Formation of uniformly sized metal droplets from a capillary jet by electromagnetic force

A new method for producing uniformly sized metal droplets is proposed. In this method, an intermittent electromagnetic pinch force is applied to a capillary jet of liquid metal to generate fluctuations of equal interval on the surface of the jet. As the fluctuations grow, the liquid metal jet breaks into small droplets whose size depends on the frequency of the intermittent electromagnetic pinch force. The breakup of the capillary jet is numerically simulated by performing multiphase fluid flow analysis with surface tracking (volume of fluid method) and electromagnetic force analysis. The simulation results agree well with the results of model experiments. The jet breaks up into uniformly sized droplets when the frequency of the intermittent force equals the frequency that corresponds to the natural disturbance wavelength of the capillary jet.     

Mathematical and physical simulation of the interaction between a gas jet and a liquid free surface

In this work a two phase 3D mathematical model was developed using the volume of fluid (VOF) algorithm, which is able to accurately describe the cavity geometry and size as well as the liquid flow patterns created when a gas jet that impinges on a liquid free surface. These phenomena are commonly found in steelmaking operations such as in the Electric Arc Furnace (EAF) and the Basic Oxygen Furnace (BOF) where oxygen jets impinge on a steel bath and they control heat, momentum and mass transfer. The model was successfully validated with measurements made on a physical model through velocity fields obtained by Particle Image Velocimetry (PIV) and high speed camera images of the cavity. Agreement between model predictions and experimental measurements is excellent in both x-velocity component of the liquid and cavity sizes. The cavity formed in the liquid by the impinging jet depends on a force balance at the free surface where the inertial force of the jet governs this phenomena, while the liquid circulation depends on also the jet inertial force of the jet, but its angle plays an important role, being the lowest angle the best choice to shear the bath and promote stronger circulation and better mixing in the liquid.     

Modeling the disintegration of modulated liquid jets using volume-of-fluid (VOF) methodology

In this study, we present the numerical investigations on the effect of finite velocity modulations imposed on an otherwise unperturbed cylindrical liquid jet issuing into stagnant gas. Sinusoidal velocity fluctuations of finite frequency and amplitude are imposed at the liquid jet inlet and the resulting liquid jet surface deformation is captured using a volume of fluid (VOF) methodology, utilizing compressive interface capturing scheme for arbitrary meshes (CICSAM) scheme. Variation of the simulation parameters, comprising of the mean liquid jet velocity, modulation amplitude and frequency grouped together using a set of non-dimensional parameters, leads to the formation of a wide gamut of reproducible liquid structures such as waves, upstream/downstream directed bells, chains of droplets similar to those observed in experiments. Elaborate tests on the effect of injection velocity and inlet jet diameter are investigated to characterize the breakup process. The computations efficiently capture the diverse flow structures generated by the evolving modulated liquid jet inclusive of several non-linear dynamics such as growth of surface waves, ligament interaction with shear vortices and its subsequent thinning process. The simulations identify the deterministic behavior of modulated liquid jets to predict liquid disintegration modes under given set of non-dimensional parameters.     

Assessment of an Eulerian CFD model for prediction of dilute droplet dispersion in a turbulent jet

The turbulent dispersion of non-evaporating droplets in an axisymmetric round jet issuing from a nozzle is investigated both experimentally and theoretically. The experimental data set has a well-defined inlet boundary with low turbulence intensity at the nozzle exit, so that droplet dispersion is not affected by the transport of nozzle-generated fluctuating motion into the jet, and is influenced solely by turbulence in the gas phase produced in the shear layer of the jet. This data set is thus ideal for testing algebraic models of droplet fluctuating motion that assume local equilibrium with the turbulence in the gas phase. Moreover, the droplet flux measurements are sufficiently accurate that conservation of the total volume flow of the droplet phase has been demonstrated. A two-fluid turbulence modelling approach is adopted, which uses the k–ε turbulence model and a simple algebraic model that assumes local equilibrium to predict the fluid and droplet turbulent correlations, respectively. We have shown that the k–ε turbulence model lacks generality for predicting the spread of momentum in jets with and without a potential core. However, in general, the model predicts the radial dispersion of droplets in the considered turbulent jet with reasonable accuracy over a broad range of droplet sizes, once deficiencies in the k–ε turbulence model are taken into account.     

