4 FINITE ELEMENT METHODS FOR FLUIDS FINITE ELEMENT METHODS FOR FLUIDS. O. Pironneau. (Université Pierre et Marie Curie & INRIA)

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1 4 FINITE ELEMENT METHODS FOR FLUIDS FINITE ELEMENT METHODS FOR FLUIDS. O. Pironneau (Université Pierre et Marie Curie & INRIA) (To appear in 1988 (Wiley)) MacDraw, MacWrite, Macintos are trade marks of Apple Computer Co. TEXis a trade mark of te American Mat. Society. TEXtures is trade mark of Blue Sky Researc Co. EasyTeX and MacFEM are copywrite softwares of Numerica Co. (23 Bd de Brandebourg BP 215, Ivry/Seine, France)

2 5 TABLE OF CONTENTS Foreword, Notation...7,9 Introduction 1. Partial differential equations of fluid mecanics. 1.1 Orientation Te general equations of Newtonian fluids: conservation of mass, momentum, energy and te state equation Inviscid flows Incompressible and weakly compressible flows Irrotational flows Te Stokes problem Coice of equations Conclusion Irrotational and weakly irrotational flows 2.1 Orientation Incompressible potential flows: introduction,variational formulation and discretisation, solution of te linear systems wit te conjugate gradient metod, nozzle computation, lift of airfoils Compressible subsonic potential flows: variational formulation, discretisation, solution by te conjugate gradient metod Transonic potential flows: generalities, numerical considerations Potential vectors Rotational corrections : Convection - diffusion penomena 3.1 Introduction: boundary layers, instability of centered scemes Stationary convections: introduction, scemes based on te metod of caracteristics, upwind scemes, te metod of streamline diffusion, Upwinding by discontinuities, upwinding by cells Convection diffusion 1: introduction, teoretical results on te convection - diffusion equation, time approximation Convection diffusion 2: discretisation of te total derivative, te Lax - Wendroff/Taylor-Galerkin sceme, streamline upwinding (SUPG) : Te Stokes problem 4.1 Statement of te problem Functional setting Discretisation : te P1bubble/P1 element, te P1isoP2/P1 element, te P2/P1 element, te non conforming P1 / P0 element, te non conforming P2 bubble / P1 element, comparison... 13

3 6 FINITE ELEMENT METHODS FOR FLUIDS 4.4 Numerical solution of te linear systems : generalities, solution of te saddle point problem by conjugate gradient Error estimates : abstract setting, general teorems, verification of te inf-sup condition Oter boundary conditions: slip boundary conditions, boundary conditions on te pressure and on te rotational. 5. Te Navier-Stokes equations 5.1 Introduction Existence, uniqueness, regularity: te variational problem, existence, uniqueness, regularity, beavior at infinity, Te case ν = Discretisation in space: generalities, error estimates Discretisation in time: semi-explicit discretisation, implicit and semiimplicit discretisations, solution of te nonlinear system Discretisation of te total derivative Simulation of turbulent flows Euler, compressible Navier-Stokes and te sallow water equations. 6.1 Introduction Te compressible Euler equations: statement of te problem, 2D tests, existence, a few centered scemes, some upwind scemes Te compressible Navier-Stokes equations: introduction, an example, an implicit sceme for weakly compressible flows, Euler-based scemes Te Sallow water equations: introduction, a velocity-dept sceme, A flux-dept sceme, comparison Conclusion References :...13 Appendix : A finite element program for fluids on te Macintos Index...13

4 ... Te Unseen grew visible to student eyes, Explained was te immense Inconscient s sceme, Audacious lines were traced upon te void; Te Infinite was reduced to square and cube. Sri Aurobindo Savitri. Book 2,Canto XI 7 FOREWORD. Tis book is written from te notes of a course given by te autor at te Université Pierre et Marie Curie (Paris 6) in 1985, 86 and 87 at te Master level. Tis course addresses students aving a good knowledge of basic numerical analysis, a general idea about variational tecniques and finite element metods for partial differential equations and if possible a little knowledge of fluid mecanics; its purpose is to prepare tem to do researc in numerical analysis applied to problems in fluid mecanics. Suc researc starts, very often wit practical training in a laboratory; tis book is terefore ciefly oriented towards te production of programs; in oter words its aim is to give te reader te basic knowledge about te formulation, te metods to analyze and to resolve te problems of fluid mecanics wit a view to simulating tem numerically on computer; at te same time te most important error estimates available are given. Unfortunately, te field of Computational Fluid Dynamics as become a vast area and eac capter of tis book alone could be made a separate Masters course: so it became necessary to restrict te discussion to te tecniques wic are used in te laboratories known to te autor, i.e. INRIA, Dassault Industries, LNH/EDF-Catou, ONERA...Moreover, te selected metods are a reflection of is career (and is antiquated notions?). Briefly, in a word, tis book is not a review of all existing metods. In spite of tis incompleteness te autor wises to tank is close collaborators wose work as been presented in tis book and opes tat it would not

5 8 FINITE ELEMENT METHODS FOR FLUIDS urt tem to ave teir results appear in so partial a work: MM C. Bernardi, J.A. Désidéri, F. Eldabagi, S. Gallic, V. Girault, R. Glowinski, F. Hect, C. Parès, J. Périaux, P.A. Raviart, P. Rostand, and J.H. Saiac. Te autor wises to tanks is colleagues too, wose figures are reproduced ere: Mrs C. Begue and M.O. Bristeau, MM B. Cardot, J.M. Dupuy, F. Eldabagi, J. Hasbani, F. Hect, B. Mantel, C. Parès, P. Rostand, J.H. Saiac,V. Scoen and B. Stoufflet as well as te following institutions AMD-BA, PSA, SNCF and STCAN wit special tanks to AMD-BA for some of te color pictures. Te autor also wises to tank warmly MM P.G. Ciarlet and J.L. Lions for teir guidance for te publication of tis work, G. Arumugam and R. Knowles for teir assistance in te translation into Englis and finally Mrs C. Demars for typing te script on MacWrite TM. Tis book as been typeset wit TEX; te translation from MacWrite to TeXture TM as been made wit EasyTeX TM. New Deli Sept 4, 1987

6 9 NOTATION : References: A number in brackets, ex (1), refers to te equation (1) of te current capter. Te notation (1.1) refers to te equation (1) of te capter 1. Te numbers in square brackets like [2] indicate a reference in te bibliograpy. Vectors, matrices, scalar products... Unless stated oterwise te repeated indices are summed. Tus, u.v = u i v i denotes a scalar product of u wit v. For all vectors u, v and for all tensors of order 2 A, B we denote : (u v) ij = u i v j ; tra = A ii ; A : B = tr(ab T ), A =(traa T ) 1 2 Differential operators : Te partial derivatives wit respect to t or x j are denoted respectively v, t = v t v i,j = v i ; v,j = v x j x j Te classical grad, div, curl and laplacian operators are denoted respectively p,.v, v, v

7 10 FINITE ELEMENT METHODS FOR FLUIDS Te operator (nabla) applies also to vectors and tensors... for example.a = A ij,i ( v) ij = v j,i ( and tus u. v = u i v, i ) Little functions: O(x): Any function obeying O(x) c; o(x): Any function obeying lim x 0 o(x) / x =0. Geometry : Te domain of a PDE is in general denoted by Ω, wic is a bounded open set in R n (n=2 or 3); its volume element by dx and te boundary by Ω or often Γ. Te domain is, in general sufficiently regular (locally on te one side of te boundary wic is Lipscitz continuous) for us to define a normal n(x) for almost all x of te boundary Γ, but we admit domains wit corners. Te tangents are denoted by τ in 2D and τ 1,τ 2 in 3D; te curvilinear abscissa is noted by s and te element lengt (or area) is denoted by dγ. Ω stands for te area or volume of Ω. diam(ω) is its diameter. Function Spaces P k = space of polynomials of degree less tan or equal to k. C 0 (Ω) = space of continuous functions on Ω. L 2 (Ω) = space of square integrable functions. We note te scalar product of L 2 by (, )andtenormby 0 or 0,Ω : (a, b) = a(x).b(x); 0 =(a, a) 1 2 Ω H 1 (Ω) = Sobolev space of order 1.We note its norm by 1 or 1,Ω. H 1 0 (Ω) = spaces of H 1 (Ω) functions wit zero trace on Γ. H k (Ω) n = Sobolev space of order k wit functions aving range in R n. Finite Elements: A triangulation of Ω is a covering by disjoint triangles (tetraedra in 3D) suc tat te vertices of Ω, te boundary of te union of elements, are on te boundary of Ω. Te singular points of Ω must be vertices of Ω. A triangulation is regular if no angle tends to 0 or π wen te element size tends to 0. A triangulation is uniform if all triangles are equal. An interpolate of a function ϕ in a polynomial space is te polynomial function ϕ wic is equal to ϕ on some points of te domain (called nodes ). A finite element is conforming if te space of approximation is included in H 1 (in practice continuous functions). Te most frequently used finite element in tis book is te P 1 conforming element on triangles (tetraedra). An element is said to be lagrangian ( oters may be Hermite..) if it uses only values of functions at nodes and no derivatives.

8 INTRODUCTION 11 INTRODUCTION. Computational fluid dynamics (CFD) is in a fair way to becoming an important engineering tool like wind tunnels. For Dassault industries, 1986 was te year wen te numerical budget overtook te budget for experimentation in wind tunnels. In oter domains, like nuclear security and aerospace, experiments are difficult if not impossible to make. Besides, te liquid state being one of te four fundamental states of matter, it is evident tat te practical applications are uncountable and range from micro-biology to te formation of stars. At te time of writing, CFD is te privilege of a few but it is not ard to foresee te days wen multipurpose fluid mecanics software will be available to non-numerical engineers on workstations. Simulations of fluid flows began in te early 60 s wit potential flows, first incompressible or compressible ypersonic, ten compressible transonic (cf. Ritcmeyer - Morton [1]). Te calculations were done using finite difference or panel metods (cf Brebbia[2]). Aeronautic and nuclear industries ave been te principal users of numerical calculations. Te 70 s saw te first implementation of te finite element metod for te potential equation and te Navier-Stokes equations (Cung[],Temam[3], Tomasset[4]); also during tat time te development of finite difference metods for complex problems like compressible Navier-Stokes equations continued (see Peyret-Taylor [5] for example). Recent years ave seen te development of faster algoritms for te treatment of 3D flows (multigrid, (Brandt[6], Hackbus[7]), domain decomposition (Glowinski [7]), vectorization (Woodward et al[8]),te development of specialized metods to reac certain objectives (spectral metods, (Orszag [9]), particle metods (Corin [10]) metods of lattice gas (Frisc et al [11]) for example ) and te

9 12 FINITE ELEMENT METHODS FOR FLUIDS treatment of problems wicaremoreandmorecomplexliketecompressible Navier-Stokes equations wit interaction of sock-boundary layers, Knutsen boundary layers (rarefied gases, see Brun[ ]), free surfaces...and yet in spite of te apparent success te day to day problems like flow in a pipe, in a glass of water wit ice, in a river, around a car still remain unsolvable wit today s computers. Simulation of turbulence, Rayley-Benard instabilities... lie still furter in te future. Suffice it to say tat te subject will ave to be studied for a good number of years yet to come! Wy restrict ourselves only to finite element metods? many reasons; te first because one can not explain everyting in 200 pages, te next because te autor as more experience wit tis metod tan wit oter metods but most importantly because if one needs to know only one metod ten tis is te best one to know ; in fact, tis is te only tecnique (in 1987) wic can be applied to all equations witout restriction on te domain occupied by te fluid. We ave also cosen to study low degree finite elements because practical flows are often singular (sock, turbulence, boundary layer) and to do better wit iger order elements one as to do a lot more tan just doubling te degree of approximation (Babuska[], Patera[]) Te capters of tis book more or less reflect te istorical discovery of te metods. But since tis book addresses itself to specialists in numerical analysis, capter 1 recalls te main equations of fluid mecanics and teir derivations. Capter 2 treats of irrotational flow, i.e. te velocity of te flow is given by te relation u = ϕ (1) were ϕ, te flow potential, is a real valued function. Incompressible potential flows provide an occasion for us to recall te iterative metods used to solve Neumann problems by te finite element metod: φ ϕ =0inΩ; = 0 on Γ (2) n Te transonic potential equation is also treated in Capter 1 :.[(1 ϕ 2 ) 1 γ 1 ϕ] =0inΩ; φ = 0 on Γ (3) n andsolvedbyanoptimizationmetod. Tis capter ends wit te description of a potential correction metod for weakly irrotational flows. So it deals wit a metod of solution for Euler s stationary system: γ p.ρu =0,.[ρu u]+ p = 0,.[ρu( γ 1 ρ u2 )] = 0 in Ω, (4)

10 INTRODUCTION 13 were u.n is given on all of Γ and ρ,u and p are given on parts of Γ according to te signs of u.n ± (γ p ρ )1/2. Capter 3 deals wit convection - diffusion penomena, i.e. equations of te type : ϕ +.ϕu k ϕ = f in Ω, (5) t wit ϕ given on te boundary and at an initial time. Te metods of upwinding and artificial viscosity are introduced. In capter 4 we give some finite element metods to solve te generalized Stokes problem : αu u + p = f,.u =0inΩ, (6) wit u given on te boundary. Tis problem wic is in itself useful for low Reynolds number flows is also utilized as an intermediate step for te resolution of Navier-Stokes equations. Te capter 5 sows ow one solves te Navier-Stokes equations using te metods given in capters 3 and 4. Some models of turbulence as well as te metods for solving tem are given. Finally, in capter 6 we give a few metods to solve te compressible Euler equations ((7)-(9) wit µ = ξ = κ = 0) and te Navier-Stokes equations : ρu t ρ +.ρu =0 (7) t +.(ρu u)+ p η u ( η + ξ) (.u) =0 3 (8) t [ρu2 2 + p γ 1 ]+.{ρu[u2 2 + γ p ]} γ 1 ρ (9) =.{ κ R p ρ +[η( u + ut )+(ξ 2 η)i.u]u} + f.u 3 wit u,p given on te boundary, ρ given on part of te boundary and at an initial time. In tis same capter, we also consider te case of Saint-Venant s sallow water equations because tey form an important bidimensional approximation to te incompressible Navier-Stokes equations and tey are of te same type as te compressible Navier-Stokes equations. Te book ends wit an appendix describing a computer program, written by te autor to illustrate te course and used to solve some of te problems studied. Unfortunately, tis software makes extensive use of te Toolbox of teapplemacintos TM and it is not portable to oter macines.

11 14 FINITE ELEMENT METHODS FOR FLUIDS A FEW BOOKS FOR FURTHER READING [1] A. Baker:Finite element computational fluid mecanics. McGraw Hill, [2] J.P. Benque, A. Haugel, P.L. Viollet:Engineering applications of ydraulics II. Pitman, [3] G. Cavent, G. Jaffrey: Matematical metods and finite elements for reservoir simulations. Nort-Holland, [4] T. Cung: Finite element analysis in fluid mecanics. McGraw Hill, 1978 [5] C. Cuvelier, A. Seigal, A. van Steenoven: Finite element metods and Navier-Stokes equations. Matematics and its applications series, D. Reidel Publising co, [6] G. Datt, G. Touzot: Une présentation de la métode des éléments finis. Editions Maloine, [7] V. Girault, P.A. Raviart:Finite element metod for Navier-Stokes equations. Springer Series SCM, 5, [8] R. Glowinski: Numerical metods for nonlinear variational problems. Springer Series in comp. pysics [9] R. Gruber, J. Rappaz: Finite element metods in linear magnetoydrodynamics. Springer series in comp. pysics, [10] C. Hirsc: Numerical computation of internal and external flows. Wiley, [11] T.J.R. Huges: Te finite element metod. Prentice Hall, [12] C. Jonson:Numerical solution of PDE by te finite element metod. Cambridge university press, [13]R.Peyret,T.Taylor: Computational metods for fluid flows. Springer series in computational pysics [14] F. Tomasset: Implementation of te finite element metod for te Navier- Stokes equations. Springer Series in comp. pysics

12 INTRODUCTION 15 CHAPTER 1 : SOME EQUATIONS OF FLUID MECHANICS 1. ORIENTATION. Te aim of tis capter is to recall te basic equations of fluid mecanics as well as te approximation principles wic we propose for teir numerical simulations. For more details, see for example Anderson [4], Bacelor [10], Landau-Lifcitz [141], Panton [187]. 2. GENERAL EQUATIONS OF NEWTONIAN FLUIDS : 2.1 Equation of conservation of mass. Let ρ(x, t) be te density of a fluid at a point x at time t ; let u(x, t) be its velocity. Let Ω be te domain occupied by te fluid and O a regular subdomain of Ω. To conserve te mass, te rate of cange of mass of te fluid in O, ( O ρ)/ t, as to be equal to te mass flux across te boundary O of O, O ρu.n, ( n denotes te exterior normal at O and te surface element) Using Stokes formula Figure 1.1 : A volume element of fluid O. O.v = O v.n (1) and te fact tat O is arbitrary, we get immediately te equation of conservation of mass ρ, t +.(ρu) =0 (2) 2.2 Equation of conservation of momentum Now let us write Newton s equations for a volume element O of te fluid.

13 16 FINITE ELEMENT METHODS FOR FLUIDS A particle of te fluid at x at an instant t will be at x + u(x, t)k + o(k) at instant t + k ; its acceleration is terefore 1 lim k 0 k [u(x + u(x, t)k, t + k) u(x, t)] = u, t +u j u,j (3) Te external forces on O are : - Te external forces on a unit volume (electromagnetism, Coriolis, gravity...): O f - Te pressure forces: Opn -Te viscous constraints due to deformation of te fluid (σ is a tensor):. σ n. Newton s laws are, terefore, written ρ(u,t + u u) = O O or, by using Stokes formula ρ(u,t + u u) = wic gives : O O f (pn σ n), O O (f p +.σ ), ρ(u,t + u u)+ p.σ = f. To proceed furter we need a ypotesis to relate te stress tensor σ wit te velocity of te fluid. By definition of te word fluid, any force, even infinitesimal, applied to O sould produce a displacement, tus σ being te effect due to te viscosity of te fluid, it must depend on te derivatives of te velocity and tend to zero wit tem. Te ypotesis of Newtonian flow is a linear law relating σ to u,(i being te unit tensor) : σ = η( u + u T )+(ξ 2η )I.u (4) 3 η and ξ are te first and second viscosities of te fluid. From (1), (3) and (4) we derive te equation of conservation of momentum ρ(u,t + u u)+ p.[η( u + u T )+(ξ 2η )I.u] =f. (5) 3 It is easy to verify tat te equations can also be written as

14 INTRODUCTION 17 ρ(u, t +u u) η u ( η + ξ) (.u)+ p = f (6) 3 because. u = u and. u T = (.u). Taking into account te continuity equation (2), we could also rewrite (6) in a conservative form : ρu t +.(ρu u)+ p η u (η + ξ) (.u) =f. (7) 3 Remark 1 : One can sow (see Ciarlet [55], for example) tat one cannot put some arbitrary relation between σ and u or else te equations are no longer invariant under canges of reference frame (translation and rotation), tat is to say tat tey depend on te system of coordinates in wic tey are written. Tus, at least in 2D, (4) is relatively general and in fact te only relation possible wen one assumes tat σ depends on u for an isotropic fluid. Let us give examples of fluids wic do not obey te above law : - Pasty fluids (near to a state of solidification) - micro-fibers in fluids (some biomecanics fluids wit long molecules), - Mixture of fluid-particles (blood, for example), - Rarefied gases. 2.3 Equation of conservation of energy and te state equation. Lastly, an equation called conservation of energy can be obtained by writing te total energy of a volume element O(t) moving wit te fluid. We note tat te energy E is te sum of te work done by te forces and te amount of eat received. Te energy of a volume element O is te sum of potential energy ρe and te kinetic energy ρu 2 /2, O ρ( u2 /2+e). Tework done by te forces is : O u.f + O uσ.n O pn.u. If tere is no source of eat (combustion...) ten te amount of eat received (lost) is proportional to te gradient flux of te temperature θ : κ θ.n. We terefore get te equation : O d [ρ(e + u2 dt O(t) 2 )] = = u.f O O {[ρ(e + u2 2 )],t +.[uρ( u2 2 + e)]} O Wit (1) and Stokes formula, we obtain : [u(σ pi)+κ θ)n u2 [ρ(e + )] +.(u[ρ(u2 t e)+p]) =.(uσ + κ θ)+f.u (8)

15 18 FINITE ELEMENT METHODS FOR FLUIDS For an ideal fluid, C v and C p being te pysical constants, we ave e = C v θ (9) and te equation of state p = Rθ (10) ρ were R is an ideal gas constant. Wit γ = C p /C v = R/C v +1wecanwrite (9) as follows : Wit (4), (8) becomes : e = p ρ(γ 1). (11) t [ρu2 2 + p γ 1 ]+.{u[ρu2 2 + γ p]} + f.u (12) γ 1 =.{κ θ +[η( u + u T )+(ξ 2 3 η)i.u]u} Tis equation can be rewritten as a function of temperature θ,t + u θ +(γ 1)θ.u κ θ (13) ρc v = 1 [.u 2 (ξ 2η/3) + u + u T 2 η ρc v 2 ]. By introducing entropy : s = R γ 1 log p ρ γ (14) -(12) can be rewritten in te form (Cf. Landau-Lifcitz [141 p. 236]): ρθ( s t + u s) =η 2 u + ut 2 +(ξ 2 3 η).u 2 + κ θ (15) Remark 2 : Some values of te pysical constants ρ(g/cm 3 ) η(g/cm.s) ξ κ(cm 2 /s) γ R(cm 2 /s 2. o C) air = water = Te system (2), (5), (10), (12), is called compressible Navier-Stokes equations. Wit suitable boundary and initial conditions, it is complete in te sense

16 INTRODUCTION 19 tat we ave 6 unknown scalars (ρ, u, p, θ) and 6 scalar equations ; Matsumura -Nisida[168] and Valli [231] ave sown tat te system is well posed wit regular initial conditions (ρ 0,u 0,p 0,θ 0 ) satisfying (10) and regular boundary conditions : - u, θ given on te boundary, p calculated from (10) - ρ given on te part of te boundary in wic u.n < 0. A number of numerical simulations ave been made (mostly using finite differences) but tey are extremely onerous and limited to quasi non-stationary laminar flows. For tese reasons and also wit a view towards getting partial analytical solutions, te fluid mecanicists ave proposed some approximations to te complete system ; it is, in general, tese approximated systems tat we solve numerically. 3. INVISCID FLOWS. If we neglect te loss of eat by termal diffusion (κ = 0) and te viscous effects (η = ξ = 0) te simplified equations are known as te compressible Euler equations. So equation (15) becomes : s + u s = 0 (16) t ence s is constant on te lines tangent at eac point to u (stream lines). In fact a stream line is a solution of te equation : and so x (τ) =u(x(τ),τ) (17) d s x i s(x(t),t)= dt x i t + s t = s, t +u s =0. (18) If s is constant and equal to s 0 constant at time 0 and if s is also equal to s 0 on te part of Γ were u.n < 0, ten from (17) we see tat (18) as an analytical solution s = s 0. Note owever tat (18) as no meaning if u as a sock. In any case, experiments sow tat s decreases troug socks. Finally tere remains a system of two equations wit two unknowns : were ρ,t +.(ρu) = 0 (19) ρ(u, t +u u)+ p = f (20) p = Cρ γ γ 1 s0 (C = e R ). (21)

17 20 FINITE ELEMENT METHODS FOR FLUIDS Te problems (2)(5)(12) and (19)-(21) will be studied in capter 6 but an algoritm for finding te stationary solutions of (19)-(21) will also be given in Capter INCOMPRESSIBLE OR WEAKLY COMPRESSIBLE FLOWS. In te case wen ρ is practically constant (water for example or air wit low velocity) we can neglect its derivatives. Ten (2), (6) become te incompressible Navier Stokes equations :.u = 0 (22) u,t + u u + p ν u = f (23) were ν = η/ρ is te reduced viscosity of te fluid and p/ρ and f/ρ ave been replaced by p and f. Te same approximation can be applied to equations (19)-(20) and we obtain (22)-(23) wit ν = 0, known as te incompressible Euler equations..u = 0 (24) u,t + u u + p = f (25) Te Navier-Stokes and Euler s equations will be studied in capter 5. Remark 3. Wit (22)(23) or (24)(25) one can no longer use (21) because te system becomes over determined. Tis ambiguity can be explained by studying (19)(21)wen te velocity of sound (dp/dρ) 1/2 tends to infinity. One can sow (Klainerman -Majda [165]) tat if C tends to infinity in (21) ten p goes to infinity but p remains bounded. Tus te pressure variation continues to ave a pysical interpretation in (22)(23) or (24)(25) but p no longer represents te pysical pressure. An equation for te temperature θ can be obtained from (13) by assuming ρ constant; if f =0weave: θ t + u θ κ ρc v θ = ν 2C v u + u T 2 (26) An intermediate approximation, called weakly compressible, in between te complete system (2), (5), (12) (compressible Navier-Stokes ) and incompressible Navier-Stokes (22)-(23), is interesting because it exibits te yperbolic caracter of te underlying acoustics in (2), (5), (12) and (19)(21). Assume tat ξ = η = κ = 0, wic implies s constant and so pρ γ constant; under te assumption tat ρ oscillates around a value ρ 0 one gets :

18 INTRODUCTION 21 ρ, t +u ρ + ρ 0.u =0 u, t +u u + Cρ 0γ 1 ρ =0. (27) If also u and ρ ρ 0 are small, ten (27) is close to wic gives ρ, t +ρ 0.u =0; u, t +Cρ 0γ 1 ρ = 0 (28) ρ, tt Cρ 0γ ρ =0. (29) Finally, if one is interested in termal convection problems in te fluid (wic appens as water is eated in a pan, for example) one could assume tat ρ is quasi constant in (7) and could neglect all variations of ρ except of f/ρ in (5) were f is gravity. Ten one obtains te Rayleig-Benard equations :.u =0 u, t +u u + p ν u = gθe 3 (30) θ, t +u θ κ θ = ν 2C v u + u T 2 were e 3 is te unit vector in te vertical direction. 5. IRROTATIONAL FLOWS : We could try to determine if, for suitable boundary conditions, tere exist solutions of equations satisfying u = 0; (31) tese solutions are called irrotational. But as (22) implies te existence of ϕ (x,t) suc tat tese solutions are also called potential. Using te identities : u = ϕ (32) u = u + (.u) (33) u u = u ( u)+ ( u2 2 ) (34) we say tat (24)-(25) ave suc solutions for f =0ifϕ is a solution of te Laplace equation : ϕ =0, (35)

19 22 FINITE ELEMENT METHODS FOR FLUIDS wit (32) and p = k 1 2 ϕ 2. (36) Tis type of flow is te simplest of all. It will be studied in capter 2. In te same way, wit (32), (27) implies ρ, t + ϕ ρ + ρ 0 ϕ = 0 (37) (ϕ, t ϕ 2 + γcρ 0γ 1 ρ) = 0 (38) If we neglect te convection term ϕ ρ tis system simplifies to a nonlinear wave equation : ϕ, tt c ϕ ϕ 2, t = d(t) (39) werec=γcρ 0γ is related to te velocity of te sound in te fluid. Finally, we sow tat tere exist stationary potential solutions of (19), (20), (21) wit f = 0. Using (34), (19) could be rewritten as follows : ρu u + ρ u2 2 Taking te scalar product wit u, weobtain u.[ρ u2 2 Also, on taking into account (21) + p =0. (40) + p] = 0 (41) Or u.(ρ u2 2 + ργ 1 Cγ ρ) = 0 (42) uρ. ( u2 2 + C γ γ 1 ργ 1 ) = 0 (43) So, te quantity between te parentesis is constant along te stream lines, tat is we ave ρ = ρ 0 (k u2 2 ) 1 γ 1 (44) Indeed a system like u ξ = 0 is integrated exactly like (16) and gives ξ constant if it is constant on te part of te boundary were u.n < 0. Tus if ρ 0 and k are constant at te entrance boundary of Ω (u.n 0), if all te streamlines intersect te entrance boundary of Ω and if u is non zero, ten (44) olds, ten (40) implies tat u is parallel to u and, at least in 2 dimensions, tis implies

20 INTRODUCTION 23 tat u derives from a potential (i.e. (31)). From (2) and (44) we deduce te transonic potential flow equation. Tis problem will be studied in capter 2..[(k ϕ 2 ) 1 γ 1 ϕ] = 0 (45) Remark 4: Equation (35) is consistent wit (45) since we can return to it by assuming ρ constant (Cf (44)). 6. THE STOKES PROBLEM. Let us come back to te system (19)-(21) and let us rewrite it in non dimensional form. Let U be a caracteristic velocity of te flow under study (for example one of te non omogeneous boundary conditions ). Let L be te caracteristic lengt (for example te diameter of Ω) and T a caracteristic time (wic is a priori equal to L/U).Let us put u = u U ; x = x L ; t = t T (46) Ten (15) and (16) can be rewritten as x.u = 0 (47) ( L TU )u, t +u x u + U 2 x p ( ν LU ) x u = f L U 2 (48) So, if we put T = L/U,p =p/u 2 ν = ν (49) LU ten (38)-(39) is te same as (15)-(16) wit prime variables.te inverse of ν is called Reynolds number. Let us give some examples : (unitsmks) U L ν Re = UL ν micro organism glider sailboat(sell) 0, car airplane tanker

21 24 FINITE ELEMENT METHODS FOR FLUIDS Wen ν >> 1, ν u dominates u u and u t Stokes problem : in (48); it becomes te ν u + p = f (50).u = 0 (51) Experience sows tat (50)-(51) is indeed an excellent approximation of (15)- (16) for low Reynolds number flows in dimension tree ; in dimension two te Stokes problem migt not ave solution in an unbounded domain. We recall in tis context tat a two dimensional flow is not an infinitely tin flow, but a flow invariant under translation in a spatial direction and wose velocity is perpendicular to te invariant direction.. Te Stokes problem will be studied in capter CHOICE OF EQUATIONS. Given a fluid system to simulate numerically, wat equations need to be cosen? Unfortunately, tere is no automatic rule ; one seeks, in general te simplest system compatible wit te penomenon. Let us take te case of an aircraft wing. Te first ting tat an engineer wises to know is evidently te resultant of te fluid forces on te wing ; tat is te lift and te drag. Te lift being somewat unrelated to te viscosity we can integrate te transonic equation to calculate it (cf Capter 2). But if tere is a lot of vorticity ten we ave to take te Euler equation, compressible if te velocity if large, incompressible if not. On te oter and, tere is no cance to calculate te drag wit Euler s equations; we ave to integrate te Navier-Stokes equations (in te boundary layer or in te wole domain). Te problem becomes more complicated if te flow is detaced ; toug te lift is not a viscous penomenon, te generation of eddies near te trailing edge is fundamentally a penomenon related to te viscosity. Te oscillation of te lift around its average value cannot be approximated correctly unless te Navier-Stokes equations are solved. To conclude, te coice of a system to solve is beyond te scope of tis book and tis capter gives only some examples. A toroug discussion wit a pysicist or wit an engineer is terefore indispensable before embarking on numerical simulations wic are often long and expensive. 8 CONCLUSION: Te compressible Navier Stokes equations describe a large number of penomena wic from te matematical point of view belong to 3 categories of PDE s : Elliptic: Stokes equations, incompressible potential flow. Parabolic : Temperature propagation, convection-diffusion. Incompressible Navier-Stokes equation.

22 INTRODUCTION 25 Hyperbolic: Euler s equations, wave equations of te acoustics of a fluid, convection. We state also, tat tere does not exist a universal sceme to approximate numerically tese tree types of equations, wic explains wy eac subsystem enumerated above and its numerical integration is treated in a separate capter of tis book.

