A new model for two-layer liquid-gas stratified flows in pipes with general cross sections

TL;DR

This paper introduces a comprehensive mathematical model for two-layer gas-liquid stratified flows in pipes with arbitrary cross sections, incorporating hydrostatic and ideal gas laws, and analyzes its properties and numerical solutions.

Contribution

The work presents a novel coupled model for immiscible two-layer flows with general cross sections, including hyperbolic analysis and numerical validation.

Findings

Model accurately captures gas-liquid interactions in pipes.

Numerical tests demonstrate stability and convergence.

Applicable to fluids with significant density differences.

Abstract

In this work, we derive a new model for immiscible two-layer gas-liquid stratified flows in pipes with general cross sections. The bottom layer is occupied by an incompressible fluid in liquid phase with hydrodynamics based on a hydrostatic pressure, following a shallow water approximation. The top layer is occupied by a compressible gas, following an ideal gas law leading to conservation of mass, momentum and energy. The two subsystems are linked through non-conservative products, representing momentum and energy exchanges between layers. The hyperbolic properties of the resulting model are analyzed, including the derivation of entropy inequalities, and the approximations of eigenvalues of the corresponding coefficient matrix. Numerical tests are included to demonstrate the merits of the model and the numerical approximations, including well-balancedness, Riemann problems, and perturbations and convergence toward steady states at rest. Besides simulations of water and air where the density difference between layers is significant, a case where such difference is not so pronounced (like gas and liquid hydrogen) is also shown.