Numerical study of the drag force, interfacial area and mass transfer in bubbles in a vertical pipe

Abstract:  A systematic numerical study of drag force and mass transfer in gravity-driven bubbles in a vertical pipe is performed at Re∼O(100−1000). This research employs a parallel multi-marker unstructured conservative level set method for interface capturing, avoiding the numerical coalescence in bubble swarms. The finite volume method discretizes transport equations on 3D collocated unstructured meshes. Unstructured flux limiter schemes solve the convective term of transport equations, preserving the numerical stability at high Reynolds numbers and high-density ratios. Thermodynamic equilibrium is assumed at the interface for the concentration of chemical species. The hydrodynamics and mass transfer of bubbles are validated against classical correlations from the literature. Finally, direct simulations are executed to develop new correlations for the drag force, normalized bubble surface and Sherwood number for bubbles in a vertical pipe.

Keywords: Bubbles, Drag force, Mass transfer, Unstructured conservative level - set method (UCLS), Unstructured flux limiters, Unstructured meshes, Finite - volume method

References:  

• Néstor Balcázar-Arciniega, Joaquim Rigola, Carlos D. Pérez-Segarra, Assensi Oliva, Numerical study of the drag force, interfacial area and mass transfer in bubbles in a vertical pipe, Chemical Engineering Journal, Volume 495, 2024, https://doi.org/10.1016/j.cej.2024.153124

A level-set model for mass transfer in bubbly flows

Abstract:  A level-set model is presented for simulating mass transfer or heat transfer in two-phase flows. The Navier-Stokes equations and mass transfer (or heat transfer) equation are discretized using a finite volume method on a collocated unstructured mesh, whereas a multiple marker level-set approach is used for interface capturing in bubble swarms. This method avoids the numerical coalescence of the fluid particles, whereas the mass conservation issue inherent to standard level-set methods is circumvented. Furthermore, unstructured flux-limiter schemes are used to discretize the convective term of momentum transport equation, level-set equations, and chemical species concentration equation, to avoid numerical oscillations around discontinuities, and to minimize the numerical diffusion. A convection-diffusion-reaction equation is used as a mathematical model for the chemical species mass transfer at the continuous phase. Because the mathematical analogy between dilute mass transfer and heat transfer, the same numerical model is applicable to solve both phenomena. The capabilities of this model are proved for the diffusion of chemical species from a sphere, external mass transfer in the buoyancy-driven motion of single bubbles and bubble swarms. Results are extensively validated by comparison with analytical solutions and empirical correlations from the literature.

Keywords: Bubbly flow, Heat transfer, Mass transfer, Unstructured meshes, Finite-volume method, Level-set method, Flux limiters

References:  

• Néstor Balcázar-Arciniega, Oscar Antepara, Joaquim Rigola, Assensi Oliva, (2019) A level-set model for mass transfer in bubbly flows, International Journal of Heat and Mass Transfer, Volume 138,  335-356.  https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.008 

DNS of Drag-Force and Reactive MassTransfer in Gravity-Driven Bubbly Flows

Abstract:  Mass transfer processes in bubbly flows are relevant in both nature and industry. Multiple examples arise from the so-called unit operations of chemical engineering, where bubble columns are used for separation processes, or as chemical and biochemical reactors. In this context, the main motivation of this research is to compute drag-force coefficients and mass transfer coefficients in gravity-driven bubbly flows, using direct numerical simulation of Navier–Stokes equations with a conservative level-set (CLS) [2,15] methodology for interface capturing in bubble swarms [9].

Keywords: Mass transfer · Bubbly flow · Vertical channel · Flux-limiters · Un-structured meshes · Level-set method · Finite volume method· High-Performance Computing.

References:  

• Balcázar, N., Antepara, O., Rigola, J., Oliva, A. (2020). DNS of Drag-Force and Reactive Mass Transfer in Gravity-Driven Bubbly Flows. In: García-Villalba, M., Kuerten, H., Salvetti, M. (eds) Direct and Large Eddy Simulation XII. DLES 2019. ERCOFTAC Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-030-42822-8_16 

DNS of Mass Transfer from Bubbles Rising in a Vertical Channel

Abstract:  This research presents Direct Numerical Simulation of mass transfer from buoyancy-driven bubbles rising in a wall-confined vertical channel, through amultiple markers level-set method. The Navier-Stokes equations and mass transfer equation are discretized using a finite volume method on acollocated unstructured mesh, whereas a multiple markers approach is used to avoid the numericalcoalescence of bubbles. This approach is based on a conservative level-set method. Furthermore, unstructured flux-limiter schemes are used to discretize the convective term of momentum equation, level-set advection equations, and mass transfer equation, to improve the stability of the solver in bubbly flows with high Reynolds number and high-density ratio. The level-set model is used to research the effect of bubble-bubble and bubble-wall interactions on the mass transfer froma bubble swarm rising in a vertical channel with a circular cross-section.