23 26 FINITE ELEMENT METHODS FOR FLUIDS CHAPTER 2 IRROTATIONAL AND WEAKLY IRROTATIONAL FLOWS 1. ORIENTATION We ave seen in capter 1 tat it is possible (wit suitable boundary conditions) to find irrotational solutions ( u = 0) to te general fluid mecanics equations under te following ypoteses : -inviscidflow - no vorticity generated by discontinuities (socks) or by te boundaries. In tis capter, we will study finite element approximations of tose equations. Te first part deals wit te Neumann problem, were we recall also te finite element metod. In te second part, we deal wit subsonic compressible flows, wic will be solved by an optimization metod. We ten extend te metod to transonic flows. Finally in te tird and fourt parts we study te resolution of te problem in terms of stream function and a rotational correction metod based on Helmoltz decomposition of vector fields. 2.INCOMPRESSIBLE POTENTIAL FLOWS 2.1 Generalities. If te density ρ is constant, te viscosity and termal diffusion are negligible and if te flow does not depend on time ten te velocity u(x) and pressure p(x) are given for all points x Ω of te fluid by.u =0 (1) u =0 (2) p = k 1 2 u2 (3) were k is a constant if te flow is uniform at infinity. On te boundary Γ = Ω in general te normal component of te velocity, u.n, isgivenby:

24 INTRODUCTION 27 u.n = g on Γ (4) We remark tat (2) implies te existence of a potential ϕ suc tat : u = ϕ (5) So te complete system is simply rewritten as a Neumann problem. ϕ =0 in Ω (6) ϕ = g n on Γ (7) Example 1. Calculation of low velocity laminar flow troug a nozzle. Figure 2.1. A divergent nozzle Example 2 Calculation of te flow around a symmetric wing (te non-symmetric i.e. lifting case will be studied later). Figure 2.2 A symmetric airfoil Remark We could ave used (1) to say tat tere exists ψ suc tat Ten, te complete system becomes u = ψ (9) ψ =0 in Ω (10) ψ.n = g on Γ (11) If te flow is bidimensional (invariant by translation in one direction) ten (10)-(11) become : ψ = 0 (12) s ψ(x(s)) = g(x(γ)dγ x(s) Γ (13) We study tis metod at te end of te capter. Note tat in 3 dimensions, te operator in (10)-(11) is not strongly elliptic and tat ψ is a vector toug ϕ is a scalar Variational formulation and discretisation of (6)-(7)

25 28 FINITE ELEMENT METHODS FOR FLUIDS if Proposition 1 Let χ L 2 (Ω)wit non zero average on Ω. If g is regular, (g H 1/2 (Γ))and Γ g = 0 (14) ten problem (6)-(7) is equivalent to te following variational problem : ϕ w = gw w W (15) Ω ϕ W = {w H 1 (Ω); Γ Ω χw =0} (16) Proof : We see, in multiplying (6) by w, and integrating on Ω wit te elp of Green s formula tat : If we use (6) and (7) in (17) we get ϕ w = Ω ϕ ( ϕ)w + Γ n w = ϕ w. (17) Ω Ω Γ gw w H 1 (Ω) (18) Putting w = 1 we get (14). As (18) doesn t cange if we replace ϕ by ϕ + constant and w by w + constant, we can restrict ourselves to W. Te converse can be proved in te same way ; from (15) wit te assumption tat ϕ is regular (in H 2 (Ω)) so as to use (17), we get : ( ϕ)w + ( ϕ g)w =0 w W (19) Ω Γ n By taking w to be zero on te boundary, we deduce tat : ( ϕ)w =0 w suc tat χw =0, Ω and so, from te teory of Lagrange multipliers, tere exists a constant λ suc tat Note also tat (19) implies Ω ϕ = λχ in Ω. (20) ϕ = g on Γ (21) n Finally, (17) wit w = 1 (20), (21) and (14) imply tat te constant in (20) is zero.

26 INTRODUCTION 29 Remark W given by (16) is equivalent to H 1 (Ω)/R. One could ave set up te problem in H 1 (Ω)/R at te start but tis presentation allows a natural construction for te discretisation of W. Besides, it is interesting to note tat if (14) is not satisfied, ϕ is te solution of (20) instead of (6). Proposition 2 If Ω is a bounded domain wit Lipscitz boundary and if g is in H 1/2 (Γ) ten (15) as a unique solution. Proof : In W and wit bounded Ω, Poincaré s inequality is satisfied: ϕ 2 C ϕ 2. (22) Ω Ω So te bilinear form associated wit (15) is W elliptic : ϕ min{c 1, 1} ϕ 2 1 (23) Ω As W is a closed subspace of H 1 (Ω) te linear map w gwdγ Γ is continuous. Te result follows from Lax-Milgram teorem (cf Ciarlet [55] Strang-Fix [223 ], Lions-Magenes [155]). Proposition 3 Let {W } be a sequence of internal approximations (W W ) of W, suc tat w W {w }, w W suc tat w w 1 0 wen 0 (24) Ten te solution ϕ W of ϕ w = Ω Γ gw w W (25) converges strongly in H 1 (Ω) to ϕ, te solution of (15)-(16). Proof : Using (15) wit w and subtracting from (24), we obtain (ϕ ϕ) w =0 w W (26) Ω wic sows tat ϕ is a solution of te problem

27 30 FINITE ELEMENT METHODS FOR FLUIDS min { (w ϕ) 2 } (27) w W Ω and so if ψ is an approximation of ϕ in te sense of (24) we ave (ϕ ϕ) 2 (ψ ϕ) 2 (28) and ence te proof (Cf. (22)). Ω Proposition 4 If tere exist α and C(Ω) suc tat in addition to (24) we ave Ω w w 1 C α w α+1 (29) ten ϕ ϕ 1 C α ϕ α+1 (30) Remark If Ω is convex, we ave also ϕ ϕ 0 C α+1 ϕ α+1. (31) Proof : Te error estimate (30) can easily be deduced from( 28) (29) and (22). To prove (31) one as to use te duality argument of Aubin-Nitsce (see Ciarlet [56]) Lagrangian triangular finite elements (also called P 1 conforming): Ω is divided into triangles (tetraedra in 3D) {T k } 1...K suc tat - T k T l =, or 1 vertex, or 1 wole side (resp. side or face) wen k l - Te vertices of te boundary of T k are on Γ - Te singular points of Γ (corners) are on te boundary Γ of T k. We note tat Ω = T k, Γ = T k, {q i } N 1 are te vertices of te triangles, and is te longest side of a triangle : = max {i,j,k:q i,q j T k } qi q j (32) H = {w C 0 (Ω ):w T P α } (33) W = {w H : χw =0} Ω (34) were P α denotes te space of polynomials in n variables of degree less tan or equal to α (Ω R n )andc 0 te space of continuous functions.

28 INTRODUCTION 31 If Ω is a polygon ten W W and proposition 3 applies. If moreover, we assume tat all te angles of te triangles are bounded by θ 1 > 0andθ 2 <π wen 0 ten te proposition 4 can be applied. Wit (33) W is of finite dimension, say M-1. Let {w i } 1...M 1 be a basis of W ;ifwewriteϕ in tis basis, we ave ϕ (x) = M 1 1 ϕ i w i (x) (35) and if we replace w by w j in (25) we get a (M 1) (M 1) linear system. AΦ =G (36) wit Φ = {ϕ 1..ϕ M 1 }, A ij = w i w j, G j = gw j (37) Ω Γ Example 1 Wit α = 1 a basis of W can easily be constructed. Let {λ k i (x)} i=1...n+1 denote te barycentric coordinates of x in T k wit respect to its n + 1 vertices ten te number of basis functions is N 1 (i.e. M = N) tat is one less tan te total number of vertices {q} i=1...n. Let w i be te canonical basis function of Lagrangian elements of order 1: w i (x) =λ k i (x) ifk is suc tat x T k, (q i T k ) (38) =0 oterwise ten w i (x) =w i (x) Ω χwi (x) Ω χ i =1..N 1 (39) is a basis of W. Remark : As w i differs from w i only by a constant, tanks to (14), one can use eiter w i or w i in (37). It makes no difference. Te effect of tis construction is simply to pull out a function w i to construct a basis. Tis procedure as te inconvenience of distinguising a vertex from oters (since we ave pulled te w i associated to it) and sometimes introduces a local numerical error around te pulled vertex because te linear system is poorly conditioned tere. We will see below tat te conjugate gradient metod avoids tis inconvenience. Figure 2.3 : Example of numerical singularity obtained by pulling a basis function ; we remark in te top middle a distortion of isovalue lines.

29 32 FINITE ELEMENT METHODS FOR FLUIDS Example 2 Wit α = 2 te number of basis functions is equal to te number of vertices q i plus te midpoints of te sides (edges) minus one. We can also construct it by (39) but wit λ i k replaced by λi k (2λi k 1) for te basis functions associated to vertices and by 4λ i k λj k for te basis functions associated to te midpoints of te sides (edges) Resolution of te linear system by te conjugate gradient metod. Te linear system (36) obtained by te finite element metod as a peculiar structure wic we must exploit to optimize te resources of te computer. Te conjugate gradient metod (cf. Polak [193], Lascaux-Teodor [142], Luenberger [162] for example) makes use of te sparse structure of te linear system to speed te solution wen properly preconditioned. Tis algoritm applies also to positive semi-definite linear systems; tus we can construct te system (36), (37) wit all te basis functions w i. Te solution obtained is more regular (cf. figures 3, 4). Figure 2.4: Result obtained by te conjugate gradient metod witout removing te basis function. Let us recall te conjugate gradient metod briefly. Notation: Let x R N be an unknown suc tat Ax = b,werea R N N,A T = A, b R N. Let C be a positive definite matrix and <, > C te scalar product associated wit C: <a,b> C = a T Cb; a C =(a T Ca) 1 2 (40) Algoritm 1 0. Initialization: Coose C R N N positive definite (preconditioning matrix ), ɛ small positive, x 0 suc tat Cx 0 ImA (0 for example), and set g 0 = 0 = C 1 (Ax 0 b), n=0. 1 Calculate : ρ n = <gn, n > C nt A n (41) x n+1 = x n + ρ n n (42)

30 INTRODUCTION 33 g n+1 = g n ρ n C 1 A n (43) γ n = gn+1 2 C g n 2 C (44) n+1 = g n+1 + γ n n (45) 2 If g n+1 C <ɛ,stopelseincrementn by 1 and go to 1. Note tat C is never inverted and tat y = C 1 z is a sort and notation for solve Cy = z Remark 1 Tis algoritm can be viewed as a particular case of a more general algoritm used to find te minimum of a function (see below). Here it is applied to te computation of te minimum of E(x) =x T Ax/2 b T x. One can verify tat i) ρ n given by (1) is also te minimum of E(x n + ρ n ) ii) g n+1 given by (43) is also te gradient of E wit respect to te scalar product associated wit C, tat is: g n+1 = C 1 (Ax n+1 b) (46) Proof : x n+1 = x n + ρ n n <=> Ax n+1 = Ax n + ρ n A n C 1 (Ax n+1 b) =C 1 (Ax n b)+ρ n C 1 A n Remark 2 Te only divisions in te algoritm are by nt A n and g n C. Te first one could be zero if te kernel of A is non empty wereas te second is non zero by construction; so to prove tat te algoritm is applicable even if det(a) = 0 we ave to prove tat nt A n is never zero. But before tat let us sow convergence. Lemma jt A k =0, j <k (47) <g k,g j > C =0, j <k (48) <g k, j > C =0, j <k (49) Proof : Let us proceed by te metod of induction. Assuming tat te property is true for j<k n, let us prove tat (47)(48)(49) are true for all j<k n +1.

31 34 FINITE ELEMENT METHODS FOR FLUIDS i) Multiplying (43) by j and using (47) and (49): <g n+1, j > C =<g n, j > C ρ n <C 1 A n, j > C =0 jt A n =0, (50) if j<n;ifj = n ten it is zero by (41). ii) Using (45): <g n+1,g j > C =<g n+1, j γ j j 1 > C = 0 (51) by i) above. iii)finally, again from (45) we ave, if j<n: n+1t A j =(g n+1 + γ n n ) T A j = g n+1t A j +0=g n+1t C (gj+1 g j ) ρ j =0 (52) were we ave used (47) to get te second equality and (43) for te last one. If j = n, we ave to use te definition of γ n. Let us sow tat γ n is also equal to g n+1t A n / nt A n. We ave : g n+1t A n nt A n = <gn+1 g n,g n+1 > C <g n+1 g n, n > C = gn+1 2 C <g n, n > C (53) We ave used (43) for te first equation and (48) and (49) for te second. We leave to te reader te task of sowing tat (47),(49) are true for k =1. We end te proof by sowing tat nt A n is never zero. Indeed if n is te first time it is zero ten we ave: so by (41) 0= nt A n, n ImA n =0 g n = γ n 1 n 1 = g n 1 ρ n 1 C 1 A n 1 γ n 1 n 2 =<g n 1 n 1 > ρ n 1 n 1T A n 1 =0 but by ypotesis n 1 is not zero. Corollary Te algoritm converges in N iterations at te most. Proof : Since te g n are ortogonal tere cannot be more tan N of tem non zero. So at iteration n = N, if not before, te algoritm produces g n =0tat is C 1 (Ax n b) =0.

32 INTRODUCTION 35 Coice of C However it is out of question to do N iterations because N is a very big number. One can prove te following result : Proposition 5: If x is te solution and x k is te computed solution at te k t iteration, ten <x k x,a(x k x ) > C 4( µc A 1 µ C A +1)2k <x 0 x,a(x 0 x ) > C were µ C A is te condition number of A (ratio between te largest and smallest eigenvalues of Az = λcz ) in te metric introduced by C. Proof : see Lascaux-Teodor [9], for example. One can also prove superconvergence results, tat is te sequence {x k } converges faster tan all te geometric progressions (faster tan r k for all r) but tis result assumes tat te number of iterations is large wit respect to N. We can easily verify tat if x 0 =0andC = A we get te solution in one iteration. So tis is an indication tat one sould coose C near to A. Wen we do not ave any information about A, experience sows tat te following coices are good in increasing order of complexity and performance: C ij = A ii δ ij (54) C ij = A ij, i, j i j < 2 (55) C ij =0, oterwise. C = incompletetely factorized matrix of A. (56) We recall te principle of incomplete factorization (Meijerink-VanderVorst [172], Glowinski et al [97]): We construct te Coleski factorization L of A (= L L T ) and put L ij =0 if A ij =0, L ij = L ij oterwise. One can also construct directly L instead of L by putting to 0 all te elements of C wic correspond to a zero element of A, during te factorization (cf [10]) but ten te final matrix may not be positive definite. Proposition 6 If A is positive semi definite, b is in te image of A, and Cx 0 is in te image of A, ten te conjugate gradient algoritm converges towards te unique solution x of te linear system Ax = b; wic verifies Cx Im A.

33 36 FINITE ELEMENT METHODS FOR FLUIDS Proof : From (42) and (45) we see tat Cx n,cg n,c n ImA Cx n+1,cg n+1,c n+1 ImA Te property is tus proved by induction. 2.4 Computation of nozzles If Ω is a nozzle we take, in general, φ zero on te walls of te nozzle and g = u.n, u constant at te entrance and exit of te nozzle. Te engineer is interested in te pressure on te wall and te velocity field. One sometime solves te same problem wit different boundary conditions at te entrance Γ 1 and at te exit Γ 2 of te nozzle : ϕ =0 in Ω, ϕ Γ1 =0, ϕ Γ2 = constant, Tesamemetodapplies. ϕ n Ω Γ 1 Γ 2 =0. (57) Figure 2.5 : Computation of flow in a nozzle. Te triangulation and te level lines of te potential are sown Computation of te lift of a wing profile Te flow around a wing profile S corresponds, in principle, to flow in an unbounded exterior domain, but we approximate infinity numerically by a boundary Γ at a finite distance ; so Ω is a two dimensional domain wit boundary Γ = Γ S. One often takes u constant and g Γ = u.n, g S = 0 (58) Unfortunately, te numerical results sow tat wit tese boundary conditions, te flow generally goes around te trailing edge P.AsP is a singular point of Γ, ϕ(x) tends to infinity wen x P and te viscosity effects (η and ζ) are no longer negligible in te neigborood of P (see figure 6). Te modeling of te flow by (1)-(2) is not valid and (2) as to be replaced by (ω constant) : u = ωδ Σ were δ Σ is te Dirac function on te stream line Σ wic passes troug P. Figure 2.6 Flow around an airfoil; Σ is te wake.

34 INTRODUCTION 37 One takes ten Ω Σ as te domain of computation. So we ave to add a boundary condition on Σ. Since Σ is a stream line : ϕ n Σ =0. (59) But as u is continuous along Σ we also ave : ϕ σ Σ + = ϕ σ Σ. by integrating tis equation on Σ we obtain : σ (ϕ Σ + ϕ Σ ) = 0 i.e. for some constant βϕ Σ + ϕ Σ = β (60) were β is a constant wic is to be determined wit te elp of (59) written on P, or better by : ϕ(p + ) 2 = ϕ(p ) 2 (61) wic comes out to be te same but is interpreted as a continuity condition on te pressure. It can be proved using conformal mappings tat wit (60)-(61) te solution ϕ does not depend on te position of Σ (wic is not known a priori) but wit (59) tis is not te case: te solution depends on te position of Σ. So we solve ϕ =0 in Ω Σ (62) ϕ Σ + ϕ Σ = β (63) ϕ(p + ) 2 = ϕ(p ) 2 (64) ϕ n Ω = g; (65) equation (64) wic is called te Joukowski condition. It deals wit te continuity of te velocity and also of te pressure. To solve (62)-(65) a simple metod is to note tat te solution of (62)- (63)-(65) is linear in β : ϕ(x) =ϕ 0 (x)+β(ϕ 1 (x) ϕ 0 (x)) (66) - ϕ 0 is te solution of (62),(65) and (53) wit β =0, i.e. ϕ 0 ϕ =0, n Γ = g, ϕ continuous across Σ - ψ = ϕ 1 ϕ 0 is te solution of (62),(65) wit g = 0 and (63) wit β =1, i.e.

35 38 FINITE ELEMENT METHODS FOR FLUIDS ψ =0, ψ n Γ =0, ψ Σ + ψ Σ =1, ψ Σ + = ψ Σ Te variational formulation of tis second problem is: find ϕ W p 1 ψ w =0 w W p 0 ; Ω W p β = {ψ H1 (Ω) : ψ Σ + ψ Σ = β, ψ Σ + = ψ Σ } suc tat We find β by solving (64) wit (66) : it is an equation in one variable β. We can sow tat te lift C f (te vertical component of te resultant of te force applied by te fluid on S) is proportional to β : were ρ is te density of te fluid. C f = βρ u (67) Practical Implementation : In practice we can use P 1 finite elements toug te Joukowski condition requires tat we know ϕ at te trailing edge ; wit P 1, ϕ is piecewise constant and so te triangles sould be sufficiently small near te trailing edge. In [11] a rule for refining te triangles near te trailing edge (to keep te error O()) :can be found te size of te triangles sould decrease to zero as a geometric progression as tey approac P and te rate of te geometric progression is a function of te angle of te trailing edge. Experience sows tat one can apply te condition (61) by replacing P + and P by te triangles wic are on S and ave P as a vertex. Figure 2.7 Configuration of te triangles near te trailing edge. Figure 2.8 : Example of numerical results : a) equipotentials b) triangulation around te neigborood of te profile c) streamlines wit lift d) and witout lift. Anoter metod, wic is no doubt better from te teoretical point of view, but very costly in terms of computer programming time is to include a special basis function in place of te one associated wit P,torepresentte singularity. It can be sown tat ϕ(x) is like x P π/(2π β) in te neigborood of te trailing edge, were β is te angle of te trailing edge. One can find te complete proof in Grisvard [102], and we give ere only an intuitive justification.

36 INTRODUCTION 39 Let us work in polar coordinates wit origin at P and θ = 0 corresponding tooneedgeofs and θ =2π β for te oter. Let f be te limit of ϕ(r, θ)/r m (m not necessarily an integer), i.e. Te equations for ϕ are: ϕ(r, θ) =r m f(θ)+o(r m ) ϕ n ϕ S = r 1 θ (r, 0) = 0 f (0) = 0 ϕ n ϕ S + = r 1 θ (r, 2π β) =0 f (2π β) =0 ϕ = r 1 r r ϕ r + r 2 2 ϕ θ 2 =0 m2 f(θ)+f (θ) =0 Te general solution of te last equation being f(θ) =a sin(m(θ θ 0 )) te first two conditions give : θ 0 =0, m(2π β) =π tat is m = π/(2π β). We ave terefore in te neigborood of P: π π ϕ(r, θ) =ar (2π β) sin[ (2π β) θ]+o(r π wic suggests taking a basis function associated wit P as w (r, θ) =r π (2π β) sin[ π (2π β) θ]i p(r) (2π β) ) were I p (r) is a smoot function wic is equal to unity near P and zero far from P. Results using tis tecnique can be found in Dupuy [70], for example. 3.POTENTIAL SUBSONIC STATIONARY FLOWS : 3.1 Variational formulation One still assumes tat te effects of viscosity and termal diffusion are negligible but does not assume tat te fluid is incompressible. If te flow is stationary, irrotational at te boundary and beind all socks if any, ten one can solve te transonic potential equation Te velocity is still given by.[(k 1 2 ϕ 2 ) 1 γ 1 ϕ] =0 in Ω (68)

37 40 FINITE ELEMENT METHODS FOR FLUIDS te density by u = ϕ (69) ρ = ρ 0 (k 1 2 ϕ 2 ) 1 γ 1 and te pressure by p = p 0 ( ρ ρ 0 )γ (70) Te constants p 0,ρ 0 are usually known at te outer boundary. Te boundary conditions are given on te normal flux ρu.n rater tan on te normal velocity: (k 1 2 ϕ 2 ) 1 γ 1 = g n on Γ (71) To simplify te notation, let us put ρ( ϕ) =ρ 0 (k 1 2 ϕ 2 ) 1 γ 1 (72) Proposition 7 Problem (44)-(45) is equivalent to a variational equation. Find ϕ W = {w H 1 (Ω); χw =0} (were χis any given function wit non zero mean on Ω)suc tat (k 1 2 ϕ 2 ) 1 γ 1 ϕ w = gw w W ; (73) Ω Moreover, any solution of (49) is a stationary point of te functional E(ϕ) = (k 1 Ω 2 ϕ 2 ) γ γ γ 1 g (74) γ 1 Γ Proof : To prove te equivalence between (68) and (73)-(74) we follow te procedure of proposition 1. Let us consider now Ω Γ We ave : E(ϕ + λw) = (k 1 Ω 2 (ϕ + λw) 2 ) γ γ γ 1 γ 1 Γ g(ϕ + λw) E, λ (ϕ + λw) = γ (k 1 γ 1 Ω 2 (ϕ + λw 2 ) 1 γ γ 1 (ϕ + λw) w gw γ 1 Γ

38 So all te solutions of (49) are suc tat and now te result follows. INTRODUCTION 41 E, λ (ϕ + λw) λ=0 =0 Proposition 8 If b<(2k(γ 1)/(γ +1)) 1/2 ten E defined by (70) is convex in W b = {ϕ W : ϕ b} (75) Proof : Let us use (71) to calculate E, λλ in te direction w : γ 1 d 2 E γ dλ 2 λ=0 = ρ( ϕ) w 2 1 ρ( ϕ) 2 γ ( ϕ. w) 2 Ω (γ 1) Ω = ρ( ϕ) 2 γ w 2 (k 1 2 ϕ 2 [1 + 2 γ 1 ( ϕ. w ϕ w )2 ] Ω Ω ρ( ϕ) 2 γ w 2 (k 1 2 ϕ 2 (1 + 2 γ 1 )) (k 1 2 b2 ) 2 γ b 2 γ +1 γ 1 (k 2 γ 1 ) w 2 0 Corollary 1 If b<(2k (γ 1)/(γ +1)) 1/2 te problem min (k 1 ϕ W b Ω 2 ϕ 2 ) γ γ γ 1 gw (76) γ 1 Γ admits a unique solution. Proof : W b is closed convex, E is W elliptic,convex, continuous. Corollary 2 If te solution of (76) is suc tat ϕ b at all points, ten it is also te solution of (69) Discretisation Proposition 9 Let b<(2k (γ 1)/(γ +1)) 1/2 and let

39 42 FINITE ELEMENT METHODS FOR FLUIDS W b = {ϕ W : ϕ b} (77) Assume tat W b and W b satisfy (24) wit strong convergence in W 1, ; let us approximate (75) by min (k 1 ϕ W b Ω 2 ϕ 2 ) γ γ γ 1 gϕ. (78) γ 1 Γ If {ϕ } are solutions in W 1, ten ϕ ϕ te solution of (76). Proof : As ϕ is bounded by b, one can extract a subsequence wic converges weakly in W 1, (Ω) weak *. Let ψ be its limit. Let ϕ be te solution of (76) and Π ϕ te interpolate of ϕ in te sense of (24). Let E be te functional of problem (78). Ten since W b W b we ave : E(ϕ) E(ϕ ) E(Π ϕ). Te weak semi-continuity of E and te fact tat ϕ is te solution of (78) implies tat any element w of W b is te limit of {w },w W b 1, in W (lim 0 w w 1, =0) E(ϕ) E(ψ) lim inf E(ϕ ) lim E(Π ϕ)=e(ϕ) (79) but ϕ is te minimum and so E(ϕ) =E(ψ). (80) Te strong convergence of ϕ towards ϕ in H 1 (Ω) is proved by using te convexity and te W b -ellipticity of E Resolution by conjugate gradients : To treat te constraint ϕ <b te simplest metod is penalization. We ten solve : min E (ϕ ) (81) ϕ W E (ϕ )= (k 1 Ω 2 ϕ 2 ) γ γ γ 1 gϕ + µ [(b 2 ϕ 2 ) + ] 2 γ 1 Γ Ω were µ is te penalization parameter wic sould be large. Te penalization is only to avoid te divergence of te algoritm if in an intermediate step te bound b is violated. If te solution reaces te bound b in a small zone ten Corollary 2 doesn t apply any more; (experience sows tat outside of tis zone te calculated solution is still reasonable); but in tis case it is better to use augmented Lagrangian metods (cf Glowinski [95]).

40 INTRODUCTION 43 Figure 2.9: Computed result of a transonic nozzle Te figure represents te isovalues of ϕ, obtained by solving (81). Taking into account te fact tat W is of finite dimension, (81) is an optimization problem witout constraint wit respect to coefficients of ϕ on a basis of W : ϕ (x) = N 1 1 ϕ i w i (x) min {ϕ E (ϕ ) 1..ϕ N 1} To solve (81), we use te conjugate gradient metod wit a preconditioning constructed from a Laplace operator wit a Neumann condition. Let us recall te preconditioned conjugate gradient algoritm for te minimization of a functional. Algoritm 2 (Preconditioned Conjugate Gradients) : Problem to be solved: min E(z) (82) z R N 0. Coose a preconditioning positive definite matrix C; cooseɛ>0 small, M a large integer. Coose an initial guess z 0. Put n =0. 1. Calculate te gradient of E wit respect to te scalar product defined by C; tat is te solution of If g n C <ɛstop else if n = 0 put 0 = g 0 else put 3. Calculate te minimizer ρ n solution of Cg n = z E(z n ) (83) γ = gn 2 C g n 1 2 (84) C n = g n + γ n 1 (85) Put min ρ {E(z n + ρ n )} (86) z n+1 = z n + ρ n n (87) If n<m increment n by 1 and go to 1 oterwise stop. Proposition 10

41 44 FINITE ELEMENT METHODS FOR FLUIDS If E is strictly convex and twice differentiable, algoritm 2 generates a sequence {z n } wic converges (ɛ = 0,M = + ) towards te solution of problem (82). Te proof can be found in Lascaux-Teodor [142] or Polak [193] for example. Let us recall tat convergence is superlinear (te error is squared every N iterations) and tat practical experiments sow tat N iterations are enoug or even more tan adequate wen we ave found a good preconditioner C. Wit a good preconditioner, te number of iterations is independent of N. Tis is te case wit problem (81) wen C is te matrix A constructed in (37). Te linear systems (83) could be solved by algoritm 1. As in te linear case, te algoritm works even if te constraint on te mean of ϕ is active provided tat te gradient g n is projected in tat space. Gelder s Algoritm [86]: Te fixed point algoritm, introduced by Gelder [86], wen it converges gives te result very fast :.[(k 1 2 ϕn 2 ) 1 γ 1 ϕ n+1 ]=0 in Ω (88) (k 1 2 ϕn 2 ) 1 ϕn+1 γ 1 = g on Γ (89) n It is conceptually very simple and easy to program. Once discretised by finite elements (88), (89) gives a linear system at eac iteration, but te convergence is not guaranteed; owever experience sows tat it works well wen te flow is everywere subsonic. Te coefficients depend on te iteration number n ;so one sould avoid direct metods of resolution. If one solves (88) by algoritm 1 (conjugate gradient) ten it becomes a metod wic from a practical point of view is very near to algoritm TRANSONIC POTENTIAL FLOWS : 4.1 Generalities : If te solution of (75) is suc tat ϕ = b almost everywere on I Ω, measure(i) > 0 ten we cannot solve (68) by (75). From pysics, we know tat wen te modulus of te velocity ϕ is superior to [2k(γ 1)/(γ +1)] 1/2 te flow is supersonic. If we linearize (68) in te neigborood of ϕ, we get :.[ρ δϕ 1 γ 1 ρ2 γ ϕ δϕ] = 0 (90) One can sow by te same calculation as (74) tat tis equation is locally elliptic in te subsonic zone and locally yperbolic in te supersonic zone. Furtermore numerical experience sows tat (68) as many solutions in general C 0 but not C 1 ( ϕ is discontinuous), te discontinuities are were ϕ is

42 INTRODUCTION 45 equal to te speed of sound. We ave to impose an additional condition to avoid te jumps in te velocities (socks) going from subsonic to supersonic in te flow direction. We ave seen in Capter 1 tat te entropy s is constant if η, ζ, κ are zero ; but tis is no longer valid wen te velocities are discontinuous ; wen η is small, for example, η u becomes big. It is sown in Landau-Lifcitz [141] tat te entropy created by te sock is proportional to te tird power of te jumps in te velocity across te sock in te flow direction : te entropy being increased troug socks it can only be produced by supersonic to subsonic socks. An original metod for imposing tis condition (called an entropy condition) as been proposed by Glowinski [95] and studied by Festauer et al [77], Necas [180] : Given 2 positive constants b and M, let WM b = {ϕ H 1 (Ω) : ϕ w M w w 0 (91) w H 1 (Ω), Ω ϕ b, Te variational inequality in (91) means tat Ω Ω χϕ =0} ϕ M in Ω (92) wic effectively avoids all positive jumps of dϕ/dx in te x-direction if φ is smoot in te y-direction. Results leading towards te existence of solutions for (68)(71)(92) can be found in Kon-Morawetz [135]. Te following property is a key to sow tat subsequences converge in W b M. Proposition 11 Te space W b M is compact convex in H1 (Ω). Proof (Festauer et al [77]) Te convexity of WM b is clear ; let us sow tat it is compact. Let us define G n (w) = ( ϕ n. w + Mw) (93) Ω G(w) = ( ϕ w + Mw) (94) Ω If ϕ n ϕ in H 1 (Ω) weak* ten G n GinH 1 (Ω) weak* and as G n (w) 0 w we ave, from a lemma of Murat [178], G n G strongly in W 1, (Ω) weak* But (ϕ n ϕ) 2 o =(G n G)(ϕ n ϕ) 0 (95)

43 46 FINITE ELEMENT METHODS FOR FLUIDS Te functional used for te computation of subsonic flows is no longer convex in te supersonic zones; one must find anoter functional. A possibility is to minimize te square of te norm of te equation: min.[(k 1 ϕ ϕ Γ H0 1(Ω) W M b 2 ϕ 2 ) 1 γ 1 ϕ] 2 1 (96) Te coice of a suitable norm is very important to insure existence of solution and a fast algoritm. Here H 1 is clearly a good coice but it requires ϕ Γ to be known (Diriclet boundary conditions). Wit Neumann conditions (also wit mixed conditions) one can solve: min { ɛ 2 dx : (97) ϕ W b M Ω ( ɛ, w) =(ρ( ϕ) ϕ, w) gw w H 1 (Ω} Indeed if we let ɛ be defined by Γ ɛ =.[(k 1 2 ϕ 2 ) 1 γ 1 ϕ], ɛ H 1 0 (Ω) ten (96) can be written equivalently as min { ɛ 2 }. ϕ ϕ Γ H0 1(Ω) W b M Ω A finite element discretisation of (97) is min { ɛ 2 : (98) ϕ W b M Ω ( ɛ, w )=(ρ( ϕ ) ϕ, w ) gw w W } It sould be noted tat tis problem always as a solution since it is te minimization of a positive differentiable function in a finite dimensional bounded space. But ϕ will be an approximation to te solution of te transonic problem only if ɛ 0; and te set of points on wic te constraints of W b M are active must tend to a set of measure zero. In Glowinski [95] details of te conjugate gradient solutions of tis problem can be found. We refer to Lions [154] for a toroug study of optimal control problems. Γ 4.2. Numerical considerations Practically, experience sows tat te constraint (92) is always saturated and so te calculated solution depends on M (Cf. Bristeau et al [44]) and tat te coice of M is tus critical.

44 INTRODUCTION 47 In te actual computation one may prefer to use upwinding or artificial viscosity to take care of te entropy condition. Upwinding will be studied in te following capter. Let us give a simple version due to Huges [115]. Equation (49) is approximated by Ω (k 1 2 ϕn 2 d ) 1 γ 1 ϕ n+1 w = Γ gw w W ϕ n+1 W (99) were W is, for example, a P 1 conforming triangular finite element approximation of W and were ( ϕ ) d is calculated on te triangle T j(k), nearest to T k upwind of te flow instead of calculating it on te triangle T k ( ϕ n ) d Tk = ϕ n Tj(k) Numerical results wit tis metod can be found in Huges [115] or Dupuy [70] for example. Artificial viscosity will also be studied in capter 3 ; one can add viscous terms to te equation (tey were tere in te Navier-Stokes equations and because of tem te entropy can only grow). Following Jameson [121] (see also Bristeau et al.[44]), we consider Ω (k 1 2 ϕn 2 ) 1 γ 1 ϕ n+1 w + Ω = Γ [ s [(u n2 c 2 ) + ] ϕ n+1 gw ρ n ]w dx (100) were te u n is an upwind approximation of ϕ n,cis te speed of sound and ρ n is an upwind approximation of (k- (1/2) ϕn 2 ) 1/γ 1 ; f/ s stands for (u/ u ) f, te derivative in te direction u ; tis upwind approximation can be constructed as above or by ( ϕ n ) d Tk =(1 ω) ϕ n Tk + ω ϕ n Tj(k) were ω is a relaxation parameter. Finally, notice tat (99) and (100) are Gelder-type algoritms to solve underlying nonlinear equations; oter metods can be used suc as te H 1 least-square metod presented above ((96)-(98)) or GMRES, a quasi Newton algoritm wic will be studied in capter 5 ; results for tese can be found in Bristeau et al [44].

