Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

TL;DR

This paper demonstrates that the 1D two-fluid model for stratified incompressible flow conserves energy and introduces a finite volume scheme that preserves this property discretely, applicable to various duct geometries.

Contribution

It extends energy conservation analysis to the 1D two-fluid model with non-conservative pressure terms for arbitrary duct shapes and develops a compatible finite volume discretization.

Findings

The continuous 1D two-fluid model satisfies an energy conservation equation.

The proposed finite volume scheme conserves discrete energy in simulations.

Numerical results confirm energy conservation in the discrete scheme.

Abstract

We show that the one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the model. This result extends upon earlier work on the shallow water equations (SWE), with the important difference that we include non-conservative pressure terms in the analysis, and that we propose a formulation that holds for ducts with an arbitrary cross-sectional shape, with the 2D channel and circular pipe geometries as special cases. The second novel result of this work is the formulation of a finite volume scheme for the TFM that satisfies a discrete form of the continuous energy equation. This discretization is derived in a manner that runs parallel to the continuous analysis. Due to the non-conservative pressure terms it is essential to employ a staggered grid, which requires careful consideration in defining the discrete energy and energy fluxes, and the relations between them and the discrete model. Numerical simulations confirm that the discrete energy is conserved.