Anton Smolianski

Numerical Modeling of Two-FluidInterfacial Flows

UNIVERSITY OF JYVASKYLAJYVASKYLA 2001

ABSTRACT

Smolianski, AntonNumerical Modeling of Two-Fluid Interfacial FlowsJyvaskyla: University of Jyvaskyla, 2001, 109 p.(Jyvaskyla Studies in ComputingISSN 1456-5390; 8)ISBN 951-39-0929-8Finnish summaryDiss.

The present work is devoted to the study on unsteady flows of two immiscibleviscous fluids separated by free moving interface. The goal of the present workis to elaborate a unified strategy for numerical modeling of all kinds of two-fluidinterfacial flows, having in mind possible interface topology changes (like mergeror break-up) and realistically wide ranges for physical parameters of the problem.

The presented computational approach essentially relies on three basic com-ponents: finite element method for spatial approximation, operator-splitting fortemporal discretization and level-set method for interface representation. Finiteelement discretization is based on variational formulation of the problem and,thus, allows to naturally incorporate discontinuous material coefficients and sin-gular interface-concentrated forces. The use of finite elements permits to local-ize the interface precisely, without introduction of any artificial parameters likeinterface thickness. We also show that interface normal and curvature can berecovered with the second-order accuracy after applying a gradient averagingtechnique; that allows us to compute accurately the surface tension force. Fortemporal discretization we employ an operator-splitting, thus, separating all ma-jor difficulties of the problem. This approach enables us, in particular, to imple-ment equal-order interpolation for the velocity and pressure. In order to modelthe phenomena involving interface topology changes we make use of the level-set approach, the finite element implementation of which brings some additionalbenefits as compared to the standard finite difference level-set realizations. Weintroduce also a simple mass-correction procedure allowing to maintain an opti-mal, second order accurate mass conservation.

Diverse numerical examples including simulations of bubble dynamics, bi-furcating jet flow and Rayleigh-Taylor instability are presented to validate theproposed computational method.

Keywords: two-fluid interfacial flow, Navier-Stokes equations, discontinuous co-efficients, singular force, free moving boundary, finite element method, operator-splitting, level-set approach

Author’s Address Anton SmolianskiDepartment of MathematicalInformation TechnologyUniversity of JyvaskylaP.O. Box 35, FIN-40351 JyvaskylaFinland

E-mail: antsmol@mit.jyu.fi

Supervisors Docent Heikki HaarioDepartment of MathematicsUniversity of HelsinkiFinland

Professor Pekka NeittaanmakiDepartment of MathematicalInformation TechnologyUniversity of JyvaskylaFinland

Professor Timo TiihonenDepartment of MathematicalInformation TechnologyUniversity of JyvaskylaFinland

Reviewers Professor Olivier PironneauLaboratory of Numerical AnalysisUniversity Paris 6France

Professor Sergey RepinV.A. Steklov Institute of Mathematicsin St.-PetersburgRussian Academy of SciencesRussia

Opponent Doctor Bertrand MauryLaboratory of Numerical AnalysisUniversity Paris 6France

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Prof. Pekka Neittaanmaki andProf. Timo Tiihonen (University of Jyvaskyla, Finland) for their support and forgiving me the opportunity to work at the Laboratory of Scientific Computing,where I have enjoyed a friendly atmosphere and access to excellent research fa-cilities. I am also deeply indebted to my late supervisor, Prof. Valery Rivkind, forspecifying the general direction of my research.

I am extremely grateful to my adviser, Doc. Heikki Haario (University ofHelsinki, Finland), for his continuous support and encouragement. I would alsolike to acknowledge a fruitful collaboration with Dr. Dmitri Kuzmin (Universityof Dortmund, Germany) and thank him for thorough reading of my manuscript.

I am very thankful to Prof. Olivier Pironneau (University Paris 6, France) andProf. Sergey Repin (V.A. Steklov Mathematical Institute, Russia) for reviewingthe manuscript and giving encouraging feedback.

This work was financially supported by COMAS Graduate School of the Uni-versity of Jyvaskyla, by the Academy of Finland and by TEKES Technology De-velopment Center.

Finally, I would like to express my deepest appreciation to my parents fortheir support throughout my life and to my wife Tanya for her patience and un-derstanding.

Jyvaskyla, February 2001

Anton Smolianski

CONTENTS

1 INTRODUCTION 91.1 Numerical methods for interfacial flows . . . . . . . . . . . . . . . . 101.2 Computational strategy and thesis outline . . . . . . . . . . . . . . . 20

2 MATHEMATICAL MODEL 222.1 Physical assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Equations and interfacial conditions . . . . . . . . . . . . . . . . . . 232.3 Weak and classical formulations . . . . . . . . . . . . . . . . . . . . 29

3 DEVELOPMENT OF THE COMPUTATIONAL METHOD 333.1 Discretization of the Navier-Stokes equations . . . . . . . . . . . . . 33

3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.2 Operator-splitting approach . . . . . . . . . . . . . . . . . . . 353.1.3 Navier-Stokes convection step . . . . . . . . . . . . . . . . . 433.1.4 Viscous diffusion step . . . . . . . . . . . . . . . . . . . . . . 473.1.5 Projection step . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Approximation of the interface . . . . . . . . . . . . . . . . . . . . . 533.2.1 Level-set approach . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Level-set convection step . . . . . . . . . . . . . . . . . . . . 553.2.3 Reinitialization step . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 Level-set correction step . . . . . . . . . . . . . . . . . . . . . 613.2.5 Approximation of the interface normal and curvature . . . . 633.2.6 Evaluating the interfacial force and density/viscosity fields 67

3.3 Summary of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . 693.4 Stability issues and time scales . . . . . . . . . . . . . . . . . . . . . 70

4 NUMERICAL RESULTS 724.1 Static bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2 Rising bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Breaking bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Merger of two bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 Rayleigh-Taylor instability . . . . . . . . . . . . . . . . . . . . . . . . 864.6 Bifurcating jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 CONCLUSIONS 92

BIBLIOGRAPHY 94

YHTEENVETO (FINNISH SUMMARY) 109

1 INTRODUCTION

Fluid flows with free moving surfaces or interfaces can be roughly divided intofour general classes: bubbles/drops, jets, waves and films. Each class encom-passes a large number of real-life physical phenomena having a great importancein diverse industrial applications. For example, bubble dynamics is of particu-lar interest for chemical engineering, as bubbly flows are the core of bubble col-umn chemical reactors; propulsion of liquid-metal jets constitutes the main partof metal forming processes; ocean waves are under thorough investigation in ma-rine and coastal engineering, and liquid film flows are frequently encountered incoating and drying processes during paper or polymer production.

It is worth noting that all above mentioned classes of fluid flows are, in essence,two-fluid flows, since even in the case when the second fluid is a gas (e.g., air) itsdynamics cannot be neglected, with only few exceptions. Thus, in general, wehave to deal with the flows of two immiscible fluids separated by their naturalinterface rather than with one-liquid free-surface flows.

Experimental results are usually supposed to be the major source of informa-tion on the behaviour of the physical process at hand. However, in many casesof free-surface/interfacial fluid flows the physical time and length scales are sosmall that any reliable experimental observations become extremely expensiveor impossible. Then, numerical modelling turns out to be the only tool allow-ing to investigate the physical phenomenon qualitatively and, sometimes, evenquantitatively.

The goal of the present work is to elaborate a unified strategy for numeri-cal modelling of all kinds of two-fluid interfacial flows, having in mind possi-ble interface topology changes (like merger or break-up) and realistically wideranges for physical parameters of the problem. There are several intrinsic difficul-ties, a correct treatment of which essentially determines the success of the entiremethod. First, large jumps of fluid density and viscosity across the interface areto be properly taken into account in order to satisfy the momentum balance in thevicinity of the interface. Since the surface tension force plays very important rolein the interface dynamics, the influence of this force should be accurately evalu-ated and incorporated into the model. Next, a sharp interface resolution has to bemaintained, including the cases of interface folding, breaking and merging. Fi-nally, mass conservation is of primary importance for any fluid flows, especiallyfor interfacial ones.

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All these issues are addressed, and special techniques are proposed for theirtreatment, which enable to construct the desired computational method.

1.1 Numerical methods for interfacial flows

There is a vast amount of literature devoted to numerical methods for free sur-face/interface fluid flows. As the comprehensive overviews containing a largenumber of references we would mention the papers by Anderson et al. [3], Cu-velier and Schulkes [32], Floryan and Rasmussen [56], Hou [86], Scardovelli andZaleski [165], Tsai and Yue [197] and the book by Shyy et al. [171].

In order to systematize the knowledge on existing methods their clear clas-sification is definitely required. Without loss of generality, we may say that themost popular way is to divide all numerical algorithms for fluid flows into Eule-rian, Lagrangian and mixed Eulerian-Lagrangian. Eulerian methods are charac-terized by a coordinate system that is stationary in the laboratory frame of refer-ence. The fluid travels between different computational cells, in contrast to theLagrangian methods, where each computational cell always contains the samefluid elements. Thus, Lagrangian methods are characterized by a coordinate sys-tem that moves with the fluid. The mixed Eulerian-Lagrangian methods rely onboth Lagrangian and Eulerian concepts. This classification is very reasonable todescribe the way of modeling of fluid flow, but does not contain any informationon approaches to modeling the interface motion. In this respect, there exists an-other commonly-used classification which treats all methods as either interface-tracking or interface-capturing. In the interface-tracking method the interface(free surface) is explicitly tracked along the trajectories of fluid particles in purelyLagrangian manner, which gives rise to the frequent use of interface-tracking incombination with Lagrangian or with mixed Eulerian-Lagrangian methods. Theinterface-capturing method is characterized by a reconstruction of the interfacefrom the properties of appropriate field variables, e.g., fluid fraction or density.The latter classification clarifies the geometrical part of interfacial-flow model-ing, that is the issues related to the interface motion, but leaves unclear other keypoints of an algorithm.

It seems that any computational method for free-surface flow consists of thefollowing main ingredients: (i) flow modeling, (ii) interface modeling, and (iii)modeling of flow–interface coupling. This information is already sufficient togain an insight into particular method, but there are still two important compo-nents of an algorithm to be included in the list of “principle classification factors”.First of them is the spatial discretization, which strongly influences the interfacerepresentation and, to a large extent, determines the last significant algorithmicalcomponent: flow equations solver. Under the latter we do not mean a methodof resolving a linear algebraic system but a strategy for the treatment of intrinsicdifficulties (nonlinearities, constraints) inherent to the fluid flow equations.

Collecting together the main parts of a numerical modeling procedure for in-terfacial flows, we arrive at the classification:

(1) flow modeling: Eulerian, Lagrangian, mixed Eulerian-Lagrangian, mappingmethod

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(2) interface modeling: tracking, capturing

(3) flow–interface coupling: integrated, segregated

(4) spatial discretization: meshless, FDM, FVM, FEM, others

(5) flow equations solver: integrated, segregated

The points (1) and (2) have been discussed, (4) is quite transparent, and onlythe terms “integrated” and “segregated” have to be explained in (3) and (5). Theirmeaning in connection with the flow–interface coupling is apparently simple: insegregated approach the flow is first computed with the “frozen” interface and,then, a new position of the interface is found using the last computed flow vari-ables; in integrated approach the flow variables and new interface position aresought simultaneously. In respect to the flow equations solver the term “segre-gated” means using a variant of operator-splitting, which makes it possible totreat all or some of the flow features (like convective nonlinearity, viscous diffu-sion, incompressibility) in a separate manner. Within the “integrated” frameworkthe system of flow equations is solved as a whole.

Any combinations of the techniques mentioned in (1)–(5) are, in principle,possible and may be found in literature, but, in the rest of this section we willfocus our attention on the most popular computational methodologies, brieflyreviewing their pros and cons.

A. Lagrangian methodsWe start with the Lagrangian methods, since, at the first glance, they seem to bebest suited for the problems with varying interfaces or domain boundaries. Thesemethods are naturally combined with the interface-tracking and have the follow-ing obvious advantages: (i) they permit material interfaces to be specifically de-lineated and precisely followed, (ii) they allow interface boundary conditions tobe easily applied and (iii) the nonlinear convective term in momentum equationis absent. The two main problems with the Lagrangian interface-tracking meth-ods are mesh tangling and numerical inaccuracy due to highly irregular meshes(see figure 1, left). Thus, in their original form these methods are suitable only forsimulation of small interfacial deformations.

As appropriate reference we could mention here the article of Hirt et al. (1970)[81], where the finite-volume method (FVM) was combined with segregated ap-proach for flow equations solver. The purely Lagrangian flow description withinterface-tracking was used also by Kawahara and co-workers (cf. [129], [79]),who advocated the finite-element method (FEM) together with fractional-stepsegregated algorithm for flow equations treatment. The same strategy but withintegrated method for the system of flow equations was introduced in Shopov etal. [170].

A.1. Free Lagrangian methodsTo get rid of severe mesh distortion within the Lagrangian framework, two ap-proaches have been mainly used: remeshing/rezoning algorithms and meshlessparticulate methods. The former approach relies on the introduction of a newgrid and subsequent transfer of information from the old scrambled mesh to thenew one. The method of interpolation between the meshes may be quite arbitrary,

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FIGURE 1 (left) strictly Lagrangian interface-tracking; (center) Free Lagrangianinterface-tracking; (right) Lagrangian meshless particulate SPHmethod.

and the meshes may be arbitrary as far as the number of cells, their geometry, andtopology are concerned. Particularly, the mesh topology may be changed so thatmesh points become added, deleted or reconnected. This latter way of remeshingis often referred to as Free Lagrangian method (figure 1, center). In combinationwith the interface-tracking the Free Lagrangian method was used, for instance,by Crowley [30], Fritts and Boris [58], Fyfe et al. [59]. Despite their seeming suit-ability for moving interface problems, the grid-based Lagrangian methods havetwo crucial disadvantages: (i) they do not permit to handle changes of interfacetopology (unless very sophisticated ad hoc algorithms are involved) and (ii) theyrequire to perform frequent remeshing, which may be prohibitevely expensiveand unreliable, especially in three dimensions.

A.2. Meshless particulate methodsThe second popular approach allowing to circumvent the problem of mesh tan-gling is to use so-called particulate methods, which abandon grid completely (seefigure 1, right). This group of methods invokes the discrete representation of vis-cous flow phenomena with a finite number of interacting particles. Each particlehas a set of attributes, such as mass, position, velocity, momentum, and energy.The state of the fluid system is defined by the attributes of the finite ensembleof particles and the system evolution is defined by the laws of interactions ofthe particles. These laws are constructed so that fluid molecular forces are sim-ulated. The feature attractive from the moving boundaries point of view is thefact that the particles are explicitly associated with different materials, and thusthe interfaces between these materials can be easily followed. The Boltzmannlattice-gas algorithms fall into the category of particulate methods (see Benzi etal. [13], Rothman and Zaleski (1994) [154], Rothman and Zaleski (1997) [155] andreferences therein). The major uncertainties affecting these methods are the fol-lowing: (i) whether modeling of the interparticle forces and the assumed viscositymodels are physically realistic and (ii) how to properly model the interfacial jumpconditions in the presence of strong density/viscosity discontinuity and surfacetension.

Another type of meshless particulate methods is the Smoothed Particle Hy-drodynamics (SPH), in which smoothing kernels are used to interpolate physical

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FIGURE 2 Arbitrary Lagrangian-Eulerian (ALE) method with interface-tracking.

quantities over their discrete pointwise values and to compute the spatial deriva-tives (see Monaghan [122], Morris [123] and references therein). Though SPH hasallowed to obtain realistic simulations of free-surface phenomena including thesurface tension effect, it still seems to have some problems with (i) accuracy ofthe approximation of flow variables (there is a trade-off for smoothing kernelsbetween improving the interpolation and adding a numerical diffusion) and (ii)modeling of high density and viscosity ratios at the interface. Additionally, itis worth noticing that particulate methods do not hold the property of the grid-based Lagrangian methods to represent the interface accurately, as the particlesclustering in some regions of the flow domain may imply insufficient resolutionof some other regions.

B. Mixed Eulerian-Lagrangian methodsB.1. Segregated flow–interface treatmentAfter considering the Lagrangian methods for flow modeling, it is logical to moveon to the mixed Eulerian-Lagrangian approaches, since they are essentially closeto the Lagrangian description, at least with respect to the interface motion. One ofthe most cited early papers within the Eulerian-Lagrangian framework is due toHirt et al. (1974) [83], in which the algorithm called ALE (arbitrary-Lagrangian-Eulerian) was proposed. Each computational cycle of the algorithm consists ofthree distinct phases: (i) an explicit Lagrangian calculation, except mesh verticesare not moved, (ii) an iterative adjustment of pressure and velocity fields to thenew time level (implicit calculation), followed by motion of the mesh verticesto their new Lagrangian position, and (iii) rearrangement of the mesh to a newconfiguration if necessary (see figure 2).

The rezoning (third phase) occurs by letting the mesh move with respect tothe fluid in a prescribed manner, where in the extreme cases the mesh followsthe fluid in a Lagrangian manner (no grid adjustment) or is kept fixed (Eule-rian calculation). The interface is tracked by following the Lagrangian motion ofvertices aligned initially with the interface. Thus, the algorthm bears a consider-able resemblance with pure Lagrangian rezoning methods possessing their main

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advantages and drawbacks, as far as the interface treatment is concerned. In par-ticular, the remarkable shortcoming is limited interface deformations because ofthe necessity to maintain a fixed topology of the grid. However, the flexibilityin dealing with the motion of mesh vertices makes the ALE-type methods veryattractive for free-surface flow simulations, and the algorithms of this type weresuccessfully used by Bansch [8], Belytschko and Flanagan [12], Donea et al. [45],Hughes et al. [88], Keunings [98], Maury and Pironneau [119], Ramaswamy [143],Ramaswamy and Kawahara [144], Yamamoto and Kawahara [207]. All theseworks relied on the finite-element method and on the segregated treatment offlow–interface coupling.

The ALE methodology was exploited also in Hansbo [76] and in Tezduyaret al. [191], [192], where space-time finite-element method was combined withleast-squares type stabilization, thus, amounting to the integrated solver for thesystem of flow equations; the interface was tracked in a Lagrangian way.

Though computational results obtained with diverse ALE-based algorithmsare very good, the changes of interface topology lie beyond the capabilities of themethod and the complexity of implementation seems to be rather high, especiallyin 3D.

B.2. Integrated flow–interface treatmentThere is a special group of methods based on Lagrangian-Eulerian conception ofmesh movement and on the fully coupled (“integrated”) treatment for the sys-tem “flow variables – interface”. These methods were proposed in the works ofRuschak [157] and Saito and Scriven [164] for steady free-surface flows, and thenextended to unsteady flows with free moving boundaries in Christodoulou andScriven [25], Cuvelier [31], Engelman and Sani [49], Kheshgi and Scriven [99]. Inthe work [25] an elliptic mesh generator was advocated for producing curvilinearboundary-conforming grid, while in the others an algebraic generation of meshwas employed. A similar strategy has been recently presented in Sackinger etal. [162]. The generation of interface-fitted mesh at each time step with the cor-rection of advective velocity taking account of a grid motion is a commonplace forall Lagrangian-Eulerian methods, but integrated approach to the flow–interfacecoupling is something to be discussed here. In this approach the system of flowequations with free-surface boundary conditions is discretized as a whole in re-spect to the flow variables and to some functional representation (parametriza-tion) of the interface. The resulting system of the nonlinear algebraic equations isthen solved using a Newton or quasi-Newton iterative procedure. The approachprovides very fast (quadratic as compared to linear for segregated method) con-vergence towards the steady-state solution, however, for purely transient prob-lems there remain some open questions: (i) whether the iterative process withineach time step should always converge to some “fixed point” (having in mindthe lack of uniqueness of the solution for certain ranges of physical parameters),(ii) how to find a good initial approximation for the Newton iteration, (iii) howto calculate efficiently the Jacobian matrix of discrete nonlinear operator, and (iv)whether it really makes sense to treat so accurately the coupling “flow variables– interface” on each time step, while the time discretization error of the entireprocess usually dominates.

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C. Eulerian methodsC.1. Surface-trackingEulerian methods are used in combination with either interface-tracking or inter-face-capturing approach. The former approach can be further decomposed intosurface-tracking and volume-tracking, the peculiarities of which we are going toconsider below.

Surface tracking methods represent an interface as a series of interpolatedcurves through a discrete set of points on the interface. At each time step, theinformation about the location of the points and sequence in which they are con-nected is saved. The points are then moved according to an interface evolutionequation. The information regarding location as well as orientation and curva-ture of the interface is explicitly available during the whole calculation process.

There are two general forms of surface-tracking methods: (i) the points aresaved as a sequence of heights above a given reference line, (ii) the points followa parametric representation (see figure 3, left). The first approach fails if the in-terpolated curve becomes multivalued, which strongly limits a practical utility ofthat method.

FIGURE 3 (left) Eulerian method with surface-tracking; (center) Eulerianmethod with volume-tracking; (right) Eulerian method with interface-capturing.

The main advantage of surface-tracking methods is their ability to resolve fea-tures of the interface that are smaller than the cell spacing of the Eulerian grid onwhich the interface is overlaid. The main disadvantages are the following: (i) itis very difficult to handle merging and folding interfaces (this requires reorder-ing the interface points and can result in a significant logical programming andcomputational overhead) and (ii) the points can accumulate in one segment ofthe interface, leaving other segments without enough resolution.

