Conversions between kinetic and surface energy in periodically forced multiphase turbulence

Abstract

In multiphase flows, kinetic and interfacial energies coexist, and their mutual conversion can strongly influence the overall energy balance. However, in statistically steady flows these energy reservoirs remain constant, making such conversions undetectable. For them to be observed, a degree of unsteadiness must be introduced, here provided by the deliberate use of a fluctuating time-periodic input of kinetic energy into the system. The main focus of the present work is on the dynamical cycle connecting energy injection, conversion, and dissipation which we explore using numerical simulations of multiphase homogeneous isotropic turbulence, subjected to periodic forcing. The database includes various Reynolds and Weber numbers and volume fractions in the dense regime. To interpret and replicate the observed dynamics, we reformulate the \textit{Ka-Pi-bara} model of \cite{Bos2026} (an extension of the $k$--$ε$ model) in terms of total energy (the sum of kinetic and surface energy), which we further enhance by adding equations for the surface energy and its destruction. This model accurately captures a key feature of turbulence: non-equilibrium effects, seen as the phase lag between kinetic energy and its rate of dissipation, which are found to operate also in multiphase flows. Linearizing the model highlights the various relevant time scales of the system and provides predictions of how different observables are coupled and respond to the energy input. In particular, the model predicts that fluctuations of surface energy and its destruction are in phase, in good agreement with numerical simulations. Therefore, unlike kinetic energy, surface energy remains in equilibrium, indicating the absence of a surface energy cascade.

Figures (5)

• Figure 1: Predictions using the $k$--$\epsilon$ model (top panel, (a, b)) and the Ka-Pi-bara model (bottom panel, (c, d)) in single phase flows. Symbols represent the numerical simulations while the lines corresponds to the model. Full lines represent solutions of the non-linear system while dashed lines correspond to solutions of the linearized system. The colours from dark to light represents an increasing Reynolds number $R_\lambda = 43,~72,~120$. The kinetic energy $E_k$ is normalised by $F^0 \alpha_F T_f$ while the terms of its balance equation are normalised by $F^0 \alpha_F$.
• Figure 2: Effect of Weber number. From dark to light : $We = 8.33, ~12.5, ~25$. Panel (a) shows $E_k^\prime, ~E_s^\prime$ (normalized by $F^0 \alpha_F T_f$) and panel (b) represents $F^\prime, ~\dot{E}_k^\prime, ~\dot{E}_s^\prime, ~\epsilon$ (normalized by $F^0 \alpha_F$). Results for single-phase flow at same $Re$ are displayed with filled symbols.
• Figure 3: Effect of the volume fraction. From dark to light : $\alpha = 12.5, ~25, ~50\%$. Panel (a) shows $E_k^\prime, ~E_s^\prime$ (normalized by $F^0 \alpha_F T_f$) and panel (b) represents $F^\prime, ~\dot{E}_k^\prime, ~\dot{E}_s^\prime, ~\epsilon$ (normalized by $F^0 \alpha_F$). Results for single-phase flow at same $Re$ are displayed with filled symbols.
• Figure 4: Effect of the Reynolds number. From dark to light : $Re = 645, ~1612, ~4166$. Panel (a) shows $E_k^\prime, ~E_s^\prime$ (normalized by $F^0 \alpha_F T_f$) and panel (b) represents $F^\prime, ~\dot{E}_k^\prime, ~\dot{E}_s^\prime, ~\epsilon^\prime$ (normalized by $F^0 \alpha_F$).
• Figure 5: Effect of the forcing period. From dark to light : $T_f = 0.75, ~1.0, ~1.5$. Panel (a) shows $E_k^\prime, ~E_s^\prime$ (normalized by $F^0 \alpha_F T_f$) and panel (b) represents $F^\prime, ~\dot{E}_k^\prime, ~\dot{E}_s^\prime, ~\epsilon^\prime$ (normalized by $F^0 \alpha_F$).