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

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

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

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 ‘halfexplicit’ 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 prese

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed. In addition, a constrained singular value decomposition is employed which enforces global momentum conservation. The resulting `velocity-only' ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (one-time) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases. Furthermore, it is shown that explicit Runge-Kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier-Stokes equations

A discretization method for the incompressible Navier–Stokes equations conserving the secondary quantities kinetic energy and vorticity was introduced, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy