Benchmark numerical simulations of segmented two-phase flows in microchannels using the Volume of Fluid method
Introduction
This paper presents benchmark simulations for the analysis of segmented flows. Such flows are ubiquitous in so-called “digital microfluidics”, where streams of discrete droplets and bubbles at small capillary number are convected through microchannel networks [1], [2]. Numerical simulations, here as always, will have to resolve the most important flow features, which in this field are (1) resolving the thin film that separates the bubbles and droplets from the confining walls, relevant for most transport processes [3] and (2) the breakup of liquid threads and the associated topological changes that occur in such networks [4], [5].
One of the key questions in segmented flow simulations is how to model the dynamic interface between two immiscible fluids. Numerical techniques for fluid interfaces [6], [7] can be divided into two categories: Lagrangian and Eulerian. Lagrangian methods such as moving-mesh [8], [9], [10], [11] or front-tracking [12], [13], [14] accurately resolve the shape of the interface and are for instance ideally suited to capture the thin lubricating film around steadily moving bubbles or droplets (Fig. 1a). It is, however, complicated to apply Lagrangian methods to problems with large interface movement and topological changes. Such problems are typically modeled by Eulerian methods, which naturally handle complex topological changes. Examples of Eulerian methods include the diffuse interface method [15], [16], the Level Set (LS) method [17], [18], the Volume of Fluid (VOF) method [19], [20], [21], and the Lattice-Boltzmann method [22]. Its robustness and ease of implementation and parallelization together with the ability to conserve mass and render reasonably sharp interfaces explains why VOF is implemented in many well-known commerical and open source CFD packages, such as Fluent [23], CFX [24], CFD-ACE+ [25] and OpenFOAM [26].
Despite the popularity of the VOF method, there are issues when applying this method to surface-tension-dominated flows in microchannels. VOF does not resolve the interface location with sub-grid resolution. As a consequence, the thin lubricating films can only be resolved at significant numerical cost, and the fine details close to the exact moment of breakup and coalescence cannot be resolved directly. A second issue of the VOF method is the presence of parasitic currents, which originate from errors in calculating the curvature of the interface and from an imbalance between the discrete surface tension force and the pressure-gradient terms [6], [27], [28], [29]. These errors propagate dramatically into the velocity field at small capillary numbers. Parasitic currents can be reduced by using a different, additional field variable (e.g. a level set function [30] or height function [31], [32]) used only to calculate the curvature. Allowing this function to be smooth one can accurately calculate curvature, but only at the expense of significant numerical cost and difficult parallelization. A less complicated approach to reduce the parasitic currents is applying a smoother to the VOF function in the interfacial region [33], [34], [35]. Smoothing leads to a less steep gradient of the VOF function, and hence improves the accuracy in the calculation of the curvature. As a consequence, parasitic currents decrease significantly without much increase in computational time.
In this study, we present well-characterized benchmark cases that allowed us to find optimal approaches to using the VOF method for segmented flows. The three benchmark cases are shown in Fig. 1: the steady motion of 2D bubbles in a straight channel, the formation of bubbles in 3D T-junctions and the breakup of droplets in 2D and 3D T-junctions. All these cases have been extensively studied theoretically and experimentally, so in all cases we can compare to known expected values. The paper is organized as follows. A short description of the VOF method as implemented in the interFoam solver in OpenFOAM-1.6 [36], including details of the sharpening and smoothing method, is followed by a standard stationary bubble test and a simple 2D breakup test to find the optimal parameters for the method. We then use these optimal parameters in the benchmark cases. In each of these cases, we compare to time-resolved experimental data of the fluid interface, and provide a detailed quantification of small remaining errors in the calculations. As we show below, our numerical simulations show a good agreement with experimental data and experimentally-validated analytical models.
Section snippets
Governing equations
In the VOF method, the transport equation for the VOF function, α, of each phase is solved simultaneously with a single set of continuity and Navier–Stokes equations for the whole flow field. Considering the two fluids as Newtonian, incompressible, and immiscible, the governing equations can be written as:where U is the fluid velocity, p the pressure, f the gravitational force, and volumetric representation of the surface tension
Interface sharpening
To evaluate the influence of the interface sharpening coefficient , we simulated the relaxation of a 2D, stationary, circular droplet from a square initial shape, in the absence of the gravity. The fluid properties are similar to those considered by Brackbill et al. [37]: background density , viscosity μ = 1 mPa s, density ratio , viscosity ratio and surface tension . Differently from the test case in Brackbill et al. [37], the diameter of the relaxed droplet was
Motion of bubbles in a straight two-dimensional channel
The first benchmark case we consider is the steady motion of a non-wetting air bubble through a straight two-dimensional channel. The thickness of the lubricating film, b, separating the bubble from the channel walls is typically two orders of magnitude smaller than the channel width; hence, capturing this film and resolving the flow inside it is a computationally demanding task.
We simulated the motion of a droplet of length l0 = 200 μm in a fixed straight 2D microchannel of width w = 100 μm and
Conclusions
We have shown that the movement, formation and breakup of confined bubbles and droplets in microfluidic systems can be predicted in very good agreement with experimental data and theory using VOF simulations. The test cases we used, all fully documented and accompanied by extensive experimental validation, form a rigorous set of benchmarks for the ability of a numerical fluid simulation to handle large interfacial tension, topological changes and large separation of characteristic length and
Acknowledgement
The authors gratefully acknowledge the financial support from STW and IROP-OSPT, The Netherlands.
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