Interface Advection Research Articles

Pressure-equilibrium semi-implicit solver for real fluids

In this study, a semi-implicit pressure-based scheme which can suppress the spurious pressure oscillations is developed for real fluid flows. Conservation properties relevant to the proposed scheme are investigated through one-dimensional numerical simulations of the nitrogen interface advection. The efficiency and validity of this scheme are carefully examined with two- and three-dimensional numerical simulations of cryogenic nitrogen jets. From the one-dimensional simulation results, the conservation properties are well conserved as the past studies. The two-dimensional simulation results show that the developed scheme can reduce the computational cost by more than 92% (13–14 times faster) compared with the conventional density-based explicit solver employing the double flux model. It is also found that, through the three-dimensional simulation results, the developed scheme can predict the structure of the nitrogen jet under both transcritical and supercritical conditions, while taking into account the effects of real fluid properties on the mixing and spatiotemporal evolution of this jet, with excellent accuracy.

Numerical Algorithms for Divergence-Free Velocity Applications

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.

A consistent methodology to transport a passive scalar with the geometric Volume-of-Fluid method isoAdvector

This paper presents a new methodology allowing the discretisation of phase specific transport equations within the Volume-of-Fluid interface capturing method framework. The method uses a sharp interface algorithm to compute the transport of species concentration. The interface reconstruction and advection is provided by the geometric advection scheme isoAdvector[1] implemented in the OpenFOAM® library. When discretising the transport equation for species, lack of consistency with the free surface advection scheme can lead to numerical errors, causing conservation or boundedness issues. This work addresses the issue of consistency in convective transport of species and is divided in two parts. First, a new interpolation procedure is used to compute face values from cell-centered values. Then, the diffusive operator of the transport equation is corrected. Finally, a set of test cases are presented to validate the transport equation's consistency with the free surface advection. Species transport across the interface is not part of the scope of this article, however, this methodology can further be used to study mass transfer at the gas-liquid interface using additional mass source terms that are not discussed here.

Comparison of methods for curvature estimation from volume fractions

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. A shape-independent test of the curvature estimation is introduced through the use of randomly oriented and shaped paraboloids as reference interfaces. The coupling of the curvature estimation, interface transport, and Navier–Stokes surface tension force is evaluated through the measurement of spurious velocities in simulations of stationary and translating droplets. These studies find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost when errors are introduced in the volume fraction field either artificially or through the interface advection and reconstruction steps.

Numerical investigation of bubble pinch-off dynamics

While several mechanisms have been conjectured for the bubble pinching-off process, the underlying mechanism is still poorly understood, leaving the connection between the observed kinematics and the amplitude of the sound generated during the process unclear. Despite some experimental studies reporting the kinematics and the corresponding amplitude of the bubble sound emissions simultaneously (Nelli et al., AFMC, 2022), minimal numerical studies have been conducted to supplement these experiments. We report a numerical study by reproducing incompressible bubble pinch-off interface kinematics. Numerical simulations were performed by the volume-of-fluid (VoF) method, which keeps track of each phase fraction using a solution of the interface advection equation. Simulations are in good agreement on the kinematics at pinch-off despite several challenges associated with high-speed, high-distortion-rate interfacial dynamics.

Extending the isoAdvector Geometric VOF Method to Flows in Porous Media

We consider the interfacial flow in and around porous structures in coastal and marine engineering.* During recent years, interfacial flow through porous media has been repeatedly simulated with Computational Fluid Dynamics (CFD) based on algebraic Volume Of Fluid (VOF) methods [1] [2]. Here, we present an implementation of a porous medium interfacial flow solver based on the geometric VOF method, isoAdvector [3] [4]. In our implementation, the porous medium is treated without resolving the actual pore geometry. Rather, the porous media, pores, and rigid structure are considered a continuum and the effects of porosity on the fluid flow are modelled through source terms in the Navier-Stokes equations, including Darcy-Forchheimer forces, added mass force and accounting for the part of mesh cells that are occupied by the solid material comprising the skeleton of the porous medium. The governing equations are adopted from the formulation by Jensen et al. [1]. For the interface advection using isoAdvector, we also account for the reduced cell volume available for fluid flow and for the increase in the interface front velocity caused by a cell being partially filled with solid material. The solver is implemented in the open source CFD library OpenFOAM®. It is validated using two case setups: 1) A pure passive advection test case to compare the isolated advection algorithm against a known analytical solution and 2) a porous dam break case by Liu et al. [5] where both numerical and experimental results are available for comparison. We find good agreement with numerical and experimental results. For both cases the interface sharpness, shape conservation as well as volume conservation and boundedness are demonstrated to be very good. The solver is released as open source for the benefit of the coastal and marine CFD community (code repository https://github.com/InterFlowers/porousInterIsoFoam) and as of OpenFOAM-v2112 the new functionality is integrated in the official interIsoFoam solver. * This article is an updated version of the conference paper Missios et al. 2022 [6] presented at the Marine2021 conference.

