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

Transcription

1 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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3 A mi familia,

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5 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)

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7 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

8 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.

9 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 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.

11 Contents 1 Introduction and basic model Introduction Classification of metods for flows wit interfaces Organization Notation Issues Two fluid Navier Stokes equations Te (one fluid) Navier Stokes equations Te two fluid Navier Stokes equations Basic discretized problem Stabilized problem Matrix version of te problem Material properties approximation Mixed boundary conditions on curved walls Te Level Set equation Interface Capturing Tecniques Implementation of te level set metod Reinitialization Coupling between te flow equations and te Level Set An enriced pressure two-pase flow model Discontinuous Gradient Pressure Sape Functions

12 12 CONTENTS 2.2 Numerical Examples Two fluid cavity D vertical cannel Slosing problem Conclusions A free surface model ALE description of te Navier Stokes equations FM-ALE free surface model Eulerian simplified free surface model Numerical examples Two fluid cavity D vertical cannel Slosing problem Two computationally demanding examples D dam-break wave interacting wit a circular cylinder D Green water problem Conclusions Pressure Segregation Metods Pressure correction metods Fractional Step (non Predictor Corrector) scemes Predictor Corrector sceme Te Pressure Scur Complement approac Velocity correction metods Te Discrete Pressure Poison Equation Approximation of DM 1 G Fractional step sceme Predictor corrector sceme Stabilized Sceme

13 CONTENTS Remarks on te ASGS and non split OSS stabilized cases Open boundary conditions Numerical examples Driven Cavity Flow beind a cylinder Convergence test Results wit te rotational form Conclusions Mould Filling Introduction Free surface monolitic model Hollow mecanical piece Alloy weel Sovel Results wit te FM-ALE model Free surface velocity correction model Enriced pressure two pase flow monolitic model Hollow mecanical piece Weel Sovel Enriced pressure two pase flow velocity correction model Hollow mecanical piece Weel Sovel Conclusions Conclusions Acievements Open lines of researc

14 14 CONTENTS

15 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 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 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

17 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 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],

19 1.1. INTRODUCTION 19 among oter references) and oter metal forming processes 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 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

21 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 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 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,

23 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 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 Two fluid Navier Stokes equations 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 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 },

25 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 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 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)

27 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 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 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

29 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 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.

31 1.2. TWO FLUID NAVIER STOKES EQUATIONS 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 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

33 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 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

35 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 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

37 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