finite element methods on moving meshes for free surface...

Finite element methods on moving meshes

for free surface and interface flows

Dissertation

zur Erlangung des akademischen Grades

doctor rerum naturalium(Dr. rer. nat.)

von M.Sc. Sashikumaar Ganesangeb. am 08.10.1976 in Salem, India

genehmigt durch die Fakultat fur Mathematikder Otto-von-Guericke-Universitat Magdeburg

Gutachter:Prof. Dr. rer. nat. habil. Lutz TobiskaProf. Dr.-Ing. Jurgen Schmidt

Eingereicht am: 26.06.2006Verteidigung am: 31.07.2006

Acknowledgements

I would like to express my deep and sincere gratitude to my advisor, professor Lutz Tobiskafor his invaluable guidance and continuous support throughout my PhD studies. His greatknowledge, personality and experience have inspired me to work with him.

I am grateful to professor Jurgen Schmidt for his valuable ideas and fruitful discussionson the computational results presented in this research work.

I would also like to thank junior professor Gunar Matthies for many interesting dis-cussions on implementing techniques. I extend my thanks to professor Volker John for hisinvaluable ideas and discussions at the earlier stage of my research work.

I am also grateful to Dr. Teodora Mitkova for her help at the department and for hereditorial comments. I am very thankful to Dr. Walfred Grambow for providing uninter-rupted computing service and technical assistance. I would like to extend my thanks to allof my colleagues at the department for their timely help and encouragement.

I would like to thank my family and friends for their encouragement and interest. Mybig and special thanks goes to my wife Sangeetha for her support and patience.

This work has been financially supported by the German Research Foundation (DFG)through the Graduiertenkolleg 828, and is greatfully acknowledged.

Sashikumaar Ganesan

Institute for Analysis and Numerical MathematicsFaculty of MathematicsOtto-von-Guericke-UniversityMagdeburg Germany

Contents

1 Introduction 1

2 Mathematical Modelling 92.1 Free surface flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Freely oscillating droplet . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Impinging droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Droplet impinging on a hot surface . . . . . . . . . . . . . . . . . . 152.1.4 Weak formulation in dimensionless variables . . . . . . . . . . . . . 16

2.2 Two-phase flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Rising bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Finite Element Methods for Stationary Stokes Equations 293.1 Finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Finite elements on triangles . . . . . . . . . . . . . . . . . . . . . . 313.1.3 Global nodal functionals . . . . . . . . . . . . . . . . . . . . . . . . 333.1.4 Finite element space . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Finite element methods for interfacial flows . . . . . . . . . . . . . . . . . . 353.2.1 Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Spurious velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Numerical Methods for Time-Dependent Domains 434.1 Methods for time-dependent domain . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Fixed Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.2 Moving grid methods . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Approximation of the curvature . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Interpolated cubic spline technique . . . . . . . . . . . . . . . . . . 534.2.2 Laplace-Beltrami operator technique . . . . . . . . . . . . . . . . . 55

5 Efficient Solutions of Free Surface and Interface Flow Problems 615.1 Discretisation in space and time . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1.1 Temporal discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 61

i

ii Contents

5.1.2 Linearisation techniques . . . . . . . . . . . . . . . . . . . . . . . . 655.1.3 Spatial discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Mesh Velocity based on ALE mapping . . . . . . . . . . . . . . . . . . . . 675.2.1 Advection of boundaries (construction of Bn

h,n+1 ) . . . . . . . . . . 685.2.2 Inner points displacement (construction of An

h,n+1) . . . . . . . . . 705.3 Axisymmetric formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Numerical Results 776.1 Spurious velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Freely oscillating droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Impinging droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3.1 Comparisons with experiments . . . . . . . . . . . . . . . . . . . . . 876.3.2 Influence of the slip coefficient . . . . . . . . . . . . . . . . . . . . 906.3.3 Influence of impact velocity on the flow dynamics . . . . . . . . . . 1006.3.4 Influence of the surface tension on the flow dynamics . . . . . . . . 102

6.4 Liquid droplet impinging on a hot surface . . . . . . . . . . . . . . . . . . 1046.5 Rising bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Summary 115

Nomenclature

CD drag coefficient

do initial diameter of the bubble/droplet

g gravitational constant

I given time

L characteristic length

MF mass fluctuation

p pressure

q pressure test function

r radial coordinate (√

x2 + y2)

r0 initial radius of the bubble/droplet

S sphericity

t time

T temperature

U characteristic velocity

Urise rise velocity

Wdmaxmaximum wetting diameter

MFmaxmaximum mass fluctuation

u liquid velocity vector

v vector test function

w domain velocity vector

D(u) velocity deformation tensor

S(u, p) dimensionless stress tensor

T(u, p) stress tensor

ν, n normal vector

τ tangential vector

iii

Bi Biot number

Ca capillary number

Eo Eotvos number

Fr Froude number

Pe Peclet number

Re Reynolds number

We Weber number

βǫ slip number

θ contact angle

δ damping factor of the oscillating droplet

ρ density

µ dynamics viscosity

ω frequency of the oscillating droplet

K sum of the principle curvatures

σ surface tension of the liquid

A ALE mapping

Ω computational domain

Φ meridian domain

Chapter 1

Introduction

Fluid flows with moving interfaces are often encountered in many scientific and engineeringapplications. Performance of fuel tanks in space, coating of solid substrates with liquids,spray cooling, film boiling and crystal growth are a few applications. If a fluid flow containsan interface and the interface is between a gas and a liquid, then the flow is often refer toas a free surface flow. The position of the interface is determined by the capillary force,which results from the balance between the normal stress and the surface tension on theinterface.

In general, these flow problems are described by the time-dependent incompressibleNavier-Stokes equations together with capillary boundary conditions. A capillary boundarycan be viewed as a balance of forces on the interface. In numerical simulations of these flows,apart from field variables such as velocity and pressure, the moving interface itself has to bedetermined. These type of flow problems are generally called free boundary value problems(or) interface/multiphase flow problems. Tracking or capturing the interface, accuratelyincorporating the surface tension on the interface and guaranteeing the mass conservationare some of the challenges in the interface flow computations. Apart from these difficulties,the discontinuity in the physical variables such as density and viscosity across the interfaceincreases the complexity of computations. It turns out that the development of robustand efficient numerical schemes for computing fluid flows with unsteady motion of movinginterfaces is a challenging problem in computational fluid dynamics (CFD) field.

This research work is concerned with the robust and efficient numerical simulation offree surface and interface flows. As examples of free surface flows, we will consider afreely oscillating liquid droplet and an impinging liquid droplet on a uniformally heatedhorizontal solid surface with/without heat transfer effects. We are interested to study theinfluence of the impact velocity, surface tension, slip coefficient and heat transfer on the flowdynamics of the impinging droplet. A rising bubble due to the boyancy force is consideredas an example of interface flows. Precise incorporation of the surface tension and materialparameters are key factors in computing these flow problems. We will develop a numericalscheme, which is based on moving, interface resolving meshes for accurate simulations ofthese flows.

1

1 Introduction

Oscillating and impinging droplets

Oscillating droplet

A freely oscillating liquid droplet, which oscillates around its equilibrium shape with zerogravity is considered as a first example of free surface flows. The entire boundary of thefreely oscillating liquid droplet is a free surface, and the liquid motion is solely driven bythe capillary forces. Furthermore, initially the liquid is assumed to be at rest. After theshape of the droplet is perturbed from the equilibrium, the droplet starts to oscillate dueto the imbalance of forces on the free surface. The fluid flow is governed by the timedependent Navier-Stokes equations with capillary and kinematic boundary conditions.

Impinging droplet

A liquid droplet impinging on a solid surface is considered as a second example of freesurface flows. We will compute the sequence of spreading and recoiling process of thedroplet after impinging on a horizontal solid surface. This problem exemplifies the generalproblem of moving contact line. Using the no-slip boundary condition on the Liquid-Solidinterface, an unbounded stress singularity could occur at the moving contact line, wherethree phases (liquid, solid and gas) intersect, see Figure 1.1. This singularity is also calledkinematic paradox. This difficulty has been addressed by several authors [38, 39, 75]. To

θ

gas liquid

solid

Figure 1.1: Droplet Deformation.

remove this singularity, different type of slip boundary conditions have been proposed inthe literature, see for an overview, [20]. Among them, the Navier-slip boundary conditionis widely accepted, but it introduces an empirical slip coefficient. The unknown coefficientis also called a momentum transfer coefficient [39]. A variety of expressions has beenproposed by several authors for the slip coefficient, see for e.g., [33, 37, 69]. We will studythe influence of the slip coefficient on the flow dynamics of the impinging droplets.

Contact angle

The contact angle is another important property of a liquid droplet which is determinedby the material properties of the liquid, solid and gas phases, see Figure 1.1. Based on the

2

contact angle, liquid droplets can be classified into wetting and non-wetting liquid droplets.This property plays a significant role in industrial applications. For instance, a smallcontact angle is desired in spray coolings, whereas a large contact angle is desired on selfcleaning materials. Thus, the inclusion of the contact angle in the numerical computationsis essential for producing physically acceptable solutions. In the sequence of spreadingand recoiling processes, often the contact angle hysteresis occurs on real surfaces. Thedifference between the advancing contact angle θa and the receding contact angle θr isgenerally referred to as the contact angle hysteresis. The advancing contact angle θa isthe largest angle just before the spreading starts and the receding contact angle θr isthe smallest angle just before the recoiling starts. Surface roughness, inhomogeneity andcontaminations are a few reasons for the occurrence of the hysteresis, see [9].

The first numerical simulation of a water droplet impacting a flat plate using a fixed gridMarker-and-Cell (MAC) method has been reported in [34]. Effects of both surface tensionand viscosity have been neglected in their simulations. A finite element deformable gridbased on Lagrangian approach for the droplet has been used in [24] without consideringthe wetting effect, and later with the wetting effect in [23]. These authors have obtained agood agreement between their experimental and computational results. Since there is noconstraint on mesh movement in the Lagrangian approach, the distortion of the meshesbecomes exceedingly large and often remeshing is needed. Several numerical studies havebeen made for the droplet impinging problem using a fixed grid based Volume-of-Fluidmethod, see for instance [74], and references therein. In all of these studies, often the freeslip condition on the entire liquid-solid interface or at the contact line has been used toremove the stress singularity.

Droplet impinging on a hot surface

In addition with the conservation of mass and momentum equations, we will consider theenergy equation for the heat transfer in the droplet. Further, our study accounts theMarangoni convection, which occurs due to the temperature dependent surface tensionand non-zero temperature gradient for the shear stress boundary condition on the freesurface. We assume that all material parameters such as density, viscosity and thermalconductivity in the droplet are independent of the temperature. Furthermore, the basicassumption of our model is that the density, viscosity and thermal conductivity in the gasphase are negligible compared with the liquid phase. In this initial study, the temperatureof the solid surface is assumed to be well below the boiling point temperature, and thusthe Leidenfrost effect does not occur. We will study the heat transfer in the droplet andthe influence of the heat transfer on the flow dynamics of the droplet. It has been reportedin [21] that the heat transfer slow downs the spreading process by creating counteractflows.

3

1 Introduction

Two-phase flows

Rising bubble

In multiphase flows, each interface separates two immiscible fluid. In general, these twofluids will have different densities and viscosities, and therefore these physical parame-ters will be discontinues across each interface. As a special example of multiphase flows,we consider a rising bubble problem with large jumps in the density and viscosity. Weconsider a spherical bubble, which is inside another immiscible quiescent liquid. Initially,the bubble is at rest. The density of the bubble is assumed to be (much) smaller thanthat of the surrounding liquid. Thus, the bubble starts to rise due to the buoyancy force.The buoyancy-driven motion of a bubble through a quiescent fluid has been studied bothexperimentally and numerically by several authors. Based on the experiment results, clas-sifications of the shape and the behaviour of rising bubbles have been reported in [12].Experimental data, correlations of rise velocity and shape for a rising bubble in a vis-cous liquid have been presented in [5]. Numerous simulations have been made by severalauthors using a boundary-integral, finite difference and level-set methods, see for exam-ple [50, 59, 67]. We will simulate the rising bubble with interface resolving, moving finiteelement meshes.

One of the main problem in the interface flow simulations is the generation of spuriousvelocities subject to external local forces [26, 28]. This non-physical velocity is also calledparasitic currents. Various sources such as the approximation of the incompressibilityconstraint, the approximation of the curvature and the approximation of the interface areresponsible for this phenomenon. The flow dynamics of the fluid strongly depends on themagnitude of spurious velocities near the interface. In particular, unphysical movementsof the interface could be generated by spurious velocities. Thus, a numerical scheme,which suppresses spurious velocities has to be used in computations of interface flows toget accurate and physically acceptable solutions. It has been studied in [26] for differentfinite element discretisations on interface resolving and non-resolving meshes in two-spacedimensions.

Numerical scheme

Discretisation

We use a finite element method to approximate the solution of all considered problems. Itis more flexible than other methods such as finite differences and finite volumes, especially,in deforming, complex domains. To achieve a high accuracy and stability, we prefer asecond order inf-sup stable finite element pair for approximating the velocity and pressure.The time discretisation is made with the strongly A-stable fractional step-ϑ scheme.

4

Advection of interfaces

In time dependent moving interface flow problems, the position of the interface is a partof the solution process. Thus, an additional method is needed to determine the positionof the interface. Several techniques have been proposed in the literature to capture/trackinterfaces, see for an overview [71]. Based on the computation of the fluid velocity, all thesetechniques can be classified into two classes: (i) fixed grid and (ii) moving grid methods.Each method has its own advantages and disadvantages. Among fixed grid methods,Marker-and-Cell (MAC), Volume-of-Fluid (VOF), Level Set (LS) and Front-Tracking (FT)are a few popular techniques used in interface flow simulations.

The MAC method, where marker particles are used to identify each fluid, is the oldestand most popular method for computing multiphase flows. In the volume-of-fluid method(VOF), a “volume-of-fluid or marker function” is used to identify each fluid phase. The“volume-of-fluid function” gives the volume fraction of one of the fluids in each cell ofthe finite difference or finite element meshes. Although the volume fraction is uniquein a cell, the representation of the interface is not unique. Furthermore, an accuratecalculation of the local curvature of the interface from the volume fraction to incorporatethe surface tension is rather difficult. Recent developments such as a higher order algorithmfor advecting the volume function and local mass preserving technique have increased theaccuracy and applicability of this method.

Another popular and very flexible method is the Level Set method, which was intro-duced by Osher and Sethian [53]. A continuous zero level set function is used to representan interface. Several interfaces can be represented by a single level set function. Topologicalchanges in the domain can be handled by this approach in a natural way. The numericalerror obtained while solving the advection equation causes difficulties in preserving themass conservation. A coupled level set method together with the Volume-of-Fluid hasbeen also proposed in [66] for better mass conservation.

The front tracking method is another popular approach for tracking interfaces. Differentvariants of this method have been proposed in the literature. For example, use a separatefront markers for the interface, and align the fixed background grid only near the interfaceto follow the front markers. Another variant of this method is use a fixed backgroundmesh for computing flow variables and track the interface by using a separate grid of onedimension lower. In this variant, to track the interface the velocity field from the fixedbackground grid has to be interpolated to the one dimension lower front grid, see for anoverview [71].

In all fixed grid techniques, interfaces are not resolved by the meshes, which are usedfor the computation of the flow variables. This is the main drawback in fixed grid meth-ods and poses difficulties to include the surface force and the material properties, whencomputing the flow variables. Often, the continuum surface force (CSF) technique [7]is used to include the surface force in fixed grid techniques. The basic idea in the CSFtechnique is to include the surface force over a small region near the interface using aregularised Dirac Delta function. For including discontinuous material parameters differ-ent techniques such as defining the material property as a function of shortest distance

5

1 Introduction

from the interface, using a steep gradient to translate jumps and defining the materialparameters by using a smoothed Heaviside function have been proposed in the literature.However, these additional techniques induce numerical errors in the solution, see for in-stance [70]. Another difficulty in the interface non-resolved meshes is the generation ofspurious velocities [26]. To reduce spurious velocities, very fine meshes are needed (at leastnear the moving interface). Furthermore, guarantying the mass conservation in each phaseseparately is also a difficult problem in fixed grid methods due to the unavoidable spuriousvelocities. Therefore, we prefer an alternative approach in our computations.

We use a deforming, moving grid technique to handle time-dependent domains. Inparticular, we use the so-called Arbitrary Lagrangian Eulerian (ALE) approach. In thisapproach all interfaces/boundaries are resolved by the meshes and move with the fluidas in the Lagrangian approach. One of the main advantages in the ALE approach is theinner mesh points can be displaced in an arbitrarily prescribed way. This avoids the quickdistortion of meshes and remeshing almost. Since all interfaces are resolved by meshesin the ALE approach, the inclusion of surface force and different material parameters ofdifferent fluids are straight forward.

The curvature term on the interface can be handled by several ways. For instance, inthe level set method the curvature can be replaced by using the level set function withoutcalculating the curvature separately. In the front tracking and the ALE method, thecurvature can be calculated by using an interpolated cubic spline. Another technique isincluding the curvature in the weak form of the model equations using Laplace-Beltramioperator. This technique was introduced in the finite element context by Dziuk [19], whichis based on replacing the curvature by the Laplace-Beltrami operator, and then integrationby parts. It avoids the explicit calculation of the local curvature, and further the curvaturecan be treated semi-implicitly. Furthermore, the contact angle in moving contact lineproblems can be included in the weak form canonically.

Axisymmetric formulation

In our numerical study most of the considered problems are rotational symmetric. In a 3D-axissymmetric problem, we first derive the weak form of the model problem in three spacedimension with Cartesian coordinates, and then transform all bilinear and trilinear formsinto the cylindrical coordinates after incorporating all boundary conditions. This avoidsseparate boundary condition in cylindrical coordinates and the same model equations canbe used in two- and three-dimensions. However, for the well-posedness of bilinear andtrilinear form in 3D-axissymmetric, the solution spaces have to be considered as weightedSobolev spaces.

Structure of the thesis

In Chapter 2, we present the governing equations of the freely oscillating and impingingdroplet problem. We discuss a choice of the type of boundary condition on the liquid-solid interface and the importance of the contact angle in the spreading droplet problems.

6

Next, the governing equations of a droplet impinging on a heated surface are presented.The weak form of all these equations is derived after rewriting it into a dimensionless form.Further, the governing equations for a rising bubble problem are presented in this Chapter.The weak form of the rising bubble problem is derived in such a way that both phases aregoverned by a single set of equations with variable material properties.

Chapter 3 gives a short overview over finite element methods. We discuss the construc-tion of a few finite elements and a few finite element spaces with global nodal functionals.Next, we present a stable finite element discretisation of stationary Stokes equations anda choice of inf-sup stable finite element pairs for Stokes and Navier-Stokes equations. Tostudy the effect of spurious velocities in interface flows, further we present a static bubbletest problem in this Chapter.

In Chapter 4, we recall a few interface capturing/tracking techniques such as VOF, LS,FT and ALE methods for handling time dependent domains. We discuss the reformulationof all of our considered model problems into an ALE form, and the geometric conservationlaw (GCL) for the ALE mapping in the finite element context. The interpolated cubic splineand the Laplace-Beltrami operator technique for handling the curvature are presented inmore detail. The inclusion of the contact angle in the impinging droplet using the Laplace-Beltrami operator technique is also presented in this Chapter.

Discretisation of the time dependent Navier-Stokes equations in space and time is pre-sented in Chapter 5. A few time discretisation schemes and linearisation techniques arediscussed. We present a few methods for advecting the boundary/interface points and apractical construction of the ALE mapping. The reformulation of a three dimensional weakform (in Cartesian coordinate) of rotational symmetric problem into a 3D-axisymmetric(cylindrical coordinate) weak form is presented in this Chapter.

In Chapter 6, we present the computational results for all our considered model prob-lems. First, we present the numerical results for the static bubble problem which areobtained using a continuous and discontinuous pressure approximations. Next, the com-putational results for the oscillating bubble problem are presented. The obtained resultsare compared with the existing computational and theoretical results. Next, the shape ofthe impinging droplet at different instances are visualised for different sets of data. Thecomputationally obtained wetting diameters, dynamic contact angles, kinetic energies areplotted as a function of time. Our numerical results are both qualitatively and quanti-tatively compared with experimental results. An array of computations is performed tostudy the influence of the slip coefficient, impact velocities and the surface tension on theflow dynamics. Effects of the heat transfer on the shape and the wetting diameter of annon-isothermal spreading droplet are also presented. Computational results for the risingbubble problem is presented in this chapter. The shape of the rising bubble and flow di-rection at different instances are visualised. Further, a few quantities such as sphericity,rise velocity kinetic energy and drag force of the bubble are plotted as a function of time.

Finally, in Chapter 7 we end up with a summary based on the results presented in theprevious Chapters.

7

Chapter 2

Mathematical Modelling

Mathematical models for free surface and interface flow problems are presented in thisChapter. We consider a freely oscillating and an impinging droplet problems as examplesof free surface flows. In the interface flow, we consider a rising bubble problem. Mod-elling difficulties in all these problems are addressed in this Chapter. Furthermore, thedimensionless form and the weak form of the governing equations are derived.

2.1 Free surface flows

In this section, we begin with the mathematical description of a freely oscillating droplet,which is a typical example of free surface flows. Next, we present a mathematical model fora liquid droplet impinging on a solid surface. Computations of the impinging droplet prob-lem is more complicated but equally more application oriented. Prescribing the boundarycondition at the contact line and incorporating the solid surface properties into the modelare more difficult tasks in the impinging droplet problem. These type of problems are oftencalled moving contact line problems. We discuss different possibilities to overcome thesedifficulties. Furthermore, the mathematical model for a droplet impinging on a hot surfaceis presented.

2.1.1 Freely oscillating droplet

We consider an oscillating viscous liquid droplet surrounded by a static ambient gas, in zerogravity. Thus, the entire liquid boundary of the oscillating droplet is a free surface. Initially,the shape of the droplet is perturbed from its equilibrium shape. Due to the imbalance ofsurface force, the droplet starts to oscillate around its equilibrium. We neglect effects ofall other external forces such as gravity, adjacent gas pressure etc.

Governing equations

The fluid flow in the time interval (0, I) is described by the time dependent Navier-Stokesequations in a time dependent domain Ω(t) ⊂ R

3, together with the kinematic and force

9

2 Mathematical Modelling

balancing boundary conditions on the free surface ΓF (t) = ∂Ω(t). Here, t ∈ (0, I) denotesthe time variable. The governing model equations are given by

∂u

∂t+ (u · ∇)u− 1

ρ∇ · T(u, p) = 0 in Ω(t) × (0, I),

∇ · u = 0 in Ω(t) × (0, I),

νF · T(u, p) · νF = σK on ΓF (t) × (0, I),

τ i,F · T(u, p) · νF = 0 on ΓF (t) × (0, I),

(2.1)

for i = 1, 2, where u denotes the velocity of the fluid, p the pressure in the fluid, ρ the fluiddensity (assumed to be a constant), σ the surface tension coefficient (also assumed to be aconstant) and K is the sum of the principal curvatures. Further, νF denotes the outwardunit normal vector and τ i,F , i = 1, 2, are tangential vectors on the free surface ΓF (t). Thestress tensor T(u, p) for Newtonian incompressible fluids is given by

T(u, p)i,j := 2µD(u)i,j − pδi,j, D(u)i,j =1

2

(

∂ui

∂xj+∂uj

∂xi

)

, i, j = 1, ..., 3, (2.2)

where µ denotes the dynamic viscosity, D(u) the velocity deformation tensor and δi,j theKronecker delta. Furthermore, the initial domain Ω(0) has to be provided and the kine-matic condition

u · νF = w · νF on ΓF (t) × (0, I) (2.3)

has to be satisfied, i.e., the normal velocity of the fluid at the interface should be equalto the normal velocity of the interface. Here, w is the interface velocity. In addition, weneed to specify an initial velocity u(x, 0) = u0(x). For computations in section 6.2, we useu0(x) = 0, i.e., the droplet is in rest at time t = 0.

2.1.2 Impinging droplet

In this section, we present the mathematical model for the fluid flow of a liquid dropletimpinging on a solid surface. In the impinging droplet problem, we are interested tosimulate the fluid flow of a liquid droplet impinging perpendicularly on a solid surface.Our simulation starts at an instant that the liquid droplet comes into contact with thesolid surface. Simulations are made until the prescribed time or the droplet comes intothe equilibrium position after a sequence of spreading and recoiling processes. Apartfrom other difficulties associated in computations of free surface flows, the main difficultyin the spreading droplet problem is the correct prescription of the boundary conditionon the liquid-solid interface. In general, this difficulty arises in all moving contact lineproblems [75, 39, 65, 76], and we will discuss it in more detail.

To slip? or not to slip?

The no-slip boundary condition that the velocity of the liquid relative to the solid surfaceis zero at the liquid-solid interface, is an assumption. However, it is commonly used in the

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free−slippartial slipno−slip

solid surface ǫ

ǫ = 0 0< ǫ < ∞ ǫ = ∞

Figure 2.1: Interpretation of the Navier slip-length ǫ.

fluid dynamics under normal conditions. Even though, the no-slip condition works well indescribing the macroscopic fluid flow, it cannot be used in moving contact line problems.By using the no-slip condition, the model leads to an non-integrable force singularity atthe contact line, where the liquid-solid and liquid-gas interfaces intersect, see for e.g. [38].A few authors calls this singularity “kinematic paradox” [4].

Several boundary conditions have been proposed in the literature to relieve this singu-larity, see for an overview [20]. Among them, the so-called Navier-slip boundary conditionhas been more often used and is widely accepted. It reads:

u · νF = 0, u · τ i,S = −ǫµ(τ i,S · T(u, p) · νS), on ΓS(t) × (0, I), (2.4)

for i = 1, ..., d − 1, where d is the dimension of the considered problem, νS and τ i,S arethe unit normal and tangential vectors on the liquid-solid interface. Here ǫµ is the slipcoefficient. The unit of the stress is kg/(m·s2) and the velocity is m/s. Thus, from thedimensional analysis, the unit of the slip coefficient ǫµ should be of ǫµ−1

ǫ , where ǫ andµǫ have the unit of a length and a dynamic viscosity, respectively. The first conditionin (2.4) is the no penetration boundary condition, that is, the fluid cannot penetrate animpermeable solid and thus the normal component of the velocity is zero. The secondcondition is the slip with friction boundary condition, that is, on the liquid-solid interface,the tangential velocities of the fluid are proportional to their corresponding tangentialstresses. Depending on the choice of ǫ, one gets different boundary conditions as follows:

1. no-slip if ǫ=0,

2. slip with friction if 0< ǫ <∞ ,

3. free slip if ǫ=∞.

For an interpretation of the slip-length, see Fig.2.1. For example, in case of shear flow, i.e.,partial slip, ǫ can be interpreted as the fictitious distance to the solid surface as shown inFig.2.1 (middle). For more detailed description of the slip boundary condition, see [43].

