Multi-phase modelling

Introduction

Gas-liquid multiphase flows are important in natural and industrial processes. Some examples include bubbles and droplets, free surfaces, liquid jets, sprays, boiling flows among others. This type of flows are usually called interfacial flows, since the contact of immiscible fluids or phases in motion produces a thin region that separates them called interface. In detail, interfacial flows are governed by the Navier-Stokes equations in the variable-density incompressibility limit, as well as energy conservation equation and mass conservation equation for heat transfer and mass transfer processes. Additionally, they require the solution of a set of equations that describes the interface topology as it moves due to the velocity field.

To accurately solve gas-liquid multiphase flows, the Navier-Stokes equations are discretized to conserve mass, momentum, and energy. In detail, finite-volume schemes, suitable for 3D collocated unstructured meshes, adaptive-mesh-refinement, and moving meshes, have been developed and implemented in the parallel CFD platform TermoFluids. These methodologies have been designed in the framework of interface-capturing methods such as level-set method [1,3,5], volume-of-fluid method [6,7], coupled level-set/volume-of-fluid method [4] and multiple marker level-set approach [2,13,18,19], on three-dimensional unstructured meshes.

In this context, our research objectives are twofold: First, the development of finite-volume based numerical methods, for DNS of gas-liquid multiphase flows on three-dimensional unstructured meshes [1,2,4,5,6,7,18], and their coupling with adaptive-mesh-refinement [22,10] and moving-mesh [11,12] strategies.  Second, these numerical models are used to research the hydrodynamics, heat transfer and mass transfer in bubbly flows [1-5,11-12,13,14-20,22-26], gas-liquid jets [8,9], and free surface flows [1,8,21,28].

1.  Numerical methods and interface capturing

Multiple methods have been developed in the last decades for DNS of two-phase flows with sharp interfaces, e.g., level-set, volume-of-fluid, front-tracking, among others.  Although the idea behind these methods is similar, their numerical implementations may differ greatly. Each method has advantages and disadvantages, and it is difficult to select a single approach as the best for all the range of applications.  As a consequence, the development and improvement of interface capturing methodologies, their extensions to three-dimensional unstructured meshes, and their applications on the computation of gas-liquid multiphase flows is an intense field of investigation over the last years.   Our research group has been working on the development and implementation of level-set method [1,3,5], volume-of-fluid method [6,7], coupled level-set/volume-of-fluid method [4] and multiple marker level-set approaches [2,5], in the framework of three-dimensional unstructured meshes and finite-volume discretizations on collocated meshes.  As general remarks, our numerical approaches are based on the fractional-step projection method (Chorin, 1967) to solve the pressure-velocity coupling, unstructured flux-limiters schemes for the convective terms as introduced in [1,5,14], central-difference schemes for diffusive terms, and least-squares method for gradient evaluation.  Surface tension force is solved in the framework of the Continuous surface-force model (Brackbill, 1992), extended to the multiple marker level-set approach [2] and variable surface tension [5,23].  In recent works, complex interfacial physics has been included, for instance thermocapillary [5,23], and interfacial heat and mass transfer [5,14,16]. 

2.  DNS of gas-liquid multiphase flows

Bubbly flows
Bubbly flows play an important role on industrial processes such as steam generators in nuclear power plants, and also in the so-called unit operations of the chemical engineering (distillation, absorption, extraction, bubble columns, bubble reactors).   The development of supercomputers has promoted Direct Numerical Simulation (DNS) in combination with High-Performance Computing (HPC) as another means to perform non-invasive and controlled numerical experiments.  As examples, Figure (1) illustrates the rising motion of a swarm of bubbles in a vertical channel [13,18,19,25], carried out by means of a multiple marker level-set methodology [2,5,13,18,19].  Figure (2) illustrates the motion of a swarm of bubbles in a fully periodic domain [14,16,38] as well as the thermocapillary motion of a set of droplets [5,23].  Simulations like those illustrated in Fig. (1) were performed as part of the Tier-0 PRACE (Partnership for Advanced Computing in Europe) supercomputing project: “Direct Numerical Simulation of Gravity Driven Bubbly Flows” (10th-Call, March 2015 up to March 2016 22M hours at the supercomputer MareNostrum III, ref. 2016153612) [4,5,13,18,19,23,25].  Currently, our research group is working on the execution of the Tier-0 PRACE supercomputing project:  “DNS of bubbly flows with interfacial heat and mass transfer” (14th-Call, March 2017 up to March 2018, 18M hours at the supercomputer MareNostrum IV, ref. 2014112666).  The reader is referred to the references [2,5,13,14,16,18,19] for further technical details.

