The double-mass model of drop deformation and secondary breakup

In the paper, a new analytical model of drop deformation and secondary breakup is presented. The model is a direct extension of the TAB (Taylor Analogy Breakup) model of O’Rourke and Amsden [P. O’Rourke, A.A. Amsden, The TAB method for numerical calculation of spray droplet breakup, SAE Paper No 872089, 1987] [9]. The drop is represented by the system of two masses connected by a spring, allowed to oscillate and move along a specified axis. Two versions of the model are analyzed: linear – offering analytical solution for drop oscillations, and nonlinear – defined in terms of parameters with clear, physical interpretation, and more interesting from the point of view of applications. Conditions of stability of a drop subjected to impulsive acceleration by ambient flow are discussed and a new criterion is introduced including droplet Weber number, Ohnesorge number and density ratio. The role of the density ratio proves to be prominent for large Ohnesorge numbers or when the drop density approaches the ambient density.     

Simulation of falling droplet by the lattice Boltzmann method

In this paper the lattice Boltzmann method (LBM) is employed to simulate deformation and breakup of a falling drop under gravity. First the two-phase LBM is applied to verify the Laplace law for static drops. In order to further verify the model, relaxation of a square droplet with two different viscosities is conducted. Then deformation and breakup of a falling drop for some range of Eotvos and Ohnesorge numbers are investigated. It is seen that at relatively low Eotvos numbers, where the surface tension force is dominant, the drop deforms slowly and reaches a steady state without breakup. At higher Eotvos numbers gravitational force prevail over the surface tension force and the drop distorts more. Breakup of the drop can be seen for a large enough Eotvos value. On the other hand the stabilizing effect of the Ohnesorge number, which is the ratio of viscous stresses and surface tension, is shown. It is found that at higher Ohnesorge values, the viscous forces are dominant and the drop tends to maintain its original shape.     

Mathematical Modeling and Simulation of Antibubble Dynamics

In this study, we propose a mathematical model and perform numerical simulations for the antibubble dynamics. An antibubble is a droplet of liquid surrounded by a thin film of a lighter liquid, which is also in a heavier surrounding fluid. The model is based on a phase-field method using a conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier and a modified Navier-Stokes equation. In this model, the inner fluid, middle fluid and outer fluid locate in specific diffusive layer regions according to specific phase filed (order parameter) values. If we represent the antibubble with conventional binary or ternary phase-field models, then it is difficult to have stable thin film. However, the proposed approach can prevent nonphysical breakup of fluid film during the simulation. Various numerical tests are performed to verify the efficiency of the proposed model.     

Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray–Scott Model

Two different types of instabilities of equilibrium stripe and ring solutions are studied for the singularly perturbed two‐component Gray–Scott (GS) model in a two‐dimensional domain. The analysis is performed in the semi‐strong interaction limit where the ratio O(?^?2) of the two diffusion coefficients is asymptotically large. For ?→ 0 , an equilibrium stripe solution is constructed where the singularly perturbed component concentrates along the mid‐line of a rectangular domain. An equilibrium ring solution occurs when this component concentrates on some circle that lies concentrically within a circular cylindrical domain. For both the stripe and the ring, the spectrum of the linearized problem is studied with respect to transverse (zigzag) and varicose (breakup) instabilities. Zigzag instabilities are associated with eigenvalues that are asymptotically small as ?→ 0 . Breakup instabilities, associated with eigenvalues that are O(1) as ?→ 0 , are shown to lead to the disintegration of a stripe or a ring into spots. For both the stripe and the ring, a combination of asymptotic and numerical methods are used to determine precise instability bands of wavenumbers for both types of instabilities. The instability bands depend on the relative magnitude, with respect to ?, of a nondimensional feed‐rate parameter A of the GS model. Both the high feed‐rate regime A=O(1) , where self‐replication phenomena occurs, and the intermediate regime O(?^1/2) ?A?O(1) are studied. In both regimes, it is shown that the instability bands for zigzag and breakup instabilities overlap, but that a zigzag instability is always accompanied by a breakup instability. The stability results are confirmed by full numerical simulations. Finally, in the weak interaction regime, where both components of the GS model are singularly perturbed, it is shown from a numerical computation of an eigenvalue problem that there is a parameter set where a zigzag instability can occur with no breakup instability. From full‐scale numerical computations of the GS, it is shown that this instability leads to a large‐scale labyrinthine pattern.     

Applications of sensitivity analysis for probit stochastic network equilibrium

Network equilibrium models are widely used by traffic practitioners to aid them in making decisions concerning the operation and management of traffic networks. The common practice is to test a prescribed range of hypothetical changes or policy measures through adjustments to the input data, namely the trip demands, the arc performance (travel time) functions, and policy variables such as tolls or signal timings. Relatively little use is made, however, of the full implicit relationship between model inputs and outputs inherent in these models. By exploiting the representation of such models as an equivalent optimisation problem, classical results on the sensitivity analysis of non-linear programs may be applied, to produce linear relationships between input data perturbations and model outputs. We specifically focus on recent results relating to the probit Stochastic User Equilibrium (PSUE) model, which has the advantage of greater behavioural realism and flexibility relative to the conventional Wardrop user equilibrium and logit SUE models. The paper goes on to explore four applications of these sensitivity expressions in gaining insight into the operation of road traffic networks. These applications are namely: identification of sensitive, ‘critical’ parameters; computation of approximate, re-equilibrated solutions following a change (post-optimisation); robustness analysis of model forecasts to input data errors, in the form of confidence interval estimation; and the solution of problems of the bi-level, optimal network design variety. Finally, numerical experiments applying these methods are reported.     

Update to the MUSIG model in ANSYS CFX for reliable modelling of bubble coalescence and breakup

The MUSIG (Multiple Size Group) model in the commercial CFD code ANSYS CFX is a population balance approach for describing binary bubble coalescence and breakup events. It is widely used in the simulation of poly-dispersed bubbly flows. The purpose of this work is to identify the internal inconsistencies in the discrete method that is applied for the solution of the population balance equation in MUSIG, and to propose an internally consistent one for discretising the source and sink terms that result from bubble coalescence and breakup. The new formulation is superior to the existing ones in preserving both mass and number density of bubbles, allowing arbitrary discretisation schemes and is free of costly numerical integrations. The numerical results on the evolution of bubble size distributions in bubbly flows reveal that the inconsistency in the original MUSIG regarding bubble breakup is non-negligible for both academic and practical cases. The discussion on the effect of internal inconsistency as well as updates to the model presented in this work are necessary and important for calibration of bubble coalescence and breakup models using the MUSIG approach.     

Error analysis: a tool for selecting travel demand models

In developing travel demand models it is generally assumed that the base-year data used in developing the parameters, as well as the forecasted data to be used as independent variables for the design year, are of acceptable quality. The purpose of this paper is to present the application of error propagation theory in assesing the predictive quality of one type of travel demand forecasting model (multinomial logit models) and to demonstrate how error considerations can be used as a tool for identifying the optimal model. The general conclusions of this study are that: (1) it is indeed possible to quantify errors in dependent variables in logit models as a consequence of errors in independent variables; and (2) error consideration can be used as a tool for identifying the optimal model from a set of candidate models. Further research is recommended to develop better insights into the phenomenon of error propagation so that the consideration of errors can be a factor in decisions on model selection.     

Modeling and simulation of severe slugging in air-water systems including inertial effects

A mathematical model and numerical simulations corresponding to severe slugging in air-water pipeline-riser systems are presented. The mathematical model considers continuity equations for liquid and gas phases, with a simplified momentum equation for the mixture. A drift-flux model, evaluated for the local conditions in the riser, is used as a closure law. In many models appearing in the literature, propagation of pressure waves is neglected both in the pipeline and in the riser. Besides, variations of void fraction in the stratified flow in the pipeline are also neglected and the void fraction obtained from the stationary state is used in the simulations. This paper shows an improvement in a model previously published by the author, including inertial effects. In the riser, inertial terms are taken into account by using the rigid water-hammer approximation. In the pipeline, the local acceleration of the water and gas phases are included in the momentum equations for stratified flow, allowing to calculate the instantaneous values of pressure drop and void fraction. The developed model predicts the location of the liquid accumulation front in the pipeline and the liquid level in the riser, so it is possible to determine which type of severe slugging occurs in the system. A comparison is made with experimental results published in literature including a choke valve and gas injection at the bottom of the riser, showing very good results for slugging cycle and stability maps. Simulations were also made assessing the effect of different strategies to mitigate severe slugging, such as choking, gas injection and increase in separation pressure, showing correct trends.     