11


Although, a number of expressions has been proposed for this fictitious slip length inthe literature [13, 38, 75], an exact mathematical expression is missing. A more complicatenon-linear form of the slip length has been also proposed for a Newtonian liquid in molecularlength scale [69]. The experimental evidences show that the slip length varies for differentflow at different situations. A large number of published experimental results report thatthe range of the slip length ǫ is from 10−9m to 10−6m, see [47].

Alternatively, in [39], Huh and Scriven have been suggested that to remove the singu-larity one can use a local slip boundary condition, that is, allow the fluid to slip only in theneighbourhood of the contact line and indeed, it does, see for example [24]. However, thislocal slip boundary condition induces discontinuity at the intersection of the no-slip andthe local slip boundary. To overcome this difficulty one has to use some smooth functionsover the discontinuity region.

To study the influence of different forms of the slip parameter ǫµ on the flow field,we present a general mathematical model in the next section for the spreading problem,which contains ǫµ. A wide array of calculations is performed for different values of ǫµ, seesection 6.3, for more details.

Governing equations

After impinging a solid surface, the sequence of spreading and recoiling processes of aliquid droplet in the time interval (0, I) is described by the time dependent incompressibleNavier-Stokes equations in a time-dependent domain Ω(t) ⊂ R

3, t ∈ (0, I) together withthe kinematic, force balancing and Navier slip boundary conditions on the liquid-gas ΓF (t)and the liquid-solid ΓS(t) interfaces, where ∂Ω(t) = ΓF (t) ∪ ΓS(t). The mathematical

Figure 2.2: Cross sectional representation of an impinging droplet on a solid surface.

12


model is given by

∂u

∂t+ (u · ∇)u− 1

ρ∇ · T(u, p) = ge in Ω(t) × (0, I),

∇ · u = 0 in Ω(t) × (0, I),

u · νF = w · νF on ΓF (t) × (0, I),

νF · T(u, p) · νF = σK on ΓF (t) × (0, I),

τ i,F · T(u, p) · νF = 0 on ΓF (t) × (0, I),

u · νS = 0 on ΓS(t) × (0, I),

u · τ i,S + ǫµ(τ i,S · T(u, p) · νS) = 0 on ΓS(t) × (0, I),

(2.5)

with given Ω(0) and u(0) = (0, 0, uimp), for i = 1, 2. Here, g is the gravitational constant,e is an unit vector in the opposite direction of the gravitational force and uimp is the impactspeed of the droplet. Furthermore, νF , νS and τ i,F , τ i,S, i = 1, 2, denote the outwardunit normal and tangential vectors on the corresponding interfaces as prescribed in Fig.2.2.In comparison with the oscillating droplet model (2.1), we have additional boundary con-ditions on the liquid-solid interface and the gravitational force in the spreading dropletproblem. To complete the model (2.5), the solid surface properties have to be included inthe model and this can be achieved by including the contact angle into the model at thecontact line.

Contact angle

The angle formed between the liquid-gas interface and the liquid-solid interface at thecontact line is conventionally defined as the contact angle in moving contact line prob-lems. At the equilibrium state, a water droplet on a glass surface produces small contactangles whereas, the same droplet produces a large contact angle on wax surface as shownin Figure 2.3. This wetting phenomena plays a vital role in industrial applications, forexample, a large contact angle is desired on self cleaning materials and a small contactangle is desired in spray cooling systems.

In thermodynamic equilibrium state, the contact angle is related to the interfacialtensions of the liquid-gas σlg, solid-liquid σsl and solid-gas σsg through the Young’s equation

cos(θ) =σsg − σsl

σlg.

In general, the contact angle θ in the Young’s equation is referred to as a static or equi-librium contact angle, i.e., θ = θe. At the equilibrium state, the equilibrium contact angleθe is unique for the considered gas, liquid and solid material phases. However, when thecontact line moves, the contact angle deviates from the equilibrium contact angle (θe),and the difference between the advancing θa and receding θr is referred as a contact angle

13


Figure 2.3: Different contact angles of a same liquid on different solid surfaces at equilib-rium state.

liquid

(a) (b)

recoiling

spreading

θd

θd

u−u

θa

θrθw

Figure 2.4: a) representation of the dynamic contact angle θd and the microscopic contactangle θw, b) representation of the velocity dependent advancing θa and receding θr contactangles.

hysteresis. The advancing contact angle θa is the largest angle just before the spreadingstarts and the receding contact angle θr is the smallest angle just before the recoiling starts,see figure 2.4(b). Surface roughness, inhomogeneity, at least on submicroscopic scale andcontaminations are a few reasons for the occurrence of hysteresis, [9, 44].

The contact angle measured in experiments is often referred as the apparent dynamiccontact angle θd, [44]. Furthermore, the microscopic contact angle θw, which is measured atthe molecular level, as shown in figure 2.4(a), differs from θd. In general, θw is inaccessiblefrom experiments [74]. Since we have different contact angles θe, θd and θw, it is not clearwhich one has to be used in computations. In mathematical models, the contact anglecondition is included at the contact line, whereas, θd is not really measured at the contactline. Therefore, in computations the use of θd seems to be inappropriate. Since θw isinaccessible, often the contact angle θ is associated with the equilibrium contact angle θe

or some analytical relations θ = θe(Ca,ucl, ǫµ), see for example, [13, 36].In our mathematical model, the contact angle inclusion is associated with the approx-

imation of the curvature of the free surface which is presented in section 4.2.2. Different

14


numerical simulations are performed with θ = θe to study the influences of the contactangle on the flow dynamics.

2.1.3 Droplet impinging on a hot surface

We consider a droplet with atmospheric temperature impinging on an uniformly heatedhorizontal surface. In this problem, we are interested to study the thermocapillary effectsover the free surface. We assume that the density, viscosity and thermal conductivity areconstants in the liquid. Furthermore, the basic assumption of our model is that the density,viscosity and thermal conductivity in the gas phase are negligible compared with the liquidphase. In addition to the Navier-Stokes equations we use the energy equation to describethe thermal flow.

Marangoni convection

Marangoni convection is a convection driven by variations in the surface tension along theliquid-gas interface ΓF (t). These variations may occur due to changes in the local interfacetemperature T on the liquid-gas interface ΓF (t), as the surface tension is dependent onthe T . We assume that the surface tension σ(T ) is linearly related to the local interfacetemperature T by

σ(T ) = σsa − C1(T − Tsa) (2.6)

where, Tsa is the saturation temperature, σsa is the surface tension coefficient at the sat-uration temperature and C1 > 0 is the negative of rate of change of surface tension withtemperature [21]. This relation induce thermocapillary effects through the balance of shearstress with surface tension gradients along the liquid-gas interface ΓF . Furthermore, sur-face tension variations on the normal stress boundary condition is also considered in thisstudy. we neglect the natural convection, which is driven by buoyant forces within theliquid as in [35], since

Gr

Re2 ≪ 1

in our considered test problems. Here, Gr is the Grashof number and Re is the Reynoldsnumber. However, the natural convection effects can also be included in the model by theBoussinesq approximation [3].

Governing equations

The fluid flow in the droplet is governed by the Navier-Stokes equation as in (2.5) butwith Marangoni convection. The heat transfer in the droplet is described by the energy

15


equation. The governing equations for the coupled model problem are

∂u

∂t+ (u · ∇)u − 1

ρ∇ · T(u, p) = ge in Ω(t) × (0, I),

∇ · u = 0 in Ω(t) × (0, I),

u · νF = w · νF on ΓF (t) × (0, I),

νF · T(u, p) · νF = σ(T )K on ΓF (t) × (0, I),

τ i,F · T(u, p) · νF = τ i,F · ∇σ(T ) on ΓF (t) × (0, I),

u · νS = 0 on ΓS(t) × (0, I),

u · τ i,S + ǫµ(τ i,S · T(u, p) · νS) = 0 on ΓS(t) × (0, I),

∂T

∂t− λ

cpρ∆T + (u · ∇)T = 0 in Ω(t) × (0, I),

−λ ∂T∂νF

= αF (T − T∞) on ΓF (t),×(0, I),

T = Tg on ΓS(t),×(0, I),

(2.7)

with given Ω(0), u(0) = (0, 0, uimp) and T (0) = T∞. Here, λ is the thermal conductivityand cp is the specific heat of the liquid. Here, αF is the convection heat transfer coefficienton the liquid-gas interface, T∞ is the temperature of the surrounding gas, and Tg is a giventemperature. In this initial study, we used one of the possibilities of a boundary conditionon the liquid-solid interface for the energy equation.

2.1.4 Weak formulation in dimensionless variables

In this section we derive the dimensionless weak form of the mathematical models proposedin the previous sections. Since the fluid flow in the model (2.1) can be considered as aspecial case of (2.5) for ΓS = ∅ and g = 0, we concentrate only on the weak formulationof (2.5) and (2.7).

Dimensionless form of the Navier-Stokes equations

To write the Navier-Stokes equations (2.5) in a dimensionless form, we introduce the scalingfactors L and U as characteristic length and velocity, respectively. Furthermore, we definethe dimensionless variables as

x =x

L, u =

u

U, w =

w

U, t =

tU

L, I =

IU

L, p =

p

ρU2. (2.8)

16


Using these dimensionless variables in the stress tensor T(u, p) in (2.2), we introduce adimensionless stress tensor S(u, p) such that

T(u, p) =2µU

LD(u) − ρU2pI

= ρU2

(

2µ

ρULD(u) − pI

)

= ρU2

(

2

ReD(u) − pI

)

= ρU2S(u, p).

Using the same variable transformation in the momentum equation (first equation in (2.5)),we get

U2

L

∂u

∂t+U2

L(u · ∇)u− ρU2

ρL∇ · S(u, p) = ge.

Multiplying this equation byL

U2and omitting the tilde afterwards, we obtain the dimen-

sionless form of the Navier-Stokes equations as

∂u

∂t+ (u · ∇)u−∇ · S(u, p) =

1

Fre in Ω(t) × (0, I),

∇ · u = 0 in Ω(t) × (0, I).(2.9)

The boundary conditions in (2.5) are also transformed into a dimensionless form:

νF · S(u, p) · νF =KWe

on ΓF (t) × (0, I),

τ i,F · S(u, p) · νF = 0, u · νF = w · νF on ΓF (t) × (0, I),

u · νS = 0 on ΓS(t) × (0, I),

u · τ i,s + βǫ(τ i,s · S(u, p) · νs) = 0 on ΓS(t) × (0, I),

(2.10)

for i = 1, 2. Furthermore, the initial velocity becomes u(0) = (0, 0, uimp)/U , and in thespreading droplet problem we use U = uimp to get u(0) = (0, 0, 1). Here, the dimensionlessnumbers (Reynolds, Weber, Froude and slip, respectively) are defined by

Re =ρUL

µ, We =

ρU2L

σ, Fr =

U2

Lg, βǫ = ǫµρU.

As we mentioned earlier, the dimensionless form of the freely oscillating droplet prob-lem (2.1) is exactly the same as (2.9) but with zero right hand side and without theboundary conditions on the liquid-solid interface ΓS(t) in (2.10).

17


Weak form of the Navier-Stokes equations

Now, we derive the weak form of the dimensionless equations (2.9) with the boundaryconditions (2.10). To simplify the notation, we use the subscript t to represent the timedependency, for example, Ωt for Ω(t).

Let L2(Ωt) and Hm(Ωt), m ≥ 1 be the usual Lebesgue and Sobolev spaces. We defineQ = L2(Ωt) as a pressure space and

V := v ∈ H1(Ωt)3 : v · νS = 0 on ΓS(t). (2.11)

as a velocity space for (2.9). As a consequence, the no penetration boundary conditionu · νS = 0 on liquid-solid interface ΓS(t) will be satisfied in both the ansatz and testspaces. We include all other boundary conditions in the weak formulation. To get the weakformulation of the time dependent Navier-Stokes equations, we multiply the momentumand mass balance equations (2.9) by test functions v ∈ V and q ∈ Q, respectively, andintegrate over Ωt. By applying the Gaussian theorem for the stress tensor, we get

-

∫

Ωt

∇ · S(u, p) · v dx

=

∫

Ωt

S(u, p) : ∇v dx−∫

Γt

v · S(u, p) · ν dγ

=

∫

Ωt

1

2S(u, p) : ∇v dx+

∫

Ωt

1

2S

T (u, p) : ∇v dx−∫

Γt

v · S(u, p) · ν dγ

=

∫

Ωt

S(u, p) : D(v) dx−∫

Γt

v · S(u, p) · ν dγ

=2

Re

∫

Ωt

D(u) : D(v) dx−∫

Ωt

p∇ · v dx−∫

Γt

v · S(u, p) · ν dγ .

(2.12)

Note that the symmetry of the tensor S(u, p) has been used. In order to include the liquid-gas and liquid-solid boundary conditions in the weak form, we split the boundary integralterm as an integral over ΓSt

and over ΓFt,

−∫

Γt

v · S(u, p) · ν dγ = −∫

ΓSt

v · S(u, p) · νS dγS −∫

ΓFt

v · S(u, p) · νF dγF . (2.13)

Using the orthonormal decomposition, we split the test function v in the liquid-solid inte-gral part of equation (2.13) as

v = (v · νS)νS +

2∑

i=1

(v · τ i,S)τ i,S.

18


Then, using the definition (2.11) and slip with friction boundary condition, the liquid-solidintegral term can be written as

−∫

ΓSt

v · S(u, p) · νS dγS

= −∫

ΓSt

(v · τ i,S)τ i,S · S(u, p) · νS dγS −∫

ΓSt

(v · νS)νS · S(u, p) · νS dγS

=1

βǫ

∫

ΓSt

(u · τ i,S)(v · τ i,S) dγS

(2.14)

for i = 1, 2. Again, we split the test function v in the liquid-gas integral term of theequation (2.13) and then by using corresponding boundary conditions, we get

−∫

ΓFt

v · S(u, p) · νF dγF

= −∫

ΓFt

(v · τ i,F )τ i,F · S(u, p) · νF dγF −∫

ΓFt

(v · νF )νF · S(u, p) · νF dγF

= − 1

We

∫

ΓFt

(v · νF )K dγF .

(2.15)

Weak form of the oscillating droplet

Note that in the case of a freely oscillating droplet (2.1), the gravitational constant andthe additional terms arising from the liquid-solid interface boundary condition vanish.Furthermore, the velocity space for the oscillating droplet becomes V = H1(Ωt). Hence,the weak form of the oscillating droplet reads:

For given Ω(0) and u(0), find (u(t), p(t)) ∈ V ×Q such that

(

∂u

∂t,v

)

+ a(u;u,v) − b(p,v) + b(q,u) = f(K,v), (2.16)

for all v ∈ V and q ∈ Q. Here,

a(u;u,v) =2

Re

∫

Ωt

D(u) : D(v) dx+

∫

Ωt

(u · ∇)u · v dx

b(q,v) =

∫

Ωt

q ∇ · v dx.

f(K,v) =1

We

∫

ΓFt

(v · νF ) K dγF .

19


Weak form of the spreading droplet

Using the derivations (2.12) - (2.15), the weak form of the impinging droplet problem (2.9)with boundary conditions (2.10) reads:

For given Ω(0) and u(0), find (u(t), p(t)) ∈ V ×Q such that(

∂u

∂t,v

)

+ a(u;u,v) − b(p,v) + b(q,u) = f(K,v), (2.17)

for all v ∈ V and q ∈ Q. Here,

a(u;u,v) =2

Re

∫

Ωt

D(u) : D(v) dx+

∫

Ωt

(u · ∇)u · v dx

+1

βǫ

∫

ΓSt

(u · τ i,S)(v · τ i,S) dγS,

b(q,v) =

∫

Ωt

q ∇ · v dx,

f(K,v) =1

Fr

∫

Ωt

e · v dx+1

We

∫

ΓFt

(v · νF ) K dγF .

Dimensionless form of the energy equation

In addition to the dimensionless variables (2.8), we define the dimensionless temperatureas

T =T − T∞Tsa − T∞

. (2.18)

Using these dimensionless variables in (2.7) and omitting the tilde afterwards, we get

U(Tsa − T∞)

L

(

∂T

∂t+ (u · ∇)T

)

− (Tsa − T∞)

L2

λ

cpρ∆T = 0 in Ω(t) × (0, I)

Multiplying the equation by L/U(Tsa − T∞), we get the dimensionless energy equation as

∂T

∂t+ (u · ∇)T − 1

Pe∆T = 0 in Ω(t) × (0, I). (2.19)

The thermal boundary conditions in (2.7) on the liquid-gas and liquid-solid interfaces aretransformed into a dimensionless form:

− ∂T

∂νF= Bi T on ΓF (t) × (0, I)

T =Tg − T∞Tsa − T∞

on ΓS(t) × (0, I).

(2.20)

20


Here, Pe and Bi denote the dimensionless Peclet and Biot numbers, given by

Pe =LUcpρ

λ, Bi =

αFL

λ.

Furthermore, the initial condition for the velocity and temperature become u(0) = (0, 0, 1)with U = uimp and T (0) = 0. Rewriting the fluid’s normal stress boundary condition(in (2.7)) on the free surface into a dimensionless form, we get

νF · S(u, p) · νF =1

ρU2L

(

σsa −C1

Tsa − T∞

(

(Tsa − T∞)T + T∞ − Tsa

)

)

K

=1

We

(

1 − C1

σsa

(

T − 1)

)

K.

Note that the relation C1 = ∂σ/∂T has been used. Omitting the tilde, we get the normalstress boundary condition for the liquid on the free surface as

νF · S(u, p) · νF =1

We

(

1 − C1

σsa(T − 1)

)

K on ΓF (t) × (0, I). (2.21)

Here, We is the Weber number with the surface tension coefficient σsa. Similarly, rewritingthe fluid’s shear stress boundary condition on the free surface into a dimensionless form,we get

τ i,F · S(u, p) · νF = − C1

ρU2Lτ i,F · ∇T

= − C1

σsaWeτ i,F · ∇T .

Omitting the tilde, we get the shear stress boundary condition for the liquid on the freesurface as

τ i,F · S(u, p) · νF = − C1

σsaWeτ i,F · ∇T. (2.22)

In comparison with the isothermal spreading droplet’s boundary conditions (2.10), here,we have an additional factor in the curvature term and a non-zero shear stress on the freesurface. These additional terms are coupled with the temperature and the gradient of thetemperature. Therefore, instead of (2.15), for a droplet impinging on a hot surface we havethe free surface integral as

21


−∫

ΓFt

v · S(u, p) · νF dγF

= −∫

ΓFt

(v · τ i,F )τ i,F · S(u, p) · νF dγF −∫

ΓFt

(v · νF )νF · S(u, p) · νF dγF

=C1

σsaWe

∫

ΓFt

(v · τ i,F )(τ i,F · ∇T ) dγF

− 1

We

(

1 − C1

σsa(T − 1)

)∫

ΓFt

(v · νF )K dγF .

(2.23)

Weak form of the energy equation

The weak form of the energy equation is obtained by multiplying it with a test functionψ ∈ V,

V := ψ ∈ H1(Ωt) : ψ = 0 on ΓS(t),and integration by parts. In particular, the diffusive term of the energy equation becomes

− 1

Pe

∫

Ωt

∆T · φ dx =1

Pe

∫

Ωt

∇T : ∇φ dx− 1

Pe

∫

ΓFt

∂T

∂νFφ dγF

=1

Pe

∫

Ωt

∇T · ∇φ dx+Bi

Pe

∫

ΓFt

T φ dγF .

(2.24)

Hence, the weak form of the energy equation reads:

For given Ω(0) and T (0), find T (t) ∈ V such that(

∂T

∂t, φ

)

+ aT (u, T, φ) + bT (T, φ) = 0, (2.25)

for all φ ∈ V (Ωt). Here,

aT (u, T, φ) =1

Pe

∫

Ωt

∇T · ∇φ dx+

∫

Ωt

(u · ∇) T φ dx,

bT (T, φ) =Bi

Pe

∫

ΓFt

T φ dx.

2.2 Two-phase flows

A mathematical model for a two-phase flow problem is presented in this section. Weassume that at least one phase is a liquid and the other phase may be a gas or an another

22

2.2 Two-phase flows

Ω1

Ω2

ΓD

ΓD

ΓNΓN

ΓLL

ν

τ

Figure 2.5: The computational domain of the rising bubble problem in two-space dimen-sion.

immiscible liquid. In general, the physical variables such as pressure, density and viscosityexhibit jumps across the interface which increase the complexity of the problem. In ourstudy, we consider a rising bubble in a liquid medium which has large jumps in theirmaterial coefficients and rises due to the buoyancy force. The key idea in a multi phaseflow simulation is to use a single set of equation for all phases in the flow, with variablematerial coefficients, see for example [71].

2.2.1 Rising bubble

The computational domain is chosen in such a way that the bubble Ω2(t) ⊂ R3, is com-

pletely surrounded the liquid phase Ω1(t) ⊂ R3 and the boundary of bubble ΓLL is closed,

where ΓLL is the interface between Ω1(t) and Ω2(t). Let us define Ω(t) := Ω1(t) ∪ Ω2(t).Figure 2.5 shows a cross section of the computational domain Ω(t) and its boundaries.Furthermore, we assume that the bubble is in rest at time t = 0, i.e., u(0) = 0.

Governing equations

In the rising bubble problem, the fluid flow in the entire domain is described by the timedependent incompressible Navier-Stokes equations but with different material propertiesin two phases. The free slip boundary condition is imposed on the vertical boundary ofthe outer liquid phase, whereas, the no-slip condition is used on horizontal boundaries.Furthermore, the kinematic and force balancing boundary conditions are imposed on the

23


interface between the two phases. The governing equations of the fluid flow are given by

ρk

(

∂u

∂t+ (u · ∇)u

)

−∇ · (Tk(u, p)) = ρkge in Ωk(t) × (0, I),

∇ · u = 0 in Ωk(t) × (0, I),

[|u|] = 0, τ i · [|T(u, p)|] · ν = 0 on ΓLL(t) × (0, I),

ν · [|T(u, p)|] · ν + σK = 0, u · ν = w · ν on ΓLL(t) × (0, I),

u = 0 on ΓD,

u · νN = 0, νN · (T1(u, p)) · τ i,N = 0 on ΓN ,

(2.26)

for i = 1, 2 and for k = 1, 2, where, the stress tensor Tk(u, p) is defined by

Tk(u, p) = 2µkD(u) − pI.

Here, ρk and µk are the density and the dynamic viscosity of the respective phases. Fur-ther, νN and τ i,N denote the unit outward normal and tangential vectors of the verticalboundaries ΓN , e an unit vector in the opposite direction of the gravitational force, and [|·|]the jump across the interface ΓLL. For example, the jump of a function w : Ω1 ∪ Ω2 7→ R,which has traces on the common boundary ΓLL is defined by

[|w|] := (w|Ω1)|ΓLL

− (w|Ω2)|ΓLL

.

2.2.2 Weak formulation

Dimensionless form

We use the characteristic values of the outer domain Ω1 to write the equations (2.26) intoa dimensionless form. Let U∞ and L be the characteristic velocity and length scale. Wedefine the dimensionless variables as

u =u

U∞

w =w

U∞

x =x

L, t =

tU∞

L, I =

IU∞

L, p =

p

ρ1U2∞

. (2.27)

Using these dimensionless variables in the stress tensor T1(u, p), we get

T1(u, p) = ρ1U2∞

(

2µ1

ρ1U∞LD(u) − pI

)

= ρ1U2∞

(

2

R1D(u) − pI

)

= ρ1U2∞S1(u, p).

Similarly, for the stress tensor T2(u, p), we get

T2(u, p) = ρ1U2∞

(

2µ2

ρ1U∞LD(u) − pI

)

= ρ1U2∞

(

µ2

µ1

2

R1D(u) − pI

)

= ρ1U2∞

(

2

R2D(u) − pI

)

= ρ1U2∞S2(u, p).

24

2.2 Two-phase flows

Here, Sk(u, p), k = 1, 2 are dimensionless stress tensors of the respective phases Ωk. Fur-thermore the dimensionless numbers

R1 =ρ1U∞L

µ1R2 =

µ1

µ2

ρ1U∞L

µ1=µ1

µ2R1

are used in the above derivations. Applying the dimensionless variable transformation onthe momentum equation (the first equation in (2.26)) and omitting the tilde afterwards,we get

ρkU2∞

L

(

∂u

∂t+ (u · ∇)u

)

− ρ1U2∞

L∇ · (Sk(u, p)) = ρkge, for k = 1, 2.

Note that the above equation is derived in such a way that the coefficient in front of thestress tensor term is the same for both phases. By writing it in this way, boundary con-ditions on the interface can be easily incorporated in the weak formulation. Furthermore,the interface boundary condition becomes,

ν · [|S(u, p)|] · ν = − σ

Lρ1U2∞

K, for k = 1, 2.

Taking U∞ =√Lg in the above two equations and dividing the momentum equation by

ρ1g, we get the dimensionless form of the coupled system as

ρk

ρ1

(

∂u

∂t+ (u · ∇)u

)

−∇ · (Sk(u, p)) =ρk

ρ1

e in Ωk(t) × (0, I)

∇ · u = 0 in Ωk(t) × (0, I)

[|u|] = 0, τ i · [|S(u, p)|] · ν = 0 on ΓLL(t) × (0, I)

ν · [|S(u, p)|] · ν = − 1

EoK, u · ν = w · ν on ΓLL(t) × (0, I)

u = 0 on ΓD,

u · νN = 0, νN · (S1(u, p)) · τ i,N = 0 on ΓN ,

(2.28)

for i = 1, 2 and k = 1, 2. Here, the dimensionless Eotvos number is defined as

Eo =ρ1U

2∞L

σ=ρ1gL

2

σ.

The Eotvos number is also called the Bond number and a few authors define the Eotvosnumber alternatively with the density difference (ρ1 − ρ2) between two phases instead ofthe outer phase density ρ1.

25


Weak form of the rising bubble problem

We define Ω1,t := Ω1(t), Ω2,t := Ω2(t) and let Ωt := Ω1,t ∪Ω2,t be the whole computationaldomain. Furthermore, to derive the weak form of (2.28) we define Q = L2

0(Ωt) be a pressurespace, and to include the Dirichlet type boundary conditions in both ansatz and test spaceswe define the velocity space as

V := v ∈ H1(Ωt)3 : v · νN = 0 on ΓN and v = 0 on ΓD. (2.29)

All other boundary conditions are incorporated in the weak formulation. It should benoted that the pressure is not fixed by the boundary conditions and it is defined up to aconstant. Thus, we use the pressure space as L2

0(Ωt) instead of L2(Ωt). First, we multiplythe momentum and mass equations by test functions v ∈ V and q ∈ Q, respectively, andintegrate over the domain Ωt. Then integrating by parts over the sub-domains Ω1(t) andΩ2(t) separately, we can incorporate the non-Dirichlet type boundary conditions. Analogto (2.12), the stress tensor term in the domain Ω1,t becomes

−∫

Ω1,t

∇ · S1(u, p) · v dx =

∫

Ω1,t

S1(u, p) : D(v) dx−∫

ΓLL

v · S1(u, p) · ν dγ

=2

R1

∫

Ω1,t

D(u) : D(v) dx−∫

Ω1,t

p ∇ · v dx−∫

ΓLL

v · S1(u, p) · ν dγ.(2.30)

Similarly, the stress tensor term in the domain Ω2,t becomes

-

∫

Ω2,t

∇ · S2(u, p) · v dx

=2

R2

∫

Ω2,t

D(u) : D(v) dx−∫

Ω2,t

p ∇ · v dx+

∫

ΓLL

v · S2(u, p) · ν dγ. (2.31)

Note that in the above derivation all other boundary integrals except over ΓLL vanish dueto the free slip condition in (2.28) and the definition of the velocity space (2.29). Aftersumming up the two equations (2.30) and (2.31), the interface integral term becomes

∫

ΓLL

v · [|S(u, p)|] · ν dγ = − 1

Eo

∫

ΓLL

Kν · v dγ.

