A coupled level set and volume of fluid method with a re-initialisation step suitable for unstructured meshes

Highlights

• The use of a numerical scheme for re-initialising the level set function for unstructured and structured meshes is described.

• The re-initialisation step is utilised based on the neighbouring cells, without defining co-ordinate directions x, y and z.

• The distance function is updated by advecting the volume fraction.

• Coupling level set method with a new volume of fluid method, which was recently developed and implemented in OpenFOAM.

Abstract

This paper presents a coupling method of the level set and volume of fluid methods based on a simple local-gradient based re-initialisation approach that evaluates the distance function depending on the computational cell location. If a cell belongs to the interface, the signed distance is updated based on a search in the neighbouring cells and an interpolation procedure is applied depending on the local curvature or the sign of the level set function following [41]. The search algorithm does not distinguish between the upwind and downwind directions and hence it is able to be used for cells with an arbitrary number of faces increasing the robustness of the method. The coupling with the volume of fluid method is achieved by mapping the volume fraction field which is advected from the isoface evolution at a subgrid level. Consequently, the coupling with the level set approach is utilised without solving the level set equation. This coupled method provides better accuracy than the volume of fluid method alone and is capable of capturing sharp interfaces in all the classical numerical tests that are presented here.

Introduction

In implicit methods for calculating the interface between two fluids flowing in a fixed mesh, the interface is captured using a scalar field advected in space. The scalar field (marker) has to be intrinsically connected to the absence or presence of the liquid phase. These methods are easily extended into three-dimensions but might require fine meshes to resolve the interface. The same limitation holds for front-tracking methods. Here, we are interested in interfaces for multiphase flows such as bubbles, droplets, and jets for liquid/liquid and liquid/gas interactions. The most commonly cited implicit methods are the volume of fluid method (VOF [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and the level set (LS) methods [12], [13], [14], [15], [16], [17], [18], [19], [20]. The level set formulation is utilised by transporting a continuous function, as in the VOF method. The level set method has been developed for an accurate representation of complex interface and boundaries for a wide range of applications including among others the areas of shape optimisation [21], computer graphics [22], medical imaging [23], grid generation [24], seismology [25], and superconductors [26]. For fluid interfaces, and particularly in the two-phase flows considered in this paper, the interface of the fluid is defined by the zero-level of a signed distance function and the level set method provides an accurate representation of the curvature of the interface. One common characteristic of this method and VOF is that the user does not need to modify the method regardless of the complexity of the geometry since both VOF and level set adjust naturally to any topological changes. One of the main differences between the two methods, is the transition from one fluid to the other, which in the level set method occurs gradually rather than as in the volume of fluid approach where the interface exists in a one-cell layer in between the two fluids.

Despite its efficiency in calculating the interface, the level set method has the shortcoming that mass conservation is not guaranteed. This barrier can be overcome by coupling the method with the volume of fluid approach which is conservative, with the level set being highly accurate. This idea was implemented first by Bourlioux [27] and Sussman and Puckett [28] giving a new method, the coupled-level-set-volume-of-fluid (CLSVOF) approach. Use of the CLSVOF showed that advecting both the volume fraction and distance functions can conserve mass increasing the accuracy of VOF, and providing the basis for different variations of the level set method which have been used in chemical process, aerospace and automotive industries.