Keywords: Mass transfer · Bubbly flow · Vertical channel · Flux-limiters · Un-structured meshes · Level-set method · Finite volume method· High-PerformanceComputing.

References:  

• Balcázar-Arciniega N., Rigola J., Oliva A. (2019), DNS of Mass Transfer from Bubbles Rising in a Vertical Channel. In: Rodrigues J. et al. (eds) Computational Science – ICCS 2019. ICCS 2019. Lecture Notes in Computer Science, vol 11539. Springer, Cham. 

DNS of the wall effect on the motion of bubble swarms

Abstract:  This research presents a numerical study of the gravity-driven motion of single bubbles and bubble swarms through a vertical channel, using High-Performance Computing (HPC) and Direct Numerical Simulation (DNS) of the Navier-Stokes equations. A systematic study of the wall effect on the motion of single deformable bubbles is carried out for confinement ratios CR = {2,4,6} in both circular and square channels, for a broad range of flow conditions. Then, the rising motion of a swarm of deformable bubbles in a vertical channel is researched, for void fractions α = {8.3%, 10.4%, 12.5%} and CR = {4, 6}. These simulations are carried out in the framework of a novel multiple marker interface capturing approach, where a conservative level-set function is used to represent each bubble. This method avoids the numerical and potentially unphysical coalescence of the bubbles, allowing for the collision of the fluid particles as well as long time simulations of bubbly flows. Present simulations are performed in a periodic vertical domain discretized by 2 × 106 control volumes (CVs) up to 16.6 × 106 CVs, distributed in 128 up to 2048 processors. The collective and individual behavior of the bubbles are analyzed in detail.

Keywords:  bubble swarm, level-set method, unstructured meshes, DNS, HPC

References:  

• Balcázar, N., Castro, J., Rigola, J., Oliva, A., (2017). DNS of the wall effect on the motion of bubble swarms. Procedia Computer Science 108, Pages 2008-2017. https://doi.org/10.1016/j.procs.2017.05.076

A multiple marker level-set method for simulation of deformable fluid particles

Abstract:  A novel multiple marker level-set method is introduced for Direct Numerical Simulation of deformable fluid particles (bubbles and droplets), which is integrated in a finite-volume framework on collocated unstructured grids. Each fluid particle is described by a separate level-set function, thus, different interfaces can be solved in the same control volume, avoiding artificial and potentially unphysical coalescence of fluid particles. Therefore, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide. The proposed algorithm is developed in the context of the conservative level-set method, whereas, surface tension is modeled by the continuous surface force approach. The pressure–velocity coupling is solved by the fractional-step projection method. For validation of the proposed numerical method, the gravity-driven impact of a droplet on a liquid–liquid interface is studied; then, the binary droplet collision with bouncing outcome is examined, and finally, it is applied on simulation of gravity-driven bubbly flow in a vertical column.

Keywords:  Level-set method, Multiple marker, Bubbles, Droplets, Two-phase flow

References:  

• Balcázar, N., Lehmkuhl, O., Rigola, J., Oliva, A. (2015). A multiple marker level-set method for simulation of deformable fluid particles, International Journal of Multiphase Flow 74, pp 125 -142.  doi: http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.04.009

Tetrahedral adaptive mesh refinement for two‐phase flows using conservative level‐set method

Abstract:  In this article, we describe a parallel adaptive mesh refinement strategy for two‐phase flows using tetrahedral meshes. The proposed methodology consists of combining a conservative level‐set method with tetrahedral adaptive meshes within a finite volume framework. Our adaptive algorithm applies a cell‐based refinement technique and adapts the mesh according to physics‐based refinement criteria defined by the two‐phase application. The new adapted tetrahedral mesh is obtained from mesh manipulations of an input mesh: operations of refinement and coarsening until a maximum level of refinement is achieved. For the refinement method of tetrahedral elements, geometrical characteristics are taking into consideration to preserve the shape quality of the subdivided elements. The present method is used for the simulation of two‐phase flows, with surface tension, to show the capability and accuracy of 3D adapted tetrahedral grids to bring new numerical research in this context. Finally, the applicability of this approach is shown in the study of the gravity‐driven motion of a single bubble/droplet in a quiescent viscous liquid on regular and complex domains.

