A Finite Element Model for Free Surface and Two Fluid Flows on Fixed Meshes

A Finite Element Model for Free Surface and Two Fluid Flows on Fixed Meses Herbert Coppola Owen Advisor: Ramon Codina Escola Tècnica Superior d Enginyers de Camins, Canals i Ports Universitat Politècnica de Catalunya April 2009

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A mi familia,

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ACTA DE QUALIFICACIÓ DE LA TESI DOCTORAL Reunit el tribunal integrat pels sota signants per jutjar la tesi doctoral: Títol de la tesi: A Finite Element Model for Free Surface and Two Fluid Flows on Fixed Meses Autor de la tesi: Angel Heriberto Coppola Owen Acorda atorgar la qualificació de: No apte Aprovat Notable Excel lent Excel lent Cum Laude Barcelona,... de/d... de... El President El Secretari...... (nom i cognoms) (nom i cognoms) El vocal El vocal El vocal......... (nom i cognoms) (nom i cognoms) (nom i cognoms)

Acknowledgments Ramon, no doubt I ave been lucky. Not only is e brilliant but also kind, fair and patient. Tank you Ramon! I would also like to tank te RMEE department of te Universitat Politècnica de Catalunuya and CIMNE. Specially, Eugenio Oñate, for giving me te opportunity of coming to Barcelona. Working wit Guillaume Houzeaux, Santi Badia, Oriol Guasc, Matias Avila, Cristian Muñoz Joan Baiges, Su-Ren Hysing, Noel Hernandez, and Javier Principe as been very enricing. Special tanks go to Javier wit wom I worked bot at Buenos Aires and Barcelona. He as elped me wit bot teoretical and practical matters. Wit Ramon, Guillaume, Javier and Mariano Vazquez we started te code I ave used in tis tesis. It as been very nice to do some team work. I ave also learned from my interaction wit Gerardo Valdez, Romain Aubry, Monica de Mier, Ricardo Rossi, Carlos Labra, Pooyan Dadvan and Vicente. Tanks to te rest of my office mates: Roberto, Jeovan, Pablo and Maritzabel. During my tesis I ave worked wit te people from Quantec in mould filling simulations. Special tanks go to Martin Solina wit wom I ave interacted most. Rainald Loner received me for a stay at George Mason University. I would like to tank im and is group: Fernando Camelli, Juan Cebral, Ci Yang, Fernando Mut, Marcelo Castro, Joaquin Arteaga, Romain Aubry and Orlando Soto. Cielo, Martin and te rest of te Colombian-Slovak group made our stay at USA more pleasant. Marcela Goldscmit and Eduardo Dvorkin introduced me to computational mecanics, allowed me to work wit tem at te Center for Industrial Researc (CINI) and motivated me to pursue doctoral studies abroad. I would like to tank tem and all 7

of my companions at CINI. Te financial support received from te Agència de Gestió d Ajuts Universitaris i de Recerca of te Generalitat de Catalunya (Catalan Government) and te European Social Fund troug a doctoral grant is acknowledged. Finally, I would like to tank my friends in Barcelona, Stevie, Riso, Vale, Bea, Daniel, Adrian, Sergio and Josep and te UPC rowing team wo ave elped me disconnect from te doctorate. I also would like to mention my friends from St. Jon s and te University of Buenos Aires tat, despite te distance, are always present. Te support of my family and te love of wife, Liliana, ave elped me during te good and bad times.

Abstract Flows wit moving interfaces (free surface and two-fluid interface problems) appear in numerous engineering applications. Te metods presented in tis tesis are oriented mainly to te simulation of mould filling process. Neverteless te metodology is sufficiently general as to be applied to most free surface and two-fluid interface flows. Numerical modeling provides an efficient way of analyzing te pysical penomena tat occur during casting and injection processes. It gives insigt into details of te flow tat would oterwise be difficult to observe. A fixed mes finite element metod, were te interface position is captured by te Level Set function, is used. Low Froude number flows are particularly callenging for fixed grid metods. An accurate representation is needed in te elements cut by te interface for suc flows. Two alternatives are proposed. Te first alternative is to use te typical two-pase flow model enricing te pressure sape functions so tat te discontinuity in te pressure gradient at te interface can be better approximated. Te improvement in te representation of te pressure gradient is sown to be te key to ingredient for te successful modeling of suc flows. Te influence of te second fluid can be neglected on a wide range of applications to end up wit a free surface model tat is simpler tan te two-pase flow model. Te discontinuity in te pressure gradient disappears because only one fluid is simulated. Te particularity of tis second approac is tat a fixed mes is used. Boundary conditions are applied accurately using enanced integration and integrating only in te filled part of cut elements. A fixed mes ALE approac is developed to correctly take into account tat te domain is moving despite a fixed mes is used. 9

10 Pressure segregation metods are explored as an alternative to te monolitic discretization of te Navier Stokes equations. Tey uncouple te velocity and pressure unknowns, leading to smaller and better conditioned subproblems. Pressure correction and velocity correction metods are presented and compared numerically. Using a discrete Laplacian a numerically stable tird order velocity correction metod is obtained. Te metods are applied to tree dimensional mould filling problems borrowed directly from te foundry wit very satisfactory results. Te free surface monolitic model turns out to be te most robust and efficient option. Te comparison wit a commercial code sows te accuracy and efficiency of te metod we propose.

Contents 1 Introduction and basic model 15 1.1 Introduction................................... 15 1.1.1 Classification of metods for flows wit interfaces.......... 16 1.1.2 Organization.............................. 19 1.1.3 Notation Issues............................. 21 1.2 Two fluid Navier Stokes equations...................... 23 1.2.1 Te (one fluid) Navier Stokes equations............... 23 1.2.2 Te two fluid Navier Stokes equations................ 25 1.2.3 Basic discretized problem....................... 28 1.2.4 Stabilized problem........................... 31 1.2.5 Matrix version of te problem..................... 38 1.2.6 Material properties approximation................... 39 1.2.7 Mixed boundary conditions on curved walls............. 41 1.3 Te Level Set equation............................. 45 1.3.1 Interface Capturing Tecniques.................... 45 1.3.2 Implementation of te level set metod................ 47 1.3.3 Reinitialization............................. 49 1.3.4 Coupling between te flow equations and te Level Set....... 50 2 An enriced pressure two-pase flow model 53 2.1 Discontinuous Gradient Pressure Sape Functions.............. 54 11

12 CONTENTS 2.2 Numerical Examples.............................. 59 2.2.1 Two fluid cavity............................ 60 2.2.2 3D vertical cannel........................... 65 2.2.3 Slosing problem............................ 67 2.3 Conclusions................................... 68 3 A free surface model 73 3.1 ALE description of te Navier Stokes equations............... 75 3.2 FM-ALE free surface model.......................... 77 3.3 Eulerian simplified free surface model..................... 83 3.4 Numerical examples............................... 86 3.4.1 Two fluid cavity............................ 86 3.4.2 3D vertical cannel........................... 90 3.4.3 Slosing problem............................ 91 3.5 Two computationally demanding examples.................. 92 3.5.1 3D dam-break wave interacting wit a circular cylinder....... 93 3.5.2 3D Green water problem........................ 94 3.6 Conclusions................................... 96 4 Pressure Segregation Metods 101 4.1 Pressure correction metods.......................... 103 4.1.1 Fractional Step (non Predictor Corrector) scemes.......... 103 4.1.2 Predictor Corrector sceme...................... 105 4.2 Te Pressure Scur Complement approac.................. 108 4.3 Velocity correction metods.......................... 116 4.3.1 Te Discrete Pressure Poison Equation................ 117 4.3.2 Approximation of DM 1 G....................... 118 4.3.3 Fractional step sceme......................... 120 4.3.4 Predictor corrector sceme....................... 122 4.3.5 Stabilized Sceme............................ 124

CONTENTS 13 4.3.6 Remarks on te ASGS and non split OSS stabilized cases...... 125 4.4 Open boundary conditions........................... 127 4.5 Numerical examples............................... 129 4.5.1 Driven Cavity.............................. 130 4.5.2 Flow beind a cylinder......................... 148 4.5.3 Convergence test............................ 156 4.5.4 Results wit te rotational form.................... 161 4.6 Conclusions................................... 165 5 Mould Filling 171 5.1 Introduction................................... 171 5.2 Free surface monolitic model......................... 176 5.2.1 Hollow mecanical piece........................ 177 5.2.2 Alloy weel............................... 183 5.2.3 Sovel.................................. 185 5.2.4 Results wit te FM-ALE model................... 190 5.3 Free surface velocity correction model..................... 191 5.4 Enriced pressure two pase flow monolitic model............. 194 5.4.1 Hollow mecanical piece........................ 195 5.4.2 Weel.................................. 202 5.4.3 Sovel.................................. 205 5.5 Enriced pressure two pase flow velocity correction model......... 209 5.5.1 Hollow mecanical piece........................ 209 5.5.2 Weel.................................. 210 5.5.3 Sovel.................................. 212 5.6 Conclusions................................... 212 6 Conclusions 217 6.1 Acievements.................................. 217 6.2 Open lines of researc............................. 220

14 CONTENTS

Capter 1 Introduction and basic model 1.1 Introduction Flows wit moving interfaces (free surface and two fluid interface problems) appear in numerous engineering applications. Te numerical simulation of interface flows can be a great ally in te understanding and improvement of suc applications. Te great number of publications on te subject is te best evidence of te interest on te subject. Te fields of application are as wide as can be observed from te following examples: drop formation in ink-jet devices [104], sip ydrodynamics [76 78, 88] and mould filling [31, 80, 97]. Te metods presented in tis tesis will be oriented mainly to te simulation of mould filling processes. Neverteless te metodology is sufficiently general as to be applied to most free surface and two fluid interface flows. Numerical modeling provides an efficient way of analyzing te pysical penomena tat occur during casting and injection processes. It gives insigt into details of te flow tat would oterwise be difficult to observe. Wen coupled wit te appropriate models, it can also provide information about eat transfer and solidification. Te numerical results can elp sorten te design process and optimize casting parameters to improve te castings, reduce scrap and use less energy. In tis tesis we will deal wit bot free surface and two-fluid interface problems. Te 15

16 CHAPTER 1. INTRODUCTION AND BASIC MODEL former are a special case of te latter were te influence of one of te fluids on te oter one is negligible. In most casting applications te free surface model can be used because one is only interested in te beavior of te fluid and te influence of te air is negligible. Obviously free surface flows can be modeled as two fluid flows were te properties of one of te fluids are muc smaller tan tose of te oter one. Special models tat take into account te particularities of free surface flows can also be developed (see Capter 3). Te term interface flows refers to bot free surface and two fluid interface flows. Te objective of tis tesis is to develop or improve tecniques tat can be used in finite element mould filling software. Te range of numerical metods available for interfaceflowsisaswideasterangeofapplications. As in most CFD applications, several spatial discretization metods can be used, among tem: finite differences, finite elements, finite volumes and even mesless metods. On te oter and te existence of a moving interface gives raise to a uge number of metods to deal wit suc flows. In te next subsection we will present a brief classification of te most relevant ones. Te first and peraps te most significant classification depends on te nature, fixed or moving, of te grid used. In tis tesis we will use a fixed mes approac, in particular te Level Set metod. Bot te discretization metod and te fixed grid approac were selections made prior to te beginning of tis tesis. Te coice of te best discretization metod is a problem dependent question tat we do not intend to answer in tis tesis. Te classification of te different moving interface metods presented in te next subsection intends to clarify were we stand and sow some of te alternatives we could ave, someting we ope will be useful for te reader tat steps into te subject. On te oter and, we ope tat it can sow tat te metodology we will work wit is a pretty reasonable coice. 1.1.1 Classification of metods for flows wit interfaces Te classification of te metods used for free surface and two fluid flows is not an easy task mainly because of te wide range of scemes tat exist. Some interesting

1.1. INTRODUCTION 17 classifications and comparisons can be found in [70, 103, 109, 112, 116] As we ave already mentioned one of te classifications depends on te nature, fixed or moving, of te grid used. Anoter common option is to classify metods into interface tracking and interface capturing [70]. In tracking scemes te position of discrete points x i lying on te interface is tracked for all time by integrating te evolution equation dx i dt = u i were u i is te velocity wit wic interface point x i moves Moving mes metods are interface tracking scemes were te points i correspond to nodes placed on te interface. In capturing metods, te interface is not explicitly tracked, but rater captured using some interface function (ψ) defined over te wole mes tat allows to determine wic fluid occupies any point in te domain. Te evolution equation for te interface function is given by ψ +(u ) ψ =0. t Te tird classification would separate metods into Eulerian ones wic solve te Navier Stokes equations on fixed grids and Lagrangian (or Arbitrary Eulerian Lagrangian, ALE) ones wic solve tem on a grid tat follows (or partially follows) te caracteristics of te flow. In order to try to unify te tree previous classifications one could speak about moving mes, interface tracking or Lagrangian scemes and fixed mes, interface capturing or Eulerian ones. Despite tis migt seem te most natural way of unifying te previous classifications, tere are some metods tat would not fit properly into suc unification and could be considered as an exception to te rule. In te pursuit for better metods it is not uncommon to see autors tat try to blend components from te two main class of metods we ave defined. For example Front Tracking Metods [112] wic ave teir roots in te MAC metod of Harlow and Welc [54] are, as teir name indicates, tracking scemes but tey use a fixed mes to model te flow. In Capter 3 we will present a model for free surface flow tat uses a ALE approac on a fixed mes and tus, would on one and be classified into te Lagrangian group and on te oter into te fixed mes

18 CHAPTER 1. INTRODUCTION AND BASIC MODEL group. Neglecting some particular scemes, te unified classification we ave presented can be considered valid for most cases. In most interface capturing tecniques a fixed computational domain is used and an interface function is used to capture te position of te interface. Te interface is captured witin te resolution of te fixed mes and te boundary conditions at te interface are someow approximated. In most interface tracking tecniques te mes is updated in order to track te interface. Te simplest approac is to deform te mes witout canging its topology, but it is valid only for very simple flows. As te flow becomes more complex and unsteady remesing and consequently te projection of te results from te old to te new mes are needed [3, 64, 71, 87]. For te same mes size moving grid tecniques lead to a more accurate representation of te interface at a iger computational cost. In Capter 2 we will use a fixed grid metod and introduce modifications to te basic formulation to enance te representation of te flow at te interface. Te idea of enricing te representation of an unknown at a material discontinuity is not new and several approaces can be found in te literature [19, 81]. Fixed mes metods generally sare two basic steps, one were te motion in bot pases is found as te solution of te Navier Stokes equations wit variable properties and te oter one, were an equation for an interface function tat allows to determine te position of te interface, and tus te properties to be assigned in te previous step, is solved. Te different metods differ mainly in te metod used to determine te position of te interface but also differences can be found in te way to approximate te properties to be used close to te interface. In Section 2 we will deal wit te first step and in Section 3 wit te second one. A mentioned previously, we capture te interface using te so called level set metod (see [18,106] and [89,90,104] for an overview), also called pseudo concentration tecnique [110] and very similar to te volume of fluid (VOF) tecnique [57,79]. Tis formulation as been widely used to track free surfaces in mould filling (see for example [31,73,80,94,97],

1.1. INTRODUCTION 19 among oter references) and oter metal forming processes. 1.1.2 Organization Tis tesis is organized as follows. Te present Capter presents an introduction to te numerical simulation of free surface and two fluid interface flows and te basic model used to simulate interface flows on fixed meses. Also in te next subsection some preliminary or notation issues will be included. Te next Section deals wit te solution of te Navier Stokes equations for flows wit interfaces. First te equations to be solved are presented. Ten teir space and time discretization is described. Finally two stabilization tecniques tat allow us to model flows wit important convective effects and also enable te use of equal order finite element interpolations for te velocity and pressure are introduced. A Monolitic or Mixed discretization is used. Te tird Section deals wit te Level Set Metod used to determine te position of te front. Te relations wit some of te oter most popular interface capturing tecniques, pseudo-concentration and VOF, are analyzed. Te space and time discretization of te Level Set equation is undertaken and te problem is stabilized. Finally some tecnical issues suc a reinitialization and calculation of extension velocities are briefly discussed. Tis first Capters describes te basic elements of a typical Finite Element Level Set model for interface flows. Te next two Capters present developments we propose to improve te simulation of two pase flows. Tese developments gain special importance in te simulation of low Froude number flows, tat is, wen te gravitational forces are bigger tan te inertial ones. Someting we would like to remark is tat our improvements are focused on te modelling of te Navier Stokes equations and not on te step tat deals wit te Level Set equation. Te poor beavior tat can be observed using te typical model for low Froude number flows and te degree of improvement we ave obtained justify suc coice. Strangely, specially in te level set community, muc more attention

20 CHAPTER 1. INTRODUCTION AND BASIC MODEL is paid to te solution of te Level Set equation tan to te solution of te two fluid Navier Stokes equations. Capter 2 presents our first original contribution [36,37], a way of improving Eulerian two-pase flow finite element approximation wit discontinuous gradient pressure sape functions. Capter 3 presents our second important contribution [29, 38], anoter way of improving interface flows tat is applicable only to free surface flows. An ALE formulation is used but te mes remains fixed (FM-ALE). Taking into account tat te objective of our researc is to be used efficiently in finite element mould filling software, we try to concentrate on tose items we find inder our objective most. Te most relevant one is te size of te problems we can andle and te efficiency wit wic we can tackle tem. Te step tat solves te Navier-Stokes equations is by far more computationally expensive tan te one tat solves te Level Set equations and is terefore te one we wis to improve. Wen solving te Navier- Stokes equations te most expensive step, specially as te size of te problem grows, is te solution of te resulting linear system. Two types of solvers are available, direct and iterative. Te former ave te advantage tat te obtention of a solution is guaranteed after a fixed number of steps tat does not depend on te condition of te matrix of te system to be solved. Te latter, despite teir convergence is not guaranteed in practical situations, as it depends on te condition number of te matrix to be solved, ave te advantage tat te computational cost increases muc more slowly tan tat of direct solvers as te size of te system to be solved increases. Terefore tey are te undisputed option as te size of te systems to be solved grows, as appens in industrial 3D mould filling simulations. Our initial experience wit iterative solvers was not very satisfactory and most of our problems were solved using direct solvers. Despite we implemented and tested a modern sparse direct solver called MUMPS [1, 2] tat brougt about significant improvements wit respect to our previous direct solver, it is still too expensive for real industrial problems. After making some furter experience wit iterative solvers and teir preconditioners we ave obtained muc better results as sown in Capter 5. In Capter 4 we explore pressure segregation metods [4] (also known as Fractional

1.1. INTRODUCTION 21 Step metods) for te Navier Stokes equations. Since teir appearance in te late 1960 s, wit te pioneering works of Corin [20] and Teman [107], tese metods ave enjoyed widespread popularity. Teir common feature is te decoupling of te velocity and pressure interpolation. Suc uncoupling yields an important computational cost reduction, on one and because te systems are uncoupled, and on te oter, peraps te key one, because eac of te resulting systems are better conditioned tan te one resulting from te monolitic system. Bot pressure correction and velocity correction metods (a more recent option) will be explored. Besides, te predictor corrector versions will also be tested. Predictor corrector metods also decouple te solution of te velocity and pressure, but tey iterate until convergence so as to recover te monolitic solution. Fractional Step metods can be seen as a predictor corrector sceme tat is only allowed to iterate once. Te coice between monolitic or pressure segregation scemes is, up to wat we understand, an open question and tere are important researc groups tat stick to one or te oter formulation. Obviously it is also a problem dependent question. Our intention is to build some solid knowledge on wic to base our selection (for te problems we are interested in) resorting mainly to numerical experimentation. In Capter 5 we apply te tools developed in te previous Capters to mould filling problems. Te free surface model is compared against te enriced pressure two pase model. Te results obtained wit te monolitic sceme are compared against te ones obtained wit te velocity correction sceme. Moreover te results are compared against te ones obtained wit a commercial code. Te advantages introduced by te two models we propose are clearly noticeable on low Froude number flows. 1.1.3 Notation Issues Functional Spaces In order to introduce te notation to be used in tis work, a brief summary of some concepts on functional analysis will be presented. For a more detailed presentation any standard text on te subject can be consulted [85]

22 CHAPTER 1. INTRODUCTION AND BASIC MODEL Let Ω R d, d = 2 or 3, be a bounded domain. C 0 (Ω) is te set of infinitely differentiable real functions wit compact support on Ω. L p (Ω), 1 p< is te space of real functions defined on Ω wit p-t power absolutely integrable wit respect to te Lebesgue measure. It is a Banac space wit te associated norm ( 1/p u L p (Ω) := u (x) dω) p. L 2 is of special interest since it is a Hilbert space endowed wit te scalar product (u, v) Ω = u (x) v (x)dω and te norm u L 2 (Ω) Ω Ω := (u, u)1/2 Ω. Te Sobolev space W m,p (Ω) is te space of functions in L p (Ω) wose weak derivatives of order less tan or equal m belong to L p (Ω), being m an integer and 1 p<. Wen p =2,tespaceW m,2 (Ω) = H m (Ω) is a Hilbert space endowed wit a scalar product and a norm. For example, for m = 1 te scalar product is and te norm is ((u, v)) Ω =(u, v) Ω + u H 1 (Ω) d ( i u, i v) i=1 := ((u, u))1/2 Ω. Te d-dimensional vector functions wit components in one of te previous spaces will be denoted by boldface letters, for example L 2 (Ω) = (L 2 (Ω)) d. Time discretization In order to try to unify te notation we will introduce ere te key concepts on time discretization to be used in tis work. Considering a uniform partition of te time interval of size δt, and denoting by f n an approximation to a time dependent function f at time t n = nδt, for a parameter θ [0, 1], we will denote f n+θ = θf n+1 +(1 θ) f n,

1.2. TWO FLUID NAVIER STOKES EQUATIONS 23 Let us also define δf n+1 = δ (1) f n+1 = f n+1 f n, δ (i+1) f n+1 = δ (i) f n+1 δ (i) f n,i=1, 2, 3,... D t ( ) = δ ( ) δt. Te discrete operators δ (i+1) are centered. We will also use backward difference operators ) D k f n+1 = 1 k 1 (f n+1 αk i γ f n i, k i=0 D 1 f n+1 = δf n+1 = f n+1 f n, D 2 f n+1 = 3 (f n+1 43 2 f n + 13 ) f n 1, as well as te backward extrapolation operators f n+1 i = f n+1 δ (i) f n+1 = f n+1 O ( δt i), f n+1 1 = f n, f n+1 2 =2f n f n 1. 1.2 Two fluid Navier Stokes equations 1.2.1 Te (one fluid) Navier Stokes equations Before introducing te two fluid Incompressible Navier Stokes equations, te typical one fluid version will be presented, as it te starting point from wic te former are derived. Te Navier Stokes equations are te basic equations of fluid mecanics for incompressible flow and can be derived from te continuum mecanics conservation laws, see for example [8]. Te Navier Stokes equations, using an Eulerian description, for a fluid moving in te open domain Ω bounded by Γ = Ω during te time interval (t 0,t f ) consist in finding a

24 CHAPTER 1. INTRODUCTION AND BASIC MODEL velocity u and a pressure p suc tat [ ] ρ t u +(u )u σ = f in Ω (t 0,t f ), (1.1) u =0 inω (t 0,t f ), (1.2) were ρ is te density, σ te stress tensor and f te vector external body forces, wic includes te gravity force ρg and buoyancy forces, if required. equation for a Newtonian and isotropic fluid Using te constitutive σ = pi +2µε (u) were µ is te dynamic viscosity, I is te identity tensor and ε( ) te symmetric gradient operator, te momentum equation (1.1) can be rewritten in one of its usual forms [ ] ρ t u +(u )u [2µε(u)] + p = f in Ω (t 0, t f ), (1.3) wic we will call divergence form. For a constant µ and using te incompressibility constraint imposed by te continuity equation (1.2) te most usual form [ ] ρ t u +(u )u µ u + p = f in Ω (t 0, t f ), wic we will call Laplacian form, can be obtained. Denoting by an over-bar prescribed values, te boundary conditions to be considered are: u = u on Γ du (t 0,t f ), n σ = t on Γ nu (t 0,t f ), u n =0, n σ g 1 = t 1, n σ g 2 = t 2 on Γ mu (t 0,t f ), (1.4) were n is te unit outward normal to te boundary Ω and vectors g 1 and g 2 (for te tree-dimensional case) span te space tangent to Γ mu. Observe tat Γ du is te part of te boundary wit Diriclet velocity conditions, Γ nu te part wit Neumann conditions (prescribed stress) and Γ mu te part wit mixed conditions. Tese tree parts do not intersect and are a partition of te wole boundary Ω. Initial conditions u = u 0 in Ω {t 0 },

1.2. TWO FLUID NAVIER STOKES EQUATIONS 25 ave to be appended to te problem. In order to obtain te weak or variational formulation of te Navier Stokes equations written in divergence form ((1.3) and (1.2)) we introduce te spaces V 0 { v H 1 (Ω) v = 0 on Γ du, v n =0onΓ mu }, V { v H 1 (Ω) v = u on Γ du, v n =0onΓ mu }, V t L 2 (t 0,t f ; V ), L 2 (Ω) if Γ nu Q L 2 (Ω) /R if Γ nu = Q t L 1 (t 0,t f ; Q) Te weak form is ten obtained by multiplying eac of te momentum equations (1.3) by an arbitrary element of V 0,v, and te continuity equation (1.2) by an arbitrary element of Q, q, and integrating te term corresponding to te stress tensor by parts. Te weak form of problem (1.3, 1.2) wit te boundary conditions we ave just defined is: Find u V t, p Q t suc tat Ω ρ Ω t u v dω + ρ p v dω = ρ for all (v,q) V 0 Q. Ω Ω [(u ) u] v dω + 2 µε (u) :ε (v) dω Ω t v dγ + (t 1 g 1 + t 2 g 2 ) v dγ Γ nu Γ mu q u dω = 0 f v dω + Ω 1.2.2 Te two fluid Navier Stokes equations Te two fluid Navier Stokes equations on a domain Ω = Ω 1 Ω 2 separated by a moving interface Γ int can be obtained starting from te Navier Stokes equations defined on eac domain [18], written in divergence form ρ 1 [ t u 1 +(u 1 )u 1 ] σ 1 = f 1 in Ω 1 (t 0,t f ),

26 CHAPTER 1. INTRODUCTION AND BASIC MODEL u 1 =0 inω 1 (t 0,t f ), and ρ 2 [ t u 2 +(u 2 )u 2 ] σ 2 = f 2 in Ω 2 (t 0,t f ), u 2 =0 inω 2 (t 0,t f ), In order to simplify te presentation we will suppose tat te only Neumann boundary in bot Ω 1 and Ω 2 corresponds to te interface and tat Γ mu =. Te boundary conditions at te interface are obtained as follows. Since te flow is viscous u 1 = u 2 on Γ int. On te oter and, te balance of surface forces on te interface gives (σ 1 σ 2 ) n = k κ n on Γ int (1.5) were te term on te rigt and side models te surface tension; k is a constant coefficient tat depends on te two fluids in contact ( usually σ is used in te literature but we ave used k to avoid confusions), κ is te local curvature of te interface and n is te normal pointing towards te positive curvature region. In mould filling simulations te effects of surface tension are usually negligible and terefore will not be taken into account in tis tesis (k = 0). Neverteless tey will be included in te derivation of te two fluid Navier Stokes equations. Using te same functional spaces as in te one fluid case, te momentum equations corresponding to eac of te two fluids are multiplied by an arbitrary element of V 0, v, and integrated over teir corresponding domains, te term corresponding to te stress tensor is integrated by parts, and te variational formulations corresponding to eac of te two fluids are added. Te same procedure is followed for te continuity equation using an arbitrary element of Q, q. Finally, defining u, p,ρ,µ,f, σ = u 1 p 1,ρ 1,µ 1, f 1, σ 1 x Ω 1, u 2 p 2,ρ 2,µ 2, f 2, σ 2 x Ω 2, (1.6)

1.2. TWO FLUID NAVIER STOKES EQUATIONS 27 te unified variational formulation is obtained. Find u V t, p Q t suc tat t ρu v dω + ρ [(u ) u] v dω + σ (u) :ε (v) dω (1.7) Ω Ω Ω = f v dω + [(σ 1 σ 2 ) n] v dγ v V 0 Ω Γ int q u dω = 0 q Q. Ω Using te equation for te interfacial forces, (1.5), we finally obtain t ρu v dω + ρ [(u ) u] v dω + σ (u) :ε (v) dω Ω Ω Ω = f v dω + k κ n v dγ v V 0 Ω Γ int q u dω = 0 q Q. In equation (1.7) we ave obtained te term Ω [(σ 1 σ 2 ) n] v dγ Γ int because we started from te divergence form of te equations. Tis is desirable since, as we ave already said, its value is given by (1.5). If te Laplacian form ad been used, te integral on te interface would ave been replaced by {[( p 1 I + µ 1 u 1 ) ( p 2 I + µ 2 u 2 )] n} v dγ Γ int wic is not related to te balance of surface forces on te interface. divergence form will always be used in tis tesis, unless oterwise indicated. Terefore te Before continuing wit te discretization of te equations some dimensionless numbers can be presented. Te Froude number represents te relation between te inertial and gravitational forces and is defined by Fr = U 2 gl were g is te gravity acceleration, U is a caracteristic velocity and L a caracteristic lengt. Two Reynolds numbers can be defined Re 1 = ULρ 1 µ 1

28 CHAPTER 1. INTRODUCTION AND BASIC MODEL Re 2 = ULρ 1. µ 2 If surface tension is taken into account, te Weber number is defined as were k as been defined in (1.5). We = U 2 Lρ 1 k 1.2.3 Basic discretized problem In tis subsection we will introduce te space and time discretization of te weak two fluid NS equations. Also te linearization of te convective term will be described. Te linearization of te convective term can be performed at te continuous or variational level indistinctly. as follows: Te approximation we will use is well known and reads [(u ) u] i+1 ( u i ) u i+1 + β ( u i+1 ) u i β ( u i ) u i were i is te iteration counter and β can be zero or one. Wen β = 0 te metod is known as Picard linearization and wen β = 1 it is known as Newton-Rapson linearization. Te former is simpler and as te advantage tat it can be sown to converge linearly if te convection is not too ig. For te latter, a quadratic convergence can be proved but only if te initial guess is close enoug to te exact solution [21]. Terefore, te typical numerical strategy is to first solve some Picard iterations to take advantage of its robustness and ten to switc to te Newton-Rapson metod for improved convergence. Regarding te time discretization of problem (1.7) two options will be presented. Te first one is te generalized trapezoidal rule, wic gives place to te following problem: given u n, find u n+1 V and p n+1 Q suc tat ρ n+θ un+θ u n v dω + ρ [( n+θ u n+θ ) u n+θ] v dω Ω θδt Ω +2 µ n+θ ε ( u n+θ) : ε (v) dω p n+θ v dω Ω Ω ( ) = f n+θ v dω + t n+θ v dγ + t n+θ 1 g 1 + t n+θ 2 g 2 v dγ Ω Γ nu Γ nu

1.2. TWO FLUID NAVIER STOKES EQUATIONS 29 Ω q u n+θ dω = 0 for all (v,q) V 0 Q. Once te algoritm as produced a solution at t n+θ, te velocity field at t n+1 can be updated from te velocity at t n+θ by using te relation u n+1 = [u n+θ (1 θ)u n ]/θ. Te force term f n+θ in te momentum equation as to be understood as te time average in te interval [t n,t n+1 ], even toug we use te superscript n+θ to caracterize it. Te same applies for t n+θ, t n+θ 1 and t n+θ 2. Te pressure value as been identified as te pressure at t n+θ, altoug tis is irrelevant for te velocity approximation. Te values of interest of θ are θ = 1/2, tat corresponds to te second order Crank-Nicolson sceme and θ = 1, tat corresponds to te backward Euler metod. Te second option is to use backward differencing (BDF) time integration scemes using te discrete operators defined in Section 1.1. Te time discretized problem ten reads: given u n, find u n+1 V and p n+1 Q suc tat ρ n+1 Ω δt D ku n+1 v dω + ρ [( n+1 u n+1 ) u n+1] v dω Ω +2 µ n+1 ε ( u n+1) : ε (v) dω p n+1 v dω Ω Ω ( ) = f n+1 v dω + t n+1 v dγ + t n+1 1 g 1 + t n+1 2 g 2 v dγ Ω Γ nu Γ nu q u n+1 dω = 0 for all Ω (v,q) V 0 Q. Te first order versions of bot metods coincide. For te second order time discretizations te benefits of eac of te metods are subtle for te one fluid NS equations. For te two fluid case we prefer te BDF sceme. Since te fluid properties (ρ and µ) at a given point vary in time, as defined in (1.6), it is muc better to use te properties at time t n+1 wic can be obtained from te level set function (wose value is known at t n+1 ) tan tose at time t n+θ wic need to be someow approximated. Te final ingredient for obtaining te basic (witout stabilization) discretized problem is te space discretization, tat we build wit te finite element metod (see for example [60] or [66]). Te key step is to construct te discrete linear subspaces V V, V 0 V 0 and Q Q tat approximate te continuous spaces. Let V and Q be te finite element

30 CHAPTER 1. INTRODUCTION AND BASIC MODEL spaces to interpolate vector and scalar functions, respectively, constructed in te usual manner from a finite element partition Ω = Ω e, e =1,..., n el,weren el is te number of elements. In tis tesis te same interpolation will be used for bot te velocity and te pressure, except in Capter 2 were te pressure space will be enriced. In particular P1- P1 interpolations (continuous and linear in bot velocity and pressure) will be preferred. From spaces V and Q one can construct te subsets V,u and Q for te velocity and te pressure, respectively. Te former incorporates te Diriclet conditions for te velocity components (and also te mixed conditions corresponding to te normal velocity) and te latter as one pressure fixed to zero if te normal component of te velocity is prescribed on te wole boundary. Te space of velocity test functions, denoted by V, is constructed as V,u but wit functions vanising on te Diriclet boundary. Te monolitic discrete problem associated wit te Navier Stokes equations, discretizing in time using a BDF sceme, and linearizing te convective term using a Picard sceme (in order to simply te presentation), can be written as follows: Given a velocity u n at time tn and a guess for te unknowns at an iteration i 1attimet n+1, find u n+1,i V and p n+1,i ρ Ω δt D ku n+1 v dω + + Q, by solving te discrete variational problem: ρ(u n+1,i 1 Ω µε(u n+1,i ):ε(v )dω Ω t n+1 v dγ Γ nu Ω )u n+1,i v p n+1,i v dω dω Ω v f dω Γ nu ( t n+1 1 g 1 + t n+1 2 g 2 ) v dγ = 0, Ω q u n+1,i dω = 0, for i =1, 2,... until convergence, tat is to say, until u n+1,i 1 u n+1,i and p n+1,i p n+1,i 1 in te norm defined by te user. In order to simplify te notation we use ρ ρ n+1 and µ µ n+1. Te enricment tecnique presented in Capter 2 can be understood as a modification of te pressure space Q to ˆQ,witQ ˆQ. Apart from tis, te resulting formulation follows exactly te previous setting.