For the overview of early works on surface-tracking methods, the paper byHyman [89] may be consulted. The later works using surface-tracking approachare due to the Glimm group (see Glimm et al. (1986) [61], Glimm et al. (1988) [62]),where finite element approximation was used with locally adaptive grid, and dueto the Tryggvason group ([201], [50], [51]), where some specific algorithm wasproposed allowing to handle merging interfaces in 3D (see also [63]). The Tryg-gvason group employed the finite difference method and segregated approachfor the system of flow equations. Among the recent works on surface-tracking

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we could mention the book by Shyy et al. [171], in which another algorithm forhandling interface topology changes may be found, the paper of Popinet andZaleski [137], relying on the finite volume discretization, and the thesis of Torn-berg [195], where the finite element method is used. In all these works, tracking ofthe interface is computationally segregated from the calculation of flow variables.

C.2. Volume-trackingVolume-tracking methods do not store a representation of the interface but recon-struct it whenever necessary. The reconstruction is done cell by cell and is basedon the presence of marker quantity within the cell. The marker particles are onlyused to show which cells contain fluid (or some particular fluid in the case ofmulti-fluid flow). These particles are moved with a fluid velocity in a purely La-grangian manner, which gives rise to the notion of “volume-tracking” (see figure3, center).

The first Eulerian volume-tracking algorithm for free-surface flows seems tobe the marker-and-cell (MAC) method of Harlow and Welch [77]. This approachuses fixed uniform mesh, on which the flow equations are approximated by thefinite-difference method and then resolved in a segregated fashion using either apressure Poisson equation or some version of velocity/pressure-correction algo-rithm (see, e.g., [18]). The free surface is given by those cells which both containfluid, i.e. marker particles, and are adjacent to an empty cell. Thus, the orientationof the free surface inside a particular cell is not obtained. The main advantagesof the method are: (i) it can treat any number of fluids, (ii) it can treat interfacessubject to large distortions, and (iii) it can simulate interacting interfaces. Theproblems associated with the method are as follows: (i) the method does not giveany details about the exact location, orientation, and curvature of the interface,(ii) the particles may accumulate in portions of the grid leaving other portionsnot well resolved, (iii) the method is computationally expensive because it effec-tively requires a double grid system (Eulerian and marker particles), and (iv) it isdifficult to impose the boundary conditions on the interface.

Despite the above mentioned drawbacks, the MAC method became very pop-ular owing to its logical simplicity and flexibility in handling large interfacial de-formations. The approach was extended and strenghtened by many researchers;we would mention here the paper of Hirt and Cook [82], where the pressure-correction segregated algorithm was proposed as a flow solver, the work of Nohand Woodward [128] with the presentation of the improved algorithm for inter-face reconstruction, and the paper by Ramshaw and Trapp [145] containing theefficient algorithm for accurate treatment of fluid convection. For more recentalgorithm using marker particles idea the work by Glowinski et al. [65] may beconsulted, in which the segregated fractional-step approach was utilized for thesystem of flow equations, and the finite element method was employed for thespatial discretization. In the paper of Nakayama and Mori [124] the MAC-typemethod was used also in combination with finite element approximation andwith segregated (pressure-correction) approach for the flow equations.

It is worth noting that, like in other methods using purely Eulerian way offlow modeling, the treatment of flow–interface coupling is performed in a segre-gated manner within the volume-tracking algorithms.

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C.3. Interface-capturingIn the interface-capturing methods the interface is reconstructed from the prop-erties of suitable field variables, such as fluid fractions. Namely, the interface isrepresented as either a discontinuity line of some characteristic function (“dis-continuous approach”) or a zero-level set of some implicit function (“continu-ous approach”). That function obeys pure transport equation, which states thatthe interface is a material line propagating with the fluid. Thus, in contrast tothe Lagrangian motion of particles in interface-tracking methods, the interface-capturing approach relies on the advection of some field variable through fixedEulerian grid. The interface is recovered from current distribution of that fieldvariable, which explains the term “interface-capturing” (see figure 3, right). Thisgroup of methods is sometimes called also “interface-embedding” or “volume-tracking”, but we reserved the latter term for the interface-tracking algorithmsusing marker particles spread over the fluid volume (see previous section C.2).

C.3.1. Discontinuous approachThe first algorithm of this type was suggested by Hirt and Nichols [84] and iscalled volume-of-fluid (VOF) method. The method defines a function which isequal to unity at any point occupied by fluid (or by one of the fluids for two-fluid flow) and zero elsewhere. Thus, the interface is a discontinuity line, andthe discontinuous function satisfies the pure convection equation with the fluidvelocity as an advective velocity.

A VOF-type algorithm generally consists of two parts: a propagation step anda reconstruction step. The first step should be done with a great care, since theadvection of discontinuous function poses a serious problem for numerical meth-ods. The reconstruction step also requires special attention, as the location, orien-tation and curvature of the interface directly affect the approximation of viscousstress and of surface tension force at the interface. In the original VOF algorithmthe simple line interface calculation (SLIC) (see [128]) method was used for theinterface reconstruction, yielding only first-order accuracy in determining the in-terface location; later, the piecewise linear interface construction (PLIC) methodwas proposed (Ashgriz and Poo [5], Puckett et al. [139], Rider and Kothe [148],Rudman [156]) which is second-order accurate.

The VOF interface-capturing methods are a commonly used numerical tool infree-surface hydrodynamics due to the following main reasons: (i) they can eas-ily treat reconnection or merger of interfaces, (ii) they preserve mass in a naturalway, and (iii) they can be relatively simply extended to three-dimensional prob-lems. The major shortcomings of the methods are: (i) the necessity to advect adiscontinuous VOF-function, (ii) the difficulties in determining the precise loca-tion of the interface as well as the interface normal and curvature, (iii) numericalsmearing of the interface details and of the interfacial boundary conditions.

The VOF-type algorithms usually employ a segregated treatment for the sys-tem ”flow variables – interface” and finite difference or finite volume approx-imation methods with fixed grids. For the review on state-of-the-art VOF-likemethods one may be pointed to the papers by Rudman [156] and by Scardovelliand Zaleski [165]. To mention some other references, we would cite first the workby Brackbill et al. [15], which is remarkable due to the continuum surface force(CSF) approach proposed to include the surface tension into the right-hand side

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of momentum equation rather than into the interfacial jump conditions. The CSFmethod has been used by many authors, see e.g. Williams et al. [205] for thereview of most recent developments, Nakayama and Shibata [125] for the finiteelement implementation of the VOF-CSF method. A variant of CSF approach,the so-called ”continuum surface stress” method, was considered in Lafaurie etal. [105] within finite volume framework and in Wu et al. [206] with finite elementmethod. The VOF interface-capturing method has been exploited in the absenceof surface tension effect by Jeong and Yang [90] (finite-element segregated algo-rithm), Kawarada and Suito [96] (finite-difference, segregated, pressure Poissonequation algorithm), Kelecy and Pletcher [97] (finite-volume, segregated, artifi-cial compressibility method), Pan and Chang [132] (finite-volume, segregated,artificial compressibility algorithm), Puckett et al. [139] (finite-difference, segre-gated, pressure-correction algorithm for the flow equations), Vincent and Calta-girone [202] (finite-volume, segregated, pressure-correction method).

C.3.2. Continuous approachIn contrast to the representation of the interface as a discontinuity line within dis-continuous interface-capturing framework, in the continuous approach the inter-face is defined as a zero level set of some continuous function. The immediateadvantages of such representation over the VOF-type interface definition are: (i)considerable simplification of the interface convection problem (as far as convect-ing a continuous function is much easier than convecting a discontinuous one)and (ii) convenient expressions for the interface normal and curvature, allowingfor natural extensions of these quantities off the interface all over the domain.In addition, the continuous interface-capturing approach shares with the VOFmethod its strength in handling multiple interfaces and in easy extension to 3D.Unfortunately for the continuous approach, there remain two major drawbacks ofthe discontinuous one, namely, the inaccuracy in determining the interface loca-tion and numerical smearing of the boundary information at the interface. Addi-tionally, mass conservation is usually worse than in VOF-like methods. However,in principle, the implementation is very similar to that of the methods of discon-tinuous approach (for instance, the flow–interface coupling is usually treated ina segregated manner, and spatial approximation is done on fixed grids).

The first work on the continuous interface-capturing algorithms seems to bethe paper by Dervieux and Thomasset [42], where the interface was defined as azero level set of a continuous ”pseudo-density” function, and integrated finite el-ement method was used for the solution of flow equations. As in VOF approach,two distinct steps are connected with the interface propagation: advection of the”pseudo-density” function and its reinitialization. This second step was foundnecessary to prevent numerical instabilities related to the interface motion; aftereach convection step the ”pseudo-density” was reinitialized to be a signed dis-tance with respect to the interface.

Later, the development of continuous interface-capturing evolved in two paral-lel ways: one based on the notion of ”pseudo-concentration” function (see Thomp-son [194], Dhatt et al. [43], Lewis et al. [111], Medale and Jaeger [120], Lock etal. [115], Lewis and Ravindran [112]) and the other based on the ”level-set ap-proach” (Osher and Sethian [131], Sussman et al. (1994) [183], Chang et al. [20],Sussman and Smereka [182], Zhang et al. [210], Zhao et al. [211], Sussman et

19

al. (1999) [179], Tornberg [195], also the book by Sethian [168] and referencestherein). The former way has been elaborated mostly for metal industry appli-cations; it relied on the finite element approximation, the surface tension wasneglected (except [115]) and the interface topology changes were excluded. Thus,the full strength of the approach could not be demonstrated. The works on level-set method have been devoted to diverse physical applications (see [168] for alarge collection of problems treated with the level-set method). Particularly, mostof the works cited above have been focused on the study of bubble/drop dynam-ics; they utilized the finite difference approximation (except [195] based on finiteelements), and special attention was paid to modeling the surface tension effectsand interface breakup/merger phenomena.

D. Mapping methodsIn the mapping method the physical irregularly shaped flow domain is trans-formed onto a fixed regularly shaped computational domain. The mapping func-tion appears explicitly as one of the unknown functions and has to be determinedtogether with the field variables. The transformation is done on each time step,thus reducing the problem to a fixed domain problem within each time step. Themapping methods are essentially close to the mixed Eulerian-Lagrangian algo-rithms with numerical grid generation: the one-to-one transformation of physicaldomain onto the computational domain uniquely defines the mapping of a fixedcomputational grid onto some interface-fitted adaptive grid in physical domain;the latter grid, thus, evolves like in adaptive mesh Eulerian-Lagrangian methods.

The main advantage of the mapping approach is the ability to maintain sharpresolution of the interface. Major drawbacks include the applicability only to thegeometries that do not lead to singular mappings, and high computational costdue to the necessity of resolving strongly nonlinear equations with coefficientsdepending on the transformation Jacobian.

The early works on mapping method were devoted to steady free-surfaceflows; they used a simple algebraic mapping, segregated approach to flow–inter-face coupling (usually, in the form of a Picard iteration), and relied on the finiteelement (Rivkind (1977) [149], Rivkind (1980) [150], Nitsche [126]) or on the finitedifference (Ryskin and Leal [158], [159], [160], Christov and Volkov [26]) meth-ods. Later, the mapping method was employed for computation of unsteadyfree-surface flows; see, e.g., Kang and Leal [93] using finite differences and themapping generated by an elliptic equation, Takizawa et al. [186] based on finitedifferences and on non-orthogonal curvilinear coordinates, Liu and Ikehata [114]and Volkov [203] exploited finite differences and algebraically generated map-ping.

We have considered some of the most popular methods suitable for simulatinggeneral viscous interfacial flows. For each method its main advantages and draw-backs have been pointed out, and some of the most representative references havebeen listed.

Finally, it is worth noting that many good numerical approaches (for exam-ple, boundary integral methods and vortex methods) have been left beyond thescope of the present overview, as they apply to simplified forms of flow equationsonly, and, thus, do not allow to model an interplay of fluid convection, viscousdiffusion and capillary forces in the flow of two immiscible fluids.

20

1.2 Computational strategy and thesis outline

The detailed comparison of diverse numerical methods for interfacial flows ledus to some specific choice of basic components of the numerical modeling strat-egy. First, we choose purely Eulerian approach, since it enables us to use a fixedstructured grid on a fixed computational domain. The particular advantages ofhaving a fixed grid have been discussed in preceding section. Second, we rely onthe interface-capturing in order to be able to deal with complex interfacial mo-tions including interface merger, folding and break-up. In particular, the level-setapproach is taken in the present work. Next, we employ the operator-splittingapproach that immediately yields a segregated treatment of not only the flow–interface coupling but also of different parts of the problem, which correspond todifferent physical processes. This is computationally very advantageous, as thespecialized numerical scheme can be used for each part of the problem’s operator,and, instead of one very large problem, we have to resolve a sequence of smallersubproblems.

Finally, we choose the finite element method for spatial discretization. Thismethod has a number of strengths not inherent to other methods of spatial dis-cretization: (i) global variational (weak) formulation which assumes the minimal,natural regularity necessary for existence of the unique solution, (ii) natural in-corporation of coefficient discontinuities and singular forces into the numericalscheme, (iii) natural incorporation of gradient (stress) boundary conditions intothe scheme, (iv) capability of local adaptivity of the approximation. The oftenmentioned property of easy approximating a complicated geometry is possessedalso by the finite volume method. Although some of the aforementioned features(i)–(iv) are exhibited by other types of spatial discretization, not solely by the fi-nite element method, the combination of all these features seems to be inherentto the finite element approximation only. We will actively exploit all these prop-erties of the finite element method during the development of the computationalalgorithm.

The rest of the thesis is organized as follows. In Chapter 2 we carefully derivethe mathematical model for unsteady viscous two-fluid interfacial flow. We dis-cuss the physical assumptions forming the physical model of the problem, thenconsider in detail a derivation of the interfacial conditions. The variational andclassical formulations of the problem as well as the question of the problem solv-ability are discussed in section 2.3. Chapter 3 is devoted to the construction of thecomputational method. We start with the brief overview of main techniques forthe numerical solution of the Navier-Stokes equations, then address the operator-splitting approach and its implementation for the Navier-Stokes system. We thor-oughly consider possible velocity and pressure approximations, and pay a spe-cial attention to the comparison of our method for accounting the surface tensionforce with other existing techniques. Section 3.2 addresses the interface approx-imation by the finite element level-set approach. We show that the continuousfunctional representation typical for the finite element method allows us to obtainobvious benefits as compared to the classical finite difference level-set approach.Particularly, we present very simple reinitialization and correction procedureswhich guarantee the optimal, second-order accuracy of mass conservation. Insubsection 3.2.5 the second-order accurate approximations of the interface nor-

21

mal and curvature are proposed, making possible an accurate evaluation of thesurface tension force. Evaluating that force and density/viscosity coefficients isaddressed in subsection 3.2.6. The stability issues strongly affecting the time-stepsize are considered in section 3.4. In Chapter 4 we present diverse numericalexamples of the algorithm performance, including bubble dynamics, Rayleigh-Taylor instability and jet flow. Finally, in Chapter 5 we draw conclusions anddiscuss some possible directions for further research.

2 MATHEMATICAL MODEL

2.1 Physical assumptions

A. FlowWe consider an unsteady laminar flow of two immiscible fluids. Both fluids areassumed to be viscous and Newtonian. Moreover, we suppose that the flow isisothermal, thus neglecting the viscosity and density variations due to changesof a temperature field. We assume also that the fluids are incompressible. Thevalidity of this assumption is affected by several factors, the most important ofwhich is the condition for Mach number to be smaller than, approximately,

(see Batchelor [9, 3.6] for thorough discussion on the incompressibility assump-tion). That condition is satisfied in all cases of our interest, since we deal withessentially subsonic flows. Presuming in addition the fluids to be homogeneous,we may infer that the densities and viscosities are constant within each fluid.However, the density as well as viscosity is different for two different fluids. Torealize how these physical parameters change from one fluid to the other we haveto consider in some details the notion of interface between the fluids.

B. InterfaceThe nature of the interface between two fluids has been the subject of extensiveinvestigation for over two centuries. Young, Laplace, and Gauss, in the early partof the 1800s, considered the interface between two fluids to be represented as asurface of zero thickness (“sharp interface”) endowed with physical propertiessuch as surface tension. In these investigations, which were based on static ormechanical equilibrium arguments, it was assumed that physical quantities suchas density or viscosity were, in general, discontinuous across the interface. Phys-ical processes such as capillarity occuring at the interface were represented byboundary conditions imposed there (e.g. Young’s equation for the equilibriumcontact angle or the Laplace-Young equation relating the jump in pressure acrossan interface to the product of surface tension coefficient and curvature).

In the second half of 19th century, Poisson, Maxwell and Gibbs recognizedthat the interface actually represented a rapid but smooth transition of physicalquantities between the bulk fluid values. Gibbs introduced the notion of a di-viding surface in order to develop the equilibrium thermodynamics of interfaces.The idea that the interface has a non-zero thickness (i.e. it is diffuse) was devel-

23

oped in detail by Lord Rayleigh and by van der Waals at the end of 19th century.A bit later, Korteweg proposed a constitutive law for the capillary stress tensorin terms of the density and its spatial gradients. These original ideas have beendeveloped further and refined over the past century (see Anderson et al. [3] forrecent review on diffuse-interface methods).

The main disadvantage of the diffuse-interface approach is the uncertaintywith the interface thickness (transition region), which has to be defined empiri-cally to make the model closed. In fact, the classical Young-Laplace-Gauss me-chanical approach representing the interface as a surface of zero thickness mayfail only when the interfacial thickness is comparable to the length scale of thephenomenon being examined (see [3]). The major examples are: (i) the flow of anear-critical fluid (interface thickness becomes infinite as the critical temperatureis approached), (ii) the motion of a contact line along a solid surface, and (iii) theflows involving changes in the interface topology. The first case requires the toolsof statistical thermodynamics and is beyond the scope of our investigation. Thesecond case is also quite specific if the fluid motion in the vicinity of the contactline is of primary importance; if not, some techniques (e.g., based on the partial-slip condition) may be used to correctly recover the general picture of fluid andinterface motion. Concerning the flows with topological changes of the inter-face, we may cite Scardovelli and Zaleski [165, p. 574]: ”... indeed, in that casethe macroscopic impact of microscopic physics may be limited. In other words,the macroscopic interface motion may be relatively less dependent on interfacephysics. This is because universal macroscopic solutions that lead to a singularityin finite time may be found...”. Thus, in spite of continuing controversy relatedto the plausibility of the sharp-interface model predictions, there is a strong hopefor realistic simulations of interfacial topology changes with this approach.

In light of the above, we utilize the sharp-interface (zero interfacial thickness)approach; the density and viscosity have, therefore, a jump discontinuity at theinterface (see, e.g., Lamb [107], Batchelor [9]). We assume that the interface hasa surface tension. We also suppose that there is no mass transfer through theinterface (i.e. the interface is impermeable), and there are no surfactants presentin the fluids (hence, there is no species transport along the interface). Under suchconditions we do not have to consider the variations of surface tension coefficientin tangential to the interface direction, i.e. the solutocapillary Marangoni effect(the thermocapillary Marangoni effect has been excluded by the assumption onisothermal character of the flow). Therefore, the surface tension coefficient maybe assumed constant.

2.2 Equations and interfacial conditions

Suppose that the motion of two viscous immiscible fluids under our investigationis confined to some box (a parallelepiped in 3D, a rectangle in 2D). The boundaryof the box can be physical (e.g., the walls of a container), artificial (if we considera flow in unbounded domain) or partly artificial. In fact, we can always restrictourselves to some bounded region of interest and consider, then, the flow in thatregion only. On the other hand, this enables us to avoid dealing with asymptoticsat infinity and to make the problem more tractable from computational view-

24

point. We denote the boundary of the box by , the domains occupied with thefluids by and and the interface between the fluids by ( ,where is the boundary of ), see figure 4. Let also be the entireregion occupied with the fluids, i.e. the interior of the box ( ). Thedomains and may be multiply connected, and the interface may intersectthe box boundary .

2

Ω

Ω

Ω

Σ

Γ

1

2

FIGURE 4 Sketch of a two-fluid flow configuration.

Taking into account the physical assumptions considered above, we may assertthat the flow of each fluid is governed by the incompressible Navier-Stokes equa-tions

! " $#&%(' #*),+-#/.01+2)4357698 : ' +<;= ! ?>@ (1)

+2)A#BDC in EF HG (2)

Here #39IJ%K: is the velocity of fluid, ;3?IJ%K: is the pressure, 8 3L+M# ' 35+M#/:N: isthe deformation rate tensor, ! O3PQ R: is the density of i-th fluid, 6SO39Q R: isthe dynamic viscosity of i-th fluid, and g is the acceleration of gravitational field(constant).

Obviously, the model described by the system (1)–(2) is incomplete. Thus, asa next step we derive the differential mass and momentum balance conditions onthe interface.

Interfacial conditionsSince we have assumed that the interface possesses such important physical prop-erty as the surface tension, it is necessary to find an explicit mathematical expres-sion for the surface tension effect. The latter may be derived by summing thetensile forces acting on an interfacial fluid element. The net tensile force, or sur-face force, is then automatically given as a sum of forces normal and tangentialto the interface (see Batchelor [9, 1.9], Brackbill et al. [15]).

