[Paper Review] A Volume-of-Fluid method for variable-density, two-phase flows at supercritical pressure

Jordi Poblador-Ibanez, William A. Sirignano|arXiv (Cornell University)|Mar 2, 2021

Fluid Dynamics and Heat Transfer113 references26 citations

TL;DR

This paper presents a novel Volume-of-Fluid (VOF) method for simulating variable-density, two-phase flows at supercritical pressure with phase change, using a split VOF approach that ensures machine-precision mass conservation and handles non-divergence-free liquid velocity and interfacial mass transfer. The method achieves accurate resolution of local thermodynamic equilibrium (LTE) and jump conditions at the interface, enabling simulation of simultaneous vaporization and condensation in n-decane/oxygen systems under supercritical conditions, validated through capillary wave and jet atomization benchmarks with high-resolution interface tracking.

ABSTRACT

A two-phase, low-Mach-number flow solver is created and verified for variable-density liquid and gas with phase change. The interface is sharply captured using a split Volume-of-Fluid method generalized for a non-divergence-free liquid velocity and with mass exchange across the interface. Mass conservation to machine-error precision is achieved in the limit of incompressible liquid. This model is implemented for two-phase mixtures at supercritical pressure but subcritical temperature conditions for the liquid, as it is common in the early times of liquid hydrocarbon injection under real-engine conditions. The dissolution of the gas species into the liquid phase is enhanced, and vaporization or condensation can occur simultaneously at different interface locations. Greater numerical challenges appear compared to incompressible two-phase solvers that are successfully addressed for the first time: (a) local thermodynamic phase equilibrium (LTE) and jump conditions determine the interface solution (e.g., temperature, composition, surface-tension coefficient); (b) a real-fluid thermodynamic model is considered; and (c) phase-wise values for certain variables (e.g., velocity) are obtained via extrapolation techniques. The increased numerical cost is alleviated with a split pressure-gradient technique to solve the pressure Poisson equation (PPE) for the low-Mach-number flow. Thus, a Fast Fourier Transform (FFT) method is implemented, directly solving the continuity constraint without an iterative process. Various verification tests show the accuracy and viability of the current approach. Then, the growth of surface instabilities in a binary system composed of liquid n-decane and gaseous oxygen at supercritical pressures for n-decane is analyzed. Other features of supercritical liquid injection are also shown.

Motivation & Objective

• To develop a robust numerical framework for simulating variable-density, two-phase flows at supercritical pressure, where classical two-phase assumptions break down.
• To address the challenge of accurate interface tracking and mass conservation in flows with phase change and non-divergence-free liquid velocity.
• To incorporate local thermodynamic equilibrium (LTE) and jump conditions for temperature, composition, and surface tension at the interface.
• To enable simulation of simultaneous vaporization and condensation at different interface locations due to enhanced gas dissolution.
• To validate the method against established benchmarks for incompressible two-phase flows and extend it to supercritical conditions.

Proposed method

• A split Volume-of-Fluid (VOF) method is generalized to handle non-divergence-free liquid velocity and interfacial mass transfer, ensuring machine-precision mass conservation in the incompressible limit.
• The method employs a pressure Poisson equation (PPE) solver using a Fast Fourier Transform (FFT) technique, eliminating iterative pressure correction and reducing numerical cost.
• Local thermodynamic equilibrium (LTE) is enforced at the interface, with jump conditions applied to temperature, composition, and surface-tension coefficient.
• Real-fluid thermodynamic models are used to describe phase behavior, with phase-wise variables (e.g., velocity) reconstructed via extrapolation techniques in under-resolved regions.
• The interface is reconstructed using the Piecewise Linear Interface Calculation (PLIC) method to maintain sharpness and reduce numerical diffusion.
• A curvature-based under-resolution detection algorithm identifies thin liquid structures or gas pockets where conflicting extrapolations may occur, with special treatment applied to maintain time-step stability.

Experimental results

Research questions

• RQ1 How can a VOF method be extended to handle variable-density, two-phase flows at supercritical pressure with phase change and non-divergence-free liquid velocity?
• RQ2 Can machine-precision mass conservation be achieved in the incompressible limit when mass transfer occurs across the interface?
• RQ3 How do local thermodynamic equilibrium (LTE) and jump conditions affect the interface solution in supercritical two-phase flows?
• RQ4 What numerical challenges arise in simulating supercritical flows with simultaneous vaporization and condensation, and how can they be addressed?
• RQ5 To what extent does the proposed split VOF and FFT-based PPE solver maintain accuracy and efficiency in complex, variable-density, low-Mach-number flows?

Key findings

• The method achieves machine-precision mass conservation in the incompressible limit, even with interfacial mass transfer, validating the robustness of the split VOF formulation.
• The simulation of a standing capillary wave shows excellent agreement with the analytical solution of Prosperetti, confirming accurate surface tension and interface dynamics.
• For incompressible planar jet atomization, the VOF-based simulation reproduces the deformation pattern of a previously validated LS-based code, with sharper interface resolution and better capture of high-curvature features.
• The method successfully captures the growth of surface instabilities in a binary n-decane/oxygen system at supercritical pressure, showing early breakup due to reduced surface tension and enhanced mixing.
• The use of FFT-based PPE solving eliminates iterative pressure correction, significantly reducing computational cost while maintaining accuracy in low-Mach-number flows.
• The model demonstrates the coexistence of vaporization and condensation at different interface locations, driven by enhanced gas dissolution into the liquid phase under supercritical conditions.