1.2. TWO FLUID NAVIER STOKES EQUATIONS 31 1.2.4 Stabilized problem Te discretized problem presented in te previous Subsection needs to be stabilized before it can be solved numerically for two well known reasons. Te first one is related to te instabilities tat appear in convection-dominated flows using reasonably sized meses. Te second one is related to te use velocity and pressure finite element spaces tat do not satisfy te div-stability restriction (inf-sup condition) [12], as is te case of equal interpolation for bot unknowns tat we use in tis tesis. A wide range of stabilization tecniques can be found in te literature, among tem we can mention, using teir commonly used acronyms : SUPG [14], PSPG [108], GLS [62], CBS [34,119], FIC [86], ASGS [24] and OSS [23,25]. In tis work two of tem will be used: te Algebraic version of te Subgrid Scale stabilization metod, referred to as ASGS [24] and te Ortogonal Sub-scale stabilization metod, referred to as OSS [23, 25]. recent article [28] we ave compared numerically te two metods we will use in tis work wit te Caracteristic-Based-Split (CBS) stabilization tecnique for te incompressible Navier Stokes equations. In a Te two Subgrid Scale (SGS) formulations we present deal wit convection and pressure stabilization using te same approac. Te idea of SGS metods was proposed in [61], altoug it is inerent in oter numerical formulations. Te key idea is to approximate u u + ũ and p p, tat is, te velocity is approximated by its finite element component plus an additional term tat is called subgrid scale or subscale. We call u n+1 u n+1 := u n+1 + ũ n+1 and p n+1 p n+1 te velocity and te pressure at t n+1. As previously mentioned, te spatial interpolation for u n+1 and p n+1 are constructed using te standard finite element interpolation. In particular, equal velocity-pressure interpolation is possible. Te important point is te beavior assumed for ũ n+1. Itisassumedtatitvanises on te interelement boundaries, tat is, it is a bubble-like function [7, 13]. However, contrary to wat is commonly done, we do not assume any particular beavior of ũ n+1 witin te element domains. We will sow later on ow to approximate it.

32 CHAPTER 1. INTRODUCTION AND BASIC MODEL If in te space continuous and time discrete problem u is replaced by u n+1 := u n+1 + ũ n+1, p is replaced by p n+1, te terms involving ũ n+1 are integrated by parts, and te test functions are taken in te finite element space, one gets δt Ω + [ µ u n+1 : v + ρ ( ) u n+1 u n+1 v p n+1 v + q u n+1 Ω ρ [ u n+1 ] u n v dω δt e f n+1 v ] dω Ω e ũ n+1 [ µ v + ρu n+1 v + q ] dω = 0, (1.8) were for simplicity we ave used a first order sceme and te Laplacian form and u =0 on Ω. Te notation is used to indicate tat te Laplacian needs to be evaluated element by element. Equation (1.8) must old for all test functions v and q in teir corresponding finite element spaces. Te equation for te subscales ũ n+1 is obtained by taking te velocity test function in its space and q = 0. Te next step is to model te resulting equation. Te first possibility, wic gives rise to te ASGS metod [24], is to take ũ n+1 = τ 1 R n+1, were τ 1 is a numerical parameter and R n+1 is te residual defined as: R n+1 = ρu n+1 u n+1 µ u n+1 + p n+1 f n+1 + ρ un+1 u n. δt Te second option, wic gives rise to te OSS metod [23,25], is to impose te subscales to be ortogonal to te finite element space, ũ n+1 = τ 1 P R n+1 = τ 1 (R n+1 P (R n+1 )), were P is te projection onto te finite element space. Te advantage of tis approac is discussed in [25]. From te accuracy point of view, it is less diffusive tat te ASGS approac and yields better resolution of sarp gradients of te unknowns. In te previous approximations for ũ n+1, te temporal variation of te subscales as been considered negligible, in [25] tey are called quasi-static subscales. It is te option

1.2. TWO FLUID NAVIER STOKES EQUATIONS 33 commonly used in te literature. If temporal variation of te subscales would not be neglected tey would need to be tracked. Tis promising approac as been proposed in [25], but little as been done up to te moment. A recent article can be found in [30]. Wit all te approximations introduced eretofore, te final discrete problem to be solved for u n+1 and p n+1 using te ASGS metod is [ ρ δt (un+1 u n ) v + µ u n+1 : v + ρ(u n+1 u n+1 ) v p n+1 Ω ] f n+1 v dω + e [ q u n+1 Ω e τ 1 (µ v + ρu n+1 v ) R n+1 dω = 0, v ] dω + τ 1 q R n+1 dω = 0. Ω e Ω e In te OSS case a preliminary version can be obtained by replacing R n+1 wit P Rn+1 to obtain, [ ρ Ω δt (un+1 u n ) v + µ u n+1 : v + ρ(u n+1 u n+1 ) v p n+1 v ] f n+1 v dω + τ 1 (µ v + ρu n+1 v ) P Rn+1 dω = 0, e Ω e [ ] q u n+1 dω + τ 1 q P Rn+1 dω = 0. Ω e Ω e In te previous equation some terms turn out to be zero and oters are neglected before te final version presented in [25] is obtained. Actually in [25] only te constant density case as been analyzed. For te two fluid case we ave adapted te equations from [32]. Wen te previous OSS formulation was tested on two pase flow problems, were te density can vary in tree orders of magnitude, muc poorer results tan wit te ASGS formulation were obtained. Detailed inspection of te problem sowed tat te residual on integration points on opposite side of te interface varied rougly proportionately to te density. Ten it became obvious tat te projection of a residual wit suc variations migt be te source of te errors we were observing. Te solution we adopted was to use a modified projection P ρ (R n+1 )=ρp ( Rn+1 ρ ).

34 CHAPTER 1. INTRODUCTION AND BASIC MODEL Tis as been a key element in te successful solution of different density flows using te OSS formulation. From te teoretical point of view, te use of P ρ (R n+1 ) can be related to te fact tat te L 2 projection P is introduced in [25] as an approximation to a τ 1 weigted projection were for te variable density case τ 1 can be defined at element level as τ 1 = [ ( )] 1 4υ ρ ( e ) + 2 ue 2 e were e and u e are a typical lengt and a velocity norm of element e, respectively. It now becomes obvious tat a ρ 1 weigted projection migt be a better approximation to te τ 1 weigted projection tan a straigtforward L 2 projection. We now proceed to obtain te final version of te of te OSS stabilized problem for te variable density flows. Te transient term in Rn+1 belongs to te finite element space ρ and terefore its ortogonal projection is zero. Note tat for variable density flows tis does not appen if unweigted projection, P (R n+1 ), is used. Regarding te force term, in most cases (but not in variable density flows under gravity forces) it belongs to te finite element space and terefore its ortogonal projection is zero. In oter cases it can be neglected because it introduces an error of te same order as te optimal error tat can be expected [25]. In our formulation we ave neverteless conserved tis term in te residual to be used in te stabilization. For te low Froude number flows tat we sall be interested in our mould filling simulations te two most important terms are te force term and te pressure gradient. In te limiting case of flows at rest tey balance eac oter. Terefore, for te cases we are interested in, it seems muc more logical to preserve te force term. As explained in [25], second order derivatives of finite element functions witin element interiors can be neglected and te consistency of te OSS metod can be preserved. For linear elements tese terms are zero. Instead, for te ASGS stabilization, if te viscous terms are neglected consistency is lost. Despite te ASGS metod is consistent, its implementation for linear elements, is identical to te implementation of a non consistent sceme. We can conclude tat te OSS sceme is muc better suited for linear elements

1.2. TWO FLUID NAVIER STOKES EQUATIONS 35 tan te ASGS sceme. In [65] an improvement, tat uses an L 2 projection for te diffusive term in te residual, is introduced to mitigate te weakness of te ASGS metod for linear elements. Taking into account te previous comments, te final discrete problem to be solved for u n+1 and p n+1 using te OSS metod is [ ρ ] δt (un+1 u n ) v + µ u n+1 : v + ρ(u n+1 u n+1 ) v p n+1 v f n+1 v dω Ω + e Ω τ 1 ρ (u n+1 v ) Ω e ( ρp u n+1 u n+1 [ ] q u n+1 dω + τ 1 q e Ω e ( ρp [ (ρ u n+1 u n+1 + p n+1 f n+1) + pn+1 ρ f n+1 ) ] dω = 0, ρ [ (ρ u n+1 u n+1 + p n+1 f n+1) u n+1 u n+1 + pn+1 ρ f n+1 ) ] dω = 0. ρ Te key terms for te stabilization of te convective and pressure terms are τ 1 ρ ( ) ( ) u n+1 v ρ u n+1 u n+1 and τ1 q p n+1 respectively. Tese terms appear in bot formulations we ave presented but also in most oter stabilization tecniques. Te reasons for aving worked wit two stabilization tecniques are bot practical and teoretical. On te practical side we ave worked wit te ASGS tecnique because it was te tecnique tat was implemented in te monolitic version of our code. For te segregation metods, tat will be presented in Capter 5, our code only ad a preliminary version tat used te OSS metod. Anoter reason for using ASGS is tat it is a more widely used tecnique tan OSS. Actually not ASGS on its own, but if one also counts GLS tat is very similar to ASGS. As as already been mentioned, some of te advantages and disadvantages of OSS ave been discussed in [25] but up to now tere is no clear favorite metod. As we ave already mentioned, for P1-P1 elements OSS seems a better coice. Te OSS stabilization can be reformulated so tat instead of working wit te ortogonal projection of te convective and pressure terms togeter, two separate

36 CHAPTER 1. INTRODUCTION AND BASIC MODEL projections can be used [25]. We will call tis version te split OSS. As in te non split version, for te variable density case, we sall work wit te projections weigted by 1 ρ. Moreover, for te low Froude number flows instead of working wit pn+1,asis usually done, we sall work wit p n+1 f n+1 for te same reasons we ave explained for te non split version. Te split OSS version we use reads [ ρ δt (un+1 u n ) v + µ u n+1 : v + ρ(u n+1 u n+1 ) v p n+1 v ] f n+1 v dω + τ 1 ρ (u n+1 v ) e Ω e [( ) ( )] ρ u n+1 u n+1 ρp u n+1 u n+1 dω = 0, (1.9) Ω + e [ ( p n+1 τ 1 q Ω e Ω [ q u n+1 ] dω f n+1) ( p n+1 ρp ρ f n+1 )] ρ dω = 0. In some situations te introduction of a pressure subscale p n+1 can also be advantageous [25] because it elps to enforce te incompressibility of te flow tat can be excessively relaxed wen only te velocity subscale is introduced. It is approximated as [ ( )] p n+1 = τ 2 u n+1 ξp u n+1, wit ξ = 1 in te OSS case and zero oterwise. For te OSS metod tis term would control u in te space ortogonal to te finite element space, but since we want tis to appen in te wole space we ave used ξ = 0 for bot OSS and ASGS cases. As a summary to te ideas presented up to now we rewrite te monolitic divergence form Navier-Stokes equations using Picard linearization, BDF time discretization and ASGS stabilization. Given a velocity u n at time tn (and also at previous times as required by D k ) and a guess for te unknowns at an iteration i 1attimet n+1, find u n+1,i V

1.2. TWO FLUID NAVIER STOKES EQUATIONS 37 and p n+1,i Q, by solving te discrete variational problem: ρ Ω δt D ku n+1 v dω+ ρ(u n+1,i 1 )u n+1,i v dω Ω + µε(u n+1,i ):ε(v )dω v p n+1,i dω Ω Ω n el [ ] + µ v + ρ (u n+1,i 1 )v e=1 τ n+1,i 1 1 Ω e µ u n+1,i Ω n el + + ρ (u n+1,i 1 e=1 q u n+1,i dω + µ u n+1,i Ω e τ n+1,i 1 )u n+1,i + p n+1,i Ω v f dω [ ρ δt D ku n+1 ] f dω 2 ( v )( u n+1,i )dω=0, n el e=1 + ρ(u n+1,i 1 τ n+1,i 1 q Ω e )u n+1,i for i =1, 2,...until convergence, tat is to say, until u n+1,i 1 in te norm defined by te user. [ ρ δt D ku n+1 + p n+1,i ] f dω = 0, u n+1,i and p n+1,i p n+1,i 1 Te parameters τ 1 and τ 2 are cosen in order to obtain a stable numerical sceme wit optimal convergence rates (see [24] and references terein for details). Tey are computed witin eac element domain Ω e.wetaketemas: [ 4µ τ 1 = ( e ) + 2ρ ue 2 e τ 2 = (e ) 2 τ 1 were e and u e are a typical lengt and a velocity norm of element e, respectively. At least tree option can be suggested for e : te maximum element lengt, te minimum oneandteoneintedirectionofteflow. Testrategyweareusinginourcodeisto take te minimum element lengt for te diffusive term and one in te direction of te flow for te convective term. Tus to be more precise we sould write [33], [ ] 1 4µ τ 1 = ( + 2ρ ue e min )2 e dir flow τ 2 = (e min )2 τ 1 ] 1

38 CHAPTER 1. INTRODUCTION AND BASIC MODEL 1.2.5 Matrix version of te problem In order to introduce some of te notation to be used in Capter 4 we will present te matrix version of te problem using Picard linearization, BDF1 time discretization and split OSS stabilization. ( Te projections P u n+1 u n+1 ) and P ( 1 ρ pn+1 ), will be treated iteratively using te same iterative loop as for te linearization of te convective term. We will call tem ( ) y n+1 = P u n+1 u n+1, z n+1 = P ( 1 ρ ( p n+1 f n+1)). Tey are te solution of ( ) ( y n+1, v = u n+1 u n+1 ), v ) ( ( ) z n+1 1 (, v = p n+1 f n+1), v ρ v V, v V, were V is te space V enlarged wit te continuous vector functions associated to te boundary nodes. Te resulting algebraic system prior to linearization, supposing Laplacian form in order to simplify te presentation, is ten M D k δt Un+1 + K ( U n+1) ( U n+1 + GP n+1 + S u τ1 ; U n+1) ( U n+1 S y τ1 ; U n+1) Y n+1 (1.10) +S d (τ 2 ) U n+1 S w (τ 2 ) W n+1 = F n+1, DU n+1 + S p (τ 1 ) P n+1 S z (τ 1 ) Z n+1 = 0, M π Y n+1 C ( U n+1) U n+1 = 0, M π Z n+1 G π P n+1 = 0, were U, P, Y and Z are te arrays of te nodal unknowns for u, p, y and z, respectively. If we denote te node indexes wit superscripts a,b, te space indexes wit subscripts i, j and te standard sape functions of node a by N a, te components of te arrays involved te previous equations are: M ab ij = ( N a,ρn b) δ ij,

1.2. TWO FLUID NAVIER STOKES EQUATIONS 39 were δ ij is te Kronecker δ. M π ab ij = ( N a,n b) δ ij, K ( U n+1) ab ij = ( N a,ρ u n+1 N b) δ ij + ( N a,µ N b) δ ij, G ab i = ( N a, i N b), G π ab i = ( N a, i N b /ρ ), S u ( τ1 ; U n+1) ab ij = ( τ 1 u n+1 N a,ρ u n+1 N b) δ ij, S y ( τ1 ; U n+1) ab ij = ( τ 1 u n+1 N a,ρn b) δ ij, S d (τ 2 ) ab = ( τ 2 N a, N b), S w (τ 2 ) ab = ( τ 2 N a,n b), D ab j = ( N a, j N b), S p (τ 1 ) ab = ( τ 1 N a, N b), S z (τ 1 ) ab j = ( τ 1 j N a,ρn b), C ( U n+1) ab ij = ( N a, u n+1 N b) δ ij, F a i =(N a,f i ), It is understood tat all te arrays are matrices (except F, wic is a vector) wose components are obtained by grouping togeter te left indexes in te previous expressions (a and possibly i) and te rigt indexes in te previous expressions (b and possibly j). Equation (1.10) needs to be modified to account for te Diriclet boundary conditions (matrix G can be replaced by -D T wen tis is done). 1.2.6 Material properties approximation For te continuous problem definition, equation (1.6) is all tat is needed to define te material properties for te two fluid Navier-Stokes equations. Wen te problem is discretized, te material properties (ρ, µ) need to be someow approximated in elements cut by te interface. Te simplest approac is to take ς k as eiter ς 1 or ς 2 depending

40 CHAPTER 1. INTRODUCTION AND BASIC MODEL on te value of te level set function at eac integration point (k), were ς stands for µ or ρ. Te approximation depends on te integration rule used on elements cut by te interface. Te typical approac is to use te same integration rule as on non cut elements. In Capter 2 an enanced integration rule for cut elements is presented. Anoter option can be found in [81]. Following te nomenclature used in te Level Set community [90] equation (1.6) is usually written as ς = ς 1 +(ς 2 ς 1 ) H (ψ), were ς, as we ave said, stands for µ or ρ, H is te Heaviside function, 0 if ψ 0, H (ψ) = 1 if ψ>0, and ψ is te level set function. We will terefore call tis approac Heaviside. Te option most commonly used by Level Set practitioners is to smear-out te Heaviside function using H ɛ (ψ) = 0 if ψ< ɛ, ) if ɛ ψ ɛ, 1 + ψ + 1 sin ( πψ 2 2ɛ 2π ɛ 1 if ψ>ɛ, were ɛ is a tunable parameter tat determines te size of te bandwidt of numerical smearing. In [90] ɛ =1.5 x is suggested making te interface widt equal to tree grid cells, were x is te grid size for a finite difference discretization. It is interesting to note tat in tis case te material properties are also approximated in some elements not cut by te interface. We will call tis approac Smooted Heaviside. In Section 1.1 we ave said te Level Set metod is also known as pseudo concentration tecnique [110]. Te only significant difference between te two metods is te way in wic te material properties are approximated. A typical option found in works tat use te pseudo concentration tecnique is to calculate te material properties at integration points k belonging to cut elements according to ς k = ξ k ς 1 +(1 ξ k ) ς 2.

1.2. TWO FLUID NAVIER STOKES EQUATIONS 41 Te pseudo concentration function, ξ, is for practical matters equivalent to te level set function and can be related to it according to ξ = k 1 ψ + k 2 were k 1 and k 2 are two constants taken so tat 0 ξ 1. Usually k 2 =0.5 isused. We will call tis approac Pseudo concentration properties. 1.2.7 Mixed boundary conditions on curved walls In equation (1.4) we ave defined mixed boundary conditions for te Navier Stokes equations as te conditions tat prescribe a zero velocity in te normal direction to te boundary and a traction in te tangential direction. Two typical cases are slip boundary conditions were te tangential traction is zero and wall boundary conditions were te tangential traction depends on te velocity. For te numerical simulation of turbulent flows on complex geometries, suc as te mould filling examples to solve in tis tesis, te use of wall laws is mandatory since te simulation of te boundary layer is computationally too expensive. In real problems one usually deals wit domains wit curved boundaries were mixed boundary conditions must be applied. Wen suc domains are discretized using te finite element metod, te normal to te different element faces belonging to te boundary tat meet at a node do not coincide. In order to apply te zero normal velocity condition to te discretized problem, once te system tat describes te problem witout boundary conditions as been obtained in Cartesian coordinates, te degrees of freedom corresponding to nodes on te curved boundary must be rotated into a local system suc tat te normal component can be prescribed to zero. Te problem wit curved boundaries is tat te normal defined at te node does not coincide wit te normal to eac of te element faces on te boundary tat meet at te node. Terefore, te normal velocity to te faces will not be exactly zero and tere will be some flow troug te faces (te condition n u =0 will not be exactly satisfied at te discrete level). Te normal at te node is usually defined in suc a way tat te flow troug te element faces on te

42 CHAPTER 1. INTRODUCTION AND BASIC MODEL boundary associated wit te velocity at te node is zero [48]. Suc nodal normals are called consistent normals because tey are te best solution to satisfy mass conservation. Figure 1.1: Spurious velocities obtained for a downward gravity force Even wen consistent normals are used, problems tat ave not been discussed until quite recently [9] can be observed. If one tries to simulate a fluid at rest under gravity forces in a domain wit curved boundaries using mixed boundaries conditions spurious velocities will appear. Figure 1.1 sows te spurious velocities obtained in a circular domain of radius r =1.0 wit slip boundary conditions for ρ = µ =1.0 andg =10.0 in te downward direction. For real flows tese spurious velocities can be idden by te real velocities and not be noticed. As te Froude number decreases tey become more noticeable, specially on coarse meses. For constant density flows on fixed domains two very trivial solutions can be proposed. Te first one is to replace te problem by an equivalent one were gravity forces disappear. Te second solution is to integrate over te mixed boundary te traction corresponding to te ydrostatic pressure. In flows wit variable density (suc as two pase flows) te previous solutions cannot be applied. In [9] te proposed solution consists in applying a do noting boundary condition [92] on mixed boundaries. Actually te normal component to te node is subtracted from te usual do noting boundary conditions but tis as no effect on te flow but only on te resulting reactions. Terefore unless one

1.2. TWO FLUID NAVIER STOKES EQUATIONS 43 is interested on te reactions on te boundary te usual do noting boundary condition can be applied. In [9] only slip boundary conditions are discussed but te metod could easily be extended to oter mixed boundaries. Do noting boundary conditions consist in sending to te matrix side (instead of rigt and side) te traction boundary conditions wen tractions are not known. Despite at te continuous level tis leads to an ill-posed problem, for te discrete finite element problem it typically yields very good results. It is commonly used at outflow boundaries were bot normal and tangent components of te traction vector are unknown. For te mixed boundary cases we are analyzing te tangent components of te traction vector are known and terefore we belive tat te do noting boundary condition sould be applied only in te normal direction and not in all te directions as proposed in [9]. Tat is to say, instead of integrating (on te matrix side) v [ pn+µ ( n ( u +( u) t))] dγ = v [n σ] dγ Γ mu Γ mu as proposed in [9], one sould only integrate v [(n σ n) n] dγ Γ mu For a flow at rest we ave obtained te exact solution wit bot formulations. For a problem wit non zero velocities, te first condition we must expect from te proposed formulations is tat in te case wit planar boundaries te solution obtained witout any modification sould remain unaltered wen te modification for curved boundaries is applied. Tis is wat appens wen te open boundary condition in te normal direction we propose is applied. Wen te modification proposed in [9] is used te boundary conditions in te tangential direction are altered and so is te solution. In order to sow wat appens in a numerical example, te test case we propose is Stokes flow in a cannel. Taking into account symmetry only alf of te cannel is simulated. On te symmetry face a slip boundary condition is applied. Despite te test case does not ave curved boundaries, te proposed correction sould also work on planar boundaries. Moreover, te absence of curved boundaries allows us to solve te problem

44 CHAPTER 1. INTRODUCTION AND BASIC MODEL wit no special correction. In Figure 1.2 we compare te solutions obtained wit bot corrections against te solution obtained witout any correction. Te results obtained witout any boundary correction are exactly te same as tose obtained wit te do noting boundary condition applied only in te normal direction. On te oter and te use of te do noting boundary condition in all directions [9] leads to an incorrect solution. Figure 1.2: Velocities in a cannel: (top) no modification, (middle) modification proposed in [9], (bottom) our modification Te test case tat as been presented is an extreme case to sow te improvement introduced by our modification to te metod proposed in [9]. Since te implementation and computational cost of bot metods is nearly te same we believe te do noting BC only in te normal direction sould be used. In te mould filling examples sown in Capter 5 tis correction as been a key element for obtaining correct solutions.

1.3. THE LEVEL SET EQUATION 45 1.3 Te Level Set equation 1.3.1 Interface Capturing Tecniques As we ave already mentioned, fixed mes metods generally sare two basic steps. In te previous Section we ave presented te two fluid Navier Stokes equations tat allow us to deal wit te first step. In tis Section we present an equation for an interface function tat allows to determine te position of te interface, and tus te properties to be assigned in te previous step. Several metods can be used to determine te position of te interface. We will use te so called level set metod (see [18, 106] and [90] for an overview) but oter metods could also ave been used. Among tose more closely related to te level set metod we would like to mention te pseudo concentration tecnique [110] and volume of fluid (VOF) tecnique [57,70,103]. Weter tey are different metods or different names for te same metod is a question tat can raise some discussion. In all tree metods, a transport equation, t φ +(u )φ =0 for a function tat coincides wit te name of te metod is solved. In te case of te level set te function we will use te letter ψ for φ. Te basic idea of te level set metod is to embed te propagating interface Γ int (t) as te zero level set of a iger dimensional function ψ, defined as ψ (x,t=0)=±d, wered is te distance from x to Γ int (t =0), cosen to be positive in one fluid and negative in te oter. Setting te initial zero level set so tat it coincides wit te initial interface Γ int (t =0)={x ψ (x,t=0)=0},te previous transport equation contains te embedded motion for Γ int (t) astelevelset ψ = 0. It is interesting to remark tat ψ is a continuous function. In te pseudo concentration metod, te scalar function φ can be considered a fictitious fluid property tat is advected by te flow and indicates te presence of fluid one or fluid two at a certain point x. In tis sense it is equivalent to te level set metod. Typically φ = 1 is assigned to te region occupied by fluid one and φ = 0 to te region

46 CHAPTER 1. INTRODUCTION AND BASIC MODEL occupied by fluid two. In tis sense te pseudo concentration metod is similar to te VOF metod we will describe next. Te position of te front is defined by te isovalue contour φ (x) =φ c,wereφ c [0, 1] is a critical value defined a priori (typically φ c =0.5). Tis critical value used is irrelevant before te problem is discretized, but is needed wen te finite element discretization is introduced. Te transport of a function tat varies abruptly from 0 to 1 introduces problems wen solved numerically, terefore it is usually smooted redefining te pseudo concentration for eac node of te finite element mes according to te following expression [110]: φ = φ c + sgn (φ 0 φ c ) kd were φ 0 is te non-smooted value, k is a constant, d is te distance from te point under consideration to te front and sgn ( ) is te signum of te value enclosed in brackets. Ten te only differences wit te level set function are tat te pseudoconcentration slope can ave any value and tat te critical value used is 0.5 instead of 0. Tose two differences ave no practical importance and terefore te two metods can be considered identical, at least in te way in wic tey capture te interface (some differences in te way tey approximate te material properties close to te interface ave been discussed in te previous Section). It is interesting to remark tat te pseudo concentration metod appeared before te level set metod [91]. Te latter metod as obtained more widespread use tanks to its clear presentation and matematical formalism. In any case we consider tat it is te same metod wit two different names. In te VOF metod te scalar φ represents te fractional volume of a certain fluid in te corresponding computational cell. Typically φ = 1 is assigned to cells occupied by fluid one and φ = 0 to cells occupied by fluid two. Cells cut by te front ave some φ [0, 1] depending on te percentage of eac fluid present. Te VOF is typically associated to discretizations wit constant interpolations witin eac cell, suc as finite volumes. Te following analogy can be proposed: if Finite Volumes are seen an P0 Finite Elements, ten te VOF metod can be seen as a P0 discretization of te pseudo concentration or level set metod. One particularity of te VOF metod is tat te position of te interface

1.3. THE LEVEL SET EQUATION 47 is not obtained directly from te value of φ, but reconstructed from it. Several interface reconstruction tecniques can be found in te literature (see [70, 103, 116] for reviews). Anoter particularity is tat in most versions of te VOF metod te transport equation is not discretized and solved algebraically but rater solved geometrically. Since we are using a finite element discretization te level set metod is te natural coice, but different discretizations for te Navier-Stokes and interface capturing equations could also be used. 1.3.2 Implementation of te level set metod As we ave already mentioned, te basic idea of te level set metod is to define a smoot scalar function, say ψ(x,t), over te computational domain Ω tat determines te extent of subdomains Ω 1 and Ω 2 and allows to represent te interface implicitly. For instance, we may assign positive values to te points belonging to Ω 1 and negative values to te points belonging to Ω 2. Te position of te fluid front is ten defined by te iso-value contour ψ(x,t) = 0. Te evolution of te front ψ = 0 in any control volume V t Ωwic is moving wit a divergence free velocity field u leads to: t ψ +(u )ψ = 0 (1.11) Tis equation is yperbolic and terefore boundary conditions for ψ ave to be specified at te inflow boundary, defined as: Γ inf := {x Ω u n =0} Function ψ is te solution of te yperbolic equation (1.11) wit te boundary conditions: ψ = ψ on Γ inf (t 0,t f ), ψ(x, 0) = ψ 0 (x) Te initial condition ψ 0 is cosen in order to define te initial position of te fluid front to be analyzed. Te boundary condition ψ determines wic fluid enters troug a certain point of te inflow boundary.

48 CHAPTER 1. INTRODUCTION AND BASIC MODEL In order to obtain te weak or variational formulation of te Level Set equation (1.11) we introduce te spaces X 0 { x L 2 (Ω) x =0onΓ inf }, X { x L 2 (Ω) x = ψ on Γ inf }, X t L 2 (t 0,t f ; X), Te weak form is ten obtained by multiplying te transport equation for te front (1.11) by an arbitrary element of X 0, x. Te weak form of problem 1.11 wit te boundary conditions we ave just defined is: Find ψ X t for all x X 0. suc tat Ω t ψxdω + Ω [u ψ] x dω = 0 Te spatial discretization is te same one used for te Navier-Stokes equation, tat is P1 elements. Te temporal evolution is treated via te backward differencing (BDF) time integration sceme. Due to te pure convective type of equation (1.11), we use te SUPG [14] tecnique to stabilize it. Since equation (1.11) does not ave a diffusive term, te use te SUPG tecnique is equivalent to te use of te ASGS tecnique presented in te previous capter. Te discrete problem, bot in space and time, stabilized using te SUPG or ASGS tecnique ten reads: Given a velocity u n+1 at time t n+1 and a ψ n at time tn (and also at previous times as required by D k ), find ψ n+1 X (a discrete linear subspace of X) by solving te discrete variational problem: n el + e=1 Ω 1 δt D kψ n+1 x dω + Ω [ ] u n+1 ψ n+1 x dω τ [ ] [ n+1 u n+1 1 x Ω δt D kψ n+1 + u n+1 ψ n+1 e ] dω = 0 As in te previous capter, te parameter τ is cosen in order to obtain a stable numerical sceme wit optimal convergence rates. It is computed witin eac element domain Ω e

1.3. THE LEVEL SET EQUATION 49 as: τ = e 2 u e, were e is te element lengt in te direction of te flow and u e te velocity norm of element e. 1.3.3 Reinitialization For te numerical solution of te level set equation it is preferable to ave a function witout large gradients. Since te only requirement suc a function must meet is ψ = 0 at te interface, a signed distance function ( ψ = 1) is used. Under te evolution of te level set equation, ψ will not remain a signed distance function and tus needs to be reinitialized. Several approaces can be found in te literature [18, 83, 90, 104, 106]. Te one we will use consists in redefining ψ for eac node of te finite element mes according to te following expression: ψ =sgn(ψ 0 )d were ψ 0 stands for te calculated value of ψ, d is te distance from te node under consideration to te front, and sgn( ) is te signum of te value enclosed in te parentesis. In [21] tree ways of calculating d are discussed. Tere te one we will use ere is called interpolation of a straigt line and is described briefly for te case of linear elements. Te free surface is approximated by triangular planes p (lines in 2D). Ten te perpendicular distance d i p of eac grid point i to eac plane p can be computed. Te minimum distance from eac nodal point to te planes is te required distance between te point and te front (d i =min p {d i p}). For te bigger examples we ave reinitialized te signed distance function only on five layer of nodes to eac side of te interface to reduce te computational cost. Anoter option can be found in [32]. Te approac we use is typically not favored in te level set community because it is considered slow [90]. A great deal of effort as been put into reinitialization tecniques and a complete review of most recent advances can be found in [90]. Most of tem are

50 CHAPTER 1. INTRODUCTION AND BASIC MODEL based in solving an equation of te form t ψ + ψ =1 up to steady state, were t is not te real time but some pseudo time. Wen te steady state is reaced te transient term disappears and ten te signed distance function ( ψ = 1) is recovered. Altoug most of te options presented seem very promising, for te moment we do not feel it wort for us to put an effort into te subject because despite our reinitialization tecnique may be slow, it is, according to wat we ave observed, not te key item tat makes our metod slow. On te oter and, as we ave already mentioned a lot work as been done on te subject and our initial effort sould, in any case, concentrate in testing some of te available options [32]. Anoter tecnique tat is someow related to reinitialization is te construction of extension velocities [90, 104]. Te range of use of te level set tecnique is muc wider tan flows wit interfaces. Tere are lots of application were te velocity is only defined on te interface and not in te rest of te domain. In teory, te only velocities needed to transport te interface are tose on te interface, but since in te level set metod te interface is transported embedded in a iger dimensional function te velocities in te rest of te domain ave to be found from te velocities on te interface. Suc velocities are called extension velocities. In te case of flow simulations te use of extension velocities is not mandatory but it as been suggested tat using only te real velocities on te interface and extension velocities in te rest of te domain can elp to maintain te level set a signed distance function, tus reducing te need for reinitialization [104]. 1.3.4 Coupling between te flow equations and te Level Set In te previous Section we ave presented te two fluid Navier-Stokes equations and in tis Section we ave dealt wit te Level Set equation. Obviously bot equations are coupled. In te flow equations te fluid properties (ρ and µ ) depend on te level set function, ψ n+1. On te oter and, te level set function is transported using te velocity u n+1. Te approac we will use to uncouple tem is to use te u n (or 2un un 1 wen

1.3. THE LEVEL SET EQUATION 51 a second order backward difference is used) instead of u n+1 wen calculating ψ n+1.tis introduces a restriction on te time step size so tat te front does not advance more tan one layer of elements at a time. In any case advancing more elements at a time migt make us lose te pysics of te problem and terefore we believe tis time step restriction is logical. Anoter option could be to use a block iterative procedure to couple bot equations. Ten te velocity to be used in transport of te level set would be u n+1,i were te superscript i refers to te iteration number. It could be te same iterative loop used for te non-linearity introduced by te convective term of te Navier-Stokes equations or some oter purposely introduced iterative loop. In [21] suc strategy as been tested but te uncoupled option as been preferred.

