Abstract
Data from direct numerical simulations (DNS) of disperse bubbly flow in an upward vertical channel are used to develop a new second-moment closure for bubble-induced turbulence (BIT) in the Euler–Euler framework. The closure is an extension of a BIT model originally proposed by Ma et al. (Phys. Rev. Fluids, vol. 2, 2017, 034301) for two-equation eddy-viscosity models and focuses on the core region of the channel, where the interfacial term and dissipation term are in balance. Particular attention in this study is given to the treatment of the pressure–strain term for bubbly flows and the form of the interfacial term to account for BIT. For the latter, the concept of an effective BIT source is proposed, which leads to a significant simplification of the modelling work for both the pressure–strain correlation and the interfacial term itself. The anisotropy of bubbly flow is analysed with the aid of the anisotropy-invariant map obtained from the DNS data, and a parameter governing this issue is established. The complete second-moment closure is tested against reference data for different bubbly channel flows and a case of a bubble column. The agreement achieved with the DNS data is very good and the performance of the new model is better than obtained with the standard procedure. Furthermore, the model is shown to be robust and to fulfil the requirements of realizability.
Highlights
Disperse turbulent bubbly flows occur in a very large number of situations encountered in process engineering, energy technology, environmental flows, etc
A full second-moment closure (SMC) for bubble-laden flows in the framework of the Euler–Euler (EE) approach was proposed for the channel centre
Several important conclusions based on direct numerical simulations (DNS) data for bubbly flow were obtained that led to the development of a new bubble-induced turbulence (BIT) model at the secondmoment level (5.3)
Summary
Disperse turbulent bubbly flows occur in a very large number of situations encountered in process engineering, energy technology, environmental flows, etc. Over the last two decades, there has been widespread work on supplementing singlephase two-equation eddy-viscosity models with specific source terms representing BIT (Lopez de Bertodano, Lahey & Jones 1994; Morel 1997; Troshko & Hassan 2001; Colombo & Fairweather 2015; Ma et al 2017) These models take the influence of bubbles into account by including additional source terms in the balance equations for both k, the turbulent kinetic energy (TKE), and ε, the turbulent dissipation rate, or another equivalent parameter. Employing DNS of a much larger number of bubbles and more realistic physical parameters, such as the density ratio and bubble Reynolds number, Santarelli, Roussel & Fröhlich (2016) computed the budget terms in the TKE equation and proposed a BIT closure based on a priori evaluations. Beyond the new model itself, the paper furnishes a systematic procedure that is of general use for this type of modelling
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