Impact of wettability on interface deformation and droplet breakup in microcapillaries

The objective of this research paper is to relate the influence of dynamic wetting in a liquid/liquid/solid system to the breakup of emulsion droplets in capillaries. Therefore, modeling and simulation of liquid/liquid flow through a capillary constriction have been performed with varying dynamic contact angles from highly hydrophilic to highly hydrophobic. Advanced advection schemes with geometric interface reconstruction (isoAdvector) are incorporated for high interface advection accuracy. A sharp surface tension force model is used to reduce spurious currents originating from the numerical treatment and geometric reconstruction of the surface curvature at the interface. Stress singularities from the boundary condition at the three-phase contact line are removed by applying a Navier-slip boundary condition. The simulation results illustrate the strong dependency of the wettability and the contact line and interface deformation.

Moments-based interface reconstruction, remap and advection

We present a new moment-of-fluid (MOF2) interface reconstruction method. It uses the zeroth, first, and second moments of the fragment of material inside a cell of the mesh to reconstruct a convex material polygon or a union of convex polygons that approximate the respective material fragment. The new method requires information about the material moments only for the cell under consideration. The MOF2 method allows to exactly reproduce several convex shapes: corners, filaments, and some concave shapes: cell-complements to corners and filaments.Interface reconstruction is formulated as a local (for each cell), non-linear, equality constrained optimization problem, which does not require additional communication and allows for an efficient parallel implementation.We present an extensive set of test problems, both for interface reconstruction on a single cell, and for reconstruction of a variety of shapes on a variety of meshes.We describe how to perform two-material advection using the MOF2 method and present the results for the classical advection tests.We also show the examples of material interface remapping needed in the framework of multi-material arbitrary Lagrangian-Eulerian methods, and give a brief description of a procedure that can be used to update the material moments on the Lagrangian stage of those methods.

Extension of generic two-component VOF interface advection schemes to an arbitrary number of components

We propose a new and simple method to extend any two-fluid dimensionally split VOF schemes on Cartesian meshes to N-fluid problems in 2D and 3D. The method is symmetric by permutation of the fluids, so that it is independent of the ordering of materials, and guarantees natural properties of the volume fractions. The method is called Renormalized VOF or ReVOF, since it relies on a new renormalization algorithm to post-process N independent calls to two-fluid numerical fluxes. Proof that the algorithm finishes is given. Various numerical test cases for advection, rotation and distorsion of three to five fluids in 2D are presented, along with a 3D example.

Assessment of a Point-Cloud Volume-of-Fluid method with sharp interface advection

Multiphase flows are ubiquitous in nature and industry. In order to fully describe multiphase systems, simulation approaches must be able to handle the interfaces separating the different phases properly. The Volume of Fluid (VOF) approach is widely used by researchers and engineers due to its intrinsic ability to conserve volume and handle large interface topology changes. A common problem that occurs in VOF Methods relates to the calculation of interfacial tension forces in the momentum equations while resolving a sharp interface. Common methods based on the gradients of the volume fraction field may lack accuracy due to the curvature and normal vector estimates through an abrupt transition. To address these points, the present work introduces a new proposal, where the VOF method based on a cloud of points for interface curvatures computation (PC-VOF) is extended by the coupling with the sharp-interface advection algorithm isoAdvector, implemented in the open source suite OpenFOAM®. The coupled method performs better than the ones implemented in the original solvers in a number of benchmark cases from the literature. It is shown to significantly reduce the spurious currents and also presents more stable and accurate results, especially for irregular triangular meshes.

A diffuse-interface compact difference method for compressible multimaterial elastic–plastic flows

In this paper, a novel high-order numerical method is proposed for the simulation of compressible multimaterial solid–fluid problems involving elastic–plastic solid behaviors. The solid, liquid, and gas behaviors are described by an Eulerian interface-capturing model derived from hyperelasticity theory. The deformations in each component of the solids are tracked using a single deviatoric strain tensor. The numerical algorithm employs a tenth-order compact finite difference scheme and a fourth-order Runge–Kutta time-stepping scheme. To prevent numerical oscillations near the material interface, a mechanical equilibrium assumption is introduced for the elastic–plastic problems, and a localized artificial diffusivity method satisfying this assumption is designed and applied in the numerical algorithm. Numerical tests in one and two dimensions are shown to demonstrate the accuracy and feasibility of the proposed approach. In particular, the accuracy and numerical resolution of the method are verified using problems with analytical solutions. Numerical tests of interface advection problems demonstrate that our method preserves the velocity, pressure, and elastic stress equilibria at the material interface and prevents overshoots in the rest of the domain. The two-dimensional Richtmyer–Meshkov instabilities between two elastic or two elastic–plastic solids are simulated to verify the suitability of the numerical method for solving problems involving large deformations. Impact of a 2D Taylor bar to a rigid wall is also simulated to test the robustness of the numerical method.