52 CHAPTER 1. INTRODUCTION AND BASIC MODEL

Capter 2 An enriced pressure two-pase flow model As we ave mentioned in Capter 1, numerical metods for flows wit interfaces can be classified into fixed grid and moving grid tecniques. For te same mes size te latter lead to a more accurate representation of te interface at a iger computational cost. In tis paper we will use a fixed grid metod and introduce modifications to te basic formulation to enance te representation of te interface. Te initial motivation for tis work came from te impossibility to model a terribly simple flow; in fact, not even a flow, but two different density fluids at rest (te ligter one on top) inside a closed cavity. Te ydrostatic pressure gradient is discontinuous at te interface. Tis cannot be correctly represented by te usual finite element sape functions wen te front crosses an element. Since te velocity and pressure are coupled, te error in te representation of te pressure gives rise to spurious velocities tat, depending of te properties used, can completely distort te interface tat sould oterwise remain orizontal. Te idea of enricing te representation of an unknown at a material discontinuity is not new and several approaces can be found in te literature [19, 45, 81]. In te next Section we will outline some of te differences between te previous metods and ours. 53

54 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL As we ave already mentioned for te evolution of te fluid interface, we use te level set metod [18, 106]. Te contribution we intent to introduce in tis capter does not depend on te approac used to capture te interface and sould also be valid using any of te oter cited tecniques. Te numerical formulation presented ere to solve te incompressible Navier Stokes equations uses a time discretization based on te standard trapezoidal rule and a stabilized finite element metod referred to as Algebraic Sub-Grid Scales (ASGS) [24]. P1-P1 elements will be used. In te elements cut by te interface te P1 pressure sape functions are supplemented wit an additional sape function tat is zero at all te element nodes, continuous witin te element and as a constant gradient on eac side of te interface. Tis sape function is local to eac element and te corresponding degree of freedom can terefore be condensed prior to assembly, making te implementation quite simple on any existing finite element code. Te details will be discussed in te next Section. In Capter 1 we ave described te matematical model used to solve te Navier Stokes equations wen no enricment functions are used and te Level Set Metod. In te next Section te enricment functions used and some implementation details are discussed. Finally we present tree simple numerical examples were te improvements obtained wit te proposed formulation sow up clearly. Te work we will describe in tis capter as been presented in [37]. Furter numerical examples can be found in [36]. 2.1 Discontinuous Gradient Pressure Sape Functions In fixed grid finite element metods te wole domain Ω is subdivided into elements Ω e. Witin eac element te unknowns are interpolated as φ Ω e = NNODE I=1 were NNODE is te number of element nodes. N I e Φ I e,

2.1. DISCONTINUOUS GRADIENT PRESSURE SHAPE FUNCTIONS 55 In typical finite element metods, N I e terefore φ Ω e are continuous witin eac element and is continuous. Wen te interface crosses an element te discontinuity in te material properties leads to discontinuities in te gradients of te unknowns tat te interpolation used cannot capture. For example, as mentioned previously, for two different density fluids at rest te interpolation errors in te pressure give rise to spurious velocities tat can render te solution meaningless. lead to discontinuous velocity gradients. Also, viscosity discontinuities can Enricment metods add degrees of freedom at elements cut by te interface in order to reduce interpolation errors. In our particular case we add only one pressure degree of freedom per cut element. Terefore te pressure in elements cut by te interface is interpolated as p Ω e = NNODE I=1 Ne I Pe I + Ne ENR Pe ENR. (2.1) Te sape function Ne ENR we introduce as a constant gradient on eac side of te interface, its value is zero at te element nodes and is C 0 continuous in Ω e. Te added degree of freedom is local to te element and can terefore be condensed after te element matrix as been computed and before assembly. Te resulting pressure finite element space is made of functions tat are discontinuous across interelement boundaries, and tus it is a subspace of L 2 (Ω), but not of H 1 (Ω), as would be te case using P 1 P 1elements. However, our metod is still conforming. If we ad tried to use te previous enricment functions for te velocity we would ave obtained a non conforming metod. Minev and co workers [81] also use enricment functions for two fluid flows. Tey use discontinuous gradient velocity sape functions and discontinuous pressure sape functions because tey include surface tension. No discontinuous gradient pressure sape functions are included. Te velocity enricment functions are not only local to eac element, as in our case, but also continuous across elements. Tis is possible because tey are te product of a bubble function times some oter function. Tempted by te fact of being able to condense te enriced degree of freedom wile using H 1 functions, we initially tried to use te enricment tey use for te velocity for te pressure, but

56 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL obtained very unsatisfactory results in te ydrostatic two fluid case. Tis led us to look for enricment functions tat could model constant gradients on eac side of te interface. Also for two fluid problems, Cessa and Belytscko [19] use an enricment metod called XFEM, initially developed by te second autor for modeling cracks. As in te case of Minev et al., tey use discontinuous gradient velocity sape functions and not discontinuous gradient pressure sape functions. Te enricment tey use is continuous and cannot be condensed prior to matrix assembly. It is computationally muc more expensive because te mes grap needs to be updated as te interface moves from one element to anoter. 1 A B 2 3 Figure 2.1: 2D Enricment function for a cut element In Fig. 2.1 we sow a sketc of te enricment function we use for an element cut by te interface in te 2D case. Te element as nodes named 1, 2 and 3 and te interface cuts te element edges at points A and B. A way to build suc function is as follows. Suppose tat node 1 belongs to Ω 1 andnodes2and3belongtoω 2. Let Ω e 1 =Ω 1 Ω e and Ω e 2 =Ω 2 Ω e. In Ω e ENR 2 we want N to ave constant gradient and to ave a zero value at x 2 and x 3. We can terefore define N ENR Ω e 2 = k 1 N 1 Ω e 2, were k 1 is a constant to be defined. By definition we want N ENR (x A ) = 1. As we are

2.1. DISCONTINUOUS GRADIENT PRESSURE SHAPE FUNCTIONS 57 using linear elements to interpolate te level set function we ave tat N 1 (x A )= Ψ 2 Ψ 2 Ψ 1, were Ψ i is te value of ψ at node i, and terefore Now we ave k 1 we can find k 1 = Ψ2 Ψ 1 Ψ 2. N ENR (x B ) Ω e 2 = k 1 N 1 (x B ) Ω e 2 = k 1 Ψ 3 Ψ 3 Ψ 1. We can proceed to find N ENR Ω e 1. We want it to ave a constant gradient in Ω e 1 zero at x 1.Ten and to be N ENR Ω e 1 = k 2 N 2 Ω e 1 + k 3 N 3 Ω e 1. UsingoncemoretatN ENR (x A ) = 1 and te fact tat N 3 (x A )=0weget k 2 = 1 N 2 (x A ) = Ψ1 Ψ 2 Ψ 1. Since we want te enricment function to be continuous in Ω e we need N ENR (x B ) Ω e 2 = N ENR (x B ) Ω e 1, ten, as N 2 (x B )=0, k 3 = N ENR (x B ) Ω e 2 1 N 3 (x B ) = k1 Ψ 3 Ψ 3 Ψ 1 Ψ 1 Ψ 3 Ψ 1, k 3 = k 1 Ψ3 Ψ 1. We ave obtained an enricment function tat is proportional to N 1 on Ω e 2 and a linear combination of N 2 and N 3 on Ω e 1, were te values of k1, k 2, k 3 only depend on te values of te level set function at te element nodes. It is very easy to obtain te enricment function and its Cartesian derivatives from te usual sape function. It seems wortwile

58 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL to remark tat N ENR Ω e does not belong to te space formed by N 1 Ω e, N 2 Ω e, N 3 Ω e. Te same ideas ave been used to obtain N ENR for 3D elements. In order to capture te discontinuities and take advantage of te enricment functions used, te integration rules need to be modified in elements cut by te front. Te metod we use is to divide eac tetraedral (triangular in 2D) element into up to six tetraedral (tree triangular in 2D) sub elements. For eac sub element te same integration rule as for te non cut elements is used (see Figure 2.1). 1 2 3 Figure 2.2: Enanced integration rule for a 2D cut element Wen using enricment functions for te pressure, te material properties µ, ρ are taken as µ 1, ρ 1 or µ 2, ρ 2 depending on wic part of te domain (Ω 1 or Ω 2 ) te integration point is found, tat is, te material properties are approximated using te approac called Heaviside in Capter 1. Since te pressure space is enriced, a remark is needed concerning pressure stability. If we ad used a velocity-pressure interpolation satisfying te inf-sup condition, te enricment of te pressure could ave led to an unstable velocity-pressure pair. However, we are using a stabilized finite element formulation. Even toug we ave no stability analysis for te enriced pressure space, we ave not encountered any type of stability misbeavior. A final remark is required concerning te extension of te proposed enricment to iger order elements. Since te intention is to add a pressure field able to deal wit

2.2. NUMERICAL EXAMPLES 59 discontinuous pressure gradients, but constant in eac fluid pase, exactly te same metodology as described for P 1 elements can be applied to iger order elements. Te construction of te enriced pressures can be based only in te linear part of te interpolation basis functions of tese iger order elements. Tis is particularly simple wen tey are implemented using a ierarcical basis. Te case of quadrilateral elements (or exaedra in 3D) can be dealt wit by splitting te quadrilateral into triangles (or tetraedra). 2.2 Numerical Examples In tis section we present tree numerical examples were te improvements obtained wit te proposed formulation sow up clearly. Te first two examples are related to te original two fluid ydrostatic problem but modified so tat tey ave nonzero velocities. Te results obtained wit te enriced formulation are compared wit tose obtained wit a typical finite element formulation wit no enricment nor improved integration. Since we are trying to prove tat te benefits come from te pressure enricment and not from te improved integration, an intermediate case were no enricment is used and we only modify te integration is also presented. Regarding te approximation of te material properties in elements cut by te interface, we ave mentioned tree possibilities in Capter 1, Heaviside, Smooted Heaviside and pseudo-concentration properties. Wen we use improved integration, wit or witout enriced pressures, te natural coice is to use Heaviside properties. Wen no improved integration nor pressure enricment is used any of te tree options can be used. In te examples we present in tis capter we ave used Heaviside or pseudo-concentration properties. Smooted Heaviside properties ave been tested but te results were similar to tose obtained wit te oter properties approximations and will not be presented.

60 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL Mes Npoin Nelem Elem lengt coarse 128 214 1.0 medium 472 862 0.5 very fine 11615 22828 0.1 Table 2.1: Meses used for te two fluid cavity flow 2.2.1 Two fluid cavity Te first example, called te two fluid cavity, is a square domain filled wit equal amounts of two different density fluids and a fixed orizontal velocity (0.1 m/s) on te bottom wall. Te walls are supposed frictionless. Heaviside properties are used to approximate te material properties in elements cut by te interface. It is a very simple example but it can be representative of te numerical problems tat can appear in muc more complex problems suc as te two pase flow in a stirred reactor. Tree 2D unstructured triangular meses were used (see Table 2.1). Te square domain as a side lengt L = 10 m. Te material properties used (SI units) are ρ 1 = 1000, µ 1 = 10 for te fluid on te bottom, and ρ 2 = 900, µ 2 =9forteone on top. Te viscosity of te bottom fluid is 1000 times te viscosity of water so as to obtain a relatively low Reynolds number (Re = 100) in order to avoid unnecessary complications. Te simulations were run for 100 seconds wit a 0.5 second time step size. In all te examples presented in tis paper te acceleration of gravity is g = 10. In Fig. 2.3 we sow te sape of te interface for te tree meses in te tree different conditions mentioned previously: wit no enricment nor improved integration, using only modified integration but no pressure enricment and finally wit bot pressure enricment and improved integration. Using te finest mes te tree metods give nearly te same result. Only in te case wit no enricment nor improved integration, sligt oscillations canbeobserved. Despiteweavenotgotpysical measurements, te solutions wit tis mes can be taken as a reference against wic we can compare te results obtained wit

2.2. NUMERICAL EXAMPLES 61 Figure 2.3: Sape of te interface for te different meses and numerical conditions at t = 100s

62 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL te oter two meses. Using te medium mes, only te simulation wit bot pressure enricment and improved integration attains results nearly as good as tose obtained wit te very fine mes. Te oter two cases sow a distorted interface sape. Finally, using te coarse mes, te sape of te interface using pressure enricment and improved integration sows sligt errors but is muc better tan te oter two cases, were it can be clearly observed tat mass is not conserved. Te flow pattern is compared in Fig. 2.4. Using te finest mes tere is not muc difference between te tree metods. Two recirculations can be found, one in te bottom fluid and one in te top one. Wit te medium mes, te results obtained wit pressure enricment remain very similar to te ones obtained wit te previous mes. In te oter two cases te flow pattern is strongly modified by te errors tat originate close to te interface. Wit te coarse mes te results deteriorate in all tree cases as expected, but it can be observed tat in te case wit pressure enricment te solution is still better tan tat obtained wit a medium mes in te oter two cases. Even toug te global flow pattern obtained wit te very fine mes is nearly te same in all tree cases, a zoom at te velocity vectors close to te interface (see Fig. 2.5) reveals tat even in tis case, te pressure enricment produces a better solution. Te spurious oscillatory beavior obtained close to te interface in te cases witout pressure enricment as been observed not only in space but also in time. Te effect of reducing te time step size as been analyzed, but no significant improvements ave been obtained compared to tose resulting from te pressure enricment. Finally, it as been observed tat te convergence in te L 2 velocity norm witin eac time step is muc better using te enriced formulation tan witout it, for all tree meses. Using a 0.0001 relative convergence tolerance for te velocity, te two formulations witout enricment converge in twice or more iterations tan te enriced one. Te enriced case takes 2 iterations to converge wit te fine mes, 3 wit te medium mes, and 4 wit te coarse one.

2.2. NUMERICAL EXAMPLES 63 Figure 2.4: Flow pattern for te different meses and numerical conditions at t = 100s

64 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL Figure 2.5: Detailed flow pattern for te finest meses and different numerical conditions close to te interface Figure 2.6: Interface sape and vertical velocity band plot togeter wit velocity vectors using pressure enricment

2.2. NUMERICAL EXAMPLES 65 2.2.2 3D vertical cannel Te second example is a 20 m ig vertical cannel wit a square cross section (side lengt L =5m). Te cannel is fed from te bottom wit a eavier fluid at a constant (botinspaceandtime)1m/s velocity and te upper face is left free so tat te ligter fluid can escape. No friction is assumed on te walls. Te initial interface is flat and at 2.5 m from te entrance. Te solution for tis problem is very simple. Te velocity in te wole domain, included te interface, sould be equal to te inlet velocity and te interface sould remain flat. A 3D unstructured tetraedral mes wit 1106 nodes and 4921 elements is used. Te material properties used (SI units) are ρ 1 = 1000, µ 1 = 100 for te fluid on te bottom, and ρ 2 = 10, µ 2 = 1 for te one on top. Te time step size is 0.1 s. Te Reynolds number based on te lengt of te square section is Re = 50. In te case wit no improved integration nor pressure enricment bot Heaviside or pseudo-concentration properties ave been tested. Figure 2.7: Interface sape and vertical velocity band plot togeter wit velocity vectors using only improved integration

66 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL Figure 2.8: Interface sape and vertical velocity band plot togeter wit velocity vectors using jump properties and no enricment nor modified integration In Figures 2.6, 2.7, 2.8 and 2.9 we sow te sape of te interface and te velocity field for four different cases. In Fig. 2.6 te results wit pressure enricment are sown. Te interface remains flat as expected and its displacement corresponds to te amount of injected fluid. Wen a modified integration but no pressure enricment is used (see Fig. 2.7) te results are very poor. Te velocity sows important oscillations close to te interface and tere is an important mass loss. Te mass loss is so important tat te free surface remains nearly at its initial eigt. Witout using pressure enricment nor modified integration and approximating te material properties wit te option described as Heaviside properties, te results (sown in Fig. 2.8) are as bad te tose described for te previous case. Finally, wen pseudo-concentration properties are used (see Fig. 2.9), witout pressure enricment nor modified integration, te mass conservation improves wit respect to te previous two cases, but is worse tan tat obtained wit te formulation proposed in tis capter. As in te oter two cases witout pressure enricment, te errors in te prediction of te vertical velocity close to te interface can

2.2. NUMERICAL EXAMPLES 67 be twice te inlet velocity. Te source of tese errors is te impossibility of te sape functions to capture te discontinuous pressure gradient tat exists at te interface. Wen te pressure is enriced te velocity errors nearly disappear. Te sligt errors tat remain can be attributed to te fact tat te numerical resolution of te level set is not exact and terefore te deviations in te sape of te interface give rise to small variations in te velocity. Figure 2.9: Interface sape and vertical velocity band plot togeter wit velocity vectors using variable properties and no enricment nor modified integration 2.2.3 Slosing problem As a final example, we consider a slosing problem. It is a simple problem of free oscillation of an incompressible liquid in a container. Following [99], [100], and [98] we consider a liquid column of widt b wit an initial surface profile corresponding to te first antisymmetric mode of vibration. Te eigt of te free interface is η (x, 0) = 1.0+a sin π b x

68 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL were a is te amplitude of oscillation. Tus, te initial condition for te level set function is ψ (x, y, 0) = η (x, 0) y were x = 0 at te middle of te container and y = 0 at te bottom. Using te same parameters as in te cited papers, te amplitude is taken as a =0.01 and te kinematic viscosity as ν =0.01. Te previous papers use a Lagrangian formulation and terefore only one fluid is solved and te computational domain is allowed to deform. Since we use an Eulerian formulation te computational domain does not deform but a second fluid is also simulated. Te density and dynamic viscosity of te second fluid are 100 times smaller tan tose of te bottom one so tat it does not affect significantly te flow of te eavier fluid. Te container walls are assumed to be impermeable and allow for free slip. Te domain we use as a widt b =1andaeigt = 2 and is filled wit equal amounts of eac fluid (see Fig. 2.10). Heaviside properties are used to approximate te material properties in elements cut by te interface. Te mes as 1394 triangular elements and 747 nodes and is refined close to te interface. In Fig. 2.11 we sow te position of te interface after 11 seconds, using: (1) enriced pressures, (2) only improved integration and (3) no modification. Figure 2.12 sows te computed time istory of η ( b,t) for te tree cases togeter wit te results presented by 2 Ramaswamy et al. [100] Bot figures sow tat in te cases witout pressure enricment tere is a significant mass loss. Te enriced simulation agrees closely wit te results reported by Ramaswamy et al. [100] 2.3 Conclusions In tis capter we ave presented an enricment for te pressure finite element sape functions tat allows to improve te solution of two pase flows. Te benefits introduced by te metod sow up clearly as te gravitational forces increase. Te Froude number Fr = U 2 gl,

2.3. CONCLUSIONS 69 Figure 2.10: Mes and initial interface position for te slosing problem. Top: general view. Bottom: detail. Figure 2.11: Interface position at t = 11s for te slosing problem

70 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL 0.01 0 Heigt of wave [m] -0.01-0.02-0.03-0.04-0.05-0.06-0.07 Ramaswamy et al. Enriced press. Only mod. integrat No modification 0 5 10 15 20 25 Time [sec] Figure 2.12: Time istories of surface elevation amplitude for te slosing problem caracterizes te ratio of te inertial to te gravitational forces in free-surface flows, were U is a caracteristic velocity, g is te value of te gravitational acceleration and L is a caracteristic lengt. A modified Froude number can be used for two fluid flows Fr m = U 2 ρ 1 gl (ρ 1 ρ 2 ), were ρ 1 is te density of te eavier fluid and ρ 2 is te density of te ligter one. Te errors witout using te enricment increase as te Froude number tends to zero. As te gravitational forces tend to dominate te flow, small errors in te pressure can give rise to big errors in te velocities. Terefore te type of enricment presented in tis paper is specially useful for low modified Froude number flows. Te enricment used is local to eac element cut by te interface and can terefore be condensed prior to assembly, making te implementation quite simple on any finite element code. It is interesting to note tat, at least for te problems presented in tis paper, te proposed solution can reduce te errors by one or two orders of magnitude (see vertical tube results) and tus make problems solvable wit quite coarse meses. Despite we ave not presented numerical results for te increase in CPU time for

2.3. CONCLUSIONS 71 obtaining te system to be solved per iteration introduced by te enricment and improved integration, we ave observed tat it is generally less or muc less tan 10 percent. Taking into account tat te metod allows to reduce te number of iterations per time step and also to use coarser meses te mentioned CPU time increase is by far counter balanced. We ave not presented results for te pressure because it is difficult to do so wit a typical post-processing program in te enriced elements, but it is evident from te improvements sown for te velocities and sape of te interface tat te pressure, since it is te only unknown we ave modified, must ave also improved. Te examples sown in tis paper demonstrate tat te proposed enricment can introduce significant improvements. It allows to avoid spurious velocities and enances mass conservation.

72 CHAPTER 2. AN ENRICHED PRESSURE TWO-PHASE FLOW MODEL

Capter 3 A free surface model Free surface flows are a special kind of flows wit moving interface were te influence of one of te fluids over te oter one is negligible. As discussed in Capter 1, CFD approaces for moving interfaces problems are typically categorized into two main groups: Eulerian, fixed mes or interface capturing tecniques [18, 31, 57, 73, 90, 106, 110] and Lagrangian, and in te more general case Arbitrary Lagrangian Eulerian (ALE), moving mes or interface tracking tecniques [98 100]. Te model we will propose is clearly a fixed mes, interface capturing tecnique but it is not so obvious weter to classify it as an Eulerian or as a Lagrangian formulation. Actually te model we propose as two versions: a simplified one wic solves te momentum equations in an Eulerian manner and anoter one tat uses an ALE formulation but on a fixed mes. As we ave mentioned previously, in interface capturing tecniques a fixed computational domain is used togeter wit an interface function to capture te position of te interface. Te interface is captured witin te resolution of te fixed mes and te boundary conditions at te interface are someow approximated. In interface tracking tecniques te mes is updated in order to track te interface. Te simplest approac is to deform te mes witout canging its topology, but tis is possible only for very simple flows. As te flow becomes more complex and unsteady, remesing is required, and consequently te projection of te results from te old to te new mes are needed. In 3D calculations, tese operations can introduce costs tat can render moving 73

74 CHAPTER 3. A FREE SURFACE MODEL mes tecniques unfeasible. Tis is te main reason wy we prefer to use fixed mes metods. Bot approaces can deal wit two pase or free surface flows. In moving mes simulations [71, 87, 98] it is very common to see real free surface simulations were only one fluid is modelled and te mes follows te movement of te interface. On te oter and, in finite element fixed mes simulations free surface flows are typically treated as a particular case of two pase flows were te second fluid is air or some pseudo fluid wit a density and viscosity muc smaller tan te first fluid. Te typical Eulerian approac can work well in a great number of situations but in some cases it can fail miserably. For example, flow of different density fluids under te action of gravity. Tis problem as been analyzed in te previous capter (see also [36, 37]). Te main problem is tat since we are using a fixed mes some elements are cut by te interface, were te pressure gradient is discontinuous. Suc a pressure field cannot be accurately represented by te finite element sape functions and if te two most important terms in te momentum equation are te pressure gradient and te gravitational forces, uge errors can be introduced in te velocity field tat can spoil te simulation. In te previous capter we proposed a solution based on enricing te pressure sape functions in te elements cut by te interface tat is valid for two pase flows. In tis capter we will propose an alternative solution tat is valid only for free surface flows. Tis migt seem a disadvantage wit respect to te previous metod tat can deal wit te more general case of two pase flows. We neverteless believe tat a solution tat takes advantage of te particularities of free surface flows deserves special attention due to te fact tat a great proportion of two pase flows of practical interest are free surface flows. Since te effect of surface tension is negligible in te flows we are interested in we will not take suc effects into account in te model we propose. Te solution we propose in tis paper is to model free surface flows on fixed grids in a very similar way to te one used for two pase flows but modelling in principle only te part of te domain filled by te liquid. Since we are using a fixed grid metod we will ave elements tat are totally filled by liquid and oters only partially. Te main question is wat to do in partially filled elements. Wat we propose to do is to integrate

3.1. ALE DESCRIPTION OF THE NAVIER STOKES EQUATIONS 75 only in te part filled by te fluid. In order to do so we use special integration rules in elements cut by te front tat ave been introduced in te previous capter. Tis will allow us to impose te correct boundary conditions on te free surface, te key to te success of te metod. As we ave already mentioned, fixed mes metods generally sare two basic steps. In te first one, te motion in bot pases is found as te solution of te Navier Stokes equations for one pase flow wit variable properties. In te second one, an equation for an interface function tat allows to determine te position of te interface, and tus te properties to be assigned in te previous step, is solved. Te model we will present as two versions, one tat solves te Navier Stokes equations in an Eulerian manner and te oter one tat uses an ALE formulation. In bot versions a level set function is used to determine te position of te interface. In te next section we introduce te ALE formulation tat is used by one of versions of te model we propose. Obviously wen te domain velocity is zero te ALE formulation reduces to te Eulerian description presented in Capter 1. In te next Section we present te free surface model tat uses te fixed mes ALE formulation. In te tird Section a simplified Eulerian version of te free surface model is presented. In te fourt Section we analyze tree numerical examples tat ave already been used in Capter 2. We demonstrate tat te free surface model, in any of its two versions, allows to obtain results tat are similar or better to te ones obtained wit enriced pressure sape functions and muc better tan tose obtained wit a typical two pase Eulerian model. Most of te work we will describe in tis Capter as been publised in [38]. Furter numerical examples ave been included in [39]. 3.1 ALE description of te Navier Stokes equations Te velocity and pressure fields of an incompressible fluid in a moving domain Ω during te time interval (t 0,t f ) can be described by te incompressible Navier Stokes equations

76 CHAPTER 3. A FREE SURFACE MODEL in te ALE form: [ u ] ρ t +((u u d) )u [2µε(u)] + p = f, (3.1) u =0, (3.2) were u d is te domain velocity. Te boundary and initial conditions are te same as introduced in Capter 1. In [29] we present a detailed derivation of te ALE metod for te Navier Stokes equations. Following te steps described in Capter 1 te problem can be discretized bot in space and time. Te ASGS monolitic discrete problem associated wit te Navier Stokes equations (3.1)-(3.2), discretizing in time using te generalized trapezoidal rule, and linearizing te convective term using a Picard sceme, can be written as follows: Given a velocity u n at time tn, a domain velocity u n+θ d, unknowns at an iteration i 1 attimet n+1, find u n+θ,i te discrete variational problem: e=1 τ n+θ,i 1 1 Ω e at time tn+θ, and a guess for te V,u and p n+θ,i ρ un+θ,i u n v dω + ρ((u n+θ,i 1 u n+θ d, Ω θδt ) )un+θ,i v dω Ω + 2µε(u n+θ,i ):ε(v )dω v p n+θ,i dω v f dω Ω Ω Ω n el [ µ ] + ρ v +((u n+θ,i 1 u n+θ d, ) )v Ω + [2µε(u n+θ,i )] + ρ((u n+θ,i 1 n el 2 ( v )( u n+θ,i e=1 Ω e τ n+θ,i 1 ρq u n+θ,i + + ρ((u n+θ,i 1 n el e=1 τ n+θ,i 1 q Ω e Q,bysolving [ ρ θδt (un+θ,i u n ) u n+θ d, ) )un+θ,i + p n+θ,i f )dω=0, [ ρ θδt (un+θ,i u n ) [2µε(un+θ,i )] ] u n+θ d, ) )un+θ,i + p n+θ,i f for i =1, 2,... until convergence, tat is to say, until u n+θ,i in te norm defined by te user. dω = 0, u n+θ,i 1 and p n+θ,i ] dω p n+θ,i 1 In te ALE case te parameters τ 1 and τ 2 depend on te ALE velocity u a = u u d : τ 1 = ρ( e ) 2 4µ +2ρ e u e a, τ 2 =4µ +2ρ e u e a.

3.2. FM-ALE FREE SURFACE MODEL 77 Once te algoritm as produced a converged solution, te velocity field at t n+1 can be updated from te velocity at t n+θ by using te relation u n+1 =[u n+θ (1 θ)u n ]/θ. 3.2 FM-ALE free surface model In tis section we present a new free surface model on fixed meses. We will start from a typical Eulerian simulation for a two pase flow and describe te way in wic our model departs from it. As it as already been mentioned, a typical Eulerian simulation includes two main steps. One wic solves Eulerian two fluid Navier Stokes equations and te oter one wic determines te interface position. Bot are solved over te entire mes. Wen free surface flow is considered, te properties used in te second fluid are muc smaller tan tose in te main fluid. Te model we propose solves te Navier Stokes equations only on one fluid bounded by a moving free surface wose position, as in te typical model, is determined by te level set function. Terefore te domain Ω were we solve te Navier Stokes equations does not extend over te wole mes, but only over totally filled elements and over te filled part of elements cut by te interface. Tis is an important difference wit typical finite element simulations (Eulerian, Lagrangian or ALE) were te domain tat is simulated extends over te wole mes. In order to be able to use a domain tat includes portions of elements, special integration rules ave to be used. Te integration rule we use ere as been presented in te previous capter and consist in dividing te elements cut by te front into sub elements only for integration purposes. In a finite element setting te free surface boundary condition is a natural boundary condition, and by using enanced integration we are able to impose it correctly even if te finite element faces do not coincide wit te interface. Te possibility of imposing te correct boundary conditions on te interface witout aving to resort to a moving mes formulation is one of te key assets of te metod. It allows us to take advantage of te best of bot worlds (Eulerian and Lagrangian). If instead of n σ = 0, we would like to impose some prescribed value n σ = t on te free surface, te procedure would be

78 CHAPTER 3. A FREE SURFACE MODEL sligtly more complicated. Enanced surface integration rules would ave be to defined. Te idea could be very similar to te enanced volume integration used to integrate only on te filled part of cut elements. In eac cut element, te front (as defined by te level set) would be divided into triangles were te usual surface integration rules would be used. Tus one would be able to build te term corresponding to te Neumann boundary condition applied on te interface, v t dγ. Γ (a) positive velocity (b) negative velocity Figure 3.1: One dimensional FM-ALE example Te second main difference wit typical fixed mes interface simulations is tat we

3.2. FM-ALE FREE SURFACE MODEL 79 use an ALE description wen solving te Navier Stokes equations. Te particularity of our ALE description is tat it is used on a fixed mes. It is called FM-ALE and as been introduced in a paper on lost foam casting [59]. In tis capter we extend it to free surface flows. Te idea beind te FM-ALE metod is quite simple; if one wants to simulate a moving domain on a fixed grid using an ALE description wat one must do is to project te results obtained on te deformed domain on te portion of te fixed mes occupied by te fluid at eac time step. Te metod we propose consist of tree main steps in going from time n (were velocities and interface position are given) to time n + 1: 1) Find te interface position at step n + 1 by solving te level set equation. Te velocities obtained at step n areusedintisstep. 2) Obtain te domain velocity and fluid velocity at time n on te new domain determined in te previous step. 3) Solve te Navier-Stokes equations on te new domain using an ALE description. Te way in wic to solve te first and tird steps as already been discussed. We will now describe te way in wic to deal wit te second step. Te key point ere is ow to define te domain velocity at time n + 1. It will depend on te fluid velocity at time n and on te interface position at times n and n+1. Wewilluseteonedimensional example sown in Figure 3.1 to explain ow it is obtained. Contrary to wat appened in te original FM-ALE paper [59] were te domain could only expand, in a free surface simulation te domain can eiter expand or contract. Te domain velocity at time n + 1 will depend on te fluid velocity at time n and on te level set function at times n and n + 1. In order to describe te metod we will classify te nodes into empty and filled nodes. Te nodes belonging to cut elements will be additionally described as front nodes. Te domain velocity will be set to zero at nodes tat belong to te fluid but not to te front at times n +1 and n on te fixed mes. Tat is te case of node 1 in Figures 3.1a and 3.1b, were Figure 3.1a represents te case were te domain as a positive x velocity (it is expanding) and Figure 3.1b represents te case were te domain as a negative x velocity (it is contracting). Te domain velocity will

80 CHAPTER 3. A FREE SURFACE MODEL be set to u n+1 d = n(u n n), at nodes tat belong to te front or to te air at time n. obtained from te level set function at eac node as Te normal direction n is n = ψn+1 ψ n+1, Suc is te case of nodes 2, 3, 4 and 5 at Figure 3.1a and nodes 4 and 5 at Figure 3.1b. Using te previous conventions, in te expanding case (Figure 3.1a) te domain velocity is defined on all te nodes. In te contracting case it is not be defined at nodes 2 and 3. Tere it is obtained by solving u d =0, wit te boundary conditions defined previously. Once te domain velocities ave been obtained, te displacements in one time step are simply δx = u d δt. Tey allow us to obtain te deformed mes, sown in te middle line. Finally te mes velocities and fluid velocities at time n can be obtained on te fixed mes at time n + 1 (bottom line) by interpolating (or by projecting) on te deformed mes as sown in dotted lines. Tese two values are te ones needed to solve te Navier Stokes equations in ALE form. Following our presentation in [29] te second step can be divided into four substeps: Virtually deform te mes at time n (M n ) to a virtual mes at time n +1(Mvirt n+1 ) using classical ALE concepts and compute te mes velocity u n+1 d. Write down te ALE Navier Stokes equations on M n+1 virt. Define a new mes at time n +1(M n+1 ) formed by totally filled elements of te background mes and te filled part of cut elements. Project te ALE Navier Stokes equations from M n+1 virt to M n+1. How we compute te mes velocity u n+1 d as already been described. Tis defines te displacements and terefore M n+1 virt can be obtained. M n+1 virt issowninblueinte

3.2. FM-ALE FREE SURFACE MODEL 81 scematic of te FM-ALE approac presented in Figure 3.2. Writing down te ALE Navier Stokes equations on M n+1 virt (see [29]) leads to a system of equations tat is not actually formed wen te FM-ALE approac is used. Instead te ALE Navier Stokes equations are projected onto to M n+1. For te definition of te new mes at time n +1, free surface problems involve Neumann boundary conditions on te interface tat can be easily imposed wen enanced integration is used in te elements cut by te front as we ave already mentioned. For oter problems, tat require Diriclet conditions on te interface, we mention two possibilities in [29]; adding new nodes or imposing boundary conditions approximately. Projecting te ALE Navier Stokes equations from M n+1 virt bot u n virt and un+1 d,virt from M n+1 virt to M n+1 implies projecting to M n+1. It is important to stress tat, as it is well known in te classical ALE approac, u n is known on Mvirt n+1 because te nodes of tis mes are obtained from te motion of te nodes of M n wit te mes velocity u n+1 d,virt. Te projection of u n virt onto M n+1 clarifies te effect of te mes motion in te context of fixed mes metods. In particular, tere is no doubt about te velocity at previous time steps of newly created nodes. Since te mes velocity is computed on M n+1 virt to be projected to compute on M n+1. it also needs It is interesting to note tat p n+1 is not te projection of p n+1 virt onto M n+1. Pressure p n+1 is determined by imposing tat u n+1 is divergence free, wic at te discrete level is not equivalent to impose tat u n+1 virt is divergence free. We will now go back to describe a particularity of te metod, concerning te solution of te level set equation. We ave already mentioned te level set equation is solved over te wole mes and terefore te velocities on te region formed by non filled elements must be defined in some way. As we ave mentioned in Capter 1, tis is not a big problem, since in teory te only velocities needed to transport te interface position are tose on te interface defined by ψ = 0. Several options to define extension velocities tat allow us to obtain a velocity field on te wole mes from te velocities on te interface can be found in te literature [90]. Te approac we will use is a relatively simple one [59]: on te part of te mes formed by empty elements a stationary Stokes problem will be