Figure 1:  DNS of gravity-driven bubbly flows [2,5,13,18,19,25] in a vertical channel.  (a) Spherical bubbles.  (b) Deformable bubbles.

Figure 2:  (a) Bubbly flow in a full-periodic domain [14,16].  (b) Thermocapillary-driven motion of a swarm of droplets [5,23].

Figure 3:  DNS of droplet collision with a fluid-fluid interface without coalescence, by means of a multiple marker level-set method [1]. 

Figure 4:  DNS of binary droplet collision with bouncing outcome, by means of a multiple marker level-set method [1]. 

Figure 5:  Bouncing interaction of two deformable bubbles rising in a vertical channel [18].

Taylor bubbles, free-surfaces, and gas-liquid jets

Further efforts have been focused on the application of the numerical models to research the hydrodynamics of Taylor Bubbles [11,20], free-surfaces flows [8,21,28], gas-liquid jets [10], the development of a numerical approach for binary droplet collisions in generalized Newtonian fluids [17], and the coupling of level-set method [1,3] with immersed-boundary and moving-mesh strategies to simulate bubbles and droplets in complex geometries [12]. 

Figure 6:  Interfacial flow with topology changes.  Oblique coalescence of two deformable bubbles, computed by means of the unstructured level-set method introduced in [1].  (a)  Time evolution of the bubble shape, (b)  Velocity field.

Figure 7:   Interfacial flow with topology changes.  Impact of a drop (water) fallen down into a liquid film (air as environment fluid), computed by means of the unstructured level-set method introduced in [1].

Figure 8:  Free surface flow.  Collapse of a liquid column (water) in a rectangular container (air as environment fluid), computed by means of the unstructured level-set method introduced in

Figure 9:  Free surface flow. Oscillating water column (OWC) system (air as environment fluid), computed by means of the unstructured level-set method introduced in [1]. 

Figure 10:  3D dam break over a fixed obstacle, simulated by means of the free-surface flow solver (single phase model) on an unstructured mesh [8].

Figure 11:  3-D sphere entry in water, presented in [21]. The interaction between the free-surface and the moving solid is considered by using a IBM technique..

Figure 12:  Injection of high speed liquid and gas flows in 3D coaxial configuration at Re=10000 [10]. A notable application is the atomization of liquid propellants in combustion engines.

Figure 13:  Study of the Liquid Injection into stagnant air by means of CLS-AMR strategy [10]. Phenomenological study for variable Re, We and Oh numbers. Case of Re=800, high Oh number.

Figure 14:  Binary droplet collision at high Weber number using a level-set with lamella stabilization model [17].

Figure 15:  Binary droplet collision at high Weber number using a level-set with lamella stabilization model [17].

On International Journals:
[1] Balcázar, N., Jofre, L., Lehmkuhl, O., Castro, J., Rigola, J.,  A finite-volume/level-set method for simulating two-phase flows on unstructured grids. International Journal of Multiphase Flow 64,  September 2014, pp. 55-72.  https://doi.org/10.1016/j.ijmultiphaseflow.2014.04.008
[2] Balcázar, N., Lehmkuhl, O., Rigola, J., Oliva, A., A multiple marker level-set method for simulation of deformable fluid particles, International Journal of Multiphase Flow 74, September 2015, pp. 125-142.  https://doi.org/10.1016/j.ijmultiphaseflow.2015.04.009
[3] Balcázar, N., Lehmkuhl, O., Jofre, L., Oliva, A.,  Level-set simulations of buoyancy-driven motion of single and multiple bubbles.  International Journal of Heat and Fluid Flow 56.  pp. 91-107, 2015. http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.07.004
[4] Balcázar, N., Lehmkhul, O., Jofre, L., Rigola, J., Oliva, A. A coupled volume-of-fluid/level-set method for simulation of two-phase flows on unstructured meshes. Computers and Fluids 124, 12-29, 2016. http://dx.doi.org/10.1016/j.compfluid.2015.10.005
[5] Balcázar, N., Rigola, J., Castro, J., Oliva, A.,  A level-set model for thermocapillary motion of deformable fluid particles, International Journal of Heat and Fluid Flow, Volume 62, Part B, 2016, pp 324-343. http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.015
[6] Jofre, L., Borrell, R., Lehmkuhl, O., Oliva, A., Parallel load balancing strategy for Volume-of-Fluid methods on 3-D unstructured meshes. Journal of Computational Physics 282, 2015, pp. 269-288
https://doi.org/10.1016/j.jcp.2014.11.009
[7] Jofre, L., Lehmkuhl, O., Castro, J., Lluís Jofre, Oliva, A., A 3-D Volume-of-Fluid advection method based on cell-vertex velocities for unstructured meshes. Computers & Fluids 94. 2014, pp. 14-29.
https://doi.org/10.1016/j.compfluid.2014.02.001
[8] Schillaci, E., Jofre, L., Balcázar, N., Lehmkuhl, O., Oliva, A. A level-set aided single-phase model for the numerical simulation of free-surface flow on unstructured meshes, Computers & Fluids 140, 2016, 97-110.  http://dx.doi.org/10.1016/j.compfluid.2016.09.014
[9] Schillaci, E., Jofre, L., Balcázar, N., Antepara, O., Oliva, A.  A low-dissipation convection scheme for the stable discretization of turbulent interfacial flow. Computers & Fluids 15. 2017, 102-117. https://doi.org/10.1016/j.compfluid.2017.05.009
[10] Schillaci, E., Antepara, O., Balcázar, N., Rigola, J., Oliva, A.  A Numerical Study of Liquid Atomization Regimes by means of Conservative Level-Set Simulations.  Computers & Fluids.  2018.  https://doi.org/10.1016/j.compfluid.2018.10.017
[11] Gutiérrez, E., Balcázar, N., Bartrons, E., Rigola, J. Numerical study of Taylor bubbles rising in a stagnant liquid using a level-set/moving-mesh method. Chemical Engineering Science 164.  2017, 158-177.  https://doi.org/10.1016/j.ces.2017.02.018
[12] Gutiérrez, E., Favre, F., Balcázar, N., Amani, A., Rigola, J. Numerical approach to study bubbles and drops evolving through complex geometries by using a level set – Moving mesh – Immersed boundary method. Chemical Engineering Journal 349.  2018, 662 - 682.  https://doi.org/10.1016/j.cej.2018.05.110