Consider, as in figure 5, an element of area T7UVXWFTZY about the point IJ[ onthe interface . The interfacial element is enclosed by a curve \ having elementalarc length ]4^ . Denote by _`[ the net surface force per unit area.

25

FIGURE 5 Sketch of an interfacial fluid element of the area TZY .

The surface force exerted on the material in T,Y by the material outside of T,Y andacting across the line element , from figure 5, is equal to ]4^ , where is theunit tangent to that is perpendicular to arc length vector ( ]4^ W ) at apoint along \ . Here is the coefficient of surface tension. The net surface force onelement TZY , _O[`TZY , is found by summing all forces ]4^ exerted on each elementof arc length ]4^ ,

_[`TZY ]4^ 3 W/: <] Y-35W +=: 3SW/: T,Y 35W +(: 3SWF: for TZY C (3)

where we have used Stokes theorem. In the limit that T,Y C , we can from (3)identify _O[ 39IS[: as

_[ 3?I [K: 35W +(: 3SWF: (4)

which upon letting the differential operator work on both W and , becomes

_[ 39I [K: !" 35W +(: W# ' W 3L+$S:#*W(G (5)

The differential operator can be written as the sum of surface and normal opera-tors, + +&% ' +&' , where +&' W3PW ),+=: , so that

W + W 3L+&% ' +$': W +&%Q (6)

since W +&' C . Furthermore, by using the identities

35W +$%H: W

+&%3PW*)ZW/:/0 W 3L+$% )ZWF:J 0 W35+&% ) WF: (7)

and

26

W 35+$ : *W +$ 0 W 35W ),+(: +$% M (8)

(5) can be rewritten as

_[ 39I [K: 0 WJ3L+&% )ZW/: ' +&% M (9)

from which we can identify

_ [ 39I [`: 0 WJ3L+&%)ZW/: (10)

as the normal component of the surface force, and

_ [ 3?I [:J+&% (11)

as the tangential component of the surface force. Now we can recall that the meancurvature of the surface, , which is, by definition, equal to 3 ' : ( and are the principal radii of curvature at the considered point of the surface), mayalso be expressed as (see, e.g., [15], [36], [168])

0

35+&% )ZWF:G (12)

It is worth noting that (12) defines the signed mean curvature, namely, C atthe points where the surface is convex in the direction of the normal W . It is alsoremarkable that + %)KW in (12) can be replaced by + )KW if W is defined not only on asingle surface but in a whole space (since + ' )9W W )`35W)P+(:KW 3PW)P+=: 35W)9WF: C ).

By virtue of (12), we can rewrite (10) as

_ [ 39I [K:J SW=G (13)

Since we have assumed the absence of a variation of the surface tension coefficientalong , the tangential component (11) of the surface force should vanish, and theresulting net surface force per unit area from (9) becomes

_[ 39IS[: SW( (14)

where we have used the notation for twice the mean curvature. From(14) we can see that surface tension results in a net normal force directed towardthe center of curvature of the interface.

After we have obtained a mathematical description for surface tension, we canderive the differential balance equations at the interface.

27

Consider any point I[ of the interface separating two domains and occupied by the first and by the second fluid correspondingly. Let W be the normalto at I [ pointing from to (we suppose that is smooth). Let be thecoordinate system moving with and such that instantaneous velocity of pointI [ at time % equals zero in . Let # and # be the velocities of the first and of thesecond fluid in at time % . Consider now a small volume containing the pointI [ as an internal point (figure 6). Let be the part of the interface, whichis inside of . Denote the maximal size of in direction of W by . At time % ' ] %the particles of both fluids are located at some new positions. Let us consider asan immobile volume, while the moving material volume consisting of particleswhich were contained in at time % we denote by . So, coincided with attime % .

Ω

ΩΓ

ω

S nh

xS

2

1

FIGURE 6 A small volume about the point I/[ of the interface .

Suppose that Y-3?IJ%K: is a function continuous in everywhere except, possibly,the surface across which Y may have a jump discontinuity (i.e. Y is continuousalong ). The following formula, known as the Leibniz rule, is valid (see, e.g.,Delhaye [37], Sedov [166]):

]]R% &Y ] ]]R% Y ] ' Y # )ZW ]4^ (15)

where W is the outward normal to the surface of volume .It can be shown ([166]) that

Y ] C . Therefore,

]]R% !&Y ] % Y # ) W@]4^ (16)

where Y #" B Y#" 02Y<O#" ; Y and Y are the limiting values of Y fromcorresponding sides of the surface . The formula (16) can be found, e.g., inGurtin [75], Sedov [166] or Slattery [172].

Multiplying (16) by % and passing to the limit as C we deduce the formulafor differentiating an integral over volume shrinking to the point I [

#%$ %&#

]]R% &Y ] Y # )ZW*0 YO# )ZW= (17)

where the fact that Y # is continuous along has been used.

28

Now we are ready to derive the desired interfacial conditions at . First, wewrite down the mass and momentum balance equations for the volume in theintegral form:

]]R% ! ] C (18)

]]R% ! # ] ! ><] ' ) W ]4^' % TH39IS[/0 I : SW ] IS[K]RI (19)

where in (18) the assumption on the absence of mass transfer through the inter-face has been taken into account. In the momentum equation (19) is the stresstensor ( 0 ; ' 76F8 , 6* 6/ or 6 , is the identity tensor, ; is the pressure and 8is the deformation rate tensor defined before). The last integral in (19) representsthe surface tension effect due to interface-concentrated capillary force; T is a deltafunction of Dirac, and the expression (14) for the surface force per unit area hasbeen employed.

Further, shrinking to the point IF[ and using (17) we obtain from (18)

3 ! # 0 ! O# :/)ZW CG (20)

For the momentum balance (19) we use (16) (shrinkage of to )

% ! # # &)ZW ]H^ % &)ZW@]4^ ' % SW ]H^ and, then, dividing by and passing to the limit as C , obtain

3 ! #S # 0 ! #/O# :/)ZW 3 0 :/)ZW ' SW(G (21)

We have derived the interfacial conditions (20)–(21) in moving (local) frame ofreference connected with the point IF[ of the interface . To rewrite them instationary (global) frame of reference we have to notice that # ) W D# O) W 0 3PF R: , where # is the velocity of i-th fluid in global frame of reference, and isthe interface velocity, i.e. the velocity of interface motion in the normal direction.Hence, from (20) we have! 39#/F)ZW 0 :J ! ,39#S )ZW 0 : (22)

which is, in fact, the well known Rankine-Hugoniot condition. If there is no masstransfer across the interface, both parts of the latter equality should be zero (sincea mass flux from one side of the interface to the other is zero), which implies

D#//) W D#S ) W=G (23)

Thus, we have obtained the condition on continuity of the normal velocity at theinterface as a direct consequence of mass conservation law. From (23) we also seethat # ) W C - (the interface moves with the fluids, i.e. normal com-ponents of fluid velocities are zero in local coordinate system of the interface).

29

Hence, from (21) the following stress jump condition on the interface can be im-mediately derived:

3 0 :/)ZW SW=G (24)

This condition may be split into a normal and tangential stress jump condition

W*)43 0 :/)ZW M (25) )4396F 8 0 6 8 :/)AW C (26)

where the vectors may be any set of ] 0 independent tangent to vectors,

and ] is the dimension of space.Condition (26) indicates that the tangential stress is continuous across the in-

terface, while condition (25) shows that the jump of the normal stress at the inter-face is balanced by the capillary pressure. It is worthwhile to note that if both flu-ids are inviscid, the normal stress jump condition (25) reduces to the well knownLaplace-Young equation

;$ 0*; M (27)

indicating that the higher pressure is in the fluid medium on the concave side ofthe interface (see, e.g., Finn [55]).

2.3 Weak and classical formulations

The equations (1)–(2) and interfacial conditions (23)–(24) should be complementedwith some boundary condition on for velocity, for example,

#B C on (28)

and with the initial conditions

(29)# D# in (30)

where is the initial position of the interface determining initial shapes of thedomains and .

Below we will give a weak (variational) formulation of the problem. We usea standard notation for Sobolev spaces, and suppose that interface belongs tosome functional class if at each point I/[ of there exists a local coordinatesystem, in which the interface may be represented as a single-valued functionfrom for some vicinity of the point I/[ . The spaces of vector-functions aredenoted by boldface letters. The following functional spaces will be also used: 35Q: # 35Q: # C on 35Q: # 35Q: ] # C 3PQ: ; 35Q: ; ] DC .

30

Then the weak formulation reads: find #3?%K: 35Q: and ; 39%K: 3PQ: suchthat for almost every % 3PC : and for any 35Q: , 35Q: ! $#$% ) ] ' ! 39# )Z+-#/:/) ] ' 768 3?# :/) ) +] 0 ;35+2) :$] ! > )D] ' SW*) ]E (31) 3L+2)A#/:$] DC (32)

#3PC :JD# (33)

where # 735Q: and 3?%K: is defined as the discontinuity lineof the piecewise-constant density ! 3?IJ%K: (and of the piecewise-constant viscos-ity 6 3?IJ%K: ), moving from the initial position with the normal velocity equalto # ) W . The formulation (31)–(33) is not, in fact, a canonical weak formulation,which requires much less temporal regularity for the unknown functions and is tobe understood in the sense of distributions on 35C : . Here we consider the equa-tions as being valid at almost every time moment % , thus, gaining a resemblanceto the classical formulation (see below) with respect to time but retaining all fea-tures of a weak formulation in space (see, e.g., Quarteroni and Valli [142, 13.2]for the discussion on alternative weak formulations for unsteady Navier-Stokesequations).

First of all, we may note that having the normal W 35: , cur-vature 35: and the integral over should be understood in the senseof duality pairing between 35: and 3P: (since the trace of belongs to 35: and is a positive constant). Let us show now that this weak formulationis formally equivalent to the system of equations (1)–(2) with boundary condition(28) and interfacial conditions (23)–(24). Indeed, assuming sufficient regularity of#39%K: and ;39%K: we have from (31) after the integration by parts

$ !

" $#$% ' #*),+-# . 01+2)4357698 : ' +<;) ]' ,W )43K3`0 ; ' 76F8 :/0D3`0 ;& ' 76 8 :K: ) ]E ! > ) ] ' SW ) ]4 (34)

where some integrals have been eliminated due to zero trace of on . Thus,from here and from (32) we immediately see that # and ; obey the equations(1)–(2) in 4 and the interfacial stress jump condition (24) as well (thiscondition appears to be natural for the weak formulation). The boundary condi-tion (28) is obviously satisfied since # 3PQ: , and the interfacial condition (23)for normal velocity is also automatically satisfied as the velocity # has a uniquetrace from

35: on the interface. So, the weak formulation implies (1)–(2), (28)and (23)–(24) if the solution is regular enough, and the reverse implication maybe easily shown.

The existence and uniqueness of the weak solution for interfacial flow prob-lem is not a trivial question, owing to the strong nonlinearity caused by coupling

31

the velocity field with the interface motion. Even for free-surface flow of a singlefluid the existence and uniqueness were proved either under the assumption onsmallness of given data (body force, boundary and initial conditions) – global-in-time solvability, or for arbitrary given data but only on some finite time in-terval (with the length of the time interval being dependent on the given data),see, e.g., Beale [10], Solonnikov (1986) [175], Solonnikov (1991) [176]. For inter-facial two-fluid flow problem we may refer to Takahashi [185], where the global-in-time existence (without uniqueness) of the weak solution is proved with theassumption on smallness of given boundary data and of viscosity jump at theinterface; the surface tension effect is neglected. In Denisova and Solonnikov(1989) [40] and in Denisova [39] the local-in-time existence and uniqueness ofthe weak solution are shown, and, if some compatibility conditions for initialvelocity are satisfied and the given data are sufficiently regular, the weak solu-tion is demonstrated to possess an additional regularity, particularly, velocity# 3PC 35 :: i.e.

N # ]R% ' . This is important in-formation for estimating the rate of convergence of an approximate solution tothe exact solution of the problem.

It is useful to consider also a classical (strong) formulation of the problem,since the topology of Sobolev spaces is not quite natural for investigating theinterface regularity: the Holder spaces are the most suitable choice.

The classical formulation of our problem is as follows: at every time moment% 3PC : find the boundary 3?%K: between the domain < 39%K: occupiedwith the fluid of viscosity 6 and density ! and the domain Q,39%K: occupiedwith the fluid of viscosity 6 and density ! , as well as the velocity vector field #and the pressure field ; of those fluids, which satisfy the initial-boundary valueproblem for the Navier-Stokes equations

! " $#&% ' #*),+-# . 01+2)4357698 : ' +<;= ! ?>@ (35)

+2)A#BDC in H&F % C (36)# # /# C on (37) # C 0 0 ; ' ,6F8 )ZW W=G (38)

Here Y Y-3?I:0

Y@39I: , for any Y , is the jump across the inter-

face, and all other symbols have been defined before. In addition, the interface moves from the initial position with the normal velocity equal to # )ZW .

We see that to the derived above interfacial conditions (23)–(24) the conditionon continuity of tangential velocity at the interface is added in classical formula-tion. This condition, though does not follow from any conservation law, is justi-fied from physical viewpoint owing to effects of viscosity: the condition is akin tothe assumption that the slip velocity on a solid wall vanishes. From mathemati-cal point of view, this condition is required to guarantee the existence of a strong(continuous) velocity solution on . It is worth noting that the condition is also“hidden” in the weak formulation: since velocity belongs to

over the whole (due to the elliptic part of the Navier-Stokes system – effect of viscosity !) it has aunique trace from

3P: on the interface; this fact may be interpreted as a weak

32

form of the continuity condition for both normal and tangential components ofvelocity at the interface. The continuity of velocity on the interface will be takeninto account during the approximation of the problem.

The solvability of the classical statement of the problem in Holder spaces isinvestigated in Denisova and Solonnikov (1995) [41], where a local (in time) exis-tence of unique classical solution is proved and some additional regularity of thesolution is shown (provided the compatibilty conditions for initial velocity aresatisfied and the given data are sufficiently smooth). In particular, is shown tobelong to \ (see also Rivkind (1983) [151] where the same regularity is provedfor steady interfacial flow). This fact is very useful for the approximation of in-terface normal and curvature.

To summarize, we may write down the system of equations over the entire and of interfacial conditions on , which is to be understood in a weak sense andto be used in the sequel for the construction of numerical algorithm:

! 39I : " $#$% ' # ),+M# . 0 + )E3576 39I : 8 : ' +<; ! 3?I:`>- (39)

+2)A# C in B% C (40) # C 0 0 ; ' 76F8 ) W SW= (41)

given some boundary and initial conditions. Here ! 39I : ! in and ! in ,6 3?I:J 6F in and 6 in .The problem at hand contains several intrinsic difficulties, some of which are

typical for any model described by the incompressible Navier-Stokes equationsbut some are caused by the presence of a free moving interface. The major diffi-culties are:

– fluid convection ( nonlinearity)

– incompressibility

– density/viscosity coefficient discontinuity

– interface-concentrated capillary force

– interface convection

– influence of interface shape on flow and vice versa ( nonlinearity)

The success of numerical modeling for the problem ultimately depends on thetreatment of above mentioned difficulties. The rest of the present work is devotedto finding efficient and accurate ways of overcoming the difficulties, which has toresult in the development of a reliable computational strategy.

3 DEVELOPMENT OF THE COMPUTATIONALMETHOD

Now we proceed with the main part of the present work, that is with the construc-tion of numerical method for solving our problem. The problem contains threekey ingredients: (i) flow equations (i.e. the Navier-Stokes equations with dis-continuous coefficients and singular source term), (ii) moving interface and (iii)coupling between velocity-pressure fields and the interface (through the coeffi-cients, capillary force and interfacial advective velocity). It seems reasonable toconsider the approximation of the problem in a step-by-step manner that is essen-tially close to the operator-splitting approach (see section 3.1.2 below); namely,we study first the approximation of the Navier-Stokes system with fixed knowninterface, then we find an appropriate approximation for the interface, its nor-mal and curvature, and, finally, we consider the flow—interface coupling, i.e.evaluate the surface tension force and the density/viscosity coefficients using theconstructed approximation of the interface.

3.1 Discretization of the Navier-Stokes equations

3.1.1 Overview

As it was pointed out in section 1.1, there are two general approaches to com-puting the time-dependent Navier-Stokes equations in primitive variables (i.e.velocity-pressure) formulation: integrated and segregated. The former approachconsists in seeking the velocity and pressure simultaneously, which results invery large algebraic systems with unpleasant numerical properties (see the bookby Cuvelier et al. [33] for the discussion on integrated methods). Within the segre-gated approach the velocity and pressure calculations are decoupled; that impliesthe solution of smaller systems and, thus, significantly reduces a computationalcost. Among the segregated methods we can mention the one based on the pres-sure Poisson equation (see the book by Gresho and Sani (1998) [71]), artificialcompressibility method (see the book by Temam [190]) and very similar Uzawamethod for the augmented Lagrangian formulation of Navier-Stokes equations(see, e.g., the book by Glowinski and Le Tallec [64]), the penalty method (see,e.g., the book by Gunzburger [74]), the projection method (see, e.g., the book by

34

Quartapelle [140]), and the method based on the discrete pressure Schur comple-ment (see the book by Turek [200]).

Besides the treatment of velocity-pressure coupling the convective nonlin-earity may be used to divide all methods into two broad categories: with in-tegrated treatment for convective and Stokes parts, and with splitting the con-vection off. While the former group of methods uses some form of fixed-pointiteration, the latter category relies on the specific schemes for pure convection orfor convection-diffusion problems. A fixed-point iteration requires the knowl-edge of good initial approximation, and also the existence of unique fixed pointremains an open question, especially in the case of high-speed flow. Due to its na-ture a fixed-point iteration is well suited for obtaining a steady-state solution, butfor truly transient problems the methods based on fluid transport schemes seemto be superior. As the examples on very successful splitting the convection offthe Navier-Stokes system we would mention the transport-diffusion algorithmof Pironneau (1982) [135] and the fractional-step -scheme of Bristeau et al. [17](see also Glowinski and Pironneau [67]).

There is yet another very important issue related to the approximation of theNavier-Stokes equations, namely, the choice of the discretizations for the veloc-ity and pressure. It is known that the Navier-Stokes system can be recast in theform of a saddle point problem with the pressure playing a role of a Lagrangemultiplier for the incompressibility constraint. According to the analysis for sad-dle point problems the approximation spaces of velocity and pressure must sat-isfy the inf-sup compatibility condition also referred to as the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. If the finite dimensional spaces fail to satisfythis condition, the numerical solution can be corrupted by spurious pressuremodes. On the other hand, the condition is rather restrictive as it prevents theuse of some convenient low-order interpolations for the velocity-pressure pair.There are two ways of dealing with the LBB condition: satisfying it by chos-ing appropriate discretizations for velocity and pressure, and circumventing itby stabilizing the discrete formulation. The first approach is very attractive, asit eliminates the problem with LBB condition completely; many suitable pairsof finite dimensional spaces for velocity-pressure have been found, see, e.g., thebooks by Pironneau (1989) [136] and by Quarteroni and Valli [142] for the collec-tion of finite element velocity-pressure approximations. The stabilization of thediscrete equations can be achieved either by appropriately modifying the formu-lation (see [142, 9.4] for a survey of such techniques) or by resorting to the projec-tion schemes (see the books [190], [140], [71]). The former methods of stabilizationintroduce some artificial numerical parameter which should be properly adjustedto the scheme, while the latter possess an intrinsic stabilization mechanism. Theprojection schemes can be also viewed as some forms of operator-splitting, i.e. ofthe segregated, methods. These facts make the projection schemes very attractivefor cost-effective solution of large-scale transient problems.

We have considered three major questions related to the approximation of theNavier-Stokes system: velocity-pressure coupling, convective nonlinearity treat-ment and the choice of velocity/pressure approximation spaces. In accordancewith these observations it seems reasonable to choose for an unsteady flow prob-lem a segregated method, namely, the projection scheme, with split-off convec-

35

tion. This leads to the so-called fractional-step projection method, which treatsthe nonlinear convection, viscous diffusion and the incompressibility separately,via three subsequent algorithmical steps. Such approach was proven to be veryefficient as it allows to use specialized numerical techniques for each of threephysical phenomena mentioned above.

Below we will cite some works on the fractional-step projection method, butbefore that it is worthwhile to note again (see the discussion in section 1.2) thatwe rely on the finite element method for spatial discretization. Some of the ad-vantages of using this method will be highlighted during the approximation ofthe equations with discontinuous coefficients and singular source term. Thus, wewill focus our attention on the finite element fractional-step projection methods.

The projection method was introduced in the late 1960s by Chorin [23], [24]and Temam [188], [189] in the context of the finite-difference discretization. Later,the method was carried over to finite elements by Donea et al. (1982) [46]. Therigorous theoretical analysis of the method can be found in the books by Temam[190], Quartapelle [140] and Prohl [138] (see also references therein), while the nu-merical aspects as well as the implementation issues are addressed in Gresho [68],[69] (see also [71]) and in Turek [199], [200]. Recently the convergence and sta-bility analysis for the finite element projection method has been strengthenedby Guermond and Quartapelle [72], [73] and by Codina [28] (see also referencesherein). The first works on the fractional-step projection method (i.e. with sepa-rate treatment of the convection) seem to be due to Laval and Quartapelle [108]and to Karniadakis et al. [95]. This approach was taken later by many others;we may refer to, e.g., Achdou and Guermond [1], Glowinski et al. (2000) [66],Kjellgren [100], Kuzmin [103], Lewis et al. [111].