Computational fluid dynamics modelling of gas jets impinging onto liquid pools

Gas jets impinging onto a gas–liquid interface of a liquid pool are studied using computational fluid dynamics modelling, which aims to obtain a better understanding of the behaviour of the gas jets used metallurgical engineering industry. The gas and liquid flows are modelled using the volume of fluid technique. The governing equations are formulated using the density and viscosity of the “gas–liquid mixture”, which are described in terms of the phase volume fraction. Reynolds averaging is applied to yield a set of Reynolds-averaged conservation equations for the mass and momentum, and the k–ε turbulence model. The deformation of the gas–liquid interface is modelled by the pressure jump across the interface via the Young–Laplace equation. The governing equations in the axisymmetric cylindrical coordinates are solved using the commercial CFD code, FLUENT. The computed results are compared with experimental and theoretical data reported in the literature. The CFD modelling allows the simultaneous evaluation of the gas flow field, the free liquid surface and the bulk liquid flow, and provides useful insight to the highly complex, and industrially significant flows in the jetting system.     

Numerical simulation of internal flow fields of swirl coaxial injector in a hot environment

Based on the fractional volume of fluid (VOF), a pure Eulerian model for defining and capturing the gas/liquid interface is developed in this paper. This model can describe gas/liquid interface in high refinement, which is better than the original VOF methodology. To validate the proposed model and the algorithm, the computational code is employed to predict the flow performance in a cylindrical swirl injector under cold-flow condition, and the predicted results agree well with experimental measurements. Furthermore, the proposed model is used to simulate gas-liquid reacting flows inside a gas/liquid coaxial swirl injector operating in a hot environment. The turbulent combustion process is simulated with the k−ε−f−g model. The numerical simulation is carried out under actual operating condition of the coaxial injector. The injector performances, such as liquid film thickness, liquid film injection velocity, spray angle, pressure drop, are obtained based on the detailed information of the internal flow field. The predicted results also show that droplets are shed from the liquid film in the recess cup of the coaxial injector because of the large velocity gradient between the gas and liquid streams, and a burning area, which is characterized by high temperature, is present inside the injector.     

Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model

In this paper simulation of cavitating flow over the Clark-Y hydrofoil is reported using the large eddy simulation (LES) turbulence model and volume of fluid (VOF) technique. We applied an incompressible LES modelling approach based on an implicit method for the subgrid terms. To apply the cavitation model, the flow has been considered as a single fluid, two-phase mixture. A transport equation model for the local volume fraction of vapour is solved and a finite rate mass transfer model is used for the vapourization and condensation processes. A compressive volume of fluid (VOF) method is applied to track the interface of liquid and vapour phases. This simulation is performed using a finite volume, two phase solver available in the framework of the OpenFOAM (Open Field Operation and Manipulation) software package. Simulation is performed for the cloud and super-cavitation regimes, i.e., σ = 0.8, 0.4, 0.28. We compared the results of two different mass transfer models, namely Kunz and Sauer models. The results of our simulation are compared for cavitation dynamics, starting point of cavitation, cavity’s diameter and force coefficients with the experimental data, where available. For both of steady state and transient conditions, suitable accuracy has been observed for cavitation dynamics and force coefficients.     

Fluid forces on a transversely oscillating cylinder

The numerical solution of the unsteady two-dimensional Navier-Stokes equations is used to investigate the fluid forces experienced by a translating and transversely oscillating cylinder. Calculations are first performed in an oscillatory frequency range outside the synchronization when oscillatory-to-translational velocity ratio is 1.5 and at a fixed Reynolds number R = 10³. The object of this study is to examine the effect of increase of forced oscillation frequency on the fluid forces. The results of this study are in good agreement with previous experimental predictions.     

Modeling of van der Waals force with smoothed particle hydrodynamics: Application to the rupture of thin liquid films

The rupture of thin liquid films driven by the van der Waals force is of significance in many engineering processes, and most previous studies have relied on the lubrication approximation. In this paper, we develop a smoothed particle hydrodynamics (SPH) representation for the van der Waals force and simulate the rupture of thin liquid films without resort to lubrication theory. The van der Waals force in SPH is only imposed on one layer, i.e., the outermost layer of fluid particles, where a weighting function is deployed to evaluate the contributions of particles on or near the interface. However, to obtain an accurate hydrostatic pressure in reaction to the van der Waals force, a smaller smoothing length is used for the calculation of the weighting function than that used for SPH discretizations of the bulk fluid. The same surface particles are also used to model the surface tension. To deal with the rupture of a thin liquid film with a very small aspect ratio ε (ε = thickness/length), a coordinate transformation is introduced to shrink the length of the liquid film to achieve accurate numerical resolution with a manageable number of particles. As verifications of our physical model and numerical algorithm, we simulate the hydrostatic pressure in a stationary film and the relaxation of an initially square droplet and compare the SPH results with the analytical solutions. The method is then applied to simulate the rupture of thin liquid films with moderate and small aspect ratios (ε = 0.5 and 0.005). The convergence of the method is verified by refining particle spacing to four different levels. The effect of the capillary number on the rupture process is analyzed.     