45 48 FINITE ELEMENT METHODS FOR FLUIDS 5. VECTOR POTENTIALS : Let us return to (1), (2) and study te possibility of calculating u by u = ψ. For simplicity, we sall assume tat Ω is simply connected. Let Γ be piecewise twice differentiable and Γ i be its simply connected components. Proposition 12 : Let u L 2 (Ω) suc tat.u L 2 (Ω). Let φ H 1 (Ω)/R te solution of and ψ [H 1 (Ω)/R] 3 te solution of ( φ, w) =(u, w) w H 1 (Ω)/R (101) ten ( ψ, v)+(.ψ,.v) =(u, v) v V (102) ψ V = {v H 1 (Ω) 3 : v n Γ =0, v.n =0}, Γ i (103) u = φ + ψ (104) Proof : Te proof being rater long, we sall give only te main ideas and te details can be seen in Bernardi [30], Dominguez et al [67], Eldabagi et al [73]. First let us note tat (101) et (102) are well posed so tat φ and ψ are unique. Ten we see tat te solution of (102) satisfies.ψ = 0. In fact, by taking v = q in (102) wit q zero on Γ (because of (103)),we see : (.ψ, q) =0 tat is we almost ave q H 2 (Ω) H 1 0 (Ω) suc tat Γ q n = 0 (105) (.ψ, g) =0 g L 2 (Ω) (106) Neverteless, one can prove, witout muc difficulty, tat (105) implies.ψ = 0. Now let we see tat (101) implies and (102) implies ξ = ψ + φ u (107).ξ =0 ξ.n = n. ψ (108) ξ = 0 (109)

46 because of te following identity : (a b) =( a, b)+ INTRODUCTION 49 Γ a.(n b) (110) Besides, if τ 1,τ 2 are two vectors tangential to Γ and ortogonal to eac oter ten so tat ψ n =0 ψ.τ i =0 (ψ.τ j ),τj = 0 (111) n. ψ = (ψ.τ 1 ),τ2 +(ψ.τ 2 ),τ1 = 0; (112) te proof is completed by using te following property : ξ =0,.ξ =0, ξ.n Γ =0 ξ =0. (113) Proposition 13 : Wit te same conditions as in proposition 12, we can also use te following decomposition : Let φ be te solution in H 1 (Ω)/R of ( φ, w) = (.u, w) w H 1 (Ω)/R (114) and ψ H 1 (Ω) 3 te solution of ( ψ, v)+(.ψ,.v) =(u, v) v V (115) ψ n = q on Γ, ψ.n = 0 (116) Γ i were q is te solution of te Beltrami equation on Γ. q w = u.nw w H 1 (Γ) (117) τ i τ i i=1,2 were τ i are two ortogonal tangents to Γ. Ten Γ Γ u = ψ + ϕ Proof : Te proof is te same as tat of proposition 12, except for (111) onward. Instead of (112), we ave: n. ψ = ( 2 q τ q τ2 2 )=u.n (118) were te last equality is due to (117). Finally, we also ave ξ.n =0andso ξ =0.

47 50 FINITE ELEMENT METHODS FOR FLUIDS We ave terefore two decompositions of u. In te first one, φ/ n =u.n wereas in te second n. ψ = u.n and φ/ n =0. If.u = 0 in order tat φ 0, one as to use te second decomposition. So we see tat te potential vector resolution of incompressible 3D flow requires, in general, - te solution of a Beltrami equation on te boundary (117), - te solution of a vectorial elliptic equation (115). Remark An alternative form for (117) is q w q w n n = Γ Γ u.nw, w H 1 (Γ) 6. ROTATIONAL CORRECTIONS: We can use proposition 12 and te transonic equation (64) to construct a metod of solving te stationary Euler equations :.(ρu) = 0 (119).(ρu u)+ p = 0 (120).[ρu( 1 2 u2 + γ p )] = 0 (121) γ 1 ρ As in capter 1, we deduce from (121) tat pρ 1 γ/(γ 1) + u 2 /2 is constant on te streamlines, tat is, if x is te upstream intersection of te stream line wit Γ, we ave γ p γ 1 ρ + u2 2 = H(x )on{x : x = u(x), x(0) = x } (122) By definition of te reduced entropy S, weave p ρ γ = S (123) and S is constant on te streamlines, except across te socks. In fact, ω = usatisfies 0=.(ρu u)+ p = ρu u + p + ρ u2 2 (124) = ρu ω + ρ H ργ γ 1 S. and so ρu S = 0, but tis calculation is not valid if u is discontinuous. However we can deduce from (124) tat

48 INTRODUCTION 51 ω =(u H ργ 1 γ 1 u S) u 2 + λρu (125) were λ is adjusted in suc a way tat.ω = 0, i.e. ρu λ =.[(u 2 )(u H ρ γ 1 u S )] (126) γ 1 Let us now use proposition 12 ; we get te following algoritm : Algoritm O. Coose u continuous and calculate te streamlines 1. Calculate a new H by transporting H(x ) on te streamlines and a new S by transporting S(x ) on te streamlines and if necessary adding a jump [S] calculated from te Rankine Hugoniot conditions across te socks (conservation of [ρu], [ρu 2 + p], [u 2 /2+γp/(γ 1)ρ]). 2. Calculate ω by (125)wit ρ calculated by (122) and (123), i.e. ρ(u) = γ 1 γs (H u2 )] γ 1 (127) and λ calculated by (126). 3. Calculate ψ by resolving (102) (we sall apply (110) to calculate te rigt and side). 4.Calculateφ by solving (119), i.e. wit (127) :.[ρ( φ + ψ)] = 0 5.Putu= φ + ψ and return to 1. ρ φ n Γ = g (128) We remark tat wen ω<<1, ψ is small so all tat remains is (128) i.e. te transonic equation. Tis algoritm is terefore a rotational correction to te transonic equation. Te convergence of te algoritm is an open problem but it seems to be quite stable (see El Dabagi et al [73]) and it gives good results wen ω is not too large. From a practical point of view, te main difficulty remains in te identification of socks and applying te Rankine- Hugoniot condition in step 1 (see Hafez et al [105] or Luo [161] for an alternative metod). Finally, we see tat it requires te integration of 3 equations of te type : u ξ = f, ξ Σ given; (129) tis will be treated in te next capter. Oter decompositions ave been used to solve te Euler equations. An example is te Clebsc decomposition used by Aker et al. [ 71] and Zijl [242] u= ϕ + s ψ were ϕ, s, ψ are scalar valued functions. Extension of tese metods to te Navier-Stokes equations can be found in El Dabagi et al [73], Ecer et. al. [71], Zijl [242 ].

49 52 FINITE ELEMENT METHODS FOR FLUIDS CHAPTER 3 : CONVECTION DIFFUSION EQUATIONS 1. INTRODUCTION Te following PDE for φ is called a linear convection-diffusion equation: φ,t + aφ +.(uφ).(ν φ) =f (1) were a(x, t) is te dissipation coefficient (positive), u(x, t) is te convection velocity and ν(x, t) is te diffusion coefficient (positive). For simplicity we assume tat a = 0, as is often te case in practice. All tat follows, owever, applies to te case a>0; for one ting wen a is constant we ave: (e at φ),t = e at (φ,t + aφ) so wit suc a cange of function one can come back to te case a = 0; secondly te diffusion terms and te dissipative terms ave similar effects pysically. Tese equations arise often in fluid mecanics. Here are some examples : - Temperature equation for θ for incompressible flows, - Equations for te concentration of pollutants in fluids, - Equation of conservation of matter for ρ and te momentum equation in te Navier-Stokes equations wit φ = u altoug tese equations are coupled wit oter equations in wic one could observe oter penomena tan tat of convection-diffusion. In general, ν is small compared to UL ( caracteristic velocity caracteristic lengt ) so we must face two difficulties: - boundary layers, - instability of centered scemes. On equation (1) in te stationary regime wen ν = 0 we sall study first tree types of metods: - caracteristic metods, - streamline upwinding metods, - upwinding-by-discontinuity. Ten we will study te full equation (1) beginning wit te centered scemes and finally for te full equation, te tree above mentioned metods plus te Taylor-Galerkin / Lax-Wendroff sceme.

50 1.1. Boundary layers INTRODUCTION 53 To demonstrate te boundary layer problem, let us consider a one dimensional stationary version of (1) : were u and ν are constants. Te analytic solution is uφ,x νφ,xx =1 (2) φ(0) = φ(l) =0 (3) Wen ν/ul 0, φ(x) =( L u )[ x L e[(ul/ν)(x/l)] 1 ] (4) e (ul/ν) 1 φ (x L) u if u<0, except in x =0, φ x u if u>0, except in x = L. But te solution of (2) wit ν =0 is (x L) φ = u if we take te second boundary condition in (3) and φ = x u (5) (6) if we take te first one. We see tat wen ν/ul 0, te solution of (2) tends to te solution wit ν = 0 and one boundary condition is lost; at tat boundary, for ν<<1, tere is a boundary layer and te solution is very steep and so it is difficult to calculate wit a few discretisation points. Figure 3.1 : According to te sign of u, te solution is given by (5) or (6) wit a boundary layer to catc te violated boundary condition Instability of centered scemes : Finite element metods correspond to centered finite difference scemes wen te mes is uniform and tese scemes do not distinguis te direction of flow. Let us approximate equation (2) by P 1 conforming finite elements. Te variational formulation of (2)

51 54 FINITE ELEMENT METHODS FOR FLUIDS L is approximated by uφ,x w + ν L φ,x w,x = L w, w H 1 0 (]0,L[) (7) L 0 (uφ,x w + νφ,x w,x w )=0 w continuous P 1 piecewise (8) and zero on te boundary. If ]0,L[ is divided into intervals of lengt, itiseasytoverifytat(8)can be written as u 2 (φ j+1 φ j 1 ) ν 2 (φ j+1 2φ j + φ j 1 )=1, j =1,..., N 1 (9) were φ j = φ (j), j= 0,...,N = L/ ; φ o = φ N =0. Let us find te eigenvalues of te linear system associated wit (9), tat is, let us solve 2α λ 1 α α 2α λ 1 α... I n (λ) =det = α 2α λ 1 α α 2α λ wereweaveputα =2ν/(u). One can easily find te recurrence relation for I n : I n =(2α λ)i n 1 +(1 α 2 )I n 2 and so wen λ =2α ten I 2p+1 =0 p. Tis eigenvalue is proportional to ν, and tends to 0 wen ν 0. So one could foresee tat te system is unstable wen ν/u is very small. Tere are several solutions to tere difficulties : 1. One can try to remove te spurious modes corresponding to null or very small eigenvalues by putting constraints on te finite element space (see Stenberg [220] for example). 2. One can solve te linear systems by metods wic work even for nondefinite systems. For example Wornom- Hafez [239] ave sown tat a block relaxation metod wit a sweep in te direction of te flow allows (9) to be solved witout upwinding. 3. Finally one can modify te equation or te numerical sceme so as to obtain non-singular well posed linear systems: Tis is te purpose of upwinding and artificial diffusion. Many upwinding scemes ave been proposed (Lesaint-Raviart [149], Henric et al. [110], Fortin-Tomasset [83], Baba-Tabata [7], Huges [115], Benque

52 INTRODUCTION 55 et al. [20], Pironneau [190]... see Tomasset [229] for example). We sall study some of tem. For clarity we begin wit te stationary version of (1). 2. STATIONARY CONVECTION: 2.1. Generalities : In tis section, we sall study te problem.(uφ) =f in Ω φ Σ = φ Γ (10) were u is given in W 1, (Ω), fis given in L 1 (Ω) and Σ={x Γ:n(x).u(x) < 0} (11) Figure 3.2 :Σis te part of Γ were te fluid enters ( u.n < 0). A variational formulation for tis problem is to searc for φ L 2 (Ω) suc tat (φ, u w) = Σ wφ Γ u.n fw w H 1 (Ω) wit w Γ Σ =0. Ω Here φ Γ H 1/2 (Σ),u,.u L 2 (Ω),f L 2 (Ω) is enoug but wit more regularity tere is an explicit solution: Proposition 1 : Let X(x;s) be te solution of X = u(x), X(0) = x. (12) Let X Γ (x) =X(x; s Γ ) be te intersection of {X(x; s) :s<0} wit Σ. Ten φ(x) =φ Γ (X Γ (x))e 0 is solution of (10). Proof : s Γ.u(X(x;s))ds + 0 s Γ f(x(x; s))e u. φ = X φ = d φ(x(x; s)) = f(x(x; s)) φ.u ds 0.u(X(x;τ ))dτ s ds (13)

53 56 FINITE ELEMENT METHODS FOR FLUIDS on integration tis equation gives (13). Corollary 1 : If all te streamlines intersect Σ, ten (10) as a unique solution. Remark : Te solution of (10) evidently as te following properties : - positivity : f 0, φ Γ 0 φ 0 - conservativity :.u =0,f =0 u.nφds =0 S closed contour Ω S stability : uφ (x) f L 1 (Ω) - u.n(x Γ (x)) φ Γ (X Γ (x)) We seek, if possible, numerical scemes wic preserve te above properties. AprioritesimpleP 1 triangular conforming finite elements and (w,.(uφ )) = (f,w ) w W w Σ =0; φ W φ Σ = 0 (14) do not preserve te above properties. In fact, if Ω is a square divided into triangles wit te sides parallel to te axes and if u =(1, 0) we obtain te centered finite difference sceme, wic we know is unstable, wen φ is irregular: 2(φ i+1,j φ i 1,j )+φ i+1,j+1 φ i,j+1 + φ i,j 1 φ i 1,j 1 =6f i,j were φ i,j is te value of φ at vertex ij. Te solution of (14) cannot be unique wen.u = 0 since it is a skew symmetric linear system. Figure 3.3:A uniform triangulation and te corresponding numbering Sceme 1 (caracteristics) : One could propose te following sceme : 1) u is approximated by u,p 1 conforming, from te values of u at te vertices q i. 2) Te caracteristics {X(q i ;.)} originating from te nodes {q i } are approximated by polygonal lines k [ξ k ξ k+1 ],ξ 0 = q i, were ξ k+1 is te intersection of te line {ξ k µu (ξ k )} µ>0 wit te boundary T l of te triangle T l wic contains ξ k and ξ k ɛu (ξ k ),ɛpositive, small (see figure 3.4). 3) (13) is evaluated at x = q i and tis defines φ(q i ) 4) φ (x) is defined from φ(q i )byap 1 interpolation. Proposition 2 : φ φ C (15) were C depends on φ Γ 1,, f 1,, u 1,, f and on diam(ω).

54 INTRODUCTION 57 Proof : To simplify, let us assume tat.u = 0; te reader can build a similar proof in te general case. Let us evaluate te error along te caracteristics; let X (x;.) be te solution of approximated as in step 2 above ; ten X = u (X),X(0) = x, X X = u (X ) u(x) (u u)(x ) + u(x ) u(x) u u + u X X Using te Belman-Gronwall lemma : X X u u (e u L 1) (16) were L is te lengt of te longest caracteristic. Finally, sγ sγ φ (q i ) φ(q i ) φ Γ (X Γ (q i )) φ Γ (X Γ (q i )) + f(x )ds f(x)ds 0 0 φ Γ (X Γ X Γ )(q i ) + L f X X + s Γ s Γ f (17) As s Γ s Γ is te difference in lengt between te discrete and te continuous caracteristics, so long as te boundary is regular enoug wit te elp of (16) we obtain te result: φ (q i ) φ(q i ) C u u Numerical considerations : Tis sceme is accurate but rater costly because one as to calculate all te caracteristics starting from all te vertices. For example for a square domain wit N 2 vertices, one must cross N 2 (N 1) triangles in total wen te )ifn s denotes te total number of vertices. One could reduce te cost by first calculating tem for te q i wic are far from Σ in te direction u ten using a linear interpolation for all te nodes for wic tere exists two opposite triangles already crossed by previously computed caracteristics. velocity is orizontal, so te cost of tis computation is 0(N 3/2 s Remark (Saiac) If u = ψ and ψ is known ten X Γ can be computed from te equation ψ(x Γ (q i )) = ψ(q i ),X Γ (q i ) Γ; so tere is no need to cross te domain starting from q i.

55 58 FINITE ELEMENT METHODS FOR FLUIDS Figure 3.4:Te caracteristics are broken lines joining vertices ξ i. If two caracteristics go around a vertex q, ten te caracteristic wic starts from q may be skipped. To calculate te {ξ k } k a good algoritm is to use te barycentric coordinates {λ k+1 j } j=1..m+1 of ξ k and decompose u (ξ k )inµ k j suc tat µ k j qj = u (ξ k ) (18) ten ξ k+1 = j=1..n λk+1 j q j (n = 2 or 3) wit j µ k j = 0 (19) j λ k+1 j = ρµ k j + λk j were ρ is suc tat λ k+1 j 0, l suc tat λ k+1 l =0. Tis calculation is explained in detail in (120) Sceme 2 (upwinding by discretisation of te total derivative) A weak formulation of (10) is written as: Find φ in L 2 (Ω) suc tat (φ, u w)+ φ Γ wu.n =(f,w) w H 1 (Ω), suc tat w Γ Σ =0. Σ Let T ± δ (Ω) be a translation of Ω by δ in te direction ±u T ± δ (Ω) = {x ± u(x)δ : x Ω} To construct te sceme, we consider te following two approximations : u w(x) = 1 [w(x + u(x)δ) w(x)] δ g = δ gu.n + o(δ) Ω Ω T δ (Ω) Γ Σ Wit tese approximations (20) can be approximated in te space W of P 1 functions continuous on te triangulation Ω by te following problem :

56 INTRODUCTION 59 (φ,w ) φ (x)w (x + u(x)δ) =δ(f,w ) δ φ Γ w u.n, Ω T δ (Ω) Σ w W ; w Γ Σ =0 or, wit te convention tat w (x) =0ifx Ω: Find φ W suc tat for all w W wit w Γ Σ =0: (φ,w ) φ (x)w (x + u(x)δ)dx = δ(f,w ) δ φ Γ w u.n, Ω Σ One may prefer scemes were te boundary conditions are satisfied in te strong sense. For tis one sould start wit te following variational formulation of (10): Find φ suc tat φ Σ = φ Γ and (φ, u w)+ φwu.n =(f,w) w H 1 (Ω), w Σ =0. (20) Γ Σ Ten by te same argument one finds te following discrete problem: Find φ wit φ φ Γ W and (φ,w ) φ (x)w (x + u(x)δ)dx = δ(f,w ), Ω T δ (Ω) w W (21) wit W = {w C 0 (Ω) : w Tk P 1 k, w Σ =0}. Comments : 1. We could ave performed te same construction by using te formula :.(uφ) =u φ + φ.u = 1 (φ(x) φ(x u(x)δ)) + φ(x u(x)δ).u(x) δ and could ave obtained, instead of (21), (φ,w )= Ω φ (x u(x)δ)w (x)(1 δ.u(x))dx + δ(f,w ), w W. (22) Tis is also a good sceme, but as will become clearer te sceme (21) is conservative wereas (22) is not.

57 60 FINITE ELEMENT METHODS FOR FLUIDS 2. If we use mass lumping ( a special quadrature formula wit te Gauss points at te vertices), tat is, {q i } i being te vertices of T : T f = T (n +1) 1..n+1 f(q i ) (23) were n =2(resp.3), T te area (resp. volume) of a triangle (resp. tetraedron) (tis formula is exact if f is affine) ten te linear system from (21) and (22) becomes quasi-tridiagonal, in particular (22) becomes : φ (q i )=φ (q i u(q i )δ)(1 δ.u(q i )) + f(q i )δ (24) On a uniform grid wen u is constant, tis is Lax s sceme [144] ; it is a positive but of order 1; owever, te finite element sceme (21) is somewat better (te dissipation comes mostly from te mass lumping in te second integral) but it is more complicated because it requires te solution of a non-symmetric linear system and calculation of te integral of a product of two P 1 functions on different grids (an analysis of tese metods can be found in Bermudez et al [28]). We also see ow one can construct scemes of iger order by using a better approximation of u w. Proposition 3 : If u is constant, and if all te streamlines cut Σ, problem (21) as a unique solution Proof : If tere were 2 solutions, teir difference ɛ wouldbezeroonσandwould satisfy: ɛ w ɛ w o(i + uδ) = 0 (25) Ω Ω T δ (Ω) Taking w = ɛ, we would obtain : but ɛ 2 0,Ω ɛ 0,Ω ɛ o(i + uδ) 0,Ω T δ (Ω) (26) ɛ o(i + uδ) 2 0,Ω T δ (Ω) = Ω T +δ (Ω) ɛ (y) 2 det (x + uδ) 1 dy (27) = ɛ 2 0,Ω T + δ (Ω) So ɛ 0,Ω Ω T + (Ω) = 0 (28) δ

58 INTRODUCTION 61 wic means tat ɛ is zero on all te vertices of te triangles wic ave a non-empty intersection wit Ω Ω T + δ (Ω). Applying similar reasoning to Ω T + δ (Ω) and proceeding step by step, we can sow tat ɛ is zero everywere. Remark on te convergence : At least for te case u constant, using te same tecnique,we would get an error estimate of te type φ φ 0,Ω Ω T + δ (Ω) C(δ2 + 2 ) and so te final error is 0(δ+ 2 /δ). Te diffusivity of te scemes is a problem; Bristeau-Dervieux [40] ave studied a 2 nd order interpolation of u w wic gives better results, i.e. less dissipations, (but wic is less stable). Generalization of te previous scemes : Te first sceme corresponds to an integration along te wole lengt of te caracteristics wereas te previous sceme corresponds to a repeated integration on a lengt δ. If one wants an intermediate metod, one could proceed as follows (Benque et al. [20]) : Let D be a subset of Ω and ψ w (x) te solution of were u ψ w =0inD, ψ w D = w As for (13), it is easy to see tat D = {x D : u(x).n(x) < 0} ψ w (x) =w(x D (x)) were X is te solution of (12) and X D te intersection wit D. From (10) we deduce fψ w =.(uφ)ψ w = wφu.n + ψ w φu.n D D D + D were D + is te part of D were u.n 0. By coosing an approximation space W, ametodtocalculatex D and a quadrature formula, one could define a family of scemes of te type : + D φ u.nw ox D + w φ u.n = f(x)w(x D (x))dx D D and tis family contains te above two scemes. w W 2.4. Sceme 3 (Streamline upwinding SUPG) :

59 62 FINITE ELEMENT METHODS FOR FLUIDS Te streamline upwinding sceme (also called SUPG=Streamline Upwinding Petrov Galerkin) was proposed by Huges [116] and studied by Jonson et al. [125]. On te space W of P 1 functions continuous on te triangulation of Ω and zero on Σ, for example, one searces φ suc tat φ φ Γ W wit (.(uφ ) f,w +.(uw )) = 0 w W (29) Comments : Te idea is te following : Te finite element metod leads to centered finite difference scemes because te usual basis functions w i (te at function ) are symmetric (on a uniform triangulation) wit respect to vertices q i. So replacing te Galerkin formulation (.(uφ ),w i )=(f,w i ) i, φ = φ Γ + φ i w i (30) by a nonsymmetric Petrov-Galerkin formulation (cf. Cristie et al. [54]) (.(uφ ),w i )=(f,w i ) (31) were te w i ave more weigt upstream tan downstream ; tat is te case of te function x w i +.(uw i ) (see figure 3.5). We note also tat one could work eiter wit w i or w i in te second member. Figure 3.5 : Te P 1 basis functions and te Petrov-Galerkin functions of Huges.[ ] Anoter interpretation could also be given ; Equation (29) is also an approximation of tat is, wen u is constant.(uφ) u (.(uφ)) = f u f (32).(uφ).[(u u) φ] =f u f (33) So we ave in fact added a tensorial viscosity u u; now tis tensor as its principal axis in te direction u, tat is we ave added viscosity only in te direction of te flow. Furtermore wat is added on te left is also added on te rigt because (10) differentiated gives u (.[uφ]) = u f. Proposition 4 : i) Te sceme (29) as a unique solution ii) If u W 1, te solution satisfies te following inequality : φ φ 0,Ω c 3 2 φ 2,Ω (34)

60 INTRODUCTION 63 Proof wen.u α>0: If (29) as two solutions, teir difference ɛ satisfies.(uɛ ) Ω(.u)ɛ u.nɛ 2 dγ = 0 (35) 2 Γ Σ because (.(uɛ ),ɛ )= 1 2 ((.u)ɛ,ɛ )+ 1 u.nɛ 2 (36) 2 Γ So ɛ is zero. Let us find an error estimate : Subtracting te exact equation from te approximated equation, we obtain: (.(u(φ φ )),w +.(uw )) = 0 w W (37) By taking w = ψ φ were ψ is te interpolation function of φ, we get : (.(uw ),w +.(uw )) = (.(u(ψ φ)),w +.(uw )) (38) =((.u)(ψ φ),w ) (.(uw ),ψ φ.(u(ψ φ))) + u.nw (ψ φ) Γ Σ or using (36) and te error estimate of interpolation ( φ ψ α c 2 α,α=0, 1 and φ ψ 0,Γ c 3/2 ): 1 2 Ω(.u)w u.nw (uw ) 2 0 Γ Σ C 2.(uw ) u.nw 0,Γ Σ + C 2 (.u)w 0 ; from wic we deduce tat : u w 0,Ω <C, (u.n) 1/2 w 0,Γ Σ C 3/2 and (.u) 1/2 w 0 C 3/2, wicinturngives from wic te result follows since w 0,Ω = φ ψ 0,Ω C 3 2 (39) φ φ C ψ φ C 3 2 (40) Remark: If.u is not positive te proof uses te following argument: Let ψ = e k(x) ϕ(x) werek(x) is any smoot function. Ten wit v = e k(x) u, ψ is a solution of

61 64 FINITE ELEMENT METHODS FOR FLUIDS.(vψ) =f, but.v = e k (.u + u k), so one can coose k so as to ave.v > Sceme 4 (Upwinding by discontinuity) : If φ is approximated by polynomial functions wic are discontinuous on te sides (faces) of te elements, one can introduce an upwinding sceme via te integrals on te sides (faces), using te values of φ at te rigt or te left of te sides depending on te flow direction. Lesaint [148] as introduced one of te first metods of tis type and as given error estimates for Friedric systems. In te case of equation (10) let us searc φ in te space of functions wic are piecewise polynomials but not necessarily continuous on a triangulation T j of Ω : We approximate (10) by T W = {w : w T P k } (41) w.(uφ ) u.n[φ ]w = fw w W T (42) T T were φ is taken in W,nis te external normal to T, [φ ] denotes te jump of φ across T from upstream of T : [φ ](x) =lim ɛ 0 +(φ (x + ɛu(x)) φ (x ɛu(x))), x T (43) and T is te part of T were u.n < 0.If T intersects Γ te convention is tat φ = φ Γ on te oter side of Γ i.e. φ (x ɛu(x)) = φ Γ (x) wenx Γ. Remark 1: If.u =0andk =0(φ piecewise constant ) (42) becomes u.n[φ ]= f T i.e. φ T =[ j T φ Tj u.n + T T j T T f]/ u.n (44) T were T j is te triangle upstream of T wic sares T. In one dimension wit u =1anddiam(T )= it reduces to φ i = φ i 1 + f, for all i. One can prove tat te sceme is positive if all te streamlines cut Σ : f 0,φ Γ 0 φ 0 (45)

62 INTRODUCTION 65 Remark 2 : If te solution φ is continuous, we expect tat φ will tend to a continuous function also in (42) ; so we ave added to te standard weak formulation a term wic is small (te integral on T ). Proposition 5 : If.u =0,if all te streamlines cut Σ and if is small, (42) defines φ in a unique manner Proof : We suppose for simplicity tat φ Γ W. Let ɛ be te difference between te 2 solutions. Let T be a triangle suc tat T is in Σ. Ten (42) wit w = ɛ T on T, 0 oterwise, gives u ɛ ɛ u.n[ɛ ]ɛ = 0 (46) = 1 2 T T u.nɛ 2 + T T T u.n ɛ2 2 + T u.nɛ T ɛ T were T is te triangle upstream of T tangent to T. If T is in Σ ten (46) becomes: u.n ɛ2 T T 2 =0. So ɛ =0on T T. Tus step by step we can cover te domain Ω. Convergence : Jonson -Pitkäranta [127] ave sown tat tis sceme is of 0( k+1/2 )for te error in L 2 norm and 0( k+1 ) if te triangulation is uniform ; tis result was extended by Ritcer [24] wo sowed tat te sceme is of 0( k+1 )ifte triangulation is uniform in te direction of te flow only. Te proofs are rater involved so we give only te general argument used in [127] and we sall only sow te following partial result: Proposition 6 If k =0(piecewise constant elements), ϕ is regular,.u =0, and f=0 ten ϕ ϕ u C were a u =( T T u.n (a + a ) 2 + u.n a 2 ) 1/2 Γ As in [127], we define Γ + =Γ Σ and we introduce te following notation:

63 66 FINITE ELEMENT METHODS FOR FLUIDS <a,b> = u.n a.b, T T Γ T <a,b> S = u.n a.b. Wit tis notation te discrete problem can be written in two ways: S <ϕ + ϕ,w > + <ϕ,w > Σ =<ϕ Γ,w > Σ w W <w + w,ϕ+ > + <ϕ,w > Γ +=< ϕ Γ,w > Σ So if we add te two forms we see tat ϕ is also a solution of: w W <ϕ + ϕ,w > <w + w,ϕ+ > + <ϕ,w > Γ =2<ϕ Γ,w > Σ w W By taking w = ϕ we find tat te sceme is stable: ϕ u C ϕ Γ 0,Γ. Te reader can establis te consistency of te sceme in te same way and obtain : <ϕ + ϕ,w > <w + w,ϕ > + <ϕ,w > Γ =2<ϕ Γ,w > Σ So for all ψ in W ϕ ϕ u ψ ϕ u If ψ is a piecewise constant interpolation of ϕ one as ϕ ψ = ϕ 1 O(), so ( u.n O() 2 ) 1 2 T C and te result follows. 2.6 Sceme 5 (Upwinding by cell) Te drawback of te previous metod is tat tere are many degrees of freedom in discontinuous finite elements (in 3D tere are on te average 5 times more tetraedra, at least, tan nodes). To rectify tis, Baba-Tabata [7] associated to all conforming P 1 function a function P o on cells centered around te vertices and covering Ω.