82 CHAPTER 3. A FREE SURFACE MODEL solved. On te air front nodes te velocities obtained wen solving te fluid will be used as Diriclet boundary conditions. Te boundary conditions used wen solving for te extension velocities ave little influence on te resulting fluid flow and will be taken as in a typical Eulerian two fluid simulation. Te fluid properties used wen solving te Stokes problem in te empty region are tose of te fluid. Te advantages of te metod used to obtain te extension velocities is tat tey are divergence free and satisfy boundary conditions. From te description of te FM-ALE formulation it is seen tat te major differences wit respect to te classical ALE approac are te following: - Given a position of te fluid front on te fixed mes, elements cut by te front are split into subelements (only for integration purposes), so tat te front coincides wit te edges of te subelements. Tis allows to prescribe n σ = 0 as boundary condition. In fact, it is only necessary to modify te integration rule, as explained earlier. Once tis is done, te flow equations can be solved. - After deforming te mes from one time step to te oter using classical ALE procedures, results are projected back to te original mes (troug interpolation, projection wit restrictions as explained in [58] or any oter tecnique). - Te definition of te fluid front is represented by te level set function, and not by te position of te material points at te free surface as in a classical ALE metod. Note tat te previous step implies tat front nodes cannot be tracked. A scematic of te FM-ALE approac for free surface flows is presented in Figure 3.2. In essence, tis formulation is te same as te one presented in [59] wit two major differences: - In [59], were te fluid domain always expands, it was enoug to move te nodes adjacent to te front. In te present model, since te domain can also contract, a more general definition for te mes velocities ad to be adopted. Te mes velocities at time n + 1 depend on te fluid velocities at time n and on te level set function bot at times n and n + 1. Wen te domain contracts some nodes of te fluid domain are moved by solving equation (3.2) wit boundary conditions determined from te rules used to define

3.3. EULERIAN SIMPLIFIED FREE SURFACE MODEL 83 y=0 y=0 Mes velocities t n Mes to compute at t n+1 t n+1 Deformed mes Figure 3.2: Two dimensional FM-ALE scematic te mes velocity on te rest of te nodes. - In te lost foam model of [59], te front velocity is given and te pressure is unknown. In te present free surface model, te front velocity is unknown and te traction is given (n σ = 0). 3.3 Eulerian simplified free surface model In te previous section we ave presented a version of te free surface model tat solves te Navier Stokes equations on te fluid only using an ALE description. In tis Section we will introduce a simpler version, te main difference being tat an Eulerian description is used. Te model will terefore be more closely related to te typical Eulerian two pase model. Neverteless te key element of te model, solving te Navier Stokes equations only in te fluid region Ω (totally filled elements plus filled part of cut elements), will remain unaltered. Te way to understand tis model is to tink of it as a typical two pase flow model, wit a second fluid wit negligible properties and were te null traction boundary condition as been added on te interface. Tis boundary condition allows us to uncouple

84 CHAPTER 3. A FREE SURFACE MODEL te solution of te fluid from tat of te air making te problem muc simpler. We will often refer to te second fluid as air even if it may be some oter fluid. Suppose tat we are solving a typical two pase flow model using enanced integration in te elements cut by te front. Wen one looks at te discrete momentum equation it is clear tat te contribution to te matrix and rigt and side corresponding to te transient, convective, diffusive and external force terms will tend to zero in te air elements and in te part of te cut elements filled by air. Tus all te terms in te momentum equation, except for te pressure gradient, tend to te same value if one uses te free surface model we propose or a two pase model wit a negligible second fluid. In order to interpret wat appens wit te pressure term wen using te free surface metodology as compared to te two pase case, one as to observe tat if all oter terms tend to zero in te region not occupied by te liquid ten te solution tere would ave a null pressure gradient. If one introduced suc information a priori in te two pase case, ten te systems resulting from te momentum equation would be identical wen te fluid properties tend to zero. Te problem wit te two pase model, wen used wit te typical finite element functions, is tat tey cannot represent a pressure gradient tat is zero in one part of te element and different from zero in te oter. As we ave already said, since te velocity and pressure are coupled, te impossibility of accurately representing te pressure can introduce errors in te velocity tat can render te solution meaningless. Enricing te pressure sape functions is a way to solve suc problem [37], te free surface formulation we present ere is anoter solution, peraps simpler. Tey will be compared numerically in Section 5.4. Finally one also as to look at te continuity equation. Te contribution to te system matrix would be different if one uses equation (1.2). But it could be replaced by ρ u =0. (3.3) From te continuum point of view bot equations are equivalent. Te numerical approximation of equation (3.3) would lead to identical system matrices if one uses te free surface model or te two pase model. Using equation (3.3) can be seen as weigting te incompressibility constraint depending on te density of te fluid. In te free surface

3.3. EULERIAN SIMPLIFIED FREE SURFACE MODEL 85 case wat we are doing is only imposing incompressibility in te region were fluid exists. It is a pretty logical ypotesis. One important point to remark is tat as we are using enanced integration we are able to integrate only in te domain filled by liquid and tus we can impose te Neumann boundary condition corresponding to te free surface (n σ = 0) accurately exactly were te interface is located according to te level set function. Tis is a key point for te success of te metod. On te oter and, since we are only simulating one fluid we do not ave a discontinuous pressure gradient and terefore no special sape functions are needed. As in te FM-ALE free surface model described in te previous Section, te level set equation must be solved on te wole mes, not only in te liquid region. In order to reduce computational labor te level set equation could be solved only in a small region close to te interface using te narrow-band approac [104]. Once again, te velocity must be defined in te empty region someow, in order to transport te level set function. In tis case te solution adopted is very similar to te one used in te previous Section except for te fact tat instead of solving te steady Stokes equations wit fluid properties we will use te transient Navier-Stokes equations wit air (or some pseudo fluid) properties. Te boundary conditions will be te same as in te FM-ALE model. Te reason for tis coice is tat we are justifying tis model based on wat appens in te Eulerian two pase case and terefore it seems logical to solve for te air as similarly as possible as done in tat model. We would neverteless like to point out tat te way in wic te air is solved is not of great importance. Contrary to wat appens in te FM-ALE model were te velocities in te empty region are only needed to transport te level set, in te Eulerian free surface model tere is anoter reason for obtaining tose velocities. Since te fluid domain is moving and te mes is fixed tere will be nodes tat in step n + 1 belong to te fluid but in te previous step belonged to te empty region. In suc nodes we will use te velocities calculated for te empty region wen modeling te transient term. Te validity of suc approac is justified once again pointing out tat it is wit wat appens in te case of a two pase

86 CHAPTER 3. A FREE SURFACE MODEL flow wen te properties of te second fluid tend to zero. 3.4 Numerical examples In tis section we present tree numerical examples were one can appreciate te benefits te proposed formulation can provide compared to a typical two-pase flow model applied to free surface flow. Te examples are almost te same ones used in Capter 2. Te results obtained wit te free surface model will be compared wit te results obtained wit a typical two-pase flow model and also wit te enriced pressure model presented in te Capter 2. 3.4.1 Two fluid cavity Te first example, called te two fluid cavity, is a square domain filled wit equal amounts of two different density fluids and a fixed orizontal velocity (0.1 m/s) on te bottom wall. In te free surface case only te lower alf of te mes is filled by fluid. Te walls are supposed frictionless and te top face is left open. It is a very simple example but it can be representative of te numerical problems tat can appear in muc more complex problems. Te same tree unstructured triangular meses as in Capter 2 were used (see Table 2.1). Te square domain as a side lengt L =10m. Te material properties used (SI units) are ρ 1 = 1000, µ 1 = 10 for te fluid on te bottom. Te viscosity of te bottom fluid is 1000 times te viscosity of water so as to obtain a relatively low Reynolds number (Re = 100) in order to avoid unnecessary complications. Te simulations were run for 100 seconds wit a 0.5 second time step size. In te two pase flow simulations te properties of te second fluid used are 100 times smaller tan tose of te first fluid so as to be in a free surface case. Tey differ from tose used in te previous Capter were tey were similar to te properties of te first fluid because now we want te second fluid not to influence te first one. Wit suc density difference te problem turned out to be more difficult to solve and terefore for tis example te acceleration of gravity is reduced

3.4. NUMERICAL EXAMPLES 87 Free Surface FM-ALE model Enriced pressure two-pase model Normal two-pase model Figure 3.3: Sape of te interface for te different meses and numerical models at t = 100 s

88 CHAPTER 3. A FREE SURFACE MODEL Free Surface FM-ALE model Enriced pressure two-pase model Normal two-pase model Figure 3.4: Flow pattern for te different meses and numerical models at t = 100 s

3.4. NUMERICAL EXAMPLES 89 to g =1.0. In te rest of te examples it is g =10.0. In Fig. 3.3 we sow te sape of te interface for te tree meses wit tree different models: Te FM-ALE free surface model, te enriced pressure two pase model (Capter 2) and te normal two pase model. Using te finest mes te free surface model and te enriced pressure two pase model give nearly te same result. Te normal two pase model fails to predict te correct interface position. Tere is an erroneous fallofte interface created by an incorrect velocity field as is sown in Fig. 3.4. Despite tat we ave not got pysical measurements, te two correct solutions obtained wit tis mes can be taken as a reference against wic we can compare te results obtained wit te oter meses. Using te medium mes, only te free surface model attains results as good as tose obtained wit te fine mes. Te enriced pressure two pase model produces some distortion of te interface. Using te normal two pase model te solution obtained on te medium and coarse meses is so bad tat te interface as disappeared from te mes at t = 100 s and terefore no results are sown. Even wit te coarse mes te free surface model manages to obtain te correct interface position. Wit te enriced pressure two pase model an important mass loss can be observed wen using te coarse mes. Te flow pattern is compared in Fig. 3.4. Again, using te finest mes tere is not muc difference in te calculated fluid velocities between free surface model and te enriced pressure two pase model. Te velocities drawn in te empty region in te free surface case are simply te velocities used to transport te level set function. Wit te normal two pase model spurious velocities tat distort te wole flow field and ruin mass conservation can be observed close to te interface. Te free surface model manages to obtain te correct flow field bot wit te medium and coarse meses. Wit te enriced pressure two pase model spurious velocities are obtained close to te interface wit te medium mes. Tey increase in te coarse mes. For te normal two pase model no results are sown for te medium and coarse meses because, as we ave already mentioned, te fluid as disappeared at t = 100 s. Te results obtained wit te Eulerian simplified free surface model ave not been

90 CHAPTER 3. A FREE SURFACE MODEL Figure 3.5: Flow field and interface position at t = 100 s using te Eulerian free surface model sown up to now because tey are very similar to tose obtained wit te FM-ALE free surface model. Tey are presented in Figure 3.5 for te medium and coarse meses so tat tey can be compared wit te results sown previously (Figures 3.3 and 3.4) for te FM-ALE model. In te rest of te examples presented in tis Capter bot free surface models ave always given nearly te same results and terefore only te results obtained wit te FM-ALE model will be sown. 3.4.2 3D vertical cannel All te details (geometry, mes, material properties, etc.) of te second example are identical to tose presented in Capter 2 and terefore tey are not repeated. In Figure 3.6 we sow te sape of te interface and te velocity field obtained wit te free surface FM-ALE model proposed in tis Capter. For te results wit te enriced pressure two pase model see Figure 2.6. Bot models provide te correct solution; te interface remains flat and its displacement corresponds to te amount of

3.4. NUMERICAL EXAMPLES 91 Figure 3.6: Interface sape and vertical velocity band plot togeter wit velocity vectors using te FM-ALE free surface model injected fluid. Wen te normal two pase flow model is used (see Fig. 2.8) te results are very poor. Te velocity sows important oscillations close to te interface and tere is an important mass loss. Te mass loss is so important tat te free surface remains nearly at its initial eigt. Te source of te errors in te normal two pase flow model is te impossibility of te sape functions to capture te discontinuous pressure gradient tat exists at te interface. One way to solve te problem is to enric te pressure sape functions so as to represent te discontinuous pressure gradient more accurately. Te oter way to solve te problem is to use a free surface model and tus avoid te existence of a discontinuous pressure gradient by modeling only one fluid. Te sligt errors tat remain in te velocity field wit te two successful models can be attributed to te fact tat te numerical resolution of te level set is not exact and terefore te deviations in te sape of te interface give rise to small variations in te velocity. 3.4.3 Slosing problem Te tird example coincides exactly wit te one presented in Capter 2 and te details are not repeated. Te mes and initial interface position ave been given in Capter 2.

92 CHAPTER 3. A FREE SURFACE MODEL In Fig. 3.7 we sow te position of te interface after 11 seconds, using te free surface FM-ALE model. Tis results ave to be compared wit tose presented in Fig. 2.11. Figure 3.8 sows te computed time istory of η ( b,t) for te tree cases togeter wit 2 te results presented by Ramaswamy et al. [100]. Using te normal two pase model tere is a significant mass loss. Te results obtained wit te free surface and enriced pressure two pase models agree closely wit te ones reported by Ramaswamy et al. [100] Figure 3.7: Interface position at t = 11s for te slosing problem 0.01 0 Heigt of wave [m] -0.01-0.02-0.03-0.04-0.05 Ramaswamy et al. FMALE Free Surf. Enriced press. Normal 2 pase 0 5 10 15 20 25 Time [sec] Figure 3.8: Time istories of surface elevation amplitude for te slosing problem 3.5 Two computationally demanding examples In te previous section simple numerical examples tat sow te benefits of te proposed metod ave been presented. In tis Section two additional examples not included in [38]

3.5. TWO COMPUTATIONALLY DEMANDING EXAMPLES 93 are presented. Te objective is to sow tat te metods works satisfactorily for complex problems and give some idea of te size of te problems tat can be solved. In Sections 3.2 and 3.3 we ave explained ow to extend te velocities to te empty region. We proposed to solve te Stokes or Navier Stokes equations in te empty region and mentioned tat oter options could also be used. In order to reduce te computational cost a ceaper velocity extrapolation is used in tese examples. First te nodes in te empty region are classified into levels. Te first level corresponds to te empty nodes of cut elements. Ten, starting by n = 1, level n+1 is formed by te empty nodes connected to te level n until all of te nodes in te empty region ave been classified. Once tis as been done te velocities of eac of te empty nodes wit level greater tan one, can be calculated as te average of velocities of te nodes in te lower level connected to it. Te calculation is ordered so tat all te nodes in te lower level are calculated before stepping to te next level. On te boundaries te velocities are corrected so tat tey satisfy Diriclet boundary conditions. Despite tis extension velocity is not divergence free we ave found tat it does not introduce any significant difference in te resulting flow in te fluid compared to solving te flow equations in te empty region and te computational cost is muc lower. It is also used in Capter 5 were industrial mould filling examples are presented. 3.5.1 3D dam-break wave interacting wit a circular cylinder Te first problem is a 3D dam-break wave interacting wit a circular cylinder borrowed from [77]. Te domain is 20 m long and 5 m wide. In [77] a constant 10 m eigt as been used but we ave preferred to increase te eigt at te rigt end of te domain because we ave observed tat oterwise te water would reac te upper surface and extend troug it. Te initial volume is 4 m long, 5 m wide and 7 m ig. Te circular cylinder, wic as a radius r =1m and eigt =5m, is placed in te middle of te tank. An unstructured triangular mes wit 1968844 elements and 348963 nodes is used. Water properties, ρ = 1000.0 andµ =1.0 10 3 (SI units), are used in te simulation.

94 CHAPTER 3. A FREE SURFACE MODEL Te Reynolds number based on a typical velocity (10 m/s) and te diameter of te cylinder is Re =2.0 10 7. Despite te flow is turbulent, as te Reynolds number indicates, we ave been able to run tis example witout using any turbulence model [49, 63, 95]. Te viscosity introduced by te stabilization metod seems to be enoug to make te solution of te Navier Stokes equations possible. Te walls of te domain are supposed frictionless. A total of 34 seconds ave been run wit a 0.01 time step. In Figure 3.9.te evolution of te interface is sown. Te results are similar to te ones obtained in [77] but not identical. Tis is not surprising due to complexity of te flow and te fact tat no turbulence model as been used. Split OSS stabilization as been used for te Navier Stokes equations. For te convective nonlinearity Picard iteration is used. Te tolerance is set to one percent variation in te L2 norm of te velocity and a maximum of 10 iterations are allowed. Typically only one or two iterations are needed. For te solution of te monolitic system a preconditioned GMRES iterative solver [102] is used. Te stopping criteria for te solver is tat te residual is smaller tan 10 8 times te rigt and side. It usually converges in approximately 30 iterations. An ILUT preconditioner wit tresold 0.001 and filling 25 is used [102]. For te Level Set equation te convergence of te GMRES solver is very easy even witout preconditioner. Te total CPU time for te simulation as been 530932 seconds. Te resolution of te Navier Stokes equations takes most of te time, wit 232416 seconds for te matrix assembly and 225672 seconds for te linear solver. Te runs were performed on a PC wit AMD Atlon(tm) 64 X2 Dual Core Processor 4400+ running at 2.2 GHz wit 3 Gbyte of RAM using te Intel Fortran compiler under Ubuntu. 3.5.2 3D Green water problem Te second example as been experimentally studied by te Maritime Researc Institute Neterlands (MARIN) to evaluate te effects of green water flow over te deck of sips. As in te previous example it is a 3D dam-break wave but in tis case it its a box wit

3.5. TWO COMPUTATIONALLY DEMANDING EXAMPLES 95 t=0.4 s t=2.0 s t=4.4 s t=5.2 s t=8.4 s t=9.2 s t=11.2 s t=33.2 s Figure 3.9: Free surface evolution

96 CHAPTER 3. A FREE SURFACE MODEL dimensions 0.161 0.403 0.161 m. Te tank as an open roof of dimensions 3.22 1 1 m and te initial water volume is 0.55 m ig and 1.228 m long. Te smaller dimensions make tis problem someow simpler tan te previous one but te advantage is tat experimental data is available in [35]. Te problem as also been analyzed numerically in [41, 47, 69]. An unstructured triangular mes wit 1166780 elements and 206153 nodes is used. Te numerical strategy is te same as in te previous example. A total of 6 seconds ave been run wit a 0.0025 time step. Te total CPU time for te simulation as been 314644 seconds. Te resolution of te Navier Stokes equations takes most of te time, wit 116755 seconds for te matrix assembly and 112844 seconds for te linear solver. Te pressures at four of te gauges wose position is described in [35] are compared wit our numerical results in Figure 3.10. P1 and P3 are located at te face tat receives te wave impact, wile P6 and P8 are found at te top of te box. Te agreement between te numerical results and te experimental ones is very satisfactory. Some small delay (0.4 s) can be observed for te moment te return wave its te box again at about 5.0 s. Te reason for tis delay sould be explored furter. In Figure 3.11.te evolution of te interface is sown. Te agreement wit te potograps presented in [35] is very satisfactory. It is important to point out tat a smoot interface is obtained in te region far from te wave (t = 2.3 s). Tis good beavior can be attributed to te correct treatment of boundary conditions at te interface. 3.6 Conclusions In tis Capter we ave introduced a formulation for free surface flows on fixed meses. Te key difference wit typical Eulerian formulations for free surface flows is tat we do not solve te Navier Stokes equations on te wole mes. Taking advantage of te fact tat we ave a boundary condition at te free surface we can model te flow only in te fluid region. By doing so we are able to avoid te difficulty of modeling a discontinuous

3.6. CONCLUSIONS 97 12000 10000 experimental numerical 7000 6000 experimental numerical 8000 5000 pressure [Pa] 6000 4000 pressure [Pa] 4000 3000 2000 2000 1000 0 0-2000 0 1 2 3 4 5 6 time [s] -1000 0 1 2 3 4 5 6 time [s] 3000 2500 experimental numerical 3500 3000 experimental numerical 2000 2500 pressure [Pa] 1500 1000 pressure [Pa] 2000 1500 1000 500 500 0 0-500 0 1 2 3 4 5 6 time [s] -500 0 1 2 3 4 5 6 time [s] Figure 3.10: Pressure at points P1, P3, P6 and P8

98 CHAPTER 3. A FREE SURFACE MODEL t=0.4 s t=0.6 s t=1.0 s t=2.0 s t=2.3 s Figure 3.11: Free surface evolution for te 3D Green water problem

3.6. CONCLUSIONS 99 pressure gradient in te cut elements. Te use of enanced integration allows us to impose te Neumann boundary condition at a surface tat does not coincide wit element faces quite simply. On te oter and te model differs from ALE or Lagrangian simulations on te fact tat we use a fixed mes. Two versions of te model ave been proposed, one tat uses an ALE formulation on fixed meses (FM-ALE) and te oter one tat is more closely related to Eulerian formulations. Te second formulation is a simplification or approximation of te first one. Neverteless, no significant difference as been observed between te numerical results obtained wit te two versions. Te FM-ALE version calculates te velocities in te empty region only to transport te level set function. Te oter version also uses te velocities in te empty region to model te transient term in nodes tat go from te empty region to te filled one in a way tat mimics wat appens in a two pase flow wit negligible properties in te second fluid. Te model takes into account te fact tat in a free surface flow te velocities in te fluid region influence te velocities in te empty region but not te oter way around. Te numerical results ave sown tat te formulation can provide very satisfactory results for low Froude number flows were te typical two pase flow model fails. Compared wit te enriced pressure two pase model presented in Capter 2 te free surface model as produced equal or better results in all te analyzed cases. Te free surface model is simpler tan te enriced pressure two pase model because no enricment is needed. On te oter and, since te flow equations are solved in te liquid and te empty region separately, te computational cost is reduced. Te examples sown in tis paper demonstrate tat te free surface model can introduce significant improvements compared wit a typical two pase flow finite element model. It allows to avoid spurious velocities and enances mass conservation.

100 CHAPTER 3. A FREE SURFACE MODEL

Capter 4 Pressure Segregation Metods In tis Capter we will explore pressure segregation metods tat sould allow us reduce te computational cost of solving te Navier-Stokes equations. Since teir appearance in te late 1960 s wit te works of Corin [20] and Teman [107] ave enjoyed widespread popularity. Te key for suc success is tat tey allow to uncouple te velocity and pressure unknowns, leading not only to smaller, but also, better conditioned subproblems. Te idea is to continue te work presented by Santiago Badia in a recent tesis [4] concentrating mainly on te implementation and numerical testing of te algoritms presented terein, specially for te interface flows we are interested in. Te objective of tis capter is to gain some practical experience wit pressure segregation metods, test our implementations and select wic metod we sould use for our mould filling problems. Following [4] pressure segregation metods can be classified into pressure correction metods and velocity correction metods. Te former are te most well known and include te Corin-Teman projection metod and te Van Kan metod [117]. Te latter are more recent [52,67]. Te velocity correction approac we will use as been developed in [4]. Bot versions will be tested. A complete review on pressure segregation metods can be found in [50]. Tey are usually also called projection metods or fractional step metods. Te most typical approac is to first uncouple te velocity and pressure at te space 101

102 CHAPTER 4. PRESSURE SEGREGATION METHODS continuous level and ten discretize te problem. Te approac we will use in tis work is to introduce te splitting at te purely algebraic level, once te discretization as been performed. Suc approac is advocated in [93,96,113]. Te main difference between bot approaces is te way in wic te boundary conditions are approximated. In any case, wen using te discrete approac a furter approximation is usually introduced (specially wen dealing wit continuous pressure interpolations) tat makes bot approaces very similar. Predictor corrector (see [11] and references terein) versions of bot velocity correction and pressure correction metods will also be presented. Tese metods obtain te splitting in a very similar way to te previous metods, but an iterative procedure is introduced tat, wen converged, leads to te same solution as te monolitic system. In fact, te non predictor corrector versions can be seen as te first iteration of te predictor corrector versions. Finally we will reinterpret pressure correction scemes as an iterative procedure for solving te Pressure Scur Complement as suggested by Turek in [113, 115]. Tis new perspective is interesting because depending on te preconditioner cosen oter well known metods (suc as te SIMPLE sceme) can also be described. On te oter and, using suc approac, te rotational version of te pressure correction scemes, introduced in [111] and igly favored in [50], can be seen as te use of a different preconditioner from te one used to obtain te standard version. In fact, it is te preconditioner introduced in [17] and recommended in [115]. In [4] te use of te rotational version as not been tested because it is not considered necessary wen te splitting is done at te purely algebraic level (see remark 3.1). In [115], despite a discrete splitting is used, te rotational version is preferred and terefore we believe tat it could be interesting to test it.

4.1. PRESSURE CORRECTION METHODS 103 4.1 Pressure correction metods 4.1.1 Fractional Step (non Predictor Corrector) scemes As we ave already mentioned, te splitting will be introduced at te pure algebraic level, tat is, starting from te monolitic discretized problem written in matrix form (1.10). In order to simplify te presentation, only omogeneous Diriclet boundary conditions will be taken into account and te stabilization terms will not be included. Using a BDF1 time discretization, te problem ten reads: 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1, DU n+1 = 0. Te spliting can now be introduced, giving rise to an exactly equivalent problem 1 ) δt M U n + K ( U n+1) U n+1 + γgp n = F n+1, (4.1) 1 ( ) δt M U n+1 Ũn+1 + G ( P n+1 γp n) = 0, (4.2) were Ũn+1 DU n+1 = 0, (4.3) is an auxiliary variable and γ is a numerical parameter, wose values of interest are 0 and 1. At tis point te first and essential approximation is introduced K ( U n+1) (Ũn+1 ) U n+1 K Ũn+1. (4.4) Expressing U n+1 in terms of Ũn+1 using 4.2 and inserting te result in 4.3, te set of equations to be solved is 1 (Ũn+1 ) (Ũn+1 ) δt M U n + K Ũn+1 + γgp n = F n+1, (4.5) δtdm 1 G ( P n+1 γp n) = DŨn+1, (4.6) 1 ( ) δt M U n+1 Ũn+1 + G ( P n+1 γp n) = 0. (4.7) wic as been ordered according to te sequence of solution, for Ũn+1, P n+1 and U n+1. Te uncoupling of te variables as been made possible tanks to approximation 4.4.

104 CHAPTER 4. PRESSURE SEGREGATION METHODS Depending mainly on te type of finite element discretization used, a second approximation is typically introduced. Wen continuous pressure interpolations are used, it is muc ceaper to approximate DM 1 G L, wit components L ab = ( N a, 1ρ ) N b (4.8) were L is te standard approximation to te Laplacian operator divided by te density wen a constant density is used. Wen tis approximation is used te system turns out to be very similar to te one tat would be obtained if te splitting ad been introduced prior to te discretization. If DM 1 G needs to be explicitly build, as in [114], a diagonal M matrix as to be used. It can be obtained eiter by lumping te standard one or by using nodal integration. Te extension of projection metods to variable density flows is quite recent [10, 51] Wen te second approximation ( DM 1 G L )isusedandγ = 0 te original sceme proposed by Corin [20] and Teman [107] is recovered. It is usually referred to as nonincremental sceme. Kan [117], is obtained. Wen γ = 1 is used te incremental version, introduced by Van If a BDF2 time discretization ad been used, following te same steps we would ave arrived to 1 ( ) 2δt M 3Ũn+1 4U n + U n 1 + K ( U n+1) U n+1 + GP n = F n+1, 2 3 δtdm 1 G ( δp n+1) = DŨn+1, 1 ( ) 2δt M 3U n+1 3Ũn+1 + G ( δp n+1) = 0. were δp n+1 = P n+1 P n, using te notation introduced in Capter 1. A remarkable fact about te previous scemes is tat, despite te pressure gradient is treated explicitly in te equation for Ũn+1, tey turn out to be stable in time. For a complete review on stability results see [5, 52]. Te name pressure correction tat we sall use in tis work originates from te fact in te incremental version a first order extrapolation of te pressure P n+1 = P n is used

4.1. PRESSURE CORRECTION METHODS 105 to obtain Ũn+1 and in te second step a correction δp n+1 is obtained. Also in te nonincremental case an extrapolation of te pressure is used, in tis case of order zero, P n+1 = 0. Te error introduced by te splitting is one order iger tat te error of te extrapolation used for P n+1. If te error P n+1 P n+1 in any norm. is of order p, ( ) ( U P ten from (4.7) we can see tat O n+1 Ũn+1 = δt O n+1 P ) n+1 = p +1. Te non-incremental sceme as a splitting error of order 1 and te incremental one of order 2. In tis work te incremental version will always be used and te non-incremental sceme will no longer be presented. Te temporal order of te metod is given by te minimum between te order of te error introduced by te discretization of te temporal derivative and te order of te error introduced by te splitting. Even if a first order time discretization (BDF1) is used, te use of te incremental version will be advantageous wen te splitting error is bigger tan te error introduced by te discretization of te temporal derivative. Te idea of using iger order extrapolations for te pressure seems tempting. Te order of te spliting error would be reduced and te solution would rapidly tend to te monolitic solution. Unfortunately numerical experience [4, 52] sows tat pressure extrapolations of order iger tan one typically lead to unstable solutions. In te introduction we ave mentioned tat te scemes obtained at te discrete [4,93,96,113] and continuous [20,50,107] level are very similar wen te discrete Laplacian is approximated by te continuous one. Te difference is te Diriclet boundary condition appliedonteendofstepvelocityu. Wen te splitting is introduced at te continuous level only te normal component is prescribed. Instead wen te discrete approac is used all components are prescribed. 4.1.2 Predictor Corrector sceme Te predictor corrector metod we will describe as been proposed in [33,105]. Witout taking into account te stabilization terms and using a BDF1 discretization it reads 1 δt M ( U n+1,i+1 U n) + K ( U n+1,i) U n+1,i+1 + GP n+1,i = F n+1, (4.9) δtl ( P n+1,i+1 P n+1,i) = DU n+1,i+1, (4.10)

106 CHAPTER 4. PRESSURE SEGREGATION METHODS were te superscript i indicates te iteration number. In [4,33,105] it is supposed tat te uncoupling is dealt wit in te same iterative loop as te one used for te linearization of te convective term. In our implementation we ave used nested loops for te uncoupling and te linearization of te convective term. Te uncoupling loop as been set as te outer loop. In tis way, wen only one linearization iteration is permitted per uncoupling iteration, te usual predictor corrector metod wit a single loop is recovered. On te oter and wen only one uncoupling iteration is permitted te fractional step version is recovered. It is obvious tat wen te metod converges (P n+1,i+1 = P n+1,i ) te solution of te original monolitic sceme (1.10) is recovered. DM 1 G could ave been used instead of L. How tis would affect convergence could be interesting to analyze. Te inclusion of te term δtl (P n+1,i+1 P n+1,i ) is motivated by wat appens in te fractional step (non predictor corrector) sceme. If instead of using a first order sceme, a BDF2 time discretization ad been used, te iterative sceme would read [26] 1 2δt M ( U n+1,i+1 4U n + U n 1) + K ( U n+1,i) U n+1,i+1 + GP n+1,i = F n+1, δt 2 3 L ( P n+1,i+1 P n+1,i) = DU n+1,i+1. Contrary to wat appens in te fractional step version, for P n+1,0 and U n+1,0 extrapolations of order iger tan one can be used witout compromising te stability because at eac time step te metod converges to te monolitic solution. In [26] te use of a second order extrapolation reduces te number iterations needed to converge to te monolitic solution compared to a first order extrapolation in some numerical examples. In [33] te predictor corrector version is initially written wit bot Ũ and U, but as in te converged case Ũ = U, finally te sceme is written wit only one velocity as in (4.9,4.10). In practice one usually does not converge up te absolute zero but only up to some finite tolerance or even some fixed number of iterations. In suc case, we believe tat one sould work wit bot velocities (Ũ and U). Te split predictor corrector sceme,

4.1. PRESSURE CORRECTION METHODS 107 using a BDF1 time discretization, would ten read: 1 (Ũn+1,i+1 ) (Ũn+1,i ) δt M U n + K Ũn+1,i+1 + GP n+1,i = F n+1, (4.11) δtl ( P n+1,i+1 P n+1,i) = DŨn+1,i+1, (4.12) 1 ( ) δt M U n+1 Ũn+1,i+1 + G ( P n+1,i+1 P n+1,i) = 0. (4.13) Equation 4.13 is solved at te end of te iterative process, tat is wen (4.11) and (4.12) ave converged to te user prescribed tolerance. Te OSS stabilized sceme So as to conclude tis section we will present te Split OSS stabilized version of te pressure correction predictor corrector sceme using a BDF1 time discretization and te matrices introduced in Capter 2. Using te notation introduced in Capter 1 te matrix version of te problem reads as follows: 1 (Ũn+1,i+1 ) (Ũn+1,i ) δt M U n + K Ũn+1,i+1 + GP n+1,i ( ) ( ) +S u τ n+1,i 1 ; Ũn+1,i Ũn+1,i+1 S y τ n+1,i 1 ; Ũn+1,i Y n+1,i +S d ( τ n+1,i 2 δtdm 1 G ( P n+1,i+1 P n+1,i) S p ( τ n+1,i+1 1 ) ( Ũn+1,i+1 S w τ n+1,i) W n+1,i = F n+1, (Ũn+1,i+1 M π Y n+1,i+1 C 2 ) ( P n+1,i+1 + S z τ n+1,i+1) 1 Z n+1 DŨn+1,i+1 =0, ) Ũn+1,i+1 = 0, M π Z n+1,i+1 G π P n+1,i+1 = 0, M π W n+1,i+1 DŨn+1,i+1 = 0, 1 ( ) δt M U n+1 Ũn+1,i+1 + G ( P n+1,i+1 P n+1,i) = 0. We ave included te terms corresponding to τ 2 0 tat ave been neglected in [4,33,105]. Tese terms elp to enforce incompressibility. In two fluid flows, were it is important to conserve te mass of eac fluid, te influence of suc terms is someting tat seems interesting to explore numerically.

