Available algorithms for the initialization of volume fractions typically utilize exact functions to model fluid interfaces, or they rely on computationally costly intersections between volume meshes. Here, a new algorithm is proposed that computes signed distances and volume fractions on unstructured meshes from arbitrarily shaped surfaces, e.g. originating from experimental data. The proposed algorithm calculates signed distances geometrically near the fluid interface, approximated as a triangle surface mesh, and propagates the inside/outside information by an approximate solution of a Laplace equation. Volume fractions are computed based on signed distances, using either geometrical intersections between cells of the unstructured mesh and a sub-set of the surface mesh that represents the interface, or using a polynomial approximation and adaptive mesh refinement. Although primarily developed for multiphase flow simulations, the proposed algorithm can potentially be used for other problems that require a phase-indicator: inside/outside information with respect to an arbitrarily shaped surface on arbitrarily unstructured meshes.

The transient start-up flow solution with slip is a useful tool to verify Computational Fluid Dynamics (CFD) simulations. However, a highly accurate, open-source black box solution does not seem to be available. Our method provides a fast, automated, and rigorously verified open-source implementation that can compute the hydrodynamic eigenmodes of a two-dimensional channel flow beyond the standard floating-point precision. This allows for a very accurate computation of the corresponding Fourier series solution. We prove that all roots are found in all special cases for the general flow problem with different slip lengths on the channel walls. The numerical results confirm analytically derived asymptotic power laws for the leading hydrodynamic eigenmode and the characteristic timescale in the limiting cases of small and large slip. The code repository including test cases is publicly available here https://git.rwth-aachen.de/fricke/start-up-flow

The transient Couette and Poiseuille channel flow solutions with different Navier slip conditions on the wall boundaries require the solution of a non-linear root-finding problem. A novel algorithm is developed that automates the computation of these roots with arbitrary precision and for general input parameters. The obtained results show a significant improvement to available coefficient values for this analytic solution. A parameter variation for the slip lengths reveals a new power law for the first series coefficient that governs the stability of the solution and is essential for slip length computations from molecular dynamics simulations. Based on the new algorithm, the time scales for the asymptotic approach to the stationary solution are quantified. A double-precision implementation and an arbitrary precision implementation provide a fast, verified, and highly accurate method to obtain benchmark quality reference solutions or a module for more general applications.

Available algorithms for the initialization of volume fractions typically utilize exact functions to model fluid interfaces, or they rely on computationally costly intersections between volume meshes. Here, a new algorithm is proposed that computes signed distances and volume fractions on unstructured meshes from arbitrarily shaped surfaces, e.g. originating from experimental data. The proposed algorithm calculates signed distances geometrically near the fluid interface, approximated as a triangle surface mesh, and propagates the inside/outside information by an approximate solution of a Laplace equation. Volume fractions are computed based on signed distances, using either geometrical intersections between cells of the unstructured mesh and a sub-set of the surface mesh that represents the interface, or using a polynomial approximation and adaptive mesh refinement. Although primarily developed for multiphase flow simulations, the proposed algorithm can potentially be used for other problems that require a phase-indicator: inside/outside information with respect to an arbitrarily shaped surface on arbitrarily unstructured meshes.

We investigate meniscus shapes and capillary rise heights in glass capillaries with rectangular cross sections (4 × 0.2 mm), which we modified with established coatings to generate a range of surfaces that interact differently with water. Meniscus positions and shapes are imaged while the capillaries are rotated horizontally about their longitudinal axis in order to generate centrifugal forces opposing the capillary driven fluid propagation, i.e. volumetric forces. Changing the rotational speed allows us to balance both forces thereby bringing capillary rise to a stop. In brief, we find very good agreement of the different meniscus shapes we observe over a wide range of centrifugal accelerations (up to 191 g) with two independent simulations of the scenario. In addition, we are able to precisely measure capillary rise heights in differently modified capillaries over a range of centrifugal accelerations and correlate these values. Lastly, we mention how this system will prove useful to investigate wetting phenomena on swellable surfaces, i.e. surfaces whose properties dynamically change upon fluid contact, by providing precise control over the propagation speed of the three phase contact line.