64 INTRODUCTION 67 To eac node q i we associate a cell σ i defined by te medians not originating from q i, of te triangles aving q i as one of te vertices. Figure 3.6:Te cell σ i associated to vertex q i. Let φ be a function P 1 piecewise on a triangulation and continuous. To φ we associate a function φ, piecewise P 0 on te cells σ i by te formula φ σ i = 1 σ i φ (x) (47) σ i Conversely, knowing ψ,p0 on te σ i, one could associate to it a function ψ in P 1 by ψ (q i )=ψ σ i (48) Ten we apply te sceme 4 to φ but we store in te computer memory only φ : Tis means tat our problem is now to find φ,p 1 continuous on a triangulation, equal to φ Γ on Σ and satisfying (in te case were.u =0) u.n[φ ]= σ i f σ i i (49) were φ σ = 1 i σ i φ (x) σ i (50) To sow tat tis sceme is well posed (i.e. tat φ is uniquely defined) one must sow tat (50) as a unique solution wen φ is known because we ave already sown tat (49) as a unique solution wen all te streamlines cut Σ. Tis is difficult in general but easy to sow if all te triangles are equilateral. Ten an elementary computation of integrals gives φ = 7 φ w i + 2 σ i 9 Ω 9 σi φ i (51) and multiplying by φ i and summing, we see tat φ i φ =0 7 φ 2 σ i 9 σ i (φ i ) 2 =0 φ = 0 Ω 9 (52) i 2.7 Comparison of te scemes So far we ave presented tree classes of metods for te stationary convection diffusion equation. It is not possible to rank te metods since eac

65 68 FINITE ELEMENT METHODS FOR FLUIDS as its advantages and drawbacks: for instance, upwinding by caracteristics gives good results but is conceptually more difficult and arder to program wile SUPG is conceptually simple but peraps too diffusive if an adjustment parameter is not added. Experience as sown to te autor tat personal preferences (suc as scientific background or previous experience wit one metod) are te determining factors in te coice of a particular metod; beyond tat we can only make te following broad remarks: 1. Upwinding by te discretisation of total derivatives as been very successful wit te incompressible Navier-Stokes equations were conservativity is not critical. 2. Streamline diffusion (SUPG) as been very successful in te engineering environment because it is conceptually simple. 3. Upwinding by discontinuity as been remarkably successful wit te compressible Euler equations because it is te rigt framework to generalize finite difference upwinding scemes into finite elements. (It as even received te name of Finite Volumes in tis context wen used wit quadrangles). 3. CONVECTION-DIFFUSION 1 : 3.1. Generalities : In tis section, we consider te equation φ,t +.(uφ) ν φ = f in Ω ]0,T[ (53) φ(x, 0) = φ o (x) in Ω; (54) φ Γ = φ Γ (55) and knowing tat in general for fluid applications ν is small, we searc for scemes wic work even wit ν = 0. Evidently,ifν = 0, (55) sould be relaxed to φ Σ = φ Γ (56) If ν = 0, (53), (54), (55) is a particular case of stationary convection (eq (10)) for te variables {x, t} on Ω ]0,T[ wit velocity {u, 1}. Tus one could simply add a diffusion term to te previous scemes and get a satisfying teory. But if we use te cylindrical structure of Ω ]0,T[, we can devise two new metods, one implicit in time witout upwinding and te oter semi-implicit but unconditionally stable. We begin by recalling an existence and uniqueness result for (53) and a good test problem to compare te numerical scemes: te rotating ump. Test Problem. Consider te case were Ω is te square ] 1, 1[ 2 and were u is a velocity field of rotation around te origin

66 INTRODUCTION 69 u(x, t) ={y, x}. We take φ Γ =0,f=0and φ 0 (x) =e ( x x0 2 r 2 ) 2 x suc tat x x 0 <r, φ 0 (x) =0elsewere. Wen ν =0,φ 0 is convected by u so if we look at t {x, y, φ(x, y, t)} we will see te region {x, y : φ(x, y) 0} rotate witout deformation; te solution is periodic in time and we can overlap φ and φ 0 after a time corresponding to one turn. If ν 0 te penomenon is similar but te ump flattens due to diffusion; we may also get boundary layers near te boundary 3.2.Some teoretical results on te convection-diffusion equation. Proposition 7: Let Ω be an open bounded set wit boundary Γ Lipscitz; we denote by n its exterior normal ; te system φ,t + u φ + aφ = f in Q =Ω ]0,T[ (57) φ(x, 0) = φ 0 (x) x Ω (58) φ(x, t) =g(x, t) (x, t) Σ=((x, t) :u(x, t).n(x) < 0) (59) as a unique solution in C 0 (0,T; L 2 (Ω)) wen φ 0 L 2 (Ω), g C 0 (0,T; L 2 (Γ)) and a, u L (Q), Lipscitz in x, f L 2 (Q). of Proof : We complete te proof by constructing te solution. Let X(τ) te solution wit te boundary condition d X(τ) =u(x(τ),τ)ifx(τ) Ω (60) dτ =0oterwise X(t) =x. (61) If u is te velocity of te fluid, ten X is te trajectory of te fluid particle tat passes x at time t. Witu L (Q) problem (60)-(61) as a unique solution. As X depends on te parameters x, t, we denote te solution X(x, t; τ) ; it is also te caracteristic of te yperbolic equation (57). We remark tat : d dτ φ(x(x, t; τ),τ),τ) τ =t = φ,t (x, t)+u φ(x, t) (62)

67 70 FINITE ELEMENT METHODS FOR FLUIDS So (57) can be rewritten as : φ,τ + aφ = f (63) and integrating φ(x, t) =e t 0 a(x(τ ),τ)dτ [λ + we determine λ from (58) and (59). If X(x, t;0) Γ, ten t 0 σ f(x(σ),σ)e a(x(τ ),τ)dτ 0 dσ] (64) λ = g(x(x, t;0),τ(x)) (65) were τ(x) istetime(<t)forx(x, t; τ) toreacγ. If X(x, t;0) Ω, ten λ = φ o (X(x, t; 0)) (66) Corollary (a =.u) : Wit te ypotesis of te proposition 7 and if u L 2 (0,T; H(div, Ω)) L (Q), Lipscitz in x,problem (53), (58), (59) wit ν = 0 as a unique solution in C 0 (0,T; L 2 (Ω)). -(we recall tat H(div, Ω) = {u L 2 (Ω) n :.u L 2 (Ω)}). Proposition 8 : Let Ω be a bounded open set in R n wit Γ Lipscitz. Ten problem φ,t +.(uφ).(κ φ) =f in Q =Ω ]0,T[ (67) φ(x, 0) = φ 0 (x) in Ω (68) φ = g on Γ ]0,T[ (69) as a unique solution in L 2 (0,T; H 1 (Ω)) if κ i,j,u j L (Ω), u i,j L (Ω), f L 2 (Q), φ o L 2 (Ω), g L 2 (0,T; H 1/2 (Γ)), and if tere exists a > 0suc tat κ ij Z i Z j a Z 2 Z R n. (70) Proof : See Ladyzenskaya [138] Approximation of te Convection-Diffusion equation by discretisation first in time ten in space.

68 INTRODUCTION 71 In tis paragrap we sall analyze some scemes obtained by using finite difference metods to discretise φ/ t and te usual variational metods for te remainder of te equation. Let us consider equation (67) wit (70) and assume, for simplicity from now on and trougout te capter tat κ ij = νδ ij,ν >0, u L (Q) and.u =0inQu.n =0onΓ ]0,T[ (71) φ =0onΓ ]0,T[ (72) As before, Ω is also assumed regular. Te reader can extend te results wit no difficulty to te case u.n 0, wit te elp of paragrap 2. As usual, we divide ]0,T[ into equal intervals of lengt k and denote by φ n (x) an approximation of φ(x, nk) Implicit Euler sceme : We searc φ n+1 H o, te space of polynomial functions of degree p on a triangulation of Ω, continuous and zero on Γ suc tat for all w H 0 we ave 1 k (φn+1 φ n,w )+(u n+1 φ n+1,w )+ν( φ n+1, w )=(f n+1,w ) (73) Tis problem as a unique solution because tis is an N N linear system, N being te dimension of H 0 : (A + kb)φ n+1 = kf + IΦ (74) were A ij =(w i,w j )+νk( w i, w j ),B ij =(u n+1 w i,w j ),F i =(f,w i ), I =(w i,w j )andwere{w i } is a basis of H 0 and φ i te coefficients of φ on tis basis. Tis system as a unique solution because te kernel of A + kb is empty : 0=ψ T (A + kb)ψ = ψ T Aψ ψ =0. (75) Proposition 9 : If φ L 2 (0,T; H p+1 (Ω)) and φ,t L 2 (0,T; H p (Ω)), we ave ( φ n φ(nk,.) νk (φn φ(nk,.)) 2 0 ) 1 2 C( p + k) (76) were p is te degree of te polynomial approximation for φ n. Proof : a) Estimate of te error in time Let φ n+1 be te solution of te problem discretised in time only :

69 72 FINITE ELEMENT METHODS FOR FLUIDS 1 k (φn+1 φ n )+u n+1 φ n+1 ν φ n+1 = f n+1 ; φ n+1 Γ =0;φ o = 0 (77) te error ɛ n (x) =φ n (x) -φ(nk, x) satisfiesɛ n Γ =0,ɛ o =0and 1 k (ɛn+1 ɛ n )+u n+1 ɛ n+1 ν ɛ n+1 = k 1 0 (1 θ) 2 φ (x, (n + θ)k)dθ (78) t2 were te rigt and side is te result of a Taylor expansion of φ(nk + k, x). Multiplying (78) by ɛ n+1 and integrating on Ω makes te convection term disappear and we deduce tat ɛ n+1 ν ( ɛ n kν ɛn ) 1 2 k 2 T φ tt 0,Ω ]0,T [ + ɛ n ν. (79) So we ave, for all n: ɛ n ν k T φ,tt 0,Q (80) b) Estimation of te error in space Let ξ n = φ n φn ; by subtracting (77) from (73) we see 1 k (ξn+1 ξ n,w )+(u n+1 ξ n+1,w )+ν( ξ n+1, w ) = 0 (81) Let ψ n be te interpolation of φn ;tenξ n = ξ n + ψn φn and ξ n satisfies (we take w = ξ n+1 ) so ξ n+1 2 ν ξn o ξ n+1 o (ψ n+1 φ n+1 ψ n + φn,ξ n+1 ) (82) k(u n+1 (ψ n+1 φ n+1 ),ξ n+1 ) νk( (ψ n+1 φ n+1 ), ξ n+1 ) ξ n+1 ν ξ n ν + Ck[ p φ,t p,ω + p ( u,q + ν) φ p+1,ω ] (83) because. o. ν,ψ n+1 - ψ n is te interpolation of φn+1 φ n wic is an approximation of kφ,t. Finally we obtain (76) by noting tat φ n φ(nk) ν ξ n ν + ψ n φn ν + φ(nk) φ n ν (84) Comments : We note tat te error estimate (76) is valid even wit ν =0. Tus,we do not need any upwinding in space. Tis is because te Euler sceme itself

70 INTRODUCTION 73 is upwinded in time (it is not symmetric in n and n + 1). Te constant c in (76) depends on φ p+1,ω and tat if ν is small (67) as a boundary layer and φ p+1,ω tends to infinity (if φ Γ is arbitrary ) as ν p+1/2. Tis requires te modification of φ Γ or te imposition of: <<ν 1 1/2p. A tird metod consists of replacing ν by ν in suc a way tat << ν 1 1/2p is always satisfied. Tis is te artificial viscosity metod Leap frog sceme : Te previous sceme requires te solution of a non-symmetric matrix for eac iteration. Tis is a sceme wic does not require tat kind of operation but wic works only wen k is O(). 1 2k (φn+1 φ n 1,w )+(u n φ n,w )+ ν 2 ( [φn+1 + φ n 1 ], w ) (85) =(f n+1,w ) w H 0, φ n+1 To start (85), we could use te previous sceme. H 0 Proposition 10 : Te sceme (85) is marginally stable, i.e. tere exists a C suc tat for all k suc tat φ n o C f 0,Q (1 C u k ) 1 2 (86) k< C u. (87) Proof in te case were u is independent of t and f = 0. We use te energy metod (Ritcmeyer-Morton [197], Saiac [211] for example). Let ten S n = φ n φn k(u φn,φn+1 ) (88) S n S n 1 = φ n φn k(u φn,φn+1 but from (85), wit w = φ n+1 + φ n 1 and if f =0 + φ n 1 ) (89) φ n φ n k(u φ n,φ n+1 + φ n 1 )+kν (φ n+1 + φ n 1 ) 2 0 = 0; (90) so S n S n 1... S o. On te oter and, using an inverse inequality

71 74 FINITE ELEMENT METHODS FOR FLUIDS we get (u φ n,φn+1 ) C u φ n+1 0 φ n C 0 u 2 ( φn φn 2 0 ) (91) S n (1 C k u )[ φn φ n 2 0] (92) Convergence : As for te Euler sceme, one can sow using (86) tat (85) is 0( p + k 2 ) for te L 2 norm error Adams-Basfort sceme : It is important to make te dissipation terms implicit as we ave done for φ because te leap-frog sceme is only marginally stable (cf. Rictmeyer- Morton [197]). In te same way, if Σ (u.n 0), it is necessary to make implicit te integral on Γ Σ. For tis reason, we consider te Adams-Basfort sceme of order 3 wic is explicit wen we use mass lumping and wic as a better stability tan te leap-frog sceme. 1 k (φn+1 φ n,w )= b(φn,w ) 16 w H 0 were 12 b(φn 1,w ) b(φn 2,w ) b(φ,w )= [(u φ,w )+ν( φ, w ) (f,w )] Te stability and convergence of te sceme can be analysed as in Te θ scemes : be In a general way, let A and B be two operators and te equation in time We consider a sceme wit tree steps u,t + Au + Bu = f; u(0) = u 0 1 kθ (un+θ u n )+Au n+θ + Bu n = f n+θ 1 (1 2θ)k (un+1 θ u n+θ )+Au n+θ + Bu n+1 θ = f n+1 θ 1 kθ (un+1 u n+1 θ )+Au n+1 + Bu n+1 θ = f n+1

72 INTRODUCTION 75 An analysis of tis sceme in te finite element context can be found in Glowinski [96]. One can sow easily tat wit A = αc, B =(1 α)c, were C is an matrix N N wit strictly positive eigenvalues, ten te sceme is unconditionally stable and of order 2 in k if α =1/2 andθ =1 2/2. In te case of te convection-diffusion equation, we could take A = αν and B = u (1 α)ν (α = 1 being admissible). Corin [53], Beale-Majda [17] ave studied metods of tis type were A = ν and steps 1 and 3 are carried out by a Monte-Carlo metod, and were step 2 is integrated by a finite element metod, a finite difference metod or by te metod of caracteristics. We will use tese for te Navier-Stokes equations Adaptation of te finite difference tecniques : All te metods presented up to now in tis section ave been largely studied on regular grids in a finite difference framework. Te same tecniques can be used in a finite element context to estimate stability and errors, wit te following restrictions : - constant coefficients (u and ν constants), - uniform triangulation - influence of boundary conditions is difficult to take into account Te analysis of finite difference scemes is based on te following fundamental property: Stability + consistency convergence. To give an example, let us study te Crank-Nicolson sceme for (67) in te case were u is constant : 1 k (φn+1 φ n,w )+ 1 2 (u (φn+1 + φ n ),w )+ ν 2 ( (φn+1 + φ n ), w ) =(f n+ 1 2,w ) w H 0 Altoug te L 2 stability is simple to prove by taking w = φ n+1 + φ n, we will consider te oter metods at our disposal. By coosing a basis {w i } of H 0 we could write explicitly te linear system corresponding to te case f =0andφ Γ =0. B(Φ n+1 Φ n )+ k 2 A(Φn+1 +Φ n )=0 were B ij =(w i,w j ) A ij = ν( w i, w j )+(u w i,w j ) If we know te eigenvalues and eigenvectors of A in te metric B, i.e. te solutions {λ i,ψ i } of λbψ = Aψ, ten by decomposing Φ n on tis basis, we get (BΦ) n+1 i =(BΦ) n (1 k 2 λ i) i (1 + k 2 λ i)

73 76 FINITE ELEMENT METHODS FOR FLUIDS By asking te amplification factor F = (1-k λ i /2)/(1 + kλ i /2) to be smaller tan 1, we can deduce an interval of stability of te metod. Evidently, te smallest real part of te eigenvalues is not known but could be numerically determined in te beginning of te calculation (as in Maday et al. [164] for te Stokes problem). On te oter and, wen H 0 is constructed wit P 1 continuous functions, te Crank-Nicolson sceme on a uniform triangulation is identical to te following finite difference sceme (exercise) : 2 12 [6φn+1 i,j + φ n+1 i+1,j+1 + φn+1 i+1,j + φn+1 i,j 1 + φn+1 i 1,j 1 + φn+1 i 1,j + φn+1 i,j+1 ] + k 12 [(φn+1 i+1,j+1 φn+1 i 1,j 1 )(u 1 + u 2 )+(2u 1 u 2 )(φ n+1 i+1,j φn+1 i 1,j ) +(2u 2 u 1 )(φ n+1 i,j+1 φn+1 i,j 1 )] + νk[4φn+1 i,j φ n+1 i+1,j φn+1 i,j+1 φn+1 i 1,j φn+1 i,j 1 ] = idem in φ n by canging k to k. If tere exists solutions of te form : tey ave to satisfy φ n lm = ψn e 2iπ(lx+my) ψ n+1 [ 2 [3 + cos((l + m))+cos(l)+cos(m)] 6 +i k 6 [sin((l + m))(u 1 + u 2 )+sin(l)(2u 1 u 2 )+sin(m) (2u 2 u 1 )] + 2νk[2 cos(l) cos(m)]] = ψ n [ same factor but k k] So we ave a formula for te amplification factor. Finally, it is easy to see tat te above finite difference sceme is consistent to order 2. So we ave a presumption of convergence of 0( 2 + k 2 )forte metod on a general triangulation. 4.CONVECTION DIFFUSION 2. In tis section we analyze scemes for te convection-diffusion equation (53)-(55) wic still converge wen ν = 0 witout generating oscillations. Tis classification is somewat arbitrary because te previous scemes can be made to work wen ν =0orwenϕ is irregular. But we ave put in tis section scemes wic ave been generalized to nonlinear equations (Navier- Stokes and Euler equations for example). Let us list te desirable properties for a sceme to work on nonsmoot functions φ :

74 INTRODUCTION 77 - convergence in L norm, - positivity: ϕ>0 ϕ > 0, - convergence to te stationary solution wen t, - localization of te solution if ν =0(tatistosaytattesolution sould not depend upon watever is downstream of te caracteristic wen ν =0) Discretisation of te total derivative: Discretisation in time. We ave seen tat if X(x, t; τ) denotes te solution of ten dx (τ) =u(x(τ),τ); X(t) =x (93) dτ φ,t + u φ = τ φ(x(x, t; τ),τ) τ =t (94) Tus, taking into account te fact tat X(x, (n + 1)k;(n + 1)k) = x, we can write: (φ,t + u φ) n+1 1 = k [φn+1 (x) φ n (X n (x))] (95) were X n (x) is an approximation of X(x, (n +1)k; nk). We sall denote by X1 n an approximation 0(k2 )ofx n (x) andbyx2 n an approximation 0(k 3 ) (te differences between te indices of Xand te exponents of k are due to te fact tat X n is an approximation of X obtained by an integration over a time k ;tusasceme0(k α ) gives a precision 0(k α+1 )). For example X n 1 (x) =x u n (x)k (Euler sceme for (93)) (96) X2 n (x) =x u n+ 1 2 (x u n (x) k )k ( Second order Runge-Kutta) (97) 2 modified near te boundary so as to get Xi n (Ω) Ω. To obtain tis inclusion one can use (96) or (97) inside te elements so tat one passes from xto X n (x) by a broken line rater tan a straigt line (see (18)(19)). Tis yields two scemes for (53) : 1 k (φn+1 φ n ox1 n ) ν φ n+1 = f n+1 (98) 1 k (φn+1 φ n ox2 n ) ν 2 (φn+1 + φ n )=f n+ 1 2 (99)

75 78 FINITE ELEMENT METHODS FOR FLUIDS Lemma 1 : If u is regular and if X n i (Ω) Ω, te scemes (98) and (99) are L2 stable and converge in 0(k) and 0(k 2 ) respectively. Proof : Let us sow consider (98). We multiply by φ n+1 : φ n νk φ n+1 2 ( f n+1 k + φ n ox 1 ) φ n+1 (100) But te map x X preserves te volume wen u n is solenoidal (.u =0). So from (96) : φ n ox n = X n1 (Ω) φ n (y) 2 det[ X n 1 ] 1 dy φ n 2 0,Ω (1 + ck2 ) (101) Hence φ n verifies φ n ν c[ f 0,Q + φ o 0,Ω ] (102) To get an error estimate one proceeds as in te beginning of te proof of proposition 9 by using (95). X n i Remark : Xi n (Ω) Ω is necessary because u.n = 0. (Ω) Ω Σ. Oterwise one only needs Approximation in space. Now if we use te previous scemes to approximate te total derivative (sceme (98) of order 1, sceme (99) of order 2) and if we discretise in space by a conforming polynomial finite element we obtain a family of metods for wic no additional upwinding is necessary and for wic te linear systems are symmetric and time independent. Take for example te case of (98) : Ω φ n+1 w + kν Ω φ n+1 w = k f n+1 w + φ n (Xn 1 (x))w (x) Ω Ω w H 0 φ n+1 H 0 (103) were H 0 is te space of continuous polynomial approximation of order 1 on a triangulation of Ω, and zero on te boundaries. Proposition 11 : If X 1 (Ω) Ω, te sceme (103) is L 2 (Ω) stable even if ν =0. Proof : One simply replaces w by φ n+1 in (103) and derives upper bounds :

76 φ n = k Ω INTRODUCTION 79 Ω φ n k f n+1 φ n+1 + Ω Ω ν φ n+1 φ n+1 (104) φ n (Xn 1 (x))φn+1 (x) (k f n φ n ox n 1 (.) 0 ) φ n+1 0 ( φ n 0 (1 + c 2 k2 )+k f n+1 0 ) φ n+1 0 Te last inequality is a consequence of(101). Finally by induction one obtains φ n 0,Ω (1 + c 2 k2 ) n ( φ o 0,Ω + k f n 0,Ω ) (105) Remark : By te same tecnique similar estimates can be found for (105) but te norms on φ n et φo will be. ν (cf. (79)). Proposition 12. If H 0 is a P 1 conforming approximation of H 1 0 (Ω) ten te L2 (Ω) norm of te error between φ n solution of (103) and φn solution of (98) is 0( 2 /k + ).Tus te sceme is 0( 2 /k + k + ). Proof : One subtracts (98) from (103) to obtain an equation for te projected error: ɛ n+1 = φ n+1 Π φ n+1, (106) were Π φ n+1 is an interpolation in H 0 of φ n+1. One gets Ω Ω ɛ n+1 w + kν Ω (φ n+1 Π φ n+1 )w + νk From (107), wit w = ɛ n+1 ɛ n+1 w ɛ n ox1 n w = (107) Ω (φ n+1 Π φ n+1 ) w Ω (φ n Π φ n )ox1 n w Ω we obtain terefore 2 ν ( ɛn ν + φ n+1 Π φ n+1 ν + φ n Π φ n 0 ) ɛ n+1 ν ɛ n+1 ɛ n+1 ν ɛ n ν + C( 2 + νk)

77 80 FINITE ELEMENT METHODS FOR FLUIDS Remark : By comparing wit (103), we see tat ɛ and φ are solutions of te same problem but for ɛ,fis replaced by : 1 k (φn+1 Π φ n+1 ) ν (φ n+1 Π φ n+1 ) 1 k (φn Π φ n )ox1 n, were is an approximation of. We can bound independently te first and te last terms. By working a little arder (Douglas-Russell [69]) one can sown tat te error is, in fact, 0( + k +min( 2 /k, )). Proposition 13 : Wit te second order sceme in time (99) and a similar approximation in space one can build scemes 0( 2 + k 2 +min( 3 /k, 2 )) wit respect to te L 2 norm: (φ n+1,w )+ νk 2 ( (φn+1 + φ n ), w )=(φ n ox2 n,w )+k(f n+ 1 2,w ) (108) w H 0 ; φ n+1 H 0 were H 0 is a P 2 conforming approximation of H 1 0 (Ω). Proof : Te proof is left as an exercise. Te case ν =0: We notice tat (103) becomes φ n+1 w = φ n ox2 n w + k Tat is to say Ω φ n+1 Ω H 0 f n+1 w w H 0 (109) φ n+1 =Π (φ n ox 1 n )+kπ f n+1 (110) were Π is a L 2 projection operator in W 0. Sceme (108) becomes : φ n+1 =Π (φ n ox n 2 )+k 2 Π (f n+1 + f n ) (111) If f = 0 te only difference between te scemes are in te integration formula for te caracteristics. Notice also tat te numerical diffusion comes from te L 2 projection at eac time step. Tus it is better to use a precise integration sceme for te caracteristics and use larger time steps. Experience sows tat k 1.5/u is a good coice. Notice tat wen ν and f are zero one solves

78 INTRODUCTION 81 φ,t + u φ =0 φ(x, 0) = φ 0 (x) (112) -(sinceweaveassumedu.n = 0, no oter boundary condition is needed). Since.u = 0, we deduce from (112) tat (conservativity) φ(t, x) = φ 0 (x) t (113) Ω On te oter and, from (109), wit w =1 φ n+1 (x) = φ n ox 1 X n = φ n n1 (Ω) (y)det Xn 1 1 dy (114) Ω Ω So if det X1 n =1(wicrequiresX1 n (Ω) = Ω), one as φ n+1 (x)dx = φ n (y)dy = φ 0 (x)dx (115) Ω Ω We say ten tat te sceme is conservative. It is an important property in practice Numerical implementation problems : Ω Two points need to be discussed furter. - How to compute X n (x), - How to compute I n : I n = φ ox n w (116) Ω Ω Computation of (116) : As in te stationary case one uses a quadrature formula: I n = ω k φ (X (ξ k ))w (ξ k ) (117) For example wit P 1 elements one can take a) {ξ k } = te middles of te sides, ω k = σ k /3in2D,σ k /4in3D,were σ k is te area (volume) of te elements wic contain ξ k. b) Te 3 (4 in 3D) point quadrature formula (Zienkiewicz [241], Stroud [224]) or any oter more sopisticated formula ; but experiments sow tat quadrature formula wit negative weigts ω k sould not be cosen. Numerical tests wit a 4 points quadrature formulae can be found in Bercovier et al. [27]. Finally, anoter metod (referred as dual because it seems tat te basis functions are convected forward) is found by introducing te following cange of variable: φ ox w = φ (y)w (X 1 (y))det X 1 dy (118) Ω X (Ω)

79 82 FINITE ELEMENT METHODS FOR FLUIDS Ten te quadrature formula are used on te new integral. If.u is zero one caneventake I = ω k φ (ξ k )w (X 1 (ξk ))w k (119) It is easy to ceck tat tis metod, (109), (119), is conservative wile (109), (117) is not. Numerical tests for tis metod can be found in Benque et al. [20]. Te stability and error estimate wen a quadrature formula is used is an important open problem ; it seems tat tey ave a bad effect in te zones were u is small (Suli [226], Morton et al.[177]). Computation of X n (x) : Formula (96) and (97) can be used directly. However it sould be pointed out tat in order to apply (117), (same problem wit (119)) one needs to know te number l of te element suc tat X n (ξ k ) T l (120) Tis problem is far from being simple. A good metod is to store all te numbers of te neigboring (by a side) elements of eac element and compute te intersections {ξ k,x n (ξ k )} wit all te edges (faces in 3D) between ξ k and X n (ξ k ); but ten one can immediately improve te sceme for X by updating u wit its local value on eac element wen te next intersection is searced; ten a similar computation for (18)-(19) is made: Let {u i } be suc tat u = i u iq i, u i =0, were {q i } are te vertices of te triangle (tetraedron): Find ρ suc tat λ i = λ i + ρu i i λ i =0, λ i 0. (121) Tis is done by trial and error; we assume tat it is λ m wic is zero, so: ρ = λ m, (122) µ m and we ceck tat λ i 0, i. If it is not so we cange m until it works. Most of te work goes into te determination of te u i. Notice tat it may be practically difficult to find wic is te next triangle to cross wen X n (ξ k ) is a vertex, for example. Tis requires careful programming. Wen u u is 0( p ) te sceme is 0( p ). If u is piecewise constant one must ceck tat u.n is continuous across te sides (faces) of te elements (wen.u = 0) oterwise (121) may not ave solutions oter tan ρ =0. If u = ψ and ψ is P 1 and continuous ten u.n is continuous Te Lax-Wendroff/ Taylor-Galerkin sceme[25] Consider again te convection equation

80 INTRODUCTION 83 φ,t +.(uφ) =f in Ω ]0,T[ (123) For simplicity assume tat u.n Γ =0andtatu and f are independent of t. A Taylor expansion in time of φ gives: φ n+1 = φ n + kφ n,t + k2 2 φn,tt +0(k 3 ) (124) If one computes φ,t and φ, tt from (123), one finds φ n+1 = φ n + k[f.(uφ n )] k2 2.(u[f.(uφn )]) + 0(k 3 ) (125) or again φ n+1 = φ n + k[f.(uφ n )] + k2 2 [.(uf)+.[u.(uφn )]] + 0(k 3 ) (126) Tis is te sceme of Lax-Wendroff [144]. In te finite element world tis sceme is known as te Taylor-Galerkin metod (Donea [68]). Note tat te last term is a numerical diffusion 0(k) in te direction u n because it is te tensor u n u n. Let us discretise (126) wit H,tespaceofP 1 continuous function on a triangulation of Ω : (φ n+1,w )=((φ n,w )+k(f.(uφ n ),w ) (127) k2 2 (.(uf),w ) k2 2 (u w,.(uφ n )) w H TescemesouldbeO( 2 + k 2 )intel 2 norm but it as a CFL stability condition (Courant-Friedriscs-Lewy) (see Angrand-Dervieux [2] for a result of tis type on an O()-regular triangulation for sceme (127) wit mass lumping) k<c (128) u An implicit version can also be obtain by canging n into n +1andk into k in (126) (φ n+1,w )+k(.(uφ n+1 ),w )+ k2 2 (.(uφn+1 ),u w ) (129) = k(f,w )+(φ n,w )+ k2 2 (.(uf),w ), w H In te particular case wen.u =0andu.n Γ = 0,we ave te following result :

81 84 FINITE ELEMENT METHODS FOR FLUIDS Proposition 14 : If u and f are independent of t and.u =0,u.n Γ =0,ten sceme (129) as a unique solution wic satisfies φ n 0 φ o 0 + T f 0 + T 2 k.(uf) 0 (stability) (130) φ n φ(nk) 0 C( 2 + k 2 ) (convergence) (131) Proof : Te symmetric part of te linear system yielded by (129) multiplied left and rigt by ψ gives ψ 2 o + k2 /2 u ψ 2 wic is always positive ; tus te linear systems being square, tey ave one and only one solution. By taking w = φ n+1 in (129), one finds φ n k2 2 u φn+1 2 φ n 0 φ n k f 0 φ n k2 2.(uf) 0 φ n+1 0 (132) tus if one divides by φ n+1 o and adds all te inequalities, (130) is found. To obtain (131) one sould subtract (126) from (129) : (ɛ n+1,w )+ k2 2 (u ɛn+1,u w )+k(u ɛ n+1,w )=(ɛ n +0(k 3 ),w ) (133) were ɛ = φ φ. Let ɛ = φ ψ were ψ is te projection of φ in H wit te norm. k =( k 2 /2 u. 2 0) 1/2. Ten ɛ = ɛ +0( 2 ) and from (133), we get Te result follows. ɛ n+1 k ɛ n + C(k2 + k 3 ) (134) Remarks : 1. Te previous results can be extended witout difficulty to te case ν 0 wit te sceme: (φ n+1,w )+k(.(uφ n+1 ),w )+ k2 2 (.(uφn+1 ),u w )+ν( φ n+1, w ) (135) = k(f,w )+(φ n,w )+ k2 2 (.(uf),w ), w H o Higer order scemes : Donea also report tat very good iger order scemes can obtained easily by pusing te Taylor expansion furter; for instance a tird order sceme would be (f =0):

82 INTRODUCTION 85 φ n+1 [φ n k.(uφ n )+ k2 2.[u.(uφn )] + k2 6.[u.(u(φn+1 φ n ))]] = Te streamline upwinding metod (SUPG). Te streamline upwinding metod studied in 2.4 can be applied to (123) witout distinction between t and x but it would ten yield very large linear systems. But tere are oter ways to introduce streamline diffusion in a time dependent convection-diffusion equation. Te simplest (Huges [116]) is to do it in space only; so consider (φ n+1,w + τ.(uw )) + k 2 (.(u[φn + φn+1 ]),w + τ.(uw )) + kν 2 ( (φn+1 + φ n ), w ) kν 2 l ( (φ n+1 T l + φ n )τ.(uw )) = k(f n+ 1 2,w + τ.(uw )) + (φ n,w + τ.(uw )), w H o were T l is an element of te triangulation and u is evaluated at time (n+1/2)k if it is time dependant; τ is a parameter wic sould be of order but as te dimension of a time. Wit first order elements, φ = 0 and it was noticed by Tezduyar [227] tat τ could be cosen so as to get symmetric linear systems wen.u =0; an elementary computation sows tat te rigt coice is τ = k/2. Tus in tat case te metod is quite competitive, even toug te matrix of te linear system as to be rebuilt at eac time step wen u is time dependant. Te error analysis of Jonson [125] suggests te use of elements discontinuous in time, continuous in space and a mixture upwinding by discontinuity in time and streamline upwinding in space. To tis end space-time is triangulated wit prisms. Let Q n =Ω ]nk, (n + 1)k[, let Wo n be te space of functions in{x, t} wic are zero on Γ ]nk, (n + 1)k[ continuous and piecewise affine in x and in t separately on a triangulation by prisms of Q n ; We searc φ n wit φn φ Γ Wo n solution of [φ n,t + (uφ n )][w + (w,t + (uw )]dxdt + ν φ n. w dxdt (136) Q n Q n + φ n (x, (n 1)k +0)w (x, (n 1)k)dx = f(w + (w,t +.(uw ))dxdt Ω Q n

83 86 FINITE ELEMENT METHODS FOR FLUIDS + φ n 1 (x, (n 1)k 0)w (x, (n 1)k)dx, w Wo n Ω Note tat if N is te number of vertices in te triangulation of Q n, equation (136) is an N N linear system, positive definite but non symmetric. One can sow (Jonson et al. [127]) te following : T ( φ n φ 2 0,Ωdt) C( 2 + k 2 ) φ H 2 (Q). (137) Upwinding by discontinuity on cells: Te metod of 2.5 can be extended to te nonstationary case but tere is an approximation problem for φ wen φ is discontinuous in space. One way is to use te upwinding by cells as in 2.6 ; since te sceme will be 0() one can use a piecewise constant approximation in time, wic is similar to a discretisation of (53) ( see (49) for te notation) by (φ n+1,w i ) k σ i u.n[ φ n+1 dx]dγ + νk( φ n+1 w i )= σ i σ i =(φ n,w i )+k(f,w i ), i, φ n+1 φ Γ H o were w i is te continuous piecewise affine function associated wit te i t vertex of te triangulation of Ω. Tis sceme is used in Dervieux et al [64][65] for te Euler equations (see Capter 6).