Book chapters:
[13]  Balcázar N., Lehmkuhl O., Castro J., Oliva A. (2018) DNS of the Rising Motion of a Swarm of Bubbles in a Confined Vertical Channel. In: Grigoriadis D., Geurts B., Kuerten H., Fröhlich J., Armenio V. (eds) Direct and Large-Eddy Simulation X. ERCOFTAC Series, vol 24. Springer, Cham.  https://doi.org/10.1007/978-3-319-63212-4_15

International Conference Proceedings:
[14]  Balcázar N., Antepara O., Castro J., Oliva A., A level-set method for mass transfer in bubble swarms.  On:  Proceedings of 12th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements.  Montpellier, France, September 26-28, 2018.
[15]  Antepara O., Balcázar N., Rigola J., Oliva A., Direct numerical simulation of rising bubble with path instability.  On:  Proceedings of 12th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements.  Montpellier, France, September 26-28, 2018.
[16]  Balcázar N., Antepara O., Rigola J., Oliva A., A level-set method for interfacial heat or mass transfer in two-phase flows.  On:  6th European Conference on Computational Mechanics (ECCM 6).  11-15 June 2018.  Glasgow-UK.
[17]  Amani A., Balcázar N., Naseri A., Oliva A.  A Study on Binary Collision of GNF Droplets Using a Conservative Level-Set Method.  On:  6th European Conference on Computational Mechanics (ECCM 6).  11-15 June 2018.  Glasgow-UK.
[18]  Balcázar, N., Castro, J., Rigola, J., Oliva, A.,  DNS of the wall effect on the motion of bubble swarms.  Procedia Computer Science 108, 2017, Pages 2008-2017.  https://doi.org/10.1016/j.procs.2017.05.076
[19]  Balcázar N., Castro J., Chiva J., Oliva A., DNS of falling droplets in a vertical channel.  International journal of computational methods and experimental measurements 6.  2018.  Pages 398-410.  doi: 10.2495/CMEM-V6-N2-398-410
[20]  Gutiérrez E., Favre F., Balcázar N., Rigola J., On the solution of the problem of a drop falling against a plane by using a level set-moving mesh-immersed boundary method. International Journal of Computational Methods and Experimental Measurements 6, 2018, Pages 208-219. doi: 10.2495/CMEM-V6-N1-208-219
[21]  Schilaci E., Favre F., Antepara O., Balcázar N., Oliva A., Numerical study of an impulse wave generated by a sliding mass. International Journal of Computational Methods and Experimental Measurements 6, 2018, Pages 98-109.  doi: 10.2495/CMEM-V6-N1-98-109
[22] Antepara O., Balcázar N., Oliva A.,  A comparative study of interface capturing methods with AMR for incompressible two-phase flows.  On: proceedings of the VII International Conference on Coupled Problems in Science and Engineering, Rhodes Islad, Greece, June 12-14, 2017
[23]  Balcázar, N.,  Rigola, J., Oliva, A.,  A level-set method for thermal motion of bubbles and droplets.  7th European Thermal-Sciences Conference (Eurotherm2016).  Journal of Physics: Conference Series 745 (2016) 032113.  doi:10.1088/1742-6596/745/3/032113
[24]  E. Gutiérrez, N. Balcázar, O. Lehmkuhl, A. Oliva., On the solution of the full three-dimensional Taylor bubble problem by using a coupled Conservative Level Set - Moving Mesh method.  7th European Thermal-Sciences Conference (Eurotherm2016).  Journal of Physics: Conference Series 745 (2016) 032116.  doi: 10.1088/1742-6596/745/3/032116
