Energy-stable discretization of the one-dimensional two-fluid model

• DOI: 10.1016/j.ijmultiphaseflow.2024.104756
• Publication Date: 2024-02-07T16:35:16Z
• Authors: J.F.H. Buist; B. Sanderse; S. Dubinkina; C.W. Oosterlee; R.A.W.M. Henkes
• Keywords: Fluid Flow and Transfer Processes; Surface tension; Mechanical Engineering; Fluid Dynamics (physics.flu-dyn); General Physics and Astronomy; FOS: Physical sciences; 76T06 (Primary) 65M08, 65M12 (Secondary); Physics - Fluid Dynamics; 02 engineering and technology; Numerical Analysis (math.NA); Energy conservation; 7. Clean energy; Energy-stable scheme; 532; Dissipation; FOS: Mathematics; Two-phase pipe flow; Mathematics - Numerical Analysis; 0204 chemical engineering; Stability

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.