Furthermore, to use a single set of equations in the entire computational domain Ωt, andtreat both the fluid phases as one fluid with variable material properties, we define

(ρ(x), µ(x)) =

(ρ1, µ1) for x in Ω1,

(ρ2, µ2) for x in Ω2,and Re(x) =

R1 for x in Ω1,

R2 for x in Ω2.

Hence, the weak form of the rising bubble problem (2.26) reads:

26

2.2 Two-phase flows

For given Ω(0) and u(0), find (u(t), p(t)) ∈ V ×Q such that

(

ρ

ρ1

∂u

∂t,v

)

+ a(u,u,v) − b(p,v) + b(q,u) = f(K,v), ∀ (v, q) ∈ V ×Q, (2.32)

where

a(u,u,v) = 2

∫

Ωt

1

ReD(u) : D(v) dx+

∫

Ωt

ρ

ρ1

(u · ∇)u · v dx,

b(q,v) =

∫

Ωt

q ∇ · v dx,

f(K,v) =

∫

Ωt

ρ

ρ1

e · v dx+1

Eo

∫

ΓLL

K ν · v dγLL.

27

Chapter 3

Finite Element Methods forStationary Stokes Equations

In this Chapter, we recall some basics of finite element methods for the Stokes equationsin fixed domains as a prepration step for the Navier-Stokes equations in moving domains.The finite element method is based on a triangulation of a domain into a finite number ofelements and approximating the solution by a finite element space over this mesh. Thus,first, we construct a few finite element spaces, which are mainly used in our calculations.The emphasis is on spaces of continuous and discontinuous finite element functions in therealization of continuity across mesh borders. In particular, we restrict ourself to mappedfinite elements, i.e., finite elements, which are defined using a reference element.

Next, we present a stable finite element method for the stationary Stokes problem in afixed domain. Further, we discuss the choice of the pressure space for incompressible flowswith interfaces. Using an appropriate pressure space, we can avoid the spurious velocities,which are generated in computations due to the pressure jump across the interface. Inparticular, we discuss the origin of spurious velocities due to approximations of the pressure,the interface and the curvature.

3.1 Finite element spaces

To define an arbitrary finite element space Xh, we need two basic ingredients. The firstingredient is the local shape function space together with its local nodal functionals of Xh.The second ingredient is the continuity conditions across the element borders within thefinite element space Xh. The continuity conditions are based on local nodal functionals.In this section, first, we describe these two ingredients, and then the construction of anarbitrary finite element space.

3.1.1 Finite elements

Our finite element definition is based on the standard definition in [11]. Let

29

3 Finite Element Methods for Stationary Stokes Equations

(i) K ⊂ Rd be a domain or a cell with piecewise smooth boundary,

(ii) V be a finite-dimensional space of functions on K, say, m = dim V ,

(iii) N = N1, N2, ..., Nm be a set of linear functionals Ni, i = 1, 2, ..., m, which aredefined over the space V .

Then the triple (K, V,N ) is called a finite element .It is implicitly assumed that the set of linear functionals N is V − unisolvent in the

sense that for any given real numbers αi, i = 1, 2, ..., m, there exist a unique functionv ∈ V such that

Ni(v) = αi, 1 ≤ i ≤ m.

In particular, there exist m functions φi ∈ V, i = 1, 2, ..., m, which satisfies

Nj(φi) = δij, 1 ≤ i, j ≤ m, (3.1)

where δij is the Kronecker delta. The linear functionals Ni, i = 1, 2, ..., m, are called nodalfunctionals and the functions φi, i = 1, 2, ..., m, are called local basis or shape functions .

Mapped and unmapped finite element

We can classify finite elements into two groups, named mapped and unmapped finite ele-ments. An unmapped finite element is constructed on the original cell K of the consideredproblem while the mapped finite element is constructed on a reference cell, say K. In ourcomputations, we use finite elements, which are only defined using a reference cell and thuswe restrict our discussions to mapped finite elements. To approximate the solutions on thegiven original domain, the reference finite element (K, V , N ) on K has to be transformedto the finite element (K, V,N ) on K. The mapping FK : K → K, which transform thereference K onto the original cell K is called a reference transformation, and it is as-sumed to be bijective. The local shape function space V is transformed to the space V oflocal shape function on K by

V := φ = φ F−1K ∀ φ ∈ V ,

where F−1K : K → K. Since FK is bijective, the mapping F−1

K exists. For the simplicialcell types bijectivity is guaranteed by the condition that K is not degenerate, that is,meas(K)>0. For the conditions, which guarantee the invertability of FK on quadrilateraland hexahedral cells, we refer to [49].

For a nodal functional N ∈ N , we define

N : DN → R, N(φ) = N(φ FK) ∀ φ ∈ V,

where DN is a suitable function space for which the nodal functionals are well defined.We assume that V ⊂ DN . The set of all nodal functionals obtained by this definition aredenoted by N and it is also V -unisolvent. With the above definitions we obtain the finiteelement (K, V,N ) on K from the reference element (K, V , N ) on K.

30


a(0,0) b(1, 0)

c(0, 1)

Figure 3.1: Reference cell K for triangular finite elements.

3.1.2 Finite elements on triangles

The described finite elements are the space P disc1 of discontinuous, piecewise linear ele-

ments and the space P bubble2 of continuous, piecewise quadratic functions enriched by cubic

bubbles. The used reference cell K for these elements is shown in Figure. 3.2. We denotethe barycentric coordinates on K by λi := λi,K(X), 1 ≤ i ≤ 3, and X ∈ R

2 with respect

to the vertices of K.Often, the local nodal functionals of the finite element functions V are defined by using

point valuesNi(p) = p(Xi),

where Xi ∈ K is a given point. Alternatively, the local nodal functionals can also bedefined by weighted integral values on K, that is,

Ni(p) =1

|K|

∫

K

f(X) p(Xi)dx,

where f(X) is a polynomial function defined over K.

Discontinuous P disc1 on triangles

The set of local shape functions of this linear finite element is the polynomial space P1(K)of degree less than or equal to 1 which is spanned by the barycentric coordinates x, y,1 − x− y. Let us define the nodal functionals by weighted integrals as

Ndisc1 (p) =

1

|K|

∫

K

p(X) dX, Ndisc2 (p) =

1

|K|

∫

K

(

x− 1

6

)

p(X) dX,

Ndisc3 (p) =

1

|K|

∫

K

(

y − 1

6

)

p(X) dX,

(3.2)

31


Figure 3.2: Dots denote degrees of freedom for the P disc1 (left) and P bubble

2 (right) finiteelements on triangle.

The polynomials in the above integrals are of degree less than or equal to two. Therefore,to evaluate the integrals (3.2) we use a quadrature formulae, which is exact for secondorder polynomials in triangles. Furthermore, the set of nodal functionals

N discK = Ndisc

1 , Ndisc2 , Ndisc

3

is P1(K) -unisolvent. We choose the local shape functions in such a way that the condi-tion (3.1) is satisfied. Thus, the local shape functions are obtained as

1, x+ y/2 − 1/2, x/2 + y − 1/2.

Hence, the triplet (K, P1(K), N discK ) is a finite element of K, and we denote it by P disc

1 (K).

Continuous P bubble2 on triangles

For this quadratic element the set of local shape functions is the polynomial space P2(K)of degree less than or equal to two together with an enriched polynomial cubic bubblefunction B3(K). We denote P

bubble2 := P2(K) ⊕ B3(K) as a space of local shape functions,

which is spanned by quadratic functions

λ21, λ

22, λ

23, λ1λ2, λ2λ3, λ3λ1,

and the cubic bubble function

λ1λ2λ3.As mentioned early, we define the nodal functionals by point values,

N bubblei (p) = p(Xi), i = 1, 2, ..., 6

with

N bubblei (pbub) = 0, i = 1, 2, ..., 6,

32


for the cubic bubble function pbub. Since dim P bubble2 = 7, we define the missing nodal

functional by the integral mean value of the reference cell K, i.e.,

N bubble7 (p) =

1

|K|

∫

K

p(X) dX.

The polynomial in the above integral is of order three, and therefore we use a seven pointquadrature formulae, which is exact for polynomials of order three in triangles [62]. Then,

N bubbleK = N bubble

i : i = 1, 2, ..., 7, is a set of P

bubble2 -unisolvent nodal functionals. Furthermore, the basis functions φi(x),

for i = 1, 2, ...7, are given by

1 − x− y − 2y(1 − x− y) − 2x(1 − x− y), 4x(1 − x− y) − 20xy(1 − x− y),

x− 2xy − 2x(1 − x− y), 4y(1 − x− y) − 20xy(1 − x− y),

4xy − 20xy(1 − x− y), y − 2xy − 2y(1 − x− y), 20xy(1 − x− y),with the condition (3.1). Then, we denote the finite element (K, P

bubble2 (K), N bubble

K )shortly by P bubble

2 (K).

3.1.3 Global nodal functionals

The set of all local nodal functionals is divided by an equivalence relation into equivalenceclasses, where each class contains all local nodal functionals, which form a global nodalfunctional.

Let NKi and NK ′

j be two local nodal functionals on the cells K and K ′, respectively.Then, these two nodal functionals are in same equivalence class if and only if

NKi (ϕ|K) = NK ′

j (ϕ|K ′), ϕ ∈ C∞(U), (3.3)

where U ⊂ R2 with K ∪ K ′ ⊂ U . Likewise, an equivalence class contains all local nodal

functionals which satisfies the above equivalence relation (3.3). Then, each equivalenceclass Φ(N) form a global nodal functional and the set of all global nodal functionals isdenoted by Nh.

As an example, to construct the correlation between local nodal functionals and globalnodal functionals, let us consider a domain Ω with four cells A, B, C, D and theircorresponding local nodal functionals for each cell are shown as in Figure 3.3 (left). Theset of all equivalence classes are

NA1 , NA

2 , NB1 , N

C1 , NB

2 , ND1 , NA

3 , NC3 , NB

3 , NC2 , N

D3 , ND

2 ,Each class in the above collection forms a global nodal functional,

NΩ1 := NA

1 , NΩ4 := NA

3 , NC3 ,

NΩ2 := NA

2 , NB1 , N

C1 , NΩ

5 := NB3 , N

C2 , N

D3 ,

NΩ3 := NB

2 , ND1 NΩ

6 := ND2 ,

33


1

44

32

5 65

A B

C D

2

5

2311

3

2

A B

C3 D

21 1

2323

X2 X3

X4 X5

X1

X6

X1 X2 X3

X4 X5 X6

Figure 3.3: A domain Ω with four cells and their local nodal functionals (left) and globalnodal functionals (right).

which are shown in Figure 3.3 (right). For example, the global nodal functional NΩ2 consists

of a set of local nodal functionals NA2 , N

B1 , N

C1 , which are defined at a same point X2

even though each local nodal functional belongs to different cells. Then, the set

NΩh:= NΩ

i i = 1, 2, ..., 6,

denotes all global nodal functionals of the domain Ω.

3.1.4 Finite element space

Let K ∈ Th, be a collection of cells in the domain Ω and (K, V (K), NK), ∀K ∈ Th betheir corresponding finite elements. Then, the finite element space Xh is defined by

Xh :=

ϕ : ϕ ∈n∏

k=1

V (Ki) : N lKi

(ϕ|Ki) = Nm

Kj(ϕ|Kj

)∀N lKi, Nm

Kj∈ Φ(N), ∀N ∈ NΩh

,

where Φ(N) is the equivalence class for the global nodal functional N ∈ NΩh. Further,

N lKi

and NmKj

are local nodal functionals in NKiand NKj

, respectively. Furthermore, eachfunction vh ∈ Xh is uniquely determined by the values N(vh) of all global nodal functionalsN ∈ NΩh

which are called (global) degrees of freedom.

In practice, a finite element space does not contain a collection of arbitrary finiteelements but a same class of finite elements. For example, a collection of P disc

1 (Kn) finiteelements on triangles form a discontinuous space on Ω ⊂ R

2 as

P disc1 := v ∈ L2(Ω) : v|K ∈ P1(K) ∀K ∈ Th,

where Ki, i = 1, 2, ..., n, are n number of triangles in the domain Ω. Similarly, we candefine all other finite element spaces, see for example [48].

34

3.2 Finite element methods for interfacial flows


Several finite element spaces for approximating the solution of Stokes and Navier-Stokesflows have been proposed in the literature, see for example [14, 30, 45, 49, 55]. It iswell-known that the velocity and the pressure are strongly coupled in these equations,and therefore an inf-sup stable pair of finite element spaces has to be used [14]. How-ever, computations with unstable pairs have been also made by several authors but then,stabilisation schemes have to be used.

Apart from the stability criterion, the choice of the pressure space for the interface flowsis crucial. In general, the pressure exhibits jumps across the interface. An improper choiceof the pressure space may generate spurious velocities in computations. The approximationof the pressure is not the only source for the generation of spurious velocities but also theapproximation error of the interface and the curvature. In this section, we address theseissues and propose some remedies to avoid it.

3.2.1 Stokes problem

We begin with a brief introduction to a stable finite element discretisation of the stationaryStokes problem in the fixed domain Ω ∈ R

2 with homogeneous Dirichlet boundary condition

−∇ · T(u, p) = f in Ω,∇ · u = 0 in Ω,

u = uD on ΓD,(3.4)

where u is the velocity, p is the pressure, f is the external force and ΓD = ∂Ω is theDirichlet boundary of the domain Ω. The stress tensor T(u, p) is defined as in (2.2). Now,let us introduce the following solution spaces for the problem (3.4):

V := H1(Ω)2,

V0 := H10 (Ω)2 = v ∈ H1(Ω)2 : v = 0 on ΓD,

Q := L20(Ω) = q ∈ L2(Ω) : (q, 1) = 0.

Here, and later (·, ·) denotes the inner product in L2(Ω) or in its vector valued versions.With these solution spaces, the weak form of (3.4) reads:

Find (u, p) ∈ V ×Q such that u|ΓD= uD and

a(u,v) − b(p,v) = (f ,v) ∀ v ∈ V0

b(q,u) = 0 ∀ q ∈ Q(3.5)

35


where

a(u,v) = 2µ

∫

Ω

2∑

i,j=0

Di,j(u) : Di,j(v) dx,

b(q,v) =

∫

Ω

q ∇ · v dx,

(f ,v) =

∫

Ω

f v dx.

The Stokes equations are linear but nevertheless a special attention is needed because of theincompressibility constraint. Thus, to show the existence and uniqueness of the solution ofthe Stokes equations, we need the so-called inf-sup or “Babuska-Brezzi” condition: thereexists a constant β0 > 0 such that

supv∈V0

b(q,v)

|v|1,Ω

≥ β0||q||0,Ω ∀ q ∈ Q. (3.6)

Lemma 3.2.1 (see, Theorem 5.1 in [30]) Let Ω be a bounded and connected open sub-set of R

2 with a Lipschitz-continuous boundary ΓD. Given f ∈ H−1(Ω)2 and uD ∈H1/2(ΓD)2 such that

∫

ΓD

uD · ν ds = 0,

then there exist a unique pair (u, p) ∈ H1(Ω)2×L20(Ω) as a solution of the Stokes equations.

Proof. The proof follows from the ellipticity property of the bilinear form a(.,.) and theinf-sup condition (3.6).

Triangulation of the domain

Let us introduce a triangulation, named Th, of Ω into cells K which can be triangles orquadrilaterals in 2D. We assume that the usual regularity conditions are satisfied:

- Ω =⋃

K, K ∈ Th.

- Any two meshes K, K ′ intersects only in common faces, edges or vertices.

- Meshes of the decomposition Th resolves the boundary ∂Ω, follows from Ω =⋃

K.

Each of the element K ∈ Th is associated with a element-size hK := diam(K) and wedenote h := maxK∈Th

hK .

Finite element discretisation

Let (K, V,N Ω) be a collection of finite elements associated with the cell K of the mesh Th,and N ∂Ω be the set of all global nodal functionals associated with the boundary ∂Ω. Onthe triangulation Th, we define the following finite element spaces

Vh :=

vh ∈ H1(Ω) : vh|K ∈ V (K), K ∈ Th

Vh,0 :=

vh ∈ Vh : N(vh) = 0 N ∈ N ∂Ω

Qh :=

qh ∈ Q : qh|K ∈ Q(K), K ∈ Th

(3.7)

36


where, V (K) and Q(K) are polynomial functions on the cell K of the same or a differentdegree. Then, the discrete Stokes problem on Th with the finite element spaces Vh and Qh

reads:

Find (uh, ph) ∈ Vh ×Qh such that uh|ΓD= uD,h with

∫

ΓD

uD,h · n dγ = 0, and

ah(uh,vh) − bh(ph,vh) = (f,vh) ∀vh ∈ Vh,0,bh(qh,uh) = 0 ∀qh ∈ Qh.

(3.8)

Since we decomposed the domain Ω into a finite number of cells K, the integral will becalculated cell-wise, i.e, all integrals over Ωh are calculated as the sum over all cells K ofintegrals over K. Thus, we introduced the bilinear forms,

ah(u,v) := 2µ∑

K∈Th

(D(u),D(v))K, and bh(q,v) :=∑

K∈Th

(q,∇ · v)K .

and the mesh-dependent H1 -semi norm by

|v|1,h :=

(

∑

K∈Th

|v|21,K

)1/2

.

In order to show the unique solvability of the problem (3.8), let us relate the continuousand discrete space by the following hypotheses:

Hypothesis H1 (Approximation property of Vh). There exist an interpolation operator

rh ∈ L(H2(Ω)2;Vh) ∩ L(H2(Ω) ∩H10 (Ω)2;Vh,0)

and an integer l such that:

||v − rhv||1,Ω ≤ Chl||v||l+1,Ω ∀ v ∈ Hm+1(Ω)2, 1 ≤ l ≤ m.

Here, L(X;Y ) denotes the set of linear and continuous mappings from X to Y.

Hypothesis H2 (Approximation property of Qh). There exists an interpolation operator

sh ∈ L(L20(Ω);Qh)

such that:||q − shq||0,Ω ≤ Chl||q||l,Ω ∀ q ∈ Hm(Ω), 1 ≤ l ≤ m.

Hypothesis H3 (Uniform inf-sup condition) There exists a positive constant β1 indepen-dent of the discretisation parameter h such that

infqh∈Qh

supvh∈Vh,0

bh(qh, vh)

‖qh‖0 |vh|1,h≥ β1 > 0. (3.9)

37


Lemma 3.2.2 (see Theorem 1.8 in [30]) Let Hypotheses H1, H2 and H3 be fulfilled.Then the problem (3.8) has a unique solution (uh, ph) ∈ Vh × Qh. Furthermore, if thesolution of (3.5) is (u, p) ∈ Hm+1(Ω)2 × (Hm

0 (Ω) ∩ L20(Ω)) for some positive integer m,

then we have the error bound:

|u− uh|1,Ω + ||p− ph||0,Ω ≤ Chm (||u||m+1,Ω + ||p||m,Ω) .

In general, the usual finite element spaces together with their natural interpolation fulfilHypotheses H1 and H2. However, a suitable finite element pair has to be used to fulfilthe Hypothesis H3 (discrete inf-sup condition). The following lemma recalls a few inf-supstable finite element pairs on simplices.

Lemma 3.2.3 (Stable finite element pairs on simplices) The following finite elementpairs satisfy discrete inf-sup stable condition (Hypothesis H3) on simplices with a constantβ1, which is independent of the mesh parameter h:

(i) continuous, piecewise polynomials of degree less than or equal to k for the velocityand continuous, piecewise polynomials of degree less than or equal to k − 1 for thepressure approximation, i.e., (Pk/Pk−1), for k ≥ 2.

(ii) continuous, piecewise polynomials of degree less than or equal to k, enriched withcell bubble functions for the velocity and discontinuous, piecewise polynomials of de-gree less than or equal to k − 1 for the pressure approximation, i.e., (P bubble

k /P disck−1)

for k ≥ 2.

(iii) continuous, piecewise polynomials of degree less than or equal to k, k = 2, 3in 2D for the velocity and discontinuous, piecewise polynomials of degree less than orequal to k − 1 for the pressure approximation, i.e., (Pk/(Pk−1 + P0)), for k = 2, 3.

(iv) On macro-element meshes, continuous, piecewise polynomials of degree less thanor equal to k, k ≥ 2 in 2D and k ≥ 3 in 3D, for the velocity and discontinuous, piece-wise polynomials of degree less than or equal to k−1 for the pressure approximation,i.e., (Pk/P

disck−1) for k, k ≥ 2 in 2D and k ≥ 3 in 3D, on macro-element meshes.

The inf-sup stability proof of P bubblek /P disc

k−1 , for k = 2, 3 is given in [14]. The stabilityproofs of Pk/(Pk−1 + P0), k = 2, 3 and Pk/P

disck−1 are presented in [41, 68] and [60, 61, 77],

respectively. For a stability proof of all other finite element pairs Pk/Pk−1, Pbubblek /P disc

k−1 ,we refer to [30] and the references given there.

In all of our considered model problems in Chapter 2, the velocity becomes an input forother calculations such as free surface/interface movement and energy equation. Therefore,we prefer higher (at least second) order finite elements for the velocity with a suitablepressure approximation.

Lemma 3.2.4 (Properties of P bubble2 /P disc

1 ) On simplices the discontinuous finite ele-ment pair P bubble

2 /P disc1 guarantees the mass conservation element-wise. Furthermore, the

error bound|u− uh|1,Ω + ||p− ph||0,Ω ≤ Ch2 (||u||3,Ω + ||p||2,Ω)

holds true.

38


Proof. We have b(qh,uh) = 0 for all qh ∈ Qh. We show that in addition

b(1,uh) =

∫

Ω

∇ · uh dx =

∫

ΓD

uh ·n dγ =

∫

ΓD

uD,h · n dγ = 0.

Now b(qh,uh) = 0 for all qh, where qh ∈ L2(Ω) : qh|K ∈ Q(K). Restricting to a functionqh with supp qh ⊂ K we get

∫

K

qh ∇ · uh dx = 0 ∀ qh ∈ P1(K).

Thus, the pair P bubble2 /P disc

1 guarantees the mass conservation element-wise. The proof ofthe error estimate follows from the lemma 3.2.2 as a special case.

3.2.2 Spurious velocities

The spurious (non-physical) velocities are generate in incompressible flow problems subjectto an external local force. In the literature, these velocities are also called “parasiticcurrents”. Even in the one phase Stokes flows with the force term as the gradient of ascalar potential, the spurious velocities have been observed, see for e.g., [17, 18, 31]. Inthe finite element discretisation of one-phase Stokes flows, several techniques have beenproposed in the literature to suppress the spurious velocities and to improve the velocityerror [17, 25, 28]. The focus of our discussion is on flow problems with interfaces. Theinfluence of different finite element discretisations, approximation of the interface and thecurvature has been studied in [26] for a two-dimensional static bubble problem. Here, weextend this analysis to an axisymmetric static bubble problem.

Let us consider a liquid droplet Ω1 ⊂ R3 of radius 1 with centre at the origin (0, 0, 0).

Also, we assume that the droplet is inside an immiscible liquid domain Ω2 ⊂ R3, which is

a cylinder of radius 2. Let us denote the interface ∂Ω1 between the two immiscible liquidsby ΓF , the outer boundary of Ω2 by ΓD, i, e, ΓD := ∂Ω2 \ΓF and Ω = Ω1 ∪Ω2 as the wholecomputational domain. Figure 3.4 shows a cross section of the domain Ω. Furthermore,

Ω2

Ω1

ΓD

ΓF

n

t

Figure 3.4: Cross section of the 3D static bubble domain.

39


we neglect gravitational effects and assume that the system is in equilibrium state. Thefluid flow is governed by the stationary incompressible Stokes equations and are given by,

−∇ · Tk(u, p) = 0 in Ω,

∇ · u = 0 in Ω,

u = 0 on ΓD,

[|u|] = 0 on ΓF ,

τ i · [|T(u, p)|] · ν = 0 on ΓF ,

ν · [|T(u, p)|] · ν = σK on ΓF ,

(3.10)

for i = 1, 2 and k = 1, 2, where u is the velocity, p the pressure, K the curvature of ΓF andσ the coefficient of the interfacial tension. The jump across the interface ΓF is denotedby [| · |]. Furthermore, τ i and ν are the tangential and outward normal unit vectors withrespect to the domain Ω1, as shown in Fig. 3.4. The stress tensor Tk(u, p) in each domainΩk is given by

Tk(u, p) = 2µkD(u) − pI, k = 1, 2,

with the dynamic viscosity µk of the respective liquids. Here, I denotes the identity tensorand the velocity deformation tensor D(u) is given as in (2.2). For simplicity, assume thatσ = µ1 = µ2 = 1 and using the fact that R = R1 = R2 = 1, K = (1/R1 +1/R2) = 2, whereR1 and R2 are principle radii of the bubble. The weak form of the static bubble problemis derived by using the finite element spaces

V0 := v ∈ H1(Ω)3 : v = 0 on ΓD,

Q := L20(Ω) = q ∈ L2(Ω) : (q, 1) = 0.

After incorporating the interface boundary conditions, the weak form of (3.10) reads:

Find (u, p) ∈ V0 ×Q such that

2(D(u),D(v)) − (p,∇ · v) = 〈K,v · ν〉ΓF∀ v ∈ V0

(q,∇ · u) = 0 ∀ q ∈ Q.(3.11)

Note that, we have an additional curvature term as an external local body force. Let usintroduce the divergence-free space

W = v ∈ V : (q,∇ · v) = 0 ∀ q ∈ Q.

