VOF evaluation of the surface tension by using variational representation and Galerkin interpolation projection

doi:10.1016/j.jcp.2019.06.036

Journal of Computational Physics

Volume 395, 15 October 2019, Pages 537-562

https://doi.org/10.1016/j.jcp.2019.06.036 Get rights and content

Highlights

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VOF simulation with variational formulation and Galerkin interpolation projection.
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Surface tension term computed via CSS method with no direct differentiation.
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Surface tension term computed via CSF method by using an extended curvature.
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Tests for two and three-dimensional geometries in the static and dynamical cases.

Abstract

In this work we propose a variational approach with cell-to-point Galerkin projections for studying two-phase interface advection problems dominated by surface tension. A Volume Of Fluid (VOF) algorithm is used for tracking and locating the evolution of the two-phase interface on a Cartesian grid and a finite element numerical scheme for solving the velocity-pressure state. The velocity field that drives the evolution of this interface is computed from the weak form of the Navier-Stokes equation where the surface tension force is represented in variational form by the continuous surface force (CSF) and continuous surface stress (CSS) methods. Standard numerical approaches solve the strong form of the Navier-Stokes equations and define the CSS term by taking the divergence of the surface tension tensor. This computation of the divergence term results in a singular force which is difficult to compute when the grid is refined since the tensor is computed in a discontinuous cell-by-cell way. In this work we use the variational formulation of the Navier-Stokes equation and avoid differentiation. The tensor, which is a function of the unit normal, is evaluated over regular Sobolev spaces by using a cell-to-point Galerkin projection. This allows a regular piece-wise continuous representation of the surface tensor and the unit normal based on the VOF reconstruction. In standard approaches the CSF surface force is computed by using the curvature, which is the divergence of the unit normal. In this paper we recover the curvature with point-wise Galerkin projection avoiding direct differentiation. Tests on convergence for two and three-dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces. This method is also natural for coupling non uniform grid computation of the fluid with Cartesian grid of the VOF algorithm.

Introduction

Two-phase flows with surface tension are of great interest in many engineering applications ranging from medicine, environmental sciences to the oil and nuclear industries. They play a central role in the dynamics of droplets and bubbles since the shape and their interactions are dominated by surface tension forces that become significant when physical phenomena are analyzed at small scales. In recent years this interest has inspired a great number of algorithms with different representation of the interface such as Volume-Of-Fluid (VOF), Level-set (LS) and markers (M), for example see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and citations therein.

All these methods provide solutions to the problem of interface advection but an accurate and robust representation of surface forces remains a challenging problem with fixed grid representation [3], [10], [12], [13]. In [3] surface tension, capillarity and wetting effects are discussed in terms of the virtual–work principle and shape sensitivity, starting from first principles and reaching a variational formulation that is adequate for numerical solutions. In [13] the authors show that the numerical time-step applied to solve the equations governing this problem is directly associated with the stability of the flow simulations. The explicit implementation of the surface tension leads to restrictions on the temporal resolution caused by capillary waves. Surface tension computations have also been investigated with alternative methods such as arbitrary Lagrangian Eulerian (ALE) [14], purely Lagrangian [15], [16] and embedded Eulerian-Lagrangian approach [17].

As a matter of fact, the appearance of the so-called spurious currents is a feature of the fix grid numerical simulation of a single droplet in equilibrium in another phase with no volume forces. These currents result from unbalanced capillary forces due to surface tension and the related pressure jump across interfaces. This leads to velocity oscillations at the interface and to numerical instabilities and artificial interface breakups. For details one can see different implementation in [10], [18], [19], [20], [21], [22], [23], [24], [25] and citation therein. An attempt to eliminate the spurious currents can be found in several papers, see for example [26], [10], [11]. In [11], a pressure term is added in the projection pressure operator to balance the spurious contribution of the surface tension force. In [26], an artificial viscosity model to mitigate the impact of numerical spurious currents at fluid interfaces is introduced by applying an artificial shear stress term tangential to the fluid interface.