Numerical simulation of cavitation dynamics using a cavitation-induced-momentum-defect (CIMD) correction approach

A new unsteady cavitation event tracking model is developed for predicting vapor dynamics occurring in multi-dimensional incompressible flows. The procedure solves incompressible Navier–Stokes equations for the liquid phase supplemented with an additional vapor transport equation for the vapor phase. The novel cavitation-induced-momentum-defect (CIMD) correction methodology developed in this study accounts for cavitation inception and collapse events as relevant momentum-source terms in the liquid phase momentum equations. The model tracks cavitation zones and applies compressibility effects, employing homogeneous equilibrium model (HEM) assumptions, in constructing the source term of the vapor transport model. Effects of vapor phase accumulation and diffusion are incorporated by detailed relaxation models. A modified RNG k–ε model, including the effects of compressibility in the vapor regions, is employed for modeling turbulence effects. Numerical simulations are carried out using a finite volume methodology available within the framework of commercial CFD software code Fluent v.6.2. Simulation results are in good qualitative agreement with experiments for unsteady cloud cavitation behavior in planar nozzle flows. Multitude of mechanisms such as formation of vortex cavities, vapor cluster shedding and coalescence, cavity pinch off are sharply captured by the CIMD approach. Our results indicate the profound influence of re-entrant jet motion and adverse pressure gradients on the cavitation dynamics.     

Heat and mass transfer in annular liquid jets: I. Formulation

Heat and mass transfer phenomena in annular liquid jets are analyzed at high Reynolds numbers by means of a model derived from the governing equations that takes into account the effects of surface tension and boundary conditions at the gas–liquid interfaces and the large differences between the thermal and mass diffusivities, densities, dynamic viscosities, and thermal conductivities between gases and liquids. The model clearly illustrates the stiffness in both space and time associated with the concentration, linear momentum and energy boundary layers, and the initial cooling of the gases enclosed by the jet when, starting from a steady state where gases are injected into the volume enclosed by the jet at a rate equal to the heat and mass absorption rates by the liquid, gas injection is stopped. It is shown that, owing to the non-linear integrodifferential coupling between the fluid dynamics and heat and mass transfer processes, the pressure of the gases enclosed by the jet may vary in either a monotonic or an oscillatory manner depending on the large number of non-dimensional parameters that govern the heat and mass transfer phenomena. For the underpressurized jets considered here, it is shown that thermal equilibrium is achieved at a much faster rate than that associated with mass transfer, double diffusive phenomena in the liquid may occur, and the mass and volume of the gases enclosed by the jet may increase or decrease as functions of time until a steady equilibrium condition is reached.     

Calculation of conjugate heat transfer problem with volumetric heat generation using the Galerkin method

A mathematical model of fluid flow across a rod bundle with volumetric heat generation has been built. The rods are heated with volumetric internal heat generation. To construct the model, a volume average technique (VAT) has been applied to momentum and energy transport equations for a fluid and a solid phase to develop a specific form of porous media flow equations. The model equations have been solved with a semi-analytical Galerkin method. The detailed velocity and temperature fields in the fluid flow and the solid structure have been obtained. Using the solution fields, a whole-section drag coefficient C_d and a whole-section Nusselt number Nu have also been calculated. To validate the developed solution procedure, the results have been compared to the results of a finite volume method. The comparison shows an excellent agreement. The present results demonstrate that the selected Galerkin approach is capable of performing calculations of heat transfer in a cross-flow where thermal conductivity and internal heat generation in a solid structure has to be taken into account. Although the Galerkin method has limited applicability in complex geometries, its highly accurate solutions are an important benchmark on which other numerical results can be tested.     

Heat and mass transfer in annular liquid jets: III. Combustion within the volume enclosed by the jet

The nonlinear dynamics of and heat and mass transfer processes in annular liquid jets are analyzed by means of a nonlinear system of integrodifferential equations which account for the liquid motion and the gases enclosed by the jet. Both linear and sinusoidal heat and mass addition sources are considered to take place homogeneously within the volume enclosed by the jet's inner interface in an attempt to simulate the combustion of hazardous wastes or materials within this volume. It is shown that the liquid's temperature at the jet's inner interface increases rapidly with linear heat addition, but drops also quickly to its initial value once heat addition is ended, whereas the pressure coefficient and the volume enclosed by the jet increase until they reach a maximum value and then decrease in an oscillatory manner towards their steady values. For the case of sinusoidal heat addition, it is shown that the pressure coefficient and interfacial concentration, temperature and heat and mass fluxes oscillate in a sinusoidal manner with the same frequency as that of the sinusoidal heat source. It is also shown that mass transfer phenomena are much slower than heat transfer ones. For the case of linear mass addition, it is shown that the temperature of the gases enclosed by the jet first decreases because of dilution and then it increases until it reaches a constant value that corresponds to the same temperature for the gases and the flowing liquid. The pressure of the gases enclosed by the jet first increases because of mass addition and then slowly decreases because of mass absorption by the jet.     

Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].     

Numerical study of flow asymmetry and self-sustained jet oscillations in geometrically symmetric cavities

In this paper we present the results of numerical investigation of self-sustained oscillations of a jet confined in a symmetric cavity. This work represents an attempt to reproduce empirical observations of asymmetric flows in geometrically symmetric systems and to extend the jet flow investigations to more complex possible scenarios. A well-known example of such two-dimensional flow has been reported experimentally and reproduced numerically for simple flow [E. Schreck, M. Schaefer, Numerical study or bifurcation in three-dimensional sudden channel expansions, Comput. Fluids 29 (2000) 583–593]. It has been found that for some particular control parameter, above its critical value (bifurcation point), the jet can be deflected to either of the two sides of the cavity. In this paper we report a similar behaviour which is, however, characterized by a more complicated flow pattern. While simple flow appears only within small cavity lengths the complex flows develops with increased cavity lengths. Unlike stationary asymmetric solutions accompanied by cavity jet oscillations, as experimentally reported in e.g., [A. Maurel, P. Ern, B.J.A. Zielinska, J.E. Wesfreid, Experimental study of self-sustained oscillations in a confined jet, Phys Rev. E 54 (1996) 3643–3651], in our investigations of both simple and complex asymmetric flow we observed the slow periodical drift of the jet from one to another side of the cavity. The essential control parameters were Reynolds number Re and the ratio length to inlet width L/d. According to experiments of Maurel et al. (1996), the jet is stable and symmetric, when both L/d and Re are below certain critical values, otherwise jet oscillations appear in both experiment and our simulation (cavity oscillations regime). However, further increase of either (or both) L/d and Re leads of so called free jet type oscillations regime. This paper describes complex jet behaviour within the later oscillations regime. We believe that both simple “classical” and “our” complex stationary asymmetric solutions (as well as superimposed cavity-type and free-jet oscillations) can be explained based on physical arguments as already done in previous works. However, the origin of slow drift motion remains still to be resolved. This might be of high importance for clear distinguishing between relevant physical and numerical features in future codes developments.     

Parametric study of liquid droplets impinging on porous surfaces

This work presents a numerical simulation of the fluid dynamics of a liquid droplet during impact/absorption onto a porous medium. The main focus of this paper is on a parametric study of the influence of the governing parameters upon the fluid flow characteristics. The problem is described in a non-dimensional form, and the influence of the main governing parameters is investigated, including their variation along the range of physical configurations of interest. This procedure revealed 7 main governing parameters: Reynolds number (Re), Darcy number (Da), porosity (ε), Froude number (Fr), Weber number (We), contact angle (θ) and the ratio between pore and particle diameter size in the porous substrate (α). The results indicate that the values of Da and Re are more related to the amount of momentum dissipation due to the drag of the solid matrix of the substrate, while the values of We, α and θ can be mainly related to capillary pressure.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

The simulation of electrified liquid jets

The simulation of a two-dimensional electrified liquid jet is described. A finite difference technique is coupled with a computational fluid dynamic code to solve Poisson's equation and the Navier-Stokes equations at the electrostatic fluid-flow interface. The dynamics of free-surface electrohydrodynamic fluid flow are simulated for a dielectric fluid in an electrostatics nozzle and a conducting two-dimensional jet. The fluid flow in a nozzle is compared with and without an applied electric field, and the effects of adding a grounded conducting cylinder to the configuration is demonstrated. A set of time sequence graphs are used to illustrate the breakup of a charged jet into droplets, and the influence of viscosity on jet formation and breakup is depicted.     

Streamwise oscillations of a cylinder beneath a free surface: Free surface effects on vortex formation modes

A computational study of a viscous incompressible two-fluid model with an oscillating cylinder is investigated at a Reynolds number of 200 and at a dimensionless displacement amplitude of A=0.13 and for the dimensionless forcing cylinder oscillation frequency-to-natural vortex shedding frequency ratios, f/f₀=1.5,2.5,3.5. Specifically, two-dimensional flow past a circular cylinder subject to forced in-line oscillations beneath a free surface is considered. The method is based on a finite volume discretization of the two-dimensional continuity and unsteady Navier-Stokes equations (when a solid body is present) on a fixed Cartesian grid. Two-fluid model based on improved volume-of-fluid method is used to discretize the free surface interface. The study focuses on the laminar asymmetric flow structure in the near wake region and lock-on phenomena at a Froude number of 0.2 and for the dimensionless cylinder submergence depths, h=0.25, 0.5 and 0.75. The equivorticity patterns and pressure distribution contours are used for the numerical flow visualization. The code validations in special cases show good comparisons with previous numerical results.