The weak form (3.11) in the divergence-free space W reads:

Find u ∈W such that

2(D(u),D(v)) = 0 ∀ q ∈W,

40


where the right-hand side of (3.11) vanishes due to K = 2 and the divergence theorem.Then, the weak solution of (3.11) is know to be u = 0. Now, let us define the discretelydivergence-free space

Wh = vh ∈ Vh : (qh,∇ · vh) = 0 ∀ qh ∈ Qh,

Again, the curvature term on right-hand side vanishes if Wh ⊂ W holds and the discreteform in Qh reads: Find uh ∈Wh such that

2(D(uh),D(vh)) = 0 ∀vh ∈Wh,

which results the solution uh = 0. This indicates that if the solution space is discretelydivergence-free then the spurious velocities will not be generated. However, it is expensiveto construct divergence-free finite element spaces, see [15]. Now, we turn to the realisticsituation in which Wh 6⊂ W . The discrete solutions are sought in the following spaces,which are defined on the triangulation Th of the domain Ω,

Vh :=

vh ∈ H1(Ω)3 : vh|K ∈ V (K), K ∈ Th

Vh,0 :=

vh ∈ Vh : N(vh) = 0 N ∈ N ∂Ω

Qh :=

qh ∈ Q : qh|K ∈ Q(K), K ∈ Th

(3.12)

where, V (K) = Vl(K) and Q(K) = Vm(K) are same or different polynomial functions onthe cell K for some degree l,m ≥ 1. Then, the discrete analogues of (3.11) reads:

Find (uh, ph) ∈ Vh,0 ×Qh such that

2(D(uh),D(vh)) − (ph,∇ · vh) = 〈Kh,vh · ν〉 ∀vh ∈ Vh,0,

(qh,∇ · uh) = 0 ∀qh ∈ Qh,(3.13)

where Kh denotes an approximation of the curvature K. In general, we have p 6∈ Qh andthe exact calculation of the curvature is also not possible. Thus, the discrete solution uh

does not vanish and instead we have the error bound

|uh|1 ≤ C

(

infqh∈Qh

‖p− qh‖0 + supvh∈Vh,0

|〈Kh,vh · n〉 − 〈K,vh · n〉||vh|1

)

, (3.14)

where the second term is a consistency error introduced by the approximation of thecurvature term. The error estimate (3.14) indicates that the spurious velocities depend onthe approximation of both the pressure and the curvature.

We study the influence of different finite element discretisation on the spurious veloci-ties. The computational results are presented in the section 6.1.

41

Chapter 4

Numerical Methods forTime-Dependent Domains

In the previous Chapter, we have presented a short overview of finite element methodsfor stationary problems in fixed domains. Now, we consider the free surface and interfaceflow problems in moving domains. Apart from the flow variables, each fluid phase in thedomain or the domain itself has to be determined during the computation of these typeof flow problems. In this Chapter, first, we recall methods to handle the time dependentdomains. Next, we discuss the interpolated cubic spline and the Laplace-Beltrami operatortechniques to approximate the curvature of the interface. Finally, the inclusion of thecontact angle in the weak form of the impinging droplet problem is presented.

4.1 Methods for time-dependent domain

Several methods have been proposed in the literature to handle time-dependent domains,see for an overview [71]. Based on the treatment of the computational domain, all thesemethods can be classified into two categories: Eulerian and Lagrangian approaches. Inthe Eulerian approach, the domain is fixed throughout the computation and the fluid isallowed to flow through the domain. Therefore,

(i) there is no need to determined the domain but each fluid phase in the domain hasto be identified,

(ii) for this, boundaries of each phase has to be captured.

Contrarily, in the Lagrangian approach, the computational domain moves with the fluidand therefore, the domain itself is a priori unknown.

4.1.1 Fixed Grid Methods

The Volume-Of-Fluid, Level-Set and Front-Tracking methods are particularly successfulfor a wide range of multiphase flow problems in Fixed Grid Methods. All these methods

43

4 Numerical Methods for Time-Dependent Domains

are briefly discussed here. A new segment projection technique [70] has also been proposedfor these type of flow problems.

Volume-Of-Fluid Method

In the volume-of-fluid method (VOF), a “volume-of-fluid function” (also called “markerfunction”) is used to identify each fluid phase. The “volume-of-fluid function” gives thevolume fraction of one of the fluids in each cell of the finite difference or finite elementmeshes. Usually, in the VOF, rectangles and bricks are used in two- and three-dimensions,respectively.

Let us briefly discuss the volume-of-fluid method to capture the interface between twoimmiscible liquids, say “Liquid A” and “Liquid B”, in two space dimensions as in Fig-ure 4.1 (a). Suppose that the domain is triangulated by a square grid of side length h.Then, the volume of “Liquid A” in each cell is defined by Ci,jh

2, where 0 ≤ Ci,j ≤ 1,Ci,j ∈ [0, 1] is the volume fraction of the “Liquid A” in the (i, j)th cell. Furthermore, thevolume fraction of “Liquid B” is defined by 1 − Ci,j on the cell (i, j). If a part of theinterface lies in the cell (i, j), then we have 0 < Ci,j < 1. Eventhough the volume fractionCi,j is unique in the cell (i, j), the representation of the interface is not unique, see forexample, Fig. 4.1 (b) and (c). Several interface reconstruction algorithms such as SLIC,piecewise linear approximation are proposed in the literature to represent the interface, see[54] for more details of these algorithms.

The volume fraction Ci,j can be considered as an approximation of the characteristicfunction C, which is defined as

C(x, y) =

1 if the point (x, y) is in “Liquid A”

0 if the point (x, y) is in “Liquid B”.

(b)(a) (c)

(i,j)

B

A

Figure 4.1: Interface representation in VOF based on the volume fraction in each cell, (a)original interface, (b) Simple Linear Interface Calculation (SLIC) approach, (c) piecewiselinear approximation.

44


Once the fluid velocity u on the background mesh is calculated at a time step, the charac-teristic function C(x, y) has to be recalculated by using the transport equation

∂C

∂t+ u · ∇C = 0.

Then, the interface position is captured by one of the interface reconstruction algorithmsusing the recalculated volume fraction Ci,j. Similarly, the interface is captured throughoutthe entire time interval.

The characteristic function C(x, y) has also been used to include the material parame-ters such as density and viscosity by using

ρ = CρA + (1 − C)ρB,

µ = CµA + (1 − C)µB,

where ρA, ρB and µA, µB are the density and dynamic viscosity of the respective liquids“A” and “B”.

The main difficulty of this method lies in the calculation of the local curvature of theinterface from the volume fraction. Furthermore, guaranteeing the mass conservation sep-arately for “Liquid A” and “Liquid B” is also rather difficult in this method, see [56]. Anumber of recent developments, including a technique to improve the resolution of the in-terface with high order schemes [54] and local mass preserving schemes [56], have increasedthe applicability of this method.

Level Set Method

A continuous zero level set function is used to represent the interface in the level setmethod. The level set function φ(x, t) is constructed as a signed distance function in sucha way that φ(x, t) > 0 on one side of the interface and φ(x, t) < 0 holds on the otherside of the interface. Similarly, several interfaces can be represented by the same level setfunction φ(x, t).

As an example, let us consider an interface of an unit circle centred at the origin attime t = 0. We assume that the interface is inside a [-2, 2] square domain, which is meshedby a square grid. A signed distance level set function, which has positive values inside thecircle and negative values outside the circle at time t = 0 is defined by

φ(x, 0) = 1 −√

x2 + y2.

Here, the zero level set function φ(x, 0) = 0 represents the interface, see Figure 4.2. Giventhe fluid velocity u on the background mesh, the interface is advected by solving theadvection equation

∂φ(x, t)

∂t+ u · ∇φ(x, t) = 0,

for a new level set function at time t ≥ 0. The numerical error, which is obtained bydiscretising this advection equation causes problems to guaranty the mass conservation in

45


−2 0 2−2

02

−2

−1

0

1

xy

φ(x,

0)

−1

−1

−1

−1

−1

−1

−1

−1

0

0

0

0

x

y

−2 −1 0 1 2−2

−1

0

1

2

Figure 4.2: The level set function φ(x, 0) (left) and their contours (right).

each fluid phase. In particular, the mass loss is significant on coarse meshes [66]. Moreaccurate and robust numerical schemes (for e.g., higher order ENO, WENO) are proposedin the literature to solve this type of advection equations. In the finite element context,the mass loss can be reduced by using finer meshes. However, the computational costswould be enormously increase. Alternatively, a mass correction scheme has been used in[32]. Additionally, a stabilisation technique such as streamline diffusion (SDFEM) has tobe used for the advection equation to get a stable system [32, 70].

It is essential that φ(x, t) remains a signed distance function to guaranty the massconservation. In general, this property is not preserved during the advection and a reini-tialisation technique has been proposed in the literature to overcome this difficulty.

Recent advances in the level set method and the natural ability to handle the topologicalchanges of the interface make the method attractive.

Front Tracking Method

The basic idea behind the front tracking method is the use of separate front markers forthe interface on a fixed grid and modify the meshes near the front to fit with the interface.The method which we discuss here is a hybrid one between the front tracking and the frontcapturing methods [71]. The main idea is to use stationary meshes for the fluid flow anda separate one dimension lower moving meshes to track the interface. The front points(interface mesh points) are advected by the calculated velocity times the time step, wherethe velocity at each front point is obtained by an interpolation from the fixed backgroundmesh.

For example, in two space dimension, the interface is represented by a one dimensionalmoving “front” consisting of connected front points χF as shown in Figure 4.3. In gen-eral, a smooth function (for e.g., Legendre polynomials or interpolated cubic splines) isconstructed from the front points to calculate the curvature, see [57, 71]. Once the fluidvelocity u on the fixed background meshes is calculated, the velocity is interpolated at the

46


Fluid B

Fluid A

front

− front points

Figure 4.3: Typical representation of the interface using marker points in hybrid front-tracking methods.

front points. Then, the new position of the interface is obtained by the equation

dχF

dt= IχF

u(χF , t), (4.1)

where IχFu(χF , t) is the interpolated velocity from the background meshes to the front

points χF . The equation (4.1) can be solved by one of the methods presented in thesection 5.2.1. Since the front points are moved in a Lagrangian manner, these points mayaccumulate or the resolution becomes inadequate at some parts of the interface. Thesedifficulties can be handled by adding and deleting or redistributing the front points. Inthe front tracking method there is no need to solve any additional equation to advect theinterface. However, one has to solve some algebraic equations to construct a parameterisedcurve to calculate the curvature and redistribute the front points.

Since the interface is tracked explicitly, topological changes such as breaking and merg-ing of interfaces have to be done manually. Furthermore, the interface is advected by aninterpolated velocity and in general, this interpolated velocity is not divergence free at thefront points. Therefore, guarantying the mass conservation in each phase is very difficultwhich is another drawback in the front tracking method.

General Remarks on Fixed Grid Methods

In the previous sections, we have briefly discussed different fixed grid methods. The mainadvantage of these methods is the ability to handle the topological changes of the domain.However, guarantying the mass conservation in each fluid phase is a serious problem inthese methods. Furthermore, accurate inclusion of the surface force over the interface isdifficult in all of these methods, since the interface is not resolved by meshes.

47


The Continuum Surface Force (CSF) technique is often used to include the surfaceforce, which compress of surface tension and local curvature. The CSF technique wasfirst introduced by Brackbell et al [7]. The basic idea behind this technique is to includethe surface tension force over a small region near the interface instead of only over theinterface. For example, to include the surface force over the interface ΓLL in our risingbubble problem, the interface integral term

Iori =ρ1

Eo

∫

ΓLL

Kν · vdγLL

in (2.32) is replaced by the CSF counterpart

ICSF =ρ1

Eo

∫

Ω

Kν · v δǫ(φ(ξ))dx,

where δǫ(φ(ξ)) is a smooth Dirac delta function. In general, the distance function ξ(x) isdefined by

ξ(x) = dist(ΓLL, x) for all x ∈ Ω,

i.e., ξ(x) is the shortest distance from the interface ΓLL to the point x ∈ Ω. Then, thesmooth Dirac delta is define by

δǫ(φ(ξ)) =

φ(ξ) if |ξ| ≤ ǫ,

0 if |ξ| > ǫ.

Several variants for the kernel function φ have been used in the literature, see for e.g., [70].

Another difficulty in fixed grid methods is to include different material properties suchas density and viscosity of different fluids on the interface meshes (where a part of theinterface lies in a mesh). In general, these physical parameters are discontinuous acrossthe interface. Several techniques such as defining the material property as a function ofshortest distance from the interface, using a steep gradient to translate the jumps anddefining the material parameters by a smoothed Heaviside function, have been proposed inthe literature to overcome this difficulty, see for instance [71]. However, these smoothingtechniques induce some numerical errors in the solution [70].

4.1.2 Moving grid methods

In moving grid methods, each fluid phase in the flow uses separate and boundary-fittedmeshes, and offers potentially the highest accuracy. Lagrangian approach and ArbitraryLagrangian Eulerian approach are the two approaches used in moving grid methods. Inthe next sections, these two approaches are presented in more detail.

48


Lagrangian approach

In the Lagrangian approach, the interface is resolved by the meshes of the triangulateddomain. Once the velocity is calculated at all points χ of the computational grid, eachpoint is advected individually by their corresponding velocity as

dχ

dt= u(χ, t).

This approach has been successfully used in several computations, see for example, thecomputation of the breakup of a two-dimensional droplet [52], the examination of theinitial deformation of a buoyant bubble [64], and the impinging of an axisymmetric dropleton a solid surface [24]. The main difficulty in this approach is the quick distortion ofthe grid. Since their is no constraint on the grid points movement, the shape of thecomputational meshes will be quickly distorted and remeshing is needed, more often than inother approaches. Eventhough this method is robust and accurate, the interpolation error,which occurs while interpolating the solution from the old domain to the new remesheddomain may cause problems in mass conservation.

Arbitrary Lagrangian Eulerian (ALE) approach

An early description of the ALE approach has been given in [16]. The ALE approach isquite popular in Fluid Structure Interaction (FSI) problems. For the application to freesurface flows in the finite element context, we refer to [3, 27, 48, 51]. Here, we recall theALE approach which is based on the description in [51].

Let us denote the boundary points of all interfaces by χF . In the ALE approach, theboundary points χF are advected with their corresponding velocity, as

dχF

dt= u(χF , t)

and the inner points can be moved arbitrarily to preserve the mesh quality. Several tech-niques are proposed in the literature for the inner points displacements and a few will bediscussed in section 5.2.2.

To describe the ALE approach, we consider a scalar advection diffusion problem as anexample. Find v : Ωt × (0, I) → R, such that

∂v

∂t− ∆v + (u · ∇)v = 0 in Ω(t) × (0, I),

v = vD on ∂Ω(t) × (0, I),

v = v0 in Ω(0),

(4.2)

where u is a convection velocity, which is assumed to satisfy ∇ · u = 0. Further, vD andv0 are given functions.

49


To rewrite the equation (4.2) in ALE form, let us define a family of mappings At, whichat each time t ∈ [0, I) map a point (namely ALE coordinate) Y ∈ Ω of a reference domainΩ onto a point (namely Eulerian coordinate) X of the current domain Ωt. That is,

At : Ω → Ωt, At(Y) = X(Y, t)

for each t ∈ [0, I). We assume that the mapping At is a homeomorphic function, i.e,,

At ∈ C0(Ω) is invertible with a continuous inverse A−1t ∈ C0(Ωt). Furthermore, we assume

that the mapping

t→ X(Y, t), Y ∈ Ω

is differentiable almost everywhere in [0, I).

The function v in (4.2) has been defined on the Eulerian frame. Here, we define

v := v At, v : Ω × (0, I) → R, v(Y, t) = v(At(Y), t),

which is the corresponding function on the ALE frame. Furthermore, the time derivativeof v on the ALE frame is defined by

∂v

∂t

∣

∣

∣

Ω: Ωt × (0, I) → R,

∂v

∂t

∣

∣

∣

Ω(X, t) =

∂v

∂t(Y, t), Y = A−1

t (X).

Here,∣

∣

∣

Ωis used to indicated that the time derivative is on the ALE frame. Further, the

time derivative on the Eulerian frame is indicated by∣

∣

∣

X

. The domain velocity w is defined

by

w(X, t) =∂X

∂t

∣

∣

∣

Ω. (4.3)

Now, we apply the chain rule to the time derivative of v At on the ALE frame and obtain

∂v

∂t

∣

∣

∣

Ω=∂v

∂t

∣

∣

∣

X

+∂X

∂t

∣

∣

∣

Ω· ∇Xv =

∂v

∂t

∣

∣

∣

X

+ w · ∇Xv, (4.4)

where ∇X denotes the gradient with respect to the Eulerian coordinate. Using the aboverelation (4.4) in the equation (4.2), we get

∂v

∂t

∣

∣

∣

Ω− ∆v + (u · ∇)v − (w · ∇X)v = 0, (4.5)

which is the ALE form of the equation (4.2). Note that the time derivative in (4.5) is nowon the ALE frame and as a consequence an additional convective domain velocity termappears in (4.5). The Eulerian and the Lagrangian methods can be obtained from the ALEmethod by choosing w = 0 and w = u, respectively.

50


Weak formulation of the ALE counterpart

To derive the weak formulation of the ALE counterpart (4.5), we define a solution space

VA(Ωt) :=

ψ : Ωt × (0, I) → R, ψ = ψ A−1t , ψ ∈ H1(Ω)

VA,0(Ωt) :=

ψ : Ωt × (0, I) → R, ψ = ψ A−1t , ψ ∈ H1

0(Ω)

.

Now, the weak formulation of the equation 4.5 reads: Find v ∈ VA(Ωt) such that

∫

Ωt

∂v

∂t

∣

∣

∣

Ωψ dx +

∫

Ωt

∇v · ∇ψ dx +

∫

Ωt

((u− w) · ∇)v ψ dx = 0. (4.6)

for all v ∈ VA,0(Ωt). Note that all functions in H10 (Ω) are independent of time.

The geometric conservation law

Next, we recall the so-called Geometric conservation law (GCL) in the context of finiteelement method. Originally, the GCL condition has been used in the finite differenceand finite volume schemes. The GCL condition, which is related to the stability andaccuracy of the ALE approach governs by the mesh geometrical quantaties and the meshvelocity. However, the significance of this condition is not completely clear yet, see foran overview [51]. It has been reported in [6] that the GCL condition does not implythe stability of a numerical scheme. Furthermore, the authors proved the stability of theimplicit Euler time discretisation scheme with sufficiently small time step even though itdoes not satisfy the GCL condition according to their stability definition.

Let Th be a triangulation of the domian Ωt. Let (K, V,N Ωt) be a finite element associ-ated with the cell K ∈ Th and N ∂Ωt be the set of all nodal functionals associated with theboundary ∂Ωt. On the triangulation we define the discrete finite element spaces

Vh(Ωh,t) :=

ψh ∈ VA(Ωt) : ψh|K ∈ V (K), K ∈ Th

Vh,0(Ωh,t) :=

ψh ∈ Vh(Ωh,t) : N(ψh|K) = 0, ∀N ∈ N ∂Ωt

.

Furthermore, let [tn, tn+1] be an interval in (0, I). Then, in the finite element frameworkthe GCL condition at the discrete level for all t ∈ [tn, tn+1] can be defined as

∫

Ωtn+1

ψh dx −∫

Ωtn

ψh dx =

∫ tn+1

tn

∫

Ωt

ψh ∇ ·wh dx ∀ ψh ∈ Vh,0(Ωh,t). (4.7)

Taking ψh = 1 on a patch of elements in Ω(t), we can obtain another form of GCL condition

∫

Ωhk,tn+1

1 dx −∫

Ωhk,tn

1 dx =

∫ tn+1

tn

∫

Ωhk,t

∇ ·wh dx =

∫ tn+1

tn

∫

∂Ωhk,t

wh · n dγ,

which is more often used in finite volume schemes.

51


Lemma 4.1.1 (see, Proposition 1 in [22]) A sufficient condition for the fulfilment ofthe GCL condition (4.7) is to use a time integration scheme of order d·p−1 for the domainvelocity term, where d is the space dimension and p is the degree of the polynomial used torepresent the time evolution of the nodal displacement within each time step.

Proof. In the time interval [tn, tn+1], let us take the reference domain as Ωtn and define theALE mapping as

Ann+1 : Ωtn → Ωtn+1

.

Then, the following identity holds for all φh ∈ Vh,0(Ωh,t), t ∈ (tn, tn+1]

∫

Ωt

ψh ∇ · wh dx =

∫

Ωtn

ψh (Jcof∇·)wh dx,

where ψh := ψh(Ann+1)

−1, wh = wh(Ann+1)

−1 and Jcof is the co-factor of the Jacobian.

Here, in the interval [tn, tn+1] the function ψh is constant in time. Furthermore, in eachtime step if the nodal displacement is made with a polynomial of degree p then ∇ · wh isa polynomial of degree p− 1, and Jcof of degree (d− 1)p. Thus the term in the integral isof polynomial of degree d · p− 1. Hence, a time integration scheme of degree exactness (atleast) d · p− 1 has to be used to fulfil the GCL condition.

Remark 4.1.1 In general an explicit form

∫

Ωtn+1

ψh ∇ · wn+1h dx ≈

∫

Ωtn

ψh ∇ · wn+1h dx,

has been used in all time integration schemes, since the domain Ωtn+1is unknown a-priori

and therefore the calculation of the Jacobian Jcof is not possible.

Remark 4.1.2 Since there is no one-point integration scheme, which is exact for polyno-mial of order 2, the right side integral of the GCL condition (4.7) should be evaluated attwo (at least) intermediate points in three space dimensional problems to satisfy the GCLcondition.

4.2 Approximation of the curvature

The exact calculation of the curvature in both the fixed and moving grid methods by anexplicit formula is not possible, since the interface is known only approximately. There-fore, we need an additional technique to approximate the curvature in moving interfaceproblems. This technique should be accurate enough, otherwise spurious velocities can begenerated, see remarks in the section (3.14) and the numerical results in section 6.1. Wenote that, the curvature has to be calculated not only at the interface nodes but also atall integral points on the interface. As we have seen earlier, an accurate approximation of

52


local curvature is rather difficult in VOF method, see Figure. 4.1. In the level set method,the curvature K can be incorporated in the level set function, since

ν =∇φ|∇φ| , K = ∇ · ν = ∇ · ∇φ

|∇φ| ,

where φ is a level-set function. In the front tracking method, as we discussed earlier, thecurvature is approximated from the interpolated cubic spline, which is constructed from thefront points. The spline technique can also be used in all boundary resolved moving gridmethods. However, in the ALE approach, the Laplace-Beltrami operator technique is moreoften used [48, 51]. The Laplace-Beltrami operator technique needs only first derivativesto handle the curvature term and it can be easily incorporated with the weak form of theproblem. In this section, we discuss the interpolated cubic spline and the Laplace-Beltramioperator techniques.

4.2.1 Interpolated cubic spline technique

The basic idea behind this technique is to parametrise the interface by an interpolatingcubic spline function using the interface points, and calculate the curvature from the pa-rameterised curve or surface.

Let us consider a two dimensional problem, i.e., the interface Γ is represented by N +1nodes Xi = (ai, bi), i = 0, 1, ..., N . We assume that the interface is open, i.e., X0 6= XN .Furthermore, assume that the interface Γ can be parameterised by

γs = (x(s), y(s)), s ∈ [0, 1],

then the curvature is given by

K = − x′(s)y′′(s) − x′′(s)y′(s)

[x′(s)x′(s) + y′(s)′y(s)]3/2.

Here, x(s) and y(s) are interpolated cubic spline functions. Since the construction of y(s)is similar, we restrict our discussion to the construction of the spline function x(s) .

We discretise s ∈ [0, 1] in such a way that

0 = s0 < s1 · ·· < sN = 1,

and seek an interpolated cubic spline function x(s), which

- admits continuous, second order derivatives on the segment [0, 1], i.e., x(s) ∈ C2[0, 1],

- coincides with a cubic function in each sub interval si−1 ≤ s ≤ si, i = 1, 2, ..., N ;

- satisfies the conditions x(si) = ai, i = 0, 1, ..., N.

Then, the function x(s) is said to be an interpolating cubic spline function with respect tothe segment [0, 1].

53


Construction of a spline function

Let us construct an interpolated cubic spline function x(s) from the interface coordinates(ai, bi), i = 0, 1, ..., N . First, we define

h0 = 0, hj =[

(aj − aj−1)2 + (bj − bj−1)

2]1/2

, j = 1, 2, ..., N.

To hold sj ∈ [0, 1], we define

d =

N∑

k=0

hi, si =1

d

i∑

k=0

hk for i = 0, 1, ..., N.

Then, in each ordinate interval [aj−1, aj], j = 1, 2, ..., N , the interpolating cubic splinefunction xj(s), sj−1 ≤ s ≤ sj, (and analogously for yj(s)) is obtained from

xj(s) = aj−1φ1 + ajφ2 +mj−1rjφ3 +mjrjφ4, (4.8)

where

φ1 = 1 + (2ξ2 − 3ξ)ξ, φ2 = (3 − 2ξ)ξ2, φ3 = (1 − ξ)2ξ, φ4 = (ξ − 1)ξ2,

rj = sj − sj−1, ξ =s− sj−1

rj.

Here, mj , j = 1, 2, ..., N denote the slope, i.e., mj = x′(sj). Since the derivatives ofxj(s) are needed to calculate the curvature, the first derivative of the interpolating splineis obtained by differentiating (4.8) with respect to s. It is given by

x′j(s) = aj−1φ′1 + ajφ

′2 +mj−1rjφ

′3 +mjrjφ

′4, (4.9)

where

φ′1 =

6ξ(ξ − 1)

rj, φ′

2 =6ξ(1 − ξ)

rj, φ′

3 =3ξ2 − 4ξ + 1

rj, φ′

4 =ξ(3ξ − 2)

rj.

Similarly, the second derivative of the interpolating spline is obtained by differentiat-ing (4.9) with respect to s once again. It is given by

x′′j (s) = aj−1φ′′1 + ajφ

′′2 +mj−1rjφ

′′3 +mjrjφ

′′4, (4.10)

where

φ′′1 =

6(2ξ − 1)

r2j

, φ′′2 =

6(1 − 2ξ)

r2j

, φ′′3 =

6ξ − 4

r2j

, φ′′4 =

6ξ − 2

r2j

.

54


Instead of the slopes mj, the above relations (4.8), (4.9) and (4.10) can also be written inmoments Mj = x′′(si), see [1]. In order to determine the slopes mi, we must specify twoend (boundary) conditions

2m0 + q0m1 = c0, pNmN−1 + 2mN = cN , (4.11)

of the interface Γ, where c0 and cN are constants, which depend on the type of the endconditions. Furthermore, the continuity of x′′(s) at si results in the condition,

pjmj−1 + 2mj + qjmj+1 = 3pjaj − aj−1

hj+ 3qj

aj+1 − aj

hj+1, (4.12)

where

pj =hj+1

hj + hj+1, qj = 1 − pj , for j = 1, 2, ...,N − 1.

Then, the algebraic system of equations (4.11) and (4.12) are written as

2 q0 0 . . . 0 0 0p1 2 q1 . . . 0 0 00 p2 2 . . . 0 0 0. . . . . .. . . . . .. . . . . .0 0 0 . . . 2 qN−2 00 0 0 . . . pN−1 2 qN−1

0 0 0 . . . 0 pN 2

m0

m1

m2

.

.

.mN−2

mN−1

mN

=

c0c1c2...

cN−2

cN−1

cN

(4.13)

where ci, i = 1, 2, ..., N −1, represents the right-hand side of the equation (4.12). However,for a closed interface, i.e, for periodic splines the algebraic system (4.13) is not a tri-diagonalsystem [1].

To calculate the curvature of a moving interface by the spline technique, an algebraicsystem similar to (4.13) has to be solved at each time step. Furthermore, if the bound-ary/interface points accumulate or the resolution of the approximation becomes inadequateduring the advection of boundaries/interfaces, we redistribute/add the boundary/interfacepoints by using the interpolated cubic spline.

4.2.2 Laplace-Beltrami operator technique

In the Laplace-Beltrami operator technique, the curvature is replaced by the Laplace-Beltrami operator and use a variational form of the curvature by applying integrationby parts to the Laplace-Beltrami operator over the interface. This technique has beenintroduced into the finite element context in [19, 58] and successfully applied for freesurface flow problems with boundary adapted meshes [3, 48]. Since we do not need tosolve any additional problem, we apply this technique for the considered free surface andtwo-phase flow problems with the help of differential geometry.

55


Tangential gradient and Laplace-Beltrami operator

Let Γ be a closed, orientable, (d-1)-dimensional manifold without boundary which is em-bedded in R

d. Let γ = (γ1(χ), γ2(χ), ..., γd(χ)) be a parameterisation of Γ with local pa-rameterisations γi(χ), χ = χ(s1, s2, ..., sd−1). Here, the range of the parameters is [a, b], i.e.,si ∈ [a, b], i = 1, 2, ..d− 1, for some constants a, b. For example, the parameterised surfacefor an unit sphere is given by γ = (γ1(χ), γ2(χ), γ3(χ)) , χ = χ(s1, s2), 0 ≤ s1, s2 < 2π.

We define the tangential vectors on (d-1)-dimensional Γ by

ti :=∂γ

∂si=

(

∂γ1

∂si,∂γ2

∂si, ...,

∂γd

∂si

)

(χ) i = 1, 2, ..d− 1.

Furthermore, we assume that

qi =

(

(

∂γ1

∂si

)2

+

(

∂γ2

∂si

)2

+ ... +

(

∂γd

∂si

)2)1/2

6= 0, i = 1, 2, ..d− 1,

and define the unit tangential vectors τ i and the metric tensor (g)ij = gij on Γ by

τ i =ti

qi, and gij := τ i · τ j =

∂γ

∂si

· ∂γ∂sj

, i, j = 1, 2, ..., d− 1.

We denote gij = (g−1)ij . For a function f on Γ, define the tangential derivative ∇f of f

by

∇f := gij ∂

∂si

(f γ) ∂γ∂sj

(4.14)

and the Laplace-Beltrami operator to f by

∆f :=1√

det g

∂

∂si

(

√

det g gij ∂γ

∂sj

(f γ))

. (4.15)

Equivalent definitions

The equivalent definitions of the tangential gradient and the Laplace-Beltrami operatorcan be obtained from (4.14) and (4.15) by using the chain rule and the definition of themetric tensor. Let U be an open set such that Γ ⊂ U . For a function f : U ⊂ R

d → Rm,

the equivalent definition of tangential gradient is given by

∇f := ∇f − (∇f · ν)ν, (4.16)

where ν := (ν1, ν2, ..., νd) is the unit normal vector which is orthogonal to each unit tan-gential vector τ i, i = 1, 2, ..., d− 1. By denoting the tangential gradient component-wise

δif := (∇f)i = ∂if − (ν · ∇f)νi, i = 1, 2, ..., d,

56


the Laplace-Beltrami operator on f is defined by

∆f :=

d∑

i=1

δi(δif). (4.17)

Note that both the tangential derivative and the Laplace-Beltrami operator formally needvalues of f not only on Γ but also in its neighbourhood on U . However, one can showthat on Γ, their results are independent from any smooth extension of f . If f is a vector-valued function, then both the tangential derivative and the Laplace-Beltrami operator areapplied component-wise.

Important property

Let U be an open set in Rd with Γ ⊂ U . Then we have

∫

Γ

∇φ dA = −d∫

Γ

φKν dA

for all φ ∈ C10(U), where dA is the surface element on Γ. For ψ ∈ C2(U) and φ ∈ C2

0(U),the integration by parts

∫

Γ

φ∆ψ dA = −∫

Γ

∇φ · ∇ψ dA =

∫

Γ

ψ∆φ dA (4.18)

holds true. Furthermore, we have the identity

∆idΓF= Kν, (4.19)

where K is the sum of the principle curvatures with appropriate sign according to ν. Proofsof these identities can be found in [29, 48].

Since φ = 0 on the boundary of Γ, there are no boundary terms in the above inte-gral (4.18). However, if Γ is not closed as in impinging droplet problem, an additional careshould be taken to handle the boundary integral term. In our derivations, we handle thisterm by including the contact angle.

Laplace Beltrami operator technique for the oscillating droplet problem

After rewriting the equation (2.16) in the ALE description, we replace the curvaturein (2.16) by the Laplace-Beltrami operator. Then, we use the identity (4.18) to get

− 1

We

∫

γFt

Kν · vdΓF = − 1

We

∫

γFt

∆idΓF· v dΓF

=1

We

∫

ΓFt

∇idΓF· ∇v dγF ,

(4.20)

57


where the restriction of the mapping idΓFt: R

d → Rd onto ΓFt

is the identity. Furthermore,to rewrite the weak form (2.16) of the oscillating droplet into its ALE counterpart, weredefine their corresponding velocity and pressure spaces V and Q as

VA(Ωt) :=

v : Ωt × (0, I) → R3, v = v A−1

t , v ∈ V(Ω)

(4.21)

QA(Ωt) :=

q : Ωt × (0, I) → R, q = q A−1t , q ∈ Q(Ω)

. (4.22)

Hence, the weak form of the oscillating droplet problem reads:

For given Ω0, u(0), find (u, p) ∈ VA(Ωt) ×QA(Ωt) such that

(

∂u

∂t

∣

∣

∣

Ω,v

)

+ a(u −w;u,v) − b(p,v) + b(q,u) = f(K,v) (4.23)

for all (v, q) ∈ VA(Ωt) ×QA(Ωt). Here,

a(u;u,v) =2

Re

∫

Ωt

D(u) : D(v) dx+

∫

Ωt

(u · ∇)u · v dx,

b(q,v) =

∫

Ωt

q∇ · v dx,

f(K,v) = − 1

We

∫

ΓFt

∇idΓF· ∇v dγF .

Inclusion of a given contact angle in the impinging droplet problem

In almost all of our considered problems, we have a closed free surface or interface andtherefore the identity (4.19) can be used directly to replace curvature term. However, thefree surface of the impinging droplet problem (2.17) is not closed and the boundary of thefree surface is the contact line. Hence, an additional contact line integral term arises inthe identity (4.18). Here, we derive the contact line integral term, and then we include thecontact angle in this term.

To rewrite (2.17) in the ALE description with Laplace Beltrami operator technique, wereplace the curvature term KνF in (2.15) by the Laplace-Beltrami operator (4.19), inte-grate it by parts and get

58


− 1

We

∫

ΓFt

v · νFK dγF

= − 1

We

∫

ΓFt

∆idΓFt· v dγF

=1

We

∫

ΓFt

∇idΓFt: ∇v dγF − 1

We

∫

ζt

(γν · ∇idΓFt) · vdζ

=1

We

∫

ΓFt

∇idΓFt: ∇v dγF − 1

We

∫

ζt

νζ · vdζ,

(4.24)

since νζ · ∇idΓFt= νζ . Here, the last term in equation (4.24) is the integral over the

contact line ζt and νζ is the outward unit normal vector at the contact line with respect tothe liquid-gas interface ΓFt

. In the second term of (4.24), we decompose the test functionas

v = (v · νS)νS +

d−1∑

i=1

(v · τ i,S)τ i,S

and use the fact that v · νS = 0 on ΓStto get

∫

ζt

νζ · vdζ =

∫

ζt

(νζ · τ i,S)(v · τ i,S)dζ =

∫

ζt

cos(θ) v · τ i,Sdζ, (4.25)

since νζ · τ i,S = cos(θ). Here θ is the contact angle, where the three interfaces meet. Now,we redefine the velocity and pressure spaces V and Q of the weak form (2.17) as

VA(Ωt) :=

v : Ωt × (0, I) → R3, v = v A−1

t , v ∈ V(Ω)

(4.26)

QA(Ωt) :=

q : Ωt × (0, I) → R, q = q A−1t , q ∈ Q(Ω)

. (4.27)

to transform into ALE counterpart. After rewriting the equation (2.17) into their ALEcounterpart and including the contact angle the weak form of the spreading droplet prob-lem reads:

For given Ω0, u(0), find (u, p) ∈ VA(Ωt) ×QA(Ωt) such that

(

∂u

∂t

∣

∣

∣

Ω,v

)

+ a(u− w;u,v) − b(p,v) + b(q,u) = f(K,v) + c(θ,v), (4.28)

59


for all (v, q) ∈ VA(Ωt) ×QA(Ωt). Here,

a(u;u,v) =2

Re

∫

Ωt

D(u) : D(v) dx+

∫

Ωt

(u · ∇)u · v dx

+1

βǫ

∫

ΓSt

(u · τ )(v · τ ) dγS,

b(q,v) =

∫

Ωt

q∇ · v dx,

f(K,v) =1

Fr

∫

Ωt

e · v dx− 1

We

∫

ΓFt

∇idΓF· ∇v dγF ,

c(θ,v) =1

We

∫

ζt

cos(θ) v · τ S dζ.

Rising bubble with Laplace-Beltrami operator technique

In the rising bubble problem (2.32), the interface is closed as in the freely oscillating prob-lem. Therefore, the curvature integral in (2.32) can be replaced by a similar relation (4.20).Further, the velocity and pressure spaces (V and Q) of the weak form (2.32) are redefinedas

VA(Ωt) :=

v : Ωt × (0, I) → R3, v = v A−1

t , v ∈ V(Ω)

(4.29)

QA(Ωt) :=

q : Ωt × (0, I) → R, q = q A−1t , q ∈ Q(Ω)

(4.30)

to rewrite (2.32) into their ALE counterpart. Hence, the weak form of the rising bubbleproblem (2.32) with Laplace-Beltrami operator technique in ALE counterpart reads:

For given Ω0 and u(0), find (u, p) ∈ VA(Ωt) ×QA(Ωt) such that

(

ρ

ρ1

∂u

∂t

∣

∣

∣

Ω,v

)

+ a(u − w,u,v) − b(p,v) + b(q,u) = f(K,v), (4.31)

∀ (v, q) ∈ VA(Ωt) ×QA(Ωt), where

a(u,u,v) = 2

∫

Ωt

1

ReD(u) : D(v) dx+

∫

Ωt

ρ

ρ1(u · ∇)u · v dx,

b(q,v) =

∫

Ωt

q ∇ · v dx,

f(K,v) =

∫

Ωt

ρ

ρ1e · v dx− 1

Eo

∫

ΓLL

∇idΓF· ∇v dγLL.

60

Chapter 5

Efficient Solutions of Free Surfaceand Interface Flow Problems

The weak form of all considered model problems is obtained in the previous Chapters. Inthis chapter, we present the temporal and spatial discretisations of the considered modelproblems. Further we present the construction of the discrete ALE mapping and a fewdomain displacement techniques. Finally, we describe a procedure to transform the weakform of a problem into an axisymmetric configuration.

5.1 Discretisation in space and time

The temporal and spatial discretisation of the Navier-Stokes equations are presented in thissection. As an example, we consider the oscillating droplet problem (4.23). Discretisation ofall other considered problems can be obtained in a similar way. The temporal discretisationincludes a temporal discretisation of ALE mapping, the semi-implicit form of the curvatureand an iterative procedure for linearising the non-linear convective term in the Navier-Stokes equations.

5.1.1 Temporal discretisation

Let 0 = t0 < t1 < · · · < tN = I be a decomposition of the considered time interval [0, I].Let us define kn = tn+1 - tn, 0 ≤ n ≤ N − 1, be a sequence of time steps.

Temporal discretisation of ALE mapping

Since the ALE technique is used to handle the time dependent domain, the domain velocityhas to be provided at each instant tn to discretise the Navier-Stokes equations in time.Therefore, first, we define the discrete (in time) domain velocity which is based on thediscrete ALE mapping. To define the discrete (in time) ALE mapping, the referencedomain Ω has to be defined. The choice of the reference domain is arbitrary, and often

61

5 Efficient Solutions of Free Surface and Interface Flow Problems

the initial domain Ω0 is taken as the reference domain Ω. However, if the deformation ofthe domain Ωt is very large in time, then it is appropriate to choose the latest availabledomain as the reference domain. Therefore, we choose the previous time step domain Ωtn

as the reference domain at the time interval [tn, tn+1]. Using this, we define the discrete(in time) ALE mapping and the domain velocity at the interval [tn, tn+1] as


, w(x, t) =∂

∂t(x, t).

Several techniques have been used in the literature to construct the ALE mapping (see,e.g [48, 70]) and to calculate the inner meshes velocity if the evolution of the boundary isknown. We discuss these methods in Section 5.2 after the spatial and temporal discretisa-tion of the Navier-Stokes equations.

Temporal discretisation of Navier-Stokes equations

The proper (semi-) discrete finite element spaces for the velocity and the pressure usingALE extension in a moving domain on the time interval [tn, tn+1] are defined as

VAn(Ωtn+1) :=

ψn+1 : Ωtn+1× [tn, tn+1] → R

3,ψn+1 = ψn (Ann+1)

−1, ψn ∈ V (Ωtn)

QAn(Ωtn+1) :=

qn+1 : Ωtn+1× [tn, tn+1] → R, qn+1 = qn (An

n+1)−1, qn ∈ Q(Ωtn)

.

Clearly, Ωtn+1= An

n+1(Ωtn). Further the time derivative∂u

∂t

∣

∣

∣

Ωin (4.23) is defined on the

reference domain. Using the so-called one-step ϑ time differencing schemes, the (semi-)discretisation of (4.23) is given by:

For given fixed Ωtn , un ∈ V (Ωtn), pn ∈ Q(Ωtn), wn ∈ H1(Ωtn) and w = wn+1 ∈H1(

(Ann+1)

−1(Ωtn+1))

, find u = un+1 ∈ VAn(Ωtn+1), and p = pn+1 ∈ QAn(Ωtn+1

)) suchthat

(

u − un

kn,v

)

n

+ ϑa(u −w;u,v)n − b(p,v)n + b(q,u)n

= f(Kn+1,v) − (1 − ϑ)a(un − wn;un,v), (5.1)

for all v ∈ VAn(Ωtn+1), and q ∈ QAn(Ωtn+1

). Here, the subscript n denotes that the integralsof the bilinear and trilinear forms are evaluated in Ωtn explicitly, see Remark 4.1.1. Let usdenote Γn+1

F = (Ann+1)

−1(ΓF (tn+1)). The curvature term

f(Kn+1,v) = − 1

We

∫

Γn+1

F

∇idΓF (tn+1) : ∇v dγF ,

in the above equation (5.1) can be treated either explicitly or semi-implicitly or fullyimplicitly. The explicit form leads to a restriction on time step [3, 7], and the implicit form

62


is too complicated. Therefore, we use the following semi-implicit form

− 1

We

∫

Γn+1

F

∇idΓF (tn+1) : ∇v dγF ≈ − 1

We

∫

ΓnF

∇(idΓF (tn) + knun+1) : ∇v dγF , (5.2)

as in [3]. The additional term

d(un+1,v) := − kn

We

∫

ΓnF

∇un+1 : ∇v dγF , (5.3)

which is symmetric and positive definite, adds stability to the equation (5.1). Using theabove semi-implicit form of the curvature in (5.1), we get:

For given fixed Ωtn , un ∈ V (Ωtn), pn ∈ Q(Ωtn), wn ∈ H1(Ωtn) and w = wn+1 ∈H1(

(Ann+1)

−1(Ωtn+1))

, find u = un+1 ∈ VAn(Ωtn+1), and p = pn+1 ∈ QAn(Ωtn+1

)) suchthat

(

u− un

kn,v

)

n

+ ϑ[a(u − w;u,v)n + d(u,v)n] − b(p,v)n + b(q,u)n

= f(Kn,v) − (1 − ϑ)[a(un −wn;un,v) + d(un,v)], (5.4)

for all v ∈ V n+1(

(Ann+1)

−1(Ωtn+1))

and q ∈ Qn+1(

(Ann+1)

−1(Ωtn+1))

. The pressure termb(p,v) in the above equation (5.4) can also be replaced by

ϑb(p,v) + (1 − ϑ)b(pn,v),

but both strategies lead to the same solution, see for example, [72]. However, the fullyimplicit treatment of the incompressibility constrain b(q,u) is very important since theterm (1 − ϑ)b(q,un) on the right hand side needs initial value, which already satisfied thecontinuity equation, that is, b(q,un) = 0.

Depending on the choice of the time-stepping method, the parameter ϑ has to bedefined. For instance, ϑ = 0 for the Forward-Euler or ϑ = 1 for the Backward-Euler orϑ = 1/2 for the Crank-Nicolson methods.

ϑ = 0The Forward-Euler method is explicit and first order accurate. This method inheritsthe stability problem, which results a restriction on time steps.

ϑ = 1The Backward-Euler method is implicit, first order accurate and strongly A-stable.Even though this method is of first order accurate, it is used more often because ofits stability property.

ϑ = 1/2The second order Crank-Nicolson method is obtained for this choice of ϑ. Since thismethod is not strongly A-stable, occasionally it suffers from unexpected instabilities.

63


Euler methods are of first order accurate and unfortunately the second order Crank-Nicolson method is not strongly A-stable. Other higher order methods like implicit Range-Kutta or backward differencing multi-step methods are well-known for temporal discreti-sation of ordinary differential equations. However, these methods are not so popular fortime dependent Navier-Stokes equations because of the complexity and the storage require-ments, see [48, 72].

Another scheme, which is second order accurate and strongly A-stable is the Fractional-step-ϑ (FS) scheme. It was first proposed in [8] as an operator splitting scheme, whichseparates the non-linearity and the incompressibility. In the meantime, due to the devel-opment of fast multigrid, linear solvers and increase in the computing power, the operatorsplitting is no longer essential. Instead of splitting, we can solve the whole Navier-Stokesequations which avoids the error arising from the operator splitting.

Comparing with the one-step-ϑ schemes, the FS scheme uses three different values forϑ in each time step kn. The FS scheme is a clever combination of three one-step-ϑ schemesto get a second order strongly A-stable scheme, and thus it can be viewed as generalisationof the one-step-ϑ scheme. For the Fractional-step-ϑ scheme we define

ϑ = 1 −√

2

2, ϑ = 1 − 2ϑ, η =

ϑ

1 − ϑ, η = 1 − η.

Furthermore, we split each time interval (tn, tn+1) into three subintervals as (tn, tn1),

(tn1, tn2

) and (tn2, tn+1), where tn1

= tn + knϑ, and tn2= tn+1 − knϑ. The ALE map-

pings in these subintervals are defined by

Ann1

: Ωtn → Ωtn1, An1

n2: Ωtn1

→ Ωtn2, An2

n+1 : Ωtn2→ Ωtn+1

.

Furthermore, the ALE mapping in the time step (tn, tn+1) for FS scheme can be defined as


, Ann+1 := An2

n+1 An1

n2 An

n1.

Thus, we can use a same reference domain (Ω = Ωtn) in all subintervals. However, to get ahighest accuracy, we use the previous time (sub-) step domain as the reference domain, i.e.,Ω = Ωtn1

on [tn1, tn2

]. Correspondingly, finite element spaces for the velocity and pressurehave to be defined on each subinterval.

For example, the (semi-) discrete velocity and pressure spaces using ALE extensionon [tn, tn1

] are defined as

VAn(Ωtn1) :=

ψn1: Ωtn1

× [tn, tn1] → R

3, ψn1= ψn (An

n1)−1, ψn ∈ V (Ωtn)

QAn(Ωtn1) :=

qn1: Ωtn1

× [tn, tn1] → R, qn1

= qn (Ann1

)−1, qn ∈ Q(Ωtn)

.

The time derivative in each subinterval is handled by the same way as we discussed for (5.1).Thus, the Fractional ϑ scheme for the Navier-Stokes problem (4.23) on (tn, tn+1) reads:

64


FS 1. For given fixed Ωtn , un ∈ V (Ωtn), pn ∈ Q(Ωtn), wn ∈ H1(Ωtn) and w = wn1 ∈H1((An

n1)−1(Ωtn1

)) , find u = un1 ∈ VAn(Ωtn1), and p = pn1 ∈ QAn(Ωtn1

) such that(

u− un

ϑkn,v

)

n

+ η[a(u− w;u,v)n + d(u,v)n] − b(p,v)n + b(q,u)n

= f(Kn,v) − η[a(un − wn;un,v) + d(un,v)], (5.5)

for all v ∈ VAn(Ωtn1) and q ∈ QAn(Ωtn1

).

FS 2. Given fixed Ωtn1, un1 ∈ V (Ωtn1

), pn1 ∈ Q(Ωtn1), wn1 ∈ H1(Ωtn1

) and w = wn2 ∈H1((An1

n2)−1(Ωtn2

)), find u = un2 ∈ VAn1 (Ωtn2), and p = pn2 ∈ QAn1 (Ωtn2

) such that(

u− un1

ϑkn

,v

)

n1

+ η[a(u− w;u,v)n1+ d(u,v)n1

] − b(p,v)n1+ b(q,u)n1

= f(Kn1 ,v) − η[a(un1 −wn1;un1 ,v) + d(un1,v)], (5.6)

for all v ∈ VAn1 (Ωtn2) and q ∈ QAn1 (Ωtn2

).

FS 3. Given given Ωtn2, un2 ∈ V (Ωtn2

), pn2 ∈ Q(Ωtn2), wn2 ∈ H1(Ωtn2

) and w = wn+1 ∈H1((An2

n+1)−1(Ωtn+1

)), find u = un+1 ∈ VAn2 (Ωtn+1), and p = pn+1 ∈ QAn2 (Ωtn+1

) suchthat

(

u− un2

ϑkn

,v

)

n2

+ η[a(u− w;u,v)n2+ d(u,v)n2

] − b(p,v)n2+ b(q,u)n2

= f(Kn2 ,v) − η[a(un2 −wn2;un2 ,v) + d(un2,v)], (5.7)

for all v ∈ VAn2 (Ωtn+1) and q ∈ QAn2 (Ωtn+1

).Both the Crank-Nicolson and the fractional-step-ϑ schemes are second order. From the

computational point of view, the Crank-Nicolson is preferable since in the fractional-step-ϑ the whole Navier-Stokes equations has to be solved thrice in each time step. But, thecomputational domain in all considered model problems moves with the liquid velocity andin some models the material parameter are having large jumps. Therefore, a stable timediscretisation scheme is needed to avoid the numerical instabilities in the solution. Thus,we use the fractional-step-ϑ scheme, which is strongly A-stable.

5.1.2 Linearisation techniques

Convection term

We discuss a linearisation technique for the convection term in the time dependent Navier-Stokes equations. This technique is based on a fixed point iteration type.

As an example, consider the convection term at the time tn2on [tn1

, tn2]. The non-linear

convective term is then∫

Ωn1

un2 · ∇un2 v dx, (5.8)

65


where Ωn1:= An1

n2

−1(Ωtn2). This results a system of non-linear algebraic equations. In

general, instead of solving non-linear system, the convection term is linearised and a linearalgebraic system is solved for the Navier-Stokes equations.

In our computations, we prefer to use the fully implicit form of the convective term (5.8)with an iteration of fixed point type. The basic idea in the fixed point iteration is to iteratethe entire system with some criterion at each time step. At the time tn2

, we use an iterationun2

l 7−→ un2

l+1 based on

∫

Ωtn1

un2 · ∇un2 vn2 dx ≈∫

Ωtn1

un2

l · ∇un2

l+1 v dx (5.9)

with known un2

0 (= un1). We iterate (5.9) for a fixed number of times or the residualbecomes less than the prescribed value in each time step. It is clear that the fully implicitform with fixed point iteration cause more computational cost in each time step than otherlinearisation techniques (explicit or semi-implicit). However, the fully implicit form leadsto a very robust and stable solution, which are essential for moving interface/boundaryproblems.

5.1.3 Spatial discretisation

In the previous section, we have obtained a sequence of generalised linear stationary Navier-Stokes equations in time. In this section, we present the spatial discretisation of theseequations by using a stable finite element pairs with the discrete ALE mapping.

For the spatial discretisation we use an inf-sup stable finite element pair in all ourcomputations, as specified in the section 3.2. Here, we prescribe the finite element spaces onmoving domains using the discrete ALE mapping. In each time step kn, n = 0, 1, ..., N−1,the ALE form of the finite element spaces for the velocity and pressure on the movingdomain are given by

Vh,An(Ωh,tn+1) :=

ψh,n+1 : ψh,n+1 = ψh,n (Ann+1)

−1, ψh,n ∈ V (Ωh,tn)

Qh,An(Ωh,tn+1) :=

qh,n+1 : qh,n+1 = qh,n (Ann+1)

−1, qh,n ∈ Q(Ωh,tn)

.

We approximate the velocity and pressure by a continuous, piecewise quadratic enrichedwith bubble functions (P bubble

2 ) and discontinuous, piecewise linear functions (P disc1 ), respec-

tively (see section 3.1.2 for the construction of these elements on triangles). As mentionedearlier, this finite element pair satisfies the inf-sup condition and leads to a good local massconservation element-wise. Here, we present the discrete (both in time and space) formof the equation (5.5). The discrete form of the other two sub-steps (5.6) and (5.7) can beobtained in a similar way. The discrete form of the equation (5.5) reads:

FS 1. For given Ωh,tn , unh ∈ V (Ωh,tn), pn

h ∈ Q(Ωh,tn), wnh ∈ H1(Ωh,tn) and w = wn1

h ∈H1((An

n1)−1Ωh,tn1

) , find u = un1

h ∈ Vh,An(Ωh,tn+1), and p = pn1

h ∈ Qh,An(Ωh,tn+1) such

66

5.2 Mesh Velocity based on ALE mapping

that(

u − unh

ϑkn,vh

)

h

+ η[ah(u −w;u,vh)h + dh(u,vh)h] − bh(p,vh)h + bh(qh,u)h

= f(Knh,vh)h − η[ah(u

nh −wn

h ;unh,vh)h + dh(u

nh,vh)h] (5.10)

for all vh ∈ Vh,An(Ωh,tn+1) and qh ∈ Qh,An(Ωh,tn+1

). Here, the bilinear and trilinear formsare define in element wise. For example, a bilinear form

(φ, ψ)h =∑

K∈Th

(φ, ψ)K =∑

K∈Th

∫

K

φ ψ dK.

In order to solve the discrete equation (5.10), we have to provide the mesh velocity at alldiscrete points in the domain Ωh,tn. In the next section, we discuss a few techniques tocalculate the domain velocity.


Let us see the calculation of the domain velocity wnh in the macro time step kn = tn+1− tn.

The calculation of the mesh velocity wnh consist two steps. In the first step, we construct

the mapping

Bnh,n+1 : ∂Ωh,tn → ∂Ωh,tn+1

, Y 7→ x(Y, tn+1), Y ∈ ∂Ωh,tn ,

for the boundary/interface points on ∂Ωh,tn corresponding on their fluid velocity. Then,according to Bn

h,n+1, we construct a discrete (both in space and time) ALE mapping in theentire domain. For given

Bnh,n+1 : ∂Ωh,tn → ∂Ωh,tn+1

at the time step tn, we construct a discrete ALE mapping Anh,n+1 such that

Anh,n+1 : Ωh,tn → Ωh,tn+1

, Y 7→ x(Y, tn+1)

Anh,n+1 = Bn

h,n+1 on ∂Ωh,tn .(5.11)

In the second step, we define the displacement vector as

dΩh,tn: Ωh,tn → R

d, dΩh,tn(Y ) = x(Y, tn+1) − Y, Y ∈ Ωh,tn .

