A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems

Abstract

SUMMARYAn Eulerian-Lagrangian approach is developed for the simulation of turbulent bubbly flows in complex systems. The liquid phase is treated as a continuum and the Navier-Stokes equations are solved in an unstructured grid, finite volume framework for turbulent flows. The dynamics of the disperse phase is modeled in a Lagrangian frame and includes models for the motion of each individual bubble, bubble size variations due to the local pressure changes, and interactions among the bubbles and with boundaries. The bubble growth/collapse is modeled by the Rayleigh-Plesset (RP) equation. Three modeling approaches are considered: (a) one-way coupling, where the influence of the bubble on the fluid flow is neglected, (b) two-way coupling, where the momentum-exchange between the fluid and the bubbles is modeled, and (c) volumetric coupling, where the volumetric displacement of the fluid by the bubble motion and the momentum-exchange are modeled. A novel adaptive time-stepping scheme based on stability-analysis of the non-linear bubble dynamics equations is developed. The numerical approach is verified for various single bubble test cases to show second-order accuracy. Interactions of multiple bubbles with vortical flows are simulated to study the effectiveness of the volumetric coupling approach in predicting the flow features observed experimentally. Finally, the numerical approach is used to perform a large-eddy simulation in two configurations: (i) flow over a cavity to predict small-scale cavitation and inception and (ii) a rising dense bubble plume in a stationary water column. The results show good predictive capability of the numerical algorithm in capturing complex flow features.

Simulation of the Bubble Behaviors for Gas–Liquid Dispersion in Agitated Vessel

“…In the Euler-Lagrange approach, the fluid phase is treated as a continuum by solving the Navier-Stokes equations, and the air phase is solved by tracking a large number of particles. In spite of Shams et al (2011), Sungkorn et al (2012 and Dapelo et al (2015) used the Euler-Lagrange approach to simulate the gas behaviors in liquid. This model is not widely applied since the bubble is assumed a rigid sphere.…”

Spherical bubble dynamics in a bubbly medium using an Euler–Lagrange model

“…Even though not directly applicable to the present study, several models have also been developed and applied to the study of non-spherical bubble dynamics (Chahine, 1982, Zhang et al, 1993, Hsiao & Chahine, 2001, Choi et al, 2009, Wang & Blake, 2010. The Rayleigh-Plesset equation (Rayleigh-Plesset equation) and its small compressibility equivalent models (Herring, 1941, Gilmore, 1952, Keller & Kolodner, 1956), remain however, by far the most widely used models for a wide range of applications involving bubble dynamics such as hydrodynamic cavitating flows (Plesset, 1949, Ceccio & Brennen, 1991, Brennen, 1995, Chahine, 2009, Brennen, 2011, acoustic cavitation applications (Keller & Miksis, 1980, Hilgenfeldt et al, 1998, Moholkar et al, 1999, Prosperetti & Hao, 1999, Calvisi et al, 2007, two-phase bubbly flows (Seo et al, 2010, Shams et al, 2011, Hsiao et al, 2013b, and underwater explosion bubbles (Herring, 1941, Keller & Kolodner, 1956, Chahine & Harris, 1997, Wardlaw & Mair, 1998, Geers & Hunter, 2002, Krieger & Chahine, 2005, de Graaf et al, 2012. This is predominantly because of the success of these methods at recovering the key physics involved in each application and because of the simplicity of their implementation as compared to other models.…”

“…While homogeneous models, which are useful for low void fraction and very small bubble oscillations conditions, ignore bubble dynamics and do not require Rayleigh-Plesset equation solutions, the other two approaches do require such modeling. Eulerian-Lagrangian approaches are more appropriate for higher void fractions (Spelt & Biesheuvel, 1997, Balachandar & Eaton, 2010, Raju et al, 2011, Shams et al, 2011. In a recent work by Raju et al (2011) comparing a continuum homogeneous model (Gilmore, 1952) , an Eulerian multicomponents model (Wardlaw & Luton, 2000, Wardlaw & Luton, 2003, and an Eulerian-Lagrangian model , Chahine, 2009, Hsiao et al, 2013b it was found that high-frequency local fluctuations were only captured when an Eulerian viscous solver was coupled with a Lagrangian discrete bubble dynamics and when the microscale behavior of the field bubbles was well resolved.…”

“…Initially, the void fraction was defined in each computational cell as the ratio of the total volume of bubbles in the cell by the cell volume. However, this was found not totally satisfactory because it often resulted in difficulty with differentiation due to the non-continuous character of α across cells (Wu et al, 2010, Raju et al, 2011 Here, to smooth the discontinuities and improve stability, we apply a Gaussian gas volume distribution scheme to smoothly "spread" a bubble volume across neighboring cells as in (Kitagawa et al, 2001, Darmana et al, 2006, Shams et al, 2011, Capecelatro & Desjardins, 2013. As illustrated in Figure 1, the void fraction computation scheme for a representative cell i is obtained using the following expression: where |X i,j | is the distance between bubble j and the center of cell i, and S is the standard deviation.…”

Coupled Lagrange-Euler Model for Simulation of Bubbly Flow in Vertical Pipes Considering Turbulent 3d Random Walks Models and Bubbles Interaction Effects

“…In this approach, in addition to considering the two-way coupling, the interaction between bubbles like the collision, coalescence and breakup effects are considered as well. Examples of works following this approach are Delonij et al (1996Delonij et al ( ), R ger et al (2000, Sommerfeld et al (2003), Darmana et al (2006), Shams et al (2010), Farzpourmachiani et al (2011), Movahedirad et al (2012.…”