The main problem in the computation of surface tension is the curvature. The curvature is well defined only along the interface when this is twice differentiable. However, for algorithms based on VOF, this regularity is not available since the computation is based on the interface which is reconstructed from the discontinuous color function. In order to overcome the regularity problem continuous surface methods may be introduced. For this we refer to [1] and [2] where the authors proposed the continuous surface force (CSF) with explicit curvature computation and the continuous surface stress (CSS) with implicit curvature computation, respectively. An extensive review of these continuous surface methods can be found in [12]. In this paper we propose CSS and CSF algorithms that use a new approach based on extension functions which are computed by cell-to-point Galerkin projections. In [1], [2], [12] and similar works, the problem has been solved by using a continuous surface approach with the introduction of a mollified function depending on a small parameter ϵ. However, in the limit as ϵ tends to zero, the computed differential quantities tend to be unbounded. For these reasons we propose to use a variational framework where weak derivatives over the indicator color function can be managed with integration by parts.

The variational formulation of the Navier-Stokes equations allows the treatment of the capillary force in its original continuous tensor form. The variational formulation is proposed also in many works. In particular in [3] a variational formulation, which is adequate for numerical solutions, is proposed. However the variational formulation in [3] needs thin interfaces with high regularity to be consistent and therefore it is suitable only for LS reconstruction. Variational formulations of the surface tension, proposed in literature from the differentiation of the identity operator on the interface, is clearly very difficult with discontinuous VOF interface representations. In this paper, the introduction of the global cell-to-point Galerkin projection not only allows to extend suitable quantities for continuous surface algorithms from the interface to the volume but also a LS reconstruction from VOF color function.

We propose two variational forms of the tension force: one based on the surface tension tensor (CSS) and another based on curvature (CSF). In the former case, by using a variational approach, we avoid the derivative of the surface tension tensor, proposed for example in [2], which is unbounded in its standard classic representation. The surface tension tensor is computed by extending the unit normal and projecting the indicator phase into a more regular functional space by a cell-to-point Galerkin projection with the purpose of reproducing these forces more regularly on fixed grids. Furthermore the proposed approach does not need an explicit computation of the curvature, which is very challenging especially in three-dimensional cases. In the latter case, the curvature is computed with a surface tension form (CSF). Again, by using a variational point-to-cell Galerkin projection, we avoid the curvature calculation through direct differentiation of the unit normal. In this implementation, with given constant curvature and correct discrete space representation, we may have zero spurious currents.

Another advantage of the point-to-cell Galerkin projection algorithm is its simple implementation in both two and three-dimensional domains. It is important also to remark that the Galerkin projection from the surface interface to the volume can be used for problems with unstructured or moving grids [27], [28].

Section snippets

Variational formulation of the two-phase Navier-Stokes equations

In this paper we consider two-phase flows over two and three-dimensional open bounded domains Ω with boundary Γ. Let the reference phase 1 be contained in $Ω_{f}$ and $χ (x, t)$ be the indicator function for the reference phase defined in such a way that is one for all $x \in Ω_{f}$ and zero on $Ω - Ω_{f}$ . Over the interval of time $[0, T]$ the indicator function χ behaves like a passive scalar and satisfies the following advection equation $\frac{\partial χ}{\partial t} + u \cdot \nabla χ = 0 in Ω \times [0, T],$ where the velocity field u obeys to the incompressibility

Finite element and VOF spatial discretization

Let $Ω_{h}$ be an open bounded domain with boundary $Γ_{h}$ . The reference phase is contained in the region $Ω_{f h} \subset Ω_{h}$ whose boundary $Γ_{f h}$ consists of a chain of connected plains or segments. Let $X^{h} \subset H^{1} (Ω_{h})$ and $S^{h} \subset L^{2} (Ω_{h})$ be two families of finite dimensional sub-spaces parameterized by h that tends to zero. Also we denote by $S_{0}^{h}$ the family of finite dimensional sub-spaces that contains piece-wise constant functions. We make the usual approximation assumptions on $X^{h}$ and $S^{h}$ including the LBB condition (see,

Numerical tests

In this Section the obtained results are discussed. In particular five test cases are shown in order to analyze the different approaches capabilities. With Test 1 steady solutions for two and three dimensional droplets are reported using both CSF and CSS methods. It is shown that, with a given exact curvature value, spurious currents vanish for each considered mesh resolution. The CSS variational implementation is considered in Test 2 and Test 3, where a two-dimensional droplet translation and

Conclusions

Two variational numerical approaches for computing the surface tension term for two-phase incompressible flows in two and three-dimensional geometries have been proposed: one based on surface tensor and another based on curvature. The first algorithm computes the surface tension tensor directly from the unit normal vector and a Galerkin cell-to-point projection of the indicator function. In the framework of this Navier-Stokes variational formulation one can model the surface tension force