84 INTRODUCTION 87 CHAPTER 4 THE STOKES PROBLEM 1. POSITION OF THE PROBLEM Te generalized Stokes problem is to find u(x) R n and p(x) R suc tat αu ν u + p = f.u =0inΩ, (1) u = u Γ on Γ = Ω. (1 ) were α, ν are given positive constants and f is a function from Ω into R n. It can come from at least two sources : 1) an approximation of te fluid mecanics equations suc as tat seen in Capter 1, tat is wen te Reynolds number is small (microscopic flow, for example), ten α = 0, and in general f =0; 2) a time discretisation of te Navier-Stokes equations. Ten α is te inverse of te time step size and f an approximation of u u. In tis capter, we sall deal wit some finite element approximations of (1). More details regarding te properties of existence, unicity and regularity can be found in Ladyzenskaya [138], Lions [153] and Temam [228] and regarding te numerical approximations in Girault-Raviart [93], Tomasset [229], Girault-Raviart [93], Glowinski [95], Huges [116]. Test Problem : Te most classic test problem is te cavity problem : Te domain Ω is a square ]0, 1[ 2, α =0,ν =1,f = 0 and te boundary condition: u = 0 on all te boundary except te upper boundary ]0, 1[ {1} were u =(1., 0).

85 88 FINITE ELEMENT METHODS FOR FLUIDS Figure 4.1 : Stokes flow in a cavity Tis problem does not ave muc pysical signifigance but it is easy to compare wit solutions computed by finite differences. We note tat te solution as singularities at te two corners were u is discontinuous (te solution is not in H 1 ).W e can regularize tis test problem by considering on te upper boundary ]0, 1[ {1} : u = x(1 x) 2. FUNCTIONAL SETTING. Let n be te dimension of te pysical space (Ω R n )and J(Ω) = {u H 1 (Ω) n :.u =0} (2) J o (Ω) = {u J(Ω) : u Γ =0} (3) Let us put (a, b) = a i b i a, b L 2 (Ω) n (4) (A, B) = Ω Ω A ij B ij A, B L 2 (Ω) n n (5) and consider te problem α(u, v)+ν( u, v) =(f,v) v J o (Ω) (6a) u u Γ J o(ω) were u Γ is an extension in J(Ω) of u Γ. (6b) Lemma 1 If u Γ H 1/2 (Γ) n and if Γ u Γ.n =0 ten tere exists an extension u Γ J(Ω) of u Γ. Remarks 1. For notational convenience we assume tat u Γ always admits an extension wit zero divergence and we identify u Γ and u Γ. 2. Taking account of tis condition and witout loss of generality, we arrive by translation (v = u u Γ )attecaseu Γ =0. Teorem 1 If f L 2 (Ω) n and u Γ J(Ω), problem (6) as a unique solution. Proof :

86 INTRODUCTION 89 J o (Ω) is a non empty closed subspace of H 1 (Ω) n and te bilinear form {u, v} α(u, v)+ν( u, v) is H0 1 (Ω) elliptic. Wit te ypotesis v (f,v) continuous, te teorem is a direct consequence of te Lax-Milgram teorem. Remark We also ave a variational principle : (6) is equivalent to 1 min u u Γ Jo(Ω) 2 α(u, u)+1 ν( u, u) (f,u) (7) 2 Teorem 2 If te solution of (6) is C 2 ten it satisfies (1). Conversely, under te ypotesis of teorem 1, if {u, p} H 1 (Ω) n L 2 (Ω) is a solution of (1) ten u is a solution of (6). Proof : We apply Green s formula to (6) : (αu ν u).v = Ω Ω fv v J o (Ω) (8) Now we use te following teorem (Cf Girault-Raviart [92] for example) g.v =0 v L 2 (Ω) wit.v =0 v.n Γ =0 (9) Ω Tere exists q L 2 (Ω) suc tat g = q (10) - We recall tat ( p, v) = 0 for all v J o (Ω). So (8) implies te existence of p L 2 (Ω) suc tat αu ν u = f p (11) and te result follows. Conversely, by multiplying (1) wit v J o (Ω) and integrating we obtain (6) since v Γ =0: uv = u v and p.v = v p Ω Ω Remark 1. It is not necessary tat u be C 2 for p to exist, but ten te proof is based on a more abstract version of (9)(10). Indeed it is enoug to notice tat te linear map : L(v) :v (f,v) α(u, v) ν( u, v) Ω Ω

87 90 FINITE ELEMENT METHODS FOR FLUIDS is zero on J 0 (Ω) because te space ortogonal to J(Ω) is te space of gradients, tus tere exists a q in L 2 (Ω) suc tat L(v) =( q, v) for all v in H 1 0 (Ω)n ; ence (1) is true in te distribution sense. 2. Problem (1) can also be directly studied in te form of a saddle point problem (Cf. Girault-Raviart [92] for example), α(u, v)+ν( u, v) (.v, p) =(f,v) v H 1 0 (Ω) n (.u, q) =0 q L 2 (Ω)/R We sall use later tis formulation in te error analysis of te metods. 3. Te uniqueness of te pressure, p (up to a constant) is not given by teorem 1 but it can be sown by studying te saddle point problem. 3. DISCRETISATION Let {J } be a sequence in a finite dimensional space, J 0 = J H 1 0 (Ω)n, suc tat v J o (Ω) tere exists v J o suc tat v v 1 0wen 0 (12) We consider te approximated problem : Find u suc tat α(u,v )+ν( u, v )=(f,v ) v J o ; u u Γ J o. (13) were u Γ is an approximation of u Γ in J. Teorem 3 If J o is non empty te problem (13) as a unique solution. Proof Let us consider te problem 1 min u u J Γ o 2 {α(u,u )+ν( u u ) (f,u )} (14) Since J o is of finite dimension N, (14) is an optimization of a strictly quadratic function in N variables ; tus it admits a unique solution. By writing te first order optimality conditions for (14), we find (13). A counter example.

88 INTRODUCTION 91 Let Ω be a quadrilateral and T be te triangulation of 4 triangles formed wit te diagonals of Ω. Let J 0 = {v C 0 (Ω) : v Tk P 1 v Γ =0, (.v,q)=0 q continuous, piecewise affine on T } Tere are only two degrees of freedom for v (its values at te center) but tere are 4 constraints of wic 3 are independent, so J 0 is reduced to {0, 0}. Figure 4.2 :values of te velocity at te vertices of te first counterexample and te pressure for te second. Remarks 1. One migt tink tat te problem wit te above counter example stems from te fact tat tere are not enoug vertices. Altoug J o would not, in general, be empty wen tere are more vertices, tis approximation is not good, as we sall see, because tere are too many independent constraints. For example, let Ω be a square triangulated into 2(N 1) 2 triangles formed by te straigt lines parallel to te x and y axis and te lines at 45 o (figure 4.2). We ave also 2N 2 constraints and 2(N 2 4N) = 2N 2 degrees of freedom wen N is large. 2. Te pressure cannot be unique in tat case. Indeed wit J 0 defined as above tere exists {p } piecewise affine continuous suc tat (.v,p )=0 v P 1 continuous on T Tese are te functions p wic ave values alternatively 0, 1 and -1 on te vertices of eac triangle (eac triangle aving exactly tese tree values at te tree vertices.) Ten (.v,p ) is zero because (.v,p )= T i (.v ) Ti ( 1 3 j=1,2,3 p (q ij )) So te constraints in J o are not independent and te numerical pressure associated wit te solution of (14) (te Lagrangian multiplier of te constraints) cannot be unique. To solve numerically (13) (or (14)) we can try to construct a basis of J o. Let {v i } N i be a basis of J o. By writing u on tis basis, u (x) = 1..N u i v i (x)+u Γ (x), (15)

89 92 FINITE ELEMENT METHODS FOR FLUIDS it is easy to see by putting (15) in (13) wit v = v i tat (13) reduces to a linear system wit AU = G (16) A ij = α(v i,v j )+ν( v i, v j ), (17) G j =(f,v j ) α(u Γ,v j ) ν( u Γ, v j ). (18) However tere are two problems 1) It is not easy to find an internal approximation of J(Ω) ; tat is in general we don t ave J o J 0 (Ω), wic leads to considerable complications in te study of convergence. 2) Even if we succeed in constructing a non empty J o wic satisfies (12) and for wic we could sow convergence, te construction of te basis {v i } is in general difficult or unfeasible wic makes (15)(16) inapplicable in practice. Before solving tese problems, let us give some examples of discretisation for J o in increasing order of difficulty. Tese examples are all convergent and feasible as we sall see later. Tey are all of te type J = {v V : (.v,q) =0 q Q } J 0 = {v J : v Γ =0} Tus J 0 is determined by te coice of two spaces 1 o ) V wic approximate H 1 (Ω), 2 o ) Q wic approximate L 2 (Ω) or L 2 (Ω)/R 3 o ) and te coice of a quadrature formula. (.,.). As usual, {T k } denotes a triangulation of Ω, Ω te union of T k, {q j } j=1..ns te vertices of te triangulation, λ k j (x) tej t barycentric coordinate of x wit respect to vertices {q kj } j=1.,n+1 of te element T k. We denote respectively by Ns, te number of vertices, Ne, te number of elements, Nb, te number of vertices on Γ, Na, te number of sides of te triangulation, and Nf, te number of faces of te tetraedra in 3D. P m denotes te space of polynomials (of degree m) in n variables. We recall Euler s geometric identity :

90 INTRODUCTION 93 Ne Na+ Ns =1in2D,Ne Nf + Na Ns = 1 in3d 3.1. P1 bubble/p1 element (Arnold-Brezzi-Fortin [5]) Let µ k (x) be te bubble function associated wit an element T k defined by : µ k (x) = λ k j (x) ont k and 0 oterwise j=1..n+1 Te function µ k is zero outside of T k and on te boundary and positive in te interior of T k. Let w i (x) be a function defined by w i (x) =λ k i (x) onalltet k wic contain te vertex i and 0 oterwise Let us put : V = { j v j w j (x)+ k b k µ k (x) : v j,b k R n } (19) tat is te set of (continuous) functions aving values in R n and being te sum of a continuous piecewise affine function and a linear combination of bubbles. We sall remark tat dimv =(Ns+ Ne)n. Let Q = { j p j w j (x) : p j R} (20) tat is a set of piecewise affine and continuous functions (usual P 1 element). We ave : dim Q =Ns Figure 4.3:Position of te degrees of freedom in te P 1 bubble/p 1 element Let us put J = {v V :(.v,q )=0 q Q } (21) J o = {v J : v Γ =0}. (22) It is easy to sow tat all te constraints (except one) in (21) are independent. In fact if N is te dimension of V te constraints wic define J are of te form BV =0wereV are te values of v at te nodes. So if we sow tat

91 94 FINITE ELEMENT METHODS FOR FLUIDS (v, q )=0 v V q = constant tat proves tat Ker B T is of dimension 1 and so te ImB is of dimension N 1, i.e.b is of rank N 1 wic is to say tat all te constraints are independent except one (te pressure is defined up to a constant). Let e i be te it vector in te cartesian system. Let us take b k = δ k,i e i, v j =0andv in (19). Ten 0=(v, q )= q area(t i ) Ti x i (n +1) q = constant So we ave (n = 2 or 3): dimj =(n 1)(Ns+ Ne)n Ns+1=Ns+ nne + 1 (23) We sall prove tat tis element leads to an error of 0() in te H 1 norm for te velocity. Wereas, witout knowing a basis (local in x) one cannot solve (13) and one as to use duality for te constraints. A basis, local in x, forj is not known (by local in x, we mean tat eac basis function v i as a support around q i and it is zero far from q i ). Remark We can replace te bubble by a function zero on T k piecewise affine on 3 internal triangles of eac triangle obtained by dividing it at its center ; we obtain te same convergence result. Figure 4.4:Position of te degrees of freedom in te element P 1 /P P1 iso P2/P1 element (Bercovier-Pironneau [26]) We construct a triangulation T /2 from te triangle T in te following manner : Eac element is subdivided into s sub-elements (s = 4 for triangles and 8 for tetraedra) by te midpoints of te sides. We put V = {v C 0 (Ω ) n : v T k (P 1 ) n T k T } (24) 2 Q = {q C 0 (Ω ):q T k P 1 T k T }. (25) We construct J o by (21)-(22) Knowing tat te number of sides of a triangulation is equal to Ne+Ns 1 if n=2 (Ns+Nf-Ne-1 if n=3) we get, in 2-D : dimj =2Ns+2(Ns+ Ne 1) Ns+1=3Ns+2Ne 1 (26)

92 INTRODUCTION 95 Figure 4.5 : Degrees of freedom in te P 1 isop 2 /P 1 element. Te independence of te constraints is sown in te following way : We take v V wit 0 on all nodes (vertices and midpoints of sides) except at te middle node q l of te side a l, intersection of te triangles T k1 and T k2. Ten : 0=(v, q )=v (q l )[ q Tkj area(t kj )] j=1,2 Let q l1 and q l2 be te values of q at te two ends of a l ;ifwetakev (q l )= q l1 q l2 ten te above constraint gives q l1 = q l2. So te q wic satisfy te constraints for all v are constant. Tis element also gives an error of 0() for te H 1 norm. We know ow to construct a basis wit zero divergence usable for tis element (Hect [109]) P2/P1 element (Hood-Taylor [113]) V = {v C 0 (Ω ) n : v Tk (P 2 ) n k} (27) Q = {q C 0 (Ω ): q Tk P 1 k} (28) and of course (21)-(22). Figure 4.6 : Position of te degrees of freedom in te P 2 /P 1 element. Tis element is 0( 2 )forteh 1 error and as te same dimension (26) but it gives matrices wit bandwidt larger tan (24)-(25). Te proof of te independence constraints is te same as above. Te dimension of J is terefore: dimj =2(Ns+ Ne 1) + 2Ns Ns+1=3Ns+2Ne 1 Construction of a zero divergence basis : We know ow to construct a basis wit zero divergence usable for tis element (Hect [109]) ; figure 4.7 gives te non zero directions at te nodes of eac basis function wit zero divergence, as well as its support (set of points were it is non zero).

93 96 FINITE ELEMENT METHODS FOR FLUIDS Figure 4.7: Basis wit zero divergence for te P 2 /P 1 element. For te 3 families of functions, te notations are te following : a denotes aside,t a triangle and q a vertex. We ave drawn te triangles on wic te function is non zero and te directions of te vectors, values of tese functions at te nodes, wen tey are non zero. Since all constraints but one in J are independent ; te dimension of J is dimj = dimv dimq =2Ns+2Na Ns+1=2Ns+ Ne+ Na te last equality being a consequence of Euler s identity. We sall define tree families of vectors : -2Nsvectors, eac wit 0 on all te nodes (middle of te sides and vertices) except a vertex q were te vector is arbitrary. - Ne vectors v k, eac aving value 0 on all te nodes except on te 3 mid-points of te sides of a triangle were v k = 1..3 λ ja j t wit, for example : λ 1 =1, λ 2 = ( w3,a 1 t ) ( w 3,a 2 t ), λ 3 = ( w2,a 1 t ) ( w 3,a 3 t ) were a t denotes a unitary vector tangent to te side a and w i te basis function P 2 associated to te vertex q i. - Na vectors v non-zero only on a side a andonasidea 1 and a 2 of te 2 triangles wic contain a. If a n denotes te unitary vector normal to a we could take, for example v = λa n + µa 1 t + νa 2 t wit : λ = ( w1,a n ) ( w 2,a t ), µ = ( w3,a n ) ( w 3,a 1 t ), ν = ( w4,a n ) ( w 4,a 2 t ) We leave it to te reader to verify tat te vectors are independent and wit zero divergence in te sense of (21)(27)(28) P1 nonconforming/p0 element (Crouzeix-Raviart [60]) V = {v continuous at te middle of te sides (faces) v Tk (P 1 ) n } (29) and (21)-(22). Q = {q : q Tk P 0 } (30)

94 INTRODUCTION 97 Figure 4.8:Position of degrees of freedom for te element P 1 nonconforming /P 0. Tis element is 0() in te H 1 norm ; it is nonconforming because v is discontinuous at te interfaces of te elements. Te dimension of V is equal to twice te number of sides (resp. faces). We prove te independence of te constraints by taking v perpendicular to a side and zero at all te oter nodes: 0=(.v,q )= side q v.ndγ = lengt(side) (q Tk1 q Tk2 ) In R 2 we ave q = constant. dimj = Ne+2Ns 1 (31) We know ow to construct a local basis wit zero divergence for tis element. In dimension 2 we define two families of basis vectors of J, one wit indices on te middle of te sides q ij : v ij (q kl )=(q i q j ) δ ikδ jl q i q j q kl middle of te side {q k,q l } Γ (32) and te oter wit indices on te internal vertices q l v l (q ij )=δ li n(q ij ) q i q j q ij middle of {q i q j } (33) were n is normal to q i q j in te direct sense. Figure 4.9: Zero divergence basis for te P 1 nonconforming P 0 element. We remark tat for te two families we ave : (.v,q )= k q Tk T k v.ndx =0 q Q (34) because eac integral is zero. Moreover if Na denotes te number of sides tere are Na + Ns basis functions in tis construction ; tat number is also equal to 2Ns+ Ne 1 wic is exactly te dimension of J (cf. (31)); so we ave a basis of J.

95 98 FINITE ELEMENT METHODS FOR FLUIDS In 3-D a similar construction is possible (Hect [109]) wit te middle of te faces and te vertices but te functions tus obtained are no longer independent; one must remove all te functions v l of a maximal tree witout cycle made of edges P2bubble/P1 discontinuous element. (Fortin [80]) V = {v C 0 (Ω ) n : v (x) =w + l b l µ l (x), w Tl (P k+1 ) n, b l R n } (35) Q = {q : q T k P k } (q is not continuous), (36) k =0 or 1. Wit (21)-(22) tis element is 0( 2 )innormh 1 if k =1and 0() if k = 0. Since te constraints are independent (except one as usual) we can easily deduce te dimension of J 0. Te construction of a basis wit zero divergence for tis element can also be found in [15]. We note tat if we remove te bubbles and if k = 1 te element becomes 0(). Figure 4.10: Position of degrees of freedom for te element P 2 bubble/p 1 discontinuous. Remark One can construct witout difficulty a family of elements wit quadrilaterals by replacing in te formulae P m by Q m, te space of polynomials of degree m wit respect to eac of its variables. 3.6 Comparison of elements: Te list below is only a small subset of possible elements tfor te solution of te Stokes problem. We ave only given te elements wic are most often used. Te coice of an element is based on te tree usual criteria : - memory capacity, - precision, - programming facility. Even wit recent developments in computer tecnology te memory capacity is still te determining factor and so we ave given te elements in te order of increasing memory requirement in 3D for te tetraedra ; but tat could cange for computers wit 256 Mwords like te CRAY 2 wic could andle elements wic are now considered to be too costly! 4. RESOLUTION OF THE LINEAR SYSTEMS If we know a basis of J o and of J it is enoug to write u in tat basis, construct A and G by (17)-(18) and solve (16). Tis is a simple and efficient metod but relatively costly in memory because A is a big band matrix.

96 INTRODUCTION 99 To optimize te memory we store only te non zero elements of A in a one dimensional array again denoted by A. I and J are integer arrays were te indices of rows and columns of non zero elements are stored. m =0 Loop on i and j If a ij 0,do m = m +1, A(m) = a ij,i(m)=i,j(m)=j end of loop. m Max = m. Tis metod is well suited particularly to te solution of te linear system AU = G by te conjugate gradient metod (see Capter 2) because it needs only te following operations : given U calculate V AU; now tis operation is done easily wit array A, I and J : To get V (i) one proceeds as follows : Initialize V = 0. For m =1.. m Max do V (I(m)) = A(m) U(J(m)) + V (I(m)) 4.1. Resolution of a saddle point problem by te conjugate gradient metod. All te examples given for J o are of te type J o = {v V o :(.v,q )=0 q Q } (36) were V o is te space of functions of V zero on te boundaries. An equivalent saddle point problem associated wit (14) is min max{ 1 u u V Γ o p i 2 α(u,u )+ 1 2 ν( u, u ) (f,u )+ p i (.u,q i )} (37) 1..M were {q i } M is a basis of Q. Teorem 4 Wen J o is given by (36) and is non empty wit dimv = N,

97 100 FINITE ELEMENT METHODS FOR FLUIDS dimq = M and V o H 1 0 (Ω)n ( Ω is polygonal), te discretised Stokes problem (13) is equivalent to (37). In addition, (37) is equivalent to α(u,v )+ν( u, v )+( p,v )=(f,v ) v V o (38) (.u,q )=0 q Q (39) u u Γ V o, p Q Tese are te necessary and sufficient conditions for u to be a solution of (13) because tey are te first order optimality conditions for (14). Proof Problem (13) is equivalent to (14) wic is a problem wit quadratic criteria and linear constraints. Te min-max teorem applies and te conditions for te qualification of te constraints are verified (Cf. Ciarlet [56], Luenberger [162] Polak [193] for example). Te q i are te M Lagrangian multipliers wit M constraints in (36). Te equivalence of (37) wit (38)-(39) follows from te argument tat if L is te criteria in (37) L u i =0, L p i = 0 (40) at te optimum, were u i are te components of u expressed in te basis of V o. Tese are te necessary and sufficient conditions for {u,p } to be te solution of (37) because L is strictly convex on V o in u and linear in p i. Remarks. 1. (37)(38) can be found easily from (1); tis formulation was used in engineering well before it was justified. 2. Witout te inf-sup condition (79), tere is no unicity of {p i } i. Proposition 1 If {v i } N is a basis of V o and {q j } M a basis of Q, problem (38)-(39) is equivalent to a linear system ( A B B T 0 )( ) U = P ( ) G 0 (41) wit U being te coefficient of u expressed on te basis {u i } and P being te coefficients of p expressed on te basis {q i }. A ij = α(v i,v j )+ν( v i, v j ), i,j =1..N (42) B ij =( q i,v j ) i =1..M, j =1..N (43) G j =(f,v j ) α(u Γ,v j ) ν( u Γ, v j ) j =1..N (44) Proof

98 We note tat if u is zero on Γ we ave INTRODUCTION 101 (.u,q )= (u, q ) (46) Ten we decompose u and p on teir basis and take v = v j and q = q i in (38) and (39). Proposition 2 Problem (41) is equivalent to (B t A 1 B)P = B t A 1 G, U = A 1 BP + A 1 G (47) Proof It is sufficient to eliminate U from (41) wit te first equations. Tis can be done because A is invertible as sown in (48) : U t AU = α(u,u )+ν( u, u )=0 u =0 U = 0 (48) Algoritm 2 (solving (47) by te Conjugate Gradient Metod) 0. Initialisation: coose p 0 (=0ifnoinitialguessknown). CooseC R N N positive definite, coose ɛ<<1, nmax>>1; (p n,q n, z, g n are in te pressure space wile u n,v are in te velocity space) 1. Solve solve Au 0 = G Bp 0 (49) put g 0 = B t u 0 (= B t A 1 G) n =0. (50) Av = Bq n (51) and Cz = B t v (= B t A 1 Bq n ) (52) 2. Put set ρ = g n 2 C <q n,z > C (53) p n+1 = p n ρq n (54) u n+1 = u n ρv (so tat Au n+1 = Bp n+1 + G) (55) g n+1 = g n + ρz (56) 3. IF ( g n+1 <ɛ)or(n >nmax)thenstop ELSEPut γ = gn+1 2 c g n 2 c (57)

99 102 FINITE ELEMENT METHODS FOR FLUIDS andreturnto1witn:=n+1. q n+1 = g n+1 + γq n (58) Coice of te preconditioning. Evidently te steps (51) and (56) are costly. We note tat (51) can be decomposed into n Diriclet sub-problems for eac of te n components of u. Tis is a fundamental advantage of te algoritm because all te manipulation is done only on a matrix not bigger tan a scalar Laplacian matrix. (56) is also a discrete linear system. A good coice for C (Benque and al [20]) can be obtained from te Neumann problem on Ω for te operator, wen we use for Q an admissible approximation of H 1 (Ω) (Cf. 3.3, 3.2 and 3.1): ( φ, q )=(l, q ) q Q /R (59) C ij = ij ( q i, q j ) i, j (boundary points included). (60) Tis coice is based on te following observation : from (1) we deduce p p =.f in Ω, = f.n + ν u.n n So in te case ν 1 tere underlies a Neumann problem. Tis preconditioning can be improved as sown by Caouet-Cabard [47] by taking instead of : C (νi 1 α 1 ) 1 wit Neumann conditions on te boundary were I is te operator associated wit te matrix (w i,w j )werew i is te canonical basis function associated wit te vertex q i. In fact, te operator B T A 1 B is a discretisation of.(αi ν ) 1. Now if α =0, it is te identity and if ν =0, it is Resolution of te saddle point problem by penalization Te matrix of problem (41) is not positive definite. Rater let us consider ( )( ) ( ) A B U G B t = (62) ɛi P 0 We can eliminate P wit te last equations and tere remains : (A + 1 ɛ BBt )U = G (63)

100 INTRODUCTION 103 We solve tis system by a standard metod since te matrix A + ɛ 1 BB t is symmetric positive definite, (but is still a big matrix). Tis metod is easy to program because it can be sown tat (63) is equivalent to α(u,v )+ν( u, v )+ 1 ɛ v V o (.u,q i )(.v,q i )=(f,v ) (64) 1..M u u Γ V o It is useful to replace ɛ by ɛ/β i,β i being te area of te support of q i ;tis corresponds to a penalization wit te coefficients β i in te diagonal of I in (62). Oter penalizations ave been proposed. A clever one (Huges et al [118]) is to penalize by te equation for u wit v = q : is replaced by (.u,q )=0 (.u,q )+ɛ[( p, q ) ( u, q ) (f, q )] = 0 but te term ( u, q ) can be dropped because.u = 0 (see (129) below). 5. ERROR ESTIMATION (Te presentation of tis part follows closely Girault-Raviart [93] to wic te reader is referred for a more detailed description of te convergence of finite element scemes for te Stokes problem.) 5.1. Te abstract set up Let us consider te following abstract problem Find u J were suc tat J = {u V : b(u,q )=0 q Q } (65) a(u,v )=(f,v ) v J (66) wit te following notations : - V sub-space of V, V Hilbert space, dim V < + - Q sub-space of Q, Q Hilbert space, dim Q < + - a(, ) continuous bilinear of V V Rwit a(v, v) λ v 2 V v V (V-ellipticity) (67) - b(, ) continuous bilinear of V Q R suc tat

101 104 FINITE ELEMENT METHODS FOR FLUIDS 0 <β inf sup b(v, q) (68) q Q v V v V q Q - (, ) duality product between V and (its dual) V. We remark tat tese conditions are satisfied by te Stokes problem wit a(u, v) =α(u, v)+ν( u, v) b(v, q) = (.v, q) V = H 1 0 (Ω)n Q = L 2 (Ω)/R λ = inf{α, ν}. (, ) = scalar product of L 2 (Ω). β =1/C were C is suc tat for all q tere is a u suc tat.u = q and u 0 C q 0 Remark: Te notation in tis formulation is somewat misleading ; in fact J o and V o are now denoted J and V! Te following proposition compares (66) wit te continuous problem : a(u, v) =(f,v) v J (69) u J = {v V : b(v, q) =0 q Q} (70) 5.2. General teorems Proposition 3 If u and u are solutions of (65)-(66) and (69)-(70) respectively, and if V is non empty, ten u u V C(λ)[ inf u v V + inf p q Q ] (71) v J q Q Proof We can sow (Brezzi [39]) tanks to (68) tat (69)-(70) is equivalent to te following problem : find {u, p} V Q suc tat a(u, v)+b(v, p) =(f,v) v V (72) b(u, q) =0 q Q (73) By replacing v by v J in (72) and by subtracting (66) from (72) we get

102 INTRODUCTION 105 a(u u,v )= b(v,p p ) v J (74) Or, by taking into account (65) a(u u,v )= b(v,p q ) v J q Q (75) and for all w J a(w u,v )=a(w u, v ) b(v,p q ) (76) So (cf (67) and te continuity of a and b) λ w u V a w u V + b p q Q (77) wic implies u u V u w V + w u V (1 + a λ 1 ) w u V + λ 1 b p q Q w J q Q (78) Proposition 4 (Brezzi [39]) ten if inf q Q b(v,q ) sup β > 0 (79) v V v V q Q inf u v V (1 + b v J β ) inf u v V (80) v V Proof Letussowfirsttatifz is in te ortogonal space of J in V (denoted J )weave β b(z,q ) q Q z J z q. (81) Tis will be a direct consequence of (79) if we sow te following : z J q suc tat z solution of sup { b(z,q ) z V z q } In fact, te optimality conditions of te problem Sup are : b(y,q )+λ(z,y )=0 y V wit λ = z 3 ; since te conditions are necessary and sufficient for z to be a solution, it is sufficient to define q as a solution of b(y,q )=(z,y ) y V

103 106 FINITE ELEMENT METHODS FOR FLUIDS wic is possible because if B te operator defined by te above equation is not of maximal rank ten by taking q in te kernel, we see tat β = 0 in (79). Now let v V be arbitrary. We can write v = z + w were w J and z J. Let us sow tat z C u v for all u in J : Since z J we ave b(z,q )=b(v,q )= b(u v,q ) q Q Ten we ave from(81) z V 1 b(z,q ) β b β 1 u v V (82) q Q so w J, v V suc tat u w V u v V + z V (1 + b β ) u v V (83) Teorem 5 Let u and u be te solutions of (65)-(66) and (69)-(70) If we ave (67), (68) and (79) ten u u V C[ inf u v V + inf p q Q ] (84) v V q Q Proof Te proof follows directly from te 2 previous propositions. Teorem 6 If p and p are te pressures associated wit u and u (Lagrange multiplier of te constraints),we also ave p p Q C(λ)[ inf u v V + inf p q Q ] (85) v V q Q Proof : From (74) and its corresponding discrete equation, we deduce tat b(v,p q )=a(u u,v )+b(v,p q ) v V q Q (86) So from (79) p q Q β 1 sup v V { [a(u u,v )+b(v,p q )]/ v V } We obtain : β 1 ( a u u V + b p q Q ).

104 INTRODUCTION 107 p p Q β 1 a u u V +( b β 1 +1) p q Q (87) To study te generalized problem (64), we consider te following abstract problem were c(p, q) =(Cp,q) is a symmetric Q elliptic continuous bilinear form : Find u ɛ V suc tat a(u ɛ,v )+ɛ 1 (C 1 BT u ɛ,b T v )=(f,v ) (88) were C and B are defined by te relations (C p,q )=c(p,q ) p,q Q (89) (B T u,q )=b(u,q ) u V, q Q (90) Teorem 7 Under te ypotesis of teorem 5 and if β is independent of, we ave u ɛ u V + p ɛ p Q ɛc f 0 (91) Proof We create an asymptotic expansion of u ɛ and of pɛ So (88) implies tat u ɛ = u0 + ɛu (92) p ɛ = p0 + ɛp (C 1 BT u 0,B T v ) = 0 (93) a(u 0,v )+(C 1 BT u 1,B T v )=(f,v ), (94) a(u 1,v )+(C 1 BT u 2,B T v )=0... v V (95) By putting q = C 1 BT v,p 0 = C 1 BT u1, we see tat u0 is solution of te nonpenalized problem (66). We easily sow tat te u i are bounded independently of, and ence te teorem olds. Remark: If te inf sup condition is not verified (88) is still sensibles but te error due to penalization is 0( ɛ) (cf. Bercovier [25]) Verification of te inf-sup condition (79). Finally for all te examples in section 3 if remains only to sow te sufficient condition (79); we follow Nicolaides et al [181].

