• Researcher: Niall Bootland
  • Academic Supervisor: Andy Wathen
  • Industrial Supervisor: Chris Kees

Background

Two-phase flows arise in many coastal and hydraulic engineering applications such as the study of coastal waves, dam-break scenarios, and the design of channels and coastal structures. Here, air and water are the two phases in question and our focus has been on such hydrological models, but there are a wide variety of other scenarios where two-phase flows are of interest, such as in the oil industry or in food processing and manufacturing. However, modelling two-phase incompressible flow with a level set or volume-of-fluid formulation results in a variable coefficient Navier–Stokes system that is challenging to solve computationally. The dominating cost at each time step of a simulation is in solving the linear systems that arise. The governing system of equations is nonlinear and thus must be solved through a sequence of linear systems, further we must solve such a sequence at each time step. Thus, the efficient solution of these linear systems is crucial for practical computation.

The linear systems we must solve are large and sparse (very few entries per row). As such, iterative methods are necessary since direct methods become prohibitively slow and require large amounts of memory at such scales. It is well known that the performance of iterative methods depends highly on the properties of the linear system to be solved and as such good preconditioners are essential to gain fast and scalable solvers. A preconditioner can be thought of as a transformation of the linear system to one which is more amenable to the iterative solution method, while still allowing the solution of the original system to be easily constructed. Typically, one would like to exploit structure present in the linear systems when designing a preconditioner. In our case, the systems have a block structure (see Figure 1), so we consider block preconditioners.

Figure 1: Pictorial illustration showing the structure of the non-zero entries in an example linear system.

This project aims to provide methods which can contribute towards scalable solvers within simulations of incompressible two-phase flows. A key part lies in the choice of preconditioner when solving the linear systems. Our focus is on the development of novel block preconditioners applicable to two-phase flow.

Progress

The last 20 years has seen the introduction of several good preconditioners for systems describing Navier–Stokes flow of a single fluid. These include the block preconditioners known as PCD and LSC which are based on the idea of approximate commutators (see Elman, Silvester, and Wathen, Finite Elements and Fast Iterative Solvers: with applications in Incompressible Fluid Dynamics, Oxford University Press, 2014). However, these techniques fail to give comparable performance in their given form when applied to variable coefficient Navier–Stokes systems. Our work has provided new two-phase variants of the PCD and LSC preconditioners which incorporate the variable coefficient nature of two-phase flow [1]. These methods are shown to give comparable performance on a model problem to the original methods for a single phase and extend their utility to solving Navier–Stokes equations describing two-phase flow. We also provide results for a dynamic dam-break problem (see Figure 2). Although simpler methods often perform well for highly dynamic problems, our new two-phase PCD preconditioner provides greater robustness by better capturing information about the advective term (which provides the non-symmetric part of the equations), especially when larger time steps are permitted. 

Figure 2: Evolution of a dam-break simulation at selected time points. Blue represents water with red being the air.

In real simulations of interest to engineers in the field, boundary conditions also play a crucial role and must be taken into account within the preconditioner. We are working towards a greater understanding of the issues that arise and implementation of the necessary modifications. 

Future Work

While our work has identified promising candidate preconditioners for two-phase flow, especially in quasi-steady simulations, the best choice of preconditioner to apply may change over the course of a simulation. One would ideally like the most efficient solver at each point but ensuring robustness throughout is also important, as such one can imagine situations where multiple preconditioners might be used during the simulation. Further adaptivity may also be beneficial in the way we update the preconditioner as, once built, a given preconditioner may provide sufficient performance for several subsequent instances of the linear systems to be solved, reducing the overall work required. Such considerations have the potential to improve the overall scalability of the solution process.

Publications

[1] N. Bootland, A. Bentley, C. Kees, and A. Wathen, Preconditioners for Two-Phase Incompressible Navier–Stokes Flow. Submitted (2017).

 

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