The linearized Boltzmann equation: Sound-wave propagation in a rarefied gas

An analytical version of the discrete-ordinates method (the ADO method) is used to establish concise and particularly accurate solutions to the problem of sound-wave propagation in a rarefied gas. The analysis and the numerical work are based on a rigorous form of the linearized Boltzmann equation (for rigid-sphere interactions), and in contrast to many other works formulated (for an infinite medium) without a boundary condition, the solution reported here satisfies a boundary condition that models a diffusely-reflecting vibrating plate. In addition and in order to investigate the effect of kinetic models, solutions are developed for the BGK model, the S model, the Gross-Jackson model, as well as for the (newly defined) MRS model and the CES model. While the developed numerical results are compared to available experimental data, emphasis in this work is placed on the solutions of the problem of sound-wave propagation as described by the linearized Boltzmann equation and the five considered kinetic models. Received: November 22, 2004; revised: February 24, 2005     

Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

Development of a compressible multiphase cavitation approach for diesel spray modelling

The influence of in-nozzle phenomena including cavitation on the morphology of the spray from a diesel injector with a sharp nozzle inlet is investigated numerically. A compressible, multi-phase Volume of Fluid Large Eddy Simulation is implemented in the OpenFOAM environment. The volume fraction transport equations for liquid and gas phases are reformulated to include mass transfer source terms. These source terms are modelled with two cavitation models by Schnerr and Kunz, which are extended to eliminate non-physical mass transfer rates. Validation is carried out only for the Schnerr cavitation model due to its independence of empirical parameters. The numerical method is validated by comparing the simulated mass flow rates, pressure and velocity profiles at different cavitation conditions against published experimental data obtained using a slightly converging square channel. Favourable comparison between simulations and experiments is achieved with minor discrepancies attributable to uncertainties in fuel properties, experimental artefacts and assumptions made in numerical models. Application of the method to calculation of in-nozzle phenomena and primary breakup of a diesel spray reveals that in-nozzle flow separation, wall shear and cavitation contribute greatly to the fragmentation of the jet. Comparison of the two cavitation models shows that after the onset of complete flow detachment, the Kunz implementation predicts higher air inflow at the nozzle outlet than the Schnerr model.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

Numerical modelling of free surface flows in metallurgical vessels

A numerical model for simulating the transient behaviour of multi-fluid problems defined in 2D rectangular and cylindrical geometries is presented. The model uses a piecewise linear volume tracking scheme, and maintains sharp interfaces and captures fine-scale flow phenomena such as fragmentation and coalescence. The numerical model was applied to four problems of pyrometallurgical relevance – entrainment of matte in the flow of slag during skimming operations, splash resulting from a drop impinging on a bath, bubble rise in a liquid bath, and top-submerged gas injection. The numerical predictions are in good agreement with the published experimental results. The simulation of top-submerged gas injection showed, in detail, the phenomena of bubble formation, bubble rise, and splash drop formation and recoalescence with the bath. Data useful for engineering purposes such as pressure traces and time-averaged flow fields were obtained, allowing assessment of splash behaviour for given gas injection conditions. The numerical model has been shown to be versatile in being able to adapt to a wide range of multi-phase flow problems.     

