Conversions between kinetic and surface energy in periodically forced multiphase turbulence

This study investigates the conversion between kinetic and surface energy in periodically forced multiphase turbulence through numerical simulations and an enhanced total energy model, revealing that while kinetic energy exhibits non-equilibrium phase lags, surface energy remains in equilibrium with its destruction, indicating the absence of a surface energy cascade.

The Gist

Imagine a busy, chaotic dance floor where two types of dancers are moving: Kinetic Dancers (who represent the swirling motion of the fluid) and Surface Dancers (who represent the tension and stretching of the boundary between two liquids, like oil and water).

In a normal, steady party, these dancers move at a constant speed. The energy they use to dance (Kinetic Energy) and the energy stored in their stretched arms and connections (Surface Energy) stay the same. It's boring to watch because nothing changes.

But what happens if the DJ suddenly starts changing the beat rhythmically—speeding up and slowing down the music? This is what the scientists in this paper did. They forced the fluid to dance to a periodic beat (a rhythmic, repeating input of energy) to see how the dancers react when the music isn't steady.

Here is the story of their discovery, broken down into simple concepts:

1. The Big Challenge: The "Time Lag"

In single-phase fluids (just one type of liquid, like pure water), when you speed up the music, the dancers don't react instantly. There is a delay.

• The Beat (Energy Input): The DJ hits the drum.
• The Spin (Kinetic Energy): The dancers start spinning faster.
• The Exhaustion (Dissipation): The dancers get tired and lose energy as heat.

In a steady state, the spin and the exhaustion happen at the same time. But when the beat changes, the dancers get tired after they start spinning. This delay is called a "non-equilibrium effect." It's like pedaling a bike: you push the pedals, but the bike takes a split second to actually speed up, and it takes a moment to slow down after you stop pushing.

2. The New Twist: The "Rubber Band" Effect

In this study, they added a second type of energy: Surface Energy. Imagine the dancers are holding giant, stretchy rubber bands between them.

• When the fluid swirls fast, it stretches the rubber bands (converting motion energy into surface energy).
• When the rubber bands snap back, they pull the dancers, giving them a burst of speed (converting surface energy back into motion).

The scientists wanted to know: Does this rubber band system have its own "time lag," or does it react instantly?

3. The Discovery: Two Different Rules

By running thousands of computer simulations (like running the dance party in a super-computer), they found two very different behaviors:

• The Motion (Kinetic Energy) is Lazy: Just like in single-phase flows, the swirling motion has a delay. It takes time for the energy to travel from the big swirls to the tiny swirls before it disappears as heat.
• The Rubber Bands (Surface Energy) are Instant: Surprisingly, the surface energy and the "breaking" of the rubber bands (destruction) happen at the exact same time. There is no delay.

The Analogy:
Think of the Kinetic Energy as a heavy truck. When you hit the gas, it takes time to accelerate, and when you hit the brakes, it takes time to stop.
Think of the Surface Energy as a light switch. When you flip it, the light turns on instantly. When you flip it off, the light goes off instantly.

4. The "Cascade" Mystery

In fluid physics, there is a concept called a "cascade," where energy flows from big swirls to tiny swirls like a waterfall.

• The scientists found that Kinetic Energy has a waterfall (a cascade). It flows down from big to small scales.
• Surface Energy, however, does not have a waterfall. It doesn't cascade. It stays in equilibrium. It reacts instantly to the forces acting on it, rather than waiting for a chain reaction of smaller and smaller events.

5. The New "Recipe" (The Model)

The scientists created a new mathematical "recipe" (a model) to predict how this dance floor behaves.

• They took an old recipe used for single-phase fluids (the $k-\epsilon$ model).
• They updated it to include the "Rubber Band" energy.
• They added a new rule: The rubber bands react instantly, while the heavy truck (motion) reacts with a delay.

When they tested this new recipe against their computer simulations, it worked perfectly. It predicted exactly when the dancers would speed up, when they would get tired, and when the rubber bands would stretch and snap.