The coupling of these two approaches does, however pose challenges for the interface reconstruction and the re-initialisation procedures that have to be addressed to successfully simulate fluid flows in the case of three-dimensions or non-orthogonal meshes. In [29] a piecewise-linear interface construction/calculation (PLIC) method is described for advecting the interface, with the level set function used to calculate the curvature. The volume of fluid in a computational cell defines a plane, which is constructed by the intersection points with the cell. The signed distance function is taken as the minimum distance from a finite volume centre to an interface-plane that is defined by a stencil of cells. This VOF-PLIC approach was developed for unstructured meshes in both two and three-dimensions. A similar approach was employed in [30] where the LS-VOF coupling evaluated the level set function from the minimum distance from an arbitrary cell centroid to the zero-level. In addition, no special re-initialisation process was employed, following the geometric operation proposed in [31] (the so-called coupled volume of fluid and level set, a.k.a. VOSET, method) to calculate the level set function near the interface. The VOSET method can be applied to accurately compute the curvature and smooth discontinuous physical quantities near the interface for both structured and unstructured meshes. A different LS-VOF coupling suitable for overlapping and moving structured grids was proposed in [32] using a PLIC method for the advection of the volume of fluid approach. The interface was advected using a hybrid split, Eulerian implicit-Lagrangian explicit interface advection scheme which provided good results for the classical test of a deforming three-dimensional sphere. In [33] the idea of flux polygon reconstruction using vertex velocities was employed to evaluate the VOF function. The computed volume fraction was then corrected by a flux corrector estimated using the face velocities. The level set function was advected by a high order total variation diminishing (TVD) scheme and then re-initialised in a narrow band around the interface with a geometric procedure. In [34] the idea of the area of fluid was employed for advecting the volume fraction developing an iterative clipping and capping algorithm for the coupling of the level set and volume of fluid methods. Both the LS and VOF functions are advected by solving a transport equation for each one of them: the volume of fluid is advected employing an interface compression scheme whereas the LS function uses a van Leer TVD scheme. Despite its efficiency in calculating the interface, the LS method has the shortcoming that mass conservation is not guaranteed. This barrier can be overcome by coupling the method with VOF approach which is conservative, and the LS which is highly accurate (see [27] and [28]). In [35] a conservative LS method was developed, which has been demonstrated to conserve mass. This has been the basis of different variations of the LS method which have been used in multiphase flows [34], [36], [37], [38]. Coupling the volume of fluid with level set it is possible to combine the benefits of both methods providing an improvement in capturing of the sharp interface with a reasonable accuracy for mass conservation. The ultimate purpose of the correct advection of the level set is the accurate calculation of curvature and mixture properties, in line with the one-fluid approach.

This paper presents a novel coupled LS and VOF method capable of simulating the interface of two fluids, of different properties. The first part of the method is the re-initialisation step of the signed distance function. All traditional level set methods face the problem of finding the proper values of the signed distance function, ψ, which satisfy the Eikonal equation, $|\mathrm{\nabla }\psi |=1$. This is usually done by solving the level set equation with a high order approach in time and space to minimise the error, and re-initialising the distance function to avoid the displacement of its initial value ${\psi }_0$ [39]. In this paper a partial differential equation re-initialisation method is presented based on the works of Russo and Smereka [40] and Hartmann et al. [41] which allows the simple and efficient calculation of the distance function across the interface. The presented formulation is second-order in space and constructed for computational cells of arbitrary shape, and is tested for both structured and unstructured meshes. The initial value of the distance function ${\psi }_0$ in the coupled volume of fluid and level set methods is derived by advecting the volume fraction with either an algebraic or a geometric method. The VOF method for the research presented here, considers the motion of an isoface in a computational cell and advecting it, using the isoAdvector method proposed in [42] and implemented in the open source CFD code OpenFOAM [43]. The isoface is properly advected within a time step, estimating the volume transport across a face before moving on to the next time-step solution. The complete volume fraction advection algorithm is described in detail in the following sections. The coupling of the LS and VOF methods is developed here within OpenFOAM and is done without the need to solve the LS function equation. The approach maps the volume fraction to ${\psi }_0$ directly from the VOF step, and then corrects the signed distance function. In order to preserve its distance function character, the level set function is re-initialised in two parts. First, the distance function is calculated for the cells at the interface and is mapped to the level set function. In the second part the re-initialisation equation is solved for the cells adjacent to the cells at the interface [39], [41]. Comparisons of the VOF and the coupled LS-VOF for classical numerical tests reveal that the LS step improves the accuracy of solution and boosts the ability of the method to capture sharp interfaces.