Scale-by-scale kinetic energy budgets in multiphase turbulence

TL;DR

This work derives a generalized Kármán-Howarth-Monin equation for multiphase turbulence that includes density/viscosity variations and surface tension, and applies phase-conditioned averaging to resolve energy exchanges across scales in both liquid and gas phases. Using the Archer DNS framework with CLSVOF interface tracking, the authors quantify how surface tension acts as an energy transfer term, storing energy at large scales as surface energy and releasing it at smaller scales, while nonlinear transport and viscous dissipation adjust the cascade. They find that the overall scale-by-scale kinetic energy transfer in multiphase flows exceeds that of single-phase flows, with density contrasts enhancing small-scale activity, particularly in the gas phase, and with pressure transport playing a key role in the lighter phase. A pivotal-scale phenomenology is proposed, with a deformation-to-restoration transition around rS, and local Weber number We(r) guiding the identification of Kolmogorov-Hinze-like scales; however, rS does not consistently coincide with rH, indicating limits to Hinze-based predictions in dense, high-contrast multiphase turbulence.

Abstract

The present work aims at exploring the scale-by-scale kinetic energy exchanges in multiphase turbulence. For this purpose, we derive the Kármán-Howarth-Monin equation which accounts for the variations of density and viscosity across the two phases together with the effect of surface tension. We consider both conventional and phase conditional averaging operators. This framework is applied to numerical data from detailed simulations of forced homogeneous and isotropic turbulence covering different values for the liquid volume fraction, the liquid/gas density ratio, the Reynolds, and Weber numbers. We confirm the existence of an additional transfer term due to surface tension. Part of the kinetic energy injected at large scales is transferred into kinetic energy at smaller scales by classical non-linear transport while another part is transferred to surface energy before being released back into kinetic energy, but at smaller scales. The overall kinetic energy transfer rate is larger than in single phase flows. Kinetic energy budgets conditioned in a given phase show that the scale-by-scale transport of turbulent kinetic energy due to pressure is a gain (loss) of kinetic energy for the lighter (heavier) phase. Its contribution can be dominant when the gas volume fraction becomes small or when the density ratio increases. Building on previous work, we hypothesize the existence of a pivotal scale above which kinetic energy is stored into surface deformation and below which the kinetic energy is released by interface restoration. Some phenomenological predictions for this scale are discussed.

Figures (15)

• Figure 1: Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy for $\alpha = 25\%$. The colours from blue to red correspond to $R_\rho = 1, ~5, ~25, ~125$ while the black curve is for single-phase turbulence. (a) $\mathbb{C} \equiv \mathbb{T}$, (b) $\mathbb{C} \equiv \mathbb{L}$ and (c) $\mathbb{C} \equiv \mathbb{G}$. The dashed line represents the $r^2$ scaling.
• Figure 2: 2D slices of the vorticity magnitude (in colour, units of $u'/L$) together with the liquid-gas interface (black curves) for $\alpha=25\%$ and increasing $R_\rho$ from (a) to (d).
• Figure 3: Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy budgets for $\alpha = 25\%$. The colours from blue to red correspond to $R_\rho = 1, ~5, ~25, ~125$ as in the legend of Fig. \ref{['fig:rho_phi25_dq2']}. The black curves are for the single-phase case at same viscosity. The different lines correspond to $\langle \mathcal{T}\rangle_\mathbb{C}$, $\langle \mathcal{P}\rangle_\mathbb{C}$, $\langle \mathcal{S}\rangle_\mathbb{C}$, $\langle \mathcal{V}\rangle_\mathbb{C}$, $\langle \mathcal{F}\rangle_\mathbb{C}$, with (a) $\mathbb{C} = \mathbb{T}$, (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$. The horizontal dashed line indicates $4\langle F\rangle_\mathbb{C}$, the limit at large separations of the forcing term. Figure (b) represents the density correction terms $\langle \mathcal{C}(\boldsymbol{a}) \rangle_\mathbb{T}$. Also represented in (a) are the scales $r_H$ with lines and $r_\mathcal{S}$ with filled circles that will be described later on.
• Figure 4: Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy for $\alpha = 75\%$. The colours from blue to red correspond to $R_\rho = 1, ~5, ~25$ while the black curve is for single-phase turbulence. (a) $\mathbb{C} \equiv \mathbb{T}$, (b) $\mathbb{C} \equiv \mathbb{L}$ and (c) $\mathbb{C} \equiv \mathbb{G}$.
• Figure 5: 2D slices of the vorticity magnitude (in colour, units of $u'/L$) together with the liquid-gas interface (black curve) at $\alpha = 75\%$ and for increasing $R_\rho$ from (a) to (c).