Scale-by-scale energy transfers in bubbly flows

Abstract

Buoyancy-driven bubbly flows naturally have spatially-dependent density fields, which allow for multiple definitions of the scale-dependent (or filtered) energy. A priori, it is not obvious which of these provide the most physically apt scale-by-scale budget. In the present study, we compare two such definitions, based on (a) filtered momentum and filtered velocity (Pandey et al. 2020), and (b) Favre filtered energy (Aluie 2013; Pandey et al. 2023). We also derive a Kármán-Howarth-Monin (KHM) relation using the momentum-velocity correlation function and contrast it with the scale-by-scale energy budget obtained in (a). We find that for the volume fraction and Atwood number explored, irrespective of the definition, energy transfers due to the advective nonlinearity and surface tension are identical. However, discrepancies arise for the buoyancy and pressure contributions. We show that the Favre filtered definition is the more appropriate choice, within which buoyancy injects energy, pressure transfers energy to large scales, and both advective nonlinearity and surface tension transfer energy downscales where it is dissipated by viscosity.

Figures (8)

• Figure 1: (a) Kinetic energy time series in the statistically steady state. (b) (time-averaged) energy spectrum $E(k)$ (normalized by the unfiltered kinetic energy) in the statistically steady state for run R2. We indicate the Kolmogorov ($k^{-5/3}$) and the pseudo-turbulence ($k^{-3}$) scaling range. The $k^{-5/3}$ scaling becomes more prominent for large Galilei number ($\hbox{Ga}\ge1000)$PandeyMitraPerlekar2023ma2025. $K_D=2 { \math@atom{\pi}{ \hbox{$\m@th\pi$}} }/D$ is the wavenumber corresponding to the bubble diameter $D$ and $\tau$ is the large eddy turnover time. (c) pseudo-color plot of the magnitude of the vorticity field $|\hbox{\boldmath $\omega$}|$ in a representative 2$d$ slice containing bubbles for run R2 (the bubble-liquid interface is marked in white).
• Figure 2: Scale-by-scale budget using F1 filtering \ref{['eq:f1_contri']} (left) and the KHM relation C1\ref{['eq:khm_contri']} (right, with $K=\sqrt{3}/r$) for small Atwood number (run R1). The pressure contribution is identically zero, hence not shown here. The shading indicates one standard deviation variations. In both the plots, all the budget contributions (vertical axis) have been normalized by the mean energy injection rate $\epsilon^g\equiv \langle {\hbox{\boldmath $u$}} \cdot {\hbox{\boldmath $F$}^g} \rangle$.
• Figure 3: Scale-by-scale budget using F1 filtering \ref{['eq:f1_contri']} (left) and the KHM relation C1\ref{['eq:khm_contri']} (right, with $K=\sqrt{3}/r$) for large Atwood number (run R2). The shading indicates one standard deviation variations. The orange dashed line indicates the unfiltered buoyancy injection to convey the non-monotonic behavior of the buoyant flux. In both the plots, all the budget contributions (vertical axis) have been normalized by the mean energy injection rate $\epsilon^g\equiv \langle {\hbox{\boldmath $u$}} \cdot {\hbox{\boldmath $F$}^g} \rangle$.
• Figure 4: Scale-by-scale budget using the Favre filtering ( F2) \ref{['eq:f2_contri']} for the large Atwood number (run R2). The shading indicates one standard deviation variations. All the budget contributions (vertical axis) have been normalized by the mean energy injection rate $\epsilon^g\equiv \langle {\hbox{\boldmath $u$}} \cdot {\hbox{\boldmath $F$}^g} \rangle$. In contrast with the F1 budget (figure \ref{['fig:hga_r8_nf']}), the buoyancy term increases monotonically.
• Figure 5: The sum of the baropycnal and the buoyancy contribution to the scale-by-scale budget (run R2) for both F1 and F2 definitions.