In this section, we consider the application of fractional-step projection methodto the incompressible Navier-Stokes equations. In our case the implementationof the method is complicated by the jump discontinuity of density/viscosity co-efficients and by the presence of singular capillary force.

3.1.2 Operator-splitting approach

Here we briefly touch upon the operator-splitting method, namely, its variantknown as the Marchuk-Yanenko fractional-step scheme, then consider the Chorinprojection method for the Stokes system with interfacial jump conditions, and,finally, combine these approaches to get the fractional-step projection scheme forour problem. We discuss also the splitting of the interfacial jump conditions. Atthe end of this section, we address the spatial discretization of the velocity andpressure.

The Marchuk-Yanenko fractional-step scheme

The differential operators often admit a decomposition into a sum of componentsof simpler structure. This observation is a key issue in the operator-splitting ap-proach, since the operator components can be treated separately rather than si-multaneously; such divide-and-conquer strategy was proved to be very efficientfor handling complex physical problems. The splitting can be performed eitherat the algebraic or at the differential level. The latter variant seems to be more

36

attractive as the operator components have specific physical meaning, and, thus,corresponding mathematical techniques for their treatment may be readily found.However, such a splitting generally requires a decomposition of the boundary(interfacial) conditions as well, in order for endowing the operator componentswith consistent boundary data.

Consider a generic time-dependent partial differential equation

$% ' 3 : C in ! 35C : (42)

in GAssume that the (possibly nonlinear) differential operator

permits the follow-

ing decomposition

' G (43)

Then the Marchuk-Yanenko fractional-step scheme consists in approximating theproblem (42) on each time interval % % by a sequence of two subproblems

G ' 3 :JC in ! 39% % `:

(44)

HG ' Z3 :JC in ! 39% % `:

(45)

where

is the approximation to /3?% : C AGAGAG and . Both the prob-

lem (42) and the subproblems (44)–(45) must be supplemented by appropriateboundary conditions.

This scheme was analyzed in, for example, Marchuk (1990) [118] (see also Ya-nenko [208], Marchuk (1975) [117]). It is only first-order accurate but has goodstability and robustness properties (see Karlsen and Risebro [94]). We have tonote also that the scheme (44)–(45) is especially attractive since each of the sub-problems can be further discretized in time and in space almost independentlyof the other; thus, specialized methods can be used for each component of thedifferential operator. The main drawback of the operator-splitting scheme is thatit is not well suited for approaching steady-state solutions, unless the time step issufficiently small. However, this drawback is not so important here, as we dealwith the transient problems only.

The Chorin projection scheme for the Stokes system with discontinuous coef-ficients

We have considered the operator-splitting scheme that can be used to separate thenonlinear convection from the Stokes system. But, the latter can also be split intotwo parts: the one related to viscous diffusion and volume forces (momentumtransfer), and the other related to the pressure (incompressibility).

37

Consider the following time-dependent Stokes system:

! 39I: &#$% 01+2)43L76 3?I: 8 : ' +;= ! 39I:O>- (46)+ ) #B C in /% C (47) ## C 0 0 ; ' 76F8 )ZW SW on (48)

where (48) is the interfacial jump conditions, ! 39I: ! in and ! in ,6 3?I:J 6F in and 6S in , 8 35+M# ' 3L+-#/:N : ; all symbols have been definedin the preceding chapter.

The system (46)–(48) is supplemented by the boundary condition, for instance,

# on G (49)

In this case, the pressure is determined up to an arbitrary additive constant. Also,due to continuity equation (47), the following compatibility condition must besatisfied: *)ZW@]4 CG (50)

Here and in the sequel, we denote by W not only the unit normal to the interface (see section 2.2), but also the outward unit normal to the boundary . The exactmeaning of the symbol W will be always evident from the context.

Finally, we specify the initial condition:

#39I C :JD# 3?I: in (51)

where it is required that

+ ) # C in B # )ZW )ZW on G (52)

According to the fractional-step method of Marchuk-Yanenko considered above,we can formally split the pressure-incompressibility part off the Stokes system,thus approximating the problem by a sequence of two subproblems:

G ! 39I : $#$% 01+2)43L,6 39I: 8 : ! 39I:O> in ! 39% % K: (53) # C 0 76F8 )ZW SW on (54)#B on (55)# D# in B (56) # GHG $#$% '! 3?I: +; DC (57)

+2)A#BDC in 3?% % K: (58) # ) W CQ ; DC on ( (59)#*)ZW )ZW on (60)# # in B (61) # ;

38

where # 3?I: and ; 39I: are the approximations to #3?IJ% : and to ;3?IJ% : ( C AGAGAG ) respectively, and # # .As it has been pointed out, the splitting implies a decomposition of the bound-

ary and interfacial conditions. The second subproblem defines an inviscid flowproblem, hence, the Dirichlet-type boundary condition is imposed only for thenormal component of velocity. The same is applied to the interfacial conditionfor velocity; the continuity of the normal velocity at the interface is a direct con-sequence of the incompressibility equation (see chapter 2). The splitting of theinterfacial stress jump condition indicates that the surface tension balances thejump of viscous stress only (subproblem 1), while the pressure is simply contin-uous across the interface (subproblem 2). Such approximation of the interfacialcondition seems to be in contradiction with the fact that the pressure does have ajump discontinuity at the interface, which can be clearly seen from the Laplace-Young equation (see section 2.2) if the viscosity is very small. However, we willdemonstrate in numerical tests that chosen scheme is capable of capturing thepressure interfacial discontinuity with a good accuracy; thus, there is no needto enforce the discontinuity explicitly. We will come back to the splitting of theinterfacial stress jump condition a bit later.

It is also noteworthy that some tangential slip is permitted on , since thetangential component of velocity is not prescribed there at the second stage ofthe splitting (subproblem 2). However, this “numerical slip” is 3@%K: (see Gresho[68]), i.e. within the first-order temporal accuracy of the whole scheme.

The scheme (53)–(61) can be interpreted as a variant of Chorin projectionscheme, if we consider it after the (implicit) time discretization:

G ! 39I: # 0# -% 01+2)43L76J39I: 8 3 # :K: ! 39I:O> in (62) # C 0 768 3 # : )ZW W on ( (63)

# on G (64)

HG # 0 # -% '

! 3?I: +; C (65)

+2)A# C in (66) # )ZW C ; C on ( (67)# )ZW *)ZW on (68)

where @% % 0 % is the time step and C GAGAG as before ( @% depends, ingeneral, on , but we drop this index).

Now the projection nature of the scheme is apparent: the intermediate veloc-ity field # is not solenoidal, and the second step (65)–(68) represents the pro-jection of # onto the set of divergence-free functions with prescribed normalcomponent on the boundary. This is, in fact, an -projection, and its mathemati-cal origin may be formulated in the form of the following theorem:

Theorem. Let ! M3PQ: and ! ! almost everywhere in with a constant! C . Then, any function # 35Q: admits the unique orthogonal decomposition

# D# '! +<; (69)

39

where # 35Q:K+ ) C in B )HW C on , ; 35Q: , andthe orthogonality is meant with respect to the ! -based inner product in defined as3?#/#S :Q ! #//)A#S] for any #/ and #S from 35Q: .

Proof. See Bell and Marcus [11]. The result is a straightforward extension ofthe well-known decomposition theorem of Ladyzhenskaya [104].

It is worth noting that the resulting pressure belongs to 3PQ: and the velocitydetermined in (65)–(68) can be expected to belong to but not to

35Q: . Whilethe final velocity can be properly projected onto

3PQ: (see section 3.1.5 below),thus completely recovering the situation, the pressure has to be sought in a nar-rower space than in its “natural” 3PQ: . The convergence results (see [73], [138])show, however, that the pressure approximation remains reasonably good. Theinterpretation of the equations (65)–(68) as an -projection for the intermediatevelocity # leads to the Poisson-type equation for the pressure

0+ )E3 ! 3?I: +<; :J 0

@% +2) # in (70)

endowed with the Neumann boundary condition

W ),+<; CG (71)

The equation (70) is obtained by applying the divergence operator to (65) andusing the incompressibility constraint, while the (71) is derived by taking thenormal component of (65) and using the fact that # ) W # ) W )AW on .

The boundary-value problem (70)–(71) makes sense, as ; 35Q: ,+ ) # 35Q: (since # 35Q: provided 3L: and is sufficientlysmooth) and, thus, WM) +; 35: (see, e.g., [60, p. 28]); the compatibilitycondition

+D) # ] C is also satisfied since +D) # ] # ) W ]H )ZW@]4 C by virtue of (50).

The solution of the Neumann problem (70)–(71) is defined up to an additiveconstant. In order to fix the unique solution we may seek ; in the quotientspace 35Q: equipped with the norm ; ; (see, e.g., [60,

p. 13]).

Remark 3.1. The homogeneous Neumann boundary condition (71) is non-phy-sical and the exact pressure does not satisfy it (see Gresho and Sani (1987) [70],Orszag et al. [130], where it is concluded that the correct boundary condition forthe pressure can be derived by applying the normal component of the momen-tum equation (46) at the boundary; see also Shen and Temam [169]). This discrep-ancy gives rise to a spurious boundary layer of width 3

@%: for the pressure( 6 ! is the kinematic viscosity). Since

@% is usually small, the layer is

very narrow for most of the practical situations, and ; is indeed a reasonableapproximation to the exact pressure ;3?% : ([68], [138]).

The considered above interpretation of the second step (65)–(68) of the Chorinscheme as an -projection is often called “continuous projection”, since it dealswith the differential operators before their spatial discretization. It is noteworthythat this variant of the Chorin method has some intrinsic stabilization property;namely, it was shown by Rannacher (1992) [146] that the continuous projection

40

scheme can be viewed as a semi-explicit pressure stabilization scheme with @%playing a role of the stabilization parameter. Hence, the scheme allows to usean equal-order interpolation for the velocity and pressure under some conditionfor @% . For example, with J / (piecewise linear continuous) finite element pairfor the velocity and pressure, the time step should satisfy the following “inversestability-constraint” ([73], [138])

@% \ (72)

where is the mesh size and \ is a constant. Thus, the LBB condition can bepartially circumvented with the “continuous projection”.

There is another approach to the treatment of the second part (65)–(68) of theChorin scheme, which is called “discrete projection”. It consists in first discretiz-ing these equations in space and then applying the discrete differential operators,which yields (see, e.g., [140])

@% (73)

and 0 @% G (74)

Here

is the gradient matrix, 0 is the divergence matrix, and

is the consis-tent mass matrix incorporating the density ! . It should be noted that the matrix does not, in general, coincide with the discretized Laplace operator forthe pressure from (70). The equations (73)–(74) can be interpreted as a discreteform of a mixed method for the elliptic problem (70)–(71) with the pressure ; being the primal and 3?# 0 # : @% being the dual variable (see, e.g., Brezziand Fortin [16]). It is known that the approximation spaces for primal and dualvariables must satisfy the discrete inf-sup (LBB) compatibility condition to ensurethe unique solvability of (73)–(74) (see, e.g., [142, 7.2]). Hence, the LBB conditionis of primary importance for the “discrete projection” methods.

For the thorough comparison of the discrete and continuous projection meth-ods one may be pointed to the work by Kuzmin [103, 3.4], but here we brieflymention the main advantages and drawbacks of the schemes. The discrete projec-tion approach does not require any artificial boundary condition for the pressure(like (71)) and seeks the pressure in its natural functional space 35Q: . However,using the discrete projection one is faced with the tedious assembly of the matrix and with the discrete LBB condition. The continuous projection allowsto use an equal-order interpolation, particularly, very attractive F resultingin continuous velocity and pressure fields; it also requires to invert the standardLaplace-type operator for the pressure, which is a quite explored problem havingmany efficient ways for its solution.

In our case the velocity approximation had better be continuous at the inter-face, not only due to corresponding properties of the exact solution (see chapter2), but also because of the necessity to satisfy the interfacial stress jump condi-tion. In fact, a discontinuous approximation of the velocity would lead to infinite(delta-function) jumps on inter-element boundaries for the stress in both normal

41

and tangential to the interface directions. This may destroy the discrete forcebalance at the interface and result in numerical oscillations.

Thus, in the interfacial flow problem the velocity needs a continuous approx-imation; this fact prevents the use of LBB-stable nonconforming finite elementpairs of Crouzeix-Raviart type (see Crouzeix and Raviart [29]), which are espe-cially well suited for the discrete projection methods.

In light of the above, it seems to be sensible to choose the continuous projec-tion approach for our problem. Then there appears a small remark concerningthe splitting of the interfacial stress jump condition. Within the continuous pro-jection framework the pressure has to be viewed as a function from 35Q: ; thenthe splitting (54)–(59) of the interfacial stress jump condition (48) is, probably, theonly possible decomposition of that condition. Indeed, the other variant of thesplitting could be

0 76F8 )ZW C on (75) ; on ( (76)

however, it introduces the pressure discontinuity at the interface, and, therefore,contradicts the inclusion ; 35Q: .The fractional-step projection algorithm for the interfacial Navier-Stokes flow

Using the Marchuk-Yanenko fractional-step scheme (44)–(45) we may separatethe convective nonlinearity, viscous diffusion and incompressibility from one an-other and treat each of them with corresponding numerical technique. This split-ting procedure has been described above (see (53)–(61)) for the Stokes system, andnow we present the algorithm for the full system of Navier-Stokes (NS) equations:

G NS-convection step$#$% ' # )Z+M# C in 39% % K: (77)# # # GHG Viscous step! 39I : $#$% 01+2)43L,6 39I: 8 : ! 39I:O> in ! 39% % K: (78) # C 0 76F8 )ZW SW on # D# # G G Projection step$#$%='! 3?I: +; DC (79)

+2)A# C in 3?% % K: # )ZW CQ ; DC on (# D# # ; G

42

Here # 39I : and ; 39I : are the approximations to #3?IJ% : and to ; 39IJ% : ( C AGAGAG ) respectively, and # # . Each of three subproblems must be com-plemented by suitable boundary conditions which will be discussed in the nextsections. The general strategy consists in resolving the steps successively; more-over, each step can be treated with its own time discretization scheme employinga specific (variable) time-step size dictated by stability and/or accuracy reasons.Thus, the user-given global time step @% % 0% may be subdivided intosmaller time steps within each of the subproblems.

Spatial approximation of the velocity and pressure

It has been inferred that the velocity approximation should be continuous for theinterfacial flow problem. The pressure approximation has to be continuous aswell, if we rely on the continuous projection approach. Therefore, we may usethe simplest / finite element pair, as long as the inverse stability-constraint(72) is satisfied, or other continuous velocity/continuous pressure finite elementapproximations which are LBB-stable and, hence, allow to get rid of the constraint(72).

Consider the two-dimensional case, and denote by a regular conformingtriangulation of into triangles whose diameter is at most . In the sequel, weconsider two different types of triangulation, i.e. two different mesh patterns:“regular” and “union-jack” (see figure 7). While former is the most popular one,the numerical tests show that the interface shape may apparently be corruptedbecause of violating the symmetry with respect to the vertical and horizontaldirections; the “union-jack” mesh pattern is more attractive from this viewpointas the more symmetric one.

FIGURE 7 Mesh patterns: “regular” (left) and “union-jack” (right).

There are many nice finite-element LBB-stable pairs using two different grids,one for the velocity and the other for the pressure variable ((cross-grid )- / and( / iso )- / are the most popular choices; see, e.g., [142, 9.3]). However, weprefer to use only one computational grid for all variables (it is very advanta-geous for the interfacial flow problem, as it eliminates diverse numerical errorsconnected with inter-grid transfer near the interface); thus, we will consider con-

43

tinuous velocity-pressure approximations on a single grid with either “regular”or “union-jack” pattern type.

The pressure will be always approximated by piecewise-linear finite elementfunctions, i.e. the discrete pressure will belong to the space

; *\ 3 : ; N F (80)

where / is the space of polynomials in two variables of degree less or equal to 1.The discrete velocity will belong to the following space:

# 3 Q: # N (81)

where is defined as

(a) /

or(b)

or

(c) SFG

Here F span & ( &E H , are the area (barycentric) coordi-nates on the triangle ), and F is the space of polynomials in two variables ofdegree less or equal to 2. In the case (a) we deal with the / finite element forthe velocity and pressure, in the case (b) with the so-called “mini” finite elementintroduced by Arnold et al. [4], and in the case (c) with the “Taylor-Hood” finiteelement proposed by Taylor and Hood [187]. It is worth noting that the “mini”and “Taylor-Hood” elements are LBB-stable; the “mini” element defines the finitedimensional space containing cubic polynomials (owing to the bubble function & ), whilst the “Taylor-Hood” element is based on the quadratic polynomi-als; on the other hand, the “mini” element has only 4 degrees of freedom for eachcomponent of velocity but the “Taylor-Hood” element has 6. However, the mosteconomical element is obviously the F , and the possibility to use it is one ofthe main reasons for chosing the continuous projection method.

3.1.3 Navier-Stokes convection step

According to the fractional-step projection algorithm presented above, the con-vection step consists in solving the evolution equation for self-advecting velocityfield $#&% ' #*),+M# C in ! 39% % K: (82)

supplemented by some initial and boundary conditions. Due to the hyperbolicnature of the equation, the boundary values must be prescribed only on the in-flow part of the boundary ( I -# )4W XC , W is the outwardunit normal to ). No boundary condition is imposed on the outflow boundary . The suitable boundary condition for (82) is, therefore,

#B on G (83)

44

It is common knowledge that the conventional Galerkin method lacks stabilitywhen applied to convection problems. The possible remedies include direct uti-lization of the concept of characteristics and/or a streamline diffusion modifica-tion of the weighting functions. Such methods can be generally described as ex-act transport + projection (Johnson [91] and references therein; see also Pironneau(1982) [135]). For transient convection problems, one of the most elegant ways topreclude spurious node-to-node oscillations is the Taylor-Galerkin approach in-troduced by Donea [44]. The main advantage of this method is the absence of anyfree or adjustable parameter. The enhanced stability results naturally from an im-proved temporal approximation. The discretization in time is performed prior tothat in space and is prearranged to match the high spatial accuracy achieved bythe Galerkin formulation. Higher-order accurate versions of the Euler, leap-frogand Crank-Nicolson time-stepping algorithms were developed on the basis ofTaylor series expansions where the arising time derivatives were evaluated fromthe governing equation.

Application of the classical Taylor-Galerkin schemes to truly nonlinear prob-lems is impractical, since the mass matrix modified by a higher-order correctionterm becomes extremely complicated and, in general, dependent on the time level(see, e.g., Donea and Quartapelle [47]). An efficient third-order fractional-step al-gorithm for nonlinear multidimensional convection problems was proposed bySelmin [167] and strengthened in the two-step fourth-order schemes of Quar-tapelle and Selmin [141] (see also [140, chapter 8]). A fractional-step approachwas also employed by Safjan and Oden [163], who derived a family of high-ordersemi-implicit Taylor-Galerkin schemes. Smolianski and Kuzmin [174] obtainedexplicit multilayer Taylor-Galerkin schemes, which attain high-order temporalaccuracy by using additional time layers.

For our purposes the fourth-order scheme of Quartapelle and Selmin seemsto be well suited, as it is explicit (hence, very efficient for treatment of the nonlin-earity) and exhibits good accuracy and stability properties. Below we will brieflydemonstrate the implementation of that scheme for the Navier-Stokes convectionstep; later, the same scheme will be used for computing the interface motion.

First, we derive from equation (82) the system of convection equations for thevelocity components in Cartesian coordinates / : Z&% ' # ),+ Z DC in 3?% % K: F (84)

complemented with the corresponding boundary conditions

Z on F G (85)

Here , and ( ) are the components of the velocity vector # and of theboundary-data vector , respectively.

The fourth-order scheme of Quartapelle and Selmin (denoted in [141] by TTG-4A) reads

1 ' @% '

3@%`: (86)

' -% '

3 -%O: (87)

45

where @% % J0 % C is the local time step of the Navier-Stokesconvection problem ( @% depends, in general, on , but we omit this index), % % % % , and is some function depending on the time variable. Herewe have used the notation for the partial derivative with respect to the time.

Equations (86)–(87) represent the time-stepping method that is apparently acombination of predictor-corrector and Lax-Wendroff schemes. To continue thederivation of the Taylor-Galerkin scheme one has to substitute for and theirexpressions from the governing equation (84) (with )

0 # ),+ 0 # ),+ (0#*),+ 39#*),+-#/:F),+ ' #*) + 39# ),+ :

where is obtained by differentiating equation (84) (equation (82) has also beenused for # ).

Substitution of the above expressions into (86)–(87) leads to the followingtime-discretized equations for each velocity component EE F :

0 @% 0

# ),+ ' -% 3?# ),+M# : )Z+

' -% # ),+ 39# ) + : (88)

0 @% 0Q# ),+ '

@%3 # ),+ # :/),+ '

@% # )Z+ 3 # ),+ :G (89)

First, it is clearly seen that on both steps (88) and (89) the velocity componentscan be sought separately owing to the explicit character of the scheme. That isan advantage, since it is always more efficient to solve two smaller problems in-stead of a large one. Next, we observe that the second-order derivatives presentin (86)–(87) give rise to the appearance of second-order terms on the right-handsides of (88) and (89) (the last terms). These terms are not to be thought of as a nu-merical diffusion inherent in the scheme but only as the components of improvedtemporal approximation; their more precise meaning will be highlighted below.

