The dynamics of a Taylor bubble rising in stagnant liquids is numerically investigated using a front tracking coupled with finite difference method. Parametric studies on the dynamics of the rising Taylor bubble including the final shape, the Reynolds number (Re(T)), the Weber number (We(T)), the Froude number (Fr), the thin liquid film thickness (w/D), and the wake length (l(w)/D) are carried out. The effects of density ratio (η), viscosity ratio (λ), Eötvös number (Eo), and Archimedes number (Ar) are examined. The simulations demonstrate that the density ratio and the viscosity ratio under consideration have minimal effect on the dynamics of the Taylor bubble. Eötvös number and Archimedes number influence the elongation of the tail and the wake structures, where higher Eo and Ar result in longer wake. To explain the sudden extension of the tail, a Weber number (We(l)) based on local curvature and velocity is evaluated and a critical We(l) is detected around unity. The onset of flow separation at the wake occurs in between Ar=2×10(3) and Ar=1×10(4), which corresponds to Re(T) between 13.39 and 32.55. Archimedes number also drastically affects the final shape of Taylor bubble, the terminal velocity, the thickness of thin liquid film, as well as the wall shear stress. It is found that w/D=0.32 Ar(-0.1).
This paper implements a coupled approach to integrate the internal nozzle flow and the ensuing fuel spray using a Volume-of-Fluid (VOF) method in the finite-volume framework. A VOF method is used to model the internal nozzle two-phase flow with a cavitation description closed by the homogeneous relaxation model of Bilicki and Kestin (1990). An Eulerian single velocity field approach by Vallet et al. (2001) is implemented for near-nozzle spray modeling. This Eulerian approach considers the liquid and gas phases as a complex mixture with a highly variable density to describe near nozzle dense sprays. The liquid mass fraction is transported with a model for the turbulent liquid diffusion flux into the gas. Fully-coupled nozzle flow and spray simulations are performed in three dimensions and validated against the x-ray radiography measurements of Kastengren et al. (2014) for a diesel fuel surrogate. A standard k − ǫ Reynolds Averaged Navier Stokes based turbulence model is used in this study and the influence of model constants is evaluated. First, the grid convergence study is performed. The effect of grid size is also evaluated by comparing the fuel distribution against experimental data. Finally, the fuel distribution predicted by the coupled Eulerian approach is compared against that by Lagrangian-Eulerian spray model along with experimental data. The coupled Eulerian approach provides a unique way of coupling the nozzle flow and sprays so that the effects of in-nozzle flow can be directly realized on the fuel spray. Both experiment and numerical simulations show noncavitation occurring for this injector with convergent nozzle geometry. The study shows that the Eulerian approach has advantages over near-field dense spray distributions.
The wall effects on the axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a liquid-filled, finite circular cylinder are numerically investigated. The bulk and gas phases are considered incompressible and immiscible. The bubble motion and deformation are characterized by the Morton number Mo, Eötvös number Eo, Reynolds number Re, Weber number We, density ratio, viscosity ratio, the ratios of the cylinder height and the cylinder radius to the diameter of the initially spherical bubble (H* =H/d 0 , R*=R/d 0 ). Bubble rise in liquids described by Eo and Mo combinations ranging from (1,0.01) to (277.5,0.092), as appropriate to various terminal state Reynolds numbers (Re T ) and shapes have been studied. The range of terminal state Reynolds numbers includes 0.02T<70. Bubble shapes at terminal states vary from spherical to intermediate spherical-cap-skirted. The numerical procedure employs a front tracking finite difference method coupled with a level contour reconstruction of the front. This procedure ensures a smooth distribution of the front points and conserves the bubble volume. For the wide range of Eo and Mo examined, bubble motion in cylinders of height H*=8 and R≥3, is noted to correspond to the rise in an infinite medium, both in terms of Reynolds number and shape at terminal state. In a thin cylindrical vessel (small R*) the motion of the bubble is retarded due to increased total drag and the bubble achieves terminal conditions within a short distance from release. The wake effects on bubble rise are reduced, and elongated bubbles may occur at appropriate conditions. For a fixed volume of the bubble, increasing the cylinder radius may result in the formation of well-defined rear recirculatory wakes that are associated with lateral bulging and skirt formation. The wall effects on the axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a liquid-filled, finite circular cylinder are numerically investigated. The bulk and gas phases are considered incompressible and immiscible. The bubble motion and deformation are characterized by the Morton number ͑Mo͒, Eötvös number ͑Eo͒, Reynolds number ͑Re͒, Weber number ͑We͒, density ratio, viscosity ratio, the ratios of the cylinder height and the cylinder radius to the diameter of the initially spherical bubble ͑HBubble rise in liquids described by Eo and Mo combinations ranging from ͑1,0.01͒ to ͑277.5,0.092͒, as appropriate to various terminal state Reynolds numbers ͑Re T ͒ and shapes have been studied. The range of terminal state Reynolds numbers includes 0.02Ͻ Re T Ͻ 70. Bubble shapes at terminal states vary from spherical to intermediate spherical-cap-skirted. The numerical procedure employs a front tracking finite difference method coupled with a level contour reconstruction of the front. This procedure ensures a smooth distribution of the front points and conserves the bubble volume. For the wide range of Eo and Mo examined, bubble motion in cylinders of height H * = 8 and R ...
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