Numerical Simulation of Phase Transition with the Hyperbolic Godunov-Peshkov-Romenski Model (2403.01847v1)

Abstract: In this paper, a thermodynamically consistent solution of the interfacial Riemann problem for the first-order hyperbolic continuum model of Godunov, Peshkov and Romenski (GPR model) is presented. In the presence of phase transition, interfacial physics are governed by molecular interaction on a microscopic scale, beyond the scope of the macroscopic continuum model in the bulk phases. The developed two-phase Riemann solvers tackle this multi-scale problem, by incorporating a local thermodynamic model to predict the interfacial entropy production. Using phenomenological relations of non-equilibrium thermodynamics, interfacial mass and heat fluxes are derived from the entropy production and provide closure at the phase boundary. We employ the proposed Riemann solvers in an efficient sharp interface level-set Ghost-Fluid framework to provide coupling conditions at phase interfaces under phase transition. As a single-phase benchmark, a Rayleigh-B\'enard convection is studied to compare the hyperbolic thermal relaxation formulation of the GPR model against the hyperbolic-parabolic Euler-Fourier system. The novel interfacial Riemann solvers are validated against molecular dynamics simulations of evaporating shock tubes with the Lennard-Jones shifted and truncated potential. On a macroscopic scale, evaporating shock tubes are computed for the material n-Dodecane and compared against Euler-Fourier results. Finally, the efficiency and robustness of the scheme is demonstrated with shock-droplet interaction simulations that involve both phase transfer and surface tension, while featuring severe interface deformations.

• The paper presents a thermodynamically consistent GPR model that integrates a sharp interface approach to simulate phase transitions.
• It introduces two-phase Riemann solvers that capture interfacial mass and heat fluxes under non-equilibrium conditions.
• The study validates a hyperbolic thermal relaxation method across complex flow scenarios, improving computational efficiency in multi-phase simulations.

Numerical Simulation of Phase Transition with the Hyperbolic Godunov-Peshkov-Romenski Model

The paper presented explores the application of the Godunov-Peshkov-Romenski (GPR) model to simulate phase transition phenomena on a macroscopic scale. It presents an approach that leverages the first-order hyperbolic nature of the GPR equations to overcome challenges traditionally associated with simulating multi-phase flows, specifically when incorporating phase transitions. This work is motivated by the complex interaction between molecular dynamics at interfaces and continuum mechanics within bulk phases, particularly in the presence of non-equilibrium thermodynamic conditions.

Key Methodological Contributions

1. Integration of a Thermodynamically Consistent Solution: The paper provides a thermodynamically consistent methodology to solve the Riemann problem at the interface using the GPR model. Interfacial phenomena, significantly influenced by microscopic interactions, are addressed using a sharp interface model with level-set techniques, enabling the simulation of phase transition effectively in complex scenarios.
2. Two-Phase Riemann Solvers: Two novel solvers, the $\text{HLLP}_{mq}$ and $\text{HLLP}_{m}$, are developed to handle the multi-scale challenges associated with phase transitions. These solvers iteratively solve for interfacial mass and heat fluxes derived from phenomenological relations of non-equilibrium thermodynamics, thereby handling discontinuities at the phase interfaces.
3. Hyperbolic Thermal Relaxation: The GPR model benefits from hyperbolic formulations of thermal processes, uniquely incorporating algebraic source terms to account for irreversible effects like heat conduction. This allows for effective coupling at interfaces without the parabolic time step limitations seen in traditional models like Euler-Fourier systems.

Numerical Validation and Comparative Analysis

The paper validates its methodology through a series of computational experiments:

• Rayleigh-Bénard Convection:

The thermal relaxation formulation was compared against Euler-Fourier systems, demonstrating competitive accuracy in simulating heat-driven flows. The GPR model showed superior computational efficiency for high heat conductivities due to the absence of parabolic constraints.

• Evaporating Shock Tube Simulations:

The solvers are evaluated using an evaporating Lennard-Jones shifted and truncated potential (LJTS) fluid shock tube, with results aligning closely to molecular dynamics data. This validation underscores the solver's capability to accurately reproduce both density and temperature profiles, reflecting the predictive capacity of interfacial phenomena.

• Complex Flow Problems:

High fidelity simulations of evaporating n-Dodecane droplets interacting with shock waves were used to further illustrate the robustness of the solvers in complex, multi-dimensional flow situations. Despite discrepancies primarily due to resolution demands, the approach demonstrated feasibility in handling severe phase deformation and surface tension effects.

Implications and Future Prospects

The coupling of the GPR model with a thermodynamic closure offers a substantial advancement for simulating highly dynamic, compressible multi-phase flows involving phase changes. The paper posits that this framework can be expanded to include multi-component flows, potentially leading to validation against experimental observations and more extensive computational studies. Future research could involve enhancements in resolving challenging thermal impulse distributions and further optimizing computational efficiency.

Despite its complexity, the GPR model's distinct advantage lies in its ability to handle significant thermodynamic variances and its resistance to parabolic time step constraints, marking a significant progression in phase transition simulation methodologies. The work represents an important bridge between fine molecular dynamics scales and larger continuum models, contributing to the depth of tools available to researchers in the fields of computational fluid dynamics and thermodynamics.