A quadrature-based moment method for polydisperse bubbly flows

A computational algorithm for polydisperse bubbly flow is developed by combining quadrature-based moment methods (QBMM) with an existing two-fluid solver for gas–liquid flows. Care is taken to ensure that the two-fluid model equations are hyperbolic by generalizing the kinetic model for the bubble phase proposed by Bieseuvel and Gorissen (1990). The kinetic formulation for the bubble phase includes the full suite of interphase momentum exchange terms for bubbly flow, as well as ad hoc bubble–bubble interaction terms to model the transition from isolated bubbles to regions of pure air at very high bubble-phase volume fractions. A robust numerical algorithm to couple the QBMM approach with a gas–liquid two-fluid solver is proposed. The resulting algorithm is tested to show hyperbolicity, verified against the two-fluid model currently implemented into OpenFOAM, and validated against two sets of experiments on bubbly flows from the literature. In both cases, the computational method shows good agreement with experimental data, and improved accuracy in comparison to a two-fluid model considered for comparison purposes. The robustness of the algorithm is demonstrated on an unstructured mesh with a high superficial gas inlet velocity and source terms for coalescence and breakup. The resulting computational approach is implemented in the open-source CFD code OpenFOAM as part of the OpenQBMM project

A Gaussian moment method and its augmentation via LSTM recurrent neural networks for the statistics of cavitating bubble populations

Phase-averaged dilute bubbly flow models require high-order statistical moments of the bubble population. The method of classes, which directly evolve bins of bubbles in the probability space, are accurate but computationally expensive. Moment-based methods based upon a Gaussian closure present an opportunity to accelerate this approach, particularly when the bubble size distributions are broad (polydisperse). For linear bubble dynamics a Gaussian closure is exact, but for bubbles undergoing large and nonlinear oscillations, it results in a large error from misrepresented higher-order moments. Long short-term memory recurrent neural networks, trained on Monte Carlo truth data, are proposed to improve these model predictions. The networks are used to correct the low-order moment evolution equations and improve prediction of higher-order moments based upon the low-order ones. Results show that the networks can reduce model errors to less than 1% of their unaugmented values

A quadrature-based moment method for polydisperse bubbly flows

A computational algorithm for polydisperse bubbly flow is developed by combining quadrature-based moment methods (QBMM) with an existing two-fluid solver for gas–liquid flows. Care is taken to ensure that the two-fluid model equations are hyperbolic by generalizing the kinetic model for the bubble phase proposed by Bieseuvel and Gorissen (1990). The kinetic formulation for the bubble phase includes the full suite of interphase momentum exchange terms for bubbly flow, as well as ad hoc bubble–bubble interaction terms to model the transition from isolated bubbles to regions of pure air at very high bubble-phase volume fractions. A robust numerical algorithm to couple the QBMM approach with a gas–liquid two-fluid solver is proposed. The resulting algorithm is tested to show hyperbolicity, verified against the two-fluid model currently implemented into OpenFOAM, and validated against two sets of experiments on bubbly flows from the literature. In both cases, the computational method shows good agreement with experimental data, and improved accuracy in comparison to a two-fluid model considered for comparison purposes. The robustness of the algorithm is demonstrated on an unstructured mesh with a high superficial gas inlet velocity and source terms for coalescence and breakup. The resulting computational approach is implemented in the open-source CFD code OpenFOAM as part of the OpenQBMM project.This is a manuscript of an article published as Heylmun, J. C., B. Kong, A. Passalacqua, and R. O. Fox. "A quadrature-based moment method for polydisperse bubbly flows." Computer Physics Communications (2019). DOI: 10.1016/j.cpc.2019.06.005. Posted with permission.</p

Analysis of dispersed multiphase flow using fully-resolved direct numerical simulation: Flow physics and modeling

Fully resolved simulation of flows with buoyant particles is a challenging problem since buoyant particles are lighter than the surrounding fluid.As a result, the two phases are strongly coupled together.In this work, the virtual force stabilization technique is used to simulate buoyant particle suspensions with high volume fractions.It is concluded that the dimensionless numerical model constant $C_v$ in the virtual force technique should increase with volume fraction.The behavior of a single rising particle, two in-line rising particles, and buoyant particle suspensions are studied.In each case, results are compared with experimental works on bubbly flows to highlight the differences and similarities between buoyant particles and bubbles.Finally, the drag coefficient is extracted from simulations of buoyant particle suspensions at different volume fractions, and based on that, a drag correlation is presented.Then velocity fluctuations in the carrier phase and dispersed phase of a dispersed multiphase flow are studied using particle-resolved direct numerical simulation.The simulations correspond to a statistically homogeneous problem with an imposed mean pressure gradient and are presented for a wide range of dispersed phase volume fractions, Reynolds number based on mean slip velocity, and density ratios of the dispersed phase to the carrier phase.The velocity fluctuations in the fluid and dispersed phase at the statistically stationary state are quantified by the turbulent kinetic energy (TKE) and granular temperature, respectively.It is found that the granular temperature increases with decrease in density ratio and then reaches an asymptotic value.The qualitative trend of the behavior is explained by the added mass effect, but the value of the coefficient that yields quantitative agreement is non-physical.It is also shown that the TKE has a similar dependence on the density ratio for all volume fractions studied here other than $\phi=0.1$.The anomalous behavior for $\phi=0.1$ is hypothesized to arise from the interaction of particle wakes at higher volume fractions.The study of mixture kinetic energy for different cases indicates that low-density ratio cases are less efficient in extracting energy from mean flow to fluctuations.The ultimate objective of this study is to understand the dynamics of freely evolving particle suspensions over a wide range of particle-to-fluid density ratios.The dynamics of particle suspensions are characterized by the average momentum equation, where the dominant contribution to the average momentum transfer between particles and fluid is the average drag force.In this study, the average drag force is quantified using fully-resolved direct numerical simulation in a canonical problem: a statistically homogeneous suspension where a steady mean slip velocity between the phases is established by an imposed mean pressure gradient.The effects of particle velocity fluctuations, clustering, and mobility of particles are studied separately.It is shown that the competing effects of these factors could decrease, increase, or keep constant the drag of freely evolving suspensions in comparison to fixed beds at different flow conditions.It is also shown that the effects of clustering and particle velocity fluctuations are correlated.Finally, a correlation for interphase drag force in terms of volume fraction, Reynolds number, and density ratio is proposed. Since this drag correlation has been inferred from simulations of particle suspensions, it includes the effect of the motion of the particles. This drag correlation can be used in computational fluid dynamics simulations of particle-laden flows that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior