Pseudo-turbulence in two-dimensional buoyancy-driven bubbly flows: A DNS study
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Pseudo-turbulence in two-dimensional buoyancy-driven bubbly flows: A DNS study Rashmi Ramadugua , Vikash Pandey, and Prasad Perlekar TIFR Center for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Gopanpally, Hyderabad, 500046, India Received 3 August 2020 / Received in final form 21 October 2020 / Accepted 2 November 2020 Published online: 25 November 2020 c EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. We present a direct numerical simulation (DNS) study of buoyancy-driven bubbly flows in two dimensions. We employ the volume of fluid (VOF) method to track the bubble interface. To investigate the spectral properties of the flow, we derive the scale-by-scale energy budget equation. We show that the Galilei number (Ga) controls different scaling regimes in the energy spectrum. For high Galilei numbers, we find the presence of an inverse energy cascade. Our study indicates that the density ratio of the bubble with the ambient fluid or the presence of coalescence between the bubbles does not alter the scaling behaviour.
1 Introduction A swarm of bubbles rising under gravity generates complex spatio-temporal flow patterns, often referred to as pseudo-turbulence (PT) or bubble-induced agitation. Although the trajectory and wake of an isolated bubble depend on its viscosity and density contrast with the surrounding fluid [1–6], the statistical properties of the flow generated by the bubble swarm are found to be universal [7–11]. A key feature of PT is the power-law scaling in the energy spectrum with an exponent of −3 either in frequency f or the wave number k space [8,12–14], explained by the balance of energy production by wakes with viscous dissipation [9, 11, 15]. The key non-dimensional numbers that characterise pseudo-turbulence are the Bond number Bo ≡ δρgd2 /σ (ratio of the buoyancy forces to the surface tension forces), the Galilei number Ga ≡ ρf δρgd3 /μ (ratio of the buoyancy forces to the viscous forces), and the Atwood number At ≡ δρ/(ρf + ρb ), where ρf is the density of the ambient fluid, ρb is the bubble density, δρ ≡ ρf − ρb , g is the acceleration due to gravity, and d is the initial bubble diameter. Experiments in three dimensions (3D) at low volume fraction φ ≈ 2% show the k−3 scaling in the energy spectrum both within and in the wake of the bubble swarm for 202 ≤ Ga ≤ 396 [12]. In Hele-Shaw geometry, experiments [14, 16] at moderate volume fraction φ ≈ 5–16% and 650 ≤ Ga ≤ 1100 also show the k −3 scaling. Most numerical studies have investigated PT in 3D [11, 17,18] and found that the statistical properties of the flow are universal for a wide range of Atwood At, Bond Bo, and a
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Galilei Ga numbers. Using a scale-by-scale energy budget analysis, ref. [11] showed that both the surface tension and kinetic energy flux contribute to the net energy production at scales smaller than d. In comparison to 3D, there have been very few studies on two-dim
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