A diffuse-interface method for two-phase flows with soluble surfactants

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier-Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed.     

A conservative phase field method for solving incompressible two-phase flows

In this paper a conservative phase-field method based on the work of Sun and Beckermann [Y. Sun, C. Beckermann, Sharp interface tracking using the phase-field equation, J. Comput. Phys. 220 (2007) 626–653] for solving the two- and three-dimensional two-phase incompressible Navier–Stokes equations is proposed. The present method can preserve the total mass as the Cahn–Hilliard equation, but the calculation and implementation are much simpler than that. The dispersion-relation-preserving schemes are utilized for the advection terms while the Helmholtz smoother is applied to compute the surface-tension force term. To verify the proposed method, several benchmarks are examined and shown to have good agreements with previous results. It also shows that the satisfactions of mass conservations are guaranteed.     

A lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio

This paper reports a new numerical scheme of the lattice Boltzmann method for calculating liquid droplet behaviour on particle wetting surfaces typically for the system of liquid–gas of a large density ratio. The method combines the existing models of Inamuro et al. [T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628–644] and Briant et al. [A.J. Briant, P. Papatzacos, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion in a liquid–gas system, Philos. Trans. Roy. Soc. London A 360 (2002) 485–495; A.J. Briant, A.J. Wagner, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion: I. Liquid–gas systems. Phys. Rev. E 69 (2004) 031602; A.J. Briant, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion: II. Binary fluids, Phys. Rev. E 69 (2004) 031603] and has developed novel treatment for partial wetting boundaries which involve droplets spreading on a hydrophobic surface combined with the surface of relative low contact angles and strips of relative high contact angles. The interaction between the fluid–fluid interface and the partial wetting wall has been typically considered. Applying the current method, the dynamics of liquid drops on uniform and heterogeneous wetting walls are simulated numerically. The results of the simulation agree well with those of theoretical prediction and show that the present LBM can be used as a reliable way to study fluidic control on heterogeneous surfaces and other wetting related subjects.     

A hybrid scheme for computing incompressible two-phase flows

We propose a hybrid scheme for computations of incompressible two-phase flows. The incompressible constraint has been replaced by a pressure Poisson-like equation and then the pressure is updated by the modified marker and cell method. Meanwhile, the moment equations in the incompressible Navier-Stokes equations are solved by our semidiscrete Hermite central-upwind scheme, and the interface between the two fluids is considered to be continuous and is described implicitly as the 0.5 level set of a smooth function being a smeared out Heaviside function. It is here named the hybrid scheme. Some numerical experiments are successfully carried out, which verify the desired efficiency and accuracy of our hybrid scheme.     

A two-phase diffuse-interface model for Hele-Shaw flows with large property contrasts

A novel two-phase diffuse-interface model is used to simulate flows inside a Hele-Shaw cell. The model assumes that the two phases coexist inside the diffuse interface, with different velocities and properties. A separate equation is used to calculate the slip velocity between the two phases inside the diffuse interface. It is shown that for one-dimensional flows parallel to the diffuse interface, the results are independent of the diffuse-interface width, regardless of the magnitude of the density and viscosity contrasts between the phases. This two-phase approach is coupled with a phase-field equation for calculating the interface motion. The model is applied to a buoyancy-driven two-phase flow involving a Rayleigh-Taylor instability and validated through a comparison with available sharp-interface results. The flows and interface topology changes are investigated for large density and viscosity contrasts between the phases. The convergence of the results with respect to the interface width is examined in detail. It is shown that the two-phase model converges better than a standard diffuse-interface model that assumes the presence of a single velocity inside the diffuse interface. Remaining interface width dependencies can be attributed to the capillary stress term in the momentum equation.     

The extended finite element method for two-phase and free-surface flows: A systematic study

In immiscible two-phase flows, jumps or kinks are present in the velocity and pressure fields across the interfaces of the two fluids. The extended finite element method (XFEM) is able to reproduce such discontinuities within elements. Robust and accurate interface capturing schemes with no restrictions on the interface topology are thereby enabled. This paper investigates different enrichment schemes and time-integration schemes within the XFEM. Test cases with and without surface tension on moving or stationary meshes are studied and compared to interface tracking results when possible. A particularly useful setting is extracted which is recommended for two-phase flows. An extension of this formulation for the simulation of free-surface flows and of floating objects is proposed.     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties

This paper concerns the development of a new Cartesian grid/immersed boundary (IB) method for the computation of incompressible viscous flows in two-dimensional irregular geometries. In IB methods, the computational grid is not aligned with the irregular boundary, and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary, the so-called “cut-cells”. In this paper, we present a new IB method, called the LS-STAG method, which is based on the MAC method for staggered Cartesian grids and where the irregular boundary is sharply represented by its level-set function. This implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut-cells. We have achieved a novel discretization of the fluxes in the cut-cells by enforcing the strict conservation of total mass, momentum and kinetic energy at the discrete level. Our discretization in the cut-cells is consistent with the MAC discretization used in Cartesian fluid cells, and has the ability to preserve the five-point Cartesian structure of the stencil, resulting in a highly computationally efficient method. The accuracy and robustness of our method is assessed on canonical flows at low to moderate Reynolds number: Taylor–Couette flow, flows past a circular cylinder, including the case where the cylinder has forced oscillatory rotations. Finally, we will extend the LS-STAG method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free-stream.     

A continuum model of gas flows with localized density variations

We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows.     

A subgrid computation of the curvature by a particle/level-set method. Application to a front-tracking/ghost-fluid method for incompressible flows

A particle/level-set method is developed to capture the interface between two immiscible phases. No redistance equation is used for the level-set function which is built from an accurate cubic spline interpolation of the markers describing the interface. Mass losses which usually appear with level-set functions are drastically reduced. Interface coupling with the incompressible momentum equation is taken into account thanks to the ghost-fluid method. Indeed, the particles allow an accurate computation of the local interfacial curvatures, and the capillary part of the pressure jump computed across the interface is more accurately evaluated. Results on static drop show a large decrease of spurious currents compared to other methods. Scalar advection validations and convergence studies are carried out as well as tough test-cases involving large density ratios (typically air/water). A better agreement with literature results is shown compared to classical approaches.     

含活性剂液膜去润湿演化的稳定性特征

针对含非溶性活性剂的液膜在固体基底上的去润湿过程,基于润滑理论建立了基态和扰动态下液膜厚度和表面活性剂浓度的演化模型,应用非模态理论分析了演化过程的稳定性特征,探讨了分子间力对液膜去润湿过程的影响. 研究表明,微扰动波的引入（k=1）有利于液膜去润湿过程的稳定进行,扰动能量逐渐衰减,然而,该效果随着扰动波数的增加而显著改变,k ≥ 2时,液膜演化的稳定性反而恶化,扰动能量被逐步放大,演化呈现出非稳定特征. 增大初始液膜厚度可以有效改善液膜流动的稳定性. 范德华力放大了液膜表面的微扰动,使得液膜演化的稳定性下降;相反,Born斥力和静电斥力具有增强去润湿稳定性的作用. 关键词： 活性剂 去润湿 分离压 稳定性     

A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids

A balanced force refined level set grid method for two-phase flows on structured and unstructured flow solver grids is presented. To accurately track the phase interface location, an auxiliary, high-resolution equidistant Cartesian grid is introduced. In conjunction with a dual-layer narrow band approach, this refined level set grid method allows for parallel, efficient grid convergence and error estimation studies of the interface tracking method. The Navier–Stokes equations are solved on an unstructured flow solver grid with a novel balanced force algorithm for level set methods based on the recently proposed method by Francois et al. [M.M. Francois, S.J. Cummins, E.D. Dendy, D.B. Kothe, J.M. Sicilian, M.W. Williams, A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys. 213 (2006) 141–173] for volume of fluid methods on structured grids. To minimize spurious currents, a second order converging curvature evaluation technique for level set methods is presented. The results of several different test cases demonstrate the effectiveness of the proposed method, showing good mass conservation properties and second order converging spurious current magnitudes.     

A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid

In the present study we propose a charge-conservative scheme to solve two-phase electrohydrodynamic (EHD) problems using the volume-of-fluid (VOF) method. EHD problems are usually simplified by assuming that the fluids involved are purely dielectric (insulators) or purely conducting. Gases can be considered as perfect insulators but pure dielectric liquids do not exist in nature and insulating liquids have to be approximated using the “Taylor–Melcher leaky dielectric model” [1], [2] in which a leakage of charge through the liquid due to ohmic conduction is allowed. It is also a customary assumption to neglect the convection of charge against the ohmic conduction. The scheme proposed in this article can deal with any EHD problem since it does not rely on any of the above simplifications. An unrestricted EHD solver requires not only to incorporate electric forces in the Navier–Stokes equations, but also to consider the charge migration due to both conduction and convection in the electric charge conservation equation [3]. The conducting or insulating nature of the fluids arise on their own as a result of their electric and fluid mechanical properties. The EHD solver has been built as an extension to Gerris, a free software solver for the solution of incompressible fluid motion using an adaptive VOF method on octree meshes developed by Popinet [4], [5].     