Why Does This Matter?

This isn't just about abstract physics. This helps engineers understand:

• Fuel Injectors: How fuel breaks into tiny droplets in an engine.
• Oil Spills: How oil mixes with water in the ocean.
• Chemical Reactors: How to mix liquids efficiently.

By understanding that the "motion" and the "surface tension" react differently to changes in speed, engineers can design better systems that mix fluids more efficiently or predict how droplets will behave in turbulent environments.

In a nutshell: The paper shows that in a turbulent mix of liquids, the movement of the fluid is slow to react to changes, but the stretching of the surface between them is instant. They built a new mathematical tool to predict this behavior, proving that surface energy doesn't follow the same "waterfall" rules as regular motion.

Technical Summary

1. Problem Statement

In multiphase flows involving immiscible fluids, two primary energy reservoirs coexist: kinetic energy ($E_k$) and interfacial (surface) energy ($E_s$). While the interaction between these reservoirs is critical for phenomena like heat/mass transfer and droplet breakup, understanding their mutual conversion remains a challenge.

• The Equilibrium Issue: In statistically steady flows, both energy reservoirs remain constant, rendering the conversion processes between them undetectable in the mean energy balance.
• The Non-Equilibrium Gap: Turbulence is inherently an "out-of-equilibrium" system where energy injection, cascade, and dissipation occur with finite time delays. While non-equilibrium effects (specifically the phase lag between kinetic energy and its dissipation rate) are well-studied in single-phase flows, they are largely unexplored in multiphase turbulence where energy conversion ($E_k \to E_s$ and vice versa) occurs simultaneously with the turbulent cascade.
• Objective: The authors aim to analyze energy exchanges in unsteady multiphase flows, examine the coupling and dissipation/destruction of kinetic and surface energies, and clarify the non-equilibrium characteristics of these systems.

2. Methodology

A. Numerical Approach

The study utilizes Direct Numerical Simulations (DNS) of homogeneous isotropic turbulence in a triply periodic domain.

• Forcing: Instead of steady forcing, the system is subjected to time-periodic harmonic forcing ($F = F_0(1 + \alpha_F \sin(\omega t))$). This introduces a deliberate unsteadiness while allowing for statistical analysis via phase averaging (averaging over the forcing period $T_f$) rather than time averaging.
• Parameters: The database covers a wide range of conditions:
  • Reynolds numbers ($Re$): 645 to 4166.
  • Weber numbers ($We$): 8.33 to 25.
  • Volume fractions ($\alpha$): 12.5% to 50%.
  • Forcing periods ($T_f$): 0.75 to 1.5.
• Solver: The archer code (Navier-Stokes solver coupled with an interface-capturing method) is used, assuming equal density and viscosity for both phases to simplify the analysis and isolate surface tension effects.

B. Theoretical Modeling

To interpret the simulation data, the authors propose a modified turbulence model extending the $k-\epsilon$ framework to multiphase flows.

1. Total Energy Formulation: The model is reformulated in terms of Total Energy ($E_t = E_k + E_s$). The evolution of $E_t$ follows the single-phase energy budget ($\dot{E}_t = F - \epsilon$), treating the conversion between $E_k$ and $E_s$ as conservative.
2. Ka-Pi-bara Extension: The authors adapt the Ka-Pi-bara model (an extension of $k-\epsilon$ by Bos [1] that captures non-equilibrium phase lags) to multiphase flows.
3. New Evolution Equations:
   • Surface Energy ($E_s$): Modeled as $\dot{E}_s = (P - \chi)E_s$, where $P$ (production) is proportional to the Kolmogorov strain rate $(\epsilon/\nu)^{1/2}$, and $\chi$ (destruction) accounts for coalescence.
   • Surface Destruction ($\chi$): A fourth equation is introduced for the evolution of $\chi$, analogous to the $\epsilon$ equation in the standard $k-\epsilon$ model: $\dot{\chi} = C_s \frac{\chi}{E_s} \dot{E}_s$.
4. Linearization: The system of four coupled Ordinary Differential Equations (ODEs) is linearized around the steady state to derive transfer functions. This allows the prediction of gain (amplitude response) and phase lag of observables ($E_k, E_s, \epsilon, \chi$) relative to the forcing frequency.

