Approximation of the thermally coupled MHD problem using a stabilized finite element method

Transcription

1 Approximation of te termally coupled MHD problem using a stabilized finite element metod Ramon Codina and Noel Hernández Universitat Politècnica de Catalunya, Jordi Girona -3, Edifici C, Barcelona, Spain. ramon.codina@upc.edu, noel@cimne.upc.edu Contents Introduction 2 2 Problem statement 3 2. Initial and boundary value problem Weak form Time integration and linearization 7 3. Time discretization Linealization and block-iterative coupling Linearization of te stationary MHD problem Full coupling and block-iterative coupling Time discrete and linearized sceme Stabilized formulation for te stationary, linearized and termally uncoupled problem 4. Stability of te Galerkin approximation Te subgrid scale framework for a general CDR system of equations Stabilized finite element approximation for te linearized MHD problem Numerical analysis and design of te stabilization parameters Coercivity Optimal accuracy Convergence Final numerical sceme 20 6 Numerical examples Flow over a circular cylinder Clogging in continuous casting of steel Crystal growt Conclusions 30

2 Abstract A numerical formulation to solve te MHD problem wit termal coupling is presented in full detail. Te distinctive feature of te metod is te design of te stabilization terms, wic serve several purposes. First, convective dominated flows in te Navier-Stokes and te eat equation can be dealt wit. Second, tere is no restriction in te coice of te interpolation spaces of all te variables and, finally, flows igly coupled wit te magnetic field can be accounted for. Different aspects related to te design of te final fully discrete and linearized algoritm are also discussed. Introduction Te objective of tis work is to present a finite element metod for te approximation of te termallycoupled magneto-ydrodynamic (MHD) problem. We discuss several issues related to te time discretization, te linearization and te iterative coupling of te unknowns. However, our main concern is te design of a stabilization tecnique tat allows one to use any continuous interpolation for all te unknowns, in particular, equal interpolation is allowed. Considering first te termally uncoupled case, in principle te unknowns involved are te magnetic field, te fluid velocity and te ydrodynamic pressure. However, to enforce te divergence free condition for te numerical approximation of te magnetic field we introduce a magnetic pseudo-pressure (wose exact value sould be zero). Tis zero divergence condition is automatically satisfied at te continuous level for te transient problem if te initial magnetic field is solenoidal, but it is convenient to explicitly enforce it in te numerical approximation, especially for stationary problems. Wit te introduction of te magnetic pseudo-pressure we are left wit a system of four equations wit four unknowns. Te augmented approac used in tis work is discussed for example in [9] in te context of MHD and more recently in [5] for te Maxwell equations. Te same approac is used in [4, 29, 24, 23, 28] (see also references terein). Oter possibilities of enforcing te divergence free condition for te magnetic field are penalty strategies (see for example []) or te use of (weakly) divergence free interpolations based on Nédelec-type elements (described for example in [25, 26]). Tese elements can also be used in combination wit te augmented approac using a continuous approximation for te magnetic pseudo-pressure [28] so as to satisfy te adequate inf-sup condition between tis scalar and te magnetic field. Tis condition also olds if an equal order discontinuous interpolation is used for bot variables [7]. Neverteless, tere is also te possibility of relying on te matematical structure of te equations and to expect tat te original problem will already yield a magnetic field close enoug to solenoidal. Tis is te idea followed in [22], wic probably contains te first analysis of a finite element approximation to te MHD problem, and it is also used in [5], among oter papers. Having introduced te magnetic pseudo-pressure as a new unknown in te problem, its finite element approximation as several difficulties. First, tere is te well known compatibility condition between te approximation spaces for te velocity and te pressure, but also for te approximation spaces for te magnetic field and te magnetic pseudo-pressure. Bot conditions can be expressed in a standard inf-sup form [2]. Tere is also te problem of dealing wit situations in wic first order derivatives, bot in te Navier-Stokes equations and in te equation for te magnetic field, dominate (from te numerical point of view) te second order terms tat give an elliptic nature to te system of equations to be solved. Tese are te classical convection-dominated flow problems. Bot te compatibility condition between interpolating spaces and te oscillations found in flows dominated by convection can be overcome by using stabilized finite element metods. First approaces in tis direction can be found in [5] (witout te introduction of te magnetic pseudo-pressure) and in [24, 23] (were te magnetic pseudo-pressure is also introduced). However, anoter particular feature of te MHD problem are te couplings involved. In te magnetic problem, te coupling wit te ydrodynamic problem comes from te convective term in te equation for te magnetic field, wereas in te Navier-Stokes equations te 2

