An LES model with finite-rate phase change and subgrid spray based on a thermodynamically consistent four-equation multiphase model

This paper proposes a computationally efficient Large Eddy Simulation (LES) framework that combines a thermodynamically consistent four-equation multiphase model with a new phase-confined subgrid spray model and a finite-rate phase change model to accurately simulate complex evaporation and spray dynamics.

The Gist

Imagine you are trying to simulate a high-performance car engine or a rocket launch on a computer. To do this accurately, you have to simulate how liquid fuel turns into a fine mist (spray) and then quickly turns into gas (evaporation) to burn.

The problem is that doing this perfectly is a mathematical nightmare. If you try to track every single tiny droplet, your computer will melt. If you take too many shortcuts, the "physics" breaks, and your simulation might show liquid magically appearing out of thin air or gas behaving like a solid.

This paper introduces a new, smarter way to "cheat" at these simulations—a way that is fast enough for a computer to handle but smart enough to stay true to the laws of nature.

Here is how they did it, using three main concepts:

1. The "Four-Equation" Shortcut (The Efficient Manager)

Imagine you are managing a massive construction site with two different teams: the "Liquid Team" and the "Gas Team."

In a traditional, super-complex simulation, you would try to track the exact temperature, pressure, and speed of every single worker in both teams separately. It’s exhausting.

The researchers use a "Four-Equation Model." Instead of tracking everything separately, they assume the two teams are always in "agreement" regarding their pressure and temperature. It’s like saying, "I don't need to know exactly how much every worker is sweating; I just need to know the overall temperature of the site." This massive shortcut saves huge amounts of computing power without losing the big picture.

2. The $\Sigma$ (Sigma) Model (The "Invisible Mist" Tracker)

When fuel leaves an injector, it doesn't just stay as one big blob; it shatters into a cloud of tiny droplets. In a computer simulation, these droplets are often much smaller than the "pixels" (the grid) the computer uses to see. If the droplets are smaller than the pixels, the computer "loses" them.

To fix this, the researchers use the $\Sigma$ (Sigma) model. Think of this like a "Fog Density Map." Instead of trying to draw every individual tiny raindrop, the computer tracks a value called "surface area density."

It’s like looking at a distant forest through a window. You can't see every individual leaf, but you can see how "leafy" or "dense" the green blur is. By tracking this "leafiness" (the surface area), the computer knows exactly how much surface is available for the liquid to turn into gas, even if it can't "see" the individual droplets.

3. Thermodynamically Bounded Phase Change (The "Strict Accountant")

This is the most clever part of the paper. When liquid turns into gas (evaporation), it happens at a certain speed. In many simulations, the math can get "drunk"—it might predict that a liquid turns into gas so fast that it violates the laws of thermodynamics (like a bank account suddenly having more money than was ever deposited).

The researchers created a "Strict Accountant" for the phase change. They use a formula called the Hertz-Knudsen model, but they add a "safety rail."

Before the computer allows any mass to move from liquid to gas, the "Accountant" checks the Gibbs Free Energy (the "chemical potential"). It basically asks: "Does the math allow this much evaporation to happen based on the current temperature and pressure?" If the math tries to over-evaporate, the Accountant steps in and says, "Stop! You can only evaporate this much." This keeps the simulation "thermodynamically bounded," meaning it stays physically realistic.

The Result: A "Digital Twin" that Works

To prove this works, they tested it against real-world experiments (the ECN Spray A case). They simulated fuel being sprayed into a chamber and compared their "digital spray" to real-world measurements of how far the liquid traveled and how much gas was produced.

The verdict? Their model matched the real-world experiments incredibly well. They found a "sweet spot" where the simulation is fast enough to be useful for engineers designing better engines, but accurate enough that they can actually trust the results.

Technical Summary

This technical summary outlines the research presented in the paper "An LES model with finite-rate phase change and subgrid spray based on a thermodynamically consistent four-equation multiphase model" by Collis et al.