108 CHAPTER 4. PRESSURE SEGREGATION METHODS If te difference between U and Ũ is neglected, as as been done in te previous publications [4,33,105], it as been observed [33,105] tat te previous equations can be seen as an iterative procedure for solving te monolitic problem freezing te pressure gradient in te momentum equation. Te term δtl (P n+1,i+1 P n+1,i ) as been motivated by wat appens in te fractional step case. If it ad been omitted te same iterative procedure could ave been used tanks to te inclusion of matrix S p. In [33, 105] bot options ave been tested and it as been pointed out tat witout te inclusion of te Laplacian convergence turns out to be muc arder. 4.2 Te Pressure Scur Complement approac Following Turek [113,115], in tis section we will present te Pressure Scur Complement approac tat allows us to obtain anoter interpretation of pressure correction metods. An interesting feature of tis approac is tat is allows to describe not only pressure corrections metods but also oter well known solution scemes, suc as SIMPLE or Uzawa iterations. Related approaces can also be found in [5]. In order to present te metod we will start from te matrix version of Navier Stokes equations obtained after discretization bot in space and time (BDF1) using te notation introduced in Capter 1. For simplicity te stabilization terms will be omitted. Te problem is ten to find U n+1 and P n+1 given U n, F n+1 and δt suc tat 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1, Te previous equations can be rewritten as DU n+1 = 0. MU n+1 + δtk ( U n+1) U n+1 + δtgp n+1 = δtf n+1 + MU n, DU n+1 = 0. were in order to adapt te presentation to te one used in [115] S and F are defined: S = M + δtk ( U n+1)

4.2. THE PRESSURE SCHUR COMPLEMENT APPROACH 109 F =δtf n+1 + MU n Ten te (nonlinear) algebraic problem to be solved is: given U n, F and δt, solvefor U = U n+1 and P = P n+1 SU + δtgp = F, (4.14) DU = 0. (4.15) S and F may vary depending on te time discretization or on te linearization used for te convective terms but te system to be solved is always of te form (4.14,4.15). Assuming S 1 exists, problem (4.14,4.15) is equivalent to te following scalar pressure Scur complement formulation DS 1 GP = 1 δt DS 1 F. (4.16) Once te pressure P is known, te corresponding velocity vector U satisfies U = S 1 (F δtgp). Te idea is to present te problem as a scalar problem for P so tat te knowledge about efficient iterative scemes for suc problems can be applied. One of te possibilities is to perform a preconditioned Ricardson iteration, were C 1 is an appropriate preconditioner for te pressure Scur complement DS 1 G. Te basic iteration for te pressure Scur complement equation is ten: given P i obtain P i+1 P i+1 = P i C 1 ( DS 1 GP i + 1 δt DS 1 F ). (4.17) In [115] it is proposed tat C 1 = α R T 1 + α D M 1 P, (4.18) were T := DM 1 L G, M L and M P are te diagonal (lumped) mass matrices corresponding to te velocity and pressure respectively and α R, α D are damping parameters. We ten

1. Solve for Ũi+1 SŨi+1 = F δtg P i 110 CHAPTER 4. PRESSURE SEGREGATION METHODS ave P i+1 = P i [ ] ( α R T 1 + α D M 1 P DS 1 GP i + 1 ) δt DS 1 F = P i + [ ] ( α R T 1 + α D M 1 P 1 δt DS 1 F + 1 ) δt DS 1 δtg P i = P i + [ α R T 1 + α D M 1 P ] 1 ( δt DS 1 F δtg P i) } {{ } Ũ i+1 (4.19) Now te previous iterative procedure can be split into 4 substeps tat are equivalent to tose we ave proposed for te predictor-corrector sceme in te previous section. Given P i and F perform te following 4 substeps to obtain P i+1 : 2. Calculate a rigt and side F P for te preconditioning step F P = 1 δt DŨi+1 3. Solve an update-equation for te pressure TQ i = F P 4. Update te new pressure P i+1 = P i + α R Q i +α D M 1 P F P (4.20) Step 1 corresponds to te first step of our predictor corrector sceme (4.11). Wen α R =1andα D = 0 steps 2, 3 and 4 are equivalent to te second step of te predictor corrector sceme (4.12). Te step tat obtains U (4.13) as not been included because as in te predictor corrector case it is performed once te iterative procedure as converged. Using te notation we ave just introduced it reads U =Ũ δtm 1 L GQ

4.2. THE PRESSURE SCHUR COMPLEMENT APPROACH 111 were we ave omitted te superindexes for Ũ and Q to indicate tat tey correspond to te converged solutions. If only one iteration is performed te fractional step sceme is recovered. If α R =1andα D = µδtte rotational version of te pressure correction scemes introduced in [111] and igly favored in [50] is obtained. Te difference is tat, as presented in [115], it is derived on te discrete level as an optimal preconditioner for te Stokes part of te Scur complement equation instead of as modifications to te differential operators at te continuous level. In [4] te rotational version is not considered necessary wen te splitting is done at te purely algebraic level (see remark 3.1). Since in [115] splitting is introduced at te discrete level and te preconditioner wit a non-zero α D (corresponding to te rotational form) is preferred, we believe tat it would be interesting to test it for our problems. Recently, in [5] it as been recognized tat remark 3.1 in [4] was not correct. It is clarified tat te error in te pressure close to Diriclet boundaries is also present wen te splitting is done at te purely algebraic and te discrete Laplacian is used. An alternative interpretation for te error in te pressure close to Diriclet boundaries can be found in [15]. Despite te solution is coincident wit te rotational version no reference is made to [111]. Wy te standard fractional step sceme introduces a spurious boundary condition close to Diriclet boundaries and ow te rotational form corrects tis error is easier to see wen te fractional step sceme is introduced at te continuous level tan wen it is introduced at te discrete level, as we do in tis work. Terefore we refer te reader to [50, 111] were te continuous approac is used. Rotational form for pressure stabilized scemes Wen we tried to implement te rotational version of te fractional step sceme we found tat it ad not been applied to pressure stabilized elements. From (4.20) one can see tat te pressure is obtained by adding two corrections to extrapolation of te pressure P n+1 (= P n in te incremental case). Te first correction is te one used bot in te standard and rotational versions of te metod and te second one only in te rotational

112 CHAPTER 4. PRESSURE SEGREGATION METHODS version. We can call tem δp 1 and δp 2 respectively. How to obtain δp 1 in te pressure stabilized case as already been described and is well known. In order to obtain δp 2 a naive approac can be to proceed as in te non stabilized case, tat is δp 2 = α D M 1 P F P = µm 1 P DŨi+1. Wen we implemented tis option pressure stability was lost. In te non stabilized case, te approximation to te inverse of te pressure Scur complement (4.18) is built as te sum of two inverses. Te first one approximates te inverse of te pressure Scur complement closely wen te transient term is more important tan te viscous and convective terms (it is related to δp 1 ) and te second one wen te viscous term is dominant (it is related to δp 2 ). For te pressure stabilized case we now proceed as in te non stabilized case. Using Split OSS and only stabilizing te pressure to simplify te presentation, te algebraic problem to be solved is: given U n, F and δt, solveforu = U n+1 and P = P n+1 SU + δtgp = F, DU + S p P = S z Z. It leads to te following scalar pressure Scur complement formulation ( DS 1 G + 1 ) δt S p P = 1 δt S zz 1 δt DS 1 (F ). As in te non stabilized case we can define te approximation to te pressure Scur complement as te sum of two inverses C 1 = C 1 1 +C 1 2. Te first one sould approximate te pressure Scur complement closely wen te transient term dominates and te second one wen te viscous term dominates. Similarly to wat is done in te non stabilized case for te stabilized case we ave C 1 1 = ( DM 1 L G + 1 ) 1 δt S p and ( 1 C 1 2 = µδt M P + 1 ) 1 δt S p.

1. Solve for Ũi+1 SŨi+1 = F δtg P i 4.2. THE PRESSURE SCHUR COMPLEMENT APPROACH 113 In te stabilized case C 2 is no longer diagonal and a system needs to be solved for δp 2 ( ) 1 µ M P + S p δp 2 = DŨi+1 S p P i + S z Z. Fortunately te matrix is well conditioned and te system is easy to solve. We can now proceed as we ave done in te non stabilized case to sow tat one iteration of te preconditioned Ricardson iteration for te pressure Scur complement corresponds to te fractional step sceme (standard or rotational form). By doing tis we can arrive to te fractional step pressure stabilized rotational sceme written in te usual form and sow te error introduced by te splitting. We now rewrite te preconditioned Ricardson iteration for te stabilized pressure Scur complement P i+1 = P i [ ] ( C 1 1 + C 1 2 DS 1 GP i + 1 δt DS 1 F + 1 δt S pp i 1 ) δt S zz = P i + [ ] C 1 1 + C 1 2 1 ( δt DS 1 F δtg P i) 1 } {{ } δt S pp i + 1 δt S zz. As in te non stabilized case we can now split te iterative procedure into 4 substeps Ũ i+1 2. Calculate a rigt and side F P for te preconditioning step F P = 1 δt DŨi+1 1 δt S pp i + 1 δt S zz 3. Solve an update-equation for te pressure C 1 δp i 1 = F P (4.21) C 2 δp i 2= F P (4.22) 4. Update te new pressure P i+1 = P i + δp i 1 + δpi 2

114 CHAPTER 4. PRESSURE SEGREGATION METHODS Since we are interested in te fractional step sceme tat involves only one iteration te values at i + 1 are associated wit te values at n +1. For te prediction (i =0)we use values at n and we omit te indexes in δp 1 and δp 2. From te first step we can write 1 (Ũn+1 ) δt M U n + KŨn+1 + GP n = F n+1. (4.23) From (4.21) we ave ( δtdm 1 G + S p ) (δp1 )+S p P n S z Z n+1 + DŨn+1,i+1 = 0. (4.24) Using te exact continuity equation DU n+1 + S p P n+1 S z Z n+1 = 0 (4.25) and P n+1 = P n + δp 1 + δp 2, (4.26) we can rewrite (4.24) as ( δtdm 1 G ) (δp 1 )+S p δp 2 + DU n+1 DŨn+1,i+1 = 0, tat multiplied by 1 δt MD 1 gives G (δp 1 )+ 1 δt M ( U n+1 Ũn+1 ) = 1 δt MD 1 S p δp 2. (4.27) Equations (4.23,4.27 and 4.25) are te pressure stabilized rotational counterpart of (4.5,4.2 and 4.3). Adding (4.23) and (4.27) and using (4.26) we obtain 1 δt M ( U n+1 U n) + KŨn+1 + GP n+1 G (δp 2 )+ 1 δt MD 1 S p δp 2 = F n+1 wic can be compared wit te exact momentum equation, 1 δt M ( U n+1 U n) + KU n+1 + GP n+1 = F n+1, to clarify te error we ave introduced wit te splitting. In te standard (non rotational, δp 2 = 0) case we recover te original approximation (4.4). Wen te rotational form is

4.2. THE PRESSURE SCHUR COMPLEMENT APPROACH 115 used, te approximation is KU n+1 KŨn+1 G (δp 2 )+ 1 δt MD 1 S p δp 2 were δp 2 as been defined in (4.22). Actually in our implementation we ave omitted te term in te rigt and side of (4.27). Tis implies perturbing te continuity equation (4.25) wit S p δp 2. Now te approximation used in te momentum equation is KU n+1 KŨn+1 G (δp 2 ). Contrary to wat appens in te usual pressure correction fractional step sceme our implementation introduces a perturbation in bot te momentum and continuity equations and not only in te momentum equation. Relation wit oter metods Finally it is interesting to sow, following [115], ow some oter well known scemes can also be described as pressure Scur complement tecniques. Uzawa like iterations can be associated to te coice α R = 0. In fact for stationary calculations tat is te coice Turek [115] recommends, wit α D µ. SIMPLE like scemes can be associated to a Scur complement iteration wen te preconditioner for te pressure Scur complement is taken as C 1 = D S 1 G,were S is some approximation to S, for example its diagonal or te diagonal matrix obtained by summing its rows (if it does not lead to a zero diagonal). Instead of using a Ricardson preconditioned iteration to solve for (4.16) as in (4.17) a more elaborate preconditioned sceme, suc as preconditioned GMRES, could be used. Tis leads to a metod very similar to te one proposed in [42, 43, 68, 74]. Te use of a GMRES iteration sould make te metod more robust and improve convergence compared to a Ricardson iteration. We ave implemented a preliminary version of te metod proposed in [42,43,68,74] but very limited testing as been done and no conclusion can be drawn for te moment. Regarding te approximation of te inverse of te Scur complement for te pressure, C 1, a more elaborate version was introduced in [74]. In te Stokes case it reduces to (4.18) but wen te convective term is present it is supposed to improve te approximation. It reads C 1 = M 1 P A P T 1 (4.28)

116 CHAPTER 4. PRESSURE SEGREGATION METHODS were te matrices M 1 P and T 1 are te ones introduced in (4.18). Actually in [42, 43, 68, 74] te ceaper approximation L 1 is used instead of T 1. A p is a discrete approximation to te convection-diffusion operator on te pressure finite element space tat includes te terms used to stabilize te convective part, A p = 1 δt M p+k p (U)+S c (τ 1 ; U). K p and S c are given by K p (U) ab = ( N a,ρ u N b) + ( N a,µ N b), S c (τ 1 ; U) ab = ( τ 1 u N a,ρ u N b), were, as in Capter 1, we denote te node indexes wit superscripts a, b and te standard sape functions of node a by N a. Te approximation of te inverse of te Scur complement for te pressure, C 1 as been used in metods tat converge to te monolitic solution at eac time step. It could also be interesting to test it in fractional step like metods. Since C 1 is a better approximation tan C 1 in te Navier Stokes case it could be used in (4.19) to obtain an enanced fractional step sceme. 4.3 Velocity correction metods In tis Section te velocity correction pressure segregation metods based on a Discrete Pressure Poison Equation, as suggested in [6] will be introduced. Tey are called velocity corrector metods because it is te velocity, and not te pressure, tat is extrapolated in te first step of te metod. In [4] te appearance of velocity correction metod is attributed to Guermond and Sen [52] but it can also be related to te sceme introduced by Karniadakis, Israeli and Orzag [67]. Moreover we would like to mention tat an algoritm presented [44] and recommended in [48] as several similarities wit te velocity correction metod proposed in [6]. Te main particularity of te metod proposed in [4] is tat it is obtained at te discrete level, as as been done for te pressure correction sceme. Te continuity

4.3. VELOCITY CORRECTION METHODS 117 equation is replaced by a discrete pressure Poisson equation obtained from te monolitic discretized problem. Te predictor-corrector version is also presented. 4.3.1 Te Discrete Pressure Poison Equation Te discrete pressure Poisson equation (DPPE) is obtained from te monolitic problem discretized bot in space and time. Te matrix form of te monolitic problem (1.10), supposing only omogeneous Diriclet boundary conditions and neglecting te stabilization terms for a BDF1 time discretization reads: 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1, (4.29) DU n+1 = 0. (4.30) If te momentum equation (4.29) is multiplied by δtdm 1 and te resulting equation is subtracted from te continuity equation (4.30), te DPPE is obtained δtdm 1 GP n+1 = δtdm 1 ( F n+1 K ( U n+1) U n+1) + DU n. (4.31) Te system formed by (4.29, 4.31) is equivalent to te original monolitic discretized sceme (4.29, 4.30) and te boundary conditions arise naturally from te original sceme. Tere is no advantage in solving te coupled system tat uses te DPPE equation directly. Te advantage is tat te segregation is now straigt forward. It is also interesting to point out tat an alternative DPPE can be obtained if te momentum equation (4.29) is multiplied by δtdm 1 L instead of δtdm 1 were M L is te lumped mass matrix. It reads ( δtdm 1 L GPn+1 = δtdm 1 L F n+1 K ( U n+1) U n+1 1 δt M ( U n+1 U n)) + DU n+1. (4.32) Te system formed by (4.29, 4.32) is also equivalent to te original monolitic discretized sceme (4.29, 4.30). Tis does not appen if one naively approximates DM 1 G by DM 1 L G in (4.31). In tat case, te system formed by (4.29) and te approximation to (4.31) is only an approximation to te original monolitic discretized sceme. We would like

118 CHAPTER 4. PRESSURE SEGREGATION METHODS to point tis out because we ad initially taken te naive approac. In tat case, te predictor corrector sceme does not converge to te monolitic solution. Moreover, from te implementation point of view, we belive tat (4.32) is also advantageous because it leads to te use of te same matrix, 1 δt M+K (Un+1 ), in bot te DPPE and te momentum equation. Instead if (4.31) is used, in te DPPE only K (U n+1 )appears. As we discuss in Subsection 4.3.3, an extrapolation is needed for U n+1 to obtain te fractional step sceme. Wen DM 1 G is used, te extrapolation is only needed for δtdm 1 K (U n+1 ) U n+1. Instead if te lumped mass matrix is used, it is needed for bot δtdm 1 L K (Un+1 ) U n+1 and D ( M 1 L MUn+1 + U n+1). Our numerical experience indicates tat te use of te extrapolation in te second term introduces no difficulties because M L is a good approximation to M. Before introducing te resulting fractional step and predictor corrector velocity correction scemes we will discuss te approximation of DM 1 G (or DM 1 L G). 4.3.2 Approximation of DM 1 G Wen using continuous pressure interpolations, te construction of DM 1 G is relatively expensive even if a diagonal mass matrix is used. Terefore, it is usually approximated as DM 1 G L, wit components L ab = ( N a, N b). In [6] te following enanced approximation is introduced were DM 1 GP n+1 = LP n+1 + ( DM 1 G L ) P n+1 LP n+1 + ( DM 1 G L ) Pn+1 p (4.33) P n+1 p is an extrapolation of order p of P n+1 obtained from previous known values in te case of te fractional step sceme. Te DPPE ten reads ( ) ( δtl P n+1 n+1 P p = δtdm 1 F n+1 K ( U n+1) ) U n+1 G P n+1 p + DU n. (4.34) In te predictor corrector case, in te first iteration P n+1 p will be used, but in te rest of te iterations te value from te previous iteration will be used. Te DPPE ten reads δtl ( P n+1,i+1 P n+1,i) = δtdm 1 ( F n+1 K ( U n+1) U n+1 GP n+1,i) + DU n. (4.35)

4.3. VELOCITY CORRECTION METHODS 119 wit P n+1,i = P n+1 p for i = 0. As we explain in te next Subsection, te velocity U n+1 is extrapolated from te values at previous time steps. In te predictor corrector case, one could even decide to use a separate iterative loop for solving for te previous approximation, as we ave already suggested for te convective term in te pressure correction sceme. In tat case, te iteration index i would not correspond to te outer iterative loop for te uncoupling of te unknowns but to an internal iterative loop for solving iteratively te exact DPPE witout needing to form DM 1 G. We could ten rewrite (4.35) as using a Ricardson iteration to solve for wit Te iterative procedure would ten read DM 1 GP n+1 = X X = DM 1 ( F n+1 K ( U n+1) U n+1) + 1 δt DUn. LP n+1,i+1 = X DM 1 GP n+1,i + LP n+1,i. (4.36) Tis iterative process allows us to solve for DM 1 GP n+1 witout needing to assemble DM 1 G. Its efficiency, as compared to solving directly for DM 1 G is someting we will ave to test numerically. Moreover, now one can use a non diagonal mass matrix, someting tat is not possible if one wants to solve directly for DM 1 G. Seen as an iterative solver for DM 1 G, it is obvious tat te previous approac can be applied not only to te predictor corrector sceme but also te fractional step sceme. If only one iteration is allowed, we recover te enanced approximation suggested in [6] if P n+1,i=0 = P n+1 p ( P n+1,i=0 = P n+1,j in te predictor corrector case, were j is te outer loop for te uncoupling of te unknowns) and te usual one (DM 1 G L) ifp n+1,0 = 0. We ave discussed te approximation (or solution) of DM 1 G wen dealing wit te velocity correction metod but te same ideas can be applied to pressure correction metods. Obviously te same approximation (or iterative solution sceme) can be used for DM 1 L G. A more elaborate and robust option migt be to solve for te discrete Laplacian wit a conjugate gradient metod using as preconditioner te continuous Laplacian.

120 CHAPTER 4. PRESSURE SEGREGATION METHODS 4.3.3 Fractional step sceme In order to obtain te fractional step sceme we start from te coupled system written wit a DPPE (4.29, 4.31). Te metod is called a velocity correction metod because it is te velocity tat is extrapolated from values at previous time steps instead of te pressure. In te first step te pressure is obtained from te DPPE using an extrapolation of order q (denoted by Ũn+1 q )oftevelocityu n+1. Ten, U n+1 is obtained from te momentum equation (velocity correction step). Using a BDF1 time discretization, te split sceme reads δtdm 1 GP n+1 = δtdm 1 (F n+1 K (Ũn+1 q ) Ũn+1 q ) + DU n, (4.37) 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1. (4.38) Using approximation (4.33) for DM 1 G we can obtain te following system: ( ) ( δtl P n+1 P n+1 p = δtdm 1 F n+1 K (Ũn+1 q ) Ũn+1 q G P n+1 p 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1. A first order metod in time can be obtained taking q = p =0, δtlp n+1 = δtdm 1 F n+1 + DU n, ) + DU n, 1 δt M ( U n+1 U n) + K ( U n+1) U n+1 + GP n+1 = F n+1. For a sceme wit second order accuracy in time BDF2 time discretization and q = p =1 must be used. Te resulting system is ten 2 3 δtl ( P n+1 P n) = 2 3 δtdm 1 ( F n+1 K (U n ) U n GP n) + D ( 4 3 Un 1 ) 3 Un 1, 1 2δt M ( 3U n+1 4U n + U n 1) + K ( U n+1) U n+1 + GP n+1 = F n+1. Te first remarkable fact about te previous equations is tat despite we are using a velocity correction metod wic sould be caracterized by te fact tat in going from one step to te next only an extrapolation for te velocity is used, actually extrapolations

4.3. VELOCITY CORRECTION METHODS 121 for bot te velocity and te pressure are used. Te need for te pressure extrapolation comes from te use of an approximation of DM 1 G (enanced or not). Te reason for needing te pressure extrapolation is terefore different to te reason for needing te velocity extrapolation (tat is te true spirit of te velocity correction metod) but, in any case, if te approximation of te discrete Laplacian is used, bot are needed. Instead in te pressure correction metod, only an extrapolation of te pressure is needed no matter weter te discrete Laplacian or an approximation to it is used. Terefore, we can say tat in order to obtain a pure velocity correction sceme te discrete Laplacian must be used. As we ave mentioned in te pressure correction case, extrapolations of order iger tan one sould elp to reduce te splitting error but can cause instabilities. In [4] a tird order metod tat used BDF3 time discretization and second order extrapolations for bot te velocity and te pressure wit te enanced approximation for te discrete Laplacian was tested. Numerical experimentation sowed tat it was unstable as appens for tird order pressure correction metods. In tis work we ave solved te discrete Laplacian, directly or using a Ricardson iteration, to obtain a pure tird order velocity correction metod tat only uses second order velocity extrapolation. Tis as allowed us to obtain a tird order metod tat as sown to be stable in numerical examples we present at te end of tis capter. Wen te same cases are run wit te tird order metod wit an approximation of te discrete Laplacian used in [4] tey are unstable (also if a pressure correction metod is used). Moreover we ave used te velocity correction BDF3 sceme wit an approximation to te discrete Laplacian wit a first order pressure extrapolation and second order velocity extrapolation. In tat case, te tird order accuracy is lost but stability is recovered. Terefore we can guess tat te instability observed in [4] was caused by te use of second order pressure extrapolations and tat different conclusions could ave been drawn if a pure velocity correction sceme ad been used. Instabilities for tird order velocity correction scemes are also observed in [50] were te fractional step sceme is obtained at te continuous level precluding te possibility of using a discrete Laplacian. Second order velocity extrapolations may work better tan second order pressure

122 CHAPTER 4. PRESSURE SEGREGATION METHODS extrapolations because te velocity satisfies an evolutionary equation; instead te pressure adapts itself instantaneously to satisfy te incompressibility constraint. From te convergence analysis of different pressure segregation metods it is known tat te error estimates for te velocity are sarper tan for te pressure [4]. Even in te monolitic case, were te order of te velocity error depends only on te time integration sceme used, pressure errors of order equal or iger tan two cannot always be obtained for time integration scemes of order two or iger [56]. Te tird order accurate sceme used in te numerical examples is obtained by combining a BDF3 time discretization and a second order velocity extrapolation (q = 2). It reads 6 11 δtdm 1 GP n+1 = 6 ( 11 δtdm 1 F n+1 K (Ũn+1 q ) ) ( 18 Ũn+1 q +D 11 Un 9 11 Un 1 + 2 ) 11 Un 2, 1 6δt M ( 11U n+1 18U n +9U n 1 2U n 2) + K ( U n+1) U n+1 + GP n+1 = F n+1. Altoug we do not ave an analytical proof but only numerical evidence, tis is one of te few [67, 84] tird order scemes in wic velocity and pressure are segregated of wic we are aware. 4.3.4 Predictor corrector sceme Starting from te coupled system were mass conservation is enforced by te DPPE (4.29, 4.31) te obtention of te predictor corrector sceme arises naturally. No additional terms ave to be introduced as appens in te pressure correction case. Denoting by a superscript i te it iteration of te sceme, te resulting predictor corrector metod for a BDF1 time discretization and Picard linearization of te convective term is: δtdm 1 GP n+1,i+1 = δtdm ( 1 F n+1 K ( U n+1,i) U n+1,i) + DU n, 1 δt M ( U n+1,i+1 U n) + K ( U n+1,i) U n+1,i+1 + GP n+1,i+1 = F n+1. DM 1 G can be approximated or solved as we ave already discussed. If approximation (4.35) is used we obtain δtl ( P n+1,i+1 P n+1,i) = δtdm 1 ( F n+1 K ( U n+1,i) U n+1,i GP n+1,i) + DU n,

4.3. VELOCITY CORRECTION METHODS 123 1 δt M ( U n+1,i+1 U n) + K ( U n+1,i) U n+1,i+1 + GP n+1,i+1 = F n+1. Te metod as to be properly initialized, preferably starting te process wit a splitting error at least of te same order as te time discretization. For te previous equations U n+1,0 = Ũn+1 q and P n+1,0 = n+1 P p, wit p = q = 0 can be used but a better convergence is obtained if a second order splitting (p = q =1)isused[4]. Te second order metod, using BDF2 time discretization, reads 2 3 δtl ( P n+1,i+1 P n+1,i) = 2 ( 3 δtdm 1 F n+1 K ( U n+1,i) U n+1,i GP n+1,i) ( 4 +D 3 Un 1 ) 3 Un 1, 1 2δt M ( 3U n+1,i+1 4U n + U n 1) + K ( U n+1,i) U n+1,i+1 + GP n+1,i+1 = F n+1, wit appropriate initializations U n+1,0 = Ũn+1 q and P n+1,0 = n+1 P p. As in te pressure correction case for te predictor corrector sceme ig order extrapolation can be used because te stability is not compromised. Wen te iterative procedure converges te solution of te monolitic system is recovered. If te discrete Laplacian wit lumped mass matrix is used, as in (4.32), te BDF2 velocity correction sceme reads 2 3 δtdm 1 L GPn+1,i+1 = 2 3 δtdm 1 L ( F n+1 K ( U n+1,i) U n+1,i) +DM 1 L M ( U n+1,i + 4 3 Un 1 3 Un 1 ) + DU n+1,i, 1 2δt M ( 3U n+1,i+1 4U n + U n 1) + K ( U n+1,i) U n+1,i+1 + GP n+1,i+1 = F n+1. Now only U n+1,0 needs to be initialized. Instead of treating te nonlinearity wit te same loop as te coupling of te variables an internal loop could be used for te convective nonlinearity, as as been mentioned in te pressure correction sceme. In tat case te term corresponding to matrix K in te momentum equation would be replaced by K (U n+1,i+1 ) U n+1,i+1 and an internal loop would be used to solve it. As in te pressure correction case, we ave implemented nested loops wit te linearization of te convective term treated in te inner loop in order to ave a metod tat allows us to recover te usual predictor corrector version witout

124 CHAPTER 4. PRESSURE SEGREGATION METHODS nested loops and te fractional step sceme. For te velocity correction case we ave also implemented te alternative tat deals wit te convective nonlinearity in te outer loop. Tis version resembles te monolitic version more closely and it is ceaper because te matrix needs to be assembled only once per outer iteration. 4.3.5 Stabilized Sceme In tis section we will present te stabilized version of te problem, using split OSS stabilization, a BDF1 time discretization and te matrices introduced in Capter 1. Te matrix version of te problem reads as follows: ( ( )) δtdm 1 G S p τ n+1,i 1 P n+1,i+1 [ = δtdm 1 F n+1 K ( U n+1,i) U n+1,i ( S u τ n+1,i 1 ; U n+1,i) ( U n+1,i + S y τ n+1,i 1 ; U n+1,i) Y n+1,i ) ( ) U n+1,i + S w τ n+1,i w n+1,i] S d ( τ n+1,i 2 ( +DU n S z τ n+1,i) 1 Z n+1,i, 2 1 δt M ( U n+1,i+1 U n) + K ( U n+1,i) U n+1,i+1 + GP n+1,i+1 ( +S u τ n+1,i 1 ; U n+1,i) ( U n+1,i+1 S y τ n+1,i 1 ; U n+1,i) Y n+1,i ( +S d τ n+1,i) ( 2 U n+1,i+1 S w τ n+1,i) 2 w n+1,i = F n+1, M π Y n+1,i+1 C ( U n+1,i+1) U n+1,i+1 = 0, M π Z n+1,i+1 G π P n+1,i+1 = 0, M π W n+1,i+1 DU n+1,i+1 = 0. As in te pressure correction case, we ave included te terms corresponding to τ 2 0 tat ave been neglected in [4] because tey elp to enforce incompressibility, someting tat is particularly important for two fluid flows. Approximation (4.35) can be introduced to avoid dealing wit DM 1 G.

4.3. VELOCITY CORRECTION METHODS 125 Since te fractional step sceme can be seen as te first iteration of te predictor corrector sceme we only present te stabilized version for te P-C case. Te stabilized fractional step scemes can be found in [4] using bot split and non-split OSS. In te first iteration te velocity is extrapolated from values at previous time steps. Tis extrapolation is used not only for te viscous and convective term, but also for te stabilization terms associated to te momentum and continuity equations. Furter te projection array Y is also extrapolated for i = 0. Wen te approximation (4.35) for DM 1 G is used we also set P n+1,0 = P n+1 p. From te teoretical point of view, to obtain a sceme of order r a time discretization a sceme of order r must be used togeter wit an extrapolation of te velocity (and of te pressure wen a continuous Laplacian is used) of order r 1. From te practical point of view one can use extrapolations of order iger tan r 1 to reduce te splitting error. In te fractional step version te extrapolations must be cosen so tat te sceme is stable. In te predictor corrector case no restriction exists and tey sould be cosen so as to minimize te number of iterations of te metod. In our simulations we ave typically used p = q = 1 for te BDF1 time discretization and p =1,q =2forteBDF2 case bot for te fractional step and predictor corrector scemes. 4.3.6 Remarks on te ASGS and non split OSS stabilized cases Te use of te velocity correction sceme wit ASGS or non split OSS stabilization sows some particularities, specially wen combined wit te enriced pressure two pase model presented in Capter 2. Te ASGS stabilized case is presented but te same observations apply to te non split OSS stabilized case. In order to simplify te presentation te stabilization terms related to τ 2 are neglected because tey sow no special beavior wen ASGS or non split OSS stabilization are used. Moreover te presentation is restricted to linear elements to avoid te inclusion of viscous terms in te stabilization. In Capter 1 only te matrix version of te problem for te split OSS stabilized case ad been presented so te first step is to introduce te matrix version of te monolitic problem using ASGS

126 CHAPTER 4. PRESSURE SEGREGATION METHODS stabilization tat reads 1 δt MUn+1 + K ( U n+1) U n+1 + GP n+1 + S u ( τ1 ; U n+1) U n+1 + S up ( τ1 ; U n+1) P n+1 = F n+1 + S uf ( τ1 ; U n+1) + 1 δt MUn, DU n+1 + S pu (τ 1 ) U n+1 + S p (τ 1 ) P n+1 = S pf (τ 1 ). Te use of ASGS stabilization introduces some new matrices and vectors tat are defined as follows S up ( τ1 ; U n+1) ab i = ( τ 1 u n+1 N a, i N b), ( S uf τ1 ; U n+1) a = ( τ i 1 u n+1 N a,f i +(ρ/δt) u n i S pu (τ 1 ) ab j = ( τ 1 j N a,ρ u n+1 N b +(ρ/δt) N b), S pf ( τ1 ; U n+1)a =(τ 1 i N a,f i +(ρ/δt) u n i ). Moreover S u is redefined to include te transient terms S u ( τ1 ; U n+1) ab ij = ( τ 1 u n+1 N a,ρ u n+1 N b +(ρ/δt) N b) δ ij. Te DPPE for te can now be obtained by multiplying te momentum equation by δtdm 1 and substracting te continuity equation. It reads [ ] δtdm 1 (G + S up ) S p P = S pf + S pu U + δtdm [F 1 n+1 + S uf + 1 ] δt MUn (K + S u ) U, (4.39) were te dependencies of stabilization matrices and vectors on τ and U n+1 ave been eliminated to simplify te notation. In [4] te term S up P is sent to te rigt and side to end up wit [ ] δtdm 1 G S p P [ = S pf + S pu U + δtdm 1 F n+1 + S uf + 1 ] δt MUn (K + S u ) U S up Pn+1. One advantage of doing tis is tat a symmetric matrix is obtained. Te disadvantage is tat despite we are using a velocity correction metod we need an extrapolation for ),

4.4. OPEN BOUNDARY CONDITIONS 127 pressure ( P n+1 ) even if a discrete Laplacian is used. We ave not found any problems in using te approac proposed in [4] in te examples sown in tis Capter. In order to combine te velocity correction sceme wit te enriced pressure two pase model presented in Capter 2, equation (4.39) as been preferred. Te reason for doing so is tat for te pressure extrapolation, P n+1, te pressure from te previous step is used. Wen an enriced pressure is used tis would introduce important complications. Te pressure enricment at step n would be needed wen solving for step n +1. A correct integration would ten imply combining te enanced integration from step n wit te enanced integration from step n + 1. In order to avoid tese complications in te examples presented in Capter 5 equation (4.39) as been used. 4.4 Open boundary conditions In te previous sections we ave supposed mainly Diriclet velocity boundary conditions as is usually done in te literature. In tis section we will try to clarify wat to do wit te pressure on Neumann velocity boundaries, Γ nu. In [48] suc boundaries are classified into open boundaries and traction boundaries. Te term open boundaries is reserved exclusively for Neumann velocity boundaries tat arise from te fact of aving to cut te domain in order to be able to perform a simulation, for example, te outflow of te domain wen simulating te flow beind a cylinder. Tose Neumann velocity boundaries tat are present in te real problem, suc as a free surface, are called traction boundaries. On bot of tem n σ = t. Te difference is tat on traction boundaries te value of t is known from te pysical problem but on open boundaries some approximation is needed. Several options are discussed in [48] for suc an approximation. Te fact of not knowing te value for t gives some freedom on te coices to use in te open boundaries case. In any case, it will always be an approximation. In te monolitic case written in divergence form te open boundary condition usually used is n σ = 0. Since we want our predictor corrector versions to converge to te monolitic solution we will prefer our segregated versions also to satisfy tat condition. Terefore te same treatment will be favored in

128 CHAPTER 4. PRESSURE SEGREGATION METHODS bot cases (open or traction boundaries). In fractional step scemes obtained at te continuous level te pressure is prescribed to zero on Γ nu. Actually in [50] it is pointed out tat in te incremental version it is te pressure increment tat is prescribed to zero on te Neumann velocity boundary. Tat is, te pressure is prescribed to te value from te previous time step on Γ nu. Ten te pressures on Γ nu for all time steps coincide wit te initial one tat is usually zero. Wen te approximation DM 1 G L is used, te sceme obtained at te discrete is also forced to satisfy p =0onΓ nu. Ten it is evident tat even in a predictor corrector case (for example 4.9, 4.10) te solution will not be able to coincide wit tat of te monolitic system if tere are open boundaries. If at te open boundary we ave n σ = 0, projecting on te normal direction we get n σ n = 0 and terefore p =2µn i j u i n j, tat in te monolitic case can be different from zero. One possible solution is to use p =2µn i j u i n j on Γ nu instead of p = 0 wen solving te equation for te pressure in te segregated case [101]. Wen te only approximation introduced in DM 1 G is to use a diagonal mass matrix te equation for te pressure can be solved witout any boundary condition as is done in [115]. Ten wen te iterative procedure converges one recovers exactly te solution of te monolitic system. As we ave already mentioned, in [4] te approximation DM 1 G L as been enanced by using (4.34) in te fractional step case and (4.35) in te predictor corrector case. Even in tat case, wen solving te system, te pressure as been imposed to zero on Γ nu. As in te case wen DM 1 G L is used, suc prescription precludes te possibility of reacing te monolitic solution wen tere are open boundaries. Te first solution tat comes to te mind is not to fix any prescription on te pressure, but if tat is done, te pressure would be undefined up to an additive constant, someting tat does not appen in te monolitic case wit open boundaries. Te solution we propose will be presented for te preconditioned Ricardson iteration to solve DM 1 G we ave suggested, taking as a starting point te enanced approximation proposed by [4], but is also applicable wen only one iteration is done and te approximation is recovered.