Moving least-squares aided finite element method (MLS-FEM): A powerful means to consider simultaneously velocity and pressure discontinuities of multi-phase flow fields

We suggest a strategy to consider discontinuities in the two-phase flow calculations. Our approach comprises enhancing the shape functions of the finite element method (FEM) with the moving least-squares (MLS) interpolation functions. The new shape functions correlate a parameter at any arbitrary point to its surrounding nodes instead of an element's nodes. We name this strategy the "moving least-squares" aided "finite element method" (MLS-FEM).In a previous paper Mostafaiyan et al., we have employed the concept of the MLS-FEM to develop the PMLS (pressure shape function enhanced by the MLS interpolation functions) method, which predicts pressure discontinuities due to the surface tension forces. We take another additional step to consider velocity and pressure discontinuities in a domain with two different fluids. Therefore, we use the MLS technique to enhance the pressure (P) and velocity (V) shape functions. We name this scheme PVMLS (pressure and velocity shape functions enhanced by the MLS technique).We validate the PVMLS method with a stationary drop problem (as a benchmark). We show that the previous PMLS method fails to predict an accurate pressure or smooth streamlines by increasing the Capillary number and deviating from the unit viscosity ratio. However, the PVMLS method provides more accurate results since it considers velocity as a discontinuous parameter. Also, we compare the results of the PVMLS method at very high viscosity ratios (undeformable fluid) with a boundary-fitted single-phase problem. The reported similarities between the predicted pressure values and streamlines in both strategies demonstrate the reliability of the PVMLS method. Finally, we discuss that by employing the PVMLS method to calculate the velocity field, the interface advection will be more mass-conservative.

Numerical simulation of fluid structure interaction in free-surface flows: the WEC case

In this work we present a numerical framework to carry-out numerical simulations of fluid-structure interaction phenomena in free-surface flows. The framework employs a single-phase method to solve momentum equations and interface advection without solving the gas phase, an immersed boundary method (IBM) to represent the moving solid within the fluid matrix and a fluid structure interaction (FSI) algorithm to couple liquid and solid phases. The method is employed to study the case of a single point wave energy converter (WEC) device, studying its free decay and its response to progressive linear waves.

Interface reconstruction and advection schemes for volume of fluid method in axisymmetric coordinates

Volume of fluid (VOF) method is a sharp interface method employed for simulations of two phase flows. Interface in VOF is usually represented using piecewise linear line segments in each computational grid based on the volume fraction field. While VOF for cartesian coordinates conserve mass exactly, existing algorithms do not show machine-precision mass conservation for axisymmetric coordinate systems. In this work, we propose analytic formulae for interface reconstruction in axisymmetric coordinates, similar to those proposed by Scardovelli and Zaleski (2000) [12] for cartesian coordinates. We also propose modifications to the existing advection schemes in VOF for axisymmetric coordinates to obtain higher accuracy in mass conservation.

An efficient moment-of-fluid interface tracking method

In this paper we present an efficient numerical method for locating material interfaces in multi-fluid simulations. Starting from a detailed investigation of the moment-of-fluid reconstruction problem, we develop an efficient procedure for the interface reconstruction, based on pre-computing certain values and mapping the domain of the function describing the interface reconstruction onto these values. The interface tracking method is tested on a range of commonly-seen interface advection test problems, where it is shown to increase the computational efficiency of the moment-of-fluid method by a factor of three, with a negligible change in the accuracy of the method.

Interface-resolved simulation of the evaporation and combustion of a fuel droplet suspended in normal gravity

An interface-resolved simulation of the combustion of a fuel droplet suspended in normal gravity is presented in this work, followed by an extensive analysis on the physical aspects involved. The modeling is based on DropletSMOKE++, a multiphase solver developed for the modeling of droplet vaporization and combustion in convective conditions. A wide range of phenomena can be described by the model, including the interface advection, the phase-change, the combustion chemistry, non-ideal thermodynamics and multicomponent mixtures. To our knowledge, this is the most detailed simulation performed on this configuration, providing a useful theoretical and numerical support for the experimental activity on this field. A recent experimental work is used as a reference, in which a methanol droplet is suspended on a quartz fiber and ignited at different oxygen concentrations. The numerical analysis offers a detailed insight into the physics of the problem and a satisfactory agreement with the experiments in terms of diameter decay, radial temperature profiles and sensitivity to the oxygen concentration. The vaporization rate is affected by the thermal conduction from the fiber, due to the high temperatures involved. Moreover, the fiber perturbs the flame itself, providing quenching at its surface. The combustion physics is compared to the one predicted at zero-gravity, evidencing a lower standoff-ratio, a higher flame temperature and an intense internal circulation. The distribution of the species around the droplet shows (i) a local accumulation of intermediate oxidation products at the fiber surface and (ii) water absorption in the liquid phase, affecting the vaporization rate.

Numerical study on strong nonlinear interactions between spark-generated underwater explosion bubbles and a free surface

In this study, strong nonlinear interactions of one and two underwater explosion bubbles with a free surface are numerically studied using an axisymmetrical fully compressible three-phase homogeneous model. The bubble interfaces and free surface are captured by solving two interface advection equations of vapor and air phases. The numerical results are validated via comparisons with the experimental results. The single-bubble case is validated first, and good quantitative and qualitative agreements are achieved. Special features of the interactions, such as jet impact, toroidal bubble, and primary and secondary water spike, are observed in our numerical model. The detailed pressure and velocity contours during the interaction process are obtained and can better reveal the mechanism underlying the bubbles and free surface dynamics. Afterward, the characteristics of the motion of the free surface and the bubbles are investigated considering different non-dimensional standoff parameters. Four distinct patterns of free surface motion are identified. Next, strong interactions between two bubbles and a free surface are simulated. The essential physical phenomena, such as coalesced bubble, annular residual, and bubble splitting, are reproduced well in the representative experiments. Finally, a parametric study reveals the dependence of the center water spike on the interaction between the bubbles and the free surface.

Dynamic wetting failure in curtain coating by the Volume-of-Fluid method

In this paper, we investigate dynamic wetting in the curtain coating configuration. The two-phase Navier–Stokes equations are solved by a Volume-of-Fluid method on an adaptive Cartesian mesh. We introduce the Navier boundary condition to regularize the solution at the triple point and remove the implicit numerical slip induced by the cell-centered interface advection. We use a constant contact angle to describe the dynamic contact line. The resolution of the governing equations allows us to predict the substrate velocity at which wetting failure occurs. The model predictions are compared with prior computations of Liu et al. [C.Y. Liu, E. Vandre, M. Carvalho, S. Kumar, J. Fluid Mech. 808, 290 (2016); C.Y. Liu, M. Carvalho, S. Kumar, Chem. Eng. Sci. 195, 74 (2019)] and experimental observations of Blake et al. [T. Blake, M. Bracke, Y. Shikhmurzaev, Phys. Fluids 11, 1995 (1999)] and Marston et al. [J. Marston, V. Hawkins, S. Decent, M. Simmons, Exp. Fluids 46, 549 (2009)].

Unstructured un-split geometrical Volume-of-Fluid methods – A review

Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional geometrical operations are significantly more complex than their two-dimensional counterparts, which is confirmed by the ratio of publications with three-dimensional results on unstructured meshes to publications with two-dimensional results or support for structured meshes. Additionally, unstructured meshes present challenges in serial and parallel computational efficiency, accuracy, implementation complexity, and robustness. Ongoing research is still very active, focusing on different issues: interface positioning in general polyhedra, estimation of interface normal vectors, advection accuracy, and parallel and serial computational efficiency.This survey tries to give a complete and critical overview of classical, as well as contemporary geometrical VOF methods with concise explanations of the underlying ideas and sub-algorithms, focusing primarily on unstructured meshes and three dimensional calculations. Reviewed methods are listed in historical order and compared in terms of accuracy and computational efficiency.

An Accurate Sharp Interface Method for Two-Phase Compressible Flows at Low-Mach Regime

An accurate numerical approach is presented for computing two-phase flows with surface tension at low-Mach regime. To develop such a model, where slight compressible effects are taken into account as well as correct thermodynamical closures, both the liquid and the gas are considered compressible and described by a precise compressible solver. A low-Mach correction has been implemented to eliminate excessive numerical dissipation. The interface between two-phase flows is captured by the level set method that is considered to be sharp. The interface capturing issue of the level set method within the Eulerian framework is the key point of the two-phase flow simulations, and in this work we propose a high-order coupled time-space approach for interface advection. Several numerical test-cases have been employed to validate the present numerical approach and enlighten its good performance.