The presence of second-order terms in (88)–(89) dictates the choice of the func-tional spaces for the semi-discrete velocity components . Let 3PQ:= 35Q: on , and $ 3PQ: 35Q: C on .Then we obtain the weak formulation (only the first step of the scheme (88)–(89)is considered, as the second is treated analogously):

Find 35Q: F , such that 0

@% ] 0 # )Z+ ]'

@% @39# ),+-# :/),+ ]

' @% # ),+ 3?# ),+ :$]

$ 35Q:G (90)

46

The last term is integrated by parts as follows: # ) + 39# ),+ :$] 0 +2)43 # :# )Z+ ]' -3?# )ZWF: 39# ),+ :$]4

0 39# ),+ : 39# ),+ :$]0 -3L+ )A# : 39# )Z+ :$]' -3?# )ZWF: 39# ),+ :$]4

where the identity +2)43 :J +2) ' ),+ has been used.

Remark 3.2. The integral 39# )Z+ : 39# ),+ :$]is the most important term. It introduces streamline diffusion intrinsic to theLax-Wendroff time discretization. In contrast to the classical streamline upwindmethod, no artificial parameter is to be adjusted. The integral @3L+2)A# : 39# ),+ :$]vanishes for divergence-free velocity field. Since the spatial approximation of # is never exactly divergence-free, we retain this integral for consistency, althoughthe influence of this term is expected to be very small. Finally, the surface integral

-3?# )ZWF: 39# ),+ :$]H

prevents the arising of spurious reflections at outflow boundaries for transientcalculations ([47]).

To complete the derivation of the Taylor-Galerkin scheme we have to constructfinite element approximation of 35Q: , , and $ 35Q: :

*\ 3 : N $ on F $ \ 3 Q: N C on

where can be / or or S polynomial space (see the preceding section); $ ,F , is the corresponding finite-element approximation of the -th componentof the boundary vector .

The resulting Galerkin finite element equations can be written in the matrixform as follows:

0 @%

' @% (91)

0 @% '

@% (92)

47

where , is the vector of nodal values of the -th velocity component at

the time level % ,

is the consistent mass matrix, corresponds to the first-

order convection term (see the first integral on the right-hand side of (90)), and corresponds to the second-order correction term (see the last two integrals onthe right-hand side of (90)).

Remark 3.3. The scheme TTG-4A of Quartapelle and Selmin ([141]) possessesfourth-order temporal accuracy and the same spatial accuracy if linear elementsare used on a uniform mesh. This spatial superconvergence was shown in, e.g.,Thomee and Wendroff [193] to be a direct consequence of the presence of a con-sistent mass matrix in the Galerkin finite element method for unsteady advectionproblems. The scheme is stable when the Courant-Friedrichs-Lewy (CFL) condi-tion \

is satisfied ( \( # -% is the Courant number). The CFL condi-tion determines the size of the “local” time step @% of the Navier-Stokes convec-tion phase. In principle, we could resort to lower-order schemes, like, for exam-ple, the second-order Lax-Wendroff finite element (LWFE) scheme (see Donea etal. (1987) [48]), but those suffer from more severe stability restrictions and, thus,require smaller time steps, which may ultimately lead to significant increase incomputational time.

At the end, we would like to address some issues of solving the matrix equa-tions (91)–(92). On both stages of (91)–(92) we have to resolve the linear algebraicsystem of the form

(93)

where 0 is the vector of nodal increments. This can be done veryefficiently, since the consistent mass matrix may be well preconditioned by itsrow-sum diagonal counterpart (see Hawken et al. [78] for a thorough discussion).The following Jacobi-type iteration is frequently employed ([46], [78],[140]):

' 3 0 : C AGAGAG C (94)

where

is the lumped mass matrix obtained by row-sum diagonalization. Typ-ically, three iterations are sufficient to achieve a reasonable accuracy (see [140,chapter 8] for the analysis of iteration error).

It is important to emphasize that, with the described above explicit Taylor-Galerkin scheme and within the operator-splitting approach for the whole Navier-Stokes system, the solution of the convection step becomes trully easy: one hasto invert always the same consistent mass matrix, independently of the time mo-ment and of the interface position.

3.1.4 Viscous diffusion step

This step amounts to solving the system

! 39I: &#$% 01+2)43L76 3?I: 8 : ! 3?I:`> in 3?% % K: (95) ## C on (96)0 76F8 )ZW SW on (97)

48

endowed with appropriate initial and boundary conditions. The boundary con-ditions, in fact, are the same as for the complete problem, i.e. (49).

Before considering the discretization of problem (95)–(97), it is interesting todiscuss the modeling of the surface tension force that appears only on the vis-cous step of the fractional-step projection algorithm. There are three generalapproaches to the capillary force treatment: delta-function method, method ofbuilt-in jump conditions and variational method. Below we will consider theirpros and cons.

Delta-function methodBesides the formulation based on stress jump interfacial condition (97) there isanother way of taking into account the capillary force: to introduce it into theright-hand side of momentum equation (95) as an interface-concentrated sourceterm, using the notion of delta-function. Namely, the formulation becomes

! 3?I: $#$% 0 +2)43576 3?I: 8 :J ! 39I :`> ' STH35]4:`W in 39% % K: (98)

where is twice the mean curvature and is the surface tension coefficient, asbefore, ] is the normal distance to the interface , T is the one-dimensional Diracdelta-function, and W is the unit normal to pointing from fluid 1 to fluid 2.

The idea was first presented by Peskin [134] and subsequently employed bymany others, see, e.g., Chang et al. [20], Fogelson and Peskin [57], Sussman et al.(1994) [183], Tu and Peskin [198], Unverdi and Tryggvason [201]. A very similarapproach was proposed in Brackbill et al. [15]. The formulation with a delta-function seems very reasonable as it directly reproduces the physical meaningof surface tension force and allows to easily incorporate the force into the mo-mentum equation. However, on numerical level the delta-function has to be ap-proximated, and that becomes a source of troubles. The most popular variant ofapproximating the delta-function is the sinus-like approximation (see, e.g., [183])

T 35]4:J 3 ' 3 ] :K: if ] C otherwise (99)

where

is the smoothing parameter that apparently prescribes the “thickness” ofthe interface. The parameter is normally taken to be of the order of mesh size . Hence, the smoothing of delta-function unavoidably results in the interfacesmearing. This makes the method similar to the diffuse-interface methods men-tioned in section 2.1; interface possesses finite thickness, but the latter is here anartificial numerical parameter. Thus, the delta-function method gives rise to theerror 3 : in the interface location, which deteriorates the interface resolutionand may significantly delay the moment of interface reconnection, if any.

Method of built-in jump conditionsAnother approach consists in adjusting the numerical scheme near the interfaceso as to incorporate the interfacial jump conditions into the scheme. This makesit possible to escape the necessity of approximating a delta-function, and resultsin improved, second-order accuracy for the interface location as compared to thefirst-order of the delta-function approach. However, the implementation of themethod seems to be rather complicated; the method is quite expensive for moving

49

interface problems, since the numerical scheme must be reconstructed each timeafter the interface has moved; the questions of stability and consistency for a dis-cretization with built-in jump conditions remain open in the cases of irregular orreconnecting interfaces. The different versions of the method were considered byLeVeque and Li [109], [110] and by Fedkiw et al. [54] (see also Kang et al. [92]), andwere termed “immersed interface” and “ghost fluid” methods correspondingly.These works dealt with the finite difference approximation. Another variant ofthe method has been recently proposed by Popinet and Zaleski [137] within thefinite volume framework.

Remark 3.4. Two above mentioned methods for capillary force treatment havebeen implemented mostly with finite difference discretization (although, the del-ta-function method in combination with finite elements has been considered byNakayama and Shibata [125]). In the context of finite differences, however, thereexists another problem: an approximation of viscous term containing the deriva-tive of discontinuous viscosity coefficient. A straightforward application of finitedifference formulae will obviously lead to inconsistent and unstable discretiza-tion; two possible remedies are either careful adjustment of the discretization inthe vicinity of the interface (the method with built-in jump conditions) or smooth-ing the viscosity jump (the delta-function method). The former approach is accu-rate but costly (see the discussion above), the latter simpler but inaccurate whenused in combination with finite difference method.

Variational methodThe method relies on the weak formulation for problem (95)–(97). In section 2.3it was shown that the interfacial stress jump condition (97) is, in fact, a naturalcondition, and, after integration by parts, the surface tension force becomes auto-matically incorporated into the variational formulation: ! 39I: &#$% ) ] ' 76J39I: 8 ) ) + N ] ! 39I:O> ) ]

' SW ) ]4 G (100)

This method was used by many researchers, see, e.g., Christodoulou and Scriven[25], Lock et al. [115], Rivkind and Fridman [153], Ruschak [157], Shopov etal. [170], Tornberg [195]. It is noteworthy that (100) can be obtained also fromthe delta-function formulation (98) by multiplying the equation with weightingfunction and integrating over (see Tornberg [195]).

The variational method does not require to approximate a delta-function and,in this sense, is close to the method of built-in jump conditions (these conditionsbecome, indeed, built into the weak formulation). If the interface can be lo-calized “precisely” (i.e. with the accuracy of the discretization scheme), then asharp interface may be maintained with the variational method, and the error inthe interface location may be expected to be of the order of the discretization er-ror. Particularly, the second-order accuracy can be recovered with the piecewiselinear interpolation of all variables (velocity, pressure, interface). Another advan-tage of the variational approach consists in alleviating the singularity connectedwith the differentiation of discontinuous viscosity coefficient; in fact, this prob-lem is completely eliminated with the formulation (100). On the discrete level the

50

discontinuity may, however, pollute the numerical solution, but, for example, asimple local mesh adaptation (if the finite element method is considered) imme-diately recoveres the optimal accuracy of the scheme. That adaptation resembles,to some extent, the discretization adjustment in the finite difference/finite vol-ume method with built-in jump conditions, but is much simpler and cheaper.

Remark 3.5. The principal strength of the variational approach seems to be in thecorrect functional class chosen for seeking a solution. It is known that the velocitydoes not possess an

-regularity on the whole domain if the viscosity is dis-continuous at the interface (see the discussion in section 2.3). Therefore, neitherfinite difference nor finite volume methods are suitable for the discretization ofthe problem, as their consistency essentially relies on sufficient smoothness of theexact solution. The appropriate discretization methods seem to be those basedon the variational formulation, in which the solution is sought in

35Q: and noadditional regularity is generally supposed. Thus, we are naturally led up to theGalerkin finite-element method for the discretization of interfacial flow problems.

Consider the weak formulation (100), where at each moment of time # 35Q: # 3PQ:# on , and weighting function 35Q: 35Q: C on . We discretize it in time by one-step -scheme ! 3?I: # 0 # @% ) ] ' 76 39I : 8 39# :/) ) + N ] 0 3 0 : 76J39I: 8 3?# :/) ) + N ]' ! 3?I:`> ) ] ' SW*)D]4 (101)

Here @% % 0D% C is the local time-step of the viscous dif-fusion problem ( @% depends, in general, on , but we omit this index), % % % % . The parameter C controls the implicitness of theviscosity-term discretization: C corresponds to the explicit Euler scheme,

3PC results in implicit scheme; corresponds to the second-order Crank-Nicolson, and

to the first-order backward Euler method. The explicit treat-ment of viscous term requires very small time steps to guarantee stability (espe-cially in the case of highly viscous, creeping flow; see, e.g., Turek (1996) [199]).Hence, the implicit schemes (unconditionally stable when ) are to be pre-ferred on the viscous step for discretizing the viscosity-term (the time-steppingfor the viscous step will not remain fully implicit when we consider the wholeproblem with moving interface; the surface tension and the gravity terms will betreated explicitly, which will pose some stability restrictions on the local time-step@% ).

After chosing the finite-element discretizations for 3PQ: and

35Q: # 3 : # N # on (102)

# 3 : # N # C on (103)

where may be or or S polynomial space (see section 3.1.2), is thecorresponding finite-element approximation of the boundary vector , we obtain

51

the following fully discretized system of equations:

3 ' @% : 3 0D3 0 R: @% : ' @% ' @% G (104)

Here is the consistent mass matrix incorporating the discontinuous density

coefficient,

is the viscous diffusion matrix incorporating the discontinuousviscosity coefficient, is the gravity force vector including the discontinuousdensity, is the capillary force vector, and

is the vector of velocity’s nodal

values at the time % . The evaluation of the capillary force term and the integralswith discontinuous density/viscosity coefficients will be discussed later, at theend of section 3.2. Concerning the solution of the linear algebraic system (104),it is important to note that the system matrix is symmetric and positive-definite,hence, the conjugate gradient method preconditioned by incomplete Choleskyfactorization should work well.

3.1.5 Projection step

On this step we compute the unknown pressure ; and the final velocity # satisfying the incompressibility constraint. The coupled system of equations reads

$#$% '! 3?I: +; DC (105)

+2)A# C in 3?% % K: # ) W CQ ; DC on (plus corresponding initial and boundary conditions. The latter have been dis-cussed in section 3.1.2: they consist in prescribing the boundary values only forthe normal component of velocity.

As we have seen, after time discretization this problem can be recast in theform of a Poisson-type equation for the pressure endowed with the homogeneousNeumann boundary condition

0+ )43 ! 3?I: +; : 0

@% +2)A# in B (106)

W*) +<; DC (107)

where # is the intermediate velocity obtained on the viscous step. The finaldivergence-free velocity # is then derived as

# D# 0 @% ! 39I: +; G (108)

On the projection stage there is no stability restriction for the time-step (the time-step should be bounded from below if equal-order interpolation is used for ve-locity and pressure; see the “inverse stability-constraint” (72)). That is why thisstage can be discretized with the global time-step @% .The weak formulation of (106)–(107) reads

52

Find ; 3PQ: such that

! 39I : +<; ),+ J] 0

@% 3L+2)A# : /] 3PQ: (109)

and the weak form of (108) isFind # 35Q: such that

# 0# @% ) ] 0 ! 39I : +<; ) D] 35Q:G (110)

It is noteworthy that although the velocity # does not belong to the space 35Q: according to the -projection theorem (see section 3.1.2), it is projectedonto

3PQ: by virtue of (110). The “natural” regularity of the velocity turns out,thus, recovered.

For the finite element discretization of the pressure space 3PQ: we employ

; \ 3 Q: ; N / ;43 : C (111)

where / is the space of polynomials in two variables of degree less or equal to1, and is some node of the triangulation. The velocity space

35Q: is approxi-mated by

# 3 Q: # N (112)

where may be or or S polynomial space (see section 3.1.2), the same ason convection and viscous diffusion steps.

The linear algebraic system resulting from the finite element discretization of(109) involves a symmetric positive-definite matrix of diffusion type which in-corporates the discontinuous “diffusion” coefficient . The evaluation of corre-sponding integrals with discontinuous coefficient is done precisely as on viscousstep (see the details at the end of section 3.2). The incomplete Cholesky precon-ditioned conjugate gradient method is used for solving the pressure system.

The discrete form of (110) contains the consistent mass matrix for velocity(which has been considered on convection step) and the right-hand side vectorincluding the discontinuous coefficient . The integrals with discontinuous co-efficient are computed as before; the system is separable with respect to the veloc-ity components, and the Jacobi iteration with lumped mass matrix can be utilizedas on convection step. It is interesting to point out that the discrete form of (110)may be viewed as a gradient recovery procedure performed via an -projectionof the discrete pressure gradient onto the space of continuous functions. Suchprocedure is known to give a superconvergent averaged gradient on the meshpatterns like those considered here (see Hinton and Campbell [80], Zienkiewiczand Zhu (1987) [212], also Zienkiewicz and Zhu (1992) [213]). Thus, this projec-tion automatically improves the accuracy of the final velocity.

53

3.2 Approximation of the interface

There are two general ways of the interface representation: explicit and implicit.The former relies on the parametric description of the curve/surface in theform H3 %K: on , where is the vector of parameters and % is the time(see figure 8, left). Since within this approach we explicitly trace the coordinatesof each point belonging to the interface, the parametric description gives rise tothe interface-tracking methods considered in section 1.1. The interface is usuallydiscretized with a set of marker points, over which a piecewise polynomial orspline interpolation is employed.

Implicit representation of the interface invokes some auxiliary function whosediscontinuity line or zero-level set determines the interface. The latter case isknown in Differential Geometry as the curve/surface description via an implicitfunction

3E%K: C on , whilst the former relies on the notion of the indica-tor (characteristic) function that is equal to 1 in the first domain and to 0 in theother (see figure 8, right and center). These approaches have been called “contin-uous” and “discontinuous” respectively, and considered in section 1.1 within theinterface-capturing group of methods. If for the auxiliary function a piecewisepolynomial interpolation is used, the interface approximation is also piecewisepolynomial.

1

r = r(α, t)

I=1

I=0 Φ( r, t )=0

Γ Γ ΓΩ Ω Ω Ω2 1 2

FIGURE 8 Representation of interface: (left) explicit (parametric); (center) dis-continuous implicit; (right) continuous implicit.

Thus, theoretically both explicit and implicit representation methods allow toapproximate the interface with the same accuracy. In practice, however, it isnot always the case, since in discontinuous implicit approach the indicator step-function has to be smoothed to avoid numerical oscillations, thus, giving rise tothe interface with finite thickness (usually, of the order of mesh size). This draw-back is absent, if we use an explicit representation, and, as we are about to see,can be eliminated with the finite element version of continuous implicit approach.On the other hand, in the case of interface topology changes (merger, breakup)the explicit representation approach fails completely, and all the known reme-dies consist in very sophisticated ad hoc algorithms. At the same time, implicitapproaches (both continuous and discontinuous) are capable of capturing the in-

54

terface independently of its topology. These observations lead us to continuousimplicit representation method combined with the finite element discretization.

3.2.1 Level-set approach

Continuous implicit representation of the interface B I 39IJK%K: C % C , ] 4 , allows to compute the interface normal W and curvature in a veryconvenient way (see section 2.2 for the curvature formula):

W +

+ (113)

0 +2)ZW 0+2) +

+ (114)

where the normal and curvature turn out to be defined in the entire domain ifthe function

is defined in . This natural extension for the normal and curva-

ture off the interface proves to be very useful in computational process.There are different options for chosing the implicit function

, but the most

convenient from computational viewpoint choice seems to be the signed normaldistance to the interface. Such a variant of defining the function

was advo-

cated by Dervieux and Thomasset [42] and by Thompson [194], but fully ex-plored and justified by Osher and Sethian [131] (see also references in the book bySethian [168]) who called this method “level-set approach” (the signed distancefunction

was called the level-set function). The sign of the level-set function

is chosen to be “plus” on one side of the interface and “minus” on the other,and the modulus of the function’s gradient is everywhere equal to 1 (as it is anormal distance function). In our case we set the level-set function positive in do-main and negative in (this automatically makes the normal W point fromfluid 1 to fluid 2, see (113)). For example, if the interface is the circle with thecenter at the origin and unit radius, the level-set function can be computed as 3 : 0 ' (see figure 9).

Equation of the interface motionIt is known fact (see, e.g., Sedov [166, vol.I,chapt.7]) that the normal velocity ofthe interface, i.e. the interface velocity in normal to the interface direction, can befound as

0 + G (115)

This expression may be formally derived by computing the full first differentialof function

39IJ%K: on .If we combine the expression for the normal interfacial velocity with the equa-

tion of continuity for fluid velocity (which is a direct consequence of mass bal-ance, as shown in section 2.2)

D#*)ZW on (116)

where # is the fluid velocity, we immediately obtain the following evolution equa-tion for the level-set function $% ' 39#*)ZWF: + CG (117)

55

FIGURE 9 The level-set function 39I : 0 ' : (left) surface plot; (right)

contour plot (solid contour is the zero level set).

This equation is defined over the whole , since all variables are defined there.Substituting into (117) the expression (113) for the normal vector, we derive thefollowing pure convection equation for the level-set function

$% ' #*)Z+ C in % CG (118)

Remark 3.6. As it is clearly seen from (117), we have taken into account only thenormal component of the interface velocity to describe the interface motion. Thisis a consequence of using the implicit approach for interface representation. If werelied on the explicit (parametric) representation, the interface points would beconvected with fluid velocity, i.e. moving along the trajectories of fluid particles(along the characteristics); the motion of interface points would not be restrictedto the normal direction. However, it is evident that only the motion of interfacein its normal direction may change the interface shape; the tangential motion hasno influence on the form of interface. In addition, Hou et al. [87] and Stockieand Wetton [177] showed: it is the tangential mode of the interface motion thatcauses main stability problems. These arguments should justify our choice of thelevel-set approach.