105 108 FINITE ELEMENT METHODS FOR FLUIDS We sall do tis for te element P 1 bubble/p 1. We recall te notation : V represents H 1 0 (Ω)n, Q=L 2 (Ω)/R, V and Q are approximations of te spaces (P 1 bubble, zero on te boundary, for V and P 1 for Q, bot are assumed conforming). Te lemma 1 below does not depend on te particular form of V and Q : Lemma 1 Te inf-sup condition (79) is equivalent to te existence of a Π suc tat (C is independent of ): Π : V V (96) (v Π v, q )=0 q Q, v V (97) Π v 1 C v 1 (98) Proof If Π exists, we ave (v, q ) (Π v, q ) (v, q ) sup sup =sup (99) v V v 1 v V Π v 1 v V Π v 1 C 1 sup v V (v, q ) v 1 C 1 β q o (β = C 1 β) (100) Conversely, let Π be an element in te ortogonal of J wic satisfies ten (cf. (81)) (Π v, q )=(v, q ) q Q (101) because Π v 1 β 1 (v, q ) q 0 β 1 v 1 (102) (v, q) = (.v, q).v 0 q 0 C v 1 q 0. (103) Teorem 8 If te triangulation is regular (no angle tends to 0 or π wen tends to 0), ten te P 1 bubble/p 1 element satisfies te inf-sup condition wit β independent of. So we ave te following error estimate : u u 1 + p p o C u 2. (104) Proof

106 INTRODUCTION 109 To simplify we assume n = 2. and we sall apply lemma 1. Let v be arbitrary in H0 1(Ω)n. Let Π v be te interpolate defined by its values of te vertices q i of te triangulation. Π v(q i )=v(q i ) (106) - Here tere is a tecnical difficulty because v may not be continuous and one must give a meaning to v(q i ). We refer te reader to [12, lemma II.4.1]. Neverteless te values at te centre of te elements are cosen suc tat Π vdx = T k vdx T k T k (107) Ten we ave (Π v v, q )= [ (Π v v)dx] q Tk = 0 k T k (108) and we easily sow tat Π v 1 C v 1. (109) Remark 1. Tis proof can be applied immediately to te case were te bubble is replaced by te element cut into tree by te centre of a triangle. 2. It is not always easy to find a Π wit (96)(98); anoter metod is to prove (79) directly on a small domain made of a few elements and ten sow (Boland-Nicolaides [33]) tat it implies (79) on te wole domain. 6. OTHER BOUNDARY CONDITIONS 6.1 Boundary Conditions witout friction Te friction on te surface is given by ν( u + u T )n pn. A no-friction condition on Γ 1 is written as ν( u + u T )n pn =0onΓ 1 Ω; (110) Tis condition may be useful if Γ 1 is an artificial boundary (suc as te exit of a canal) or a free surface. Let us consider te Stokes problem wit (110) and u =0on Ω Γ 1 (111) To include (110) in te variational formulation of te Stokes problem, we ave to consider : α(u, v)+ ν 2 ( u + ut, v + v T )=(f,v) v J 01 (Ω) (112)

107 110 FINITE ELEMENT METHODS FOR FLUIDS Wen.u =0weave: and So (112) implies u u Γ J 01 (Ω) = {v J(Ω) : v =0on Ω Γ 1 } (113) (. u T ) i = u j,ij = 0 (114) ( u + u T, v T )=( u + u T, v) (115) α(u, v) ν( u, v)+ ν( u + u T )nv =(f,v) Γ 1 v J 01 (Ω) (116) One can prove witout difficulty tat (116) implies (1) and (111). Te finite element approximation of (112)-(113) and te solution of te linear system is done as before. However in te saddle point algoritm te knowledge of te pressure does not decouple te velocity components (cf (51) in Algoritm 2). 6.2 Slipping Boundary Conditions. It is also interesting to solve te Stokes problem witout tangential friction and a slip condition on te velocity : τ.[ν( u+ u T )n pn] =0, u.n =0onΓ 1 Ω; u = u Γ on Ω Γ 1 (117) were τ is a vector tangential to Ω (in 3D tere are 2 conditions and 2 tangent vectors). We treat te problem in te same manner by solving. α(u, v)+ ν 2 ( u + ut, v + v T )=(f,v) v J 0n1 (Ω) (118) u u Γ J 0n1 (Ω) = {v J(Ω) : v =0on Ω Γ 1 ; v.n Γ1 =0} (119) Here also te previous tecniques can be applied ; owever tere are some additional problems in te approximation of te normal (cf. Verfürt [233]). Some turbulence models replace te pysical no slip condition on te surface of a solid by te slip condition : au.τ +( u n ).τ = b, u.n =0onΓ 1 Ω (120) were τ is a tangent vector to Γ 1 (two vectors in 3D). We can eiter work wit te standard variational formulation, tat is :

108 INTRODUCTION 111 α(u, v)+ν( u, v)+ au.v =(f,v)+ bτ.v v J 0n1 (Ω) (121) Γ 1 Γ 1 u u Γ J 0n1 (Ω) = {v J(Ω) : v =0on Ω Γ 1 ; v.n Γ1 =0} (122) or we can also reduce it to te previous case by noting tat n. u/ τ = u.τ/r, so tat (120) is equivalent to : (a = a+1/r were R is te radius of curvature): a u.τ + τ( u + u T )n = b, u.n =0onΓ 1 Ω a condition wic can be obtained wit te variational formulation α(u, v)+ ν 2 ( u + ut, v + v T )+ au.v =(f,v)+ bv.τ (123) Γ 1 Γ 1 v J 0n1 (Ω) u u Γ J 0n1 (Ω). Finally, we note tat ( q, v) =0 q H 1 (Ω).v =0and v.n Γ =0 and so rater tan working wit J 0n1 wic is difficult to discretise, we can work wit : J 0n1(Ω) = {v H 1 (Ω) : ( q, v) =0 q L 2 (Ω); v Ω Γ1 =0} were te normals do not appear explicitly; but condition (120) is satisfied in te weak sense only. Parès [188] as sown tat tis is neverteless a good metod Boundary Conditions on te pressure and on te vorticity To simplify te presentation, let us assume tat Ω and Ω aresimply connected. Let us consider te Stokes problem wit velocity equal to u Γ on a part of te boundary Γ 2 and wit prescribed pressure p 0 on te complementary part Γ 1 of Ω. If te tangential components of te velocity are prescribed on Γ 1 (u n = u Γ n) ten te problem is well posed and we can solve it by considering te following variational formulation : α(u, v)+ν( u, v)+ν(.u,.v) =(f, v)+ p 0 v.n Γ 1 v J 0 n1 (Ω) (124) u u Γ J 0 n1 (Ω) = {v J(Ω) : v =0onΓ 2 = Ω Γ 1 ; v n Γ1 =0} (125)

109 112 FINITE ELEMENT METHODS FOR FLUIDS Te finite element approximation of tis space is carried out exactly as in capter 2, 5 If we solve α(u, v)+ν( u, v)+ν(.u,.v) =(f, v)+ d.v v J 0n1 (Ω) (126) Γ 1 u u Γ J 0n1 (Ω) = {v J(Ω) : v =0onΓ 2 = Ω Γ 1 ; v.n Γ1 =0} (127) ten we compute te solution of te Stokes problem wit, on Γ 1 : p u.n = u Γ.n, n u = d, = f.n.d, (128) n were u Γ and d are arbitrary but suc tat d.n =0. Finally, V. Girault [91] as sown tat te following problem α(u, v)+ν( u, v)+ν(.u,.v) =(f,v) v J 0n(Ω) u u Γ J 0n (Ω) = {v J(Ω) : v.n Γ =0;n. v Γ =0}, yields p u.n = u Γ.n, n. u =0, = f.n on Γ n Naturally, we can mix (125),(127) and te above condition on parts of Γ. To sum up, we can treat a large number of boundary conditions but tere are certain compatibilities needed. Tus it does not seem possible to ave on te same boundary u.n and p prescribed. Also it is not clear weter it is permissible to give all te components of u on a part of te boundary in 3D. For more details, see Begue et al.[18]. 6.4 Mixing different boundary conditions. If we want to mix te boundary condition studied in 6.2 wit tose of 6.3 we are confronted wit te coice of te variational formulation: we ave to use ( u, v) near to a boundary and ( u, v) onoter parts of te boundary! Te following formulae are satisfied for all regular u, v and all Ω aving a polygonal boundary : (.u,.v)+( u, v) =( u, v) [v u.n v.n.u] (129) Γ 1 2 ( u+ ut, v+ v T )=(.u,.v)+( u, v)+ [v u.n v.n.u] (130) Γ

110 INTRODUCTION 113 We note tat tese formulae are still true if u, v are continuous and piecewise polygonal on a triangulation because te boundary integrals contain only te tangential derivatives of u and v or te jump of te tangential derivatives of piecewise continuous polynomial functions and tose are zero at te discontinuous interfaces. Tus, (126), for example, is identical to α(u, v)+ν( u, v) ν [v u.n v.n.u + u v.n u.n.v] 2 Γ (131) =(f,v)+ d.v Γ 1 v J 0n1 (Ω) u u Γ J 0n1 (Ω) But te replacement of ( u + u T, v + v T )by( u, v) produces an additional term (.u,.v), wic is zero in te continuous case and non zero in te discrete case. Tere are furter simplifications. Suppose, for example, tat we ave to solve te following problem in Ω wit te simply connected polynomial boundary Γ = Γ 1 Γ 2 Γ 3 Γ 4. αu ν u + p = f,.u = 0 in Ω (132) u Γ1 = u Γ (133) u.n = u Γ.n; au.τ + τ. u n = b on Γ 2 (134) u n = u Γ n; p = p 0 on Γ 3 (135) u.n = u Γ.n; n u = d on Γ 4 (136) Ten we could solve : α(u,v )+ν( u, v )+a u.v (137) Γ 2 =(f,v )+ bv.τ + Γ 2 p 0 v.n Γ 3 d.v Γ 4 + (v u Γ.n v.n.u Γ ) Γ 3 Γ 4 v J u u Γ J = {v J : v Γ1 =0, v n Γ3 =0, v.n Γ2 Γ 4 =0} In fact, te extra integral is a function of u Γ and not of u because it contains only tangential derivatives of u.n on Γ 4 and on Γ 3 it contains only tangential derivatives of tangential components of u (in fact.u n u/ n ). Remark Since tese formulations apply only to polygonal domains one sould be cautious of te Babuska paradox for curved boundaries;

111 114 FINITE ELEMENT METHODS FOR FLUIDS te solutions obtained on polygonal domains may not converge to te rigt solution wen te polygon converges to te curved boundary. Figure 4.11 : Iso values of pressure on a spere in a Stokes flow. On te first figure (left) te computation was done wit te P1/P1 invalid element (oscillations can be seen) on te second figure (rigt) te P1isoP2/P1 element was used (Courtesy of AMD-BA).

112 INTRODUCTION 115 CHAPTER 5 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 1. INTRODUCTION Te Navier-Stokes equations : u, t +u u + p ν u = f (1).u =0 (2) govern Newtonian incompressible flows ; u and p are te velocity and pressure. Tese equations are to be integrated over a domain Ω occupied by te fluid, during an interval of time ]0,T[. Te data are : - te external forces f, -teviscosityν (or te Reynolds number Re), - te initial conditions at t =0: u 0, - te boundary conditions : u Γ, for example. Tese equations are particularly difficult to integrate for typical application (ig Reynolds numbers) (small ν ere) because of boundary layers and turbulence. Even te matematical study of (1)-(2) is not complete ; te uniqueness of te solution is still an open problem in 3-dimensions. Tere any many applications of te incompressible Navier-Stokes equations ; for example : - eat transfer problems (reactors,boilers...) - aerodynamics of veicles (cars, trains, airplanes) - aerodynamics inside motors (nozzles, combustion camber...) - meteorology, marine currents and ydrology. Many tests problems ave been devised to evaluate and compare numerical metods. Let us mention a few: a ) Te cavity problem Te fluid is driven orizontally wit velocity u by te upper surface of a cavity of size a (cf. figure 1). Te Reynolds number is defined by : R = ua/ν. (cf. Tomasset [229]). Te domain of computation Ω is bidimensional.

113 116 FINITE ELEMENT METHODS FOR FLUIDS Figure 5.1 b ) Te backward step : Te parabolic velocity profile (Poiseuille flow) at te entrance and at te exit section are given, suc tat te flux at te entrance is equal to te flux at te exit. Te domain is bidimensional and te Reynolds number is defined as u l /ν were l is te eigt of te step and u te velocity at te entrance at te centre of te parabolic profile (cf. Morgan et al [174]): l =3 L =22 a =1 b =1.5. Figure 5.2 Te backward step problem (l =1). c ) Te cylinder problem. Te problem is two dimensional until te Reynolds number rises above approximately 200 at wic point it becomes tridimensional. Te domain Ω is a periodic system of parallel cylinders of infinite lengt (for 3D computations one could take a cylinder lengt equal to 10 times teir diameter ). Te integral, Q, ofu.n at te entrance of te domain is given and is equal to te total flux of te flow. Te Reynolds number is defined by 3Q/2ν (cf. Ronquist-Patera [202]). Figure 5.3 In tis capter, we give some finite element metods for te Navier-Stokes equations wic are a syntesis of te metods in capters 3 and 4. We begin by recalling te known main teoretical results. More matematical details can be found in Lions [153], Ladyzenskaya [138], Temam [228] ; details related to numerical questions can be found in Tomasset [229], Girault-Raviart [93], Temam [228] and Glowinski [95]. 2. EXISTENCE, UNIQUENESS, REGULARITY.

114 2.1. Te variational problem INTRODUCTION 117 We consider te equations (1)-(2), in Q = Ω ]0,T[ were Ω is a regular open bounded set in R n, n = 2 or 3, wit te following boundary conditions : u(x, 0) = u 0 (x) x Ω (3) u(x, t) =u Γ (x) x Γ, t ]0,T[ (4) We embed te variational form, as in te Stokes problem, in te space J o (Ω) : J(Ω) = {v H 1 (Ω) n :.v =0} : (5) J o (Ω) = {u J(Ω) : u Γ =0}. (6) Te problem is to find u L 2 (O, T, J(Ω)) L (O, T, L 2 (Ω)) suc tat (u, t,v)+ν( u, v)+(u u, v) =(f,v), v J o (Ω) (7) u(0) = u 0 (8) u u Γ L 2 (O, T, J o (Ω)) (9) In (7) (, ) denotes te scalar product in L 2 in x as in capter 4 ; te equality is in te L 2 sense in t. Wetakef in L 2 (Q) and u 0,u Γ in J(Ω). We searc for u in L in order to ensure tat te integrals containing u u exist Existence of a solution. Teorem 1 Te problem (7)-(9) as at least one solution. Proof (outline) We follow Lions [153]; te outline of te proof only is given. To simplify, let us assume u Γ =0. Let{w i } i be a dense basis V s of {v D(Ω) n :.v = 0} in H s wit s = n/2; tis basis is defined by : <w i,v >= λ i (w i,v) v V s. (10) -(te λ are arranged in increasing order, te notation <>represents a scalar product in V s and D(Ω)tesetofC functions wit compact support ; if n =2,{w i } i are te eigenvectors of te Stokes problem wit α = ν =1). Let u m be te approximated solution of (7)-(9) in te space generated by te first m solutions of (10) : (u m t,wj )+(u m u m,w j )=(f,w j ) j =1..m (11) u m = v i w i (12) i=1..m

115 118 FINITE ELEMENT METHODS FOR FLUIDS u m (0) = u 0m te component of u o on w m. (13) On multiplying (11) by v j and summing we get t 1 2 um 2 o + ν u m 2 o =(f,u m ) because (v v, v) =0 v J o (Ω). (14) From (14) we deduce easily tat u m 2 o +2ν t o u m (σ) 2 dσ u om 2 o + C f 0,Q (15) wic allows us to assert tat te differential system in {v i } (11) as a solution on [O,T] and tat solution is in a bounded set of L 2 (0,T,J o (Ω)) L (O, T, L 2 (Ω)). Let P m be te projection from J o (Ω) into te sub-space generated by {w 1,..w m }. From (1) we deduce tat u m t = P m f νp m u m P m (u m u m ) (16) P m being a self-adjoint projector, its norm in time as an operator from V s in V s, is less tan or equal to 1. Te mapping ϕ ϕ being continuous from J 0 (Ω) into J 0 (Ω), u is bounded in L 2 (O, T, J 0 (Ω) ). Finally, by Sobolev embeddings, one sows tat u u (J0) C u L p (Ω) n wit p = 2 (1 1 n ) (17) Wic demonstrates tat u m t bounded in L 2 (O, T, J 0 (Ω) ) m. (18) By taking a subsequence suc tat u m converges to u in L 2 (O, T, J o (Ω)) weakly, L (O, T, L 2 (Ω)) weakly*, L 2 (Q) strongly and suc tat u m t u t in L 2 (O, T ; J 0 (Ω) ) weakly, one can pass to te limit in (11) Uniqueness Teorem 2 In two dimensions problem (7)-(9) as a unique solution. Proof Te following relies on an inequality wic olds if Ω is an open bounded set in R 2 :( φ 1 denotes te semi norm in H 1 0 : φ ( φ, φ)1/2 ). 1 1 ϕ L4 (Ω) C ϕ 0,Ω 2 ϕ 1,Ω 2 v H0 1 (Ω) (19) because (w u, w) is ten bounded by w o w 1 u 1.

116 INTRODUCTION 119 Let tere be two solutions for (7)-(9) and let w be teir difference. We deduce from (7) wit v = w : So w is zero. C 1 2 w(t) 2 + ν t o t o w o w 1 u 1 dt ν w 2 dt = t o t o (w u)wdt (20) t w 2 1 dt + C w 2 o u 2 1 dσ Teorem 3 If u 0 1 is small or if u is smoot (u L (O, T ; L 4 (Ω)) ten te solution of (7)-(9) is unique in tree dimensions. Proof : See Lions [153]. Remark It sould be empasized tat teorem 3 says tat if u is smoot, it is unique ; but one cannot in general prove tis type of regularity for te solution constructed in teorem 1 ; so its uniqueness is an open problem. Tere are important reports wic deal wit te study of te possible singularities of te solutions. For instance, it is known tat te singularities are local (Cafarelli et al. [46], Constantin et al. [59]) and we know teir Hausdorff dimension is less tan 1. o 2.4. Regularity of te solution. Tere are two important reasons for studying te regularity of te solutions of (7)-(9) : 1. As we ave seen, te study of uniqueness is based on tis regularity 2. Te error estimates between te calculated solution and exact solution depend on te H p norm of te latter. Unfortunately it is not unusual to ave u H 2 but it is rater difficult to find initial conditions wic give u H 3 (Heywood-Ranacer [111]) so it would appear useless from te point of view of classical error analysis, to approximate te Navier-Stokes equations wit finite elements of degree 2 because tey will not be more accurate tan finite elements of degree 1; owever even if tere is some trut to wat was said, tis conclusion is too asty and in fact one can sow, on linear problems, tat te error decreases wit iger order elements even wen tere are singularities (but it does not decrease as muc as it would if tere were no singularities).. In te case of two dimensions, tere exists general teorems on regularity, for example : Teorem 4 (Lions [153])

117 120 FINITE ELEMENT METHODS FOR FLUIDS If f and f, t are in L 2 (Q) 2,f(.,0) in L 2 (Ω) 2,u o in J 0 (Ω) H 2 (Ω) 2 ten te solution u of (7)-(9) is in L 2 (0,T; H 2 (Ω) 2 ). As in 3 dimensions, as one cannot sow uniqueness, te teorems are more delicate. Let us give an example of a result wic, togeter wit teorem 3 illustrates point 1. Te Hausdorff dimension d of a set O is defined as a limit (if it exists) of log N(ɛ)/log(1/ɛ) wenɛ 0, were N(ɛ) is te minimum number of cubes aving te lengt of teir sides less tan or equal to ɛ and wic cover O. (d =0ifOisapoint,d =1ifO is a curve of finite lengt, d = 2 for a surface... non integer numbers can be found in fractals (Feder [76]) Teorem 5.(Cafarelli et al.[46]) Suppose tat f is in L q (Q) 3,q>5/2, tat.f =0and tat u o.n and u Γ are zero. Let u be a solution of (7)-(9) and S be te set of {x, t} suc tat u L loc (D) for all neigboroods D of {x, t}. Ten te Hausdorff dimension in space-time of S is less tan Beavior at infinity. In general we are interested in te solution of (1)-(4) for large time because in practice te flow does not seem igly dependent upon initial conditions : te flow around a car for instance does not really depend on its acceleration istory. Tere are many ways to forget tese initial conditions for a flow ; ere are two examples : 1. te flow converges to a steady state independent of t ; 2. te flow becomes periodic in time. Moreover, te possibility is not excluded tat a little memory of initial conditions still remains ; tus in 1 te attained stationary state could cange according to initial conditions since te stationary Navier-Stokes equations as many solutions wen te Reynolds number is large (tere is no teorem on uniqueness for ν large [28]). We distinguis oter limiting states: 3. quasi-periodic flow : te Fourier transform t u(x, t) (were x is an arbitrary point of te domain) as a discrete spectrum. 4. Caotic flows wit strange attractors: t u(x, t) as a continuous spectrum and te Poincaré sections (for example te points {u 1 (x, nk),u 2 (x, nk)} n for a given x ) ave dense regions of points filling a complete zone of space (in case 1, te Poincaré sections are reduced to a point wen n is large, in cases of 2 and 3, te points are on a curve).

118 INTRODUCTION 121 In fact, experience sows tat te flows pass troug te 4 regimes in te order 1 to 4 wen te Reynolds number Re increases and te cange of regime takes place at te bifurcation points of te mapping ν u, were u is te stationary solution of (1)-(4). As in practice Re is very large (for us ν is small), 4 dominates. Te following points are under study : a) Weter tere exists attractors, and if so, can we caracterize any of teir properties? (Hausdorff dimension, inertial manifold,...cf. Gidaglia [89], Foias et al.[79] or Bergé et al [29] and te bibliograpy terein). b) Does u(x, t) beave in a stocastic way and if so, by wic law? Can we deduce some equations for average quantities suc as u, u 2, u 2... tis is te problem of turbulence modeling and we sall cover it in a little more detail at te end of te capter. (cf Lesieur [150], for example and te bibliograpy for more details). Here are te main results relating to point a). Consider (7)-(9)wit u Γ =0,findependent of t, and Ω a subset of R 2. Tis system as an attractor wose Hausdorff dimension is between cre 4/3 and CRe 2 were Re = f diam(ω)/ν (cf. Constantin and al.[59], Ruelle [206][207]). Tese results are interesting because tey give an upper bound for te number of points needed to calculate suc flows (tis number is proportional to ν 9/4 ). In tree dimensions, we do not know ow to prove tat (7)-(9) wit te same boundary conditions as an attractor but we know tat if an attractor exists and is (rougly) bounded by M in W 1, ten its dimension is less tan CM 3/4 ν 9/4 (cf.[10]) Euler Equations (ν =0) As ν is small in practical applications, it is important to study te limit ν 0. If ν = 0 te equations (7)-(9) become Euler s equations. For te problem to be meaningful we ave to cange te boundary conditions ; so we consider te following problem : u, t +u u + p = f,.u = 0 (21) u(0) = u 0 (22) u.n Γ = g (23) u(x) =u Γ (x), x Σ={x Γ: u(x).n(x) < 0}. (24) Proposition 1 If Ω is bidimensional and simply connected, if f L 2 (Q), u 0 H 1 (Ω) 2, g = u Γ.n and (u Γ ) i are in H 1/2 (Γ) and te integral of g on Γ is zero, ten (21) - (24) as a unique solution in L 2 (O, T ; H 1 (Ω) 2 ) L (O, T, L 2 (Ω) 2 ).

119 122 FINITE ELEMENT METHODS FOR FLUIDS Proof (sketc ) To simplify, we assume f =0,g =0. Since.u =0tereexistsaψ suc tat : u 1 = ψ, 2,u 2 = ψ, 1. (25) By taking te curl of (21)-(22) and by putting We see tat ω = u 1, 2 u 2, 1 = ψ (26) ω t + u ω = 0 (27) ω(0) = u 0 1, 2 u 0 2, 1 = u 0 (28) ω(x) =ω Γ (x) (derived from g and u Σ ;ereω Γ is zero) if x Σ. (29) If u is continuous, Lipscitz and g is zero, (27)-(28) can be integrated by te metod of caracteristics (Cf. Capter 3) and ω as te same smootness as tat of u 0. So we ave an estimate wic relates ω s to te Holderian norm C 0 of u (Kato [129]) ω C( u + u 0 ) (30) Moreover, from (25)-(26), (23) wit g =0gives Tus from (26), we ave ψ Γ =0, (31) u 1+s C ω s. (32) So it is sufficient to construct a sequence in te following way : u n given ω n+1 calculated by (27)-(29) ω n+1 given u n+1 calculated by (25),(26),(30) and to use te estimates (30) and (32) to pass to te limit. A complete proof can be seen in Kato [129], McGrat[170], Bardos[16]. Remark 1 Intreedimensionsweonlyaveanexistence teorem in te interval ]O, τ[,τ small. In fact, equation (27) becomes ω, t +u ω ω u =0 wic can give terms in e λt were λ are te eigenvalues of u. A number of numerical experiments ave sown te formation of singularities after a finite time but te results ave not yet been confirmed teoretically (Frisc [84], Corin [52], Sulem [225]). Remark 2

120 INTRODUCTION 123 We know little about convergence of te solution of te Navier-Stokes equation towards te solution of Euler s equation wen ν 0. In stationary laminar cases, one can prove existence of boundary layers of te form e y/ ν (y being te distance to te boundary) in te neigborood of te walls (Landau-Lifscitz [141], Rosenead [203]) ; so convergence is in L 2 at most in tose cases. 3. SPATIAL DISCRETISATION 3.1. Generalities. Te idea is simple: we discretise in space by replacing J o (Ω) by J o in (7)-(9): Find u J suc tat (u, t,v )+(u u,v )+ν( u, v )=(f,v ) v J o (33) u (0) = u 0 u u Γ J o (34) If J o is of dimension N and if we construct a basis for J o ten (33)-(34) becomes a nonlinear differential system of N equations: AU + B(U, U)+νCU = G. (35) One could use existing library programs (LINPAK[152] for example) to solve (35) but tey are not efficient, in general, because te special structure of te matrices is not used. Tis metod is known as te Metod of Lines So we sall give two appropriate metods, taking into account te following two remarks : 1 o )Ifν>>1 (33)-(34) tends towards te Stokes problem studied in Capter 4. 2 o )Ifν = 0 (33)-(34) is a non-linear convection problem in J o. In particular in 2-D te convection equation (27) is underlying te system so te tecniques of capter 3 are relevant. As we need a metod wic could adapt to all te values of ν we sallto take te finite elements in te family studied in te Capter 4 and te metod for time discretisation adapted to convection studied in Capter 3. But before tat, let us see a convergence teorem for te approximation (33), (35) Error Estimate. Let us take te framework of Capter 4: J(Ω) is approximated by

121 124 FINITE ELEMENT METHODS FOR FLUIDS J = {v V : (.v,q )=0 q Q } (36) and J o (Ω) is approximated by J o = J H 1 0 (Ω) n. We put : V o = {v V : v Γ =0}. (37) Assume tat {V o,q } is suc tat -tereexistsπ : Vo 2 = H2 (Ω) n H0 1(Ω)n V o suc tat (q,.(v Π v)) = 0 q Q v Π v C v 2,Ω. (38) -tereexistsπ : H 1 (Ω) Q suc tat q π q 0,Ω C q 1,Ω (39) inf sup q Q v V o (.v,q ) q π o v 1 β v 1 C v 0 v V 0 (inverse inequality) Remark : Tese conditions are satisfied by te P 1 + bubble/p 1 element and among oters, te element P 1 iso P 2 /P 1. To simplify, let us assume tat u o =0 and u Γ =0. (40) Teorem 5. (Bernardi-Raugel [31]) a)n=2. If te solution of (7)-(9) is in L 2 (0,T; H 1 0 (Ω)2 ) C 0 (0,T; L q (Ω) 2 ), q>2, ten problem (33)-(34) as only one solution for small satisfying u u L 2 (o,t ;H 1 0 (Ω)) C(ν) u u 0,Q (41) b) n=3. If te norm of ν 1 u in C o (0,T; L 3 (Ω) 3 ) is small and if u L 2 (0,T; H 2 (Ω) 3 ) H 1 (0,T; L 2 (Ω) 3 ) ten (34)-(36) as a unique solution wit te same error estimate as in a). Proof (sketc) Te proof is tecnical, but it is based on a very interesting idea (introduced in Brezzi et al [38]). We define te operator G : u G(u) =u u f. (42)

122 We observe tat te non stationary Stokes problem INTRODUCTION 125 u, t ν u + p = g,.u =0, u(0) = 0, u Γ = 0 (43) defines a linear map L : g u and tat te solution of (7)-(9) is noting but te zero of u u + LG(u) : u solution of (7)-(9) F (u) =u + LG(u) =0. (44) Similarly, for te discrete problem if L is te nonstationary discrete Stokes operator underlying (33)-(34): u solution of (33)-(34) F (u) =u + L G(u )=0. (45) To prove te existence of a solution for te discrete problem, we note tat te derivative of F wit respect to u, F (u ), is a linear map (because F is quadratic) and tat it is an isomorpism if v is near to u. Ten we see tat te solution is also a fixed point of v ϕ (v) =v F (v ) 1 F (v) (46) for all v near to u. For ϕ = I F (v ) 1 F (v) in te neigborood of 0 is a contraction and from a fixed point teorem we ave te existence and uniqueness of te solution. To obtain an estimate, we calculate te following identity : u u = F 1 [F.(u u ) (F (u) F (u )) + F (u)] (47) F 1 [ 1 0 F 1 So, for small we ave : Moreover [F (u ) F (u + θ(u u ))].(u u )dθ]+f (u) [sup F (u ) F (v) u u + F (u) ] v u u C F (u) (48) F (u) =u + L G(u) =u + LG(u)+(L L)G(u) =(L L)G(u) (49) but (L L)G(u) is precisely te error in te resolution of te non-stationary Stokes problem wit g = G(u). Hence te result. For more details, see [4][5] [16,IV 3] Remark 1

123 126 FINITE ELEMENT METHODS FOR FLUIDS Te proof of teorem 5 suggests tat te usual inf sup condition on te conforming finite elements in te error estimate for te non-stationary Stokes problem is necessary for Navier-Stokes equations. Remark 2. Te error estimate of teorem 5 sows tat sould be small wen ν is small. We note tat u u depends also on ν. For example in a boundary layer in e y/ ν, u u 0 ( 1 0 ν 1 e 4y ν dy) 1 2 = O(ν 1 4 ) (50) Since C(ν) 0(ν), we sould take <<ν 5/4. Tesamereasoningleadsto <<ν 5/8 if we want u u 1,Q << 1. We note tat te numbers are less pessimistic tan tose given by te attractor argument. Tis is not contradictory as tere must exist attractor modes of modulus less tan ν 5/8. 4. TIME DISCRETISATION 4.1 Semi-explicit discretisation in t Let us take te simplest explicit sceme studied in Capter 3, tat is te Euler sceme. Sceme 1: Find u n+1 wit u n+1 u Γ J 0 suc tat 1 k (un+1,v )=g(v ) v J 0, (51) g(v )= ν( u n, v )+(f n,v )+ 1 k (un,v ) (u n un,v ) (52) Evidently, k as to satisfy a stability condition of te type As for te Stokes problem (51) is equivalent to k<c(u,, ν) (53) 1 k (un+1,v )+( p,v )=g(v ) v V 0 (54) (.u n+1,q )=0 q Q ; (55) it is still necessary to solve a linear system to get u n+1 ; we use a mass-lumping formula and te P 1 isop 2 /P 1 element, wit quadrature points at te nodes {q i } i of te finer triangulation; σ i denotes te area of support of te P 1 continuous basis functions, v i, associated to node q i ; we replace (54) by

124 INTRODUCTION 127 σ i 3k un+1 i +( p,v i )=g(v i )+ σ i 3k un i 1 k (un,vi ) i (56) (.u n+1,q )=0 q Q u n+1 (x) = u i v i (x). i (57) We can now substitute u in (57) and we obtain 3k ( p,v i )( q,v i )=G(q ) σ i i. (58) i tat is a linear system of a Laplacian type. Te main drawback of tese types of metods is te stability (53). Tere exist scemes wic are almost unconditionally stable suc as te rational Runge-Kutta used by Satofuka [212]: we use To integrate Sceme 2 V, t = F (V ) (59) V n+1 = V n +[2g 1 (g 1,g 3 ) g 3 (g 1,g 1 )](g 3,g 3 ) 1 (60) were (.,.) represents te scalar product in te space of V (t) and g 1 = kf(v n ) g 2 = kf(v n cg 1 ) g 3 = bg 1 +(1 b)g 2 Tis metod is of order 2 if 2bc = 1 and of order 1 if it is not. For a linear equation wit constant coefficients, te metod is stable wen 2bc 1(forte proof, it is sufficient to calculate te amplification coefficient wen it is applied to (59) and F (V )=FV were F is a diagonal matrix). For our problem, we must add a step of spatial projection on zero divergence functions of J 0. Tat is, we solve : σ i 3k un+1 i +( p,v i )= σ i 3k vn+1 i v V 0 (61) (.u n+1,q )=0 q Q (62) Numerical experiments using tis metod for te Navier-Stokes equations discretised wit te P 1 isop 2 /P 1 finite elements can be found in Sing [215] Semi-Implicit and Implicit discretisations

125 128 FINITE ELEMENT METHODS FOR FLUIDS A semi-implicit discretisation of O(k) for (34)-(36) is Sceme 3: 1 k (un+1,v )+ν( u n+1, v )+(u n u n+1,v )=(f n+1,v )+ 1 k (un,v ) (63) AversionO(k 2 ) is easy to construct from te Crank-Nicolson sceme. Tis sceme is very popular because it is conceptually simple and almost fully implicit : yet it is not unconditionally stable and eac iteration requires te solution of a non symmetric linear system. Tus on te cavity problem wit mes and k =0.1 temetodworkswelluntilν 1/500, beyond wic oscillations develop in te flow. Let us consider te following implicit Euler sceme of order O(k) Sceme 4: 1 k (un+1,v )+ν( u n+1, v )+(u n+1 u n+1,v )=(f n+1,v )+ 1 k (un,v ) (64) v J 0. Tis sceme is unconditionally stable but we must solve a non-linear system. 4.3 Solution of te non-linear system (59). Tis non-linear system can be solved by Newton s metod, by a least square metod in H 1 or by te GMRES algoritm : Newton s Metod Te main loop of te algoritm is as follows : For p =1..pMax do: 1. Find δu wit (δu,v ) 1 k + ν( δu, v )+(u n+1,p δu + δu u n+1,p,v ) = (65) 2. Put {(u n+1,p,v ) 1 k + ν( un+1,p, v )+(u n+1,p u n+1,p,v ) (f n+1,v ) (u n,v ) 1 k } v J o δu J o (66)

126 INTRODUCTION 129 u n+1,p+1 = u n+1,p + δu (67) But ere also, experience sows tat a condition between k, and ν is necessary for te stability of te sceme because if ν is very small wit respect to, problem (59) as many branc of solutions or because te convergence conditions for Newton s metod (essien > 0) are not verified. Again for a cavity wit a and k =0.1 we can use tis sceme till Re=1000 approximately. If te term u u is upwinded, one can get a metod wic is unconditionally stable for all mes k ; tis is te object of a few metods studied below Abstract least-squares and θ sceme. Tis metod as been introduced in Capter 2, Paragrap 4.2. It consists ere of taking for u n+1 te solution of te problem min { u w 2 dx : u u Γ J 0, (68) w u Γ J o Ω 1 k (u,v )+ν( u, v )+(w w,v )=(f n+1,v )+ 1 k (un,v ) v J 0 } Te solution of tis problem by a preconditioned conjugate gradient metod, solving tree Stokes problems per iteration (calculation of te step size in te descent directions is analytic because te optimized quantity is of degree 4). Tis metod as good stability qualities. Only symmetric linear systems are solved but it is more costly tan oter metods ; in Bristeau et al [44] a set of numerical experiments can be found. To decrease te cost Glowinski [96] suggested te use of a θ sceme because it is second order accurate for some values of te parameters and because it allows a decoupling between te convection and te diffusion operators. If B is te convection operator u and A is te Stokes operator te θ sceme is: 1 kθ (un+θ u n )+Au n+θ + Bu n = f n+θ 1 (1 2θ)k (un+1 θ u n+θ )+Au n+θ + Bu n+1 θ = f n+1 θ 1 kθ (un+1 u n+1 θ )+Au n+1 + Bu n+1 θ = f n+1 So te least square metod will be used only for te central step wile te two oter steps are generalized Stokes problems. Alternatively, te convection step can be solved by oter metods suc as GMRES Te GMRES algoritm.

