Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow

• Published in: Journal of computational physics 2019-05, Vol.384, p.170-199
• Main Authors: Sanderse, B., Veldman, A.E.P.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Computational fluid dynamics; Computational physics; Computer simulation; Constraint modelling; Cylindrical tanks; Fluid flow; Gas pipelines; Incompressibility; Incompressible flow; Index-3 DAE; Liquid sloshing; Methods; Momentum; Multiphase flow; Natural gas; Poisson equation; Runge-Kutta method; Runge–Kutta; Time dependence; Time integration; Two fluid models; Two-fluid model; Volume constraint
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.02.001

New time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one-dimensional two-fluid model. The methodology is based on ‘half-explicit’ Runge–Kutta methods, being explicit for the mass and momentum equations and implicit for the volume constraint. These half-explicit methods are constraint-consistent, i.e., they satisfy the hidden constraints of the two-fluid model, namely the volumetric flow (incompressibility) constraint and the Poisson equation for the pressure. A novel analysis shows that these hidden constraints are present in the continuous, semi-discrete, and fully discrete equations. Next to constraint-consistency, the new methods are conservative: the original mass and momentum equations are solved, and the proper shock conditions are satisfied; efficient: the implicit constraint is rewritten into a pressure Poisson equation, and the time step for the explicit part is restricted by a CFL condition based on the convective wave speeds; and accurate: achieving high order temporal accuracy for all solution components (masses, velocities, and pressure). High-order accuracy is obtained by constructing a new third-order Runge–Kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time-dependent boundary conditions. Two test cases (Kelvin–Helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank) show that for time-independent boundary conditions the half-explicit formulation with a classic fourth-order Runge–Kutta method accurately integrates the two-fluid model equations in time while preserving all constraints. A third test case (ramp-up of gas production in a multiphase pipeline) shows that our new third-order method is preferred for cases featuring time-dependent boundary conditions. •A new time integration method is proposed for the one-dimensional two-fluid model.•Incompressibility constraints in the model are related to Riemann invariants and DAEs.•The new method ensures conservation, constraint consistency, and avoids order reduction.•New boundary conditions respecting both wave directions and constraints are proposed.•Liquid sloshing and gas production ramp-up can be simulated with the new method.