Conversions between kinetic and surface energy in periodically forced multiphase turbulence

TL;DR

This study investigates how kinetic and surface energies convert in unsteady multiphase turbulence by using periodic forcing and a reformulated energy model, revealing phase relationships and energy dynamics.

Contribution

It introduces a novel energy-based model capturing non-equilibrium effects and phase lag in multiphase turbulence under periodic forcing.

Findings

Surface energy remains in equilibrium, indicating no cascade.

The model accurately predicts phase lag between kinetic energy and dissipation.

Numerical simulations confirm the model's predictions.

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 $k$--$\epsilon$ 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.