The breakup dynamics of a capillary bridge on a hydrophobic stripe between two hydrophilic stripes is studied experimentally and numerically using direct numerical simulations. The capillary bridge is formed from an evaporating water droplet wetting three neighboring stripes of a chemically patterned surface. By considering the breakup process in a phase space representation, the breakup dynamics can be evaluated without the uncertainty in determining the precise breakup time. The simulations are based on the Volume-of-Fluid (VOF) method implemented in Free Surface 3D (FS3D). In order to construct physically realistic initial data for the VOF simulation, Surface Evolver is employed to calculate an initial configuration consistent with experiments. Numerical instabilities at the contact line are reduced by a novel discretization of the Navier-slip boundary condition on staggered grids. The breakup of the capillary bridge cannot be characterized by a unique scaling relationship. Instead, at different stages of the breakup process different scaling exponents apply, and the structure of the bridge undergoes a qualitative change. In the final stage of breakup, the capillary bridge forms a liquid thread that breaks up consistently with the Rayleigh-Plateau instability. (C) 2021 The Author(s). Published by Elsevier Ltd.

The occurrence of extremely thin liquid-sided concentration boundary layers at bubble or droplet interfaces for realistic, i.e., high Schmidt numbers is a severe obstacle for the numerical simulation of mass transfer processes in gas-liquid systems. This contribution provides a survey of different approaches to overcome this problem, with the main emphasis put on the approach introduced and further developed by the authors. This approach employs a nonlinear flux computation and is based on the modeling of subgrid-scale concentration profiles. Based on the latest developments, recommendations for future research are also provided.

Starting from the full continuum mechanical description of free surface flows, a model for the rise of a liquid in a capillary is derived. The derivation improves on several aspects: Firstly, the influence of a slip boundary condition on the capillary wall is added to the model. Secondly, the stationary rise height is corrected. Thirdly, a regularization is suggested that uses the mass in interface vicinity. Fourthly, the influence of the velocity field close to the contact line is added. Finally, a convection contribution arises as the analysis considers the capillary without a reservoir. To validate the model, an Arbitrary Lagrangian-Eulerian (ALE) approach is used to solve the full continuum mechanical model that serves as a reference solution. The correction for the stationary height shows excellent agreement with the continuum mechanical results, while the extended model generally improves the classical description. Increasing the viscous term to increase for additional dissipation effects leads to excellent agreement with the fully resolved validation cases. However, in the regime with rise height oscillations, the amplitudes are overestimated by the extended models. (C) 2020 Published by Elsevier Ltd.

We investigate the dynamics of spreading in a regime where the shape of the drop is close to a spherical cap. The latter simplification is applicable in the late (viscous) stage of spreading for highly viscous drops with a diameter below the capillary length. Moreover, it applies to the spreading of a drop on a swellable polymer brush, where the complex interaction with the substrate leads to a very slow spreading dynamics. The spherical cap geometry allows to derive a closed ordinary differential equation (ODE) for the spreading if the capillary number is a function of the contact angle as it is the case for empirical contact angle models. The latter approach has been introduced by de Gennes (Reviews of Modern Physics, 1985) for small contact angles. In the present work, we generalize the method to arbitrary contact angles. The method is applied to experimental data of spreading water-glycerol drops on a silicon wafer and spreading water drops on a PNIPAm coated silicon wafer. It is found that the ODE-model is able to describe the spreading kinetics in the case of partial wetting. Moreover, the model can predict the spreading dynamics of spherical cap-shaped droplets if the relationship between the contact angle and the capillary number is universal.

