A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.     

On the usage of NURBS as interface representation in free‐surface flows

When simulating free‐surface flows using the finite element method, there are many cases where the governing equations require information which must be derived from the available discretized geometry. Examples are curvature or normal vectors. The accurate computation of this information directly from the finite element mesh often requires a high degree of refinement—which is not necessarily required to obtain an accurate flow solution. As a remedy and an option to be able to use coarser meshes, the representation of the free surface using non‐uniform rational B‐splines (NURBS) curves or surfaces is investigated in this work. The advantages of a NURBS parameterization in comparison with the standard approach are discussed. In addition, it is explored how the pressure jump resulting from surface tension effects can be handled using doubled interface nodes. Numerical examples include the computation of surface tension in a two‐phase flow as well as the computation of normal vectors as a basis for mesh deformation methods. For these examples, the improvement of the numerical solution compared with the standard approaches on identical meshes is shown. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermomechanics of solids with general imperfect coherent interfaces

The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect coherent interfaces in view of their thermomechanical behavior and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Conventionally, imperfect interfaces with respect to their thermal behavior are often restricted to being either highly conducting (HC) or lowly conducting (LC) also known as Kapitza. The interface model here is general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Clearly, in extreme cases, the current model simplifies to HC and LC interfaces. A new characteristic of the general imperfect interface is that the interface temperature is an independent degree of freedom and, in general, is not a function of only temperatures across the interface. The interface temperature, however, must be computed using a new interface material parameter, i.e., the sensitivity. It is shown that according to the second law, the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. In particular, even if the temperature jump at the interface vanishes, the interface temperature may be different from the temperatures across the interface. This finding allows for a better, and somewhat novel, understanding of HC interfaces. That is, a HC interface implies, but is not implied by, the vanishing temperature jump across the interface. The problem is formulated such that all types of interfaces are derived from a general imperfect interface model, and therefore, we establish a unified finite element framework to model all classes of interfaces for general transient problems. Full details of the novel numerical scheme are provided. Key features of the problem are then elucidated via a series of three-dimensional numerical examples. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites and also thermal interface materials.     

A staggered grid‐based explicit jump immersed interface method for two‐dimensional Stokes flows

In this paper the explicit jump immersed interface method (EJIIM) is applied to stationary Stokes flows. The boundary value problem in a general, non‐grid aligned domain is reduced by the EJIIM to a sequence of problems in a rectangular domain, where staggered grid‐based finite differences for velocity and pressure variables are used. Each of these subproblems is solved by the fast Stokes solver, consisting of the pressure equation (known also as conjugate gradient Uzawa) method and a fast Fourier transform‐based Poisson solver. This results in an effective algorithm with second‐order convergence for the velocity and first order for the pressure. In contrast to the earlier versions of the EJIIM, the Dirichlét boundary value problem is solved very efficiently also in the case when the computational domain is not simply connected. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3D finite element ALE method using an approximate Riemann solution

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann‐like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann‐like problem. Two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well‐known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problem results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space

In a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier‐Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite‐element code is extremely easy with the version presented here because the new shape functions are based on the usual finite‐element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

A compressible two‐fluid multiphase model for CO2 leakage through a wellbore

This paper introduces an effectively mesh‐independent and computationally efficient model for CO₂ leakage through wellbores. A one‐dimensional compressible two‐fluid domain, representing a homogeneous air gas and a multiphase CO₂ with a jump at the interface between them, is modeled. The physical domain is modeled using the drift‐flux model, and the governing equations are solved using a mixed finite‐element discretization scheme. The standard Galerkin FEM, the partition of unity method, and the level‐set method are integrated to solve the problem. All important physical phenomena and processes occurring along the wellbore path, including fluid dynamics, buoyancy, phase change, compressibility, thermal interaction, wall friction, and slip between phases, together with the jump in density and enthalpy between air and CO₂, are considered. Two numerical examples illustrating the computational capability and efficiency of the model are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Coupled ghost fluid/two‐phase level set method for curvilinear body‐fitted grids