Then, the mesh velocity at all points in the domain Ωh,tn is obtained by the relationwn = dn/kn.

Several techniques have been proposed in the literature to construct the mappingsBn

h,n+1 and Anh,n+1, see for example [48, 51]. First, we present a few methods to construct

the boundary mapping Bnh,n+1 to move the boundary of the domain. Then, we describe two

type of methods to construct the mapping Anh,n+1 to displace the inner points for the given

boundary displacement based on the harmonic extension and the linear elastic theory.

67


5.2.1 Advection of boundaries (construction of Bnh,n+1

)

We advect the boundary/interface points Xn using the flow velocity by

dX

dt= u(X, t). (5.12)

However, one can use the relation

dX∗

dt= (u(X, t) · ν) · ν (5.13)

also to advect the boundaries. The following lemma shows that both are equivalent.

Lemma 5.2.1 Let u = u(x, t) be the divergence free velocity, that is, ∇ ·u = 0, in a timemoving domain Ωt ∈ R

2 and Γ(t) = ∂Ωt. Let ν be the outward unit normal vector on Γ(t)Then,

dX

dt= u(X, t), X(0, s) = X0(s), s ∈ P (5.14)

and

dX∗

dt= (u(X∗, t) · ν) · ν, X∗(0, s) = X0(s), s ∈ P (5.15)

are equivalent and both preserve volume of Ωt.

Proof. The solution of (5.14) gives a parameterisation of Γ(t) as

x = x(t, s), y = y(t, s), s ∈ P ∀ X = (x, y) ∈ Γ(t).

Since Ωt ∈ R2, if the area of Ωt is conserved, then the volume of Ωt is conserved. The area

of Ωt is

|Ωt| =1

2

∫

Γ(t)

xdy − ydx =1

2

∫

s∈P

x(t, s)∂y

∂s(t, s) − y(t, s)

∂x

∂s(t, s)

ds.

Now, take the time derivative and use (5.14), we get

d

dt|Ωt| =

1

2

∫

s∈P

u1∂y

∂s− u2

∂x

∂s+ x

∂2y

∂s∂t− y

∂2x

∂s∂t

ds

=1

2

∫

s∈P

u1∂y

∂s− u2

∂x

∂s+ x

∂u2

∂s− y

∂u1

∂s

ds

=1

2

∫

s∈P

2

(

u1∂y

∂s− u2

∂x

∂s

)

+∂

∂s(xu2 − yu1)

ds (5.16)

68


The first term (·) in the above integral vanishes, since

0 =

∫

Ωt

∇ · u dX =

∫

Γ(t)

u · ν dγ =

∫

s∈P

(

u1∂y

∂s− u2

∂x

∂s

)

ds.

Then, we have

1

2

∫

s∈P

∂

∂s(xu2 − yu1) ds

=1

2x(t, s)u2(x(t, s), y(t, s), t) − y(t, s)u1(x(t, s), y(t, s), t)s=s1

s=s0.

If P ∈ [0, 1], then we have X0(0) = X0(1), since Γ(0) is a closed curve. Due to the uniquesolvability of (5.14) we conclude

X(t, 1) = X(t, 0) ∀ t ≥ 0.

Thus, the second term also vanishes, and we get

d

dt|Ωt| = 0 =⇒ |Ωt| = constant = |Ω0|.

Now, using (5.15) we have to replace (5.16) by

u1 = (u · ν)ν1 and u2 = (u · ν)ν2.

Thus, for the first term in the integral (5.16) we get

(u · ν)ν1 · ν1 + (u · ν)ν2 · ν2 = (u · ν),

which means that the first term vanishes again. Further the second term vanishes too,since it is independent of the velocity field, i.e., it uses only the argument of Γ(0) is closedand the unique solvability of the (5.15). Hence, both (5.14) and (5.15) are equivalent, andin the continuous model ∇ · u = 0 in Ω(t) guarantees the conservation of volume.

Solution of advection equation

We can use a first or second order scheme to solve the equation (5.14). For example, in [70]a second order implicit Crank-Nicolson scheme,

xn+1 = xn +kn

2(un(xn, tn) + un+1(xn+1, tn+1)). (5.17)

has been used for the front tracking method. Since the points xn+1 are unknown a priori,and thus un+1(xn+1, tn+1) are not known. An iterative procedure

xn+1l+1 = xn +

kn

2(un(xn, tn) + un+1(xn+1

l , tn+1)) l = 0, 1, ..., N, (5.18)

69


has been used for (5.17) with xn+10 = xn in [70]. In each iteration l, when the position of

the interface points are moved, the velocities of the interface points un+1(xn+1l , tn+1) have

been obtained from the fixed background meshes velocity field un+1 by an interpolationwithout recalculating. However, if the position of the interface points are moved, thenthe velocity field un+1 on the background meshes could have been recalculated with theupdated density, viscosity and surface tension to get an accurate solution.

In the moving grid method like ALE, the scheme (5.17) has to be modified, since wehave only un+1(xn, tn) instead of un+1(xn+1, tn+1). Furthermore, the iterative scheme (5.18)cannot be used, unless we recalculate the flow velocity at new positions of the boundarypoints. Alternatively, we can use un+1(xn, tn+1) in the Crank-Nicolson scheme (5.17). Weobtain

xn+1 = xn +kn

2(un(xn, tn) + un+1(xn, tn+1)). (5.19)

However, it should be noted that in the ALE method, we are not really calculating un(xn)but un(xn−1) only. Then, we move all mesh points xn−1 to xn and denote un(xn−1) byun(xn), which may not be divergence-free. Especially in moving meshes, it is essential touse a divergence-free flow velocity to move the boundary points for guarantying the massconservation. Thus, we prefer as in [3, 48], the implicit Euler scheme

xn+1 = xn + knun+1. (5.20)

In particular, in these papers an update of the form

xn+1 = xn + kn

(

un+1 · ν)

ν, (5.21)

has been used for a freely oscillating droplet problem. Here, un+1 := un+1(xn, tn+1) and νis the outward normal at the corresponding boundary points xn. The choice (5.21) couldavoid the quick distortion of the mesh in case of u has large tangential velocities. However,the update of the form (5.21) is not applicable for all type of problem. Therefore, in all ourcomputations we always use (5.20) to update the boundary as well as the interface points.

5.2.2 Inner points displacement (construction of Anh,n+1

)

Suppose we displace the inner points of the domain with the fluid velocity, the distortionof the meshes will be large and the remeshing has to be done more often. To avoid this,the inner points are displaced in a prescribed way to preserve the mesh quality. Theharmonic extension and the linear elastic solid are the two techniques more often used forthe prescription of inner points displacement with respect to the boundary displacement.Here, we describe both of these techniques in detail.

Harmonic extension

In the harmonic extension, the new position of all inner points are obtained by solving thefollowing problem. Let Bn

h,n+1 : ∂Ωh,tn → ∂Ωh,tn+1be given and assume that the Bn

h,n+1 isone-to-one mapping. Then, the ALE mapping An

h,n+1 : Ωh,tn → Ωh,tn+1is the solution of

70


−∆Xn+1 = 0 in Ωtn , Xn+1 = Bnh,n+1(Y

n) ∀Y n ∈ ∂Ωtn . (5.22)

where Xn+1 = Anh,n+1X

n and Ωtn := (Anh,n+1

−1(Ωtn+1). The above equation can be solved

by a finite element function on the existing meshes in the original domain Ωh,tn. Further,the discrete problem reads:

Find Xn+1h ∈ V n

h (Ωh,tn) such that

∫

Ωh,tn

∇Xn+1h ∇ϕ = 0 ∀ϕ ∈ V n

h,0(Ωh,tn), Xn+1h = Bn

h,n+1(Ynh ) ∀Y n

h ∈ ∂Ωh,tn . (5.23)

Note that, if we use a fixed reference domain (may be the initial domain Ω0), then thematrix of the discrete problem (5.23) does not change in time, and thus the matrix hasto be assembled only once in an entire computation. However, the right-hand side will bedifferent in each time step.

The harmonic extension is well suitable for those domains, which remain convex in alltimes or the deviation from the convex is not too large. However, by choosing the referencedomain as the previous time step domain we can use the harmonic extension for the non-convex domains also. Next, we discuss the linear elasticity technique, which preserves themesh quality better than the harmonic extension.

Linear elastic solid

The construction of the ALE mapping in the elastic solid is basically different from theharmonic extension. In the elastic solid, we calculate the displacement vectors subject tothe displacement of the boundary instead of the new position vectors. The displacementvector on the domain Ωtn and the boundary ∂Ωtn are defined as follows:

Ψn : Ωtn → Rd, Ψn(Y ) = An

h,n+1(Y ) − Y, ∀ Y ∈ Ωtn

Υn : ∂Ωtn → Rd, Υn(Y ) = Bn

h,n+1(Y ) − Y, ∀ Y ∈ ∂Ωtn ,

The ALE mapping Anh,n+1 has to be determined from the following equations. Given the

boundary displacement vector Υn(xn), xn ∈ ∂Ωtn , find the Ψn(xn), xn ∈ Ωtn such that

∇ · S(Ψn) = 0 in Ωtn

Ψn(xn) = Υn(xn) on ∂Ωtn

(5.24)

where

S(φ) = λ1(∇ · φ)I + 2λ2D(φ).

To solve the above problem (5.24) by finite element methods, we use the same triangulationTh of the domain which has been used for the flow variables. Furthermore, we define the

71


discrete displacement vectors of all points in the domain Ωh,tn as

Ψnh : Ωh,tn → R

d, Ψnh(Y ) = An

h,n+1(Y ) − Y, ∀ Y ∈ Ωh,tn

Υnh : ∂Ωh,tn → R

d, Υnh(Y ) = Bn

h,n+1(Y ) − Y, ∀ Y ∈ ∂Ωh,tn .

Here, the displacement vectors of the boundary points ∂Ωh,tn are determined by using therelation (5.20) in the mapping Bn

h,n+1. The displacement vectors of the inner points arethe solution of the following discrete problem.

Find Ψnh ∈ V n

h (Ωh,tn) such that Ψnh = Υn

h on ∂Ωh,tn and∫

Ωh,tn

D(Ψnh) : D(φ) + C1

∫

Ωh,tn

∇ · Ψnh ∇ · φ = 0, ∀φ ∈ V n

h,0(Ωh,tn). (5.25)

Here, C1 = λ1/λ2 is a positive constant, which can be chosen arbitrarily for this problemand in all our calculations we use C1 = 1. The solution of (5.25) is approximated bythe continuous piecewise linear P1 triangular finite element. We refer to [48] for a-prioribounds and the regularity requirements of mappings in (5.25).

5.3 Axisymmetric formulation

In the free surface/interface flows the computational domains are time dependent, and avery fine discretisation (both in space and time) is needed to get an accurate and efficientsolution. This requirement increases the computational costs enormously in 3D. Since thedomains of all our considered problems are rotational symmetric, we use a 2D geometrywith 3D-axisymmetric configuration to get physically acceptable solutions.

Let us define the cylindrical coordinates r (the radial coordinate), φ (the azimuthalcoordinate) and z (the axial coordinate) in terms of Cartesian coordinates by

x = r cosφ, y = r sin φ, z = z,

where r is the radial distance from the origin, and φ is the counterclockwise angle fromthe x-axis, see Fig. 5.1. In terms of x and y,

r(x, y) =√

x2 + y2 and φ(x, y) = arctan(y/x), 0 ≤ φ(x, y) < 2π.

Let u = (u1, u2, u3) be the velocity vector in Cartesian coordinates and uc = (ur, uφ, uz)be the velocity vector in cylindrical coordinate system. We define the cylindrical velocitycomponents in terms of Cartesian velocity components by

u1 = ur cosφ− uφ sinφ, u2 = ur sinφ+ uφ cosφ, u3 = uz.

We assume that the cylindrical velocity components ur, uφ and uz are independent of φ,and uφ = 0. Therefore, we have

u1(x, y, z) = ur(r(x, y), z) cosφ(x, y),u2(x, y, z) = ur(r(x, y), z) sinφ(x, y),u3(x, y, z) = uz(r(x, y), z),

(5.26)

72


x

y

z

rφ

z

r

Φ

Figure 5.1: Initial computational meridian domain (shaded area) in 3D-axisymmetric con-figuration.

We also assume that the pressure does not depend on φ. Similarly, for the test functionsv = (v1, v2, v3) and q we have

v1(x, y, z) = vr cos φ, v2(x, y, z) = vr sin φ, v3(x, y, z) = vz,

q(x, y, z) = q(r(x, y), z).(5.27)

The key idea in our axisymmetric formulation is to transform the Cartesian coordinatevariables in the bilinear and trilinear forms to cylindrical coordinates. Here, we illustratethe transformation for the bilinear forms,

(u,v) =

∫ ∫ ∫

Ω

u · v dx dy dz, (5.28)

(D(u),D(v)) =

∫ ∫ ∫

Ω

D(u) : D(v) dx dy dz, (5.29)

(p,∇ · v) =

∫ ∫ ∫

Ω

p ∇ · v dx dy dz, (5.30)

with Ω ⊂ R3.

Let us consider the integral (5.28), which arises from the discretisation of the timederivative and the external body force in the Navier Stokes equations. Now, we rewritethe integral (5.28) in terms of cylindrical coordinate using (5.26) to (5.27) and get

∫ ∫ ∫

Ω

u · v dx dy dz =

∫ 2π

0

∫ ∫

Φ

(urvr cos2 φ+ urvr sin2 φ+ uzvz)r dr dz dφ

= 2π

∫ ∫

Φ

(urvr + uzvz)r dr dz, (5.31)

73


where Φ denotes the cross section of Ω in (r, z) coordinates. Next, let us consider theintegral (5.29) with the definition

D(u) =1

2

(

∂ui

∂xj+∂uj

∂xi

)

, i, j = 1, 2, 3,

where x1 = x, x2 = y, x3 = z. The partial derivatives of the velocity components in thecylindrical co-ordinate become

∂u1

∂x1

=∂ur

∂r

∂r

∂xcosφ− ur

∂φ

∂xsin φ =

∂ur

∂rcos2 φ+ ur

sin2 φ

r,

∂u1

∂x2

=∂ur

∂r

y

rcosφ− ur

x

r2sinφ =

(

∂ur

∂r− ur

r

)

sinφ cosφ,

∂u1

∂x3=

∂ur

∂zcosφ,

∂u2

∂x1=

∂ur

∂r

x

rsin φ− ur

y

r2cosφ =

(

∂ur

∂r− ur

r

)

sinφ cosφ,

∂u2

∂x2

=∂ur

∂r

y

rsinφ+ ur

x

r2cosφ =

∂ur

∂rsin2 φ+ ur

cos2 φ

r,

∂u2

∂x3

=∂ur

∂zsinφ,

∂u3

∂x1

=∂uz

∂r

x

r=∂uz

∂rcos φ,

∂u3

∂x2

=∂uz

∂r

y

r=∂uz

∂rsinφ,

∂u3

∂x3=

∂uz

∂z.

(5.32)

By using the above relations (5.32), we have

D(u) =

∂ur

∂rcos2 φ+ ur

sin2 φ

r,

(

∂ur

∂r− ur

r

)

sin φ cosφ,

(

∂ur

∂z+∂uz

∂r

)

cosφ

2

∗ ∂ur

∂rsin2 φ+ ur

cos2 φ

r,

(

∂ur

∂z+∂uz

∂r

)

sin φ

2

∗ ∗ ∂uz

∂z

,

74


where “∗” means known by symmetry. Then, we have the tensor product,

D(u) : D(v) =∂ur

∂r

∂vr

∂r

(

cos4 φ+ 2 sin2 φ cos2 φ+ sin4 φ)

+∂ur

∂r

vr

r

(

sin2 φ cos2 φ− 2 sin2 φ cos2 φ+ sin2 φ cos2 φ)

+∂vr

∂r

ur

r

(

sin2 φ cos2 φ− 2 sin2 φ cos2 φ+ sin2 φ cos2 φ)

+urvr

r2

(

+ sin4 φ+ 2 sin2 φ cos2 φ cos4 φ)

+1

2

(

∂ur

∂z+∂uz

∂r

)(

∂vr

∂z+∂vz

∂r

)

(

cos2 φ+ sin2)

+∂uz

∂z

∂vz

∂z

=∂ur

∂r

∂vr

∂r+urvr

r2+

1

2

(

∂ur

∂z+∂uz

∂r

)(

∂vr

∂z+∂vz

∂r

)

+∂uz

∂z

∂vz

∂z.

Hence, the integral (5.29) becomes:

∫ ∫ ∫

Ω

D(u) : D(v) dx dy dz

= 2π

∫ ∫

Φ

[

∂ur

∂r

∂vr

∂r+urvr

r2+

1

2

(

∂ur

∂z+∂uz

∂r

)(

∂vr

∂z+∂vz

∂r

)

+∂uz

∂z

∂vz

∂z

]

r dr dz. (5.33)

Next, we consider the divergence term (p,∇ · v). Using (5.32), we obtain

(p,∇ · v) =

(

p,∂v1

∂x1

+∂v2

∂x2

+∂v3

∂x3

)

=

(

p,∂vr

∂r+

1

rvr +

∂vz

∂z

)

.

Hence, the integral (5.30) becomes:

∫ ∫ ∫

Ω

p ∇ · v dx dy dz = 2π

∫ ∫

Φ

[

p

(

∂vr

∂r+

1

rvr +

∂vz

∂z

)]

r dr dz. (5.34)

Similarly, all other bilinear and trilinear forms can be transformed into the axisymmetricconfigurations.

Comparing the 3D-axisymmetric forms (5.31), (5.33), and (5.34) with their correspond-ing 2D planar forms, we have

r dr dz instead of dx dy in all integrals,

urvr

r2as an additional term in (D(u) : D(v)),

vr

ras an additional term in (p,∇ · v).

75


The wellposedness of the bilinear and trilinear forms require

r1

2

∂ur

∂r, r

1

2

∂uz

∂z, r

1

2

∂ur

∂z, r

1

2

∂uz

∂r,

ur

r1

2

, r1

2p ∈ L2(Φ).

Note that the requirement r−1/2ur ∈ L2(Φ) induce the boundary condition ur = 0 at theartificial boundary r = 0. In the weak formulation, there is another “hidden” boundarycondition incorporated, which becomes visible by integrating (5.33) by parts. This, reads

τ · (T(u, p)) · ν = 0 ⇐⇒ ∂uz

∂r= 0, at r = 0, (5.35)

where τ · (T(u, p)) · ν is the tangential stress tensor on the z-axis.

76

Chapter 6

Numerical Results

In this Chapter, we present computational results for all of our considered model problemsin Chapter 2 and 3. First, to show the accuracy of the numerical scheme proposed inthe previous Chapters, we present numerical results for the static bubble problem. Themagnitude of spurious velocities and their influence on the mass flux are examined forcontinuous and discontinuous pressure approximations. In the second section, simulationsof 3D-axisymmetric oscillating liquid droplet without gravity are presented. The numeri-cally obtained frequencies and damping factors are compared with the existing numericaland analytical results in the literature. The third section covers an array of computationsof isothermal and non-isothermal liquid droplets deformation after impinging on an uni-formly heated solid surface. The flow dynamics in both the isothermal and non-isothermaldroplets during the spreading and recoiling process is visualised for different sets of pa-rameters. Effects of the slip coefficient, the impact velocity and the surface tension onthe flow dynamics are studied. Furthermore, some interesting physical quantities such aswetting diameter, time taken to attain the maximal wetting diameter, dynamic contactangle and kinetic energy are computed as a function of time during a sequence of spreadingand recoiling processes.

A set of computational results for the rising bubble problem is presented in the finalsection of this Chapter. The capability of the numerical scheme is examined by consideringtwo phase flows with large jumps in their material properties. Different parameters suchas sphericity, rising velocity, kinetic energy and drag force are calculated.

The mass conservation of the numerical scheme is examined in all computations. Wehave developed the inhouse finite element package MooNMD [42] further for the proposednumerical scheme in the previous Chapters. In our simulations, triangular meshes aregenerated by the mesh generator “Triangle” [63] and all linearised algebraic systems aresolved by the direct solver UMFPACK.

77

6 Numerical Results

Φ1

Φ2

Γz

Figure 6.1: The meridian domain Φ = Φ1 ∪Φ2 (left) of the 3D-axisymmetric static bubbleproblem and its triangulation (right) at level 0. Γz denotes the artificial boundary on therotational axis.

6.1 Spurious velocities

In this section, we present numerical results for the static bubble problem to study thespurious velocities. The description of the problem (in three space dimension) is given insection 3.2.2. Here, the computations are performed after reformulating it into the 3D-axisymmetric configurations. We use a rectangle [0, 2]× [0, 4] as the computational domainΦ for this 3D-axisymmetric static bubble problem which is the plane meridian domain ofthe 3D-axisymmetric cylinder Ω. This results in a half circle of radius 1 with centre at(0, 1) as the meridian domain Φ1 for the bubble. Since the 3D-axisymmetric configurationis obtained from the weak form of the model problem, boundary conditions are naturallyincorporated in the 3D-axisymmetric configuration. On the artifical boundary Γz at therotational axis at r = 0, we use the boundary conditions (5.35) and ur = 0. The followingtwo different cases are considered in this static bubble problem: case A: the fluid flow isdescribed by the stationary Stokes equations as in section 3.2.2, and case B: the fluid flowis described by the time dependent Navier-Stokes equations by assuming the static bubbleproblem as a two phase flow in the absence of gravitational force. Since the bubble is inequilibrium state and both fluid phases are in rest, the velocity should be zero in both theStokes and time dependent Navier-Stokes cases.

In both cases we approximate the pressure using a continuous (variant (i)) and discon-tinuous pressure (variant (ii)) approximations. In the continuous approximation we usethe Taylor-Hood element, i.e., globally continuous, piecewise quadratics functions for thevelocity and globally continuous, piecewise linear functions for the pressure. In the dis-continuous approximation, we use a globally continuous, piecewise quadratics functions,enriched with a cubic cell bubble function for the velocity and globally discontinuous,piecewise linear functions for the pressure. For solving the stationary Stokes equations we

78


decomposed the meridian domain Φ using interface resolved quadrilateral meshes. Thecomputational grid is obtained by successively refining an initial coarse grid, see Figure 6.1(right) for the grid at level 0. For solving the time dependent Navier-Stokes equations, wedecomposed the meridian domain Φ using interface resolved meshes. The fractional-step-ϑscheme is used for the time discretisation. In the case B, when the interface moves dueto the spurious velocities, we use the ALE technique to handle the mesh movement. Inboth the Stokes and Navier-Stokes cases the Laplace-Beltrami technique is used to handlethe curvature term. It is necessary that iso-parametric elements have to be used with theLaplace-Beltrami technique to obtain an optimal order of convergence [26]. Therefore, inall of our computations we use iso-parametric elements.

First, we present the computational results for the 3D-axisymmetric static bubble prob-lem obtained for the stationary Stokes equations. In this test case, we have used the dy-namic viscosities µ1 = µ2 = 1 N s/m2, and the surface tension σ = 1 N/m. Spuriousvelocities generated in these computation at level 2 and 3 for the continuous pressure ap-proximations are visualised in Figure 6.2. Colours in the figure represent different valuesof the axial velocity component uz. Further each arrow in the Figure 6.2 indicates thedirection of the spurious velocity at their corresponding point, and the length correspondsto its magnitude. As we expected, the spurious velocities in the discontinuous pressure ap-proximation are almost zero and are not visualised here, since they are invisible in pictures.In the following graphs we can see this.

The velocity error ||u−uh|| = ||uh|| (since u = 0) in L2-norm, and its computationally

Figure 6.2: Spurious velocities generated in the static bubble problem with the continuouspressure approximations at level 2 (left) and 3 (right).

79

6 Numerical Results

104

105

10−8

10−6

10−4

10−2

number of unknowns

erro

r in

L2 −

norm

var 1 var 2

1 2 31

2

3

4

level

orde

r of

con

verg

ence

var 1 var 2

Figure 6.3: Velocity error in L2-norm (left) and order of convergence (right) in the staticbubble problem. Var 1: Q2/Q1 and Var 2: Q2/P

disc1 finite element pairs

104

105

10−5

10−3

10−1

number of unknowns

erro

r in

H1 −

sem

i nor

m

var 1var 2

1 2 3

0.5

1.5

2.5

level

orde

r of

con

verg

ence

var 1var 2

Figure 6.4: Velocity error in H1-semi norm (left) and order of convergence (right) in thestatic bubble problem. Var 1: Q2/Q1 and Var 2: Q2/P

disc1 finite element pairs

obtainted order of convergence for both the continuous and discontinuous pressure ap-proximations are presented in Figure 6.3. The velocity error in the discontinuous pressureapproximation is about three order better than the continuous approximation in both theL2 and H1-semi norms. Furthermore, we can show that the error for the best approxima-tion of discontinuous linear functions is of order O(

√h) [26]. Thus, the error estimate (3.14)

gives an order of convergence 1/2 in the H1−semi norm for the velocity, provided that theconsistency error is in same or of higher order. It is confirmed by our numerical results,see Figure 6.4 (right).

Next, let us see the solution of the time dependent Navier-Stokes equations for thestatic bubble problem. In this computation, we have used the density ρ1 = 1000 kg/m3,ρ2 = 1.23, kg/m3, the dynamic viscosity µ1 = 1 × 10−3 N s/m2, µ2 = 1.73 × 10−5 N s/m2

80


and the surface tension 0.073 N/m. We use two different grids: (i) a coarse grid withhmin = 0.0157 and hmax = 0.158, and (ii) a fine grid with hmin = 0.00785 and hmax =0.0969. Here, and in further computations, often we calculate the kinetic energy using

KE =

∫

Φ

r u(t) · u(t) dr dz∫

Φ

r dr dz=

U2

∫

Φ

r u(t) · u(t) dr dz∫

Φ

r dr dz, (6.1)

where U is the characterstic velocity, and u(t) is the dimensionless velocity. Another impor-tant parameter for evaluating the accuracy of a numerical scheme is the mass fluctuationfactor MF . The mass fluctuation is calculated using

MF =

∣

∣Ω0 − Ωt

∣

∣

∣

∣Ω0

∣

∣

× 100 %, (6.2)

as a function of time, where∣

∣Ωt

∣

∣ denotes the volume of Ωt at time t.

10−5

10−3

10−1

0

5

10

15x 10−3

time

kine

tic e

nerg

y

var 1 var 2 var 3 var 4

10−4

10−2

100

20

60

100

time

mas

s flu

ctua

tion

%

var 1 var 2 var 3 var 4

Figure 6.5: Kinetic energy (left) and mass fluctuation (right) in the static bubble problemdue to spurious velocities. Var 1: Q2/Q1 on coarse grid, Var 2: Q2/P

disc1 on coarse grid,

var 3: Q2/Q1 on fine grid and var 4: Q2/Pdisc1 on fine grid.

