In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

The pressure-free two-fluid model (PFTFM) is a recent reformulation of the one-dimensional two-fluid model (TFM) for stratified incompressible flow in ducts (including pipes and channels), in which the pressure is eliminated through intricate use of the volume constraint. The disadvantage of the PFTFM was that the volumetric flow rate had to be specified as an input, even though it is an unknown quantity in case of periodic boundary conditions. In this work, we derive an expression for the volumetric flow rate that is based on the demand for energy (and momentum) conservation. This leads to PFTFM solutions that match those of the TFM, justifying the validity and necessity of the derived choice of volumetric flow rate. Furthermore, we extend an energy-conserving spatial discretization of the TFM, in the form of a finite volume scheme, to the PFTFM. We propose a discretization of the volumetric flow rate that yields discrete momentum and energy conservation. The discretization is extended with an energy-conserving discretization of the source terms related to gravity acting in the streamwise direction. Our numerical experiments confirm that the discrete energy is conserved for different problem settings, including sloshing in an inclined closed tank, and a traveling wave in a periodic domain. The PFTFM solutions and the volumetric flow rates match the TFM solutions, with reduced computation time, and with exact momentum and energy conservation.

A novel pressure-free two-fluid model formulation is proposed for the simulation of one-dimensional incompressible multiphase flow in pipelines and channels. The model is obtained by simultaneously eliminating the volume constraint and the pressure from the widely used two-fluid model (TFM). The resulting ‘pressure-free two-fluid model’ (PF-TFM) has a number of attractive features: (i) it features four evolution equations (without additional constraints) that can be solved very quickly with explicit time integration methods; (ii) it keeps the conservation properties of the original two-fluid model, and therefore the correct shock relations in case of discontinuities; (iii) its solutions satisfy the two TFM constraints exactly: the volume constraint and the volumetric flow constraint; (iv) it offers a convenient form to analytically analyse the equation system, since the constraint has been removed. A staggered-grid spatial discretization and an explicit Runge-Kutta time integration method are proposed, which keep the constraints exactly satisfied when numerically solving the PF-TFM. Furthermore, for the case of strongly imposed boundary conditions, a novel adapted Runge-Kutta formulation is proposed that keeps the volumetric flow exact in time while retaining high order accuracy. Several test cases confirm the theoretical properties and show the efficiency of the new pressure-free model.