1. The Problem: Computational Complexity vs. Physical Fidelity

Modeling engineering-scale multiphase flows (e.g., fuel injection in engines or rocket propulsion) presents a significant computational challenge. There is a fundamental tension between two modeling approaches:

• Full Non-Equilibrium Models: These capture the complex physics of phase change and velocity/temperature differences between phases but are computationally prohibitive for large-scale Large Eddy Simulations (LES).
• Homogeneous Equilibrium Models (HEM): These assume instantaneous equilibrium, providing massive computational savings but failing to capture the finite-rate mass transfer essential for predicting evaporation, flashing, and cavitation in under-resolved (subgrid) regions.

Furthermore, existing finite-rate models often lack thermodynamic consistency (potentially predicting unphysical mass transfer) or require explicit tracking of droplet diameters, which is difficult in Eulerian frameworks.

2. Methodology: A Unified Multiphysics Framework

The authors propose an extension to a high-resolution, four-equation multiphase model. The methodology is built on three pillars:

A. Thermodynamically Bounded Finite-Rate Phase Change

To bridge the gap between HEM and non-equilibrium models, the authors implement a finite-rate model based on the Hertz-Knudsen kinetic theory. To ensure physical validity, they introduce a "thermodynamic bound." Instead of a direct mass transfer term, they define the rate using a relaxation timescale ($\tau$) that is constrained by the Gibbs free energy (chemical potential). This ensures that the phase change rate can never overshoot the state of thermo-chemical equilibrium defined by the HEM.

B. Phase-Confined Subgrid $\Sigma$ Spray Model

To inform the phase change model, the simulation must estimate the interfacial surface area ($\Sigma$) available for mass transfer. The authors propose a phase-confined Eulerian $\Sigma$ model. Unlike standard $\Sigma$ models that can suffer from "leakage" (where subgrid scalars diffuse into the wrong phase), this model uses a modified transport equation that confines the subgrid surface area density strictly to the gaseous phase. This allows for the accurate representation of unresolved liquid features (droplets/ligaments) within the gas.

C. Multi-Regime Spray Physics

The model distinguishes between different spray regimes to apply appropriate source terms for surface area production and destruction:

• Dense Spray: Uses models for turbulent breakup and flashing (bubble growth/nucleation).
• Dilute Spray: Uses models for secondary breakup and coalescence.
• Regime Indicators: The transition between dense/dilute and flashing/evaporation is determined via liquid volume fraction and the Jakob number.

3. Key Contributions

1. Thermodynamic Consistency: A novel implementation where finite-rate mass transfer is mathematically bounded by the chemical potential equilibrium, preventing unphysical results.
2. Leakage-Free Subgrid Modeling: A new phase-confined transport formulation for the $\Sigma$ equation that prevents subgrid surface area from diffusing into the liquid phase.
3. Integrated LES Framework: A complete, high-resolution numerical strategy that couples compressible Navier-Stokes equations with subgrid spray and finite-rate phase change, optimized for engineering-scale simulations.

4. Results and Validation

The framework was validated using the Engine Combustion Network (ECN) Spray A case (n-dodecane injection).

• Non-Evaporating Case (Spray Dynamics): The model showed excellent agreement with experimental data regarding Projected Mass Density (PMD), Projected Surface Area (PSA), and Sauter Mean Diameter (SMD). This confirmed that the subgrid $\Sigma$ model and turbulent liquid flux closures accurately capture spray breakup and spread.
• Evaporating Case (Phase Change): The fully coupled model successfully predicted liquid penetration length and gas penetration length. The agreement with experimental measurements (DBI and Mie-scattering techniques) demonstrates that the thermodynamically bounded finite-rate model accurately captures the complex interplay between spray atomization and mass transfer.

5. Significance

This work provides a robust, predictive tool for simulating high-pressure, high-temperature multiphase flows. By proving that a "four-equation" model (which assumes mechanical and thermal equilibrium) can be successfully augmented with finite-rate subgrid physics, the authors demonstrate that researchers can achieve high-fidelity results at a fraction of the computational cost required by full non-equilibrium models. This is highly significant for the development of more efficient internal combustion engines and advanced propulsion systems.