The breakup dynamics of a capillary bridge on a hydrophobic stripe between two hydrophilic stripes is studied experimentally and numerically. The capillary bridge is formed from an evaporating water droplet wetting three neighboring stripes of a chemically patterned surface. The simulations are based on the Volume-of-Fluid (VOF) method implemented in Free Surface 3D (FS3D). In order to construct physically realistic initial data for the VOF simulation, Surface Evolver is employed to calculate an initial configuration consistent with experiments. Numerical instabilities at the contact line are reduced by a novel adaptation of the Navier-slip boundary condition. By considering the breakup process in phase space, the breakup dynamics can be evaluated without the uncertainty in determining the precise breakup time. It is found that within an intermediate inviscid regime, the breakup dynamics follows a t-scaling, indicating that the breakup process is dominated by the balance of inertial and capillary forces. For smaller bridge widths, the breakup velocity reaches a plateau, which is due to viscous forces becoming more important. In the final stage of breakup, the capillary bridge forms a liquid thread that breaks up consistent with the Rayleigh-Plateau instability. The critical wavelength is identical to the distance between the tips of two liquid cones between which the thread is arranged. The existence of satellite droplets in a regular pattern indicates that the primary breakup process is followed by self-similar secondary breakups.

Four different numerical approaches are compared for the rise of liquid between two parallel plates. These are an Arbitrary Lagrangian-Eulerian method (OpenFOAM solver interTrackFoam), a geometric volume of fluid code (FS3D), an algebraic volume of fluid method (OpenFOAM solver interFoam), and a level set approach (BoSSS). The first three approaches discretize the bulk equation using a finite volume method while the last one employs an extended discontinuous Galerkin discretization. The results are compared to ODE models which are the classical rise model and an extended model that incorporates a Navier slip boundary condition on the capillary walls and levels at a corrected stationary rise height. All physical parameters are based on common requirements for the initial conditions, short simulation time, and a non-dimensional parameter study. The comparison shows excellent agreement between the different implementations with minor quantitative deviations for the adapted interFoam implementation. While the qualitative agreement between the full solutions of the continuum mechanical approach and the reference model is good, the quantitative comparison is only reasonable, especially for cases with increasing oscillations. Furthermore, reducing the slip length changes the solution qualitatively as oscillations are completely damped in contrast to the solution of the ODE models. To provide reference data for a full continuum simulation of the capillary rise problem, all results are made available online.

When liquid rises in a capillary, the interface has a shape close to a spherical cap when the Eotvos number is sufficiently small. This assumption is typically used when computing the stationary rise height of the liquid. Considering a continuum mechanical approach it can be shown that the exact shape of a spherical cap is not a solution to the rise problem yet a good approximation. Consequently, the assumption of a spherical cap used in Washburn‐type rise models is justified only for small Eotvos numbers. In this limit, Jurin's height has the largest error.

An arbitrary Lagrangian Eulerian (ALE) interface tracking method is applied to multiphase flows involving wetting phenomena. The method is implemented in OpenFOAM and validated for the static shape of a drop deformed by gravity influence. Results for a constant contact angle model are compared to approximate analytic solutions for the limiting cases of vanishing or infinite Eötvös numbers. Results between the two limiting cases are compared to numerical results from literature.

A VOF-based approach for the detailed computation of reactive mass transfer at rising gas bubbles is presented. Based on a local approximation of the transport equation at the fluid interface, a reactive subgrid-scale model for uncoupled reactions within the concentration boundary layer is derived. For this purpose, the convective term of the boundary layer description is modeled, reducing the governing nonlinear partial differential equation to a one-dimensional nonlinear differential equation. A numerical scheme is developed to solve the remaining differential equation and obtain the concentration gradient at the fluid interface. Mesh convergence for the resulting scheme is shown. The method is validated within the VOF code FS3D by comparing to local and integral Sherwood numbers, and Schmidt numbers up to 1000 for a 3D reactive Stokes flow regime around a rising spherical bubble (Hadamard-Rybczynski flow field). Local Sherwood numbers show significant reactive mass transfer into the liquid phase in the wake region of the rising bubble. (C) 2016 Elsevier Ltd. All rights reserved.