An efficient moving mesh spectral method for the phase-field model of two-phase flows

We develop in this paper a moving mesh spectral method for the phase-field model of two-phase flows with non-periodic boundary conditions. The method is based on a variational moving mesh PDE for the phase function, coupled with efficient semi-implicit treatments for advancing the mesh function, the phase function and the velocity and pressure in a decoupled manner. Ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method.     

A local level-set method using a hash table data structure

We present a local level-set method based on the hash table data structure, which allows the storage of only a band of grid points adjacent to the interface while providing an O(1) access to the data. We discuss the details of the construction of the hash table data structure as well as the advection and reinitialization schemes used for our implementation of the level-set method. We propose two dimensional numerical examples and compare the results to those obtained with a quadtree data structure. Our study indicates that the method is straightforward to implement but suffers from limitations that make it less efficient than the quadtree data structure.     

A new approach to sub-grid surface tension for LES of two-phase flows

In two-phase flow, the presence of inter-phasal surface – the interface – causes additional terms to appear in LES formulation. Those terms were ignored in contemporary works, for the lack of model and because the authors expected them to be of negligible influence. However, it has been recently shown by a priori DNS simulations that the negligibility assumption can be challenged. In the present work, a model for one of the sub-grid two-phase specific terms is proposed, using deconvolution of the velocity field and advection of the interface using that field. Using the model, the term can be included into LES. A brief presentation of the model is followed by numerical tests that assess the model’s performance by comparison with a priori DNS results.     

A conservative SPH method for surfactant dynamics

In this paper, a Lagrangian particle method is proposed for the simulation of multiphase flows with surfactant. The model is based on the multiphase smoothed particle hydrodynamics (SPH) framework of Hu and Adams (2006) [1]. Surface-active agents (surfactants) are incorporated into our method by a scalar quantity describing the local concentration of molecules in the bulk phase and on the interface. The surfactant dynamics are written in conservative form, thus global mass of surfactant is conserved exactly. The transport model of the surfactant accounts for advection and diffusion. Within our method, we can simulate insoluble surfactant on an arbitrary interface geometry as well as interfacial transport such as adsorption or desorption. The flow-field dynamics and the surfactant dynamics are coupled through a constitutive equation, which relates the local surfactant concentration to the local surface-tension coefficient. Hence, the surface-tension model includes capillary and Marangoni-forces. The present numerical method is validated by comparison with analytic solutions for diffusion and for surfactant dynamics. More complex simulations of an oscillating bubble, the bubble deformation in a shear flow, and of a Marangoni-force driven bubble show the capabilities of our method to simulate interfacial flows with surfactants.     

FIVER: A finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps

A robust finite volume method for the solution of high-speed compressible flows in multi-material domains involving arbitrary equations of state and large density jumps is presented. The global domain of interest can include a moving or deformable subdomain that furthermore may undergo topological changes due to, for example, crack propagation. The key components of the proposed method include: (a) the definition of a discrete surrogate material interface, (b) the computation of a reliable approximation of the fluid state vector on each side of a discrete material interface via the construction and solution of a local, exact, two-phase Riemann problem, (c) the algebraic solution of this auxiliary problem when the equation of state allows it, and (d) the solution of this two-phase Riemann problem using sparse grid tabulations otherwise. The proposed computational method is illustrated with the three-dimensional simulation of the dynamics of an underwater explosion bubble.     

Conditional methods for continuum reacting flows in porous media

A special version of conditional moment closure—PCMC—is suggested for modeling reacting flows in porous media. The model involves conditioning on a special tracer scalar, which is introduced to characterize scalar transport in the gaseous phase. (i.e., for the flow in the interparticle space or in the pores). The model accounts for interparticle variations of species concentrations and emulates diffusion in the interparticle space. Special boundary conditions that are consistent with conventional conditions at the phase interface are obtained for the PCMC model. The model is tested against complete direct simulation of a reacting flow in porous media with favourable results.