The kinetic energy (due to spurious velocities) and the mass fluctuation in the case Bof the static bubble problem are plotted in Figure 6.5. An interesting observation in thiscomputation is that spurious velocities are almost suppressed in the discontinuous pres-sure approximation. Furthermore, magnitudes of spurious velocities are numerically zeroin the entire time interval. But this is not true in the continuous pressure approxima-tion. Since spurious velocities generate enormously over time in the continuous pressureapproximation, the mass loss in the bubble is very large and the bubble starts to shrink,see Figure 6.6.

In both the Stokes and the time dependent Navier-Stokes computations of the 3D-axisymmetric static bubble problem, the discontinuous pressure approximation gives very

81

6 Numerical Results

0 1

1

2

0 1

1

2

0 1

1

2

0 1

1

2

Figure 6.6: Mass flux in the static bubble over time due to spurious velocities generatedin continuous pressure approximation.

good results (spurious velocities almost suppressed). Even on problems with large jumpsin material parameters our numerical scheme with discontinuous pressure approximationworks well without loss of accuracy. Therefore, we use P bubble

2 /P disc1 finite element pair in

all of our computations.

6.2 Freely oscillating droplet

In this section we present a set of numerical results for a freely oscillating 3D-axisymmetricdroplet. The governing equations and the boundary conditions for the fluid flow are de-scribed in section 2.1.1 in three space dimensions. The intention of this test example is toillustrate the validity of our 3D-axisymmetric numerical scheme.

Comparison with an existing simulation

We consider a 3D droplet of shape

3∑

i=1

(xi/ri)2 = 1

with initial radii r1 = r2 = 1, r3 = 1.2, and the centre at the origin. To obtain a 3D-axisymmetric configuration, we transform the weak form of this problem after incorporatingboundary conditions in a 3D-axisymmetric form as described in section 5.3. We use themeridian (half in the ellipse with semi-minor axis (radial axis) length of 1 and the semi-major axis (rotational axis) length of 1.2) as the computational domain Φ. Furthermore,the fluid is assumed to be in rest at time t = 0. The boundary conditions (5.35) and ur = 0are used on the artificial boundary along the rotational axis at r = 0. Since the meancurvature is large at the top and bottom tips of the major axis, and small at the tip of the

82


0 5 10 15

1.0

1.1

1.2

time

z max

at r

=0

0 5 10 150

0.005

0.01

0.015

time

mas

s flu

ctua

tion

(%)

Figure 6.7: Trajectory of the top tip on the z-axis (left) and mass fluctuation (right) in afreely oscillating droplet computation with Re = 300, We = 1.

1 2 3 4

5 6 7 8

Figure 6.8: A sequence of shapes and flow fields at different instances of a freely oscillatingdroplet, timings from image 1 are t = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0. Coloursrepresent the magnitude of the pressure field in the droplet.

83

6 Numerical Results

minor axis, the surface force becomes imbalanced. Due to the imbalances in forces, thedroplet starts to oscillate around its equilibrium position.

In this computation, we have used Re = 300 and We = 1. The trajectory of the toptip point on the z-axis during the 3D-axisymmetric simulation is plotted in Figure 6.7(left). A 3D simulation for the same data set has been performed in [2]. The trajectoryof the top point obtained in our 3D-axisymmetric computation is qualitatively in goodagreement with the trajectory obtained in [2]. The mass fluctuation in our simulationfor the grid with hmin = 0.02, hmax = 0.11, and 15,562 degrees of freedom, is plottedin Figure 6.7 (right). The mass fluctuation at time t = 15 is about 0.014% with 15,562degrees of freedom which shows the accuracy of the numerical scheme. The flow field inthe droplet during oscillations is visualised (arrows) in Figure 6.8. Colours in each imageof the Figure 6.8 represent the magnitude of the pressure field in the droplet, where theblue and red have the minimum and maximum values, respectively.

Next, to validate our numerical scheme quantitatively, we calculate the frequency ωnum

after n period, and the damping factor δ as given in [2]:

ωnum :=n

tn, and δ := n

√

Trmax(tn) − r(t∞)

Trmax(t0) − r(t∞)

,

where tn is the time at the nth period, Trmax(·) is the trajectory of the tip which has its

largest radius at time t = 0, and r(t∞) is the radius of the spherical droplet with thesame volume as the initial droplet. The frequency and the damping factor obtained in ourcomputation for the considered set of data are ωnum = 0.403 and δ = 0.966. These valuesare in good agreement with the values 0.406 and 0.966, respectievely, obtained in [2] atlevel 12.

Comparisons with analytical expressions

The oscillating frequency for of a freely oscillating inviscid, incompressible droplet can beapproximated from a linear stability analysis. For the three-dimensional case, the frequency

ω2 =k(k − 1)(k + 2)σ

ρ r0(6.3)

has been obtained in [10, 46]. Here, k is the order of the spherical harmonic Sk(φ), whichis used to define the initial shape of the droplet through the relation

r(φ) = r0(1 + ǫSk(φ)).

Here, r is the distance from the centre of the droplet to the free surface and ǫ is a positiveconstant. Furthermore, it has been given in [46] that a water droplet of initial radius lessthan the critical value rc = 2.3 × 10−8 m is damped aperiodically.

We consider a water droplet with three different radii: (i) r0 = 2.3 × 10−9 m, whichis less than the critical value, (ii) r0 = 1 × 10−3 m, (iii) r0 = 5 × 10−3 m. The used

84


material parameters of the water droplet are the surface tension σ = 0.074 N/m, thedensity ρ = 1000 kg/m3 and the dynamic viscosity µ = 0.001 N s/m2. These parameterstogether with L = r0 lead to three variants:

(i) Re = 2.3, We = 31 with U = 103

(ii) Re = 1000, We = 13.5 with U = 1

(iii) Re = 5000, We = 67.5 with U = 1.

Computations are made for these three variants with the mode k = 2 and ǫ = 0.3. Thetrajectories of the top tip point obtained in these three variants are shown in Figure 6.9.As predicted in the theoretical analysis, the droplet damped aperiodically in our compu-

5 15 25

0.9

1.0

1.1

1.2

time

z max

at r

=0

i ii iii

Figure 6.9: Trajectories of the top tip on the z-axis in three variants of a freely oscillatingdroplet.

variant ω ωnum δ MFmax

ii 121.1434 119.0527 0.9663 0.005

iii 10.8353 10.5129 0.9653 0.001

Table 6.1: Comparisons of theoretical and numerical frequencies, and damping factor inthree variants for a freely oscillating droplet

tations for the variant (i). Furthermore, the oscillating frequency in variant (ii) is high in

85

6 Numerical Results

comparison with variant (iii). The numerically obtained frequencies for the variants (ii)and (iii) are compared with the theoretical frequencies, see table 6.1. The damping factorsfor these two variants are also presented in table 6.1.

An array of numerical computations is made for a freely oscillating droplet using thenumerical scheme proposed in the previous chapters. The numerically obtained results areboth qualitatively and quantitatively compared with the existing numerical and analyticalresults in the literature. The results are in good agreement and in particular, our numericalscheme captures the behaviour of aperiodic damping for a droplet of radius less than thecritical value. This study clearly shows the validity and the accuracy of the numericalscheme.

6.3 Impinging droplets

In this section we present computational results for an isothermal liquid droplet impingingon a horizontal solid surface. First, we validate the proposed numerical scheme for theimpinging droplets by performing two different computations, and comparing our numericalresults with the existing experimental results. Next, we perform an array of computationswith different parameters to study the effects of the slip coefficient, the impact velocity,and the surface tension on the flow dynamics.

In all of our test cases in this section, computation starts at the time when the dropletcomes in direct contact with the horizontal solid surface. The mathematical model for thefluid flow is described in (2.5). The key fact in our numerical scheme for this moving contactline problem is the contact angle inclusion in the weak formulation, see section 4.2.2. Exceptfor the first test example, the initial shape of all other impinging droplets is assumed to bespherical while touching the solid surface. To define all the dimensionless numbers we takethe initial diameter of the droplet as a characterstic length scale L, i.e., L = d0 = 2r0, andthe impact velocity as the characterstic velocity, i.e., U = uimp. Furthermore, we assumethat the droplet impinging process is 3D-axisymmetric. Therefore, first we transform theinitial domain into the cylindrical coordinates and get

Ω0 := (r, φ, z)∣

∣ (r, z) ∈ Φ0, 0 ≤ φ < 2π,where Φ0 is the meridian of the spherical domain. After scaling, the meridian domain isgiven by

Φ0 := (r, z)∣

∣ 0 < r < 1, r2 + z2 − 1 < 0.In the impinging droplet computations, we calculate a few interesting parameters for aquantitative study. The wetting diameter is calculated in impinging droplet problemsto study the wetting effects at different situations. We define the wetting diameter in adimensionless form as

Wd =d(t)

d0

.

We calculate the dynamic contact angle θd as the angle between the solid surface and thefree surface at the contact line from the currently available geometry of time dependent

86


domain Φt. In all of our computations, we have observed a rolling motion of spreading atthe initial spreading phase. Thus, if a free surface mesh point reaches the solid surface wechange the boundary description of the corresponding edge without remeshing. All of theimpinging droplet computations are performed with the time step 0.0005 and about 18,000degrees of freedom.

6.3.1 Comparisons with experiments

Qualitative comparisons

In the first test case, we use the parameter set from [57] for a qualitative comparison of ournumerical results with their experimental results. We compare the numerically obtainedshape of the droplet with their corresponding experimental image at different instances.An interesting behaviour in this test example is the formation of a pyramidal structureduring the deformation process. A water droplet of radius r0 = 1.75×10−3 m is impinginga super-hydrophobic solid surface at a velocity u0 = 0.41 m/s. The following materialparameters are used in this simulation: the density ρ = 1000 kg/m3, the dynamic viscosityµ = 1×10−3 N s/m2, the surface tension σ = 0.0728 N/m. This set of parameters results inthe following dimensionless numbers: Re = 1435, We = 8 and Fr = 5 with the charactersticlength L = d0. We considered four variants of the slip coefficient: (i) 1/βǫ = 0, (ii) 1/βǫ = 1,(iii) 1/βǫ = 5, (iv) 1/βǫ = 100. Furthermore, the initial shape of the droplet is slightlyperturbed (to make an experimentally identical setup) as in their numerical calculations,i.e., the distance from the centre of the drop to the free surface is defined as

R0(φ) = r0(1 + 0.29 Y2(φ)),

where Y2(φ) is a spherical harmonic of order 2. Since the super-hydrophobic surface gen-erate very high contact angle, we use the equilibrium contact angle θe = 175 in ourcomputation.

The numerically obtained shape of the droplet at different instances is shown in Fig-ure 6.10. Note that each image in Figure 6.10 is a cross-section of the droplet. In eachimage, colours in the left side of the cross-section represent the magnitude and the isolinesof the axial velocity component uz, and the right side of the cross-section represent themagnitude and the isolines of pressure field in the droplet. After the droplet impact onthe solid surface, capillary waves start to propagate over the free surface. Our numericalscheme captures the main feature (pyramidal structure) in the flow dynamics of this testcase. The numerically obtained shape of the droplet is in good agreement (visibly) withtheir corresponding experimental and numerical images in [57].

Numerically obtained dimensionless wetting diameter and the dynamic contact anglewith different variants of slip coefficient are plotted in Figure 6.11. In this test case, theinfluence of slip coefficient on the wetting diameter and the flow dynamics is very small.

87

6 Numerical Results

1

2

3

4

5

6

7

8

9

10

Figure 6.10: A sequence of computationally obtained shapes of a water droplet on a super-hydrophobic surface at different instances for θe = 175, Re = 1435, We = 8, Fr = 5 and1/βǫ = 0. Timings from frame 1 are 0, 0.3165, 0.527, 0,574, 0.6913, 0.7145, 0.738, 0.785,0.832, 0.9375.

88


0.2 0.6 1

0.4

0.8

1.2

time

wet

ting

diam

eter

i ii iii iv

0.2 0.6 1

120

160

200

time

cont

act a

ngle

i ii iii iv

Figure 6.11: Wetting diameter (left) and the dynamic contact angle (right) of a waterdroplet with θe = 175, Re = 1435, We = 8 and Fr = 5, impinging on a super-hydrophobicsurface. (i) 1/βǫ = 0, (ii) 1/βǫ = 1, (iii) 1/βǫ = 5, (iv) 1/βǫ = 100.

Quantitative comparisons

Next, to compare our numerical results quantitatively with experimental results, we per-form an array of computations for a glycerin droplet impinging on a wax surface. Weuse the same value of parameters as used in [74] (Exp.1): the initial radius of the dropletr0 = 1.225 × 10−3 m, the impact velocity uimp = 4.1 m/s, ρ = 1220 kg/m3, the dynamicviscosity µ = 0.116 N s/m2, the surface tension σ = 0.063 N/m and θe = 95. Thus, weobtain the dimensionless numbers Re = 105, We = 796 and Fr = 700. In this parametric

10−1

100

101

102

0.5

1

1.5

2

time

wet

ting

diam

eter

i ii

10−1

100

101

102

60

100

140

180

time

cont

act a

ngle

i ii

Figure 6.12: Computationally obtained wetting diameter (left) and the dynamic contactangle (right) for a glycerin droplet impinging on a wax surface with θe = 95, Re = 105,We = 796 and Fr = 700.. (i) 1/βǫ = 1, (ii) 1/βǫ = 100. In experiments, Wdmax

isabout 2.25.

89

6 Numerical Results

study, we compare the computationally obtained dimensionless wetting diameter and thedynamic contact angle, with their experimental results. We perform two different compu-tations for this set of parameters with: (i) 1/βǫ = 1 and (ii) 1/βǫ = 100. The numericallyobtained values for the wetting diameter and the dynamic contact angle are presentedin Figure 6.12. In both of these computations, the wetting diameter is similar until thedimensionless time about t = 1. During this initial spreading phase, the droplet spreadsalmost by a rolling motion in both cases. After this initial phase, the rate of wetting incase (ii) increases slightly in comparison with case (i) due to the less frictional force coef-ficient (1/βǫ). The maximum dimensionless wetting diameter in case (i) is Wdmax

= 2.367at the dimensionless time t = 1.51 and in case (ii) Wdmax

= 2.197 at the dimensionlesstime t = 1.97. It is interesting to note that by increase the value of 1/βǫ, the maximumwetting diameter Wdmax

decreases but the time taken to attain the Wdmaxincreases. After

reaching the Wdmax, the droplet starts to recoil in both cases. However, in case (i) the

recoiling rate is high due to the less frictional force. Among these test cases, the case (i)fits well with the experimental value for the maximum wetting diameter and the time takento attain it, see Fig. 5 (Exp. 1) in [74].

The contact angle in case (i), initially decreases to a local minimum of about 110.Then, the contact angle starts to increase, and a local maximum is achieved at the sametime as the maximal wetting diameter Wdmax

. During the recoiling phase, the contactangle decreases rapidly, and the local minimum value become less than the equilibriumcontact angle value θe. Then, it slowly increases again to reach the equilibrium value. Inthe case (ii), we observe that the local minimum of the contact angle at the initial stage isabout 120 and at the recoiling phase is about 50. However, the case (ii) takes longer timeto reach the equilibrium contact angle value. A similar dynamic behaviour of the contactangle is observed in experiments for this set of material parameters.

These numerical studies show that our numerical scheme is capable of producing aphysically acceptable solutions. Further our scheme captures key features (capillary waves,pyramidal structure, dynamic behaviour of the contact angle) in the flow dynamics. Fromthese initial studies, we infer that the slip coefficient βǫ has less influence on large equilib-rium contact angle (non-wetting case) droplets.

6.3.2 Influence of the slip coefficient

We study the influence of the slip coefficient βǫ on the flow dynamics for both the wettingand non-wetting liquid droplets in more detail. In both of these cases, we use the same setof material parameters but with different contact angles. In the non-wetting liquid case,we use the equilibrium contact angle value of θe = 100 and in the wetting liquid case,we use θe = 10. Different forms for the slip coefficient βǫ have been used in literature.For instance the slip coefficient as a function of h, h2 and h3, where h is the height of thedroplet, have been used in [33]. Further it should be noted that the additional integral

90


term due to the slip coefficient in 2D planar case is

∫

ΓSt

1

βǫ(u · τ )(v · τ ) dγS,

and in the 3D-axisymmetric case is

2π

∫

∂ΦSt

r

βǫ

(uc · τ )(v · τ ) ds,

where uc is the velocity vector in cylindrical coordinate system. This motivates us to choosefour different variants in our computations: (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/rand (iv) 1/βǫ = 10/r, where r (=

√

x2 + y2) is a dimensionless value. We recall that if thevalue of 1/βǫ decreases, then the frictional force will also be decrease, and in the limit of1/βǫ = 0, we get the free slip condition on the liquid-solid interface.

Influence of βǫ on non-wetting liquid droplets

water droplet:

A water droplet of diameter d0 = 2.7 × 10−3 m with the density ρ = 1000 kg/m3, thedynamic viscosity µ = 1×10−3 N s/m2 and the surface tension σ = 0.073 N/m is consideredin this study. The impact velocity of the droplet is assumed to be uimp = 1.563 m/s.This results in the following dimensionless numbers Re = 4204, We = 90 and Fr = 93with the characterstic length L = d0. Since the experimental results are available forthese parameters (see, Fig. 4.18 and Fig. 4.19 in [73]), we can compare the numericalresults obtained for different slip coefficients. We perform an array of computations forthe four variants of the slip coefficient with the equilibrium contact angle θe = 100.Experimentally, this equilibrium contact angle is observed for a water droplet on a waxsurface. A sequence of images obtained in our computations at different instances duringthe spreading and recoiling process of the droplet is presented in Figure 6.13. Colours ineach image represent the magnitude of the radial velocity component ur. Shapes obtainedfor the variant 1/βǫ = 10/r are also similar (which are not shown here) to these images. Itis interesting to note that the shape of the droplet at each instance in Figure 6.13 is samefor all these variants of the slip coefficient.

After the impact, a liquid sheet (lamella) ejects in the neighbourhood of the contactline. At the beginning, the thickness of the lamella is very small and then the thicknessincreases in time during the spreading and receding phases. At the maximum wettingdiameter the kinetic energy becomes very small and the capillary force becomes a majordriving force. Thus, the droplet starts to recoil in order to obtain an equilibrium shapewith minimum surface energy.

In order to study the influence of the slip coefficient quantatively, we calculate thewetting diameter, the dynamic contact angle and the height of the droplet at the axisof rotational symmetry. Figure 6.14 represents the wetting diameter as a function of

91

6 Numerical Results

(i) (ii) (iii)

Figure 6.13: A sequence of images obtained in different simulations of a water dropletimpinging on a wax surface with θe = 100, Re = 4204, We = 90 and Fr = 93 . (i) 1/βǫ = 1,(ii) 1/βǫ = 10, and (iii) 1/βǫ = 1/r. Timings from the top: t =0.125, 0.25, 0.5, 1.0, 1.5,2.0, 2.5, 3.0.

time. As we expected the wetting diameters for variants (i) and (ii) are slightly largerin comparison with the other two variants due to the less frictional force. The dynamic

92


0 2 4 60

1

2

3

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 100

1

2

3

time

wet

ting

diam

eter

i ii iii iv

Figure 6.14: Wetting diameter of a water droplet impinging on a wax surface with θe =100, Re = 4204, We = 90 and Fr = 93. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/r and(iv) 1/βǫ = 10/r. In experiments, Wdmax

is about 3.0.

0 4 8

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

0.5 1.5

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.15: Dynamic contact angle of a water droplet impinging on a wax surface withθe = 100, Re = 4204, We = 90 and Fr = 93. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/rand (iv) 1/βǫ = 10/r.

contact angles for these variants are presented in Figure 6.15. Here, the slip coefficientplays a significant role. For less frictional variants, after the impact the contact line movesvery fast and therefore the contact angle decreases below the equilibrium value and thenincreases above the equilibrium value. This advancing contact angle remains larger thanthe equilibrium value until the recoiling starts, and in the recoiling phase the contact anglelies below the equilibrium value, see Figure 6.15. However, in higher frictional variants, thedroplet spreads like a rolling motion, and thus the contact angle remains large at the initial

93

6 Numerical Results

0.01 0.1 1 10

0.2

0.6

1.0

time

y max

at r

=0

i ii iii iv

Figure 6.16: Height of the droplet along theaxis of symmetry, Re = 4204, We = 90 andFr = 93.

1/βǫ Wdmaxt at Wdmax

MFmax

1 3.31 2.15 0.80

10 3.09 2.28 0.86

1/r 3.29 2.15 0.75

10/r 3.18 2.39 0.84

Table 6.2: Maximum dimensionlesswetting diameter Wdmax

and timetaken to attain it for Re = 4204, We =90 and Fr = 93.

spreading phase and then rapidly decrease to an equilibrium value. From the Figure 6.15,we can observe that the receding contact angle is small for these variants in comparisonwith the other two ((i) and (ii)) variants.

The height of the droplet on the axis of symmetry is plotted in Figure 6.16. There isno influence of the slip coefficient on the height of the droplet at the axis of symmetry.The maximum wetting diameter and the time to attain the maximum value are presentedin Table 6.2. The maximum wetting diameter values of variants (ii) and (iv) are very closewith the experimental results, see Fig. 4.19 in [73]. But, the other two variant values arealso well within this range.

Glycerin droplet:

Next, we study the influence of the slip coefficient for a very viscous non-wetting liquiddroplet with the same four variants for βǫ. In this case, we use a glycerin droplet of diameterd0 = 2.4 × 10−3 m impinging on a wax surface with the velocity uimp = 2.941 m/s.The following material parameters are used: the density ρ = 1220 kg/m3, the dynamicviscosity µ = 0.116 N s/m2, and the surface tension σ = 0.063 N/m. This results inthe dimensionless numbers Re = 75, We = 402 and Fr = 368 with the charactersticlength L = d0. A sequence of images obtained for different variants of slip coefficient arepresented in Figure 6.17. Colours in each image represent the magnitude of the radialvelocity component ur. As in the previous water droplet case, the influence of the slipcoefficient on the flow dynamics of the considered glycerin droplet is very small. Due tothe high viscosity, the droplet spreads slowly and there is no lamella ejected as in the waterdroplet case. Numerically obtained wetting diameters for different variants are presentedin Figure 6.18 as a function of time. Since the glycerin droplet spreads almost like arolling motion and only the approximated free surface is available, there is no smooth

94


(i) (ii) (iii)

Figure 6.17: A sequence of images obtained in different simulations of a glycerin dropletimpinging on a wax surface with θe = 95, Re = 75, We = 402 and Fr = 368. (i) 1/βǫ = 1,(ii) 1/βǫ = 10, and (iii) 1/βǫ = 1/r.

behaviour of contact angle while spreading, see Figure 6.19. After reaching the maximumwetting diameter the contact angle rapidly decreases, and during the recoiling phase thecontact angle value is below the equilibrium value. The height of the droplet at the axis

95

6 Numerical Results

0 4 8 120

1

2

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 100

1

2

time

wet

ting

diam

eter

i ii iii iv

Figure 6.18: Wetting diameter of a glycerin droplet impinging on a solid surface withθe = 95, Re = 75, We = 402 and Fr = 368. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/rand (iv) 1/βǫ = 10/r. In experiments, Wdmax

is about 2.2.

0 4 8

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

0 1 2 3

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.19: Dynamic contact angle of a glycerin droplet impinging on a solid surface withθe = 95, Re = 75, We = 402 and Fr = 368. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 10/rand (iv) 1/βǫ = 1/r.

of symmetry is presented in Figure 6.20.

The maximum wetting diameter and the time taken to attain the maximum value arepresented in Table 6.3. The numerically obtained values for the glycerin droplet case arein very good agreement with the experimental results, see Figure 4.24 in [73].

These studies for the influence of the slip coefficient on the non-wetting liquid dropletsshow that the slip coefficient of order 1 works well for the considered test cases. It should benoted that according to our form, the slip coefficient βǫ is equal to ǫµρU , see the derivation ofour dimensionaless form. Furthermore, our numerical results in terms of maximal wetting

96


0.01 0.1 1 10

0.2

0.4

0.6

0.8

time

y max

at r

=0

i ii iii iv

Figure 6.20: Height of the droplet along theaxis of symmetry, Re = 75, We = 402 andFr = 368.


MFmax

1 2.18 2.69 0.14

10 2.10 2.81 0.29

1/r 2.23 2.65 0.13

10/r 2.11 2.73 0.27



diameter and the time taken to attain the maximum value are in very good agreement withthe experimental results.

Influence of βǫ on wetting liquid droplets

water droplet

We perform an array of computations with a small equilibrium contact angle θe = 10.Experimentally, this equilibrium contact angle is observed for a water droplet on a smoothglass surface [73]. We use the same set of parameters as used in the non-wetting waterdroplet case, that is, d0 = 2.7×10−3 m and uimp = 1.563 m/s. This results in the following

Wd/2 Wd/2

Figure 6.21: Shapes of a water droplet impinging on a smooth glass with 1/βǫ = 100,θe = 10, Re = 4204, We = 90, Fr = 93 at time t =0.25 and 0.5.

97

6 Numerical Results

dimensionless numbers Re = 4204, We = 90 and Fr = 93. The computationally obtaineddroplet shapes at time t = 0.25 and 0.5 with 1/βǫ = 100 are shown in Figure 6.21. Afterthe impact, a part of the liquid ejects near the contact line and the ejected liquid partadvances without wetting the solid surface. After some time, the computation stops dueto the unphysical and irregular shape of the ejected liquid part. Next, we consider the

Figure 6.22: A sequence of images obtained in different simulations of a water dropletimpinging on a smooth glass with θe = 10, Re = 4204, We = 90 and Fr = 93 for thevariant (i) 1/βǫ = 1. Timings from the top: t =0.25, 0.5, 1.0, 1.5, 2.5, 5.0, 7.5.

98


following four variants of slip coefficient: (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/r and(iv) 1/βǫ = 10/r. A sequence of images obtained in the computation for the variant (i) atdifferent instances during the deformation process is presented in Figure 6.22.

In this wetting droplet case, recoiling effect is not observed, instead after reaching themaximum wetting diameter value a small thickness of lamella deforms back to obtain theequilibrium shape. The images obtained in our computations for the other three variantsare also similar. This can be clearly seen from the Figure 6.23, which represent differentwetting diameters as a function of time for different variants of the slip coefficient. Thereis almost no difference in the trajectory of the wetting diameter for these slip coefficient

0 4 8 12

1

3

5

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 10

1

3

5

time

wet

ting

diam

eter

i ii iii iv

Figure 6.23: Wetting diameter of a water droplet impinging on a smooth glass with θe =10, Re = 4204, We = 90 and Fr = 93. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/r and(iv) 1/βǫ = 10/r.

4 8 12

20

80

140

200

time

dyna

mic

con

tact

ang

le i ii iii iv

1 2 3

20

80

140

200

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.24: Dynamic contact angle of a water droplet impinging on a smooth glass withθe = 10, Re = 4204, We = 90 and Fr = 93. (i) 1/βǫ = 1, (ii) 1/βǫ = 10, (iii) 1/βǫ = 1/rand (iv) 1/βǫ = 10/r.