3.2.2 Level-set convection step

We have derived the equation for the level-set function

. Since the equation isa first-order hyperbolic one, we have to prescribe the boundary values of

only

on the inflow part of the boundary (as before, I #) W2C ,where W is the outward unit normal to ). Prescribing also the initial condition

56

we obtain the following problem:

$% ' # )Z+ DC in ! 35C : (119)

on (120) in G (121)

As we have seen from the derivation of equation (119), each level set of

moveswith velocity # (for a level set only the normal component of the velocity is rele-vant, see Remark 3.6). We defined the interface as the zero level set, i.e. at eachmoment of time % C the interface 39%K: is defined by

39%K: I * 39IJ%K: DC GAccording to (121), at the initial moment the interface is defined as

3PC : I 3?I:JC GEquation (119) is a particular case of the Hamilton-Jacobi equation that was stud-ied in Osher and Sethian [131]. A parabolic form of this equation (in connectionwith mean curvature flow) was investigated by many authors, see, e.g., Evansand Spruck [52] and references in Sethian [168]. Particularly, it was shown thatthe evolution of the level-set function implies a correct motion of the interface (asthe zero level set) even beyond the moment of interfacial topology change. Thetransport problem (119)–(121) has been recently studied by Maitre and Witom-ski [116] (see also references therein) who proved the unique solvability of theproblem under suitable regularity assumptions on the boundary and initial dataand on advective velocity. They also proved that the zero level set is

– independent of the initial condition (i.e. for two solutions and

havingdifferent initial conditions

and such that I 3?I:( C

I 3?I:JC the zero level sets coincide % C );– independent of the boundary condition (not really proved, but conjectured);

– non-spreading (i.e. if 4^ I 3?I: C C then 4^ I 3?IJ%K:DC C % C ).In Nochetto [127] the non-spreading of interface is referred to as the interfacenon-degeneracy property (NDP) and its tight link with the approximability of aninterfacial/free-boundary problem is emphasized. Particularly, the level-set for-mulation possesses very good interface NDP (namely, NDP of order 1 in measure:4^ , 39%K:S I `C 3?IJ%K: 3 :% C ).

All these theoretical aspects of the level-set approach make it especially attrac-tive for modeling of interfacial flow problems. Some particular advantages con-nected with the finite element implementation of the method will be discussed insubsequent sections.

To solve the problem (119)–(121) on each time interval 39% % : , C AGAGAG ,we employ the same fourth-order explicit Taylor-Galerkin scheme TTG-4A usedfor the discretization of the Navier-Stokes convection step (see section 3.1.3). The

57

velocity # that has been obtained on the projection step is taken as an advec-tive velocity; time substep size is dictated, as before, by CFL stability condition.If the piecewise linear continuous ( ) approximation is chosen for the level-setfunction

, we arrive at the fully discrete system of the form (91)–(92) as on the

Navier-Stokes convection step.

Remark 3.7. An explicit scheme with the CFL stability condition \ ( \ # @% ), particularly the scheme TTG-4A, is well suited for solving the level-

set convection problem, as this condition means that during one time step theinterface does not move farther than to the distance from its previous position.Thus, the interface does not skip any domain element, which is proved to be areliable guide in selecting the time step for interface propagation (see, e.g., Locket al. [115]).

3.2.3 Reinitialization step

At the beginning of computational process the level-set function

is initializedas the signed distance to the interface . But, as time goes, the level-set func-tion may become very steep or flat in some regions, particularly in the vicinity ofthe interface, thus losing the nice property + &

(see figure 10). This deteri-oration of level-set function is a natural consequence of the convection process,but it makes difficult an accurate determination of the interface. In the interfacialregion, where the level-set function is too flat, it is hardly possible to distinguishthe points on one side of the interface from the points on the other side, as theycan only be distinguished by the sign of the level-set function. In the vicinityof the interface, where the level-set function is too steep, the interface propaga-tion problem amounts to the transport of a nearly discontinuous function, whichrequires special discontinuity capturing (high-resolution) scheme to suppress nu-merical oscillations near the interface. Those oscillations affect not only interfaceshape but also the quality of interface normal and curvature calculations, and,moreover, the mass conservation.

steep Xflat

FIGURE 10 One-dimensional sketch of possible deterioration of the level-setfunction.

In order to cure the situation, the level-set function must be reinitialized, i.e. madeagain the signed distance function. Theoretically, this operation is justified asthe interface does not depend on the particular choice of the initial data

ifthe latter has zero level set coinciding with (see the discussion in precedingsection). Reinitialization is simply the process of replacing

by another function

that has the same zero contour as

but behaves better, and then taking this

58

new function

as the initial data to use until the next round of reinitialization.In addition, we may note that the “redistancing” should be done only in somevicinity of the interface, since the values of the level-set function far from haveno influence on the interface dynamics.

The necessity of the level-set function reinitialization in complex hydrodyna-mical problems was mentioned by Dervieux and Thomasset [42] and by Thomp-son [194], and specially emphasized by Sussman et al. (1994) [183]. In the workby Sussman et al. the following redistancing method was proposed: let

be

the level-set function obtained after the level-set convection step, then the newlevel-set function

is found from the problem

^A 3 : 3 0 + : in 3PC [ : (122) (123)

where ^A is the sign function and Z[ is the time when the solution reachesits steady state. In practice, however, it is sufficient to solve (122)–(123) until thetime to make

a distance function in the -band of computational

cells around the interface. The method does not require to localize the interface,but the computational cost of solving the nonlinear hyperbolic equation (122) israther high. Moreover, any numerical scheme for the solution of (122) shouldintroduce some numerical diffusion (some part of the diffusion comes also fromthe approximation of ^A J3 : , see, e.g., [183]), which results in small inaccuracyof the interface location and some loss of mass (see, e.g., Tornberg [195]).

The reinitialization procedure of Sussman et al. was later improved by Changet al. [20], Sussman and Fatemi [180] (see also Sussman et al. (1998) [181]), Peng etal. [133]. In the latter paper, equation (122) is solved not on the whole domain but only in some “tube” around the interface, which saves time though requiresto maintain a correct motion of the “tube” along with the interface. If the two-dimensional grid contains ! nodes, then the computational complexity of thealgorithm by Peng et al. is 3*: as compared to 3 : for the original method ofSussman et al. (this can be readily calculated taking into account that the numberof grid points along the interface is 3 : ). Another variant of reinitializationprocedure based on solving the eikonal equation + J

has been presentedby Adalsteinsson and Sethian [2]; they make use of the Sethian’s Fast MarchingMethod to solve this equation, which results in the algorithm with the complexity 3 *: .There is another approach to the reinitialization, which was termed “bruteforce” or direct method. It consists in the localization of the interface, i.e. findingits location using some interpolation technique, and subsequent straightforwardcomputation of the signed distance to the front. The sign of the new level-setfunction

at any point can be obviously taken the same as that of the old level-

set function

, thus, only the normal distance to the interface has to be computedat each grid node. The direct method was used by Chopp [22] and by Merrimanet al. [121] and then almost forgotten, since its computational complexity is 3 :for the grid. However, if we perform direct redistancing only in a neigh-borhood of , the complexity obviously falls to 3 : , and even to 3 *:when a locally refined quadtree mesh is used (see Strain [178]).

59

Despite the common opinion that the direct reinitialization is too costly, thisprocedure constitutes only a small fraction of the total computational time formany physical applications, particularly for multi-fluid interfacial flows. On theother hand, main advantage of the direct method is that it does not move theinterface (up to the numerical accuracy of the interpolation scheme). The ma-jor problems with the method seem to consist in the interpolation of the level-set function and in exact localization of the interface within the finite differenceframework. In fact, any grid function can be interpolated in many ways on finitedifference grid (see Strain [178] for the discussion of some interpolations), and itis not an easy task to find the location of zero level set for the interpolated level-set function (for example, the intersection of an arbitrary bilinear function withthe plane rectangular cell on which that function is defined can be quite com-plicated set with many possible configurations). That is, probably, why almostall works using level-set approach relied on the reinitialization procedure (122)–(123) of Sussman et al. (all the papers cited above employed finite differencediscretization on quadrilateral grids, except Tornberg [195] who utilized finiteelement approximation on a triangulation but used the same reinitialization bySussman et al.).

In the finite element method the problems discussed above may disappearcompletely. Indeed, if we use piecewise linear continuous ( ) approximation

for the level-set function

on simplicial mesh, the interface can be easily localized:

the intersection of such with each closed simplex-element is just a line segment

possibly degenerated to a point (a vertex of the element), see figure 11. In 3Dthe situation is slightly more complicated but still quite simple: the zero level setof

, found as the union of intersections of with each closed tetrahedron, is

a plane quadrilateral within every tetrahedron, with possible degeneration to atriangle, line segment (the tetrahedron’s edge) or point (the tetrahedron’s vertex);the fact that the intersection set is always plane immediately follows from the lin-earity of

on each simplex-element. It is worth noting also that the intersectionset (the zero level set within each simplex-element) is completely defined bythe intersection points of

with the element’s edges, independently of the spacedimension. Thus, using piecewise linear continuous approximation for the level-set function on simplicial mesh, we obtain unique piecewise linear representation for the interface .The localization of the interface allows to compute the integral over , i.e. toevaluate the capillary force term, and to determine the density and viscosity co-efficients (see section 3.2.6 below). It also makes a direct reinitialization simple.If we call “interfacial” all elements crossed by the interface, then the direct re-distancing consists in calculating the distance to from each vertex of everyinterfacial element, finding the neighbouring vertices for each such vertex andsubsequent repeating the procedure. In order not to reconsider the vertices atwhich

has been already reinitialized, we introduce an array-vector mask ofsize for the grid (the numbers of grid points along each dimensionare taken here equal only for simplicity); the array components are zero if thecorresponding vertices have not been involved into redistancing, and unity oth-erwise. We introduce also two array-vectors: current vertices contains numbers of

60

. ....

.... . ....

....

γh

Γh.

.....

.

FIGURE 11 Localization of the interface: zero level set of is the union of line

segments.

the vertices undergoing the reinitialization at the current step of this procedure,new vertices contains numbers of the vertices at which distance is to be computedon the next step of the reinitialization. In algorithmic form the procedure can bewritten as follows:

current vertices numbers of the vertices of interfacial elements ;mask 3 : DC ;DO F

DO

,SIZE(current vertices)! for each

-th vertex of current vertices:

IF (mask(current vertices()) C ) THEN

dist distance from the vertex with numbercurrent vertices(

) to ; (current vertices(

)) SIGN(

(current vertices()))*dist ;

mask(current vertices())

;find the neighbouring vertices for the vertex withnumber current vertices(

) (i.e. the vertices belonging

to the patch of elements with the center at theconsidered vertex) ;write into new vertices the numbers of theneighbouring vertices ;

END IFEND DOcurrent vertices new vertices ;

END DOThe computation of the distance from a point to can be accomplished sim-ply by taking a minimum of the distances from that point to each interfacial linesegment . Then, since we perform the reinitialization only in the interface’neighbourhood of half-width ( is some constant independent of , typically C ), the computational complexity of the procedure is 3 : as in the

61

method (122)–(123) (but, practically, the direct reinitialization is simpler, cheaperand more robust).

3.2.4 Level-set correction step

The reinitialization of level-set function is very important for accurate determin-ing the interface, but it cannot guarantee the mass conservation. In fact, due tothe incompressibility assumption the area (volume in 3D) of the region occupiedwith each of the two fluids must be conserved during the whole computationalprocess. However, this is not the case because of different numerical errors, andafter many time-steps or any topological change of the interface the relative de-crease/increase of the area of one of the fluids may be about several percents.That was a reason for the appearance of the schemes combining the reinitializa-tion procedure (122)–(123) considered in preceding section with some enforce-ment of mass balance (see Chang et al. [20], Sussman and Fatemi [180]).

We present here a different, very simple approach that uses an additional al-gorithmic step to explicitly enforce the mass conservation. The key observationis that the error in mass balance should be very small within one time-step, if weemploy sufficiently accurate scheme for the convection of level-set function. Themain reason for any significant losses of mass seems to be an accumulation of nu-merical errors during many time-steps; this hints at the possible way to remedythe situation by correcting the mass (i.e. the area) after every time-step.

The correction can be done easily. Suppose that we have performed the reini-tialization of the level-set function

, thus, in some vicinity of the interface,

is the signed distance function and its level sets (isolines) are equidistant (seefigure 12). The domain Q occupied with the second fluid can be defined as I

C . To correct the area of Q (it is evident that we haveto correct only one of the domains and as their union has fixed constantarea) we are not allowed to change the shape of the interface , since the shapeis naturally obtained in the course of solving the problem. The only thing wecan do is to change the zero level set, i.e. to accept as a new zero-level set someneighbouring isoline, owing to the fact that it has almost the same shape (

is a

distance function !). This operation can be accomplished simply by moving thelevel-set function upward or downward, i.e. by adding to

some signed con-

stant

, where is the distance between old and new zero-level sets (the fact

that

is a distance function is again exploited here).

Φ = Φ +

new

Γ Γ

c

c

new

new

Φ

Φ

ΩΩ

2

2

FIGURE 12 Level-set correction (solid lines are the old and new zero-level sets).

62

To find the expression for

, we denote by

the new (raised or lowered) level-set function, I

C , and utilize the well-known formula for

the first variation of a volume integral (see, e.g., Delfour [36] or Cuvelier andSchulkes [32]):

0 Q35 : ](0 ] 3

W/:/)ZW ]4 ' 3 :

]E ' 3

: (124)

where is the exact area of the region occupied with the second fluid ( is always known to us), Q35Q: is the area of . From here immediately follows

0 Q35 :

35: (125)

where 3PF: is the length of the interface . The formula (125) is accurate up to 3 : (we assume that C 3PF: ' ,

and are independent of any

physical and numerical parameters of the problem).First, it is noteworthy that if Q35Q: we obtain from (125)

C , andthe level-set function

is to be lowered, which automatically implies shrinking

of the domain Q ; if Q35 : then

C , and automatically expands.Next, we may note that the level-set correction procedure makes sense also in thecase of multiply-connected domain and in the case when is only a part ofthe boundary of .

Now we can try to justify the presented mass-correction method. The methodessentially uses two facts:

being a distance function in some vicinity of the

interface and smallness of . The former fact means that the level-set correction

should follow the reinitialization on every time-step. The smallness of modulusof the correction constant is the crucial point in the justification of the method;indeed, artificial moving of the interface is not allowed, except as the movementmagnitude is not greater than the interface interpolation error. If we assume that \ , the piecewise linear interfacial interpolation gives us the error 3 : in -norm. Hence,

should be not greater than 3 : ; below we will show thatit can be expected. It is worth noting that the effect of the level-set correction isusually almost negligible within one time-step, but the correction’s main purposeis to prevent an accumulation of numerical errors in a long run.

The major sources of the error in the interface location are the discretizationerror of the convection scheme and the discretization error of advective velocity.The Taylor-Galerkin scheme TTG-4A employed for the level-set convection is 4th-order accurate in time and has the 4th-order spatial accuracy at grid nodes (seeRemark 3.3 and also Swartz and Wendroff [184], Donea and Quartapelle [47]).Since we use -elements for the level-set function

, which immediately yieldsa piecewise linear representation for , the -norm of the error in the interfaceconvection should not be greater than 3 : , if the exact interface is smoothenough (the temporal error does not pollute the spatial accuracy, since -%J 3 :due to accuracy and stability reasons). The error coming from the discrete advec-tive velocity is also not greater than 3 : : in fact, the interface propagates with

63

the fluid velocity, whose discretization error is not worse than 3 : in -norm(as will be shown later), thus, during one time-step of size @% 3 : the errorin the interface location cannot become greater than 3 :Q) 3 : 3 : . So,the magnitude of zero-level set movement during each mass-correction can beexpected as not exceeding 3 : .Remark 3.8. Since the error of formula (125) is 3

: and is 3 : , the level-

set correction method should imply the mass conservation up to 3 : . However,in reality we deal with the function

, which is not an exact distance function butonly its piecewise linear approximation; thus, moving

upward or downwardin accordance with

we obtain a new zero-level set that may differ by 3 :

from the desired one. This implies the mass conservation error 3 : which isconsistent with the spatial accuracy of the interface approximation.

Finally, it is worthwhile to mention that the presented level-set correction pro-cedure can be interpreted as some diffusion process applied at the interface. Infact, the correction yields a motion of the interface in the normal direction, whichmay be viewed as a diffusion flux 3`0 + : of the level-set function

( +

at eachpoint of is parallel to the normal W , and the magnitude of the signed diffusioncoefficient

can be recovered using

). Thus, the level-set correction brings ad-

ditional stabilization of the interface motion, which is confirmed by numericalexperiments.

3.2.5 Approximation of the interface normal and curvature

Evaluation of surface tension force term W ) ]4 requires the computation of

the interface normal vector W and curvature . These quantities can be also of spe-cial interest as detailed indicators of the interface shape (especially the curvatureshould be controlled during the computational process). To calculate the normaland curvature we use the piecewise linear approximation

of the level-set func-tion

; thus, a straightforward implementation of the formulae W + + ,

- 0 + ) W (see section 3.2.1) gives us a piecewise constant approximation forthe normal and does not allow to approximate the curvature at all. On the otherhand, all “traditional” methods of computing the curvature (like passing a circlethrough three neighbouring interfacial points, or piecewise parabolic interpola-tion of the discrete interface) are unsuitable for modeling the interface topologychanges, as those methods use a connectivity of the interfacial segments at inter-element boundaries. To model the interfaces of arbitrary shapes, one has to dealwith the interface within each element without a knowledge to which other el-ements the interface extends. The discrete level-set function

gives such “ele-mentwise” interface representation and is, therefore, convenient for determiningthe interface and its normal and curvature.

Below we will show that an application of the gradient averaging techniqueenables to obtain a piecewise linear continuous approximation for the normal,which can be expected to differ only by 3 : in -norm from the exact normal.The curvature, then, can be computed either directly as 0 + )7W to result in apiecewise constant approximation, or by using a variational formulation for thecurvature equation to yield a piecewise linear continuous approximation; the ac-curacies of such curvature approximations are 3 : and 3 : correspondingly.

64

Suppose the following:

(a1) the exact interface \ ;(a2)

3?I: is a continuous in function having as a zero level set;

(a3) is a conforming regular triangulation of with being the triangulationparameter;

(a4) there exists such a vicinity of , , having half-width ( ),that

\ ,3 : ;(a5)

is the signed distance function within , i.e. + 7

in .According to (a5) the interface normal can be computed as W +

on (thoughall the results of this section remain valid without the last assumption, if C + 7 ' in with 4 and independent of ).

With the triangulation we associate the finite element space of linear ele-ments

\ 3 : N / having the basis / , @ . We define linear interpolant

for \ 3 Q: by setting

3 : 3 : for each node of the triangulation . Next, wedefine the columns , , of size

as

(126)

where

is the lumped mass matrix of size

obtained by the row-sumdiagonalization from the consistent mass matrix

with components

3PK : F ] ; , , is

matrix with components

39 : ]and

is a column of size

containing the nodal values of the interpolant

.

Finally, we denote byW the vector with components

39I: , ( , whichare the functions from

defined by their nodal values .Lemma. Under the assumptions (a1)–(a5) there holds

W Q0 W \

where constant \ is independent of .Proof. First, we see that each -th component of the column , , can beexpressed from (126) as

3P : 3P : but

is a diagonal matrix with entries

3PK :M O3 :$] ,] ,

which implies

3P: 3 : 3P: 39 : ] ,]

3

N F,] : N

N ,]

N 7]

N ,] N G (127)

65

Here the fact that is constant within each element has been employed, and

, are the triangles having the node with number as a common ver-tex. It is evident from (127), that at each internal node of the triangulation (i.e.not belonging to the boundary of the domain ) we obtain the components

of our approximate normal by averaging

over the patch of elements sharing

the considered node. Particularly, if all the triangles , have the samearea, we obtain the well-known formula

3 9:

N (128)

where is the node with number .Thus, the approximate normal

W is obtained, in fact, by applying to the in-terpolant

the linear gradient-averaging operator studied in

Krizek and Neittaanmaki (1984) [101], Krizek and Neittaanmaki (1987) [102]. Theform (126), using the lumped mass matrix, is, however, more convenient fromcomputational viewpoint than the form (127).

The rest of the proof is straightforward. In Krizek and Neittaanmaki (1984)[101] it was shown that for any polynomial ; ;3 F : of 2nd degree thereholds E3 ; : 3 :( +;3 : at any internal node of the triangulation (see Lemma 3.1 in [101]). Then, using the Taylor series expansion for +

in and the fact that

*\ 3 : , we immediately obtain

E3 :/01+ \ (129)

which yields the result of the Lemma.

Having computed the continuous approximate normalW we can calculate the

curvature either directly as

0 +2) W (130)

or by using the following variational formulation for the curvature equation:

F] W ),+ F]=0 3 W )ZW : ]4 (131)

which amounts to solving the system with the consistent mass matrix. ThreeJacobi iterations considered in section 3.1.3 are sufficient to obtain the solutionwith a reasonable accuracy.

To assess the accuracy of our normal and curvature approximations we con-sider a simple example of the circle with center (0.5;0.5) and radius CHG . Theexact normal is given by W for any point 3 S : located onthe circumference, the exact curvature . The exact signed distancefunction

is defined so that it is positive inside of the circle and negative outside

(hence, W + ). Then we construct a regular triangulation in the closed

domain C C , the linear interpolant

for

, and localize the interfaceto obtain the piecewise linear interface interpolation as described in section3.2.3. Let IS , be the set of points determining " (the intersection

66

points of zero-level set of

and elements’ edges), and , be the setof interfacial line segments ( " is the union of all and all IS ). We calculate thecontinuous approximation

W for the normal using (126) and two approximationsfor the curvature: discontinuous piecewise constant by (130) and continuous by (131). The error is computed in the

and in the discrete (“average”)norms:

W 0 W W 439I ?:/0 W 39IP:and

W 0 W J

<'

W 439I ?:/0 W 39IP:for

W and ,

0 0 3 I9:and

0 J

'

0 3 I9:for ( I is the midpoint of the interfacial line segment , F ).

