Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

The motion of elongated gas bubbles in vertical pipes has been studied extensively over the past century. A number of empirical and numerical correlations have emerged out of this curiosity; amongst them, analytical solutions have been proposed. A review of the major results and resolution methods based on a potential flow theory approach is presented in this article. The governing equations of a single elongated gas bubble rising in a stagnant or moving liquid are given in the potential flow formalism. Two different resolution methods (the power series method and the total derivative method) are studied in detail. The results (velocity and shape) are investigated with respect to the surface tension effect. The use of a new multi-objective solver coupled with the total derivative method improves the research of solutions and demonstrates its validity for determining the bubble velocity. This review aims to highlight the power of analytical tools, resolution methods and their associated limitations behind often well-known and wide-spread results in the literature.

The Drag Forces on a Taylor Bubble Rising Steadily in Vertical Pipes

By the adoption of a drag-buoyancy equality model, analytical solutions were obtained for the drag coefficients (CD) of Taylor bubbles rising steadily in pipes. The obtained solutions were functions of the geometry of the Taylor bubble and the gas volume fraction. The solutions were applicable at a wide range of Capillary numbers. The solution was validated by comparison with experimental data of other investigators. All derived drag formulas were subject to the condition that Bond number >4, for air-water systems.

COMPUTATIONAL FLUID DYNAMICS SIMULATIONS OF TAYLOR BUBBLE RISE IN FLOWING LIQUIDS

Systematic analysis of the effect of gravitational, interfacial, viscous and inertia forces acting on a Taylor bubble rising in flowing liquids characterised by the dimensionless Froude (Uc), inverse viscosity (Nf ) and Eötvös numbers (Eo) is carried out using computational fluid dynamic finite element method. Particular attention is paid to cocurrent (i.e upward) liquid flow and the influence of the characterising dimensionless parameters on the bubble rise velocity and morphology analysed for Nf, Eo and Uc ranging between [40, 100], [20, 300] and [−0.20, 0.20], respectively. Analysis of the results of the numerical simulations showed that the existing theoretical model for the prediction of Taylor bubble rise velocity in upward flowing liquids could be modified to accurately predict the rise velocity in liquids with high viscous and surface tension effects. Furthermore, the mechanism governing the change in morphology of the bubble in flowing liquids was shown to be the interplay between the viscous stress and total curvature stress at the interface. Keywords: Taylor bubble, finite element, slug flow, CFD, rise velocity

Dynamic mode decomposition in adaptive mesh refinement and coarsening simulations

Dynamic mode decomposition (DMD) is a powerful data-driven method used to extract spatio-temporal coherent structures that dictate a given dynamical system. The method consists of stacking collected temporal snapshots into a matrix and mapping the nonlinear dynamics using a linear operator. The classical procedure considers that snapshots possess the same dimensionality for all the observable data. However, this often does not occur in numerical simulations with adaptive mesh refinement/coarsening schemes (AMR/C). This paper proposes a strategy to enable DMD to extract features from observations with different mesh topologies and dimensions, such as those found in AMR/C simulations. For this purpose, the adaptive snapshots are projected onto the same reference function space, enabling the use of snapshot-based methods such as DMD. The present strategy is applied to challenging AMR/C simulations: a continuous diffusion–reaction epidemiological model for COVID-19, a density-driven gravity current simulation, and a bubble rising problem. We also evaluate the DMD efficiency to reconstruct the dynamics and some relevant quantities of interest. In particular, for the SEIRD model and the bubble rising problem, we evaluate DMD’s ability to extrapolate in time (short-time future estimates).

Computational simulation of a single Taylor bubble rising in a vertical column with stagnant liquid

This work presents a computational simulation of a single Taylor bubble rising in a vertical column of stagnant liquid. The computational simulation was based on the Navier-Stokes equations for isothermal, incompressible, and laminar flow, solved by using the open source software OpenFOAM. The two fluids were assumed immiscible. The governing equations were discretized by the volume-of-fluid (VOF) method and solved using the Gauss iteration method. Parametric mesh was used in order to improve the modeling of curvilinear geometry. Numerical solutions were obtained for the rise velocities and shapes of the bubbles which are in excellent agreement with experimental data and correlations from literature.