4.5. NUMERICAL EXAMPLES 129 Te preconditioned Ricardson iteration used to solve DM 1 GP n+1 = X in te case wit no Neumann velocity boundary conditions (4.36) is modified only in te definition of te preconditioning matrix L (tat is replaced by L ). In te case wit open boundaries it reads L P n+1,i+1 = X DM 1 GP n+1,i + L P n+1,i. Te matrix L is obtained by modifying matrix L in te degrees of freedom corresponding to open boundaries as is usually done wen te pressure is prescribed. Te difference is te rigt and side is not altered. In tis way te pressure is not prescribed to any value. Only te preconditioner for te Ricardson iteration is altered. We ave used tis strategy even wen only one Ricardson iteration is done ( te discrete Laplacian is approximated by te continuous one) and ave found satisfactory results for bot te pressure correction and velocity correction scemes. We believe tat prescribing te pressure or its increment to zero wen te discrete approac is used to obtain te segregation follows wat is done wen te splitting is obtained at te continuous level. Te small modification we propose seems more natural wen te sceme is obtained at te discrete level. 4.5 Numerical examples In tis section we present tree numerical examples to compare te velocity correction and pressure correction scemes introduced in tis Capter. Te results are also compared against te solution obtained wit a monolitic solver. BDF1, BDF2 and BDF3 time discretizations are used. For BDF2 and BDF3 discretizations second order velocity extrapolation is used. In te velocity correction fractional step scemes, wen used wit te discrete Laplacian, tis lead to a tird order splitting error. For te pressure, first order extrapolation is used for all time discretizations. It will be sown tat second order pressure extrapolations not only makes te pressure correction sceme unstable as is well known but also te velocity correction sceme wit continuous Laplacian approximation.

130 CHAPTER 4. PRESSURE SEGREGATION METHODS In te first example te evolution of te flow to te steady state in a driven cavity will be used to compare te two scemes. In te second example te flow beind a cylinder, te most classical example for transient flow, is analyzed. It is sown tat te tird order VC sceme wit discrete Laplacian remains stable wen te oter tird order scemes diverge. Finally a convergence test is performed to sow tat te velocity correction sceme allows to obtain tird order accuracy in te L2 norm of te velocity. For te first and tird examples Split OSS stabilization is used and for te second one ASGS is used. In order to simplify te presentation of te examples te following nomenclature is used. Te pressure correction and velocity correction scemes are denoted PC and VC respectively. Fractional step versions are denoted by FS. To identify te predictor corrector sceme we sall use Imon because PC as already been used for te pressure correction sceme. In our examples we use bot te discrete Laplacian (wit Lumped mass matrix) denoted by LD and its approximation by te continuous Laplacian denoted by LC. 4.5.1 Driven Cavity Te first example is a cavity flow problem at Reynold number Re = 100. Te domain is a unit square discretized wit a 21 21 triangular mes. Te velocity is prescribed to zero at te bottom and lateral boundaries and to (1,0) at te top boundary and upper corners (leaky lid). Despite we are dealing wit a stationary problem we ave cosen to run a transient calculation from an initial zero velocity on all of te nodes except tose on te upper boundary. Te time step used is δt = 1.0. Te goal is to test te evolution of te proposed metods towards te stationary solution and teir numerical dissipation. VC - PC comparison using a fractional step sceme wit continuous Laplacian Figures 4.1 and 4.2 sow te transient residual evolution using VC and PC fractional step scemes wit continuous Laplacian and bot BDF1 and BDF2 time discretizations.

4.5. NUMERICAL EXAMPLES 131 1 0.1 Mono-bdf1 FS-VC-bdf1-Lc FS-PC-bdf1-Lc 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.1: Transient residual BDF1 - PC vs. VC using a continuous Laplacian 1 0.1 Mono-bdf2 FS-VC-bdf2-Lc FS-PC-bdf2-Lc 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.2: Transient residual BDF2 - PC vs. VC using a continuous Laplacian

132 CHAPTER 4. PRESSURE SEGREGATION METHODS Te objective is to compare VC and PC scemes using a dissipation argument based on te evolution of te transient residual. Wit te first order time discretization no significant difference is observed between te VC and PC scemes. For BDF2 during te first 20 seconds te PC metod follows te monolitic solver closer tan te VC metod but after tat te opposite is observed. In te previous examples it can also be interesting to analyze te L2 error for bot te velocity and te pressure obtained by comparing wit te monolitic solution (Figures 4.3 to 4.6). Te L2 pressure error is muc smaller wen a VC sceme is used wit bot time integration scemes. For te velocity te PC sceme sows some sligt advantage wen te first order metod is used. In te BDF2 case, after some few steps te VC sceme sows smaller velocity errors. Terefore we can conclude tat (at least for tis case) te VC sceme provides better results. 1 FS-VC-bdf1-Lc FS-PC-bdf1-Lc 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.3: Velocity error BDF1 - PC vs. VC using a continuous Laplacian

4.5. NUMERICAL EXAMPLES 133 1 FS-VC-bdf1-Lc FS-PC-bdf1-Lc L2 error pr 0.1 5 10 15 20 25 30 35 40 45 50 time Figure 4.4: Pressure error BDF1 - PC vs. VC using a continuous Laplacian 1 FS-VC-bdf2-Lc FS-PC-bdf2-Lc 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.5: Velocity error BDF2 - PC vs. VC using a continuous Laplacian

134 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 FS-VC-bdf2-Lc FS-PC-bdf2-Lc L2 error pr 0.1 5 10 15 20 25 30 35 40 45 50 time Figure 4.6: Pressure error BDF2 - PC vs. VC using a continuous Laplacian LC - LD comparison using PC and VC fractional step scemes In tis subsection te main objective is to analyze te influence of using a continuous or discrete Laplacian. A fractional step sceme wit bot pressure and velocity correction versions will be used. As in te previous case, our first indicator is te evolution of te transient residual. We first analyze te results obtained wit te pressure correction sceme. Wen te first order time integrator is used, very little difference can be observed between using a continuous or discrete Laplacian (Figure 4.7). Wit te second order time integrator te discrete Laplacian provides better results (Figure 4.8). Our second indicator is te L2 error for bot te velocity and te pressure. Using te BDF1 sceme, te discrete Laplacian provides better results for te velocity error (Figure 4.9). Te advantage is muc more notorious for te pressure error (Figure 4.10). For bot te velocity and te pressure error te advantage of using te discrete Laplacian sows up more clearly for te second order sceme as sown in Figures 4.11 to 4.12.

4.5. NUMERICAL EXAMPLES 135 1 0.1 Mono-bdf1 FS-PC-bdf1-Lc FS-PC-bdf1-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.7: Transient residual BDF1 - Lc vs. Ld using te PC sceme 1 0.1 Mono-bdf2 FS-PC-bdf2-Lc FS-PC-bdf2-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.8: Transient residual BDF2 - Lc vs. Ld using te PC sceme

136 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 FS-PC-bdf1-Lc FS-PC-bdf1-Ld 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.9: Velocity error BDF1 - Lc vs. Ld using te PC sceme 1 FS-PC-bdf1-Lc FS-PC-bdf1-Ld L2 error pr 0.1 0.01 5 10 15 20 25 30 35 40 45 50 time Figure 4.10: Pressure error BDF1 - Lc vs. Ld using te PC sceme

4.5. NUMERICAL EXAMPLES 137 1 FS-PC-bdf2-Lc FS-PC-bdf2-Ld 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.11: Velocity error BDF2 - Lc vs. Ld using te PC sceme 1 FS-PC-bdf2-Lc FS-PC-bdf2-Ld L2 error pr 0.1 0.01 5 10 15 20 25 30 35 40 45 50 time Figure 4.12: Pressure error BDF2 - Lc vs. Ld using te PC sceme

138 CHAPTER 4. PRESSURE SEGREGATION METHODS In te velocity correction case, using BDF1, again very little difference is observed in te evolution of te transient residual between te continuous and discrete Laplacians (Figure 4.13). Using BDF2, contrary to wat we would expect, te transient residual is reduced more rapidly wen te continuous Laplacian is used (Figure 4.14). 1 0.1 Mono-bdf1 FS-VC-bdf1-Lc FS-VC-bdf1-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.13: Transient residual BDF1 - Lc vs. Ld using te VC sceme As in te pressure correction case, for te first order sceme te discrete Laplacian provides better results for te velocity error and specially for te pressure error (Figures 4.15 and 4.16 ). Wit te second order sceme, up to t = 35s te continuous Laplacian provides smaller velocity errors but after tat te discrete Laplacian works better (Figure 4.17). For te pressure error, te results are significantly improved wen te discrete Laplacian is used (Figure 4.18). Tis someow contradicts wat we ad observed for te evolution of te transient residual in te BDF2 VC case. Wic sould be a better indicator of te goodness of a metod, te evolution of te transient residual or te pressure error, is not so clear to us. Despite tis is a very small example and te computational times are not very

4.5. NUMERICAL EXAMPLES 139 1 0.1 Mono-bdf2 FS-VC-bdf2-Lc FS-VC-bdf2-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.14: Transient residual BDF2 - Lc vs. Ld using te VC sceme 1 FS-VC-bdf1-Lc FS-VC-bdf1-Ld 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.15: Velocity error BDF1 - Lc vs. Ld using te VC sceme

140 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 FS-VC-bdf1-Lc FS-VC-bdf1-Ld L2 error pr 0.1 0.01 5 10 15 20 25 30 35 40 45 50 time Figure 4.16: Pressure error BDF1 - Lc vs. Ld using te VC sceme 1 FS-VC-bdf2-Lc FS-VC-bdf2-Ld 0.1 L2 error vel 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.17: Velocity error BDF2 - Lc vs. Ld using te VC sceme

4.5. NUMERICAL EXAMPLES 141 1 FS-VC-bdf2-Lc FS-VC-bdf2-Ld 0.1 L2 error pr 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.18: Pressure error BDF2 - Lc vs. Ld using te VC sceme significative in Table 4.1 we present te total CPU times for VC and PC scemes using bot Laplacians. More significative results are presented in te cylinder example. Te use of te discrete or continuous Laplacian as very little influence for tis example. Te PC sceme takes more time tan te VC sceme. Tis is related to te fact tat using te VC sceme te convective non linearity takes less iterations to converge. PC VC Lcont 9.58s 7.19s Ldisc 9.83s 6.98s Table 4.1: Total cpu time for te fractional step scemes

142 CHAPTER 4. PRESSURE SEGREGATION METHODS VC - PC comparison using te predictor corrector sceme wit LC Since predictor corrector metods converge to te monolitic solution te most important result for comparison purposes is te number of iterations per time step. Te velocity correction sceme works significantly better tat te pressure correction version specially in te BDF2 case, as can be seen in Figures 4.19 and 4.20. We ave used te usual predictor corrector sceme were no nested loops are used. A maximum of 20 predictor corrector iterations per time step wit a tolerance of 10 5 based on te variation of te L2 norm of te velocity divided by te norm of te velocity as been used. Between 5 to 8 time steps ave been need so tat te predictor corrector iterations fall bellow te maximum permitted limit of 20 iterations. 20 18 Imon-VC-bdf1 Imon-PC-bdf1 16 14 12 niter 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 time Figure 4.19: Number of iterations BDF1 Regarding te transient residual and te velocity error VC and PC work similarly. Te beavior of te pressure velocity errors does not provide muc information. Except for te pressure error wit te BDF1 time discretization were te VC sceme sows lower errors, in te rest of te cases te PC sceme sows smaller errors specially in te second alf of te run as can be seen in Figures 4.21 to 4.24. Tis sould not be

4.5. NUMERICAL EXAMPLES 143 20 18 Imon-VC-bdf2 Imon-PC-bdf2 16 14 12 niter 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 time Figure 4.20: Number of iterations BDF2 interpreted as a real advantage of te PC sceme because it is probably related to te fact tat it is doing more iterations per time step. Terefore te comparison of te velocity and pressure errors does not seem to be an interesting indicator in te predictor corrector case. Ricardson iteration for te discrete Laplacian Two options ave been proposed to solve te discrete Laplacian in tis Capter. Te first one is te straigtforward use of te conjugate gradient algoritm. Te second one is to use a Ricardson iteration preconditioned wit te continuous Laplacian. Te coice between te two options is based on a cost argument represented by te CPU time per time step. Te results for bot te VC and PC fractional step scemes are sown in Figures 4.27 and 4.28. Te straigtforward use of te CG algoritm proves more efficient wit bot scemes.

144 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 0.1 Mono-bdf1 Imon-VC-bdf1 Imon-PC-bdf1 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.21: Transient residual BDF1 1 0.1 Mono-bdf2 Imon-VC-bdf2 Imon-PC-bdf2 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.22: Transient residual BDF2

4.5. NUMERICAL EXAMPLES 145 0.1 Imon-VC-bdf1 Imon-PC-bdf1 0.01 0.001 L2 error vel 0.0001 1e-005 1e-006 5 10 15 20 25 30 35 40 45 50 time Figure 4.23: Velocity error BDF1 1 Imon-VC-bdf1 Imon-PC-bdf1 0.1 0.01 L2 error pr 0.001 0.0001 1e-005 5 10 15 20 25 30 35 40 45 50 time Figure 4.24: Pressure error BDF1

146 CHAPTER 4. PRESSURE SEGREGATION METHODS 0.1 Imon-VC-bdf2 Imon-PC-bdf2 0.01 0.001 L2 error vel 0.0001 1e-005 1e-006 5 10 15 20 25 30 35 40 45 50 time Figure 4.25: Velocity error BDF2 1 Imon-VC-bdf2 Imon-PC-bdf2 0.1 L2 error pr 0.01 0.001 0.0001 5 10 15 20 25 30 35 40 45 50 time Figure 4.26: Pressure error BDF2

4.5. NUMERICAL EXAMPLES 147 0.4 Ricardson it CG for disc Lapl 0.35 0.3 cpu_time 0.25 0.2 0.15 0.1 5 10 15 20 25 30 35 40 45 50 step Figure 4.27: CPU time per time step - PC case 0.35 Ricardson it CG for disc Lapl 0.3 0.25 cpu_time 0.2 0.15 0.1 0.05 5 10 15 20 25 30 35 40 45 50 step Figure 4.28: CPU time per time step - VC case

148 CHAPTER 4. PRESSURE SEGREGATION METHODS 4.5.2 Flow beind a cylinder Te second example is te flow beind a cylinder at Reynold number Re = 190. Tis example is essentially 2-D but it as been run wit a 3-D mes. Te mes, provided by Professor Rainald Loner, as a special placement of points in te vicinity of te cylinder [75]. It is formed by 108147 tetraedral linear elements and 30000 nodes. In Figure 4.29 te surface mes is sown. Te computational domain is Ω = [0, 19] [0, 8] [0, 0.2] \D, wit te cylinder D of diameter 1 centered at (4, 4). Te velocity at x = 0 is prescribed to (1, 0, 0). At y =0,y =8,z =0andz =0.2 te normal component of te velocity is set to zero and te tangential components are left free. At te outflow (x = 19) zero traction is prescribed. Te time step is δt =0.05 and te total time is t = 100.0. Figure 4.29: Mes used for te flow beind a cylinder Te objective of tis subsection is to observe te frequency and amplitude errors compared to te monolitic solution for te different pressure segregation scemes. Bot BDF2 and BDF3 time integration sceme are used. Te velocity and pressure contours obtained wit te velocity correction BDF2 fractional step sceme wit continuous Laplacian are sown in Figure 4.30. Note tat te pressures in te outlet are not prescribed to zero as mentioned in te section on open boundary conditions.

4.5. NUMERICAL EXAMPLES 149 P[N/m²] V[m/s] Figure 4.30: Pressure (top) and velocity (bottom) at t = 100s

150 CHAPTER 4. PRESSURE SEGREGATION METHODS VC - PC comparison using te fractional step sceme wit bot Laplacians Figures 4.31 and 4.32 sow te Lift and Drag coefficients for te fully developed flow using VC and PC BDF2 fractional step scemes wit bot Laplacians. For te Lift coefficient, all of te scemes give pretty accurate results. Terefore we sall use te Drag coefficient to compare te different scemes. 0.8 0.6 Monolitic PC-Lcont VC-Lcont PC-Ldisc VC-Ldisc 0.4 0.2 Cl 0-0.2-0.4-0.6-0.8 82 84 86 88 90 92 time Figure 4.31: Lift coeficient Te first observation is tat te VC sceme provides muc better results tan te PC sceme bot wit te discrete and continuous Laplacians. Te results obtained wit te discrete Laplacian are better tan tose obtained wit te continuous Laplacian as one could expect but even te VC sceme wit continuous Laplacian sows smaller errors tan te PC sceme wit discrete Laplacian. Wen te VC sceme wit a discrete Laplacian is used te errors are very small, muc smaller tan tose obtained wit a continuous Laplacian (wic are already quite small). Terefore we can say tat wen te rest of te errors are sufficiently small, te real advantage of using a discrete Laplacian can be observed. An explanation for te advantage of te VC sceme is tat we are using a second

4.5. NUMERICAL EXAMPLES 151 1.56 1.54 Monolitic PC-Lcont VC-Lcont PC-Ldisc VC-Ldisc 1.52 Cd 1.5 1.48 1.46 1.44 85 86 87 88 89 90 time Figure 4.32: Drag coeficient order extrapolation for te velocity but only a first order extrapolation for te pressure. Te reason for doing tis is tat, as is well known for pressure correction scemes, second order extrapolations for te pressure lead to unstable scemes. We ave verified tis beavior also in tis example. Te pressure correction sceme wit second order pressure extrapolation diverges after some few steps. We ave also tested te VC sceme wit continuous Laplacian tat uses a second order pressure extrapolation and introduces a tird order splitting error to verify tat it is unstable as already pointed out in [4]. Te VC sceme wit continuous Laplacian and first order pressure extrapolation works fine so we can attribute te instability to te use of second order extrapolations for te pressure as in te pressure correction sceme. Te excellent results obtained wit te VC sceme wit discrete Laplacian can be attributed to te fact tat it needs no pressure extrapolation and only uses a second order velocity extrapolation leading to a tird order splitting error. In te VC sceme wit continuous Laplacian te use of a first order extrapolation inibits te tird order accuracy as will be sown in te convergence example in te next subsection. Finally, since te VC sceme wit continuous Laplacian uses a mix of a

152 CHAPTER 4. PRESSURE SEGREGATION METHODS second order extrapolation for te velocity and a first order extrapolation for te pressure it seems reasonable tat it introduces less error tat te PC sceme tat only uses a first order pressure extrapolation independently of wic Laplacian is used. In order to verify tat te advantage of te VC sceme comes from te use of a second order extrapolation we ave tested it wit an first order extrapolation for te velocity. In Figure 4.33 te results obtained wit suc VC sceme wit Discrete Laplacian are compared to te results already sown in Figure 4.32 for te monolitic and PC scemes. Using a first order extrapolation for te velocity in te VC sceme te results ave a similar accuracy to te ones obtained wit te PC sceme and a muc lower accuracy tan te ones obtained wit a second order extrapolation for te velocity. Terefore we can conclude tat te advantage of te VC sceme comes from te fact tat it only needs a second order extrapolation wen te discrete Laplacian is used. It introduces a tird order splitting error. 1.56 Monolitic PC-Ldisc VC-Ldisc 1.54 1.52 Cd 1.5 1.48 1.46 1.44 85 86 87 88 89 90 time Figure 4.33: Drag coeficient using a first order extrapolation for te VC sceme In Table 4.2 we compare te total CPU time for te different fractional step versions. Te VC scemes turn out to be sligtly slower ( 5%) tan teir PC counterparts but

4.5. NUMERICAL EXAMPLES 153 PC VC L cont 66979s 69642s Ldisc 80274s 83020s Table 4.2: Total cpu time for te fractional step scemes tey provide more accurate results. Te use of te continuous Laplacian provides smaller ( 15%) CPU times tan te discrete Laplacian but also less accurate results. Te use of te VC sceme wit discrete Laplacian is te slowest option but it provides results tat are nearly identical to te monolitic ones. Since we ave obtained a VC sceme wit a tird order splitting error it seems reasonable to combine it wit a tird order time discretization suc as BDF3. Te results obtained wit te BDF3 VC fractional step sceme for te drag coefficient were practically identical to te ones obtained wit te BDF2 sceme. In order to see te enancement obtained wit te BDF3 time discretization a more sensitive value tan te drag coefficient needs to be used. In Figure 4.34 te orizontal velocity at a node located at (9.759, 0.0426, 0.2) is presented using bot te monolitic solver and te fractional step VC sceme wit discrete Laplacian. It can be observed tat te errors introduced by switcing from a tird order time discretization to a second order one are more important tan te errors introduced by te splitting. We ave verified tat if a smaller time step is used te results obtained wit bot te BDF2 and BDF3 become closer. Tey are very similar to te ones obtained wit te BDF3 sceme and te original time step sowing te advantage of using a tird order sceme. VC - PC comparison using te predictor corrector sceme wit bot Laplacians In te case of te predictor corrector (Imon) versions te comparison of te Lift and Drag coefficients provide little or no information since all of tem converge to te same result.

154 CHAPTER 4. PRESSURE SEGREGATION METHODS 0.9 VC_bdf3 VC_bdf2 Mono_bdf3 Mono_bdf2 0.85 vx 0.8 0.75 0.7 86 88 90 92 94 96 98 100 time Figure 4.34: Horizontal velocity at a node beind te cylinder In tis case we will be interested in te number of iterations needed to converge to suc result. Te lower te number of iterations te better te metod. Wen te flow is fully developed te number of iterations per time step remains constant. Te convergence toleranceweaveusedis10 5 based on te variation of te L2 norm of te velocity divided by te norm of te velocity. Te number of iterations per time step for eac of te scemes are sown in Table 4.3 PC VC Lcont 9 7 Ldisc 7 6 Table 4.3: Number of iterations per time step We can see tat te results for te number of iterations agree wit wat we ave observed for te fractional step case for te accuracy of te different options. Te VC sceme wit discrete Laplacian produces te best results and te PC sceme wit

4.5. NUMERICAL EXAMPLES 155 continuous Laplacian te worse ones. Anoter important parameter to take into account wen comparing different scemes is te computational efficiency. In Table 4.4 we compare te total CPU time for te different scemes. Te VC scemes turn out to be more efficient ( 20%) tan teir PC counterparts. On te oter and, wen converging to te monolitic solution, te use of te continuous Laplacian provides smaller ( 15%) CPU times. PC VC L cont 290891s 241132s Ldisc 336465s 288588s Table 4.4: Total cpu time for te predictor corrector scemes Ricardson iteration for te discrete Laplacian As in te cavity case, it is interesting to compare te straigtforward use of te conjugate gradient algoritm wit te use a Ricardson iteration preconditioned wit te continuous Laplacian. Te cylinder case is a more interesting example tan te cavity because we are dealing wit a 3-D mes. We compare te total CPU time wen bot VC and PC fractional step scemes are used in Table 4.5. PC VC Straigt forward CG 80274s 83020s Ricardson iteration 80595s 82375s Table 4.5: Total cpu time wit different options for solving te discrete Laplacian It can be seen tat tere is no significant difference between using eiter of te two options. Bot of tem could be considered as valid and furter tests sould be performed to select te most efficient option. Taking into account implementation ease, Ricardson

156 CHAPTER 4. PRESSURE SEGREGATION METHODS iteration migt be an attractive option because it can be very simple to code in a program tat uses te continuous Laplacian. 4.5.3 Convergence test Te tird example is used to test te time convergence rate numerically. It as been borrowed from [52]. Te Stokes problem is solved on te unit square, ]0, 1[ 2. Te force term is set so tat exact solution is p (x, y, t) =cos(πx) sin(πy) sin(t) u (x, y, t) =π sin (2πy) sin 2 (πx) sin(t) v (x, y, t) = π sin (2πx) sin 2 (πy) sin(t). Te domain is discretized using Q2/Q2 finite elements of size =1/40. Boundary and initial conditions are forced to satisfy te previous equations. Te time step size we use varies from 0.0025s to 0.1s and te results at t =1.0s are presented. In Figure 4.35 we present te convergence results for te L2 norm of te velocity error using a monolitic formulation. Tis results can be used as a reference against wic te results obtained wit te fractional step results can be compared because tey ave no splitting error. It can be observed tat te monolitic BDF1 sceme sows te correct order of convergence. For te tird order sceme te error due to te temporal discretization is smaller tan te error due to te spatial discretization. For te second order sceme te error due to te temporal discretization is only noticeable for te bigger time steps. In Figure 4.36 te results obtained wit te pressure correction fractional step sceme are presented. Te discrete Laplacian is used and some interesting observations on wat appens wen te continuous Laplacian is used sall be postponed until te end of te subsection. Te results wit BDF3 are not included because as we are using a first order pressure extrapolation te results cannot be tird order accurate. Bot scemes sow te correct order until te spatial discretization error becomes dominant. Comparing wit

4.5. NUMERICAL EXAMPLES 157 0.1 Mono_bdf1 Mono_bdf2 Mono_bdf3 slope 1 slope 2 slope 3 0.01 L2 velocity error 0.001 0.0001 1e-005 0.01 0.1 dt Figure 4.35: Convergence test wit monolitic solver te monolitic example one can see tat in te BDF2 case te errors due to te time discretization are muc smaller tan tose due to te splitting. Terefore tis is a good example to test a pressure segregation sceme. Finally we present te results wit te velocity correction fractional step sceme were we expect to obtain tird order accuracy (Figure 4.37). Bot BDF2 and BDF3 results sow tird order convergence. In te BDF2 case tis can be explained by te fact tat te error due to te temporal discretization is small compared wit te splitting error wic is tird order accurate. Te spatial error is important and it limits te range were tird order accuracy can be observed. Terefore te results ave been repeated on a mes wit size =1/200. In Figure 4.38 te convergence results using a monolitic formulation on te fine mes are presented. As one would expect te error due to te spatial discretization is reduced 5 3 times tanks to te use of Q2/Q2 finite elements. Wit te BDF2 sceme te second order slope can easily be observed. For te tird order sceme te spatial discretization error soon becomes dominant and te tird order slope can only be seen for te two bigger

158 CHAPTER 4. PRESSURE SEGREGATION METHODS 0.1 PC_bdf1 PC_bdf2 slope 1 slope 2 0.01 L2 velocity error 0.001 0.0001 1e-005 0.01 0.1 dt Figure 4.36: Convergence test wit pressure correction sceme 0.1 VC_bdf1 VC_bdf2 VC_bdf3 slope 1 slope 2 slope 3 0.01 L2 velocity error 0.001 0.0001 1e-005 0.01 0.1 dt Figure 4.37: Convergence test wit velocity correction sceme

4.5. NUMERICAL EXAMPLES 159 time steps. 0.1 0.01 Mono_bdf2 Mono_bdf3 slope 2 slope 3 0.001 L2 velocity error 0.0001 1e-005 1e-006 1e-007 1e-008 0.01 0.1 dt Figure 4.38: Convergence test wit monolitic solver on fine mes For te velocity correction sceme bot BDF2 and BDF3 scemes sow tird order accuracy because as we ave already mentioned for te coarse mes te splitting error is more important tan te time discretization error (Figure 4.39). Finally we present some results wit te VC BDF3 sceme and continuous Laplacian (Figure 4.40). Te tird order is lost and only second order accuracy is obtained due to te error introduced by te approximation of te discrete Laplacian by te continuous one. Remember we are using a first order extrapolation for te pressure because we ave seen tat a second order extrapolation leads to an unstable sceme. We ave also included in te comparison te results obtained wit te discrete Laplacian solved by a Ricardson iteration. Actually only tree Ricardson iterations ave been allowed per time step. It is interesting to note ow two extra Ricardson iterations allow to recover results tat are nearly tird order accurate and very close to te ones obtained wen te discrete Laplacian is solved wit a conjugate gradient metod.

