[Paper Review] Linearly decoupled energy-stable numerical methods for multi-component two-phase compressible flow

Jisheng Kou, Shuyu Sun+1 more|arXiv (Cornell University)|Dec 6, 2017

Computational Fluid Dynamics and Aerodynamics2 references3 citations

TL;DR

This paper proposes two linear, decoupled, energy-stable numerical schemes for multi-component two-phase compressible flows using a realistic equation of state (e.g., Peng-Robinson). By introducing intermediate velocities and a component-wise scalar auxiliary variable (SAV) approach, the method decouples velocity-molar density coupling and solves only linear systems per time step, ensuring unconditional energy dissipation and enabling efficient, stable simulations of complex fluid systems.

ABSTRACT

In this paper, for the first time we propose two linear, decoupled, energy-stable numerical schemes for multi-component two-phase compressible flow with a realistic equation of state (e.g. Peng-Robinson equation of state). The methods are constructed based on the scalar auxiliary variable (SAV) approaches for Helmholtz free energy and the intermediate velocities that are designed to decouple the tight relationship between velocity and molar densities. The intermediate velocities are also involved in the discrete momentum equation to ensure the consistency with the mass balance equations. Moreover, we propose a component-wise SAV approach for a multi-component fluid, which requires solving a sequence of linear, separate mass balance equations. We prove that the methods preserve the unconditional energy-dissipation feature. Numerical results are presented to verify the effectiveness of the proposed methods.

Motivation & Objective

• To develop efficient numerical methods for multi-component two-phase compressible flows with a realistic equation of state, such as Peng-Robinson, which are thermodynamically consistent and avoid pressure equation construction.
• To overcome the strong coupling between velocity and molar densities in mass and momentum balance equations, which complicates numerical schemes.
• To construct linear, decoupled schemes that preserve the discrete energy-dissipation law unconditionally, ensuring numerical stability.
• To introduce a component-wise SAV approach that simplifies computation for multi-component fluids while maintaining energy stability.
• To ensure consistency between the momentum equation and mass balance equations by incorporating intermediate velocities in the discrete formulation.

Proposed method

• Utilizes the scalar auxiliary variable (SAV) approach to handle the nonlinear Helmholtz free energy density, transforming the energy-stable scheme into a linear system.
• Introduces two intermediate velocities to decouple the tight coupling between velocity and molar densities in the convection terms of mass balance equations.
• Designs a discrete momentum equation that incorporates intermediate velocities, ensuring consistency with the mass balance equations and preserving kinetic energy variation.
• Proposes a component-wise SAV formulation that allows solving separate linear mass balance equations for each component, significantly improving computational efficiency.
• Employs the NVT-based modeling framework with primal variables of moles, volume, and temperature, enabling thermodynamically consistent modeling without a pressure equation.
• Proves unconditional energy dissipation by constructing a discrete energy functional and showing its decay over time steps.

Experimental results

Research questions

• RQ1 How can energy-stable, linear, and decoupled numerical schemes be constructed for multi-component two-phase compressible flows with a realistic equation of state?
• RQ2 What is an effective way to decouple the velocity-molar density coupling in the convection terms of mass balance equations while preserving energy stability?
• RQ3 Can intermediate velocities be designed such that they are consistent with both the mass balance equations and the momentum equation in a discrete setting?
• RQ4 How can the SAV approach be extended to multi-component systems in a component-wise manner to enhance computational efficiency?
• RQ5 What is the impact of using a component-wise SAV approach on the stability and accuracy of the numerical scheme for compressible multi-component flows?

Key findings

• The proposed schemes are unconditionally energy-dissipative, as rigorously proven via a discrete energy functional that decreases at each time step.
• The method requires solving only a sequence of linear equations at each time step, significantly improving computational efficiency compared to nonlinear coupled solvers.
• The component-wise SAV approach enables efficient treatment of multi-component fluids by decoupling the Helmholtz free energy into individual component contributions.
• Numerical results demonstrate the effectiveness of the schemes in simulating complex flow patterns, including flow quivers and velocity magnitude contours, over long-time simulations.
• The use of intermediate velocities ensures consistency between the momentum equation and mass balance equations, preserving the physical consistency of the discrete system.
• The schemes are validated on a three-component system (methane, pentane, decane) using realistic Peng-Robinson parameters, showing robustness and stability across different time steps.