Mixed element FEM level set method for numerical simulation of immiscible fluids

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Abstract

A new realization of a finite element level set method for simulation of immiscible fluid flows is introduced and validated on numerical benchmarks. The new method involves a mixed discretization of the dependent variables, discretizing the flow variables with non-conforming Rannacher–Turek finite elements while using a simple first order conforming discretization of the level set field. A three step segregated solution approach is employed, first a discrete projection method is used to decouple and compute the velocity and pressure separately, after which the level set field can be computed independently.

The developed method is tested and validated on a static bubble test case and on a numerical rising bubble test case for which a very accurate benchmark solution has been established. The new approach is also compared against two commercial simulation codes, Ansys Fluent and Comsol Multiphysics, which shows that the developed method is a magnitude or more accurate and at the same time significantly faster than state of the art commercial codes.

Introduction

Modern computer hardware together with improved numerical algorithms have made computer simulations a commonly used tool by engineers today. Simulations of industrially relevant flows can now seemingly be made both quickly and easily. However, although problems involving complex physical phenomena or large geometries can be simulated, there is still room for improvement with regard to accuracy and computational efficiency. Typical applications involving two-phase flows and free interfaces, which is the main concern of this paper, can for example be to understand droplet generation in inkjet printing and spray painting processes, to study wave generation and force impact on ships and offshore structures, and design microfluidic capillary lab-on-chip devices for medical analysis.

Numerical simulation of immiscible fluid flows has come a long way since the early Marker-and-Cell method of Harlow and Welch [5] which eventually evolved into the Volume of Fluid (VOF) method by Hirt and Nichols [6]. In these approaches a scalar function for the fluid volume fraction is tracked throughout the simulation from which the interfaces are reconstructed. The level set method by Osher and Sethian [19] was alternatively designed to implicitly track interfaces by representing and embedding them as iso-contour levels of a higher dimensional function. In these Eulerian immersed interface approaches the free boundaries are allowed to move arbitrarily through a fixed computational grid and do not have to be resolved sharply. This simplifies the implementation and can lead to significant performance gains in contrast to moving mesh methods where the interfaces always are aligned with the edges of the grid cells. An approach to combine the advantages of a sharp Lagrangian interface representation with an Eulerian representation of the other flow variables is the Front Tracking method by Unverdi and Tryggvason [26]. All of these methods have their respective strengths and weaknesses and much work has been done to improve and combine methods in an effort to yield more accurate and faster algorithms.

With all this in mind a new level set methodology is introduced for simulation of immiscible fluid flows which essentially consists of combining a non-conforming finite element flow solver with a conforming level set interface tracking method. This technique, although somewhat unconventional, has resulted in a simulation code which has proven to be both significantly faster and at the same time more accurate than two major commercial computational fluid dynamics (CFD) and simulation software tools.

This paper first describes the numerical algorithms and methods necessary to realize the new simulation approach which then is applied to both standard test problems and also benchmarked against the commercial simulation codes Ansys Fluent and Comsol Multiphysics. The following section first presents an efficient method to discretize and realize a solver for the fluid flow, and also discusses how to efficiently incorporate surface tension effects. Section 3 focuses on interface tracking, deriving the level set method, discussing its discretization in space and time, level set reinitialization, and computation of normals and curvature. The flow and interface tracking algorithms are combined in Section 4 to establish a complete solution approach. Section 5 presents results from testing and validation of the developed code on both a static bubble test case, as well as a rising bubble benchmark problem for which very accurate reference solutions have previously been established. Lastly, a comparison with current commercial codes is presented in Section 6 together with a summary and conclusions.

Section snippets

Flow solver

This section discusses discretization and solution techniques for the Navier–Stokes equations which are the mathematical model equations describing flow of incompressible fluids. The task is to solve the following saddle point systemρ(x)ut+(u·)u=-p+·μ(x)u+uT+ρ(x)g,·u=0for the unknown velocity, u, and pressure, p, in a given d-dimensional domain ΩRd. For immiscible fluid flows, where one has a mixture of two or more fluids, the density, ρ, and viscosity, μ, fields will be discontinuous

Level set solver

The level set method introduced by Osher and Sethian [19] has proved to be applicable in many diverse fields such as image processing, crystal growth, inverse problems, and multiphase flow [18]. The main idea of the level set method is to embed an interface Γ(t) (represented by a curve in two dimensions or surface in three dimensions) in a higher dimensional function ϕ, that isΓ(t)=xRd|ϕ(x,t)=vls,where vls is the contour level or isosurface value implicitly representing the interface. The

Solution approach

The two previous sections have discussed the different components needed to numerically simulate two-phase flows. The interface is always implicitly coupled to the flow variables through the density, viscosity, and surface tension. The flow variables are in return coupled to the interface through the velocity field. In the solution process the order and way in which these variables are solved for needs to be considered carefully. The following section will cover the construction of an

Numerical validation

This section describes the testing undertaken to validate the simulation code implemented according to the described methodology. The modular design makes it easier to debug and validate the code by testing the solvers independently before applying them to coupled and more complex problems. The flow solver has previously been rigorously validated on several incompressible single phase flow problems [25], for example the DFG flow around cylinder benchmarks for which very accurate reference

Summary and conclusions

This paper has presented a novel mixed finite element level set method for simulation of flows with immersed interfaces. The new method combines an efficient finite element flow solver based on the non-conforming Rannacher–Turek Q1Q0 elements with a conforming Q1 level set solver to track the interfaces. The two solvers are coupled in a segregated and modular way sequentially solving for each of the dependent variables. This modular approach is computationally efficient and also allows each

Acknowledgements

The author thanks the German Research foundation (DFG) and the European Union for partially supporting the work under Grants Paketantrag PAK178 (Tu102/27-1, Ku1530/5-1), Sonderforschungsbereich SFB708, and EU STF2/41+CRIS No: 235-833.

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