A coupled ghost fluid/two‐phase level set method to simulate air/water turbulent flow for complex geometries using curvilinear body‐fitted grids is presented. The proposed method is intended to treat ship hydrodynamics problems. The original level set method for moving interface flows was based on Heaviside functions to smooth all fluid properties across the interface. We call this the Heaviside function method (HFM). The HFM requires fine grids across the interface. The ghost fluid method (GFM) has been designed to explicitly enforce the interfacial jump conditions, but the implementation of the jump conditions in curvilinear grids is intricate. To overcome these difficulties a coupled GFM/HFM method was developed in which approximate jump conditions are derived for piezometric pressure and velocity and pressure gradients based on exact continuous velocity and stress and jump in momentum conditions with the jump in density maintained but continuity of the molecular and turbulent viscosities imposed. The implementation of the ghost points is such that no duplication of memory storage is necessary. The level set method is adopted to locate the air/water interface, and a fast marching method was implemented in curvilinear grids to reinitialize the level set function. Validations are performed for three tests: super‐ and sub‐critical flow without wave breaking and an impulsive plunging wave breaking over 2D submerged bumps, and the flow around surface combatant model DTMB 5512. Comparisons are made against experimental data, HFM and single‐phase level set computations. The proposed method performed very well and shows great potential to treat complicated turbulent flows related to ship flows. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

Spurious currents in finite element based level set methods for two‐phase flow

A study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley & Sons, Ltd.     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher‐order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

A two‐phase flow interface capturing finite element method

A new interface capturing algorithm is proposed for the finite element simulation of two‐phase flows. It relies on the solution of an advection equation for the interface between the two phases by a streamline upwind Petrov–Galerkin (SUPG) scheme combined with an adaptive mesh refinement procedure and a filtering technique. This method is illustrated in the case of a Rayleigh–Taylor two‐phase flow problem governed by the Stokes equations. Copyright © 2006 John Wiley & Sons, Ltd.     

The improved space–time conservation element and solution element scheme for two‐dimensional dam‐break flow simulation

A new numerical scheme, namely space–time conservation element and solution element (CE/SE) method, has been used for the solution of the two‐dimensional (2D) dam‐break problem. Distinguishing from the well‐established traditional numerical methods (such as characteristics, finite difference, finite element, and finite‐volume methods), the CE/SE scheme has many non‐traditional features in both concept and methodology: space and time are treated in a unified way, which is the most important characteristic for the CE/SE method; the CEs and SEs are introduced, both local and global flux conservations in space and time rather than space only are enforced; an explicit scheme with a stagger grid is adopted. Furthermore, this scheme is robust and easy to implement. In this paper, an improved CE/SE scheme is extended to solve the 2D shallow water equations with the source terms, which usually plays a critical role in dam‐break flows. To demonstrate the accuracy, robustness and efficiency of the improved CE/SE method, both 1D and 2D dam‐break problems are simulated numerically, and the results are consistent with either the analytical solutions or experimental results. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

Two‐dimensional non‐Newtonian injection molding with a new control volume FEM/volume of fluid method

A new method based on volume of fluid for interface tracking in the simulation of injection molding is presented. The proposed method is comprised of two main stages: accumulation and distribution of the volume fraction. In the first stage the equation for the volume fraction with a noninterfacial flux condition is solved. In the second stage the accumulated volume of fluid that arises as a consequence of the application of the first one is dispersed. This procedure guarantees that the fluid fills the available space without dispersion of the interface. The mathematical model is based on two‐phase transport equations that are numerically integrated through the control volume finite element method. The numerical results for the interface position are successfully verified with analytical results and numerical data available in the literature for one‐dimensional and two‐dimensional domains. The transient position of the advance fronts showed an effective and consistent simulation of an injection molding process. The nondispersive volume of fluid method here proposed is implemented for the simulation of nonisothermal injection molding in two‐dimensional cavities. The obtained results are represented as transient interface positions, isotherms and pressure distributions during the injection molding of low density polyethylene. Copyright © 2012 John Wiley & Sons, Ltd.     

Mathematical model and numerical method for spontaneous potential log in heterogeneous formations

This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has beer/ shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.