[25]  Balcázar, N., Lehmkuhl, O., Castro, J., Rigola, J., Oliva, A., GRAVITY-DRIVEN MOTION OF A SWARM OF BUBBLES IN A VERTICAL PIPE.  27th International Conference on Parallel Computational Fluid Dynamics, Parallel CFD2015.  2015. 
[26] Balcázar, N.; Jofre, L.; Lehmkuhl, O.; Castro, J.; Oliva, A., A multiple marker level-set method for simulation of bubbly flows. Proceedings of the jointly organized WCCM XI, ECCM V, ECFD VI. July 20-25, 2014, Barcelona, Spain., pp. 5298-5309.
[27] Jofre, L.; Balcázar, N.; Lehmkuhl, O.; Borrell, R.; Castro, J., Direct numerical simulation of the flow over a spherical bubble in a turbulent pipe flow. Proceedings of the jointly organized WCCM XI, ECCM V, ECFD VI., July 20-25, 2014, Barcelona, Spain. pp. 5333-5343.
[28] Schillaci, E.; Balcázar, N.; Lehmkuhl, O.; Jofre, L.; Oliva, A."A free surface model for the numerical simulation of oscillating water column systems".En: Proceedings of the jointly organized WCCM XI, ECCM V, ECFD VI. pp. 5286-5297. 2014.
[29] Balcázar, N.; Jofre, L.; Lehmkuhl, O.; Rigola, J.; Castro, J.; Oliva, A., A finitevolume/level-set interface capturing method for unstructured grids: Simulations of bubbles rising through viscous liquids. Advances in Fluid Mechanics X, WIT Transactions on Engineering Sciences. pp. 239-252. 2014.
[30] Jofre, L.; Lehmkuhl, O.; Balcázar, N.; Castro, J.; Rigola, J.; Oliva, A, Conservative discretization of multiphase flow with high density ratios. Advances in Fluid Mechanics X, WIT Transactions on Engineering Sciences. pp. 153 – 164. 2014.
[31] R. Borrell, L. Jofre, O. Lehmkuhl and J. Castro. Parallelization strategy for the volume-of-fluid method on unstructured meshes. In Proceedings of the 25th International Conference on Parallel Computational Fluid Dynamics, 2013.
[32] L. Jofre, N. Balcázar, O. Lehmkuhl, J. Castro and A. Oliva. Numerical study of the incompressible Richtmyer-Meshkov instability. Interface tracking methods on general meshes. In Proceedings of the 15th International Conference on Fluid Flow Technologies, 2012.
[33] N. Balcázar, L. Jofre, O. Lehmkuhl, J. Castro and A. Oliva. Numerical simulation of incompressible two-phase flows by conservative level-set method. In Proceedings of the 15th International Conference on Fluid Flow Technologies, 2012.
[34] L. Jofre, O. Lehmkuhl, J. Castro and A. Oliva. Vof/Navier-Stokes implementation on 3D unstructured staggered meshes. Application to the Richtmyer-Meshkov instability. In Proceedings of the 7th International Conference on Computational Heat and Mass Transfer, 2011.
[35] L. Jofre, O. Lehmkuhl, J. Castro and A. Oliva. A Plic-Vof implementation on parallel 3D unstructured meshes. In Proceedings of the 5th European Conference on Computational Fluid Dynamics, 2010.

Posters and Conferences (Presenter with underline):
[36]  Balcázar N., Rigola J., Castro J., Oliva A., Poster: DNS of Gravity-Driven Bubbly Flows.  PRACEdays17.  Barcelona, Spain, May-2017.
[37]  Antepara O., Balcázar N., Castro J., Oliva A.,  Poster: DNS of rising bubble with path instability. PRACE Scientific & Industrial Conference  2018, PRACEdays18.  Ljubljana, Slovenia, May-2018.
[38]  Balcázar N., Antepara O., Rigola J., Oliva A.,  Presentation:  DNS of Interfacial Heat and Mass Transfer in Bubble Swarms.PRACE Scientific & Industrial Conference  2018, PRACEdays18.  Ljubljana, Slovenia, May-2018.