127 130 FINITE ELEMENT METHODS FOR FLUIDS Te GMRES algoritm (Generalized Minimal RESidual metod) is a quasi-newtonian metod deviced by Saad [210]. Te aim of quasi-newtonian metods is to solve non-linear systems of equations : F (u) =0, u R N Te iterative procedure used is of te type : u n+1 = u n (J n ) 1 F (u n ) were J n is an approximation of F (u n ). To avoid te calculation of F te following approximation may be used: F (u + δv) F (u) D δ F (u; v) = F (u)v. (69) δ As in te conjugate gradient metod to find te solution v of Jv = F one considers te Krilov spaces : K n = Sp{r 0,Jr 0..., J n 1 r 0 } were r 0 = F Jv 0,v 0 is an approximation of te solution to find. In GMRES te approximated solution v n used in (69) is te solution of min r 0 J(v v 0 ). v K n Te algoritm below generates a quasi-newton sequence wic {u n } opefully (it is only certain in te linear case) will converge to te solution of F (u) =0 in R N : Algoritm (GMRES) : 0. Initialisation: Coose te dimension k of te Krylov space; coose u 0. Coose a tolerance ɛ andanincrementδ, coose a preconditioning matrix S R N N. Put n =0. 1. a. Compute wit (69) rn 1 = S 1 (F n + J n u n ),wn 1 = r1 n / r1 n, were F n = F (u n )andj n v = D δ F (u n ; v). b. For j =2..., k compute rn j and wj n from r j n = S 1 [D δ F (u n ; w j 1 n were n i,j = wit n S 1 D δ F (u n ; w j n) 2. Find u n+1 te solution of w j n = rj n r j n. j 1 ) i=1 n i,j 1w i n]

128 INTRODUCTION 131 min S 1 F (v) 2 k v=u n+ 0 ajwn j = min a 1,a 2,...a k S 1 [F (u n )+ k a j D δ F (u n ; wn)] j 2 j=1 3. If F (u n+1 ) <ɛstop else cange n into n+1. Te implementation of Y. Saad also includes a back-tracking procedure for te case were u n departs too far away from te solution. Te result is a black box for te solution of system of equations wic needs only a subprogram to calculate F (u) givenu; tere is no need to calculate F. Experiments ave sown tat tis implementation is very efficient and robust. 5. DISCRETISATION OF THE TOTAL DERIVATIVE Generalities Let us apply te tecniques developed in capter 3: u, t +u u 1 k (un+1 u n ox n ) we obtain te following sceme : Sceme 5: 1 k (un+1,v )+ν( u n+1, v )=(f n,v )+ 1 k (un ox n,v ) v J 0 (70) were X n (x) is an approximation of te foot of te caracteristic at time nk wic passes troug x at time (n +1)k (X n(x) x un k). We note tat te linear system in (70) is still symmetric and independent of n. IfX n is well cosen, tis sceme is unconditionally stable, and convergent in 0(k + ). Teorem 6 : If X n (Ω) Ω, det( X n ) 1 1+Ck, (71) ten sceme 5 is unconditionally stable. Proof :

129 132 FINITE ELEMENT METHODS FOR FLUIDS Tis depends upon te cange of variable y = X n (x) in te integral φ(x X n (x))dx = φ(y)det( X n n (Ω) ) 1 dy (72) Ω From tis we obtain u n ox n 0,Ω det( X n ) 1,Ω u n 0,Ω (73) In getting a bound from (70) wit v = u n+1 and by using (73) and ypotesis (71), we obtain u n+1 2 ν u n kν u n (k f n 0 +(1+Ck) u n 0 ) u n+1 0 (74) Since u 0 u ν, we obtain u n+1 ν k f n 0 +(1+Ck) u n ν ; (75) or, after summation, u n+1 ν k f i 0 (1 + Ck) n i + u 0 ν(1 + Ck) n (76) i n and ence te result follows wit n T/k. Remark : If u n = ψ, wit ψ P 1 piecewise and continuous and if X n exact solution of is te d dτ X (τ) =u n (X (τ)) (77) ten (71) is satisfied wit C = 0. It is teoretically possible to integrate (77) exactly since X is ten a piecewise straigt line but, in practice, te exact calculation of te integral (u n oxn,v ) is unnecessarily costly ; we use a direct Gauss formula (Pironneau [190]) (u n ox,v ) = i u (X n (ξ i ))v (ξ i )π i (78) or a dual formula (Benqué et al. [22]) (u n ox,v ) = i u (ξ i )v (X n 1 (ξ i ))π i (79) 5.2. Error analysis wit P1 bubble/p1. Definition of te metod : To simplify, we assume tat u Γ =0;wecoose

130 INTRODUCTION 133 J o = {v V o :(.v,q )=0, q Q } (80) V o = {v C o (Ω) : v T P 1, T T } (81) 2 Q = {q C o (Ω) : q T P 1, T T }. (82) were T is a regular triangulation of Ω ( assumed to be polygonal) and T /2 is te triangulation obtained by dividing eac triangle (resp. tetraedron) in 3 (resp. 4) by joining te center of gravity to te vertices (fig. 4.2). We define X n (x) as te extremity of te broken line [χo = x, χ 1,..., χ m+1 = X n (x)] suc tat te χj are te intersections wit te sides (faces), χ j 1 is calculated from χ j in te direction u m (χj ) and te time covered from x to χ m+1 is k : For eac j ρ >0 suc tat χ j+1 = χ j + ρu n (χ j ) Tj T j, (83) χ j+1 χ j u n = k (χj ) Tj (84) j=0..m In practice, we use (78) or (79) wit te 3 (resp. 4) Gauss points inside te triangles, but wit tis integral approximation, te error estimation is an open problem; furtermore (71.b) is not easily satisfied, so we sall define anoter X n. Definition of X n satisfying (71) : Let ψ n be te solution of ( ψ n, w )+(.ψ n,.w )=(u n, w ), w W, (86) ψ n W = {w V : w n Γ =0, w.ndγ =0} Γ i (87) were Γ i are te connected components of Γ. Let us assume for simplicity tat Ω is simply connected. Tis problem as already been studied in paragrap 5 of capter 2 (ere owever w n is zero on Γ instead of w n ; W is non empty because Γ is a polygon. We ave seen tat (86)-(87) as one and only one solution and tat ψ n = u n. Since ψn is piecewise constant and wit zero divergence in te sense of distributions, (83)(84) enables us to compute an analytic solution of (77) if we replace u n by ψ ; ten (71) is satisfied. So we sall define X n as a solution of (77) wit ψn, and we will assume tat (u n o X n,v ) is calculated exactly. Teorem 7 : Assume Ω convex, bounded, polygonal and u W 1, (Ω ]0,T[) 3 C o (0,T; H 2 (Ω) 3 ),p C o (0,T; H 2 (Ω)). Let {u n (x)} n be te solutions in V o of

131 134 FINITE ELEMENT METHODS FOR FLUIDS 1 k (un+1, v )+ν( u n+1, v )=(f n,v )+ 1 k (un n ox,v ), v V o (88) were X nis calculated by (77) wit un replaced by ψn and ψn solution of (86), (87). Ten if u is te exact solution of te incompressible Navier-Stokes equations, (1)(4) wit u Γ =0,we ave ( u n u(., nk) νk (un u(., nk) 2 0 ) C[ + + k] (89) k were te constant C is independent of ν, affine in u 2, p 2 and exponential in u 1,. Remark 1 : Note tat (89) is true even wit ν 0 (Euler s equation) but it is necessary tat te solution of (1)-(4) remain in W 1, wen ν 0 (wic assumes tat f depends on ν). Numerically, we observe tat te metod remains stable even wit ν = 0 so long as te solution of te continuous problem is regular. Remark 2 : Te ypotesis Ω convex can be relaxed. It is only necessary in order tat (ψ ψ 2 +.ψ 2 ) 1/2 be a norm. To prove (89), we sall follow tese tree steps : -estimate ψ n - un -estimatex n - Xn -estimateu n - un were u n is te value of te exact solution at time nk. Before starting, we note tat from (1)-(4), we can deduce 1 k un+1 ν u n+1 + p n+1 = f n + 1 k un ox n +k u, tt 0(1),.u n+1 = 0 (90) u 0 given, u n+1 Γ = 0 (91) were 0(1) is a function bounded independently of k and ν. Lemma 1: X n Xn 0 ke k u n ψ n un 0 (92) Proof : From te definitions of X and X, weave d dτ (X X) = ψ (X ) u(x) = ψ (X ) u(x )+u(x ) u(x) (93)

132 INTRODUCTION 135 and so d dτ X X o (τ) ( ψ u)ox o + u(x ) u(x) o (94) ψ u o + u X X o (τ) Since X (t) =X(t) =x, we deduce (92) from te Bellman-Gronwall inequality (by te integration of (94) on te interval ]nk,(n+1)k[. Lemma 2 : ψ u 0 C u n u n 0 + u n 0 (95) Proof: Let ψ n be a solution of ( ψ n, w)+(.ψ n,.w) =(u n, w), w W (96) ψ n W = {w H 1 (Ω) n : w n Γ =0, w.ndγ =0} (97) By subtracting (96) from (86), we see Γi ( (ψ n ψ n ), w )+(.(ψ n ψ n ),.w )=(u n u n, w ) (98) Let ξ be a projection of ψ n in W, tat is, te solution in W of ( ξ, w )+(.ξ,.w )=( ψ n, w )+(.ψ n,.w ), w W (99) By taking w = ξ ψ n in (99) and by subtracting (86), we see tat (ξ ψ n 0 C u u n 0 (100) Finally, we obtain te result because u is equal to ψ and (ψ n ψ) 0 (ψ n ξ ) 0 + (ξ ψ) 0 C( u u 0 + u n 0 ) (101) te last inequality comes from (100) and from te error estimate between ξ and ψ (te error of projection is smaller tan te interpolation error). To ave existence and uniqueness for (86)+(99), te bilinear form sould be coercive, wic we can prove only for convex polygonal domains (Girault et al [93]). Corollary 1 :

133 136 FINITE ELEMENT METHODS FOR FLUIDS X n X n 0 kc( u )[ u n u n 0 + ] (102) Proof of Teorem 7 : We subtract (90) from te variational form of (88) : (u n+1 u n+1,v )+νk( (u n+1 u n+1 ), v )+k( (p n+1 p n+1 ),v ) (103) =(u n ox n u n ox n,v )+k 2 u,t (0(1),v ) Ten we proceed as for te error analysis of te Stokes problem (cf. (4.74)- (4.78)). From (103), we deduce tat for all w J o and all q Q, we ave (u n+1 w,v )+νk( (u n+1 w ), v )+k( (p n+1 q ),v ) (104) =(u n+1 w,v )+νk( (u n+1 w ), v )+k( (p n+1 q ),v ) +(u n ox n un ox n,v )+k 2 u,t (0(1),v ), v J o So (see (74) for te definition of. ν ) u n+1 w ν u n+1 w ν +k (p n+1 q ) 0 + u n ox n u n ox n 0 +k 2 u,t (105) wic gives if we coose q equal to te interpolation of p n+1 u n+1 u n+1 ν u n+1 w ν + w u n+1 ν 2 u n+1 w ν +k p n+1 2 (106) + u n ox u n ox n 0 + k 2 u,t Taking w as equal to te interpolation of u n+1,we see tat u n+1 u n+1 ν C[( 2 + νk) u n+1 2 (107) +k p n k 2 u,t + u n ox n u n ox n 0 it remains to estimate te last term ; u n ox n un ox n 0 u n ox n un ox n 0 + u n o(x n Xn ) 0 (108) u n un o 0+ u n X n Xn 0 (1 + Ck) u n un ν + C k were C and C depend upon u (cf. (102)). By using a recurrence argument, te proof is completed because (107) and (108) give

134 INTRODUCTION 137 u n+1 u n+1 ν (1 + C 1 k) u n un ν + C 2 ( 2 + k + k 2 ) (109) 6. OTHER METHODS. In practice, all te metods given in capter 3 for te convection-diffusion equation can be extended to te Navier-Stokes equations. Paragrap 5 is an example of suc an extension. One can also use te following : - Lax-Wendroff artificial viscosity metod : owever it cannot be completely explicit (same problem as in 4.1) unless one adds p t in te divergence equation (Temam [228], Kawaara [131]); - te upwinding metod by discontinuity ; tis could be superimposed wit te oter metods of 4.1 and of 4.2. In tis way we can obtain metods wic converge for all ν (cf. Fortin-Tomasset [83], for an example of tis type); - streamline upwinding metod (SUPG). 6.1 SUPG and AIE (Adaptive Implicit/Explicit sceme) As in paragrap 4.3 in capter 3 a Petrov-Galerkin variational formulation for te Navier-Stokes is used wit te test functions v + τu v instead of v : (u,t + u u + p,v + τu v )+ν( u, v ) ν l T l τu v u =(f,v + τu v ) v V 0 (.u,q )=0 q Q. Here τ is a parameter of order, T l is an element; we ave used te simplest SUPG metod were te viscosity is added in space only. Jonson [125] rigtly suggest in teir error analysis te use of space-time discretisation (see 3.4.3). A semi-implicit time discretisation gives te following sceme: ( un+1 u n k +ν( u n+1, v ) ν l + u n u n+1,v + τu n v )+( p n+1,v )+( p n,τu n v ) T l τu n v u n =(f n+1,v + τu n v ) (.u n+1,q )=0 q Q. v V 0

135 138 FINITE ELEMENT METHODS FOR FLUIDS As noted in Tezduyar [227], it is possible to coose τ so as to ave symmetric linear systems because te non-symmetric part comes from: ( un+1 k,τun v )+(u n un+1,v ) Now tis would be zero if τ = k and if u n un+1 was equal to.(u n un+1 ). So tis suggests te following modified sceme: Find u n+1 u Γ V 0 and p n+1 Q suc tat ( un+1 u n k + u n u n+1,v + ku n v )+(.u n,u n+1.v ) +( p n+1,v )+( p n,ku n v )+ν( u n+1, v ) ν l =(f n+1,v + ku n v ) v V 0 T l ku n v u n (.u n+1,q )=0 q Q. Naturally a Crank-Nicolson O(k 2 ) sceme can be derived in te same way. Similarly a semi-explicit first order sceme could be: ( un+1 u n k +ν( u n, v ) ν l + u n u n,v + ku n v )+( p n+1,v )+( p n,ku n v ) T l ku n v u n =(f n,v + ku n v ) (.u n+1,q )=0 q Q. v V 0 Still following Tezduyar [227] to improve te computing time we can use te explicit sceme in regions were te local Courant number is large and te implicit sceme wen it is small; te decision is made element by element based on te local Courant number for convection, u n T l k/ l and for diffusion νk/ 2 l, and on a measure of te gradient per element size of un ;ere l is te average size of element l. So we define n numbers by βi l (c) =max{ un k T l 1, ( U i l ul i l Ui Ω u Ω i ) j u i x j c} were Ui l (resp ul i ) is te maximum (resp minimum) of u i(x) onelementt l and similarly for Ui Ω,uΩ i on Ω instead of T l. Te constant c is cosen for eac geometry; ten if βi l (c) > 0tecontribution to te linear system of te convection terms on element l in te i t momentum equation is taken explicit (wit u n instead of un+1 ) and oterwise implicit. Similarly if νk/ 2 l is less tan one, te contribution to te matrix

136 INTRODUCTION 139 of te linear system from ν T l u, v is taken wit u = u n wit u n+1 oterwise. (explicit) and 7. TURBULENT FLOW SIMULATIONS Reynolds Stress Tensor. As we ave said in te beginning of te capter, tere are reasons to believe tat wen t, te solution of (1)-(4) evolves in a space of dimension proportional to ν 9/4. For practical applications ν (= 1/Re) is extremely small and so it is necessary to ave a large number of points to capture te limiting solutions in time. We formulate te following problem (te Reynolds problem) : Let u ν w be a (random) solution of u,t + u u + p ν u =0,.u =0inΩ ]0,T[ (110) u(x, 0) = u o (x)+w(x, ω), u Γ = u Γ (111) were w(x,.) is a random variable aving zero average. Let <>be te average operator wit respect to te law of u introduced by w. Can we calculate <u>,<u u>...? Tis problem corresponds closely to te goals of numerical simulation of te Navier-Stokes equations at large Reynolds number because, wen ν<<1, u is unstable wit respect to initial conditions and so te details of te flow cannot be reproduced from one trial to te next. Tus, it is more interesting to find <u>. One may also be interested in <u 2 > and ν< u 2 >. Only euristic solutions of te Reynolds problem are known (see Lesieur [150] for example) but let us do te following reasoning : If we continue to denote by u te average <u ν w > and by u te difference u ν w <uν w >, ten (110) becomes : u,t + u u + p ν u +.u u = (u,t + u u + u u + p ν u ) (112).u +.u = 0 (113) because u u =.(u u) wen.u =0. If we apply te operator <>to (112) and (113), we see tat u,t + u u + p ν u +. <u u >=0,.u = 0 (114) wic is te Reynolds equation and R =< u u > (115)

137 140 FINITE ELEMENT METHODS FOR FLUIDS is te Reynolds tensor. As it is not possible to find an equation for R as a function of u, we will use an ypotesis (called a closure assumption) to relate R to u. It is quite reasonable to relate R to u because te turbulent zones are often in te strong gradient zones of te flow. But ten R( u) cannotbe arbitrary cosen because we must keep (114) invariant under canges of coordinate systems. In fact, it would be absurd to propose an equation for u wic is independent upon te reference frame. One can prove (Cacon-Pironneau [50]) tat in tis case te only form possible for R is (see also Speziale [218]). in 2D. In 3D R = a( u + u T 2 )I + b( u + u T 2 )( u + u T ). (116) R = ai + b( u + u T )+c( u + u T ) 2 (117) were a, b and c are functions of te 2 nontrivial invariants of ( u + u T ). So we get.r = a +.[b( u + u T )] +.[c( u + u T ) 2 ] (118) but a is absorbed by te pressure (we cange p to p + a) andsoalawofte type R = b( u + u T )+c( u + u T ) 2 (119) as te same effect. In 2D, it is enoug to specify a function in one variable b(s) and in 3D two functions in two variables b(s, s ),c(s, s )weres and s are te two invariants of u + u T Te Smagorinsky ypotesis [216] : Smagorinsky proposes tat b = c 2 u + u T, tat is R = c 2 u + u T ( u + u T ), c = 0.01 (120) were is te average mes size used for (114). Tis ypotesis is compatible wit te symmetry and a bidimensional analysis of R ; it is reasonable in 2D but not sufficient in 3D (Speziale [219]). Te fact tat is involved is justified by an ergodic ypotesis wic amounts to te identification of <>wit an average operator in a space on a ball of radius. Numerical experiments sow tat we obtain satisfying results (Moin-Kim [173]) wen we ave sufficient points to cause u to correspond to te beginning of te inertial range of Kolmogorov Te k ɛ ypotesis (Launder-Spalding [143]): Te so-called k ɛ model was introduced and studied by Launder-Spalding [143], Rodi [200] among oters.

138 INTRODUCTION 141 Define te kinetic energy of te turbulence k and average rate of dissipation of energy of te turbulence ɛ by: ten R, k, ɛ are modeled by k = 1 2 < u 2 > (121) ɛ = ν 2 < u + u T 2 >; (122) R = 2 3 ki c k 2 µ ɛ ( u + ut ), (c µ =0.09) (123) k,t + u k c µ 2 k 2 ɛ u + ut 2 k 2.(c µ ɛ k)+ɛ = 0 (124) ɛ,t + u ɛ c 1 2k u + ut 2 k 2.(c ɛ ɛ ɛ)+c ɛ 2 2 k = 0 (125) wit c 1 = , c 2 = 1.92, c ɛ = A roug justification for tis set of equations is as follows. First one notes tat k 2 /ɛ as te dimension of a lengt square so it makes sense to use (123). To obtain an equation for k, (112) is multiplied by u and averaged (we recall tat A : B = A ij B ij ): 1 2 <u 2 >,t + <u u >: u u <u 2 > +. <p u > ν <u u > Tat is to say wit (123) <u 2 u >=0; k 2 k,t c µ ɛ ( u+ ut ): u+u k ν <u u >= <u u 2 2 >. <p u > Te last 3 terms cannot be expressed in terms of u, k and ɛ, so tey must be modelled. For te first we use an ergodicity ypotesis and replace an ensemble average by a space average on a ball of centre x and radius r, B(x, r): <u u >=< u 2 > + B(x,r) u. u n dγ. By symmetry (quasi-omogeneous turbulence) te boundary integral is small. Te second term is modelled by a diffusion: <u u 2 2 > =. <u u > k; If u 2 and u were stocastically independent and if te equation for u 2 was linear, it would be exact up to a multiplicative constant, te caracteristic time

139 142 FINITE ELEMENT METHODS FOR FLUIDS of u. Te tird term is treated by a similar argument as te first one so it is conjectured to be small. So te equation for k is found to be : k,t + u k c µ 2 k 2 ɛ u + ut 2 k 2.(c µ ɛ k)+ɛ = 0 (126) To obtain an equation for ɛ one may take te curl of (112),multiply it by u and use an identity for omogeneous turbulence: Letting ω = u, one obtains: ɛ = ν< u 2 > because 0=2ν<ω.(ω,t +(u + u ) (ω + ω ) (ω + ω ) (u + u ) ν ω ) > = ɛ,t + u ɛ+ <u νω 2 > 2ν <ω (u ω) > 2ν(< ω ω >: u +. <(ω ω )u >)+2ν 2 < ω 2 > (u ω) =ω u u ω. te term <ω (u ω) > is neglected for symmetry reasons; <u νω 2 > is modelled by a diffusion just as in te k equation; by frame invariance, < ω ω > sould also depend only on u + u T,k and ɛ terefore in 2D it can only be proportional to u + u T and by a dimension argument it must be proportional to k. Te tird term is neglected because it as a small spatial mean and te last term is modelled by reasons of dimension by a quantity proportional to ɛ 2 /k. Finally one obtains : ɛ,t + u ɛ c 1 2k u + ut 2 k 2.(c ɛ ɛ ɛ)+c ɛ 2 2 k = 0 (127) Te constants are adjusted so tat te model makes good predictions for a few simple flows suc as turbulence decay beind a grid. Natural boundary conditions could be k, ɛ givenatt=0;k Γ =0, ɛ Γ = ɛ Γ. (128) owever an attempt can be made to remove te boundary layers from te computational domain by considering wall conditions (see Viollet [235] for furter details) k Γ = u 2 c 1 2 µ, ɛ Γ = u 3 Kδ (129)

140 INTRODUCTION 143 u.n =0, αu.τ + β u.τ n = γ (130) were K is te Von Karman constant (K =0.41), δ te boundary layer tickness, u te friction velocity, β = c µ k 2 /ɛ, α = c µ k 2 /[ɛδ(b + log(δ/d))] were D is a rougness constant and B is a constant suc tat u.τ matces approximately te viscous sublayer. To compute u, Reicard s law may be inverted (by Newton s metod for example) : u [2.5log( δu ν δu )+7.8(1 e 11ν δu 11ν δu e 0.33 ν )] = u.τ So in reality α and β in (3a) are nonlinear functions of u.τ, ɛ, k. Positiveness of ɛ and k. For pysical and matematical reasons it is essential tat te system yields positive values for k and ɛ. At least in some cases it is possible to argue tat if te system as a smoot solution for given positive initial data and positive Diriclet conditions on te boundaries ten k and ɛ must stay positive at later times. For tis purpose we looks at θ = k ɛ. If D t denotes te total derivative operator, / t+u and E denotes (1/2) u+ u T 2, ten D t θ = 1 ɛ D tk k ɛ 2 D tɛ = θ 2 E(c µ c 1 )+ c µ ɛ.(k2 ɛ k) c k ɛ ɛ 2.(k2 ɛ ɛ) 1+c 2 = θ 2 E(c µ c 1 ) 1+c 2 + c µ.kθ θ +2c µ θ. ( θ k )+(c µ c ɛ ) θ2 k.kθ ɛ Because c µ <c 1 and c 2 > 1,θ will stay positive and bounded wen tere are no diffusion terms because θ is a solution of a stable autonomous ODE along te streamlines: D t θ = θ 2 E(c µ c 1 ) 1+c 2 Also θ cannot become negative wen c µ = c ɛ because te moment te minimum of θ is zero at (x, t) we will ave: θ =0, θ =0 By writing te θ equation at tis point, we obtain :

141 144 FINITE ELEMENT METHODS FOR FLUIDS θ,t = c 2 1 > 0 wic is impossible because θ cannot become negative unless θ,t 0. Similarly we may rewrite te equation for k in terms of θ : D t k c µ 2 kθ u + ut 2.(c µ kθ k)+ k θ =0 Here it is seen tat a minimum of k wit k = 0 is possible only if D t k =0at tat point, wic means tat k will not become negative. Note owever tat k may ave an exponential growt if c µ θ 2 u+ u T 2 > 2; tis will be a valuable criterion to reduce te time step size wen it appens Numerical Metods. Let us consider te following model : u,t + u u + p.[ν( u + u T )( u + u T )] = 0,.u = 0 (131) u.n =0, au.τ + ν u.τ n = b (132) were a, b are given and ν is an increasing positive function of u + u T. Tis problem is well posed ( tere is even a uniqueness of solution in 3D under reasonable ypoteses on ν (cf. Lions [153])). Te variational formulation can be written in te space of functions aving zero divergence and aving normal trace zero : (u,t,v)+(u u, v)+ 1 2 (ν( u + ut )( u + u T ), v + v T ) (133) + [au.v bv]dγ =0 Γ v J on (Ω); u J on (Ω) = {v H 1 (Ω) 3 :.u =0,v.n Γ =0} (134) By discretising te total derivative, we can consider a semi-implicit sceme, 1 k (un+1 u n ox n,v )+ 1 2 (νn ( u n+1 +( u n+1 ) T ), v +( v ) T ) (135) + (au n b)v dγ =0, v J on Γ u n+1 J on = {v V :(.v,q )=0 q Q ; v.n Γ =0} (136) were V and Q are as in capter 4 and were :

142 INTRODUCTION 145 ν n = ν( u n +( u n ) T ) (137) Tere is a difficulty wit J on and te coice of te approximated normal n, especially if Ω as corners. We can also replace J on by because J on = {v V : (v, q )=0 q Q } (138) 0=(u, q) = (.u, q)+ u.nqdγ, q.u =0, u.n Γ =0. Γ (139) Wit J on te slip boundary conditions are satisfied in a weak sense only but te normal n does not now appear. Te tecniques developed for Navier-Stokes equations can be adapted to tis framework, in particular te solution of te linear system (134) can be carried out wit te conjugate gradient algoritm developed in capter 4. However we note tat te matrices would ave to be reconstructed at eac iteration because ν depends on n. To solve (114), (123) (te equations k ɛ), we can use te same metod ; we add to (133)-(134) (k is replaced by q) +( (n+1)k nk (q n+1 [ɛ n c µ qox n n,w )+kc µ ( qn 2 q n2 2ɛ n ɛ n q n+1, w ) (141) u n+1 + u n+1 2 ](X(t))dt, w )=0, w W o +( (n+1)k nk [ c 1 (ɛ n+1 2 qn un+1 ɛ n ox n,w )+kc ɛ ( qn 2 + u n+1t ɛ n ɛ n+1, w n ) (142) 2 ɛ n + c 2 q n ](X(t))dt, w )=0, w W o Te integrals from nk to (n+1)k are carried out along te streamlines in order to stabilize te numerical metod (Goussebaile-Jacomy [99]). So, at eac iteration, we must - solve a Reicard s law at eac point on te walls, - solve (134), - solve te linear systems (141)-(142). Te algoritm is not very stable and converges slowly in some cases but it may be modified as follows.

143 146 FINITE ELEMENT METHODS FOR FLUIDS As in Goussebaile et al.[99] te equations for q ɛ are solved by a multistep algoritm involving one step of convection and one step of diffusion. However in tis case te convection step is performed on q, θ rater tan on q, ɛ. Te equation for q is integrated as follows: (q n+1,w ) (qox n n,w )+(q n+1 (n+1)k nk ( c µ 2 q n ɛ n u n + u n 2 ɛn q n ),w ) (143) +kc µ ( qn2 ɛ n q n+1, w )=0 w Q o = {w Q : w Γ =0} q k Γ Q o (144) But te equation for ɛ is treated in two steps via a convection of θ = q/ɛ tat does not include te viscous terms (θ n+ 1 2,w )+(θ n+ 1 2 (n+1)k nk 1 2 θn un + unt 2 (c 1 c µ ),w ) (145) =(θ n ox n,w )+(c 2 1,w )k were θ n = qn /ɛn. Ten ɛn+1/2 is found as ɛ n+ 1 2 = qn+1 θ n+ 1 2 and a diffusion step can be applied to find ɛ n+1 : (146) (ɛ n+1,w )+kc ɛ ( qn2 ɛ n w Q o ; ɛ n+1 ɛ n+1, w )=(ɛ n+ 1 2,w ) (147) ɛ Γ Q o Notice tat tis sceme cannot produce negative values for q n+1 and ɛ n+1 wen ( c µ ]x,x n (x)[ 2 θn un + un 2 1 θ n ) > 1 (148) because c 1 >c µ and c 2 > 1 so (145) generates only positive θ wile in (143) te coefficient of q n+1 is positive. Note tat (148) is a stability condition wen te production terms are greater tan te dissipation terms.