The results are presented in tables 1–3:

W 0 W W Q0 W C G ) C G ) C C $G<) C HG ) C C HG C<) C G ) C C HG ) C G <) C RC G <) C $G ) C ; G G

TABLE 1 The error of the normal approximation.

Q0 Q0 = C G C) C HG C) C = C $G <) C G <) C = C $G ) C G <) C = C HG C) C G C) C = C G C) C G ) C ; G C G CRC

TABLE 2 The error of the curvature approximation (discontinuous).

67

Q0 Q0 = C G ) C G C) C = C HG ) C $G ) C = C HG ) C G C) C = C G <) C G ) C = C G ) C $G ) C ; HG C G

TABLE 3 The error of the curvature approximation (continuous).

It is clearly seen that all the approximations converge monotonically to the exactquantities with the rate 3 : , ;

(only the three most refined grids with C C RC have been used to calculate the convergence rate). Theapproximate normal converges with the rate 3 : as predicted theoretically, andthe discontinuous approximation of the curvature converges as 3 : what couldbe expected; the continuous approximation of the curvature exhibits, however, a2nd-order convergence rate, which is quite remarkable, since we use only linearinterface interpolation.

Remark 3.9. We have investigated the approximation properties of the normaland curvature derived from the interpolant

of the exact level-set function

.

In reality, however, we have at our disposal only the finite element approxima-tion

obtained as a solution of the level-set convection problem. For ellipticproblems there holds an inequality of the type

+ Q01+ \ (see Krizek and Neittaanmaki (1987) [102] and references therein), using whichthe 3 : -superconvergence of the averaged gradient of

can be easily de-duced (in fact, 3 : -superconvergence, but 7 3 : for any realisticmesh sizes). An analogous result is also obtained for repeated gradient averag-ing, showing a superconvergence for the second derivatives if the exact solutionis smooth enough (see Lakhany and Whiteman [106]). We are not aware of anysimilar results for the solutions of hyperbolic (transport) equations, but there isa hope (supported by numerical experiments) that the approximate solution

is also superclose to the interpolant

of the exact solution

as in elliptic prob-

lems. It is important to note that we can expect at least \ -regularity of theexact interface (hence, of

in some vicinity of the interface) under the assump-

tion of sufficient smallness and smoothness for the given data of the problem, seethe discussion in section 2.3.

3.2.6 Evaluating the interfacial force and density/viscosity fields

After the approximations for the interface normal and curvature have been con-structed, the discrete interfacial force term W ) F]E (132)

68

can be easily computed ( stands here for either or ). Since is the union

of closed interfacial line segments , the integral (132) can be represented asa sum of the integrals over and, thus, computed elementwise. The integralover each interfacial line segment is evaluated with the help of the standard 2-point Gaussian quadrature that is exact for polynomials of 3d degree. This choiceof the quadrature formula is justified if the weighting function for the viscousmomentum equation is piecewise linear ( -elements for velocity). Numerical ex-periments show that the same quadrature rule can be utilized when -elementsare used for velocity, although the integration will not be exact. If the velocity isapproximated by -elements, the total polynomial degree of the integrand be-comes equal to 4, when , and the 3-point Gaussian quadrature exact forpolynomials of 5th degree may be exploited.

The localization of the interface (see section 3.2.3) enables us to compute accu-rately the integrals containing discontinuous density or viscosity coefficients; thatis especially important for preserving a discrete momentum balance in the vicin-ity of the interface. Define the characteristic function

3?I:J for I C for I

and the discontinuous density and viscosity fields! 39I : ! ' 39I: 3 ! /0 ! :6 3?I:J 6 ' 3?I: 396FF0 6 :GIt is evident that on the elements not crossed by the interface, the density andviscosity coefficients are simply constant, and evaluation of the correspondingintegrals can be done in a standard way. Thus, the problem is to compute theintegrals of the form

N ! 3?I:`Y@39I:$] (133)

over each triangle intersected by the interface " ( Y is a continuous within function). First, we may note that taking ! 39I : constant in (equal, e.g., to theaverage 3 ! ' ! : ) implies the error 3 : of computing the integral over ,and, since the number of such triangles crossed by is proportional to

, theerror in computing the global integral

! 39I:OY-39I :$] is 3 : for a general func-tion Y . This is insufficient accuracy, and numerical experiments indicate that thismethod is approximately equivalent to the smoothing out the density/viscositycoefficients over a few grid cells in the finite difference framework (see the discus-sion in Tornberg [195, 4]). On the other hand, a straightforward use of standardnumerical quadratures for a discontinuous integrand is strictly prohibited, as thequadratures are not convergent in such a case.

To overcome the difficulty of integrating a discontinuous function, we ad-vocate the so-called “modified quadrature” or “discontinuous integration” ap-proach that is proved to be useful in two-phase problems (see, e.g., Fachinottiet al. [53] and references herein, Tornberg [195]). The approach essentially uses

69

the fact that the discrete interface " is just a line segment within each interfacialelement . Indeed, the triangle becomes cut in two parts, a triangle and aquadrilateral. If we denote by

and the values of the characteristic function

39I : in those parts of , the integral (133) can be calculated as follows:

N ! 39I :`Y-3?I:$] 3 ! ' 3 ! F0 ! :: N Y-39I :$] ' 3 ! ' ,3 ! 0 ! :K: N N Y-3?I:$] 3 F0 : 3 ! F0 ! : N Y@39I:$] ' 3 ! ' ,3 ! F0 ! :: N Y-3?I:$] G

This expression can be readily computed, since

where the level-set function C and

C where

C (we set

positive in and negative in ), andboth integrals are over triangles; no subdivision of the quadrilateral intotriangles is required, which makes the algorithm simpler (this is especially truefor its 3D counterpart). On the discrete level the integrals over and may becomputed exactly, assuring a good evaluation of the discrete balance equationsnear the interface.

The error of the discontinuous integration method comes from the interfaceapproximation by a straight line, which yields 3 : error of the integration overa single triangle (with an error constant depending on Y@39I: ). The number ofthe triangles intersected by the interface is proportional to

, thus the error incomputing the global integral

! 39I :`Y-3?I:$] is 3 : for a general function Y .

3.3 Summary of the algorithm

Our computational approach for numerical modeling of interfacial flows can besummarized as follows:

Step 0. Initialization of the level-set function and velocity.

For each -th time-step, HAGAGAG :1. Computing of interface normal, curvature and density/viscosity fields.

2. Navier-Stokes convection step.

3. Viscous diffusion step.

4. Projection step.

5. Level-set convection step.

6. Reinitialization step.

7. Level-set correction step.

The steps 1.–7. are performed successively, and each of the steps 2.–5. may useits own local time-increment size. On each step the last computed velocity isexploited; the viscous diffusion and projection steps use the interface positionfound on the previous global time-step. It is also noteworthy that the steps 5.–7.

70

can be computed in a fully parallel manner with the step 2. The whole algorithmis very flexible; it allows, for instance, to compute unsteady interfacial Stokesflow just by omiting the Navier-Stokes convection step.

3.4 Stability issues and time scales

Each interfacial flow problem is characterized by some set of non-dimensionalparameters (criteria) as well as by some specific time scale. That time scale is ex-tremely important, as it reflects the dynamics of the considered physical process;it also serves as a guideline for selecting a time-step size in numerical simula-tions. Although the physical time scale is always dependent on the problem athand, there are some common scales connected with typical physical phenomenalike gravity, surface tension, viscosity.

Let be the length scale of the considered problem. The gravity phenomenonis well represented by the dimensionless Froude number

, where is the magnitude of the gravitational acceleration and

is the characteristicscale of velocity. If we denote by @% [ the typical time scale of a gravity-drivenphysical process, then @% [ . A characteristic time-scale should besuch that all physical effects forming corresponding non-dimensional criterioncounterbalance each other; thus, from <

we can deduce

@% [ G

The surface tension phenomenon in viscous flow is characterized by the capil-lary number \

, where 6 is the dynamic viscosity, is the coefficient

of surface tension and

is the velocity scale. If @% [ is the typical timescale of a capillarity-driven physical process, then

-% [ , and from\ @

we obtain

@% [ 6 GFinally, the viscosity is known to give a “viscous time-scale” through the non-dimensional Strouhal number /%

. Requiring /% we immediately

obtain

@% [ ! 6 GTo capture the dynamics of the modelled physical process, the numerical methodshould use a time-step not exceeding the time-scales of all involved physical phe-nomena. However, besides physical restrictions on the time-step size, there aresome constraints originated from the numerical approximation. In our compu-tational method, an example of such stability-restriction was the CFL conditionappearing on the Navier-Stokes convection and on the level-set convection steps

71

due to explicit time-discretization. The projection step does not suffer from anyupper bound for the time increment, having only the inverse stability-constraint(lower bound) for the time-step size if the equal-order interpolation is used forvelocity and pressure (this constraint is excluded by using an LBB-stable approx-imation). The viscous diffusion step possesses two stability constraints due toexplicit treatment of the gravity and surface tension forces. Those constraints canbe easily derived by the following empirical analysis (see Kang et al. [92]):

the stability restriction due to gravity can be represented in a form of CFL-

type condition

, where the corresponding gravity-induced ve-locity @% ; then, we obtain the maximal allowed time-step size as

@%

the stability restriction due to capillarity can be represented in a form ofCFL-type condition

, where the corresponding capillarity-indu-ced velocity can be calculated using the capillarity-induced acceleration ( T is the discrete delta-function included since the capillary force ap-pears only on the interface); it is easy to see that T at the interface, andthe maximal possible curvature resolved on the grid with size is ( ;thus, we obtain @%

, and finally

@%

! GThe latter condition was derived also in Brackbill et al. [15].

If an implicit time-discretization for the viscosity term is used, there are noother stability restrictions except those indicated above. Thus, the final stability-constraint for the viscous diffusion step is

@% -% [ @% [ @% @%

since @% [ is always greater than @% . It is important to note that @% and@%

observed in numerical experiments are almost 2 times larger than theirtheoretical predictions, which can be explained by the stabilizing influence of thelevel-set correction procedure.

4 NUMERICAL RESULTS

4.1 Static bubble

Our first test is a circular bubble in static equilibrium. The net surface forceshould be zero, since at each point on the bubble surface the tension force iscounteracted by an equal and opposite force at a diametrically opposed point.The correct solution is a zero velocity field and a pressure field that rises from aconstant value of ; outside the bubble to a value of ; ; ' inside thebubble, according to the Laplace-Young law ( is the bubble radius).

FIGURE 13 Spurious currents in a static bubble simulation.

The problem is characterized by three dimensionless parameters: Laplace num-ber 35<: ! 6 , density ratio ! ! and viscosity ratio 6 6 , where index“1” corresponds to the fluid surrounding the bubble, index “2” to the fluid insideof the bubble. When numerical simulations are performed, spurious currents ofamplitude are observed (see figure 13). In Lafaurie et al. [105] it was conjec-tured that the amplitude of the spurious currents must be proportional to 6 ,

73

which is equivalent to having an approximately constant value of the capillarynumber \ - 6F .

The computational domain is the square 35C : 35C : ; the bubble has the center3PC G C G R: and the radius C G . The viscosity and density ratios are equal to1, and no-slip condition is imposed everywhere on the domain boundary; thepressure is fixed by setting it zero at one of the domain corners (hence, ; C inour computations). The gravity is neglected, and the surface tension coefficient may be varied, thus yielding different Laplace numbers. The computations wereperformed until the non-dimensional time

% % @% [ C , where the physical(capillary) time scale @% [ 235<:`6 (see also Popinet and Zaleski [137]).

Table 4 illustrates the constant character of \ over a broad range of Laplacenumbers:

HG C 4G C ) C HG C<) C HG C) C HG C) C \ G ) C G C) C G ) C G ) C G <) C TABLE 4 Independence of the capillary number \ with respect

to the Laplace number (mesh size RC , ele-ments for velocity).

Table 5 shows the convergence of the non-dimensional velocity in the norm,

i.e. the decrease rate for the non-dimensional amplitude 6 of the spuriouscurrents. The experiments were performed on three different grids with C C C , and the constant time step of size C G C was always used; theLaplace number was fixed at HG C) C . Three types of elements were tested for thevelocity approximation: J , and S . The corresponding convergence rates aredepicted also on figure 14.

/ ( C G <) C G ) C HG ) C ( C G ) C G ) C G ) C ( C HG C) C $G ) C G ) C ; C G C G C G

TABLE 5 The error and the order of convergence (; ) in the

norm for the non-dimensional velocity #" 6F : compar-ing the three types of velocity elements.

It is apparent that and elements give nearly first-order convergence, whilethe order of convergence for is CHG . In fact, the origin and the decrease rate ofnumerical spurious currents can be understood, if we take into account that theonly source of singularity in the present test is the interface-concentrated capillaryforce. The analysis provided by Wahlbin [204] for elliptic problems shows that asingular force term brings a ”local” pollution to the finite element solution. Inthe considered test the problem is essentially elliptic; the pollution appears in theinterface neighbourhood of half-width and makes the rate of convergence inthe (correspondingly, in the

) norm equal only to 3 : , while the optimalrate is 3 : for linear elements. The use of quadratic elements, obviously, doesnot give any merit because of the singularity. Moreover, the J elements possess

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2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4−5.6

−5.4

−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

|log(1/h)|

log(

max

|vhµ/

σ|)

1

1

FIGURE 14 Convergence rates in the norm for the non-dimensional velocity:

“—–” - / element, “ 0) 0 ” - and “ 0 0 ” - S element.

the degrees of freedom, which are located at midpoints of element edges, i.e. oninter-element boundaries, and those degrees of freedom propagate the error offthe interface neighbourhood to surrounding elements. That, probably, explainsthe poor performance of the velocity elements in the test.

/ ( C G ) C G ) C G ) C ( C G ) C HG ) C G ) C ( C $G ) C HG) C G C ) C ; G G G

TABLE 6 The error and the order of convergence (; ) in the dis-crete norm for the non-dimensional velocity #" 6F :comparing the three types of velocity elements.

In table 6 the convergence of the non-dimensional velocity in the discrete (“av-erage”) norm is shown (the discrete norm is defined as # # E3P : ,where # 439: , = is the set of nodal values of the discrete velocity, and 3 : is the total number of velocity nodes). Since all the errors are con-centrated within an interface neighbourhood covering 3 : nodes, the rate ofconvergence in the discrete norm should be one order higher than in the

norm. This is confirmed by the numerical results, see table 6.Figure 15 illustrates the pressure behavior in dependence on the mesh size. The/ F finite element pair was used for the velocity-pressure approximation; the

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FIGURE 15 Pressure field corresponding to three different mesh sizes.

F and S / elements gave qualitatively the same results. It is remarkablethat the pressure discontinuity at the interface is maintained very well, in spite ofthe continuous approximation for the pressure. This result justifies the splittingof the interfacial stress jump condition, according to which the surface tensionacts on the pressure only indirectly (the capillary force is present in viscous mo-mentum equation, being completely separated from the pressure computation).The qualitatively good agreement with the Laplace-Young law is supported byexcellent quantitative results shown in table 7. The absolute and relative errorsin the Laplace-Young law indicate very good accuracy of the pressure inside ofthe bubble (;$ ) when the elements are employed for velocity approximation.The velocity elements give nearly the same pressure accuracy, whilst the Jelements result in an order of magnitude worse accuracy for the pressure.

;$ 0*; 0 ) CRC

= C 4G C) C C G = C 4G C) C C G C = C HG C) C C G C

TABLE 7 The absolute and relative errors in the Laplace-Young law.

4.2 Rising bubble

The next test is the simulation of a single bubble, rising in an initially quiescentfluid due to the effects of buoyancy. The problem is characterized by four dimen-sionless parameters, which we choose to be the density ratio ! ! , the viscosityratio 6F 6 , the Reynolds number 3L <: ! 6F and the Bond number ! . Here is the initial radius of the bubble, index “1” correspondsto the fluid surrounding the bubble, index “2” to the fluid inside of the bubble.The Morton number

is often used instead of the Reynolds number.The computational domain is the rectangle 35C : 35C R: ; the bubble is initially

circular, with the center 35C G C G R: and the radius XXC G . The free-slip bound-

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ary condition is imposed on the vertical walls, and the no-slip condition on thehorizontal walls; the initial velocity is zero everywhere in the domain.

Figures 16 and 17 show the evolution of the bubble shape and of the veloc-ity field, corresponding to the same density and viscosity ratios ( ! ! C ,6F 6 C ). The Reynolds number in both cases, but the Bond numberequals and

respectively. The computations were performed on the trian-gulation based on C C rectangular grid, the time-step size @%* . TheCrank-Nicolson time discretization ( CHG ) was employed for viscous diffusionterm, and the elements were used for the velocity.

It is clearly seen that when the surface tension coefficient is rather large ( < ) the bubble has an ellipsoidal shape, while smaller surface tension (

)results in a more significant deformation of the bubble. In the latter case thebubble has a shape of a “dimpled ellipsoidal-cap”. Although all experimentalresults are known for really three-dimensional bubbles, a qualitative comparisonis possible. As a main reference the book by Clift et al. [27] may be cited, wherea general diagram of bubble shapes in dependence on the Bond and Reynoldsnumbers can be also found (see figure 2.5 in [27]). The comparison enables us toconclude that our numerical bubble shapes are in a good agreement with the ex-perimental predictions. Since many researchers use the Morton number insteadof the Reynolds number, we have to calculate this parameter: the Morton numberis C G C for figure 16 and G for figure 17. Then, a detailed comparison canbe made for our figure 16 with the numerical results of Unverdi and Tryggva-son [201] (figure 3 in [201]), and for our figure 17 with the experimental resultsof Bhaga and Weber [14] (figure 3e in [14]; see also the numerical results by Chenet al. [21], figures 15 and 19). Taking into account the fact that there is no exactcoincidence of the non-dimensional parameters in the tests done by different au-thors, we may find, however, a good qualitative agreement between our resultsand those cited above.

The streamlines in the reference frame of the bubble are shown on figure 18 fortwo considered cases. It is apparent that, while the secondary vortices developinside of the bubble in any case, the vortices in the bubble wake are pronouncedonly when the bubble deformation is significant, i.e. for high Bond numbers (es-pecially remarkable fact is the formation of the recirculating region behind thebubble, see figure 18, right). The results again compare very well with the resultsof Unverdi and Tryggvason [201] (figure 4 in [201]).The next group of numerical experiments is devoted to the convergence study.We took the dimensionless parameters as in the second case above, i.e. ! ! C , 6F 6 C , * , , and compared the solutions correspond-ing to CH CH C at the time %< G C<^ . Figure 19 illustrates the shapeconvergence, figure 20 shows the convergence of the pressure (the modified pres-sure, i.e. the pressure without its hydrostatic component, is depicted). We mayconclude that with the present method the qualitatively correct picture may beobtained even on very coarse meshes with C . It is worth noting also thatthe pressure discontinuity is captured very well; this fact has been already em-phasized in preceding section.

In order to assess the convergence rate of the velocity, we use the same phys-ical parameters as before, but the grids with C C C . At the time

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FIGURE 16 Evolution of rising bubble with large surface tension ( ! ! C ,6 6 = C , * , ); triangulation is based on C Crectangular grid.

FIGURE 17 Evolution of rising bubble with small surface tension ( ! ! ( C ,6 6 C , , ); triangulation is based on RC Crectangular grid.

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FIGURE 18 The streamlines in the reference frame of the bubble ( ! ! C ,6 6 Q C , < ): (left) , (right)

; triangulationis based on C C rectangular grid.

FIGURE 19 Grid convergence test for the bubble shape ( ! ! C , 6F 6 C , , ); time % G C .

%J G C ^ we compare the solutions obtained by three different methods. The firstmethod uses discontinuous curvature approximation (see section 3.2.5) thathas only 3 : accuracy. The other major sources of the discretization error in themethod are: (i) the singular capillary force which gives “local” pollution of thefinite element solution and 3 : velocity error in the

norm but 3 : error

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FIGURE 20 Grid convergence test for the modified pressure ( ! ! C ,6 6 C , , < ); time %J G C .in the discrete norm (see preceding section), (ii) the discontinuity of viscositycoefficient which results in a “global” pollution of the finite element solution andonly 3 : accuracy in both

and norms (see, e.g., Wahlbin [204]). All othersources of the discretization error are expected to be of the order 3 : . Thus,overall accuracy of the first method should be 3 : in both considered norms.