3. Key Contributions

1. Development of a Multiphase Non-Equilibrium Model: The paper presents a novel 4-equation ODE system that couples kinetic energy, surface energy, and their respective dissipation/destruction rates. It successfully extends the Ka-Pi-bara model to account for interfacial effects.
2. Identification of Surface Energy Equilibrium: A major theoretical finding is that, unlike kinetic energy, surface energy ($E_s$) and its destruction rate ($\chi$) remain in phase. This implies that surface energy does not undergo a "cascade" to smaller scales in the same way kinetic energy does; rather, it adjusts quasi-instantaneously to the dissipation rate.
3. Quantification of Phase Lags: The study quantifies the time delays between energy injection, conversion, and dissipation in multiphase turbulence, demonstrating that these lags are significant and dependent on Reynolds number.
4. Parametric Sensitivity Analysis: The authors systematically analyze how $Re$, $We$, volume fraction ($\alpha$), and forcing frequency affect energy dynamics, providing a comprehensive database for future model validation.

4. Key Results

• Non-Equilibrium Effects:
  • In single-phase flows, the phase lag between kinetic energy ($E_k$) and dissipation ($\epsilon$) increases with Reynolds number.
  • In multiphase flows, this lag persists. The Ka-Pi-bara model accurately reproduces this lag (controlled by parameter $\gamma$), whereas the classical $k-\epsilon$ model fails (predicting in-phase behavior).
• Surface Energy Dynamics:
  • Phase Synchronization: The model and simulations confirm that fluctuations in surface energy ($E_s$) and its destruction ($\chi$) are in phase.
  • Coupling to Dissipation: At high Reynolds numbers, $E_s$ and $\chi$ become synchronized with the kinetic energy dissipation rate ($\epsilon$).
  • No Cascade: The in-phase behavior of $E_s$ and $\chi$ indicates the absence of a surface energy cascade. Surface energy is produced and destroyed locally without a significant scale-by-scale transfer delay.
• Parameter Sensitivity:
  • Weber Number ($We$): Variations in $We$ (surface tension) have a negligible effect on the dynamics. A decrease in surface tension is compensated by an increase in interfacial area density, keeping the total surface energy reservoir constant.
  • Volume Fraction ($\alpha$): Increasing $\alpha$ increases the amplitude of surface energy fluctuations while decreasing kinetic energy fluctuations. At $\alpha = 50\%$, the two energy reservoirs are comparable.
  • Forcing Frequency: The model parameters ($C_k, \gamma, C_s$) were found to vary with the forcing frequency. This suggests the current model lacks a proper two-timescale decomposition (separating large-scale forcing response from small-scale adaptation), as the model parameters should ideally be intrinsic to the flow, not the forcing.

5. Significance and Conclusion

This work provides a fundamental understanding of how energy is exchanged between kinetic and surface reservoirs in unsteady multiphase turbulence.

• Theoretical Insight: It challenges the assumption that surface energy behaves like a passive scalar or follows a turbulent cascade. Instead, it behaves as a reservoir in quasi-equilibrium with the dissipation rate.
• Modeling Impact: The proposed 4-equation model offers a computationally efficient tool (compared to full DNS) to predict energy budgets in multiphase flows. It successfully captures the non-equilibrium phase lags that are crucial for accurate engineering predictions in applications like emulsification, spray combustion, and bubble columns.
• Future Directions: The authors note that the dependence of model parameters on forcing frequency indicates a need for a more sophisticated model that explicitly separates large-scale and small-scale timescales to improve universality.

In summary, the paper establishes that while multiphase turbulence shares non-equilibrium characteristics with single-phase turbulence (phase lags in kinetic energy), the surface energy component behaves differently, remaining in local equilibrium with the dissipation rate and lacking a distinct cascade mechanism.