144 INTRODUCTION 147 CHAPTER 6 EULER, NAVIER-STOKES AND THE SHALLOW WATER EQUATIONS 1. ORIENTATION In tis capter, we ave grouped togeter te tree principal problems of fluids mecanics were te nonlinear yperbolic tendency dominates. First, we sall recall some general results on nonlinear yperbolic equations. Ten we sall give some finite element metods for te Euler equations and finally we sall extend te results to te compressible Navier-Stokes equations. In te second part of tis capter, we sall present te necessary modifications needed to apply te previous scemes to te incompressible Navier-Stokes equations averaged in x 3 : Saint-Venant s sallow water equations. 2. COMPRESSIBLE EULER EQUATIONS 2.1. Position of te problem Te general equations of a perfect fluid in 3 dimensions can be written as (cf. (1.2), (1.7), (1.12)) and, in relation to (1.12) and W,t +.F (W )=0 (1) E = ρ(e u 2 ) W =[ρ, ρu 1,ρu 2,ρu 3,E] T (2) F (W )=[F 1 (W ),F 2 (W ),F 3 (W )] T (3)

145 148 FINITE ELEMENT METHODS FOR FLUIDS F i (W )=[ρu i,ρu 1 u i + δ 1i p, ρu 2 u i + δ 2i p, ρu 3 u i + δ 3i p, u i (E + p)] T (4) p =(γ 1)(E 1 2 ρ u 2 ) (5) Te determination of a set of Diriclet boundary conditions for (1) to be well defined is a difficult problem. Evidently we need initial conditions W (x, 0) = W o (x) x Ω (6) and to determine te boundary conditions, we write (1) as : W,t + A i (W ) W =0 (7) x i were A i (W )iste5 5 matrix wose elements are te derivative of F i wit respect to W j. Let n i (x) be a component of te external normal to Γ at x. One can sow tat is diagonalizable wit eigenvalues B(W, n) = A i (W )n i (8) λ 1 (n) =u.n c, λ 2 (n) =λ 3 (n) =λ 4 (n) =u.n, λ 5 (n) =u.n + c (9) were c =(γp/ρ) 1/2 is te velocity of sound. Finally, by analogy wit te linear problem, we see tat it is necessary to give Diriclet conditions at eac point x of Γ wit negative eigenvalues. We will ten ave 0, 1, 4 or 5 conditions on te components of W according to different cases. Te important cases are: - supersonic entering flow ( u.n < 0, u.n >c) : 5 conditions for example, ρ, ρu i,p - supersonic exit flow : 0 conditions. - subsonic entering flow : 4 conditions, independent combinations of W.l i were l i is te left eigenvector associated to te eigenvalue λ i < 0. - subsonic exit flow : 1 condition (in general, on pressure) - flow slipping at te surface (u.n = 0): 1 condition, i.e.: u.n = Bidimensional test problems a) Flow troug a canal wit an obstacle in te form of an arc of circle (Rizzi-Viviand[199]) Figure 6.1 :Stationary flow around an obstacle in te form of an arc of circle. Te picture sows te geometry, te type of boundary conditions and te position of socks.

146 INTRODUCTION 149 Te eigt of te ump is 8% of its lengt. Te flow is uniform and subsonic at te entrance of te canal, suc tat te Mac number is We ave slipping flow along te orizontal boundary and along te arc (u.n = 0); te quality of te result can be seen, for example, from te distribution of entropy created by te socks. b) Flow in a backward step canal (Woodward-Colella[238]) Figure 6.2 :Sock waves configuration in te flow over a step. Tis flow is supersonic at te exit and we give only conditions at te entrance p =1,ρ=1.4,u=(3, 0). Te quality of te results can be seen in te position of te socks and of te L contact discontinuity lines. c)symmetric flow wit Mac 8 around a cylinder. (Angrand-Dervieux[2]) Figure 6.3:Positions of socks at Mac 8. In tis example, te difficulties arise because te Mac number is large, te domain is unbounded and because tere is a quasi rarefaction zone beind te cylinder Existence One sould interpret (1) in te distribution sense because tere could be discontinuities : to ave uniqueness for te problem, one must add an entropy condition. Te pysical (reduced) entropy, s, S = log p ρ γ (10) sould decrease in time wen we follow te fluid particles (t S(X(t),t))). In certain cases, (Kruzkov[136], Lions[153]) one can sow tat if te solution of (1) is perturbed by adding a viscosity ɛ W ten wen ɛ 0itconvergesto te entropy solution of (1). So te existence and uniqueness of te solution of (1) can be studied as a limit of te viscosity solution. For system (1) wit te above prescribed conditions tere exists te following partial results : - an existence teorem for a local solution by a Caucy-Kowalevska argument (cf. Lax [145])

147 150 FINITE ELEMENT METHODS FOR FLUIDS - a global existence teorem in one dimension (cf. Majda [165]) - a global existence teorem in te case were W as only one component (scalar equation) and in arbitrary dimension (Kruzkov [136],Glimm [94], Lions [153]) Some centered scemes To obtain finite element metods for equations (1), we can use te metod developed in capter 3: - artificial viscosity metods, - semi-lagrangian metods, - streamline viscosity metods. a) Artificial viscosity metods (Jameson [121]) We take for (1) an explicit discretisation in time and a P 1 conforming finite element metod for space : 1 n+1 (W W n k,v ) (F (W n ), V )+ V.F (WΓ n ).n (11) Γ (.[G(W n ) W n ],V )=0 V U ; W n+1 WΓ n U 0 were G is a viscosity tensor cosen carefully [15] and were U 0 is te space of vector valued functions, piecewise P 1 continuous, on te triangulation and wose components corresponding to tose were W Γ is known, are zero on Γ; W n Γ is obtained by replacing in W n te known values of W Γ. As viscous and convection terms are taken into account explicitly in time, tere is a stability condition on k; if one is interested only in stationary solutions, mass lumping can be used to make te sceme really explicit; in addition a preconditioner can be added to accelerate it. Te viscosity cosen by Jameson is approximately :.(G(W ) W )= 2 k ( )[a 3 p x i p W z b a2 p p + 3 W z 3 ] (12) were z is te flow direction and a and b are constants of O(1); owever it is written as a nodal sceme in te sense tat mass lumping is applied to te first term of (11) and te contribution of te diffusion term is someting like ( (W.j W.m ) (W.k W.l )) j,k T (i) m N(j) l N(k) were N(i) is te set of neigbor nodes of node i and T (i) te triangles wic contain te it vertex.

148 INTRODUCTION 151 b) Lax-Wendroff/Taylor-Galerkin Metod (Donea [68], Loner et al. [160]) We start wit a Taylor expansion W n+1 = W n + kw,t n + k2 2 W,tt n + o(k2 ) (13) From (1), we deduce tat W,tt =.( F(W ) )=.(F t w(w )W, t )=.(F w(w ).F (W )) (14) So (13) can be discretised as (W n+1,v )=(W n,v )+k(f (W n ), V ) k2 2 (F w(w n ).F (W n ), V ) (15) [kf(w n k2 Γ ).n Γ 2 F w (W n ).F (W n )n].v, V U o were U o is as in a). Here also, one could accelerate te computation by mass lumping. In Loner [158][160] spectacular results obtained using tis metod can be found. An FCT correction (Boris-Book [34]) can be added to make te sceme more robust; if W pn+1 denotes te vector of values at te nodes of te solution of (15) wit mass lumping in te first two terms, ten te vector of values at te nodes of W n+1 is constructed from W n+1 = W pn+1 + cd M 1 l (M M l )W n, were c is a constant, M is te mass matrix, M l is te lumped mass matrix and D is a flux limiter built by limiting te element contribution to M 1 l (M M l )W n in suc a way as to attempt to ensure tat W n+1 is free from extrema not found in W n or W pn+1. c) Predictor-corrector viscosity metod ( Dervieux [64].) Tis is anoter form of metod b. We start wit a step (predictor) in wic W p is piecewise constant on eac element T and calculated from W p T = T 1 [ W n αk F (W n )n ] (16) T wit α =(1+ 5)/2 (cf. Lerat and Peyret [189]). Ten, we compute W n+1 by solving 1 n+1 (W W n,v ) + ((1 β)f (W n )+βf(w p k ), V ) (17) T

149 152 FINITE ELEMENT METHODS FOR FLUIDS V.F (WΓ n )n +(G(W n ) W n, V )=0, V U Γ were β =1/2α and were (, ) denotes te mass lumping quadrature formula; G is cosen to be proportional to 2 and to te first derivatives of W n (to give more viscosity were W is nonsmoot). d) Streamline upwinding metods (SUPG). (Huges-Mallet [119], Jonson-Szepessy [128]) Te basic idea is to add te equations of te problem to te basis functions; we consider te sceme (n+1)k nk (W,t +.F (W ),V + (V,t + A i (W )V,i ))dt (18) +(W n+ W n,vn )=0 were U n± is te left or rigt limit of U(t) went nk. We discretise in time and in space in a coupled manner, (cf Capter 3) for example wit continuous P 1 approximation in space and P 1 discontinuous in time. Toug te metod is efficient numerically, (Mallet [166]) te convergence study of te algoritm suggests 3 modifications (Jonson-Szepessy [128]): - use of entropy formulation of Euler s equations, - replacing by M in (18), were M is a matrix function of W, - adding anoter viscosity called sock capturing. Te final sceme is (n+1)k nk (A ow,t + i A iw,xi,v +( i 0 A i ) 1 2 [A o V,t + i A iv,xi ])dt (19) were +(W n+ W n,vn+ )=d V V n V n = {U,P 1 continuous in x, P 1 discontinuous in t, (20) suc tat U i = 0 on te parts of Γ were W i is given } We coose a = 1, b<<1, c = 1, we put U =[U,t,U,xi ] T and we take d = a(a ow,t + A iw,xi ) W. V Ω ]nk,(n+1)k[ (b + W i ) +c W n+ W n ( W,xi V,xi ) Ω i (21) Te matrices A i are computed from te entropy function S(W )(cf(10))by A o =(S,ww ) 1,A i = A ia o.

150 INTRODUCTION 153 Te sceme is implicit in time like te Crank-Nicolson sceme. For Burger s equation, (F (W ) = W 2 /2, W aving only one component) on Ω = R, result wic converges towards te continuous solution wen 0can be found in [17] ; also if W as only one component Jonson and Szepessy ave sown tat te metod converges to te entropy solution. In te general case one can also sow tat if W converges to W,tenW satisfies te entropy condition S(W ), t + i S, w F i (W ),xi 0 e) General Remarks on te above scemes Due to te complexity of te problem, we still do not know ow to prove convergence of te sceme in a general case (except maybe SUPG). So te analysis often reduces to te study of finite difference scemes on regular grids and of a linear equation (like tat of capter 3). Te scemes wit artificial viscosity ave te advantage of being simple but to be efficient, te regions were te viscosity plays a role must be small: tis is possible wit an adaptative mes were we subdivide te elements if W is large (Loner [158], Palmério and al. [186], Bank [12], Kikuci et al. [133]). But te stability condition forces us to reduce k if diminises so more iterations are needed tan wit implicit scemes; tis requires a careful vectorization of te program and/or te use of fast quadrature formulae (Jameson [121]). Finally, te main drawback as te coice of viscosity coefficient and te time step size, one may prefer te scemes wic use upwinding metods, is not being more robust (but not necessarily more accurate). On te teoretical side, it is sometimes possible to sow tat te scemes satisify te entropy conditions in te case of convergence (Leroux [147]) Upwinding by discontinuity As in capter 3, upwinding is introduced troug te discontinuities of F (W ) at te inter-element boundaries, eiter because W is a discontinuous approximation of W, or because for all W continuous, we know ow to associate a discontinuous value at rigt and at left, at te inter-element boundary. a) General Framework (Dervieux [64], Fezoui [78], Stoufflet et al. [222]). As in capter 3, for a given triangulation, we associate to eac vertex q i a cell σ i obtained by dividing triangles (tetraedra) by te medians (by median planes). Tus we could associate to eac piecewise continuous function in a triangulation a function P o (piecewise constant ) in σ i by te formula W p σ i = 1 σ i σ i W dx (22)

151 154 FINITE ELEMENT METHODS FOR FLUIDS By multiplying (1) by a caracteristic function of σ i and by integration (Petrov- Galerkin weak formulation) we obtain, after an explicit time discretisation, te following sceme : W n+1 (q i )=W n (q i )+ k σ i F d (W p ).n σ i i (23) On σ i Γ, we take F d (W ) = F (W ) and we take into account te known components of W Γ ; and elsewere F d (W ) is a piecewise constant approximation of F (W )verifying σ i F d (W p ).n = j i Φ(W p σ,w p i σ ) j n σ i σ j (24) were Φ will be defined as a function of F (W p σ i )andf (W p σ j ); Φ(u, v) iste numerical flux function cosen according to te qualities sougt for te sceme (robustness precision ease of programming). In all cases tis function sould satisfy te consistency relation : Φ(V,V )=F (V ), for all V. (25) b) Definition of te flux Φ : Let B(W,n) R 5 5 (4 4in2D)besuctat F (W ).n = B(W, n).w W n (26) Note tat B is te same in (8) because F is omogeneous of degree 1 in W (F (λw )=λf (W )) and so B is noting but F i (W )n i. As we ave seen tat B is diagonalizable, tere exists T R 5 5 suc tat were Λ is te diagonal matrix of eigenvalues. We denote B = T 1 ΛT (27) Λ ± = diag(±max(±λ i, 0)), B ± = T 1 Λ ± T (28) B = B + B, B = B + + B (29) We can coose for Φ one of te following formulae : Φ SW (V i,v j )=B + (V i )V i + B (V j )V j (Steger-Warming) (30) Φ VS (V i,v j )=B + ( V i + V j )V i + B ( V i + V j )V j (Vijayasundaram) 2 2 (31) Φ VL (V i,v j )= 1 2 [F (V i )+F (V j )+ B( V i + V j ) (V j V i )] (Van Leer) (32) 2

152 INTRODUCTION 155 Φ OS (V i,v j )= 1 V j 2 [F (V i )+F (V j ) B(W ) dn] (0ser) (33) V i te guiding idea being to get Φ l (V i,v j ) = F (V i ) l if λ l is positive and F (V j ) l if λ l < 0. So Φ SW, for instance, can be rewritten as follows : because Φ SW (V i,v j )= 1 2 [F (V i )+F (V j )] [ B(V i ) V i B(V j ) V j ], F (V i )+F (V j )=(B + (V i )+B (V i ))V i +(B + (V j )+B (V j ))V j ; Te first term, if alone, would yield an upwind approximation. Te second, after summation on all te neigbors of i, is an artificial viscosity term. Te Van Leer and Oser scemes rely also on suc an identification ; te flux of Oser is built from an integral so tat it is C 1 continuous ; te pat in R 5 from V i to V j is cosen in a precise manner along te caracteristics so as to capture exactly singularities like te sonic points. c)integrationintime. Te previous scemes are stable up to CFL of order 1 (c w k/ < 1werec depends on te geometry); if one wants only te stationary solution, te sceme can be speeded up by a preconditioning in front of W/ t ; tese scemes are quite robust ; 3D flows at up to Mac 20 can be computed ; but tey are not precise. Scemes of order 2 are being studied. d) Spatial approximations of order 2. A clever way (due to Van Leer) to make te previous scemes second order is to replace V i and V j by te interpolates V i and V j+ defined on te edges were tey must be computed by : V i = V i +( V ) i (qj q i ) 2 V j+ = V j ( V ) j (qj q i ) 2 An upwinding is also introduced to compute te gradients (cf. Stoufflet et al. [222]). An alternative is to use discontinuous elements of order 2 or iger (Cavent-Jaffré [51]) Convergence : te scalar conservation equation case Te convergence of te above sceme is an open problem except peraps in 1D or wen W as only one component (instead of 5!); tis is te case of

153 156 FINITE ELEMENT METHODS FOR FLUIDS te nonlinear scalar conservation equations wic govern te water-oil concentration in a porous media; let us take tis as an example. Let f be in C 1 (R) and φ(x, t) R be a solution of φ,t +.f(φ) =0inQ =]0,T[ R n, φ(0) = φ o in R n (34) wit φ o aving compact support. One can sow (Kruzkov [136]) tat for φ o L (R n ) tere exists a unique solution (34) satisfying te entropy inequalities were Φ c,t +.F c 0inQ for all c (35) Φ c (φ) = φ c, F c (φ) =(f(φ) f(c))sgn(φ c) (36) Moreover, φ is also te limit of φ ɛ,ɛ 0, te only solution of φ ɛ,t +.(f(φɛ )) ɛ φ ɛ =0inQ,φ ɛ (0) = φ o (37) Finally, if BV (R n ) denotes te space of functions of bounded variations, we ave : φ o L 1 (R n ) BV (R n ) φ L (0,T; L 1 (R n )) BV (R n ) (38) In te finite difference world, te following definitions are introduced : Let {φ n i } be te values at te vertices of φn (x) (φn i = φ n (qi )); an explicit sceme φ n+1 i = H({φ n j }) i (39) is monotone if H is a non-decreasing function in eac of its arguments. A sceme is TVD (Total Variation Diminising) if φ n+1 BV φ n BV were φ n BV = q i,q j neigbors φ i φ j q i q j (40) Tese notations can be generalized to finite element scemes at least in te case of uniform meses. In [10] a quasi-monotone finite element sceme is constructed wit upwinded fluxes and a viscosity of O(), and one gets a class of metods for wic te convergence is proven. Implicit metods wit viscosity ave te advantage of being automatically stable and in te case of convergence (were all te difficulties lie) convergence occurs towards an entropy solution. For example, let us consider te following sceme :

154 INTRODUCTION k (φn+1 φ n,w )+(.f(φ n+1 ),w )+ɛ α ( φ n+1, w )=0, w V (41) were V is a space of conforming finite elements. By taking w = φ n+1, we get te stability φ n+1 2 o φ n o φ n+1 o (42) because (.f(φ ),φ ) is equal to a boundary integral on Σ wic is positive plus c φ 2 o Under rater strong convergence ypoteses, by taking w equal to te interpolation of sgn(φ c) one can sow in te limit of (41) (φ,t sgn(φ c)) + (.f(φ),sgn(φ c)) + ɛ lim α ( φ, sgn(φ c)) = 0 (43) but ( φ, sgn(φ c)) = {x φ c=0} φ n dγ > 0 so (43) implies ultimately (35). As mentionned above, SUPG+sock capturing is one case were convergence can be proved by a similar argument (and many more ingredients [128]) 3. COMPRESSIBLE NAVIER-STOKES EQUATIONS 3.1. Generalities Te equations (1.2), (1.7), (1.12) can also be written as a vectorial system in W. Wit te notations (2)-(5), we ave W,t +.F (W ).K(W, W ) = 0 (44) were K(W, W ) is a linear 2 nd order tensor in W suc tat K i,1 =0, K.,2,3,4 = η u +(ζ + η )I.u, (45) 3 K.,5 = η( u + u T )u +(ζ 2 3 η)u.u + κ (E ρ u 2 2 ) (46) Te term.k(w, W ) is seemingly a diffusion term if ρ, θ > 0 since it corresponds to η u +(η/3+ζ) (.u) inte2 nd, 3 rd and 4 t equations of (44) and to κ(c v /ρ) θ +1/ρC v [.u 2 (ζ 2η/3)+ u + u T η/2] in te last one (cf. (1.13)):

155 158 FINITE ELEMENT METHODS FOR FLUIDS W.K(W, W )= [η u +( η + ζ) (.u)]u (47) R 3 R 3 3 θ (κc v R ρ θ + θ ρ C v[.u 2 (ζ 2 η 3 )+ u + ut η 2 ]) 3 (η u 2 +( η R 3 + ζ).u 2 )+κ C v θ 2 ρ 3 max R 3 + θ min C v [.u 2 (ζ 2 η ρ max R 3 )+η 2 u + ut 2 ] 0 3 So we can tink tat te explicit scemes in step 1 to integrate (1) can be applied to (44) if te stability condition on te time step k is modified : k Cmin[ w, 2 K ] (48) were K is a function of κ, η, ζ. Tis modification is favorable to te scemes obtained by artificial viscosity but still wen te boundary conditions are different boundary layers arise boundary layers in te neigborood of te walls. Te boundary conditions can be treated in te strong sense (tey appear in te variational space V o instead of V ) or in te weak sense (tey appear as a boundary term in te variational formulation). Flow around a NACA0012 airfoil A test problem is given in Bristeau et al.[43] wic concerns a 2D flow around a wing profile NACA0012. Taking te origin of te reference frame at te leading edge te equation of te upper surface of te profile is : y = x x x x x 4, 0 x 1 Te boundary conditions at infinity are suc tat te Mac number is 0.85, te Reynolds number (uρ/η) is 500 and te angle of attack is 0 0 or Te temperatures at infinity and on te surface are given. Figure 6.5: Flow around a NACA0012. Te figure sows te position of socks and te boundary layers An example As an example, we present te results obtained by Angrand et al. [2] and Rostand [204]. Te algoritm used is a modification of (16)-(17) and te boundary conditions are treated in te weak sense. Te problem treated was te modeling

156 INTRODUCTION 159 of rarefied gas layers in te neigborood of te surface ( Knudsen layers) and so a Robin type boundary condition is applied: A(W )W + K(W, W )n = g on Γ (49) were A is a 2nd order tensor and g R 5. (If A + we obtain te Diriclet conditions and if A 0, te Neumann conditions). Let U be te space of functions aving range in R 5 continuous and piecewise P 1 on a triangulation of Ω. Problem (44) is approximated by 1 n+1 (W W n,v ) (F (W n ), V )+(K (W n, W n ), V ) (50) k + [F (W n )V n +(A(W n )W n g)v ]=0 V U Γ K possibly contains also te artificial viscosity if needed, F te upwinded flux and (, ) te mass lumping. Tis sceme is mostly used to calculate stationary states. In Rostand [204], te flow around a wing profile (2D) NACA0012 as been calculated wit te following boundary conditions : u.n =0, n.d(u).τ + Aρu.τ =0 u.d(u).n + γ θ Pr n = 0 (51) were D(u) = u + u T 2/3.u I, A is a coefficient to be cosen and Pr is te Prandtl number. We point out te following problems wic are yet to be solved : - find non reflecting boundary conditions at infinity towards downstream - find a fast sceme (implicit?) and more accurate (order 2?) - remove te pressure oscillations near te walls and te leading edges in particular; it seems tat one as to use a different approximation for p and u wen u<<c,wic makes sense from wat we know about incompressible te Navier-Stokes equations. - make te sceme a feasible approximation for small Mac number also, (someting wic does not exist at te moment). 3.3 An extension of incompressible metods to te compressible case. Since te incompressible Navier-Stokes equations studied in Capter 5 are an approximation of te compressible equations wen ρ tends to a constant, it sould be possible to use te first one as an auxiliary solver for te second one. So we sall try to associate te continuity equation and te pressure. Te complete system (44) can be written as a function of σ = logρ as sown below, Bristeau et al.[41]. Here te metod is presented wit ζ =0): σ, t +u σ +.u =0

157 160 FINITE ELEMENT METHODS FOR FLUIDS u, t +u u +(γ 1)(θ σ + θ) =ηe σ ( u + 1 (.u)) 3 (52) θ,t + u θ +(γ 1)θ.u = e σ (κ θ + F ( u)) (53) were F ( u) =η u + u T 2 /2 2η.u 2 /3(andwereθ and κ ave been normalized). An implicit time discretisation of total derivatives gives rise to a generalized Stokes system if we treat te temperature explicitly : 1 k σn+1 +.u n+1 = 1 k σn ox n, (54) 1 k un+1 ηe σ n( u n (.un+1 )) + (γ 1)θ n σ n+1 = 1 k un ox n (γ 1) θ n, (1 + 1 k (γ 1).un )θ n+1 e σn κ θ n+1 = 1 k θn ox n + e σn F ( u n ) A metod for solving tis system is to multiply te last two equations by e σ so as to obtain ασ +.u = g (55) βu a u + b σ = f δθ c θ = were a and c are constant. So following te metod of capter 4, we take te divergence of te second equation, substitute it in te first and obtain : ασ +.[(β a ) 1 (f b σ)] = 0 an equation tat we solve by using te conjugate gradient metod. Tis metod as te advantage of leading back to te metods studied in capter 5 wen te flow is quasi-incompressible, but as te drawback of giving a metod wic is non-conservative for Euler s equations wen η and κ tend to SAINT-VENANT S SHALLOW WATER EQUATIONS 4.1. Generalities Let us go back to te incompressible Navier-Stokes equations wit gravity (g = 9.81) u,t +.u u + p ν u = ge 3.u = 0 (56) and assume tat te domain occupied by te fluid as a small tickness, wic is true in te case of lakes and seas.

158 INTRODUCTION 161 Figure 6.8 : Vertical section troug te fluid domain sowing te bed of a lake and te water surface. Let z s (x 1,x 2,t) be te eigt of te surface and z f (x 1,x 2,t) te eigt of te bed. Te continuity equation integrated in x 3 = z gives zs.udz = u 3 (z s ) u 3 (z f )+ 12.( udz) = 0 (57) z f z f but by te definition of u 3 we ave dz s /dt = u 3 (z s ) so, taking into account te no-slip condition at te bottom and defining v, z by zs udz =(z s z f )v(x 1,x 2,t), z = z s z f (58) z f one can rewrite (57) as zs z,t +.(zv) = 0 (59) In (56)(a) we neglect all te terms in u 3, so te tird equation gives p z = g (60) By putting p calculated from (60) in te two first equations of (56)(a) integrated in x 3 we see (zv),t +.(zv v)+gz z ν (zv) = 0 (61) Te system (59), (61) constitutes te sallow water equations of Saint-Venant (for more details, see Benqué et al. [22], for example). For large regions of water (seas), te Coriolis forces, proportional to ω v were ω is te rotation vector of te eart must be added. Oter source terms, f, in te rigt and side of (61) could come from - te modeling of te wind effect (f constant) - te modeling of te friction at te bottom z f (f = c v v/z). If we cange scales in (59), (61) as in (1.39),we obtain z,t +.(z v ) = 0 (62) (z v ),t +.(z v v )+F 2 z z R 1 (z v ) = 0 (63) wit R = v L/ν and te Froude number F = v gz (64)

159 162 FINITE ELEMENT METHODS FOR FLUIDS If we expand (61) and divide by z,we get v,t + v v + g z νz 1 (zv) = 0 (65) If F<<1andR<<1, ten g z dominates te convection term v v and te diffusion term νz 1 zv ; we can approximate (65) (59) by, z,t +.(zv) =0 v,t + g z = 0 (66) wic is a nonlinear yperbolic system wit a propagation velocity equal to gz. If R >> 1 so tat we can neglect te viscosity term, we remark tat (59), (61) is similar to te Euler equations wit γ =2,ρ= z and an adiabatic approximation p/ρ γ = constant ; on te oter and if in (65) we set u = zv ten it becomes similar to te incompressible Navier-Stokes equations wit an artificial compressibility suc as studied by Temam [228]: u,t +.(z 1 u u)+ g 2 z2 ν (u) =0 z,t +.u =0; tis problem is well posed wit initial data on u, z and Diriclet boundary data on u only. But if we make te following approximation: z 1 (zv) = v, ten (59)(61) becomes similar to te compressible Navier-Stokes equations wit γ =1: v,t + v v + g z ν v =0 z,t +.(zv) =0; An existence teorem of te type found by Matsumura-Nisida [169] may be obtained for tese equations ; tis means tat te problem (59), (61) is well posed wit te following boundary conditions : z,v given at t =0 (67a) v Γ given Γ and z given on all points of Γ were v.n < 0 (67b) wit te condition tat z stays strictly positive. Tis fact is confirmed by an analysis of te following problem (Pironneau et al [192]): g z ν v =0.(zv) =0; wit (67b). Tis reminds us tat certain seemingly innocent approximations ave a dramatic effect on te matematical properties of suc systems. Boundary conditions for tese sallow water equations is a serious problem; ere are some difficult examples found in practical problems: - on te banks were z 0

160 INTRODUCTION 163 in deep sea were one needs non reflecting boundary conditions if te computational domain is not to be te wolesurface of te eart. - if te water level goes down and islands appear ( z 0). Test Problem : A simple non stationary test problem is to study a wave wic is axisymmetric and Gaussian in form wit 2m eigt in a square of 5m dept and 10m side returning to rest. Figure 6.9 :Computational domain (left) and vertical section troug te pysical domain Numerical sceme in eigt-velocity formulation Wen te Froude number is small, we sall use te metods applicable to incompressible Navier-Stokes equations rater tan te metods related to Euler s equations. We sall neglect te product term z v (small compared to v v) inte development of (zv) in te formulation (59), (65), tat is we sall consider te system z,t + v z + z.v = 0 (68) v,t + v v + g z ν v = 0 (69) z(0) = z o,v(0) = v o in Ω; v Γ = v Γ, z Σ = z Γ (70) For clarity, we assume tat v Γ =0. Recall tat Σ is tat part of Γ were te flux enters (v.n < 0). If D/Dt is te total derivative, (68)-(69) can also be written Dz + z.v = 0 (71) Dt Dv + g z ν v =0. (72) Dt Wit te notation of capter 3, we can discretise te system wit te semiimplicit Euler sceme 1 k (zm+1 z m ox m )+z m.v m+1 = 0 (73) 1 k (vm+1 v m ox m )+g z m+1 ν v m+1 = 0 (74) For spatial discretisation, we sall use te metods given in capter 4 for te Stokes problem : we coose Q = L 2 (Ω) and V o = H 1 o (Ω) n and we find [z m+1 ]solutionof,v m+1

161 164 FINITE ELEMENT METHODS FOR FLUIDS 1, q k (zm+1 z m )+(.v m+1,q )= 1 k (zm ox m, q z m ) q Q (75) 1 k (vm+1,w )+ν( v m+1, w )+g( z m+1,w )= 1 k (vm ox m,w ) w V o (76) We recall tat X m (x) is an approximation of X(mk, x), wic is a solution of dx dτ = vm (X(τ),τ), X((m +1)k) =x, (77) tus te boundary condition on z Σ appears in (75) in te calculation of z mox m (x), x Σ. At eac iteration in time, we ave to solve a linear system of te type were D ij = 1 k Ω ( )( ) A B V B T = D Z ( F G q i q j z m, B ij =( q i,w j ), G j = 1 k (zm ox m, qj z m ) ) (78) F j = 1 g (vm ox m,wj ), A ij = 1 kg (wi,w j )+ ν g ( wi, w j ) (79) As wit te Stokes problem, te matrix of te linear systems is symmetric but ere it is nonsingular watever may be te cosen element {V,Q }. In fact, by using (76), to eliminate v m+1, we find an equation for z m+1 : (B T A 1 B + D)Z = B T A 1 F G (80) te matrix of tis linear system is always positive definite (cf. (79)). It can be solved by te conjugate gradient metod exactly as in te Stokes problem. A preconditioner can also be constructed in te same manner ; Goutal [100] suggests : C = D kg (81) were D is given in (79) and is a Laplacian matrix wit Neumann condition (tis preconditioning corresponds to a discretisation of te operator on z in te continuous case wen ν =0). Stability and convergence are open problems but te numerical performance of te metod is good[13] A numerical sceme in eigt-flux formulation We put D = zv so tat (59)-(61) can be rewritten as

162 INTRODUCTION 165 z,t +.D = 0 (82) D,t +.( 1 D D)+gz z ν D = 0 (83) z Let us take as a model problem te case were z(0) = z o,d(0) = D o in Ω (84) D = D Γ on Γ ]0,T[ (85) In [13] te following sceme is studied : wit te same finite element space V 0 and Q 0, te functions in te space Q wic are zero on Γ, one solves 1 (Dm+1 kg z m 1 k (zm+1 z m,q )+(.D m+1,q )=0 q Q (86) v m ox m,w )+( z m+1,w )+ ν g ( Dm+1, w ) (87) z m + 1 g (vm.vm,w )=0 w V o were v m = D m/zm and were Xm is as in te previous paragrap, tat is, it is calculated wit v : X m(x) = x v m(x)k. Eac iteration requires te resolution of a linear system of te type (78) but wit D ij = 1 k (qi,q j ) A ij = 1 [ wi w j kg Ω z m + ν w i wj z m ] (88) We note tat A is no longer symmetric except wen ν = 0. Te uniqueness of te solution of (86)-(87) is no longer guaranteed except wen ν<<1wic makes it symmetric again. Wen ν 0, we can also use te following approximation ( Dm+1 z m, w ) = Ω 1 z m D m+1 D m w Ω (z m)2 zm w (89) Anoter metod can be devised by working wit z = z 2 instead of z and by replacing (z m+1 z m,q ) in (86) by (z m+1 z m,q /(2z m )). Te convergence of tis algoritm is also an open problem Comparison of te two scemes Te main difference between te eigt-velocity and eigt-flux formulations is in te treatment of te convection term in te continuity equation and in te boundary conditions. We expect te first formulation to be more stable

163 166 FINITE ELEMENT METHODS FOR FLUIDS for flows wit ig velocity. Te oter factors wic sould be considered in selecting a metod are : - te boundary conditions, - te coice of variable to be conserved - te presence of socks (torrential regime, ν = 0) ; (86) is in conservative form wereas (75) is not. 5. CONCLUSION In tis capter we ave briefly surveyed some recent developments in te field of compressible fluid dynamics and te sallow water equations. We ave omitted reacting flows (fluid mecanics + cemistry), te Rayleig Benard problem and flows wit free surfaces because a wole book would ave been necessary in place of just one sort capter; tese subjects are also evolving quite rapidly so it is difficult to write on tem witout becoming rapidly obsolete; even for compressible flows interactions between te viscous terms and te yperbolic terms is not well understood at te time of writing. Te arcitectures of computers are also canging fast; dedicated ardware for fluid mecanic problems is already appearing on te market and tis are likely to influence deeply te numerical algoritms. Finally compressible turbulence is, wit ypersonic combustion, one of te callenge for te 90 s.

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