Modeling and simulation of the CLS cryogenic system

This paper presents results pertaining to the numerical modeling of the cryogenic system at the Canadian Light Source. The cryogenic system consists of a cryostat that houses a Radio Frequency (RF) cavity used for boosting the energy of an electron beam. For consistent operation of the RF cavity, it must be kept immersed in liquid helium at a constant level with the pressure in the gas space maintained to an accuracy of ±1 mbar. An improvement to the cryostat model suggested in [3] using control volumes is described. The model and numerical method developed for the liquid helium supply and gaseous helium return lines are validated using two different cases, viz., the liquid helium flow rate from the liquid helium transfer line and the gaseous helium flow rate from the cryostat for various heater power input settings. The numerical method described here is significantly more accurate, efficient, and flexible than that used in [1] based on an iterative bisection method.     

Capillary breakup of liquid threads: a singularity-free solution

The process of capillary breakup of a thread of Newtonian liquidis considered theoretically in the simplest case where the threadis surrounded by an inviscid, dynamically passive gas. The goalis to remove the singularities inherent in the known solutionsto the problem obtained in the framework of the standard modeland explain some puzzling qualitative features of the processobserved in experiments. The analysis is based on the idea that,since the known solutions indicate that the rate at which freshfree-surface area is created tends to infinity as breakup isapproached, one has that the surface tension, whose relaxationto equilibrium is always associated with a small but finiterelaxation time, is bound to deviate from its equilibrium valuein the process of breakup. This gives rise to a surface-tensiongradient which starts to pull the liquid thread apart (the flow-inducedMarangoni effect), whilst the role of the capillary pressure-drivensqueezing of the liquid out of the neck diminishes as the surfacetension in the minimal cross-section decreases. An earlier developedtheory incorporating the interface formation process is appliedwithout any ad hoc alterations and analysed in the frameworkof the slender-jet approximation. The resulting solution issingularity-free and allows one to describe some previouslyunexplained features of experiment by Kowalewski (1996, FluidDyn. Res., 17, 121).     

The second‐order upwind finite difference fractional steps method for moving boundary value problem of oil‐water percolation

The research of the three‐dimensional (3D) compressible miscible (oil and water) displacement problem with moving boundary values is of great value to the history of oil‐gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil‐gas resources, and numerical simulation of seawater intrusion. The mathematical model can be described as a 3D‐coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of 3D‐bounded region, a kind of second‐order upwind finite difference fractional steps schemes applicable to parallel arithmetic is put forward. Some techniques, such as the change of variables, calculus of variations, and the theory of a priori estimates, are adopted. Optimal order estimates in l² norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, for model numerical method and for software development. Thus, the well‐known problem has been solved.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1103–1129, 2014     

Numerical study of the primary breakup of a plane liquid sheet with co-flowing air streams

The objective of this study is to model the primary breakup of a plane liquid sheet emerging from an air-blast nozzle. In the present work the interface compression scheme proposed by OpenCFD Ltd. [1] has been used to capture the interface between the liquid and gas. A One-equation subgrid scale (sgs) turbulent energy transport model attributed to Yoshizawa [2] is used for modeling the effects of turbulence. The set up case selected for this study is based on the experiments carried out by Mitra [3]. The 2D simulations performed in this study predict the breakup length of the plane liquid sheet in good agreement with the experimental data. Future work will involve, performing 3D simulations of the plane liquid sheet generated by the air-blast nozzle and performing comparisons of the resulting droplet characteristics with the experimental data. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuous whole-link travel time model with occupancy constraint

The continuous dynamic network loading problem (CDNLP) aims to compute link travel times and path travel times on a congested network, given time-dependent path flow rates for a given time period. A crucial element of CDNLP is a model of the link performance. Two main modeling frameworks have been used in link loading models: The so-called whole-link travel time (WTT) models and the kinematic wave model of Lighthill–Whitham–Richards (LWR) for traffic flow.In this paper, we reformulate a well-known whole-link model in which the link travel time, for traffic entering a time t, is a function of the number of vehicles on link. This formulation does not require the satisfying of the FIFO (first in, first out) condition. An extension of the basic WTT model is proposed in order to take explicitly into account the maximum number of vehicles that the link can accommodate (occupancy constraint). A solution scheme for the proposed WTT model is derived.Several numerical examples are given to illustrate that the FIFO condition is not respected for the WTT model and to compare the travel time predictions effected by LWR and WTT models.