Keywords:  adaptive mesh refinement, conservative level‐set, finite‐volume method, multiphase flows, tetrahedral elements, tetrahedral mesh

References: 

• Antepara, O, Balcázar, N, Oliva, A. (2020)  Tetrahedral adaptive mesh refinement for two‐phase flows using conservative level‐set method. Int J Numer Meth Fluids; 1-23. https://doi.org/10.1002/fld.4893

Numerical study of rising bubbles with path instability using conservative level-set and adaptive mesh refinement

Abstract:  This paper focuses on three-dimensional direct numerical simulations of rising bubbles in the wobbling regime, and the study of its dynamical behavior for Eötvös number 1  ≤  Eo  ≤  10 and Morton number 1e−11  ≤ M ≤  1e−9. The computational methodology is based on a mass Conservative Level-Set method, whereas the spatial discretization of the computational domain employs an Adaptive Mesh Refinement strategy for the reduction of computational resources. The Navier–Stokes equations are discretized using the finite-volume approach on a collocated unstructured mesh; the pressure-velocity coupling is solved using a classical fractional-step projection method. This methodology is applied to a series of verification and validation tests, which are compared with experiments and numerical results from the literature. Finally, buoyancy bubbles rising in the wobbling regime are researched at moderate to high Reynolds numbers (100 < Re < 3000). Terminal Reynolds number, drag coefficient and frequency of path oscillations are compared with empirical correlations and numerical studies from the literature. Results show the discharge of alternate oppositely-oriented hairpin vortex structures. Moreover, depending on the characteristics numbers of the system, different path features, bubble shape, and vortical structures in the wake are reported.

Keywords:  Wobbling bubble, Path instability, Direct numerical simulation, Conservative level-set, Adaptive mesh refinement

References:  

• Oscar Antepara, Néstor Balcázar, Joaquim Rigola, Assensi Oliva, (2019). Numerical study of rising bubbles with path instability using conservative level-set and adaptive mesh refinement, Computers & Fluids, Volume 187, Pages 83-97.  doi: https://doi.org/10.1016/j.compfluid.2019.04.013  

A coupled volume-of-fluid/level-set method for simulation of two-phase flows on unstructured meshes

Abstract:  This paper presents a methodology for simulation of two-phase flows with surface tension in the framework of unstructured meshes, which combines volume-of-fluid with level-set methods. While the volume-of-fluid transport relies on a robust and accurate polyhedral library for interface advection, surface tension force is calculated by using a level-set function reconstructed by means of a geometrical procedure. Moreover the solution of the fluid flow equations is performed through the fractional step method, using a finite-volume discretization on a collocated grid arrangement. The numerical method is validated against two- and three-dimensional test cases well established in the literature. Conservation properties of this method are shown to be excellent, while geometrical accuracy remains satisfactory even for the most complex flows.

Keywords:  Unstructured meshes; Level-set method; Volume-of-fluid method; Finite-volume method; Two-phase flow

References:  

• N. Balcázar, O. Lehmkuhl, L. Jofre, J. Rigola, A. Oliva, (2016).  A coupled volume-of-fluid/level-set method for simulation of two-phase flows on unstructured meshes, Computers & Fluids, Volume 124, Pages 12-29, doi:  https://doi.org/10.1016/j.compfluid.2015.10.005 

A finite-volume/level- set method for simulating two-phase flows on unstructured grids

Abstract:  The conservative level-set method for capturing the interface between two fluids is combined with a variable density projection scheme to simulate incompressible two-phase flows on unstructured meshes. All equations are discretized by using a conservative finite-volume approximation on a collocated grid arrangement. A high order scheme based on a flux limiter formulation, is adopted for approximating the convective terms, while the diffusive fluxes are centrally differenced. Gradients are computed by the least-squares approach. Physical properties are assumed to vary smoothly in a narrow band around the interface to avoid numerical instabilities. The numerical method is validated against classical advection test and two-phase flow examples including topology changes.

Keywords:  Conservative level set method, Finite volume method, Flux limiter, Incompressible two-phase flow, Unstructured grid

References:  

• Balcázar, N., Jofre, L., Lehmkuhl, O., Castro, J., Rigola, J. (2014). A finite-volume/level- set method for simulating two-phase flows on unstructured grids. International Journal of Multiphase Flow 64, pp. 55 – 72.  doi:  https://doi.org/10.1016/j.ijmultiphaseflow.2014.04.008