^{1,2,a)}, Martin Wörner

^{2,b)}and Hakan S. Soyhan

^{3}

### Abstract

The stability of a train of equally sized and variably spaced gas bubbles that move within a continuous wetting liquid phase through a straight square minichannel is investigated numerically by a volume-of-fluid method. The flow is laminar and cocurrent upward and driven by a pressure gradient and buoyancy. The simulations start from fluid at rest with two identical bubbles placed on the axis of the computational domain, the size of the bubbles being comparable to that of the channel. In vertical direction, periodic boundary conditions are used. These result in two liquid slugs of variable length, depending on the initial bubble-to-bubble distance. The time evolution of the length of both liquid slugs during the simulation indicates if the bubble train flow is “stable” (equal terminal length of both liquid slugs) or “unstable” (contact of both bubbles). Several cases are considered, which differ with respect to bubble size, domain size, initial bubble shape, and separation. All cases lead to axisymmetric bubbles with the capillary number in the range of 0.11–0.23. The results show that a recirculation pattern develops in the liquid slug when its length exceeds a critical value that is about 10%–20% of the channel width. If a recirculation pattern exists in both liquid slugs, then the bubble train flow is stable. When there is a recirculation pattern in one liquid slug and a bypass flow in the other, the bubble train flow may be stable or not depending on the local flow field in the liquid slugs close to the channel centerline. These results suggest that a general criterion for the stability of bubble train flow cannot be formulated in terms of the capillary and Reynolds number only, but must take into account the length of the liquid slug.

The supports of the Institute of Science and Technology at Sakarya University, Forschungszentrum Karlsruhe and EU LLP program are gratefully acknowledged.

I. INTRODUCTION

II. NUMERICAL SIMULATION OF BUBBLE TRAIN FLOW

A. Governing equations and numerical method

B. Validation by Bretherton’s problem

C. Computational setup and test cases for bubble train flow

III. STABILITY OF BUBBLE TRAIN FLOW

A. Time evolution of bubble velocities and mean liquid velocity

B. Time evolution of liquid slug length

IV. DISCUSSION

A. Dependence of relative velocity on liquid slug length

B. Relative bubble velocity as a function of bubble volume

C. Stability for flow of small bubbles in large domain

D. Flow structure in the liquid slug

V. CONCLUSIONS

### Key Topics

- Liquid thin films
- 28.0
- Bubble coalescence
- 24.0
- Flow instabilities
- 15.0
- Gas liquid flows
- 15.0
- Capillary flows
- 13.0

## Figures

(a) Computed steady bubble shape for the four different capillary numbers. The dashed lines denote the channel walls. The flow is from left to right. (b) Comparison of computed liquid film thickness (filled squares) with the correlation of Bretherton (Ref. 39) (solid line), the numerical results of Giavedoni and Saita (Ref. 40) (open squares), and the correlation of Halpern and Gaver (Ref. 41) (dashed line).

(a) Computed steady bubble shape for the four different capillary numbers. The dashed lines denote the channel walls. The flow is from left to right. (b) Comparison of computed liquid film thickness (filled squares) with the correlation of Bretherton (Ref. 39) (solid line), the numerical results of Giavedoni and Saita (Ref. 40) (open squares), and the correlation of Halpern and Gaver (Ref. 41) (dashed line).

Sketch of computational domain, boundary conditions, and initial bubble positions. (a) Perspective view; (b) lateral view.

Sketch of computational domain, boundary conditions, and initial bubble positions. (a) Perspective view; (b) lateral view.

Time history of the mean liquid velocity and the velocity of the two bubbles for the different runs of case A.

Time history of the mean liquid velocity and the velocity of the two bubbles for the different runs of case A.

Time history of the mean liquid velocity and the velocity of the two bubbles for the different runs of case B.

Time history of the mean liquid velocity and the velocity of the two bubbles for the different runs of case B.

Time history of the slug lengths and for the different runs of case A.

Time history of the slug lengths and for the different runs of case A.

Time history of the slug lengths and for the different runs of case B.

Time history of the slug lengths and for the different runs of case B.

Nondimensional relative velocity as function of the nondimensional slug length evaluated from the simulation results of Wörner *et al.* (Ref. 36).

Nondimensional relative velocity as function of the nondimensional slug length evaluated from the simulation results of Wörner *et al.* (Ref. 36).

Time history of the slug lengths and for cases C1 and C2.

Time history of the slug lengths and for cases C1 and C2.

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case A4 for and (b) case A0g for .

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case A4 for and (b) case A0g for .

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case B3 for and (b) case B4 for .

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case B3 for and (b) case B4 for .

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case C1 for and (b) case C2 for .

Visualization of particle trajectories (left half) and velocity vectors (right half) in moving frame of reference for plane for (a) case C1 for and (b) case C2 for .

Profiles of normalized vertical velocity in the middle of the two liquid slugs. The instants in time for cases A0g and B4 correspond to those in Figs. 9(b) and 10(b). The position in -direction is (due to the staggered grid). The dashed horizontal line corresponds to the maximum velocity of a fully developed Poiseuille profile.

Profiles of normalized vertical velocity in the middle of the two liquid slugs. The instants in time for cases A0g and B4 correspond to those in Figs. 9(b) and 10(b). The position in -direction is (due to the staggered grid). The dashed horizontal line corresponds to the maximum velocity of a fully developed Poiseuille profile.

Profiles of normalized vertical velocity in the middle of the two liquid slugs. The instants in time for cases C1 and C2 correspond to those in Figs. 11(a) and 11(b). The position in -direction is . The dashed horizontal line corresponds to the maximum velocity of a fully developed Poiseuille profile.

Profiles of normalized vertical velocity in the middle of the two liquid slugs. The instants in time for cases C1 and C2 correspond to those in Figs. 11(a) and 11(b). The position in -direction is . The dashed horizontal line corresponds to the maximum velocity of a fully developed Poiseuille profile.

(a) A pair of mesh cells and with a plane representing the interface, the liquid being below the plane. (b) Situation where face is a g/l-interface face. (c) Extension of the left and right integration domain to obtain and for this case exactly.

(a) A pair of mesh cells and with a plane representing the interface, the liquid being below the plane. (b) Situation where face is a g/l-interface face. (c) Extension of the left and right integration domain to obtain and for this case exactly.

## Tables

Summary of parameters and results of the different simulations. The values for , , and correspond to the last time step of the transient simulations.

Summary of parameters and results of the different simulations. The values for , , and correspond to the last time step of the transient simulations.

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