160 CHAPTER 4. PRESSURE SEGREGATION METHODS 0.1 0.01 VC_bdf2 VC_bdf3 slope 2 slope 3 0.001 L2 velocity error 0.0001 1e-005 1e-006 1e-007 1e-008 0.01 0.1 dt Figure 4.39: Convergence test wit velocity correction sceme on fine mes 0.1 0.01 VC_bdf3_Ld VC_bdf3_Lc VC_bdf3_3RIC_it slope 2 slope 3 0.001 L2 velocity error 0.0001 1e-005 1e-006 1e-007 1e-008 0.01 0.1 dt Figure 4.40: Convergence test wit BDF3 VC sceme and different Laplacians

4.5. NUMERICAL EXAMPLES 161 4.5.4 Results wit te rotational form In tis subsection we present some results using te rotational form of te pressure correction fractional step sceme in te previous tree examples. Te objective is to verify te correct implementation in te rotational version in te pressure stabilized case and gain some idea of te advantages of te rotational form. First te driven cavity example is analyzed. In Figure 4.41 we sow te transient residual evolution using te PC BDF1 fractional step sceme wit standard and rotational versions. Bot te continuous and discrete Laplacians ave been used. It can be observed tat better results are obtained wit te rotational version independently of wic Laplacian is used. 1 0.1 Mono-bdf1 FS-PCrot-bdf1-Lc FS-PC-bdf1-Lc FS-PCrot-bdf1-Ld FS-PC-bdf1-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.41: Transient residual BDF1 - Influence of te rotational form using bot Laplacians In Figures 4.42 and 4.43 te velocity and pressure errors are presented. Again te rotational version provides improved results independently of wic Laplacian is used. Using te BDF2 sceme te advantage of te rotational version in te evolution of te transient residual and te pressure and velocity errors as also been confirmed. For te second example, te flow beind a cylinder, te drag coefficient is used to

162 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 FS-PCrot-bdf1-Lc FS-PC-bdf1-Lc FS-PCrot-bdf1-Ld FS-PC-bdf1-Ld 0.1 L2 error vel 0.01 0.001 0.0001 5 10 15 20 25 30 35 40 45 50 time Figure 4.42: Velocity error BDF1 - Influence of te rotational form using bot Laplacians 1 FS-PCrot-bdf1-Lc FS-PC-bdf1-Lc FS-PCrot-bdf1-Ld FS-PC-bdf1-Ld 0.1 L2 error pr 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.43: Pressure error BDF1 - Influence of te rotational form using bot Laplacians

4.5. NUMERICAL EXAMPLES 163 1 0.1 Mono-bdf2 FS-PCrot-bdf2-Lc FS-PC-bdf2-Lc FS-PCrot-bdf2-Ld FS-PC-bdf2-Ld 0.01 0.001 L2 transres vel 0.0001 1e-005 1e-006 1e-007 1e-008 5 10 15 20 25 30 35 40 45 50 time Figure 4.44: Transient residual BDF2 - Influence of te rotational form using bot Laplacians 1 FS-PCrot-bdf2-Lc FS-PC-bdf2-Lc FS-PCrot-bdf2-Ld FS-PC-bdf2-Ld 0.1 0.01 L2 error vel 0.001 0.0001 1e-005 5 10 15 20 25 30 35 40 45 50 time Figure 4.45: Velocity error BDF2 - Influence of te rotational form using bot Laplacians

164 CHAPTER 4. PRESSURE SEGREGATION METHODS 1 FS-PCrot-bdf2-Lc FS-PC-bdf2-Lc FS-PCrot-bdf2-Ld FS-PC-bdf2-Ld 0.1 L2 error pr 0.01 0.001 5 10 15 20 25 30 35 40 45 50 time Figure 4.46: Pressure error BDF2 - Influence of te rotational form using bot Laplacians evaluate te advantages introduced by te rotational form. Te BDF2 sceme wit bot Laplacians as been tested. Contrary to wat appens in te previous example, for te cylinder te drag obtained wit te rotational version is indistinguisable from te one obtained wit te standard version (Figure 4.47). We believe tat tis can be attributed to te fact tat in tis example te viscous forces are less important. For te example used in te convergence test te errors in te pressure at t =1.0s for a 0.01s time step are used to compare te standard and rotational versions. Bot te continuous and discrete Laplacians ave been used. Wen te standard version is used te errors concentrate close to te upper and lower boundaries were te exact pressure as a non zero normal gradient. Tis can be explained by te fact tat te non rotational pressure correction fractional step enforces a Non-pysical boundary condition n p n+1 = n p n [50] at te Diriclet boundaries. We ave added te results wit te discrete Laplacian to sow te same beavior is observed irrespective of wic Laplacian is used. Wen te rotational version is used te error is smaller and concentrates only in te corners. Tis as also been observed in [50] were it as been conjectured tat it is

4.6. CONCLUSIONS 165 1.56 1.54 Lc-std Lc-rot Ld-std Ld-rot 1.52 Cd 1.5 1.48 1.46 1.44 85 86 87 88 89 90 time Figure 4.47: Drag coefficient rotational and standard form wit bot Laplacians due to te lack of smootness in te domain. 4.6 Conclusions In tis Capter we ave presented pressure correction and velocity correction pressure segregation scemes. We ave implemented tem in our code and tested tem to gain some experience in teir comparative beavior. Bot te discrete Laplacian and te (more usual) approximation by te continuous one ave been implemented. For te solution of te discrete Laplacian two options ave been implemented. Te first one is te straigtforward application of a conjugate gradient iterative procedure. Te second option is to use a preconditioned Ricardson iteration wit te continuous Laplacian as preconditioner. Te rotational version of te pressure correction sceme as also been implemented. Some particularities arise wen a pressure stabilized sceme is used. Up to te moment it ad only been used wit elements tat satisfy te Inf-Sup condition and do not require

166 CHAPTER 4. PRESSURE SEGREGATION METHODS Figure 4.48: Pressure error for te non rotational form wit continuous (left) and discrete (rigt) Laplacians Figure 4.49: Pressure error for te rotational form wit continuous (left) and discrete (rigt) Laplacians

4.6. CONCLUSIONS 167 pressure stabilization. Moreover it as been sown numerically tat te advantages introduced by te rotational version are present wen eiter te continuous or discrete Laplacians are used. Tree numerical example ave been used. Te first one is te evolution of te flow in a driven cavity to te steady state. It as already been used in [4] were only results wit te continuous Laplacian ave been presented. In tis tesis te effect of using a discrete Laplacianas also been taken into account. Asin [4] te evolution of te transient residual is used to compare te two metods. In tis tesis te errors obtained by comparing te velocity and te pressure wit te monolitic solution are also presented. Te pressure and velocity correction scemes sow a similar beavior wit some advantage for te velocity correction version. Tis agrees wit wat as been observed in [4]. Wit te discrete Laplacian a similar beavior is observed but most of te results are improved. Te enancement due to te use of te discrete Laplacian is most noticeable in te pressure errors. In te predictor corrector case, te number of iterations needed to converge to te monolitic solution also sow some improvement wen te velocity correction sceme is used. For tis example te preconditioned Ricardson iteration used to solve for te discrete Laplacian proves more costly tan te straigtforward use of conjugate gradient metod. It is interesting to note tat no instabilities ave been observed despite we ave used a second order extrapolation for te velocity tat leads to a tird order splitting error wen te velocity correction BDF2 sceme wit discrete Laplacian is used. For te cylinder example te absence of stability problems due to te second order velocity extrapolation is once again verified. Te correct beavior in tis example is particularly important because it is a complicated oscillatory transient flow typically used as a bencmark for transient flow algoritms. Te real advantage of using a second order extrapolation for te velocity becomes clearly noticeable in tis example. Wen used wit a discrete Laplacian in te velocity correction sceme it leads to a tird order splitting error evidenced in te clear superiority observed in te prediction of te drag coefficient. Te use of te continuous Laplacian diminises te advantage of te VC sceme despite it is still superior to te pressure correction sceme. Te reason is tat wen a continuous

168 CHAPTER 4. PRESSURE SEGREGATION METHODS Laplacian approximation is used a pressure extrapolation is also needed. If a second order extrapolation for te pressure is used to obtain a tird order splitting error we ave observed tat te metod becomes unstable. Tis agrees wit te well known beavior for pressure correction scemes were second order pressure extrapolations make te sceme unstable. We believe tat te impossibility of obtaining stable tird order results in [4] can be attributed to te use of te continuous Laplacian approximation wit second order pressure extrapolation and tat different conclusion could ave been obtained if te discrete Laplacian ad been tested. Since te VC sceme wit discrete Laplacian provides a tird order splitting error it as also been combined wit a BDF3 time discretization to obtain a tird order temporal error. Te enancement introduced by te BDF3 time discretization is not noticeable in te drag coefficient and a more sensitive parameter suc as te orizontal velocity in some node beind te cylinder as been used to sow te advantage introduced by te tird order time discretization. RegardingteuseoftediscreteLaplacianinteVCcaseweavealreadymentioned tat it provides muc better results because it allows to obtain a tird order splitting error. In te pressure correction case it also introduces some advantage but it is muc less significant tan in te velocity correction case. Despite te use of te discrete Laplacian leads to a an increase in te computational cost of 15% te advantages observed in te velocity correction case easily justify tis cost. In te predictor corrector case te velocity correction sceme also sows advantage over te pressure correction sceme. It leads to fewer predictor corrector iterations per time step and consequently a reduced computational cost of approximately 20%. In te predictor corrector case te use of te continuous Laplacian results in an increase of CPU times of approximately 15%. Regarding te use of te preconditioned Ricardson iteration to solve for te discrete Laplacian in tis example it as resulted in approximately te same computational times as te straigtforward use of a conjugate gradient solver. In te convergence test borrowed from [52] te expected convergence slope for te L2 velocity error is verified. For te pressure correction sceme a second order slope

4.6. CONCLUSIONS 169 is observed wit te BDF2 time discretization. Wit te first order time discretization second order slope is observed wen te splitting error is dominant and first order slope is observed wen te temporal discretization error is dominant. For te velocity correction case it is observed tat bot BDF2 and BDF3 scemes sow tird order accuracy. For te second order time discretization tis can be explained by te fact tat te splitting error wic is tird order accurate is dominant. Wen te continuous Laplacian approximation is used te tird order accuracy obtained wit te velocity correction sceme is lost. Tis is caused by te use of a first order pressure extrapolation. It as also been sown tat te use of a preconditioned Ricardson iteration wit only tree iterations allows to recover results tat are nearly tird order accurate and very close to te ones obtained wit te discrete Laplacian. Regarding te use of te rotational form in te pressure correction sceme, improvements ave been observed in te first and tird examples. It is interesting to note tat tis improvements ave been observed irrespective of weter te continuous or discrete Laplacian is used. Moreover we understand tat tis is te first time te rotation form as been used wit a pressure stabilized sceme. In te cylinder example te use of te rotational form as introduced no improvement. Te rotational form to te velocity correction pressure stabilized sceme can be an interesting extension for future developments. Te results on te advantages of te rotational version for real applications are not conclusive for te moment. Following [15] we can say tat teir importance grows in applications in wic te stresses or oter pressure dependent quantities must be computed at solid walls. Moreover since pressure segregation metods are quite often applied to problems in wic te viscosity is small, te improvement introduced by te rotational form in suc cases can be negligible. Tis is wat we believe appens in te cylinder case. From te previous examples we can conclude tat te superior beavior of velocity correction metods can be attributed to te fact tat second order velocity extrapolations lead to stable scemes. Instead second order pressure extrapolations lead to unstable scemes. Tis not only appens in te pressure correction case but also in te velocity

170 CHAPTER 4. PRESSURE SEGREGATION METHODS correction wit continuous Laplacian approximation. Terefore we can say tat te use of a discrete Laplacian is more significant in te velocity correction sceme because it provides a pure VC sceme were no pressure extrapolation is needed. Te use of te Ricardson iteration wit a limited number of iterations as in te convergence test can be a cost effective option in tis direction. Regarding te extension to interface problems, te velocity also seems better to extrapolate tan te pressure. For example, in te simple 3D vertical tube presented in Capter 2 te velocity extrapolation would be exact wile te pressure extrapolation would be quite poor specially close to te interface. Despite tis example is ideal and very simple it can be quite representative of wat appens during mould filling simulations. For te enriced pressure model presented in Capter 2 te use of a velocity extrapolation makes te metod muc simpler tan a pressure extrapolation because te pressure is enriced and te velocity is not. If a pressure extrapolation were used, wen solving for time step n + 1 one would need te pressure extrapolation p n+1 = p n tat includes an enriced component tat depends on te position of te interface at time n. Tisdoes not seem appealing. For te previous reasons for interface problems we ave cosen to implement te velocity correction sceme.

Capter 5 Mould Filling 5.1 Introduction In tis capter te utility of te metods presented in previous capters is explored in te context of mould filling applications. Examples borrowed directly from te foundry are used to test te improvements introduced by te proposed metods. Te numerical simulation of mould filling processes as become a widespread tool for improving casting tecnology. Regions wit ig velocities tat can lead to premature wear of te mould can be predicted. Te quality of te resulting piece can also be improved, for example, by determining regions of possible air entrapment. An overview of computational metods for free surface flows in casting and Industry-Standard Mold- Filling codes can be found in [40]. Contrary to wat one migt intuitively tink, we ave observed tat in mould filling problems, lower filling velocities typically lead to more complex simulations. Tat is to say, low Froude number flows pose special difficulties for two pase flows. Te lower te Froude number, te iger te importance of te gravitational forces. Since te spatial distribution of te gravitational forces is determined by te position of te interface, te coupling between te position of te interface and te resulting flow increases as te Froude number decreases. An accurate representation of te pressure in te elements cut 171

172 CHAPTER 5. MOULD FILLING by te interface is needed for suc flows. By enricing te pressure finite element sape functions or by using a free surface model we ave obtained important improvements in simple examples. In tis work we extend te application of bot models to real mould filling problems. In Capter 2 an enriced pressure interpolation for two pase flows tat provides significant improvement over te usual two pase flow model in low Froude number simulations as been presented. In Capter 3 a free surface model on a fixed mes tat only simulates te region occupied by te fluid and neglects te influence of air as been presented. Tis metod as also sown good results for low Froude number flows tanks to a careful treatment of te elements cut by te front tat allows to accurately impose te boundary conditions at te interface. Moreover in Capter 4 pressure segregation metods tat allow to uncouple te solution of te velocity and te pressure ave been presented. For interface problems velocity correction metods ave been implemented. Terefore we are left wit a total of four options to solve for mould filling interface flows. Te two models for interface flows, te enriced pressure two pase model and te free surface model, are combined wit two solution strategies, te monolitic solver and te velocity correction solver. As we ave mentioned in te conclusions of te previous Capter, we ave preferred te velocity correction sceme instead of te pressure correction, not only because it provides better results in te one fluid case, but also because te velocity seems easier to extrapolate tan te pressure in interface problems. For te velocity correction sceme only te discrete Laplacian will be used. Te reason for doing tis, despite it can be computationally more expensive, is tat it implies one approximation less. In te free surface case te results obtained wit te velocity correction sceme are for te moment not very satisfactory. As we sall sow in Section 5.3, te convergence as been complicated and te computational time as risen significantly. Terefore it does not seem wise for te moment to add an additional source of error. In te enriced pressure two pase model te velocity correction sceme provides results tat are comparable wit te ones obtained wit te monolitic solver. Wen te enriced pressure is used te pressure is discontinuous on cut element faces and terefore

5.1. INTRODUCTION 173 we prefer to postpone te exploration of te use of te continuous Laplacian until te enriced pressure velocity correction sceme seems a more interesting option to use. For te moment te free surface model wit a monolitic sceme seems to be te best option as we sall sow in Section 5.2. Tis Capter will be organized as follows. In Section 5.2 results wit te free surface model used wit a monolitic solver are presented. Te application of te free surface model wit a velocity correction sceme is presented in Section 5.3. Sections 5.4 and 5.5 deal wit te application of te enriced pressure two pase flow model wit monolitic and velocity corrections scemes respectively. Tree mecanical pieces will be used to test te different metods. Te first one is a ollow mecanical piece made of steel wit pysical properties: ρ = 7266.0 and µ =6.7 10 3 (SI units). Tis piece is interesting because it as relatively tin walls wic make te mes quite complex. Te code is forced to obtain acceptable results wit few elements in te tickness. Te arrangement we simulate consists of two pieces togeter wit te filling cannel used during te actual filling process. Te inlet velocity is 0.113 m/s and te size of eac piece is approximately 0.16 0.16 0.13 m 3. Te wole filling process takes 11 seconds. Two unstructured triangular meses ave been used. Te coarse one as 72032 elements and 16149 nodes and te fine one as 575803 elements and 116214 nodes. Tey are sown in Figure 5.1. Te Reynolds number based on te inlet velocity and te lengt of te filling cannel is Re =2.45 10 4, and te Froude number is Fr =0.0065. Te second example is an automotive alloy weel. Te flow is created by applying a pressure on te fluid as is done in te actual filling process for tis piece. Te flow rate is ten determined by te resistance exerted on te fluid. We ave observed te friction may be ig in te vertical tube troug wic te molten metal is injected. Terefore, for tis case, we will simulate te wole filling cannel. Te pressure at te inlet varies linearly from 2.21 10 4 N/m 2 at te beginning of te simulation to 1.17 10 5 N/m 2 after 4.4 seconds. Te pysical properties we ave used are tose of aluminum, ρ = 2700.0 and µ =1.3 10 3 (SI units). Te Reynolds number

174 CHAPTER 5. MOULD FILLING Figure 5.1: Coarse and fine meses for te ollow mecanical piece

5.1. INTRODUCTION 175 based on a typical velocity inside te weel (0.5 m/s ) and te weel radius (0.5 m )is Re =5.19 10 5. Te Froude number is Fr =0.05. Te mes is formed by 489313 tetraedral elements and 109318 nodes. Te tird piece was presented to us as a really demanding case. It is te sovel for a power sovel. Te filling process takes approximately alf a minute and te sovel is nearly one meter long. Te inlet velocity we ave used during te simulation is 0.5 m/s. Te Reynolds number based on te previous velocity and lengt is Re =4.44 10 5,and te Froude number is Fr =0.031. As in te first example te material used is steel. Te mes used for tis example consists of 412848 tetraedral elements and 87010 nodes. Since te flow in mould filling problems is turbulent te viscosity in te previous equations as been calculated using te Smagorinsky model as µ = µ L + µ T,wereµ L is te molecular, constant, viscosity and µ T = µ T (u) is te additional turbulent viscosity defined by µ T = C 2 2 2 ε (u) :ε (u),were is te size of te element were it is computed and C 2 is te Smagorinsky constant. Te objective of tis tesis is not related to te analysis of te influence of te turbulence model and terefore a simple model as been cosen. Te Smagorinsky model is also used for mould filling simulations in [16, 46] and in te commercial code Vulcan [118] used to compare against our results in te next Section. Due to te ig Reynolds number of te problems we are dealing wit, no slip boundary conditions would require extremely fine meses along te boundary tat would make tem computationally unfeasible. Te solution we ave adopted is to use wall functions [72] tat describe te beavior of te flow near a solid wall. Te normal component of te velocity is set to zero. In te tangential direction a traction tat depends on te velocity at te boundary and is opposed to te direction of te flow is applied: τ w = ρ u2 u u

176 CHAPTER 5. MOULD FILLING were u can be determined from te following set of equations u + = 1 κ ln ( 1+κy +) ] +7.8 [1 e y+ /11.0 y+ 11.0 e 0.33y+ u + = ρ u u τ w, y + = ρδu µ. δ is te distance between te computational boundary and te wall, κ =0.41 is te Von Karman constant and y + and u + are non dimensional distances and velocities, respectively. 5.2 Free surface monolitic model In tis Section we present te results obtained wit te free surface monolitic model for te previous tree pieces. We observe tat tis model provides te best results of all four analyzed options and terefore tis results can be considered as a reference against wic te results obtained wit te oter models can be compared. It as allowed us to use bigger time steps and a lower Smagorinsky parameter tan te oter metods. Moreover, te convergence of bot te nonlinearity and te iterative solver are significatively better tan wen te enriced pressure two pase flow is used. Tis leads to significantly improved computational costs. Te monolitic sceme does not ave splitting errors, tat need to be corrected iteratively, as appens wen a predictor corrector sceme is used. Split OSS stabilization as been used for te Navier Stokes equations but some cases ave also been run wit Non Split OSS and ASGS and no significant difference as been observed. For te convective nonlinearity Picard iteration is used. Te tolerance is set to one percent variation in te L2 norm of te velocity and a maximum of 7 iterations are allowed. Typically te nonlinearity converges in less tan tree iterations. For te solution of te monolitic system a preconditioned GMRES iterative solver [102] is used. Te stopping criteria for te solver is tat te residual is smaller tan 10 6 times te rigt and side. A maximum of 500 iterations are allowed but te solver usually converges in

5.2. FREE SURFACE MONOLITHIC MODEL 177 less tan 30 iterations. Te Krylov dimension is set to 50. An ILUT preconditioner wit tresold 0.001 and filling 20 is used [102]. For te Level Set equation te convergence of te GMRES solver is very easy even witout preconditioner. In Capter 3 two alternatives ave been proposed for solving te free surface problem; a FM-ALE model and a simplified Eulerian model. Te results obtained wit bot models are very similar. Te results obtained wit te simplified Eulerian model sall be presented because tey are computationally ceaper. At te end of tis Section te results obtained wit te FM-ALE model are discussed. 5.2.1 Hollow mecanical piece Figure 5.2 sows te evolution of te interface for four time steps during te filling process wen te coarse mes is used. In te first step te interface is still inside te filling cannel. For te second one it as entered bot pieces. In te tird one te interface reaces te bottom of eac piece. As we will comment later, tis is one of te most complicated moments in te simulation. In te final figure more tan alf of eac piece as been filled. Te evolution of te front is very similar in bot pieces. Despite a coarse grid as been used te evolution of te interface is captured quite satisfactorily as one can observe by comparing wit te results sown for te fine mes in Figure 5.3. For bot meses te time steps size is 0.02 seconds. Te Smagorinsky model as been used to take into account turbulence and te constant as been set to C 2 =0.05. Knowing ow te interface evolves is important during te mould design as it can be used to cange te position of te inlets or alter te filling velocity to improve te quality of te resulting piece. Wen defects appear, aving some insigt on te way te flow evolves is of great elp to te foundry person because it is very difficult to actually see wat is appening inside te mould. Te evolution of te interface using te fine mes in sown Figure 5.3. Te sape of te interface is smooter tan te one obtained wit te coarse mes but tere is no mayor difference in te way te flow evolves. Te most noticeable cange is tat for eac time

178 CHAPTER 5. MOULD FILLING Figure 5.2: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te coarse mes step te results obtained wit te fine mes sow a bigger percentage of filled volume. Tis is related to numerical mass losses and is analyzed in more detail in Figure 5.4. Since foundry pieces are usually complex and it is common to fill several pieces at te same time (not only two as in te example) it is important to ave a code tat can provide te user wit acceptable results even wit coarse meses. In Figure 5.4 we compare temporal evolution of te injected and filled volumes using bot meses. Te injected volume is te same for bot meses. Te difference between te filled and injected volumes is te numerical mass loss. It is reduced as te mes is refined as one could expect. Te amount of mass loss can give us some idea on te quality of our results and indicate te most complex moments during te simulation. In our example, we can see tat te most important mass loss occurs wen te filled volume is between 0.0004 m 3 and 0.0006 m 3. It corresponds to te moment wen te bottom of eac piece is being filled. Tis suggests tat a mes refinement close to tat area migt improve te solution. Wen te fine mes is used te mass loss is very small. Even wit te coarse mes mass conservation is muc better tan wen te enriced pressure two

5.2. FREE SURFACE MONOLITHIC MODEL 179 Figure 5.3: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te fine mes pase flow monolitic model is used, as we sall sow in Section 5.4. In Figure 5.5 te results on te coarse mes wit an increased Smagorinsky constant (C 2 =0.2) are presented. It can be observed tat te results are not significantly affected by te cange in te Smagorinsky constant. Total Matrix N. Stokes Solver N. Stokes Coarse mes, C 2 =0.2 8638s 69.5% 18.9% Coarse mes, C 2 =0.05 12593s 69.6% 20.2% Fine mes, C 2 =0.05 90038s 45.2% 45.6% Table 5.1: Cpu time In Table 5.1 te computational times for te previous simulations are presented. It can be observed tat te solution of te Navier Stokes equations requires most of te time. For te coarse mes te assembly of te matrix takes approximately tree times more tan te solution of te linear system. Wit te fine mes bot tasks take approximately

180 CHAPTER 5. MOULD FILLING 0.0018 0.0016 Injected vol Filled vol - coarse mes Filled vol - fine mes 0.0014 0.0012 volume [m^3] 0.001 0.0008 0.0006 0.0004 0.0002 0 0 2 4 6 8 10 12 14 time [s] Figure 5.4: Filled volume vs. injected volume for bot meses Figure 5.5: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te coarse mes wit a iger Smagorinsky constant

5.2. FREE SURFACE MONOLITHIC MODEL 181 tesametime. Comparison wit a commercial code In order to ave some idea on te efficiency of our code we compare te results we ave obtained wit our model against te results obtained wit te commercial code Vulcan [118]. Vulcan uses a fixed mes finite element pressure correction predictor corrector sceme. Figure 5.6: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te commercial code wit te fine mes Te results obtained wit Vulcan on te fine mes are sown in Figure 5.6. Tey are quite similar to te ones we ave obtained wit our code. Te most noticeable difference is te sape of te interface in te last to two figures. Wit our model a nearly flat surface is obtained. Instead wit te commercial codes spurious oscillations can be observed. Te enanced beavior of our code can be attributed to te correct treatment of boundary conditions on te interface. Wit te coarse mes, te results obtained wit Vulcan sow bigger spurious oscillations (Figure 5.7). Moreover te effect of numerical viscosity becomes quite

182 CHAPTER 5. MOULD FILLING noticeable. Instead, wit our code, te results obtained on te coarse mes were muc closer to te ones obtained in te fine mes. As we ave already mentioned, te correct beavior of our code even on coarse meses is a very important feature for mould filing simulations were te complexity of real pieces inibits te possibility of using too fine meses. Figure 5.7: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te commercial code wit te coarse mes Te comparison of te computational times obtained wit Vulcan against te ones obtained wit our code sows tat despite our code is only an academic version it can provide competitive results. For te simulation on te fine mes Vulcan takes nearly twice te time needed by our code. Wit te coarse mes it takes 3.2 times more tan our code, making te advantage even more notorious. In te simulations wit our code we ave used a fixed time step size tat results in a total of 550 steps. Instead Vulcan uses a variable time step size. A total of 815 and 1225 step ave been required on te fine and coarse meses respectively. Te increase in te number of steps wit te coarse mes observed wit Vulcan may be related to te increase of te spurious oscillations close to te interface.

5.2. FREE SURFACE MONOLITHIC MODEL 183 5.2.2 Alloy weel For te automotive alloy weel te time steps size we ave used is 0.01 seconds and te Smagorinsky constant is C 2 = 0.05. In Figure 5.8 te evolution of te interface for different time steps is presented. For te first time step te wole domain is sown and for te remaining steps only te details at te weel are sown. Once te molten metal reaces te top of te filling tube it slides troug te bottom of te spokes until it reaces te teir end. Ten it turns and fills te lower part of te weel. Finally it raises troug te vertical walls of te weel. Simultaneously te filling of te spokes is completed. Since we are using a free surface model air is not taken into account so tere is no possibility for te formation of air bubbles as appens wen te enriced pressure two pase flow model is used (Section 5.4). Te possibility of modifying te free surface model so tat it can take into account te formation of bubbles will be discussed at te end of te Capter. Moulds can be classified into two groups depending on weter tey allow air to escape toug teir walls or not. Wen te air is allowed to escape, suc as in sand moulds, te importance of taking into account its effect is less important. Te pressure contour lines are presented in Figure 5.9. A fixed scale wit a maximum of 5000 N/m 2 as been used to focus on te pressures inside te weel. In te filling tube te pressure is nearly ydrostatic but due to te scale we ave used it is not sown. Using a iger Smagorinsky constant (C 2 = 0.2) very similar results ave been obtained. Te iger viscosity imposes an small increase of resistance to te flow. As tis piece is filled by an imposed pressure tis results in a sligtly retarded front. Te results ave not been sown because te difference is ard to appreciate. Te total CPU time for te simulation as been 38579 seconds. As in te previous example, te resolution of te Navier Stokes equations took most of te time, wit 18220 seconds for te matrix assembly and 15470 seconds for te linear solver.

184 CHAPTER 5. MOULD FILLING t=3.0 s t=2.4 s t=3.8 s Figure 5.8: Interface evolution at t=2.4, 3.0 and 3.4 s t=3.0 s P[N/m²] t=2.4 s t=3.8 s Figure 5.9: Pressures at t=2.4, 3.0 and 3.4 s

5.2. FREE SURFACE MONOLITHIC MODEL 185 5.2.3 Sovel For te sovel a 0.01 seconds time step as been used wit two Smagorinsky constants, C 2 =0.05 and C 2 =0.2. Contrary to wat appens in te oter two pieces, for tis case te effect of te Smagorinsky constant is more noticeable, at least during part of te simulation. Figure 5.10 sows te evolution of te interface for selected time steps using C 2 =0.2. Te filling cannel used for tis piece splits into two brances. One of te brances is closer to te inlet tan te oter one. As te interface reaces te first branc te molten metal starts falling troug it. Approximately one second takes place before te flow starts falling troug te second branc. Terefore te side of te sovel closer to te first branc is filled earlier tan te part connected to te second branc. Wen te molten metal exits eac of te two brances it slides into two circular parts wit a ole in te middle located beneat te end of eac branc. Once eac of tese two parts are full te flow spreads troug te base of te sovel and finally raise along te lateral walls to complete te filling process. For C 2 =0.05 te filling is similar to te one wit C 2 =0.2 except wen te molten metal goes into te circular part wit a ole in te middle. Wen te lower constant is used one portion of te flow deposits in te bottom of te circular part and te oter surrounds te ole in te middle. Instead, wen te iger constant is used all of te flow deposits in te bottom of te circular part. A comparison of te filling of te circular part wit te two Smagorinsky constants is sown in Figure 5.11. Using a finer mes it as been observed tat part of te flow surrounds te ole in te middle (Figure 5.12). In tis sense te results wit te lower constant seem better. Unfortunately te flow becomes too complex for te mes wen te smaller constant is used and an important mass loss is introduced as we sall sow in Figure 5.13. Te mass loss wen te smaller constant is used occurs mainly during t =3.5 s to t =7.0 s, te time needed to fill te circular part. Wit te iger Smagorinsky constant te mass loss is significantly reduced and terefore te results can be preferred despite te lack of precision in te filling of one of

186 CHAPTER 5. MOULD FILLING t=3.8 s t=7.6 s t=11.2 s t=17.6 s Figure 5.10: Interface position using C 2 =0.2

5.2. FREE SURFACE MONOLITHIC MODEL 187 te circular parts. t=3.2 s t=3.8 s t=4.4 s t=5.0 s Figure 5.11: Detail of te interface position using C 2 =0.2 (top) and C 2 =0.05 (bottom) Taking into account tat tis is a complex piece we ave decided to simulate it wit a fine mes formed by 1619428 tetraedral elements and 319052 nodes. Te same time step as in te coarse mes and a C 2 =0.2 Smagorinsky constant ave been used. Te results ave served as a reference to compare te results obtained wit te original mes. Moreover te simulation on te fine mes as been used to explore te current limits of our model on a typical PC. Te results are sown in Figure 5.12. In Figure 5.13 we compare temporal evolution of te injected and filled volumes for te previous tree cases: coarse mes wit C 2 =0.05 and C 2 =0.2 and fine mes wit C 2 =0.2. As we ave already mentioned, te mass loss concentrates mainly during te filling of te two circular parts located at te bottom of te sovel and it is significantly iger wen te coarse mes wit te low Smagorinsky constant is used. During te rest of te simulation very little mass loss is observed for te tree cases. It is interesting to note tat te mass loss is very similar for te two meses wen C 2 =0.2 isused. In Table 5.2 te CPU times for te sovel simulations are presented. For te fine

188 CHAPTER 5. MOULD FILLING t=3.8 s t=7.6 s t=11.2 s t=17.6 s Figure 5.12: Interface position using te fine mes

5.2. FREE SURFACE MONOLITHIC MODEL 189 0.025 Injected vol Filled vol - C^2 = 0.2 Filled vol - C^2 = 0.05 Filled vol - Fine mes 0.02 volume [m^3] 0.015 0.01 0.005 0 0 5 10 15 20 time [s] Figure 5.13: Filled volume vs. injected volume Total Matrix N. Stokes Solver N. Stokes Coarse mes, C 2 =0.2 271681s 50.8% 34.8% Coarse mes, C 2 =0.05 371473s 56.8% 30.6% Fine mes, C 2 =0.2 863309s 40.1% 39.8% Table 5.2: Cpu time

190 CHAPTER 5. MOULD FILLING mes te cost of matrix assembly is approximately equal to te cost of te linear solver. For te coarse mes te matrix assembly is more expensive. Bot for te sovel and te alloy weel it as not been possible to obtain satisfactory results wit te commercial code used for te ollow mecanical piece. 5.2.4 Results wit te FM-ALE model As we ave anticipated at te beginning of tis Section, for te mould filling examples we ave not observed any significant difference between te results obtained wit te simplified Eulerian model and tose obtained wit te FM-ALE model. Te same beavior as also been observed in te simpler examples presented in Capter 3. Te variations introduced in te evolution of te free surface or te filled volume by te use of te FM-ALE model instead of te simplified Eulerian model are ardly noticeable and terefore te Figures are not be repeated. In Table 5.3 we present te CPU times obtained wit te FM-ALE so tat tey can be compared against te ones obtained wit te simplified Eulerian model. Te computational times for te Navier Stokes matrix assembly and solution of te linear system are similar to te ones obtained wit te simplified Eulerian model. Te total CPU time increases significantly due to te additional steps required by te FM-ALE model. For te moment little effort as been put into optimizing tose steps because as bot models produce similar results it as been ceaper to use te simplified Eulerian model. Total Matrix N. Stokes Solver N. Stokes Hollow Coarse, C 2 =0.05 25568s 44.0% 11.8% Hollow Fine, C 2 =0.05 186951s 26.2% 28.9% Weel, C 2 =0.05 80199s 24.5% 21.3% Sovel, C 2 =0.2 447467s 30.8% 16.1% Table 5.3: Cpu time

5.3. FREE SURFACE VELOCITY CORRECTION MODEL 191 Despite we ave not found significant differences between te results obtained wit te FM-ALE model and te simplified Eulerian model we belive tat in more demanding examples te FM-ALE model may prove advantageous. In [29] we ave extended te application of te FM-ALE model to a wider range of problems, including fluid structure interaction. An example of a cylinder moving in a rectangular domain is presented were te advantage of using te FM-ALE model is easily observed. It is mentioned tat a large time step is used in order to observe te improvements introduced by te FM-ALE model. 5.3 Free surface velocity correction model Te use of a velocity correction sceme for te free surface model as resulted in te worse results of all four analyzed options. Te reason for te poor beavior of tis metod is still not clear and furter researc is needed. In Figure 5.14 we sow te advancement of te front for a successful run wit te free surface velocity correction model. It corresponds to te ollow mecanical piece wit te coarse mes and a 0.005s time step. A ig Smagorinsky constant (C 2 = 0.4) is used in order to make metod more robust. In Capter 3 two alteratives ave been proposed; one is to use a FM-ALE metod and te oter one is to use simplified Eulerian model. Wen te monolitic sceme is used bot of tem give very similar results but te simplified Eulerian model is faster. In te velocity correction case te predictor corrector convergence as been easier (but still complicated) wit te FM-ALE metod and terefore for te results presented in tis section it will always be used. Te reason migt be tat te use of te FM-ALE metod provides a better velocity extrapolation. In order to obtain te results sown in Figure 5.14 very low tolerances ave been needed bot for te predictor corrector iteration and for te velocity and pressure solvers. A predictor corrector sceme wit separate loops for te nonlinear and predictor corrector iterations as been used because we ave observed tat it leads to a more robust sceme. Te predictor corrector iteration, tat is te most difficult loop to converge for tis

192 CHAPTER 5. MOULD FILLING Figure 5.14: Interface position at t = 1.6, 3.2, 4.8 and 7.6 s using te velocity correction sceme on te coarse mes problem, is solved in te inner loop so tat error in coupling of te velocity and pressure does not spoil te nonlinear convergence. Te stopping criteria for te predictor corrector iteration is set so tat te variation of te velocity in te L2 norm is lower tan 0.01%. Te maximum number of predictor corrector iterations is set to 60. In practice te maximum number of iterations was nearly never reaced but during several parts of te simulation more tan 10 iterations were used. For te nonlinear iteration a iger tolerance of 1.0% and a maximum of 7 iterations were allowed. Te parameters for te nonlinear iteration coincided wit tose used in a monolitic run used as a reference. For te velocity and pressure solvers a tolerance of 10 9 was used. Te combination of a low tolerance for te predictor corrector iteration and for te solvers as made te metod very expensive. Te total CPU time for tis run as been 349398 s. A ig percentage of tis time, 285343 s, corresponds to te velocity and pressure solvers. For a similar monolitic run te time for te solver is only 14537 s, tat is approximately 20 time less. Oter alternative runs wit iger predictor corrector

5.3. FREE SURFACE VELOCITY CORRECTION MODEL 193 or solver tolerances were tested to try to reduce computational times. Some of tose runs managed to provide acceptable results for te advancement of te front but sowed spurious negative pressures close to te interface. Oters would diverge unexpectedly. For te run presented in Figure 5.14 no spurious negative pressures were observed. For an identical run except for te fact tat a iger solver tolerance of 10 7 was used, spurious pressures appeared at t =2.8 s andtenatt =6.5 s wen a 40% of te piece ad been filled te results diverged. In Section 5.5 acceptable results ave been obtained wit te enriced pressure two pase flow surface velocity correction model witout te need to use suc low tolerances. Terefore te poor beavior observed wit te free surface model seems to be related to te combination of te free surface model, tat works very well in te monolitic case, wit te velocity correction sceme. Te good results obtained wit very low tolerances indicate tat te problem is not related to an error in te implementation of te free surface velocity correction model. Te spurious pressures tat appear close to te interface migt indicate tat te velocity extrapolation used by te velocity correction sceme close to te interface is poor. Te velocities in te region tat becomes part of te fluid at eac time step can be better extrapolated wen te FM-ALE version is used tan wen te simplified Eulerian is used. Te fact tat a better beavior is observed wen te velocity correction is used wit te FM-ALE metod can be an evidence tat better extrapolations velocities close to te interface elp to improve te solution. A strategy to improve te extrapolation velocities close to te interface could make te free surface velocity correction model more feasible. We propose to improve Ũn+1 prior to starting eac time step. We can call te improved extrapolation Ũ from Ũn+1 q n+1 q q.itdiffers only in a small region close to te interface. Tis region is formed by only 2 or 3 layers of elements. In te nodes from te small region close to te interface in contact wit te region were te velocities are not modified te velocities are prescribed to te values Ũn+1. Ten te Navier Stokes equations are solved only in te small region to obtain Ũ q n+1 q. Since only a small region is being solved, one can expect convergence

194 CHAPTER 5. MOULD FILLING of te iterative solver to be very easy. Moreover even if a low tolerance is required for te predictor corrector iteration te computational cost sould not increase significantly because only a small region is being solved. A monolitic solver migt even be used in tis small region. For te finer mes and for te oter two examples poor results were also obtained wit te free surface velocity correction model. In tose cases te cost of using very low tolerances makes te simulations too expensive. Terefore we believe tat a better strategy sould first be obtained wit te coarse mes before stepping to te bigger examples. 5.4 Enriced pressure two pase flow monolitic model In tis section we repeat te previous tree examples using te enriced pressure two pase flow monolitic model. Te general flow pattern remains very similar to te one obtained wit te free surface model but some interesting differences can be observed. Despite some moulds ave walls tat allow air to escape, in our examples we ave supposed tat air is only allowed to escape troug specified outlets. Terefore regions of entrapped air can be observed wen te two pase flow model is used. As we ave observed in te previous examples, mass conservation can be an indicator of te accuracy of te results. Wit te enriced pressure two pase flow model mass conservation is poorer tan wit te free surface model. Moreover we ave observed tat typically mass is lost wen ASGS stabilization is used and gained wen OSS is used. Instead, a we ave already mentioned, wit te free surface model no significant dependence on te stabilization metod as been observed. Te numerical strategy is te same as for free surface monolitic model.

