Numerical simulations of the dynamics of Taylor bubble in the presence of small-dispersed bubbles

Abstract

This numerical study investigates the dynamics around a single Taylor bubble rising with small-dispersed bubbles in a vertical pipeline. The Volume-of-fluid (VOF) method in Ansys Fluent 18.1 is used to track the Taylor bubble, and the Discrete Particle Method (DPM) is used to track the small-dispersed bubbles, for the following parameter range; $812 \leqslant \text{Re} \leqslant 5684$, ${\kern 1pt} Eo = 94$ and $\log (Mo) = - 10.6$. The coupling strategies for both the continuous gas–liquid phase as well as the discrete bubble phase are all accounted for in the numerical calculations. The dispersed bubbles significantly affect the behaviour of the rising Taylor bubble in numerous ways. The terminal velocity increases with an increase in the number of dispersed bubbles. The rise velocity is, however directly correlated with the deformation of the nose of the bubble. The flow dynamics around the nose of the Taylor bubble are stronger for low velocities due to lower inertia. The computations also demonstrated that there are both acceleration and deceleration effects on the Taylor bubble. Our predictions are in quantitative agreement with published experimental results by Cerqueira and Paladino (Int. J. Multiph. Flow, vol. 133, p. 103,450, Dec. 2020).

Appendix: Dimensionless analysis for the system of bubble rising in a viscous liquid

The following dimensionless variables were introduced in the modelling system for a better understanding of the physics:

$$x_i^* = \frac{{x_i}}{{D_B}};\,{u^* } = \frac{u}{{\sqrt {g{D_B}} }};\,{t^* } = \frac{t}{{D_B^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}g - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}};\,{\rho^* } = \frac{\rho }{{\rho_L}};\,{p^* } = \frac{p}{{{\rho_L}g{D_B}}};\,{\mu^* } = \frac{u}{{\mu_L}};\,{\kappa^* } = \frac{\kappa }{{{D_B} - 1}}$$

(22)

where D_B is the equivalent diameter of the large bubble, which can be defined using the bubble volume (V_B) as D_B = (6 × V_B/π)^1/3. Hence, the dimensionless governing equations for the fluids in the continuous domain is represented by:

$$\frac{{\partial {\alpha_c}}}{\partial t} + \nabla \cdot \left( {{u^* }{\alpha_c}} \right) = 0$$

(23)

$$\frac{{\partial {\rho^* }{u^* }}}{{\partial {t^* }}} + \nabla \cdot \left( {{\rho^* }{u^* }{u^* }} \right) = - \nabla {p^* } + \frac{1}{Ar}\nabla \cdot \left[ {{\mu^*}\left( {\nabla {u^*} + \nabla {u^{*T}}} \right)} \right] + \frac{1}{Eo}{\kappa^*}\nabla \Phi + \left( {{\rho^*} - 1} \right)\frac{g}{\left| g \right|} + \frac{{{F_{DC}}}}{{{\rho_L}g}}$$

(24)

Archimedes number and Eotvos number are then introduced in the new set of governing Eq. (24) as:

$$Ar = \frac{{{\rho_L}{g^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}D_B^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}}{{\mu_L}};\,Eo = \frac{{{\rho_L}gD_B^2}}{\sigma }$$

(25)

Based on the above formulation, the problem is characterized by the following four dimensionless parameters, namely the density ratio (ρG/ρL), the viscosity ratio (μG/μL), the Archimedes number (Ar) and the Eotvos number (Eo), which can be used as the input parameters for the problem setup. The Morton number is also essential in the characterization of two-phase flow systems:

$$Mo = \frac{g\mu_L^4}{{{\rho_L}{\sigma^3}}} = \frac{{E{o^3}}}{{A{r^4}}}$$

(26)

It is important to note that most experimental studies present the Reynolds number based on the terminal velocity of the Taylor (U_B) as follows:

$$\text{Re} = \frac{{{\rho_L}{D_B}{U_B}}}{{\mu_L}}$$

(27)

The terminal velocity can also be represented in a dimensionless form as the Froude number, Fr = U_B/(gD_B)^½. Now, we can correlate the Reynolds number with the Archimedes number as: Re = Ar × Fr..