[Paper Review] Taylor bubble motion in stagnant and flowing liquids in vertical pipes. Part I: Steady-states

Habib Abubakar, Omar K. Matar|arXiv (Cornell University)|Sep 19, 2021

Fluid Dynamics and Mixing51 references4 citations

TL;DR

This study numerically investigates steady-state Taylor bubble dynamics in vertical pipes using a Galerkin finite-element method to solve the axisymmetric Navier-Stokes equations with interface tracking. Key findings show that surface tension effects become negligible above Eo ≈ 100, and viscous effects are significant only below Nf ≈ 80, with distinct shape transitions in the film and bottom regions driven by curvature-viscosity interactions.

ABSTRACT

Taylor bubbles are a feature of the slug flow regime in gas-liquid flows in vertical pipes. Their dynamics exhibits a number of transitions such as symmetry-breaking in the bubble shape and wake when rising in downward-flowing and stagnant liquids, respectively, as well as breakup in sufficiently turbulent environments. Motivated by the need to examine the stability of a Taylor bubble in liquids, a systematic numerical study of a steadily-moving Taylor bubble in stagnant and flowing liquids is carried out, characterised by a dimensionless inverse viscosity ($Nf$), and Eötvös ($Eo$), and Froude ($Fr$) numbers based on the centreline liquid velocity, using a Galerkin finite-element method. A boundary-fitted domain is used to examine the dependence of the steady bubble shape on a wide range of $Nf$ and $Eo$. Our analysis of the bubble nose and bottom curvatures shows that the intervals $Eo = \left[ 20,30 ight)$ and $Nf=\left[60,80 ight)$ are the limits below which surface tension and viscosity, respectively, have a strong influence on the bubble shape. In the interval $Eo = \left(60,100 ight]$, all bubble features studied are weakly-dependent on surface tension. This is Part I of a two-part publication in which its companion paper (Abubakar & Matar, 2021) reports the results of a linear stability analysis of the steady-states discussed herein.

Motivation & Objective

• To systematically analyze the steady-state shape and hydrodynamic features of Taylor bubbles in stagnant and flowing liquids in vertical pipes.
• To identify the critical ranges of dimensionless numbers (Nf, Eo, Fr) where surface tension and viscous forces significantly influence bubble morphology.
• To map the parameter space for wake formation and bubble bottom shape under varying liquid flow conditions.
• To establish the foundation for a subsequent linear stability analysis of axisymmetric solutions (companion paper).

Proposed method

• A Galerkin finite-element method is employed to solve the steady-state Navier-Stokes equations in a boundary-fitted computational domain.
• The interface between gas and liquid is tracked using a kinematic update algorithm to accurately capture the evolving bubble shape.
• Dimensionless numbers Nf (inverse viscosity), Eo (Eötvös), and Fr (Froude, based on centerline liquid velocity) are used to characterize the flow regime.
• Steady-state solutions are computed across a wide range of Nf ∈ [20, 100], Eo ∈ [10, 100], and Uₘ ∈ [-1, 1] to explore the influence of liquid flow direction and speed.
• Hydrodynamic features such as frontal curvature, film thickness, and bottom shape are extracted and analyzed for shape transitions.
• A flow pattern map is generated to classify regions of parameter space based on wake existence and bottom curvature (concave/convex).

Experimental results

Research questions

• RQ1 At what Eötvös number (Eo) does surface tension cease to significantly influence Taylor bubble shape and hydrodynamics?
• RQ2 What is the critical inverse viscosity (Nf) below which viscous forces dominate the bubble morphology, particularly in the film and bottom regions?
• RQ3 How does imposed liquid flow (upward or downward) alter the steady-state shape of the bubble nose and bottom?
• RQ4 What are the conditions under which a wake forms behind the Taylor bubble, and how do Nf and Eo influence this?
• RQ5 How do curvature and viscous stresses interact to deform the bubble nose and bottom regions?

Key findings

• Surface tension has a strong influence on Taylor bubble shape for Eo ∈ [10, 30], with a distinct transition observed at Eo ≈ 20–30, beyond which its effect diminishes.
• For Eo > 100, all studied hydrodynamic features (nose shape, film thickness, wake) are weakly dependent on surface tension, indicating the onset of the inertia-dominated regime.
• Viscous effects are significant for Nf ≤ 80; below this range, a bulge emerges in the liquid film near the bubble bottom, which propagates toward the nose as Nf decreases.
• The normalised frontal radius of curvature indicates that Nf ∈ (60, 80] is the critical range where viscosity strongly influences the bubble nose shape.
• The interaction between viscous stress and interfacial curvature is the primary mechanism modifying the shape of the bubble nose and bottom regions.
• Downward liquid flow causes the bubble nose to flatten and the bottom to become more convex, while upward flow sharpens the nose and increases bottom concavity; at high downward speeds, nose and bottom shapes converge, suggesting loss of distinct topological features.