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

The one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes suffers from an ill-posedness issue: it is only conditionally well-posed. This results in severe linear instability for perturbations of vanishing wavelength, and non-convergence of numerical solutions. This issue is typically only examined from the perspective of linear stability analysis. In order to analyze this long-standing problem from a nonlinear perspective, we show the novel result that the TFM (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the TFM. 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 result of this work is a new 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, and this distinguishes the discretization from existing energy-conserving discretizations of the SWE. Numerical simulations confirm that the discrete energy is conserved.