Euler-Euler large eddy simulations of the gas–liquid flow in a cylindrical bubble column

In this work Euler-Euler Large Eddy Simulations (LES) of dispersed turbulent gas-liquid flows in a cylindrical bubble column are presented. Besides, predictions are compared with experimental data from Vial et al. 2000 using laser Doppler velocimetry (LDV). Two test cases are considered where vortical-spiral and turbulent flow regimes occur. The sub-grid scale (SGS) modelling is based on the Smagorinsky kernel with model constant Cs = 0.08 and the one-equation model for SGS kinetic energy. The emphasis of this work is to analyse the performance of the one-equation SGS model for the prediction of bubbly flow in a three-dimensional high aspect ratio bubble column (H D ⁄ ) of 20 and the investigation of the influence of the superficial gas velocity using the OpenFOAM package. The model is compared with the Smagorinsky SGS model and the mixture k − ε model in terms of the axial liquid velocity, the gas hold-up and liquid velocity fluctuations. The bubble induced turbulence and various interfacial forces including the drag, lift, virtual mass and turbulent dispersion where incorporated in the current model. Overall, the predictions of the liquid velocities are in good agreement with experimental measurement using the one-equation SGS model and the Smagorinsky model which improve the mixture k − ε model in the core and near-wall regions. However, small discrepancies in the gas hold-up are observed in the bubble plume region and the mixture k − ε model performs much better. The numerical simulations confirm that the energy spectra of the resolved liquid velocities in churn-turbulent regime follows the classical -5/3 law for low frequency regions and close to -3 for high frequencies. More details of the instantaneous local flow structure have been obtained by the Euler-Euler LES model including large-scale structures and vortices developed in the bubble plume edge.


Introduction
Bubbly gas-liquid flows in multiphase reactors are important for many industrial processes, for instance in the chemical, biochemical, or environmental industries and have advantageous characteristics in mass and heat transfers. In bubble column reactors, the gas phase is dispersed in the form of tiny bubbles in a continuous liquid phase using a gas distribution device. The complex interplay between operating conditions, the gas-liquid interfacial area, bubble size, bubble rise velocity, turbulence in the liquid phase, and bubble-bubble interactions lead to extensive range of flow regimes and complex flow structures. Furthermore, as the bubbles rise in the column, they induce pseudo-turbulence in the liquid phase. Several numerical studies of these types of flows have been  Table 1 gives a summary of previous works of gas-liquid flows in bubble column reactors in a chronological manner. For instance, Zhang et al. (Zhang et., 2006)

The flow equations
The two-fluid model is built up on the spatial filtering for LES or conditional averaging for RANS of the conservation equations of mass and momentum. In this approach, both phases, the continuous liquid phase and the dispersed gas phases, are modelled as two interpenetrating continua. In LES cases, it is assumed that the filtered equations are used to compute the large-scale lengths while the effect of unresolved turbulent scales are modelled using a sub-grid model. In the present work, the flow is assumed to be adiabatic, without the consideration of the interfacial mass transfer between the air and the water phases.
The present formulation closely follows the procedure outlined by Weller 2005, where the mass and momentum equations for the phase are given by Here is the volume fraction of each phase, is the phase resolved velocity, and eff represents the effective stress tensor usually decomposed into a mean viscous stress and turbulent stress tensor for phase as (1) where is the turbulent kinetic energy of phase , is the identity tensor, and ff is the effective viscosity of phase . The effective viscosity of the liquid phase is obtained through the summation of the molecular viscosity, the shear-induced turbulent viscosity, and the bubble-induced turbulent viscosity and is formulated in the present study using two models: (a) the Smagorinsky model proposed by There is still no complete agreement on the closures or the combination to be used at best. The drag force (per volume) for the liquid phase is estimated as (10) where refers to the drag force coefficient and is calculated according to the Schiller-Neumann The SGS component of those forces will be neglected except the turbulent dispersion force which can be estimated using the modelled SGS energy in the one-equation model. The turbulent dispersion force proposed by Lopez de Bertodano et al. (1994) is adopted. It is modelled as Several turbulent dispersion coefficients , required to obtain good agreement with experimental measurements, were tested. For the one-equation and mixture − models we use = 0.6.

Numerical simulation set-up
The numerical simulations were carried out in a cylindrical bubble column with uniform aeration. The geometry of the current bubble column reactor is the same as used by Vial et al. (2001) and OpenFOAM 3.0.1. The solver is based on a finite volume formulation to discretise the model equations which has shown to be stable for transient calculations (Weller, 2005). The first-order bounded implicit Euler scheme is adopted for the time integration, the gradient terms are approximated with a linear interpolation, the convective terms are discretized with second-order upwind scheme, and the diffusive terms are interpolated with the Gauss linear orthogonal scheme. we employ the PIMPLE algorithm to solve the pressure-velocity coupling where the pressure equation , that is the mesh size must be at least 50% larger than the bubble diameter for Eulerian-Eulerian simulations. Fig. 1

The one-equation SGS and Smagorinsky model
The resolved axial liquid velocity is presented in Fig. 1a, it can be seen that there is no significant change in the prediction between the medium and the fine mesh. In order to understand that how the

Instantaneous flow
The instantaneous flow structure in 3D bubble columns was classified, based on visual study of (Chen et al., 1994;Lin et al., 1996)

Conclusions
Euler   Table 3. Experimental and numerical centerline axial fluctuations of the liquid velocity