The second method eliminates the global pollution produced by the discon-tinuity of viscosity coefficient. It is done by virtue of a very simple local meshadjustment. Namely, the piecewise linear interface is considered as an anothergrid-line; thus, each triangle crossed by the interface becomes cut into two parts:a triangle and a quadrilateral; that quadrilateral is further divided into two tri-angles by drawing its shortest diagonal. This procedure is described in detail byLock et al. [115]. Definitely, the simplicity of this procedure must be paid off: theminimal angle in one of the three newly formed triangles may become infinitelysmall. However, it is possible to show that, due to the regularity of the originaltriangulation, the maximal angle in each of those triangles will be always boundedaway from . Hence, the “maximum angle condition” should be always satisfiedwith the local mesh adjustment, which guarantees a validity of standard interpo-

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lation estimates (see, e.g., Zenisek [209, p.391]). Two other drawbacks related toviolating the minimal angle condition are the deterioration of the LBB-stabilityand the increasing of the condition number of the final algebraic system (see, e.g.,Rannacher (1999) [147]); we did not observe the former problem (probably, ow-ing to the very local character of the grid degeneracy), but did observe the latter(though, the increasing of the condition number did not have a dramatic effect).

To summarize, the only significant source of the discretization error in the sec-ond method is the insufficent accuracy of the curvature approximation, and nu-merical results confirm that the accuracy in the discrete norm remains 3 : .The third method combines the local mesh adjustment of the second methodwith the improved approximation

for the curvature (see section 3.2.5); thatimproved approximation has 3 : accuracy. Hence, the discretization error inthe discrete norm may be 3 : in the third method.

Since our scheme has first-order temporal accuracy, we kept the time-stepsmall enough to estimate the spatial accuracy of the velocity approximation. Weused the time-step size @%< and the velocity elements. The results arereported in tables 8 and 9.

We see that the first method, based on discontinuous curvature approximationand without local mesh adjustment, has a nearly first-order convergence in both and discrete norms, as could be expected. The second method, using dis-continuous curvature approximation and local grid adjustment, gives also first-order convergence of the velocity in both norms; the bottleneck of the methodis, probably, the curvature approximation having only 3 : accuracy. The thirdmethod, using continuous curvature approximation and local mesh adjustment,exhibits the second-order convergence in the discrete norm, as could be ex-pected, and the convergence rate 3 : in the

norm, which indicates thatthe local mesh adjustment eliminates not only the global but partially the localpollutions caused by the viscosity discontinuity and by the singular force respec-tively. It is important to note that the 3 : convergence rate in the average-normfor the velocity is allowed by the

-regularity of the exact velocity in the subdo-mains and (see the discussion in section 2.3).

method I method II method III # Q0 # HG C <) C G ) C G ) C # 0# G ) C G C ) C G ) C

G G G C; C G C G G TABLE 8 Measured order ; of velocity convergence in the

norm for the three different methods ( DC G ): method I– discontin. curvature /no mesh-adjustment, methodII – discontin. curvature /local mesh adjustment,method III – contin. curvature

/local mesh adjust-ment.

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method I method II method III # 0 # G ) C G C ) C $G ) C # 0 # G ) C $G ) C G ) C

G HG G ; C G G G TABLE 9 Measured order ; of velocity convergence in the dis-

crete norm for the three different methods ( C G ):method I, II, III as in table 8.

The following set of numerical tests is related to the mass conservation issue. Wetook again the physical parameters ! ! C , 6F 6 C , , @ ,the elements for velocity and the triangulation based on C C rectangulargrid. At the time % G C<^ we compare the result obtained without reinitializa-tion and level-set correction steps with the one computed with reinitialization–correction procedure. The bubble shapes are depicted on figure 21, and figure 22shows the relative change of bubble mass (i.e. of bubble area) as a function oftime.

FIGURE 21 The bubble shapes ( ! ! C , 6F 6 C , , ;time %B G C ; 7C ): (left) without reinitialization–correction,(right) with reinitialization–correction.

It is clearly seen from figure 21 that without reinitialization and correction stepsthe bubble lost not only a considerable part of its mass but also its shape: thenumerical oscillations in the level-set function caused the formation of artificialholes within the bubble. The necessity of the reinitialization–correction procedureis also confirmed by figure 22. The maximal relative error in the bubble mass on

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FIGURE 22 Relative change of bubble mass vs. time ( ! ! C , 6F 6 C , B , ; D C ): “ 0)0 ” - without reinitialization–correction, “—–” - with reinitialization–correction.

the considered time interval C is about without reinitialization–correctionand C G with the reinitialization–correction procedure.

Figure 23 illustrates the grid convergence for the relative change of bubblemass when the reinitialization and correction steps are included. The maximalrelative error in the bubble mass on the time interval C is about

for each ofthe three computations corresponding to C C C . This confirms the 3 : accuracy of mass conservation, as theoretically predicted in section 3.2.4.The conservation of bubble mass is, indeed, very good: for = RC the maximalrelative error is about C G , and for C it is C G C .

4.3 Breaking bubble

In this test we use the same initial conditions, boundary conditions, the den-sity and viscosity ratios as in preceding section, but take two different pairs ofReynolds and Bond numbers: < CRC , C C and < C C , C . Forsuch Bond numbers the bubble should become ”skirted” in accordance with theexperimental predictions (see figure 2.5 in Clift et al. [27]), which can be seen onfigures 24 and 25. Later, the skirt breaks off due to the action of the vortices in thebubble wake, and the remaining part of the bubble rapidly develops a spherical-cap shape. This is also in a full agreement with experimental observations (seeHnat and Buckmaster [85]), but, in reality, it happens at much smaller Reynoldsnumbers (about

C CRC ) than those used in our experiments. That is, probably,

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FIGURE 23 Relative change of bubble mass vs. time ( ! ! C , 6F 6 C , , ; with reinitialization–correction): “ 0 0 ” - C , “ 0) 0 ” - C , “—–” - C .

due to the essentially three-dimensional behavior of a real skirted bubble (see alsoSussman and Smereka [182]). However, a qualitative picture of bubble break-upseems to be captured well even with the presented two-dimensional computa-tions.

Our results compare well with the 2D numerical results of Sussman et al.(1994) [183], who employed the finite-difference level-set method, and with the2D numerical predictions of Baker and Moore [7], who advocated the boundaryintegral method for an inviscid gas bubble.A typical dependence on time for the relative change of bubble mass is shown onfigure 26. The fluctuation in about appears precisely at the moment of the bub-ble break-up and seems to be unavoidable, because of a limited grid-resolutioninherent to any numerical experiment. However, the method quickly recovers thebubble mass, and the relative error of mass conservation becomes a small fractionof one percent.

4.4 Merger of two bubbles

As in preceding section, we consider the rectangular domain 35C : 3PC R: withtwo circular bubbles inside; the center of the first bubble is 35C G G C : and its radiusis equal to C G , the center of the second bubble is 35C G CHG R: and the radius is CHG .Thus, the bubbles have a common axis of symmetry, and the initial distance be-tween them equals

of the radius of the largest bubble. We take zero velocity

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FIGURE 24 Break-up of rising bubble; C C , @ CRC , ! ! C , 6F 6Q C , triangulation is based on rectangular grid with C , ele-ments for velocity, @% .

field at the initial moment and the same boundary conditions as in sections 4.2,4.3. The dynamics of the bubbles, to a large extent, depends on the initial distancebetween them and on the magnitude of the surface tension. If the surface tensionis high enough (correspondingly, the Bond number is small), no merger happens,the bubbles develop nearly ellipsoidal shapes and rise separately (see, e.g., Un-verdi and Tryggvason [201]). Hence, in order to simulate a merger process, wetake comparably small surface tension coefficient and, to get the merger earlier, asmall initial distance between the bubbles. Our non-dimensional parameters forthis test are , C , ! ! C , 6F 6 C (Reynolds and Bondnumbers are based on the diameter of the large bubble).

Figure 27 illustrates the merger process. Since the small bubble is located veryclose to the large one, this lower bubble turns out to be in the wake of its upper“neighbour” and rises faster than that. In the process, two opposite signed vor-tices are created in the wake of the large bubble. This produces a lower pressure

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FIGURE 25 Break-up of rising bubble; CRC , < C , ! ! C , 6F 6 C , triangulation is based on rectangular grid with C , ele-ments for velocity, @% .

region behind the large bubble and generates flow streaming into the symme-try line of the flow. As a result, the front portion of the small bubble becomesnarrower and sharper. At time %Q CHG R , we see that the head of the small bub-ble almost catches up with the bottom of the large bubble. In the next moment% CHG , the two bubbles merge into a single bubble. At this time, the interfaceconjunction forms a cusp singularity that is rapidly smoothed out by viscosityand surface tension.

The results compare well with the computations by Chang et al. [20], by Torn-berg [195], and by Unverdi and Tryggvason [201]. The results compare favor-ably with the numerical predictions by Delnoij et al. [38] who found a qualitativeagreement with available experimental data.

The dependence on time for the relative change of total bubble mass is shownon figure 28. The fluctuations in about appear at the moment of the bubblemerger. However, owing to the reinitialization–correction procedure, the methodquickly recovers the total bubble mass and keeps the error of mass conservationwithin the predicted 3 : -accuracy.

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FIGURE 26 Breaking bubble: relative change of bubble mass vs. time; < CRC , < CRC , ! ! C , 6F 6 C , = C .4.5 Rayleigh-Taylor instability

The Rayleigh-Taylor instability, associated with the acceleration of a heavy fluidinto a light one under the action of a gravitational field, is generic to a widerange of physical phenomena, and many numerical simulations have been per-formed (see Baker et al. [6], Daly [34], Dervieux and Thomasset [42], Harlow andWelch [77], Tryggvason and Unverdi [196]). Chandrasekhar [19] analyzed theproblem by means of the linear theory, which is applicable in the early stages ofinstability development.

In our simulations we took the data as in Puckett et al. [139] (the same datawere used by Bell and Marcus [11] and by Popinet and Zaleski [137]). Namely, thecomputational domain is the rectangle 35C : 35C E: , the viscosities of both fluidsare equal to 6/ 6 ) C 3 ) ^,: , the fluid densities are ! @ G and ! C G , and the surface tension coefficient is zero. Unfortunately,the magnitude of the gravitational acceleration is not indicated in the last citedpapers, so we used the value G C ^ in our tests. The initial interface shapeis given by the function 3 : 4G C ' C G C 3L : (see figure 29), initial velocityfield is zero, and the boundary conditions are no-slip on the top and the bottomwalls and free-slip on the vertical walls.Figure 30 shows the development of the instability. As the heavy fluid penetratesthe light fluid, the interface begins to roll up along the sides of the spike givingthe characteristic mushroom shape. This phenomenon, known as the Kelvin-Helmholtz instability, is due to the development of short wavelength perturba-

87

FIGURE 27 Merger of two rising bubbles; , C , ! ! C ,6 6 C , triangulation is based on rectangular grid with = C ,F elements for velocity, @% .

tions along the fluid interface and parallel to the main flow. Despite the differ-ence in the magnitude of gravity, which results in different instability growthrates, our numerical predictions compare well with those of Puckett et al. [139]and of Popinet and Zaleski [137]. The maximum mass fluctuation is approxi-mately C G C , which is better than in front-tracking based simulations of [137]but slightly worse than in volume-of-fluid based computations of [139] (although,our grid was

G times coarser).Figure 31 illustrates the instability development in the presence of small amountof surface tension. The surface tension coefficient was chosen to be smaller thana critical value for which the flow is stabilized. According to Chandrasekhar [19],for an initial perturbation whose wavelength is , this critical value is given byJ 3 ! 0 ! :

GIn our case

, hence HG ) C ; we took C G C . The regularizing effectof surface tension is clearly seen from figure 31: the surface tension decreases

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FIGURE 28 Merger of two bubbles: relative change of total bubble mass vs. time; , C , ! ! C , 6F 6 C , C .

FIGURE 29 Initial configuration for Rayleigh-Taylor instability problem; the in-terface is given as 3 :J HG C ' C G C 35 : .

the growth rate of the Rayleigh-Taylor instability, delays the development of thesecondary, Kelvin-Helmholtz instability, and makes the interface smoother andmore compact. This is in a good agreement with numerical observations of Locket al. [115].

89

FIGURE 30 Rayleigh-Taylor instability with *C ; triangulation is based on rect-angular grid with = C , elements for velocity, @% .

FIGURE 31 Rayleigh-Taylor instability with B C G C ; triangulation is based onrectangular grid with = C , J elements for velocity, @% .

4.6 Bifurcating jet

This test concerns the high-speed propagation of a horizontal viscous jet in a vis-cous medium. The jet evolution is considered in the absence of gravity, but inthe presence of surface tension. The geometry consists of the rectangular domain3PC : 3PC : and the jet inflow boundary H3 : VC C G C G .

90

The initial velocity field is zero; the inflow velocity is horizontal and its magni-tude evolves in time as $3 0 : . The prescribed boundary conditions are theoutflow (zero tangential and normal stress) at the right side, the inflow at the jetinflow part of the boundary, and the no-slip on the rest of the domain’s boundary.

We assume that both fluids have equal viscosity 6 . Then, the problem ischaracterized by three non-dimensional parameters: the density ratio ! ! , theReynolds number ! ] 6 and the Weber number ! ] , whereindex “1” corresponds here to the fluid around the jet, index “2” to the jet fluid, is the magnitude of the inflow velocity and ] is the inflow diameter. Accord-ing to the values of these parameters, diverse regimes of jet flow can be observed(see Lin and Reitz [113]). Figure 32 illustrates the jet evolution with ! ! C , CRCRC and C , which approximately corresponds to the so-called sec-ond wind-induced regime. This is a high jet-velocity regime with relatively smallsurface tension coefficient. As the flow starts, the interface front expands veryquickly and undergoes the interfacial stress fluctuations which cause the unstablegrowth of short-length waves. Those waves could be suppressed by the surfacetension, but the interface is nearly flat in the vicinity of its frontal point, hence,the surface tension (that is proportional to the interface curvature) becomes al-most negligible there; this results in the jet bifurcation. In our computations thejet break-up happened approximately at time %- C G ^ at the distance C G ]downstream, which is in excellent agreement with the predictions of Danaila andBoersma [35]. Generally, the jet unbroken length is a function of the dimension-less parameters of the problem. At later time, the jet branches start to roll updue to the Kelvin-Helmholtz instability, and multiple secondary bifurcations aswell as drop formations can be observed (see figure 32, bottom row). The qualita-tive behavior of the jet in our numerical simulation compares satisfactorily withexperimental results, see Lin and Reitz [113].

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FIGURE 32 Jet bifurcation; C CRC , C , ! ! ( C , triangulation isbased on rectangular grid with C , elements for velocity,@%J .

5 CONCLUSIONS

We have presented the computational approach that is well suited for numericalsimulation of practically all kinds of unsteady two-fluid interfacial flows. Theapproach essentially relies on three basic components: finite element method forspatial discretization, operator-splitting for temporal discretization and level-setmethod for interface representation. The proposed numerical method is purelyEulerian and based on a fixed regular mesh. That is very advantageous from theviewpoint of both computational reliability and cost-effectivity, as the mesh doesnot have to be reconstructed on each time step, and also from the viewpoint ofimplementing the efficient preconditioners for resulting linear systems, since themesh possesses very simple regular structure.

The interface capturing approach employed in the method easily allows to fol-low the interface deformation even beyond the moment of the change of interfa-cial topology. Particularly, the combination of the level-set method with the finiteelement discretization enables to maintain sharp interface, without introducingany artificial parameters like interfacial thickness. The second-order accuracy ofthe interfacial normal and curvature approximations are shown to be readily at-tainable with the method, despite only piecewise linear interface representation.That is very important for modeling the interface-concentrated surface tensionforce. Using discontinuous integration technique and accurately found locationof the interface, the jump discontinuities of the density and viscosity coefficientsare properly incorporated into the scheme. This is a principal strength of thefinite element discretization, that the weak formulation, on which the discretiza-tion is based, allows to naturally build in coefficients discontinuities and singularinterface-concentrated forces. We have demonstrated that, using a simple localmesh adjustment at the interface, it is possible to obtain second-order accuratevelocity approximation, while without the adjustment only first-order spatial ac-curacy of velocity can be observed. The whole method has first-order temporalaccuracy that seems to be sufficient, since the spatial errors usually dominate incomplex interfacial flows. It is important to add that the special cost-effectivereinitialization–correction procedure guarantees an optimal, second-order accu-rate mass conservation for the method.

The operator-splitting invoked for time discretization splits the entire stronglynonlinear problem into the sequence of simpler subproblems corresponding todifferent physical processes. This allows to utilize specialized numerical tech-

93

niques for treatment of different physical phenomena and results in algorithmi-cally simple and very robust computational method. In particular, the methodhas been successfully tested for very wide range of the Reynolds number (fromthe Stokes limit up to

C ) and of density and viscosity ratios (up to C ). The

Chorin’s projection method, used as a part of the operator-splitting scheme, en-ables us to employ equal-order interpolation for both velocity and pressure; thus,all unknowns of the problem (velocity, pressure, interface) can be approximatedby continuous piecewise linear functions. Finally, it is noteworthy that the methodis easily extendible to three-dimensional case.

The performance of the proposed computational approach has been assessedin diverse tests and benchmark problems. However, a rigorous theoretical anal-ysis of the method has not been done yet. In fact, the convergence analysis forthe approximations to viscous free-surface/interfacial flows seems to be an al-most unexplored field. To our knowledge, there are very few works devotedto this topic. Rivkind (1980) [150] studied the steady Navier-Stokes equationsfor two-fluid interfacial flow and obtained some a priori error estimates using\ finite elements for the velocity and for the interface approximations (see alsoRivkind (1991) [152]). Nitsche [126] investigated the steady free-surface Stokesflow and derived a priori error estimates for the finite element solution to theproblem. Later, Saavedra and Scott [161] considered a model free-boundary prob-lem that had many features of a free-surface problem for a viscous liquid; theysucceeded to show the optimal-order a priori estimates using only linear \ fi-nite elements. Recently, Smolianski [173] obtained a priori error estimates forthe finite element solution to the steady coupled Navier-Stokes/heat-conductionfree-boundary problem, though employing \ elements for the velocity and forthe interface approximations.

All aforementioned works dealt with the steady problems only and used ratherobsolete algorithms for interface modeling. It is even more important that in allthese works very strong assumptions on the uniqueness and regularity of the ex-act solution were made. In unsteady interfacial flows, we have at our disposalonly local-in-time solvability results (see the discussion in section 2.3); for a fi-nite time and realistic values of physical parameters, we cannot, generally, expecteven uniqueness of the exact solution (this is especially true in the case of anyinterfacial topology changes). Thus, a priori error estimates for an approximatesolution, which are obtained under some artificial assumptions on the exact so-lution, seem to be of little practical utility. The sensible directions of theoreticalresearch in this field could be, probably, a stability analysis (see recent work byBansch [8]) and a posteriori error estimation for simplified subproblems like, forexample, the steady Stokes system with discontinuous coeffcients and singularsource term (of course, that will not be a rigorous a posteriori error analysis ofthe entire problem but will still provide a necessary information for the adap-tivity of approximation). A more sophisticated local mesh adjustment at the in-terface and multigrid/multilevel approach could further improve the presentedcomputational method.

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YHTEENVETO (FINNISH SUMMARY)

Tyossa on tarkastelaan kahden sekoittumattoman viskoosin nesteen virtauksiatapauksessa, jossa nesteiden valilla on vapaa liikkuva pinta. Tavoitteena on luodayhtenainen strategia erilaisten kahden nesteen rajapintavirtausten numeeriseenmallinnukseen, jossa mukana on mahdollisesti myos reunan topologian muutok-sia (kuten yhdistymisia ja hajoamisia) seka realistinen maara fysikaalisia parame-treja.

Esitetty laskennallinen lahestymistapa perustuu kolmeen paaosaan: aarellis-ten elementtien menetelmaan laskenta-alueessa, operaattorin ositusmenetelmaandiskretisoinnissa ajan suhteen seka level-set-menetelmaan vapaan rajapinnan esit-tamisessa. Aarellisten elementtien menetelma perustuu ongelman esittamiseenheikossa muodossa, joka ottaa huomioon materiaaliparametrien epajatkuvuudetja rajapinnalla esiintyvat singulaariset voimat. Kayttamalla elementtimenetelmaavoidaan rajapinta paikallistaa tarkasti ilman keinotekoisten parametrien, kutenrajakerroksen paksuus, kayttoa. Tyossa osoitetaan myos, etta rajapinnan nor-maali ja kaarevuus voidaan laskea toisen asteen tarkkuudella gradientin keskiar-voistusmenetelmalla; nain pintajannitysvoimat pystytaan laskemaan tarkasti. Ai-kadiskretisointiin kaytetaan operaattorin ositusmenetelmaa, jonka avulla alku-perainen tehtava jaetaan pienempiin osatehtaviin. Erityisesti tama lahestymistapamahdollistaa samanasteisen approksimaation nopeudelle ja paineelle. Rajapin-nan topologian muutosten hallitsemiseen kaytetaan level-set-lahestymistapaa ele-menttimenetelmaan perustuvalla toteutuksella, joka antaa lisahyotya verrattu-na differenssimenetelmaan perustuvaan level-set-toteutukseen. Tyossa esitetaanmyos yksinkertainen massan korjausalgoritmi, joka mahdollistaa toisen asteentarkkuudella optimaalisen massan sailymisen.

Tyossa ehdotetun laskennallisen menetelman tueksi esitetaan lukuisia numee-risia esimerkkeja sisaltaen kuplan dynamiikan simuloinnin, jakaantuvan suihku-virtauksen ja Rayleigh-Taylor-epastabiilisuuden.