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.
💡 Analysis
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.
📄 Content
Flows of immiscible fluids are common in both natural and industrial systems. In many cases, the interfacial surface area between the two phases is a critical parameter, controlling phenomena such as heat and mass transfer. Yet, predicting the interface surface area requires understanding the complex coupling between the flow and the interface which remains a major challenge. This interplay becomes clear when considering energy balances: the interface stores surface energy through surface tension, coexisting with kinetic energy [2][3][4][5]. A key open question is how these different forms of energy interact.
Recent studies [6][7][8][9][10][11][12][13][14] have gradually converged toward a coherent framework for describing energy exchanges in multiphase turbulence. In this picture, a portion of the kinetic energy injected at large scales cascades to smaller scales, much like in single-phase flows. Concurrently, when surface tension is present, part of the injected energy is also converted into interfacial or surface energy as interfaces become progressively distorted and fluid structures break up. At smaller scales, interfaces may relax, fluid structures could also coalesce, converting surface energy back into kinetic energy. Both the turbulent cascade and energy conversion processes continue down to the smallest scales, where kinetic energy is ultimately dissipated into heat.
Multiphase flows are thus characterized by a complex interplay of energy injection, cascade, conversion, and dissipation. At steady state, both kinetic and surface energy remain constant, and energy conversions vanish, reflecting an equilibrium between the two reservoirs and between energy injection and dissipation, as in single-phase flows. In turbulent flows, however, steady state can only be defined statistically, after averaging over long times. Instantaneously, spatial averages remain non-stationary, making turbulence a textbook example of an “out-of-equilibrium” system, where the injected energy requires finite time to cascade across scales before being dissipated [see e.g. [15][16][17]. Non-equilibrium effects are then manifested in particular in a time delay between fluctuations of kinetic energy and its rate of dissipation.
Because in multiphase turbulence, energy conversion occurs alongside energy transfer, it can potentially lead to atypical out-of-equilibrium behaviours that remain largely unexplored. To date, only Mukherjee et al. [6] have revealed an intriguing dynamical cycle linking kinetic energy injection, conversion into surface energy, and dissipation into heat. This study pursues three main objectives: (i) to analyse energy exchanges in unsteady multiphase flows, (ii) to examine the coupling and dissipation/destruction of kinetic and surface energies, and (iii) to clarify the out-ofequilibrium characteristics of multiphase turbulent flows for both forms of energy.
Here, we deliberately keep the flow in a non-stationary regime while retaining the possibility of meaningful statistical analysis. This is achieved by considering periodically forced turbulence, for which phase averaging can be employed in place of classical time averaging. Such a configuration has been widely used, in particular to investigate nonequilibrium effects [18,19] or to explore the role of intermittency of the large scales [20,21]. They also provide a convenient benchmark for one-point and two-point closure models [see, e.g., 1, 22-24, and references therein].
Periodically forced flows are known to exhibit two asymptotic responses. At very low forcing frequencies, the system has sufficient time to adjust to variations in energy input and therefore evolves through a sequence of locally steady states; this corresponds to the static limit. Conversely, at very high forcing frequencies, the turbulent cascade acts as a low-pass filter, so that dissipation does not respond to changes in the energy input, defining the frozen limit. While these extreme regimes are relatively straightforward to predict, more interesting dynamics arise at intermediate forcing frequencies, which is the regime of primary interest in the present study.
We conduct numerical simulations of multiphase turbulence driven by harmonic forcing in a triply periodic domain using a standard Navier-Stokes solver coupled with an interface-capturing method. The resulting flow is homogeneous and isotropic. To interpret and reproduce the numerical observations, we propose a model that accounts for the coupling between kinetic and surface energies, as well as their respective dissipation or destruction mechanisms. This model builds upon a recent extension of the k-ϵ framework, referred to as the Ka-Pi-bara model [1], which was developed to capture non-equilibrium effects in single-phase flows.
The extension to multiphase flows proceeds in two steps. First, we reformulate the model in terms of the total energy, defined as the sum of kinetic and surface energies, in order to incorporat
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