Principios generales

The numerical simulation techniques available for predicting turbulent gas-liquid bubbly flows are still in need of substantial improvement. Although an extensive amount of research has been performed in the past, the simultaneous existence of physical phenomena (bubble dynamics, bubble coalescence and breakup, bubble size distribution) spanning a wide range of length scales makes the modelling of gas-liquid bubbly flows extremely complex. In addressing this multiscale complexity, until recently one of the key challenges has been the limitation in available computational resources. This has led to engineers having to make significant assumptions and simplifications, neglecting certain physical effects or limiting the size of the computational domain. These difficulties, coupled with limitations in the experimental measurement techniques available, have resulted in many physical aspects of bubbly flows still being poorly understood. This lack of understanding, and modelling limitations, negatively affects the design and the performance of multiphase equipment such as chemical reactors.

Therefore, this thesis is motivated by the necessity to advance numerical modelling of such flows, and to improve understanding of the underlying physics of turbulent gas- liquid bubbly flows. To this end, numerical simulations are performed to study dispersed and dense bubbly flows by varying the bubble volume fraction. In the flow, the size of the bubbles can undergo continuous change due to breakup and coalescence. The size of the bubbles determines the interfacial area density that drives exchanges of mass,

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momentum and energy between the phases. Therefore, accurate description of the bubble size and its distribution is of paramount importance for the accurate simulation of bubbly flow behaviour. In view of this, a significant portion of the work is focused on extending Lagrangian modelling capabilities to account for bubble breakup and coalescence in a truly four-way coupled fashion.

Specifically, the objective of this thesis is to develop a comprehensive and robust numerical approach for turbulent gas-liquid bubbly flows based on the Eulerian- Lagrangian method and LES. The overall model is used to obtain detailed information and insight into the hydrodynamics of bubbly flows at the small scale, and to quantify their impact on large scale processes with only moderate computational requirements. The modelling techniques used in this study include the following elements:

• The continuous liquid phase is modelled using the LES method, where only the largest and most energetic turbulence scales are resolved, and the small scales are modelled with a SGS model. Specifically, the dynamic Smagorinsky SGS model (Germano et al., 1991; Piomelli and Liu, 1995) has been used to model the effects of the sub-filter scales on the resolved field. It has been shown that LES, when the SGS influence on bubble dispersion is correctly accounted for, can reproduce the results of DNS-based predictions with reasonable accuracy and computational efficiency for turbulent two-phase flows (Delnoij et al., 1997; Deen et al., 2001; van den Hengel et al., 2005; Lau et al., 2014). Recently, Schutte et al. (2015) have demonstrated that the properties of particle agglomerates formed in such flows change when a two-way coupling model is considered rather than one-way coupling. In contrast, no difference was noticed when LES was employed rather than DNS. The work of Schutte et al. (2015), therefore, demonstrates that eddy-resolving simulations (LES and DNS) can successfully capture particle-particle and particle-turbulence interactions.

• The trajectories of individual microbubbles are computed in a Lagrangian framework under the action of gravity, buoyancy and hydrodynamic forces (drag, pressure gradient, shear-lift and added-mass forces), see Maxey and Riley (1983). A set of closure relations for these inter-phase hydrodynamic forces was carefully chosen. The Schiller Naumann drag correlation (Cliff et al., 1978) is used for the drag coefficient, the Legendre and Magnaudet (1998) correlation for the lift coefficient, and the Fukagata et

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al. (2001) correction coefficient for drag modification due to a wall. Impact of SGS velocity fluctuations on microbubble acceleration is considered with a stochastic Markov method (Bini and Jones, 2007; Bini and Jones, 2008). It should be noted that the Eulerian- Lagrangian approach adopted in this work considers bubbles as pointwise objects with a size smaller than the grid spacing and does not resolve the gas-liquid interface.

• Two-way coupling between the continuous fluid phase and the microbubbles is implemented by including, in the Navier-Stokes equations, momentum source terms due to the dispersed phase (Schwarzkopf et al., 2011; Lain et al., 2002; Bini and Jones, 2008). The source terms are calculated by time and ensemble averaging of the bubble trajectories for each control volume.

• A CFD code based on the Eulerian-Lagrangian framework previously applied to particle simulations (Bini and Jones, 2007; Njobuenwu, 2010; Njobuenwu and Fairweather, 2014) is modified to handle bubbly flows and extended to account for bubble coalescence and breakup. In doing so, a deterministic bubble-bubble collision model based on Hoomans et al. (1996) hard-sphere collision model is employed. An efficient collision search algorithm based on virtual cells is also implemented. After a collision is detected, the Prince and Blanch (1990) film drainage model is adopted for the description of bubble coalescence. This method is selected due to its accuracy in predicting experimental results (Darmana et al., 2006; Chesters, 1991) and its compatibility with bubbly flows considered using the Lagrangian framework. For bubble breakup, the model of Martinez-Bazan et al. (1999) is adopted, again in the Lagrangian framework. This choice was motivated by the fact that it has an extensive theoretical basis and its results are comparable with experimental data (Lasheras et al., 2002; Liao and Lucas, 2009).

• First, the CFD model developed is validated using DNS solutions for gas-solid turbulent channel flows at shear Reynolds numbers 𝑅𝑒_𝜏 = 150 (Marchioli et al., 2008), 𝑅𝑒_𝜏 = 300 (Marchioli and Soldati, 2007) and 𝑅𝑒_𝜏 = 590 (Moser et al., 1999). In order to study both the effect of bubble-bubble interactions (collision and coalescence) and the effects of bubble-fluid interactions (turbulence modulation and breakup), four kinds of simulations are addressed: (1) a turbulent channel flow with bubbles (bubble size, 𝑑_𝑏= 110, 220 and 330 𝜇𝑚) at three shear Reynolds numbers (𝑅𝑒𝜏 = 150, 300 and 590) under

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a one-way coupled assumption; (2) a turbulent channel flow with bubbles, but without collisions using two-way coupling; (3) a turbulent channel flow with bubble-bubble interaction (four-way coupled case), and (4) a turbulent channel flow with bubble breakup. For the latter case, four shear Reynolds numbers (𝑅𝑒_𝜏 = 150, 300, 590 and 2000) are considered, and an additional fluid system with a refrigerant at 𝑅𝑒𝜏 = 1154. The differences between cases (1) and (2) quantify the effects of bubble-fluid interactions, whilst the differences between cases (2) and (3) quantify the effects of the bubble-bubble interactions. Differences between cases (3) and (4), if any, quantify the effects of bubble breakup.

• The CFD model is used to carry out comprehensive and detailed sensitivity studies of microbubble dynamics, turbulence modulation and microbubble coalescence and breakup with respect to channel flow orientation and bubble size. Specifically, different channel orientations are considered, with no gravity, horizontal and vertical downward and upward flow conditions considered.