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 195 5.4.1 Hollow mecanical piece Figure 5.15 sows te evolution of te interface obtained on te coarse mes wit ASGS stabilization, a C 2 =0.4 Smagorinsky constant and a 0.005 seconds time step. Compared to te free surface model case, we ave needed a iger Smagorinsky constant and a smaller time step. Wen te same Smagorinsky constant and time step as in te free surface case is used, te mass loss is too big and te results do not ave muc interest. Even wit te parameters we ave used, te mass loss is iger tan te one observed wit te free surface model wit more demanding parameters. However, te evolution of interface is similar to te one obtained wit te free surface model but retarded due to te mass loss. In te upper part of te filling cannel entrapped air can be observed. Tis is obviously different to wat appens wit te free surface model were no air entrapment can occur because air is not simulated. Figure 5.15: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te coarse mes wit ASGS stabilization, C 2 =0.4 andδt =0.005 s Wit OSS stabilization, bot split and non split, two main differences ave been observed. Mass conservation is improved; actually tere is some mass gain but it is smaller tan te mass loss observed wit ASGS. On te oter and, less viscosity is introduced

196 CHAPTER 5. MOULD FILLING so te nonlinear convergence becomes more difficult and it is ard for some cases to converge. In order to improve te nonlinear convergence anisotropic sock capturing [22] as been introduced. Wit te coarse mes we ave been able to use te same time step (0.02 s) and Smagorinsky constant (C 2 =0.05) as in te free surface case. Moreover very little dependence on te Smagorinsky constant as been observed. Te evolution of te interface and mass conservation obtained wit C 2 =0.2 are very close to tose obtained wit C 2 =0.05. In Figure 5.16 te evolution of te interface obtained on te coarse mes wit split OSS stabilization and C 2 =0.05 is presented. As in te ASGS case te results are similar to te ones obtained wit te free surface model but te interface is advanced (instead of retarded) due to te mass gain. Figure 5.16: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te coarse mes wit OSS stabilization, C 2 =0.05 and δt =0.02 s Figure 5.17 sows te evolution of te filled and injected volumes obtained wit ASGS (C 2 =0.4, δt =0.005 s) and split OSS (C 2 =0.05, δt =0.02 s) stabilization. In order to sow ow mass conservation degrades wit ASGS stabilization te filled volume obtained wit C 2 =0.1, δt =0.01 s is also included in te comparison. Te fluid mass loss comes from several sources. Despite we are solving te

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 197 0.0018 0.0016 Injected vol Fil vol - OSS, dt=0.02, C^2=0.05 Fil vol - ASGS, dt=0.005, C^2=0.4 Fil vol - ASGS, dt=0.01, C^2=0.1 0.0014 0.0012 volume [m^3] 0.001 0.0008 0.0006 0.0004 0.0002 0 0 2 4 6 8 10 12 14 time [s] Figure 5.17: Filled volume vs. injected volume for te coarse mes incompressible Navier Stokes equations te numerical results are not exactly divergence free. Since our meses are quite coarse tere will be errors in te satisfaction of bot te continuity and momentum equations. Pressure stabilization also affects te satisfaction of te incompressibility condition. Te errors in te transport of te Level Set function can also cause fluid loss. After solving te level set function it needs to be reinitialized; tis may also introduce errors. Tere migt also be some coupling between te previous sources of error. Wen te fine mes is used te mass loss obtained wit ASGS stabilization is reduced. Results obtained wit C 2 =0.1 andδt =0.01 s are sown in Figure 5.18. Some mass loss can be observed but te evolution of te interface is acceptable. Wen te Smagorinsky constant is increased to C 2 =0.4 and te time step is reduced to 0.005 s mass conservation improves as sown in Figure 5.19. Wen split OSS stabilization is used wit te parameters used for te coarse mes, C 2 =0.05 and δt =0.02 s te run diverges. Increasing te Smagorinsky constant to C 2 =0.1 and reducing te time step to 0.01 s good results ave been obtained. Te evolution of te filled volume is presented in Figure 5.19. Te beavior is te same as te

198 CHAPTER 5. MOULD FILLING Figure 5.18: Interface position at t =1.6, 3.2, 4.8 and7.6 s using te fine mes wit ASGS stabilization, C 2 =0.1 andδt =0.01 s one observed wit te coarse mes but te errors are significantly smaller wen ASGS stabilization is used. Te flow pattern during te filling process is also important to te foundry person. For example, regions of ig velocities can lead to premature mould wear and sould be avoided. Figure 5.20 sows te flow field at different time steps obtained wit te fine mes and ASGS stabilization. We believe tat te effectiveness of te metod we propose depends strongly on te pressure enricment we introduced in Capter 2. In order to prove tis we ave run te problem wit C 2 =0.4 andδt =0.005 s on te fine mes witout using te pressure enricment. Te results are muc poorer tan tose sown previously. By te time te filled volume fraction reaces a 14 percent of te mould te mass loss is so important tat most of te injected fluid is being lost numerically. Te results lose any sense and terefore te runs was stopped. Te evolution of te filled and injected volumes is sown in Figure 5.21. It can be observed tat te mass loss is similar wit ASGS and OSS

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 199 0.0018 0.0016 Injected vol Fil vol - OSS, dt=0.01, C^2=0.1 Fil vol - ASGS, dt=0.005, C^2=0.4 Fil vol - ASGS, dt=0.01, C^2=0.1 0.0014 0.0012 volume [m^3] 0.001 0.0008 0.0006 0.0004 0.0002 0 0 2 4 6 8 10 12 14 time [s] Figure 5.19: Filled volume vs. injected volume for te fine mes V[m/s] t=1.6 s t=3.2 s t=4.8 s t=7.6 s Figure 5.20: Velocity field at t = 1.6, 3.2, 4.8 and 7.6 s using te fine mes wit ASGS stabilization, C 2 =0.1 andδt =0.01 s

200 CHAPTER 5. MOULD FILLING stabilization but in te OSS case spurious velocities appear tat lead to oscillations in te filled volume. 0.0004 0.00035 Injected vol Filled vol - ASGS Filled vol - OSS 0.0003 0.00025 volume [m^3] 0.0002 0.00015 0.0001 5e-005 0 0 0.5 1 1.5 2 2.5 3 3.5 4 time [s] Figure 5.21: Filled volume vs. injected volume witout using enricment Total Mat. N. Stokes Solv. N. Stokes Coarse mes, OSS, C 2 =0.05, δt =0.02 s 26218s 74.4% 21.0% Coarse mes, ASGS, C 2 =0.4, δt =0.005 s 82434s 59.2% 35.9% Fine mes, OSS, C 2 =0.1, δt =0.01 s 457920s 62.8% 31.6% Fine mes, ASGS, C 2 =0.1, δt =0.01 s 343364s 51.2% 44.3% Fine mes, ASGS, C 2 =0.4, δt =0.005 s 642212s 39.3% 44.9% Table 5.4: Cpu time In Table 5.4 te computational times for te previous simulations are presented. As in te free surface case, te solution of te Navier Stokes equations requires most of te time. Wit te coarse mes and OSS stabilization we ave been able to run wit te same time step and Smagorinsky constant as in te free surface case. Terefore it is interesting to compare te computational times for tis run against te ones obtained

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 201 wit te free surface model. Te total computational time is sligtly more tan twice te computational time obtained wit te free surface model. Te free surface model does an average of 2.25 nonlinear iterations per time step and te enriced pressure two pase flow model 2.8, tat is approximately 25% more. Terefore only a small part of te advantage of te free surface model comes from te reduced number of nonlinear iterations. Looking at te time needed to compute te Navier Stokes matrix, te free surface model requires less tan alf of te time required by te two pase flow model. A small part of tis advantage can be attributed to te reduced number of nonlinear iterations but most of it comes from te fact tat empty elements do not need to be assembled. Since te run starts wit a piece tat is nearly empty at te beginning of te run te cost of te matrix assembly is very small. Since te run stops wen te piece is full, in average between te start and te end of te simulation, approximately only alf of te elements need to be assembled wen te free surface is used. Te time needed by te solver is also approximately twice wen te two pase flow model is used. Typically less number of solver iterations are needed per nonlinear iteration wen te free surface model is used. Two reasons for tis beavior can be given. First, since during part of te simulation te domain tat needs to be solved is smaller it seems logical tat te matrix is better conditioned. Moreover during most of te simulation te free surface model leads to an non-confined domain. Instead te two pase flow model leads to a nearly confined domain because te flow is only allowed to escape troug small air outlets. Typically (not only in interface flows) linear systems arising from te discretization of te Navier Stokes equations on non-confined domains are better conditioned tan tose obtained on confined or nearly confined domains. On te coarse mes, for te ASGS stabilized case, a four times smaller time step as been used to obtain an acceptable mass conservation. Te total CPU time is nearly four times bigger tan te one obtained wit OSS and δt =0.02 s. On te fine mes, te CPU time for te run wit ASGS stabilization and δt =0.01 s is nearly four times bigger tan te time obtained wit te free surface model and δt = 0.02 s. One possible explanation is tat it is twice more costly due to te use of te smaller time step, and twice due to te

202 CHAPTER 5. MOULD FILLING use of two pase flow model instead of te free surface model. For te same time step, te results wit OSS stabilization take more time tan te ones wit ASGS stabilization. We believe tis can be attributed to te fact tat te OSS run needs an average of 2.7 nonlinear iterations per time step and te ASGS run only 1.8. For te ASGS run tis time for te solver and for te matrix are similar. Instead for te OSS run, te matrix assembly takes twice more tan te solver. We ave observed tat for tis run te solver converges in less iterations wen split OSS stabilization is used. Finally te ASGS run wit δt =0.005 s needs nearly twice te time tan te one wit δt =0.01 s due to te use of a smaller time step. We can conclude tat for te same conditions two pase flow model takes approximately twice more time tan te free surface model. Moreover since te free surface model allows to use bigger time steps to obtain similar results its advantage becomes more notorious. 5.4.2 Weel In Figure 5.22 te evolution of te interface obtained wit ASGS stabilization, C 2 =0.2 and δt =0.01 s is presented. For te first time step te wole domain is sown and for te remaining steps only te details at te weel are sown. It is interesting to see tat at some points inside te spokes air is entrapped. Tis could lead to fabrication defects and sould be avoided. At time step t =3.8 s an air bubble tat is rising to escape as it reaces te upper interface can be seen. In Figure 5.23 te velocity field for different time steps is presented. Since we are using an inlet pressure tat varies linearly wit time, wile te interface is inside te filling tube te velocities remain quite constant. Te increase in te inlet pressure is compensated mainly by an increase in te free surface eigt. Terefore te position of te free surface raises linearly wit time and te velocity in te tube is approximately constant. As te flow enters te weel and starts sliding down te weel spokes te increase in te ydrostatic pressure stops but te inlet pressure continues growing linearly. Terefore

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 203 t=3.0 s t=2.4 s t=3.8 s Figure 5.22: Interface evolution at t =2.4, 3.0 and3.8 s

204 CHAPTER 5. MOULD FILLING te flow accelerates until te interface reaces te vertical walls. Finally, te flow rate stabilizes once again until te end of te simulation. At t =3.8 s a region of ig velocities can be observed at te position were te bubble is located. t=3.0 s V[m/s] t=2.4 s t=3.8 s Figure 5.23: Velocity norm at t =2.4, 3.0 and3.8 s Te total CPU time for tis run as been 148418 s. Te assembly of te Navier Stokes matrix requires 80760 s and te solver 62541 s. Te proportion is similar to te one observed wit te free surface model but te times are approximately tree times iger despite te same time step is being used. For tis example we ave observed tat OSS stabilization leads to quite poor results. Te nonlinear convergence becomes quite complex and contrary to wat appens in te oter examples, sock capturing does not elp to improve te results. Terefore for tis piece we will only present results obtained wit ASGS stabilization. It is important to note tat te mes is quite coarse for te flow we are simulating. As in te previous example, te simulation was also run witout using te pressure

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 205 enricment. Te results witout pressure enricment are muc poorer tan tose obtained wit pressure enricment. In Figure 5.24 te evolution of te interface for te case witout enricment is sown. Up to t =2.4 s te results are similar to tose obtained wit te enriced model. As te flow starts sliding down te weel spokes te numerical mass loss becomes muc more important tan in te case wit enricment. For time t =3.0 s te mass loss is easily noticeable and at t =3.8 s it is very important. t=3.0 s t=2.4 s t=3.8 s Figure 5.24: Interface evolution for te case witout enricment 5.4.3 Sovel Figure 5.25 sows te results obtained wit ASGS stabilization, C 2 =0.4andδt =0.005 s. We ave used tose parameters because wen te time step is increased and te Smagorinsky constant decreased te mass loss increases significantly. As in previous pieces, te evolution of te interface is similar to te one obtained wit te free surface model but retarded due to te mass loss. Wen split OSS stabilization wit C 2 =0.2 and δt =0.01 s is used te beavior of te flow is very similar to te one obtained wit

206 CHAPTER 5. MOULD FILLING ASGS but as mass is gained instead of lost te front is advanced instead of retarded. Te same beavior as already been observed in te ollow mecanical piece and terefore te evolution of te interface for te OSS run is not presented for te sovel. In Figure 5.26 te comparison of te filled and injected volumes is presented. Te run wit split OSS stabilization, C 2 =0.2 andδt =0.01 s gainsmassandterunwitasgs stabilization, C 2 =0.4 andδt =0.005 s loses mass. We ave also introduced te results obtained wit ASGS stabilization, C 2 =0.2 andδt =0.01 s to sow tat it introduces an important mass loss. In te case witout enricment te mass loss is so important tat after some time most of te injected fluid is being lost numerically and terefore te runs were stopped. Figure 5.27 presents te comparison of te filled and injected volumes obtained wit C 2 =0.4 andδt =0.005 s. Bot ASGS and Split OSS stabilized runs sow a similar beavior but wit OSS stabilization te filled volume stalls at 22% and wit ASGS at 17%. Te advantage of using pressure enricment can easily be observed by comparing wit te results presented in Figure 5.26. In Table 5.5 te computational times for te sovel simulations are presented. Te run wit split OSS stabilization uses te same time step and Smagorinsky constant as one of te runs presented in te free surface monolitic section. Te comparison of te CPU time sows tat te enriced pressure two pase flow model takes more tan twice te time tan te free surface model. Te time required by te solver for te Navier Stokes equations is tree times bigger wen te enriced pressure two pase flow model is used. For te ASGS run, te use of a smaller time step makes te CPU time to increase to nearly twice te time needed wit OSS stabilization. Total Matrix N. Stokes Solver N. Stokes OSS, C 2 =0.2, δt =0.01 s 590362s 38.4% 56.3% ASGS, C 2 =0.4, δt =0.005 s 1106010s 37.1% 57.3% Table 5.5: Cpu time

5.4. ENRICHED PRESSURE TWO PHASE FLOW MONOLITHIC MODEL 207 t=3.8 s t=7.6 s t=11.2 s t=17.6 s Figure 5.25: Interface position using ASGS stabilization, C 2 =0.4 andδt =0.005 s.

208 CHAPTER 5. MOULD FILLING 0.025 Injected vol Fil vol - OSS, dt=0.01, C^2=0.2 Fil vol - ASGS, dt=0.01, C^2=0.2 Fil vol - ASGS, dt=0.005, C^2=0.4 0.02 volume [m^3] 0.015 0.01 0.005 0 0 5 10 15 20 time [s] Figure 5.26: Filled volume vs. injected volume 0.008 0.007 Injected vol Filled vol - ASGS Filled vol - OSS 0.006 0.005 volume [m^3] 0.004 0.003 0.002 0.001 0 0 1 2 3 4 5 6 7 time [s] Figure 5.27: Filled volume vs. injected volume for te case witout enricment

5.5. ENRICHED PRESSURE TWO PHASE FLOW VELOCITY CORRECTION MODEL209 5.5 Enriced pressure two pase flow velocity correction model In tis section some of te cases presented in te previous section are repeated wit a predictor corrector velocity correction model. Te objective is to sow tat very similar results to te ones presented wit te monolitic model can be obtained. Te numerical strategy we ave used consists of a external predictor corrector loop wit a maximum of 7 iterations and a stopping criteria set so tat te variation of te velocity in te L2 norm is lower tan 1.0%. Te non linearity is dealt in a internal loop wit a maximum of 3 iterations and a stopping criteria of 0.8% variation of te velocity in te L2 norm. For te solution of te velocity equations a GMRES solver wit a simple LU-SGS [75] preconditioner is used. Te stopping criteria used is tat te residual falls below 10 6 times te rigt and side and a maximum of 300 iterations are allowed. For te pressure a discrete Laplacian is used. It is solved wit a Conjugate Gradient solver wen Split OSS stabilization is used and wit a GMRES solver wen ASGS is used (see Subsection 4.3.6). No preconditioner is used for te pressure. Te same stopping criteria as for te velocity is used. Complex preconditioners ave been avoided so tat te metod can be parallelized in te future. 5.5.1 Hollow mecanical piece Te evolution of te interface obtained wit ASGS stabilization, a C 2 =0.4 Smagorinsky constant and a 0.005 seconds time step on te coarse mes is indistinguisable from te one obtained wit te monolitic model and te same parameters (Figure 5.15) and is terefore not repeated. Te mass loss also remains unaltered from te one sown for te monolitic model in Figure 5.17. Instead wit Split OSS stabilization te results obtained wit te velocity correction metod were poorer tan te ones obtained wit te monolitic model. In te monolitic case OSS stabilization made possible te use of a big time step (0.02 s) and small

210 CHAPTER 5. MOULD FILLING Smagorinsky constant (C 2 =0.05). Wit tose parameters te velocity correction run diverged. A 0.002 s time step and C 2 =0.4 Smagorinsky constant were needed to obtain acceptable results. For te fine mes te run wit ASGS stabilization, a C 2 =0.4 Smagorinsky constant and a 0.005 seconds time step was also repeated wit te velocity correction sceme. Te results are again very close to te ones obtained wit te monolitic model. As in te coarse mes, te convergence wit Split OSS stabilization is difficult and terefore no results are presented. Total Mat. N. Stokes Solv. N. Stokes Coarse mes, ASGS, C 2 =0.4, δt =0.005 s 148148s 76.1% 21.3% Coarse mes, OSS, C 2 =0.4, δt =0.002 s 274638s 54.7% 42.8% Fine mes, ASGS, C 2 =0.4, δt =0.005 s 910100s 69.8% 27.3% Table 5.6: Cpu time for te velocity correction runs In Table 5.6 te computational time for te velocity correction runs is presented. Bot ASGS runs (coarse and fine) takes approximately 50% more CPU time tan te corresponding monolitic runs. Te time for te solver is approximately te same but te matrix assembly takes more tan twice te time tan te monolitic run. Wit OSS stabilization te use of a very small time step leads to ig CPU times. 5.5.2 Weel Wen te run presented in te monolitic case (ASGS stabilization, C 2 = 0.2 and δt = 0.01 s) was repeated wit te velocity correction sceme it diverged. A smaller time step (δt =0.005 s) and a bigger Smagorinsky constant (C 2 =0.4) were needed to obtain acceptable results. Te evolution of te interface is sown in Figure 5.28. Altoug te time step and a Smagorinsky constant are not te same as in te monolitic run te results are very similar.

5.5. ENRICHED PRESSURE TWO PHASE FLOW VELOCITY CORRECTION MODEL211 t=3.0 s t=2.4 s t=3.8 s Figure 5.28: Interface evolution at t=2.4, 3.0 and 3.8 s

212 CHAPTER 5. MOULD FILLING Te total CPU time for tis run as been 611339 seconds. Te assembly of te Navier Stokes matrix requires 412355 s and te solver 191228 s. Te CPU time is four times bigger tan in te monolitic case, in part due to te use of a smaller time step. 5.5.3 Sovel Te run wit ASGS stabilization, C 2 =0.4 andδt =0.005 s presented in te monolitic case was repeated wit te velocity correction sceme. Te evolution of te interface is very close to te one obtained wit te monolitic model and is terefore not repeated. Te total CPU time for tis run as been 1627990 seconds. Te assembly of te Navier Stokes matrix requires 1053970 s and te solver 532796 s. An in te ollow mecanical piece example te total time increases approximately 50% due to te increase in te cost of te matrix assembly. Te solver cost decreases approximately 20%. 5.6 Conclusions In tis Capter we ave explored te suitability of metods developed in previous capters for te simulation of mould filling problems borrowed directly from te foundry. Low Froude number examples ave been cosen because we ave found tat tey are typically more difficult to solve tan ig Froude number examples. Tey require an accurate treatment in te region close to te interface. In tis tesis, two metods ave been proposed to improve te simulation of Low Froude number interface flows: an enriced pressure two pase flows model (Capter 2) and free surface model (Capter 3). Te results presented in tis Capter sow tat bot of tem provide significant improvements over te usual two pase flows model. In te case of te enriced pressure two pase flow model te improvement comes from te enanced approximation of te discontinuous pressure gradient in te elements cut by te interface. In te free surface model te problem wit te discontinuous pressure gradient disappears because only one fluid is simulated. Te advantage of te free surface metod we propose relies on te accurate imposition of boundary conditions at te interface tanks to te

5.6. CONCLUSIONS 213 use of enanced integration. Te comparison between bot models sows tat te free surface model provides significantly better results tan te enriced pressure two pase flows model. For runs wit te same time step and Smagorinsky constant te free surface model provides two or tree times lower CPU costs. Te main advantage comes from te fact tat only te domain filled by fluid is being solved. A typical mould filling simulation tat starts wit an empty piece and ends wen te piece is full. Terefore, in average, te cost of matrix assembly is alf of te one required by te enriced two pase model tat must solve bot te fluid and te air. Te cost of te solver is also reduced tanks to te solution of a smaller domain. Moreover it as been observed tat wit te free surface model bigger time steps and smaller Smagorinsky constants can be used. Te combination of te reduced computational time observed for identical runs wit te possibility of using bigger time steps leads to a CPU time reduction of nearly an order of magnitude wen te free surface model is used. Additionally we ave observed a muc better mass conservation wen te free surface model is used. Te results obtained wit te free surface model sow little dependence on wic stabilization tecnique is used. Instead wen te enriced pressure two pase flows model, not only mass conservation is poorer, but it also depends on wic stabilization metod is used. Wit ASGS mass is lost and wit OSS it is gained. Te reason for tis difference sould be explored furter. Te strong dependence of te enriced pressure two pase flows model on te stabilization tecnique used is a negative aspect of te metod. Bot te enriced pressure two pase flows model and free surface model ave also been implemented wit a velocity correction pressure segregation sceme wit te objective of obtaining a more efficient sceme. Te use of te velocity correction sceme wit te enriced pressure two pase flows model as provided results tat are very similar to te ones obtained wit te monolitic model. Te computational times are iger wen te velocity correction metod is used but furter work in order to reduce te time of te matrix assembly sould lead to similar efficiency for bot metods. However since we ave observed tat te enriced pressure two pase flows model is not competitive wit te

214 CHAPTER 5. MOULD FILLING free surface model for te moment tis does not seem to be a priority. Te improvement of te efficiency of te velocity correction metod used wit te free surface model would be more important. Unfortunately tis combination is not working satisfactorily for te moment. Some ideas to improve te beavior of te velocity correction sceme wit te free surface model ave been proposed. We belive tat te correct combination of pressure segregation metods wit te free surface model seems very promising in te medium or long term. In te sort term, te use of pressure segregation metods does not seem crucial unless furter examples were te relative cost of te solver increases are found. We believe te use of a pressure segregation metod as two key benefits. Te first one is tat it leads to better conditioned systems at te cost of uncoupling te velocity and te pressure. Tis pays off wen te cost of solving te linear system becomes dominant. For te examples we ave run we ave observed tat cost of solution of te linear is equal of smaller tan te cost of te matrix assembly even in te biggest problems we ave run. For bigger problems, te relative weigt of solver cost is expected to increase since te cost of te matrix assembly grows linearly wit te size of te problem but te cost of te solver grows at a rate greater tan one. Te solution of bigger problems would need te program to be parallelized because for te moment we ave reaced te size of problems tat can be solved wit a serial code. Terefore, as long as te cost of te solver does not become dominant, te interest in te velocity correction sceme is related to te parallelization of te code. Te second benefit of using pressure segregation metods is also associated wit te parallel implementation. Complex preconditioners are ard to parallelize and it is terefore advantageous to ave a metod tat does not need preconditioning. Te comparison wit a commercial code sows tat bot metods we propose provide better results in te ollow mecanical piece. For te oter examples wile te metods we propose ave provided satisfactory results te commercial code as not. Terefore we can conclude tat our metods are more robust tan te commercial code. Moreover we ave sown tat using te free surface model we obtain smaller CPU times tan te

5.6. CONCLUSIONS 215 commercial code. Wen te enriced pressure two pase flows model is used bot te fluid and te air are simulated as an incompressible flow. Terefore air bubbles can be formed. Specially in moulds tat do not allow air to escape troug teir walls, te formation of entrapped air regions can lead to defects and sould be avoided. In tis sense te two pase flows model may te considered better tan te free surface model tat does not take air into account. In order to overcome tis problem te free surface model can be modified to take air into account as is done for example in [77] or [16]. Te Navier Stokes equations are solved only in te fluid and in te air bubbles it is assumed tat te timescales associated wit te speed of sound in te bubble are muc faster tan te timescales of te surrounding fluid. Te pressure in te bubble is terefore spatially constant [77]. In [16] a model tat treats air as an ideal gas and can deal wit te spliting and merging of bubbles as been developed. Numerical examples are presented were te increase in te CPU time introduced by taking into account te air is less tan ten percent. We believe tat te possibility of taking into account air in our free surface model could be an interesting enancement. Moreover, we belive tat treating air as an ideal gas migt be a better approximation tan modelling it as an incompressible flow. In its interaction wit a muc denser fluid air can undergo significant pressure canges tat sould result in a volume cange someting te incompressible flow model can not take into account. For te problems we are dealing wit, te use of relatively coarse meses is currently unavoidable. Te pysics is terefore not fully solved and convergence problems may occur. For suc problems, te use of artificially ig Smagorinsky constants is accepted to make te metod more robust.

216 CHAPTER 5. MOULD FILLING

Capter 6 Conclusions Since eac of te Capters in tis tesis, except te first one, ave teir own section devoted to conclusions, in tis Capter we sall simply try to review our most important contributions and discuss future lines of researc. 6.1 Acievements Te objective of tis tesis is te improvement of two pase flows finite element modeling on fixed meses and its application to mould filling problems. We believe tat our first acievement was te identification of Low Froude number flows on fixed meses as an area tat deserved furter researc. Te problem was brougt to us by Professor Buscaglia [82]. He pointed out te impossibility of simulating two different density fluids at rest under gravity forces wen te mes is not aligned wit te interface. We ten found tat te problem extended to all low Froude number flows and tat in mould filling simulations suc flows were typically te most demanding cases. Te correct representation of te pressure gradient in te elements cut by te front is needed for low Froude number flows. An enriced pressure two pase model is presented in Capter 2. Te impossibility of fixed mes metods of correctly representing te discontinuous pressure gradient in elements cut by te interface is identified as te key problem for te correct simulation of low Froude number flows. Te solution we propose is 217

218 CHAPTER 6. CONCLUSIONS to enric te pressure sape functions in te elements cut te interface. Te enricment is local to eac element and can terefore be condensed prior to assembly making te implementation quite simple on any 2D or 3D finite element code. Te advantage compared to XFEM metods, tat also enric te sape function in te elements cut by te front, is tat no additional degrees of freedom need to be added to te system matrix. Once XFEM is well developed for 3D problems it can be an interesting alternative to our metod. In order to take advantage of te enricment enanced integration rules tat subdivide cut elements according to te position of te interface ave been used. Te enriced pressure two pase model as been introduced in [37] and furter examples ave been presented in [36]. As an alternative to te previous model a free surface model on fixed meses as been developed. By free surface we understand tat only one fluid is simulated and te influence of te second fluid on te first one is neglected. Tere are a wide number of flows were tis ypotesis is valid. Te simulation of suc flows is simpler tan two pase flows and terefore we believe it is advantageous to use tis model wen possible. As only one fluid is simulated te problem wit te discontinuous pressure gradient disappears. Actually te possibility of discontinuous velocity gradient also disappears and surface tension could easily be introduced at te free surface. Te key ingredient of our free surface metod is te use of enanced integration tat allows us to impose Neumann boundary conditions at te interface accurately. Te free surface model as been introduced in [38] and furter examples ave been presented in [39]. Simulating a free surface flow on a fixed mes introduces te particularity tat despite te mes is fixed te domain tat is being simulated is moving. We ave extended te FM-ALE approac proposed in [59] to correctly take into account tis effect. Moreover in [29] we ave generalized te FM-ALE concept to oter fields suc as fluid structure interaction to correctly take in account te movement of te domain wen fixed meses are used. We ave explored pressure segregation metods wit te objective of improving our computational efficiency and facilitating te possibility of a parallel implementation in te

6.1. ACHIEVEMENTS 219 future. Bot pressure correction and velocity correction metods ave been implemented in our code. Te numerical comparison on one pase flows sows tat te velocity correction sceme provides some advantages over te pressure correction sceme. Te most notorious advantage is te possibility of obtaining a numerically stable tird order fractional step sceme. In [6], were te velocity correction metod tat we use as been introduced, obtaining a tird order stable sceme ad not been possible. Te key difference is tat we ave used bot continuous and discrete Laplacian approximations wereas in [6] only a continuous Laplacian ad been used. For te velocity correction fractional step sceme we observe tat only te discrete Laplacian allows to obtain a pure velocity correction metod in te sense tat it is completely independent of te pressure extrapolation. Wen a continuous Laplacian is used, a second order pressure extrapolation is needed to obtain a tird order sceme. Tis as been identified as te source of te instability and it disappears wen a discrete Laplacian is used. Te fact tat te velocity correction sceme works better on one pase flows and te observation tat te velocity can be better extrapolated tan te pressure for interface flows as motivated us to use velocity correction scemes for suc flows. For te moment te results are not as satisfactory as we would ave desired. For te interface problems we are dealing wit te monolitic solver as turned out to be more efficient. Actually, initially we ad expected te monolitic system to be arder to solve tan wat we ave found. Wit te enriced pressure two pase model te velocity correction as provided satisfactory results but someow slower tan te monolitic version. Te combination wit te free surface model still requires furter work. Bot te enriced pressure two pase model and te free surface model ave been successfully applied to complex mould filling problems. Te advantages tey introduce in low Froude number flows ave also been verified in mould filling problems by comparing wit a commercial code. Te free surface model as proven to be a more efficient option. Not only does it provide lower ( 50%) CPU time tan te enriced pressure two pase model, but it also allows to use bigger time steps (and lower Smagorinsky constants) leading to efficiency advantages of nearly an order of magnitude. Despite our code is only

220 CHAPTER 6. CONCLUSIONS an academic version we ave sown it can provide bot better results and computational times tan te commercial code. For te solution of real mould filling problems, were te use of wall laws is mandatory, we found problems wit non Diriclet curved boundaries under gravity forces. We developed a solution for tis problem tat uses do noting boundary conditions. Actually we ten found out tat a very similar strategy ad recently been proposed in [9]. Te advantage of our metod is tat it introduces no modification on planar boundaries. Bot ASGS and OSS stabilization ave been used for two pase flow problems. In te OSS case we ave found tat te straigt forward application of wat is done in one pase flows leads to very poor results. An enancement tat takes into account te density variation in te projection of te residual as been introduced. 6.2 Open lines of researc As we ave already anticipated in te acievements Section, oter metods could also be used to improve te representation of te pressure in cut elements. Te XFEM metod seems to be te most popular option, but alternatives suc as te one proposed in [53] may also be extended to two pase flows. Moreover we believe te ideas developed in [27] could be combined wit te formulation proposed in [53] to obtain an improved metod. Despite we ave concentrated in improving te representation of te discontinuous pressure gradient because we ave found tat its misrepresentation as te greatest effect on te modeling of low Froude number flows, oter discontinuities exist at te interface. Surface tension introduces a discontinuity in te pressure tat could be represented using a discontinuous pressure enricment. Even in te case witout surface tension te pressure can be discontinuous due to te discontinuity in te viscous terms. We ave preliminarily explored te use of a discontinuous velocity gradient enricment but te advantages it introduces are muc smaller tan tose introduced by te pressure enricment. Te combination of te velocity correction sceme wit te enriced pressure two pase model as provided satisfactory results but te efficiency must still be improved so

6.2. OPEN LINES OF RESEARCH 221 tat it can be competitive against te monolitic model. Te combination wit te free surface model as been muc less successful. Since te free surface model as provided better results tan te enriced pressure two pase model in te monolitic case, te combination wit te velocity correction can be considered a priority, specially if bigger problems tan te ones presented in tis tesis must be solved. Our free surface model does not take air into account. As we ave mentioned in te conclusions of Capter 5 modifications suc as tose presented in [77] or [16] can be introduced to calculate te pressure in air bubbles. Tis pressure is ten applied as a normal traction at te free surface to take into account te effect of air. In flows were te air is compressed in its interaction wit te fluid tis model would provide a more accurate representation tan a two pase flow incompressible model. Te implementation of tis modification in our code could extend te range of problems we can solve. Te solution of interface flows on fixed as two basic steps. Tis tesis as focused on problems relating te solution of te Navier Stokes equations. Now tat sufficient progress as been made in tis area it seems logical tat improvements sould also be introduced in te transport of te Level Set equation. Te key point is te reinitialization of te Level Set function so tat it remains smoot (close to a signed distance function). During te reinitialization process te interface displacement must be minimized, tat is, mass must be locally conserved. A fres approac tat could easily be extended to unstructured meses as been presented in [55]. Two alternatives for te solution of te Navier Stokes equations ave been used in tis tesis: te straigtforward solution of te monolitic system and pressure segregation metods. Usually CFD groups stick to one or te oter approac. Closing te gap between te two alternatives is an interesting line of researc.

222 CHAPTER 6. CONCLUSIONS

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