit (MED) is performed. The real gas equation is used to calculate the properties of the working fluid. It is assumed that mixing of the primary and entrained flows occurs at constant pressure inside the constant-area section of the thermo-compressor. The effects of variations in operating conditions such as motive steam pressure and temperature and suction and discharge temperatures on thermo-compressor performance are studied. Various numerical simulations based on the two-fluid multiphase flow formulation are performed. Based on the numerical simulations, a correlation for the calculation of mixing efficiency of motive and entrained flows inside thermo-compressors is proposed. Quantitative validation of data from the proposed mathematical algorithm with industrial data is also presented.

Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity.Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency.The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model.The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary.

Disturbance amplification in boundary layers over thin wall films

In single-fluid boundary layers, streaks can amplify at sub-critical Reynolds numbers and initiate early transition to turbulence. Introducing a wall film of different viscosities can appreciably alter the stability of the base flow and, in particular, the transient growth of the perturbation streaks. The formalism of seminorms is used to identify optimal disturbances which maximize the kinetic energy in the two-fluid flow. An examination of optimal growth over a range of viscosity ratios of the film relative to the outer flow reveals three distinct regimes of amplification, each associated with a particular combination of the eigenfunctions. In order to elucidate the underlying amplification mechanisms, a model problem is formulated: An initial value problem is solved using an eigenfunction expansion and is used to compute the evolution of pairs of eigenfunctions. By appropriately selecting the pair, the initial value problem qualitatively reproduces the temporal evolution of the optimal disturbance, and provides an unambiguous explanation of the dynamics. Two regimes of transient growth are attributed to the evolution of the interface mode along with free-stream vortical modes; the third regime is due to the evolution of the interface and a discrete mode. The results demonstrate that a lower-viscosity film can effectively reduce the efficacy of the lift-up mechanism and, as a result, transient growth of disturbances. However, another mechanism of amplification of wall-normal vorticity arises due to the deformation of the two-fluid interface and becomes dominant below a critical viscosity ratio.

Effect of Magnetic Field on Kelvin-Helmholtz Instability in a Couple-Stress Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface

Kelvin-Helmholtz instability (KHI) appears in stratified two-fluid flow at surface. When the relative velocity is higher than the critical relative velocity, the growth of waves occurs. It is found that magnetic field has a stabilization effect whereas the buoyancy force has a destabilization effect on the KHI in the presence of sharp inter-face. The RT instability increases with wave number and flow shear, and acts much like a KHI when destabilizing effect of sheared flow dominates. It is shown that both of ablation velocity and magnetic field have stabilization effect on RT instability in the presence of continued interface. In this paper, we study the effect of magnetic field on Kelvin-Helmholtz instability (KHI) in a Couple-stress fluid layer above by a porous layer and below by a rigid surface. A simple theory based on fully developed flow approximations is used to derive the dispersion relation for the growth rate of KHI. We replace the effect of boundary layer with Beavers and Joseph slip condition at the rigid surface. The dispersion relation is derived using suitable boundary and surface conditions and results are discussed graphically. The stabilization effect of magnetic field takes place for whole waveband and becomes more significant for the short wavelength. The growth rate decreases as the density scale length increases. The stabilization effect of magnetic field is more significant for the short density scale length.

Three-dimensional modeling of fluid dynamics and heat transfer for two-fluid or phase change flows

A numerical algorithm based on the VOF method was proposed to solve two-phase/two-fluid flow problems by the authors recently. Different from other surface reconstruction approaches, the interface between the two fluids is represented by a continuous surface as the contour surface of the VOF function. It is robust and easy to implement. In this article, the method is extended to deal with three-dimensional flows. Also considered in the study is the heat transfer in the flow with or without phase change. In order to validate the method, a variety of flow problems are under consideration and comparison with empirical correlations is made. The method is first tested by considering a benchmark case with prescribed 3-D velocity field. It is then applied to the problem of bubble rising in a quiescent liquid, followed by investigating the splashing of a drop impacting on a liquid film. To assess the heat transfer model for phase change, the boiling flow emerging from a superheated planar film is under investigation. The last one considered is a jet flow caused by injection of high-temperature octane into a water container.