99

6 Numerical Results

values. In all of these four variants, after the droplet impacting the surface, the dynamiccontact angle rapidly decreases to a minimum value about 30, see Figure 6.24. However,in variant (ii) and (iv) the contact angle slowly attains this minimum value in comparisonwith the other two variants due to the high frictional slip coefficient. The height of the

0.01 0.1 1 10

0.2

0.5

0.8

time

y max

at r

=0

i ii iii iv

Figure 6.25: Height of the droplet on the axisof symmetry, Re = 4204, We = 90 and Fr =93.


MFmax

1 4.75 10.88 0.27

10 4.98 9.52 0.43

1/r 4.85 11.02 0.27

10/r 4.62 10.75 0.42



droplet at the axis of symmetry as a function of time is presented in Figure 6.25 for all ofthese variants. As in the non-wetting test cases the slip coefficient has no influence on thisheight. The maximum dimensionless wetting diameter and the time taken to attain themaximum value are presented in Table 6.4.

This study for the influence of the slip coefficient on wetting liquid droplets shows thatthe slip coefficient in the order of 1 has no much influence on the flow dynamics. However,unphysical shapes (in comparison with experiments) of the droplet have been observed fora large large value of 1/βǫ. Furthermore, these results reflects the precise incorporation ofthe contact angle in the numerical scheme and the capability of our scheme to handle asmall equilibrium contact angle droplets.

6.3.3 Influence of impact velocity on the flow dynamics

Now, we study the influence of the impact velocity on the flow dynamics of a waterdroplet impinging on a wax surface. In this study, we consider a droplet of diameterd0 = 2.7 × 10−3 m impinging on the wax surface with different impact velocities. Thefollowing material parameters are used in this study: the density ρ = 1000 kg/m3, thedynamic viscosity µ = 0.001 N s/m2, the surface tension σ = 0.073 N/m, the equilib-rium contact angle θe = 100 and the slip coefficient 1/βǫ = 1/r. Four variants of im-pact velocities: (i) uimp = 0.824 m/s, (ii) uimp = 1.165 m/s, (iii) uimp = 1.563 m/s and(iv) uimp = 2.084 m/s are considered in this study. These impact velocities together withthe set of material parameters give the four variants:

100


(i) Re = 2216, We = 24, Fr = 26

(ii) Re = 3134, We = 50, Fr = 52

(iii) Re = 4204, We = 90, Fr = 93

(iv) Re = 5604, We = 160, Fr = 164

The dimensionless wetting diameters obtained from these four variants are presented inFigure 6.26 as a function of time. Note that the Weber number We depends quadraticallyand the Reynolds number Re linearly on U = uimp. Therefore, if the impact velocity

1 3 5 70

1

2

3

4

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 10

1

2

3

4

time

wet

ting

diam

eter

i ii iii iv

Figure 6.26: Influence of the impact velocity on the wetting diameter as a function of timefor a water droplet on a wax surface with θe = 100. (i) uimp = 0.824 m/s, (ii) uimp =1.165 m/s, (iii) uimp = 1.563 m/s and (iv) uimp = 2.084 m/s. In experiments, Wdmax

isabout 2.5 for (ii), 3.0 for (iii) and 3.8 for (iv).

increases, then the maximum wetting diameter is also expected to increase. This behaviouris clearly seen in Figure 6.26. Figure 6.27 represents different dynamic contact anglesobtained for these four variants. In all these variants, we obtained the general behaviourof the dynamic contact angle, that is, the advancing angle is larger than the equilibriumangle and the receding angle is smaller than the equilibrium angle, see Figure 6.27. Themaximum wetting diameter and the time take to attain it are tabulated in Table 6.5 forall these variants. These values are in very good agreement with the experimental results,see Figure 4.19 in [73].

These studies of the influence of the impact velocity show that when the impact velocityincreases the wetting diameter and the time taken to attain it are also increasing. Further,these studies show that our numerical scheme produce stable and physically acceptable so-lutions even for large Reynolds numbers.

101

6 Numerical Results

2 4 6

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.27: Dynamic contact angle for differ-ent impact velocities of a water droplet withθe = 100 on wax surface.

Re We Wdmaxt at Wdmax

2216 24 2.57 1.39

3134 50 2.85 1.81

4204 90 3.28 2.25

5604 160 3.87 2.53


and timetaken to attain for different impact ve-locities of a water droplet on wax.

6.3.4 Influence of the surface tension on the flow dynamics

Low Reynolds number

To study the influence of the surface tension on the flow dynamics we consider differentliquids (hypothetic) with a fixed Reynolds number Re = 2216 and Fr = 26. In this study,we consider the following four variants: (i) We = 10, (ii) We = 50, (iii) We = 100 and(iv) We = 200. The computationally obtained wetting diameter over a time for all ofthese variants are presented in Figure 6.28. For a large value of surface tension coefficientthe maximal wetting diameter is expected to become smaller. This behaviour is clearlydisplayed in Figure 6.28. Due to a large surface tension in variant (i), the deformation of the

2 4 6 80

1

2

3

4

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 10

1

2

3

4

time

wet

ting

diam

eter

i ii iii iv

Figure 6.28: Wetting diameter of a liquid droplet with θe = 100, 1/βǫ = 1, Re = 2216 andFr = 26. (i) We = 10 (ii) We = 50 (iii) We = 100 (iv) We = 200.

102


1 3 560

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.29: Dynamic contact angle of a liq-uid droplet with θe = 100, 1/βǫ = 1, Re =2216 and Fr = 26. (i) We = 10 (ii) We = 50(iii) We = 100 (iv) We = 200.

We Wdmaxt at Wdmax

MFmax

10 – – –

50 2.87 1.75 0.20

100 3.29 2.35 0.31

200 3.67 3.04 0.56

Table 6.6: Maximum wetting diameterWdmax

and time taken to attain it for aliquid droplet with θe = 100, 1/βǫ =1, Re = 2216 and Fr = 26.

droplet is much smaller. However, we observed a splitting effect at the top of the dropletin the computation. Thus, we stopped the computation, since the topological changes inthe droplet has not been considered in this study. The effect of surface tension on thedynamic contact angle is presented in Figure 6.29. A general behaviour of the contactangle hysteresis is observed in all of these variants. The maximum wetting diameter andthe time to attain it are tabulated in Table 6.6 for all these variants. We can observe thatthe time taken to attain the maximal wetting diameter is also increasing when the surfacetension decreases.

Large Reynolds number

Next, we study the influence of the surface tension on flow dynamics for the same variantsas in the previous study but with a large Reynolds number Re = 5604 and Froude numberFr = 164. Figure 6.30 represent wetting diameters obtained from different computationsfor theses four variants. As in the low Reynolds number case a splitting effect is observedin variant (i). The dynamic contact angle and the maximum wetting diameter for thesevariants are presented in Figure 6.31 and Table 6.7, respectively. The maximal wettingdiameter and the time to attain it for different cases are presented in Figure 6.32.

These studies for the influence of the surface tension with small and large Reynoldsnumbers show that when the surface tension increases the wetting diameter and the timetaken to attain it are decreasing. Furthermore, the influence of Reynolds number on thewetting diameter is small for small Weber numbers. But if the Weber number increases(i.e., decreasing the surface tension), the influence of the Reynolds number becomes large.Further the contact angle hysteresis is observed in all cases irrespective of impact velocityand surface tension.

103

6 Numerical Results

1 3 50

2

4

time

wet

ting

diam

eter

i ii iii iv

0.01 0.1 1 10

2

4

time

wet

ting

diam

eter

i ii iii iv

Figure 6.30: Wetting diameter of a liquid droplet with θe = 100, Re = 5604 and Fr = 164.(i) , We = 10 (ii) We = 50 , (iii) We = 100 (iv) We = 200.

1 3 560

100

140

180

time

dyna

mic

con

tact

ang

le i ii iii iv

Figure 6.31: Dynamic contact angle of a liq-uid droplet with θe = 100, Re = 5604 andFr = 164. (i) We = 10 (ii) We = 50(iii) We = 100 (iv) We = 200.

We Wdmaxt at Wdmax

MFmax

10 – – –

50 2.86 1.74 0.27

100 3.35 2.45 0.28

200 4.03 3.00 0.46

Table 6.7: Maximum wetting diameterWdmax

and time taken to attain it fora liquid droplet with θe = 100, Re =5604 and Fr = 164.

6.4 Liquid droplet impinging on a hot surface

In this section, an array of computations is performed for a liquid droplet impinging ona hot solid surface. Here, we study the heat transfer in the droplet and the influence ofthe heat transfer on the fluid flow. The main focus of this study is to illustrate that ournumerical scheme captures the behaviour of heat transfer effects on the flow dynamics.In our model, the surface tension is considered as a function of temperature. Furtherthe Marangoni convection is included in the model. In general, the temperature serves asensitive control on spreading by creating a counteract flows that slow down the spreading.

104

6.4 Liquid droplet impinging on a hot surface

40 120 200

2.6

3

3.4

3.8

4.2

We

max

Wd

i ii iii

40 120 200

1.5

2.5

3.5

We

t at m

ax W

d

i ii iii

Figure 6.32: Maximal wetting diameter (left) and the time to attain it (right) for differentcases. (i) Re with respect to We for water droplet, (ii) Fixed Re = 2216 for hypotheticliquid and (iii) Fixed Re = 5604 for hypothetic liquid.

In particular, the heat transfer prevents the zero equilibrium contact angle droplets spreadto infinity [21]. Since the temperature near the contact line region is large in comparisonwith the other part of the free surface, effects of the heat transfer in the fluid flow isexpected to be large in this region. In both test cases, we assume that Tw = 332.4 K,Ts = 373 K and T∞ = 283 K (temperature of the solid surface, the saturation and theatmospheric, respectively). Further in this initial study, we use C1 = 1 in the surfacetension temperature relation (2.6). Note that a stabilisation technique has to be used forthe scalar energy equation if 1/Pe ≪ 1 or the convection term ((u − w) · ∇u) is large.However, the focus of this work is to study the heat transfer effects on fluid flow, andtherefore we use small Peclet numbers to overcome the stability problem in the energyequation.

In the first test case, with the characterstic length L = r0, the following dimensionlessnumbers have been used : Re = 27, We = 24, Fr = 5, Pe = 27, Bi = 8.7 × 10−6

and 1/βǫ = 1000. Here, a large value of 1/βǫ is used to show that it works for dropletswith small wetting diameter droplets. The computationally obtained shape of the 3D-axisymmetric droplet at different instances without and with heat transfer effects is plottedin Figure 6.33 (i) and (ii), respectively. The isolines in (i) and (ii) represent the pressure andtemperature fields in the droplet, respectively. As we expected, effects of the temperaturenear the contact line region are large in comparison with other parts of the free surface.At later stage, a rim like structure is developed near the contact line and it counteractswith the spreading. This process slow down the wetting rate and reduce the maximalwetting diameter. These effects can be clearly seen in Figure 6.34, which represent thewetting diameter in time. However, the influence of the heat transfer on the kinetic energyis small.

We perform another set of computations with different sets of dimensionless numbers:

105

6 Numerical Results

Re = 53, We = 97, Fr = 20, Pe = 53 and Bi = 8.7 × 10−6. The computationally obtainedshapes of droplets for this test case at different instances without and with heat transfer arevisualised in Figure 6.33 (iii) and (iv), respectively. The isoline in (iii) and (iv) representthe pressure and temperature fields in the droplet, respectively. The wetting diameter andthe kinetic energy for this test case are presented in Figure 6.35. We observe a similareffect of the heat transfer as in the previous test case. Our numerical scheme captures themain feature in the non-isothermal spreading of liquid drops on a hot surface, which showsits applicability.

This numerical study illustrates that the heat transfer slow down the spreading rate and

case 1 (i) case 1 (ii) case 2 (i) case 2 (ii)

Figure 6.33: Sequence of images of liquid droplets with θe = 50 obtained in differentsimulations. (i) without heat transfer effect, (ii) with heat transfer effect. Isolines in (i)and (ii) represent the pressure and temperature fields, respectively.

106

6.5 Rising bubble

the maximum value of the wetting diameter, and further our numerical scheme capturesthis behaviour well. Furthermore, for a large value of 1/βǫ, unphysical effects have notbeen observed on the shape of the droplet with small wetting diameter. However, a largedifference in the advancing and receding contact angle is observed.

6.5 Rising bubble

In this section we present computational results for the rising bubble problem described insection 2.2.1. The shape of the bubble and a few interesting quantaties such as sphericity,rise velocity, drag force coefficient and kinetic energy are calculated in these computations.We compare the shape of the bubble obtained at different instances with the experimentallypredicted shapes.

Computations are performed for two different test cases. The initial geometrical con-figurations in both test cases are identical and are based on the benchmark parametersin [40]. However, the benchmark computations in [40] have been made for 2D planar caseand here we perform computations on 3D-axisymmetric configurations. We consider a half-meter radius, two-meter high cylinder, which contains a spherical bubble of radius 0.25 mwith centre (0.5, 0.5) as the initial geometrical configuration Ωt. Now, we reformulate theinitial Cartesian geometrical configuration into a 3D-axisymmetric cylindrical geometricalconfiguration. This results in a half-meter wide, two-meter high rectangle, which containsa half circle of radius 0.25 m with centre at (0.5, 0.5) as the initial computational (merid-ian) domain Φt, see Figure 6.1 for a similar configuration. On the artificial boundary atr = 0, the boundary conditions (5.35) and ur = 0 are imposed. Furthermore, we assumethat both the liquid and bubble are in rest at time t = 0, i.e., u0 = 0.

In the first test case, we use the following material parameters: the density ρ1 =

0 1 2 3

0.4

0.8

1.2

1.6

time

wet

ting

diam

eter

i ii

0 1 2 30

0.6

1.2

1.8

time

kine

tic e

nerg

y

i ii

Figure 6.34: Wetting diameter (left) and kinetic energy (right) of a liquid droplet impingingon a heated solid surface with Re = 27, We = 24, Fr = 5, Pe = 27, Bi = 8.7 × 10−6.(i) without heat transfer, (ii) with heat transfer.

107

6 Numerical Results

0 1 2 3 0

1

2

time

wet

ting

diam

eter

i ii

0 1 2 30

0.6

1.2

1.8

time

kine

tic e

nerg

y

i ii

Figure 6.35: Wetting diameter (left) and kinetic energy (right) of a liquid droplet impingingon a heated solid surface with Re = 53, We = 97, Fr = 20, Pe = 53 and Bi = 8.7 × 10−6.(i) without heat transfer, (ii) with heat transfer.

1000 kg/m3, ρ2 = 1 kg/m3, the dynamic viscosity µ1 = 10 N s/m2, µ2 = 0.1 N s/m2,the coefficient of surface tension σ = 19.6 N/m and a reduced gravitational constantg = 0.98 m/s2. Using the characterstic length scale L = 1 m and the characterstic velocityU =

√Lg, we get the dimensionless numbers Re = 99 and Eo = 500. Further in the

computation we have used the time step ∆t = 0.001. Note that in this test case we haveρ1/ρ2 = 1000 and µ1/µ2 = 100.

The computationally obtained shape of the rising bubble at different instances for thisset of parameters is plotted in Figure 6.36. Eventhough, the density ratio is very large

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

Figure 6.36: A sequence of images of the rising bubble at different instance for Re = 99and Eo = 500: t = 0.5, 1.0, 1.5, 2.0, 2.5 .

108

6.5 Rising bubble

Figure 6.37: Flow direction in the rising bubble at different instance for ρ1/ρ2=1000,µ1/µ2=100, Re = 99 and Eo = 500 (colours represent the axial velocity component uz).t = 0.5, 1.0, 1.5, 2.0, 2.5 .

in this test case, the deformation and the rise of the bubble are not so large because ofthe reduced gravitational constant and very high surface tension. The shape of the bubbleobtained from our computations is in good agreement with the experimental predictions[12] (Figure 2.5 on page number 27) for the shape regimes of bubbles. Note that in [12],the Reynolds and Eotvos numbers are defined in a different form and according to theirdefinition we get Re = 35 and Eo = 125 for this computed set of parameters. The flowfield in the fluid and the bubble is visualised in Figure 6.37. Colours in each image of theFigure 6.37 correspond to the axial velocity component uz and arrows indicate the flowdirection.

In the second test case, we consider the same initial geometrical configuration as inthe previous test case. The set of material parameters used in this second test case isthe density ρ1 = 1000 kg/m3, ρ2 = 10 kg/m3, the dynamic viscosity µ1 = 10 N s/m2,µ2 = 1 N s/m2, the coefficient of surface tension σ = 24.5 N/m and a reduced gravitationalconstant g = 0.98 m/s2. This set of parameters results in the following dimensionlessnumbers Re = 99 and Eo = 40. The computationally obtained shape of the bubble atdifferent instances for this set of parameters is plotted in Figure 6.38. The deformationof the bubble in this second test case is very small in comparison with the first test casebecause of a low density ratio and a very high surface tension. The shape of the bubblelies well in the experimentally predicted shape regime in [12]. Note that according to thedefinition in [12], we get Re = 12.38 and Eo = 2.48 for this set of parameters. The flowdirection and the magnitude of the axial velocity component uz in the flow are visualised

109

6 Numerical Results

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

0 0.5

0.4

1.2

2

Figure 6.38: Rising bubble droplet at different instances for ρ1/ρ2=100, µ1/µ2=10, Re = 99and Eo = 40, t = 0.5, 1.0, 1.5, 2.0, 2.5. t = 0.5, 1.0, 1.5, 2.0, 2.5 .

Figure 6.39: Flow direction in the rising bubble at different instance for ρ1/ρ2=100,µ1/µ2=10, Re = 99 and Eo = 40. t = 0.5, 1.0, 1.5, 2.0, 2.5 (colours represent theaxial velocity component uz).

110

6.5 Rising bubble

in Figure 6.39 with arrows and colours, respectively.Next, we calculate a few parameters such as sphericity, rise velocity, kinetic energy and

drag force coefficient for both test cases.

Sphericity

The sphericity of the bubble is calculated using

SΩ2,t=Ae

A=

surface area of volume-equivalent sphere

surface area of the bubble.

This implies that if the bubble is in spherical shape, then we get SΩt= 1 and if the bubble

deforms then, SΩtdecreases. In the 3D-axisymmetric configuration the volume and the

surface area of a spherical bubble are calculated from the meridian domain Φ2 using

surface area = 2π

∫

ΓF,Φ2

1 r ds, volume = 2π

∫

Φ2

1 r dR.

Rise velocity

The mean velocity in the bubble Ω2 can be defined as

UΩ2=

∫

Ω2

u dX

∫

Ω2

1 dX

=

U∞

∫

Ω2

u dX

∫

Ω2

1 dX

=

U∞

∫

Φ2

u r dS

∫

Φ2

1 r dS

.

This definition gives the mean velocity of both the radial (ur) and axial (uz) velocity com-ponents. In general, the velocity component, which is directed parallel to the gravitationalforce is referred to as the rise velocity. Thus, the rise velocity (after omitting the tilde) inthe bubble can be computed by

Urise =

U∞

∫

Φ2

uz r dS

∫

Φ2

1 r dS.

Further if the rise velocity reaches a stationary value then, this value is called a terminalvelocity.

The sphericity of the bubble obtained from both of these test cases is shown in Fig-ure 6.40 (left). The large deformation of the bubble in the first test case can be clearlyseen in this graph. The sphericity of the bubble in case 1 starts to reduce rapidly from theunity at the time about t = 0.5 s and reducing until the time about t = 2 s. However, thesphericity of the bubble in the second test case is slightly deviating from the unity aroundthe time region t = 1 s and remains a constant there after, i.e., the bubble rises without

111

6 Numerical Results

0 1 2 3

0.1

0.2

0.3

0.4

time

rise

velo

city

case 1case 2

0 1 2 3

0.7

0.8

0.9

1

time

sphe

ricity

case 1case 2

Figure 6.40: Sphericity (left) and rise velocity (right) of a rising bubble. Case (1) ρ1/ρ2 =1000, µ1/µ2=100, Re = 99 and Eo = 500, Case (2) ρ1/ρ2 = 100, µ1/µ2=10, Re = 99 andEo = 40.

deforming there after. This effect can also clearly seen in Figure 6.38. The computationallyobtained rise velocity in the bubble for both test cases is plotted in Figure 6.40 (right) as afunction of time. The rise velocity in case 1 is slightly larger in comparison with the case 2until it reaches the maximum rise velocity. Then, it starts to decrease when the bubbledeforms. In case 2 the rise velocity increases until the time about t = 0.75 s and it reacha stationary value, i.e., a terminal velocity. An interesting observation is that the risevelocity in case 1 is less than the rise velocity of the case 2 after the time about t = 0.8 s.

Drag force

The influence of the viscous and pressure forces on the bubble can be studied by studying

FΩ2=

∫

ΓF

ν · T1(u, p) dγ = 2πρ1U2∞

∫

ΓF,Φ

ν · S1(u, p) r ds.

Here, we used the viscosity from the outer liquid and the normal ν on the interface points di-rected from the bubble to the outer liquid. For the definition of dimensionless variables, seesection 2.2.2. The above definition of FΩ2

gives two force components in 3D-axisymmetricmeridian domain Φ2. In general the force in the major flow direction is called drag force.The major flow in the rising bubble problem is in the axial direction.

Hence, the drag force is defined as

DF,Ω2= 2πρ1U

2∞

∫

ΓF,Φ

1

R1

(

∂ur

∂z+∂uz

∂r

)

ν0 +

(

2

R1

∂uz

∂z− p

)

ν1

r ds

The drag force together with the rise velocity give the drag coefficient CD as

CD =2DF,Ω2

ρ1r0U2rise

.

112

6.5 Rising bubble

0 1 2 3

−5

−3

−1

time

drag

coe

ffici

ent

case 1case 2

1 2 3 0

0.1

0.2

0.3

time

kine

tic e

nerg

y

case 1case 2

Figure 6.41: Kinetic energy and drag force coefficient in a rising bubble. (1) ρ1/ρ2 = 1000,µ1/µ2=100, Re = 99 and Eo = 500, (2) ρ1/ρ2 = 100, µ1/µ2=10, Re = 99 and Eo = 40.

Further we calculate the kinetic energy using (6.1) with the characteristic velocity U∞.The kinetic energy in the bubble obtained from these computations is plotted in Fig-

ure 6.41 (left) as a function of time. In the case 1, the kinetic energy increases rapidly untilthe time of about t = 0.5 s and then decreases when the bubble starts to deform (whenthe sphericity decreases). In the case 2, the kinetic energy increases at the initial phaseto a local maximum value at about t = 0.8. Then, it slightly reduce during the sphericitydecreases and attain a stationary value. The drag force coefficient on the bubble in bothtest cases is presented in Figure 6.41 (right). A same value of drag force coefficient actson the bubble in both test cases until the time about t = 0.5. Once the bubble deforms incase 1 the drag force coefficient starts to decrease, whereas it remains a constant in case 2.

This study shows that the numerical scheme is capable of solving two phase flow prob-lems with large jumps in material parameters. Since we used interface resolved meshes,the surface force and the jumps in material parameters are handled very precisely. Themass fluctuation in these two test cases is about 0.0012%, which reflects the accuracy ofthe numerical scheme.

113

Chapter 7

Summary

The main aim of this work was to develop a robust and efficient numerical scheme forcomputing flows with free surfaces and interfaces. In particular, this scheme is used tosimulate the sequence of spreading and recoiling process of a liquid droplet impingingon a horizontal surface, and to study the flow dynamics of the droplet. The impingingdroplet problem exemplifies the problem of moving contact line, which is one of the hottesttopic in the field of computational physics. To illustrate the robustness of the developednumerical scheme, we applied the numerical algorithms in addition to the problem of afreely oscillating droplet, a droplet impinging on a hot solid surface and a rising bubble.

The fluid flow of all our considered problems is described by the time dependent incom-pressible Navier-Stokes equations in a time dependent domain with appropriate boundaryconditions. For the moving contact line problem, we have used the Navier-slip boundarycondition on the liquid-solid interface to avoid the kinematic paradox, which could occurif we use the usual no slip boundary condition. Further, the heat transfer is described bythe energy equation. In the rising bubble two-phase flow problem, we have derived theweak form in such a way that the fluid flows in both phases are described by a single setof equations with variable material parameters.

Our intention was to develop a finite element based numerical scheme for simulatingthese type of flows. Therefore, the basics of finite element methods, the construction ofa few finite elements and finite element spaces have been presented. To guarantee thestability and the accuracy, we preferred an inf-sup stable finite element pair of order two.Furthermore, the finite element pair should suppress spurious velocities on these interfaceflows. Thus, to study the effects of different finite discretisations on spurious velocities, wehave presented a static bubble test problem. Another ingredient of the numerical schemewhich is essential for these interface flows is an interface capturing/tracking technique. Wehave recalled a few available techniques. All of these techniques can be classified into twoclasses: (i) fixed grid (interface unresolved meshes), and (ii) moving grid (interface resolvedmeshes) methods. Several advantages and difficulties in each of these techniques have beendiscussed. Since the surface force, density and viscosity can be included more precicely ininterface resolved meshes, we have chosen a moving grid method. In particular, we usedthe Arbitrary Lagrangian Eulerian (ALE) approach, which avoids quick distortion of the

115

7 Summary

meshes, and thus avoids frequent remeshing. Furthermore, the surface tension and thelocal curvature play a significant role in free surface and interface flows. In our numericalscheme, we used the Laplace-Beltrami operator technique to handle the curvature term.An advantage of using this technique is that we can include the contact angle at the movingcontact line in the weak formulation of the model. Spatial and temporal discretisations formoving domains using finite elements and fractional step ϑ scheme have been presented.Further, an iteration of fixed point type to linearise the non-linear convection term inthe Navier-Stokes equations has been discussed. Construction of the ALE mapping anda few mesh moving techniques have been also discussed for the moving domain problems.The transformation of a weak formulation from the Cartesian coordinate system to thecylindrical coordinate system for 3D-axisymmetric problems has been presented.

The computational results for all these model problems are obtained on their respective3D-axisymmetric domains. The numerical studies on spurious velocities show that the dis-continuous pressure approximation suppress the spurious velocities and conserves the masswithout fluctuation. The computational results such as shapes, frequencies and dampingfactors of freely oscillating 3D-axisymmetric droplets are compared with the analytical andthree dimensional numerical results. An array of computations has been performed for theimpinging 3D-axisymmetric droplets, and the results are compared with the experimentresults. The proposed numerical scheme captures the key features such as pyramidal struc-tures, dry-out, contact angle hysteresis and shapes in computations of impinging droplets.In computations, a large value of 1/βǫ produces unphysical effects and difficulties, espe-cially for droplets with large wetting diameter. Although computations with a large valueof 1/βǫ produce physically acceptable solutions for droplets with small wetting diameter,a large difference in the advancing and receding contact angle has been observed. How-ever, the influence of 1/βǫ on the flow dynamics for droplets with a very large equilibriumcontact angle is very small. In these numerical studies, the results with the slip coefficientterm (1/βǫ) of order 1 fits well with the experiment results.

In the 3D-axisymmetric rising bubble problem, an array of computations is performedwith different sets of parameters, and the shapes are compared with the experimentallypredicted shape regimes. Furthermore, a few interesting parameters such as the sphericity,rise velocity and drag forces of the rising bubble have been computed.

The maximum mass fluctuation in each of these computations is computed, and is lessthan 1 %.

116

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