3 coupling wit te magnetic problem comes from Lorentz s force. Our objective is to design a stabilized finite element metod tat takes tese couplings into account. Te stabilized finite element metod presented ere is based on te two-scale decomposition of te unknowns into teir finite element component and a subscale tat cannot be captured by te finite element space. Te format tat we follow of tis idea was introduced in [8]. In particular, te version for systems we employ ere was already presented in [6]. A first version of our formulation, considering only te stationary and termally uncoupled problem, can be found in [9]. Te formulation is first designed for linear problems, and terefore our first concern is to devise a linearization tecnique for te fully coupled problem. For simplicity, we consider a fixed point metod. Among te different possibilities, we identify te only one tat leads to a linearized problem tat is coercive, and tus guarantees existence and uniqueness of solution. Tis fixed point metod is often used, but rarely justified. It is for tis linearized problem tat we propose a stabilized finite element metod based on te subgrid scale concept. Te important point is ow to approximate te subgrid scales. We use te simplest approac of taking tem proportional to a projection of te residual of te finite element approximation multiplied by te so called matrix of stabilization parameters. We consider two possibilities for te projection. Te first is to take it as te identity (at least wen applied to te residual of te finite element solution), and te second is to consider tis projection as te ortogonal to te finite element space. Te first option leads to a classical residual based stabilized finite element metod, wereas te second was termed ortogonal subscale stabilization formulation (OSS) in [7], were it is fully developed for incompressible flows. A toroug numerical analysis for te stationary and linearized problem can be found in [8]. Te design of te matrix of stabilization parameters is solely based on te stability and convergence analysis of te problem. Tis analysis will be presented in a situation as simple as possible, trying to avoid matematical tecnicalities. It is not te purpose of tis paper a deep numerical analysis of te formulation to be presented, but to present it wit a sound motivation. Te resulting formulation differs from te one proposed in [24] bot in te structure of te stabilizing terms (no attempt is made tere to account neiter for convection-dominated situations nor for te coupling effects) and in te design of te stabilization parameters. It also differs form te metod proposed in [5] in te inclusion of te magnetic pseudo-pressure and in te design of te stabilization parameters. Te paper is organized as follows. Te problem to be solved is presented in te Section 2, including its strong and its variational forms. Issues not directly related to te finite element approximation are treated in Section 3, were a simple time integration sceme is described and linearization possibilities are discussed, starting wit te identification of te only feasible fixed-point iteration for te termally uncoupled MHD problem and ten including te termal coupling. Te stabilization metod is proposed and fully analyzed for te linearized stationary MHD problem in Section 4. Te sceme we finally propose is written in Section 5. Numerical examples are presented in Section 6 and conclusions are finally drawn in Section 7. 2 Problem statement 2. Initial and boundary value problem Let Ω R d (d = 2 or 3) be a domain were we want to solve te termally coupled MHD problem during te time interval [0, T ], te termal coupling being modeled troug Boussinesq s assumption. Te unknowns of te problem are te fluid velocity u : Ω (0, T ) R d, te pressure p : Ω (0, T ) R, te magnetic induction (wic we will simply call magnetic field) B : Ω (0, T ) R d, te magnetic pseudo-pressure r : Ω (0, T ) R and te temperature ϑ : Ω (0, T ) R, 3

4 wic are solution of te system of partial differential equations: t u + u u ν u + ρ p µ m ρ ( B) B + gβϑ = f f + g[ + βϑ r ], () u = 0, (2) t B + µ m σ ( B) (u B) + r = f m, (3) B = 0, (4) t ϑ + u ϑ k t ϑ ρc p ρc p µ 2 mσ B 2 2µ f S u 2 = Q. ρc p (5) In tese equations, ρ is te fluid density, µ f te fluid viscosity, ν = µ f /ρ is te kinematic viscosity, g te gravity acceleration vector, β te termal expansion coefficient, f f te body force of te flow motion, ϑ r a known reference temperature, µ m te magnetic permeability, σ te conductivity, f m a forcing term for te magnetic field (zero in te applications), k t te termal conduction coefficient, c p te specific eat at constant pressure, S is te symmetrical gradient operator and Q te eating source. From now on, we consider tat f f contains g[ + βϑ r ]. All pysical properties will be considered constant. In order to write te boundary conditions for problem ()-(5), let us consider te disjoint splittings Ω = Γ E,u Γ N,u = Γ E,B Γ N,B = Γ E,ϑ Γ N,ϑ. Subscript E refers to essential boundary conditions in te variational form to be presented, wereas N refers to natural boundary conditions. Te second subscript indicates te variable to wic te condition is applied. If we denote wit an overbar prescribed values, te boundary conditions to be considered for all time t (0, T ] are: On Γ E,u : u = 0. (6) On Γ N,u : pn + νn u = t. (7) On Γ E,B : n B = 0, r = 0. (8) On Γ N,B : n B = B, n ( B) = J. (9) On Γ E,ϑ : ϑ = 0. (0) On Γ N,ϑ : k t n ϑ = q. ρc p () In tese equations, n denotes te unit external normal to te boundary. Note tat essential boundary conditions ave been considered all omogeneous. Tis simplifies te writing of te problem (te functional spaces were te solutions belong will be linear instead of affine). Note also tat te boundary condition (7) does not correspond to te prescription of te pysical traction, but it is te natural condition associated to te way te viscous term in () as been written. To complete te definition of te problem we need to add initial conditions of te form u = u 0, B = B 0 and ϑ = ϑ 0, all olding in te spatial domain Ω at t = 0. Remark Te forcing term f m, introduced for generality, as to be divergence free. Likewise, te initial field B 0 must be also solenoidal. If one takes te divergence of (3) and uses te boundary conditions (8), it turns out tat r = 0. However, from te numerical point of view te introduction of r will be very useful to enforce te zero divergence condition (4) wile keeping te correct functional setting of te problem, as we will see in te next subsection. Even toug we will work wit te dimensional form of te problem ()-(5) to igligt te way it 4

5 needs to be scaled, te equations can also be written in terms of te following dimensionless numbers: Re := lu ν, Pe := lu κ, Reynolds number, Péclet number, Pr := ν κ, Prandtl number, Gr := β g l3 δϑ ν 2, Grasof number, Re m := µ m σlu, Magnetic Reynolds number, σ Ha := Bl ρν, S := Hartmann number, B2, Coupling number, µ m ρu 2 were l is a caracteristic lengt of te problem, U a caracteristic velocity, B a caracteristic magnetic field and δϑ a caracteristic temperature difference, usually computed from temperature boundary values wen tese are not zero. Tese numbers are obviously not independent. For example, Pe = RePr, Ha = (ReRe m S) /2. Te data for te numerical examples presented in Section 6 will be given in terms of tese numbers. 2.2 Weak form To write te weak form of ()-(5) wit te boundary conditions (6)-(), let v, q, C, s and ψ be te test functions for u, p, B, r and ϑ, respectively. We consider tem time-independent (time will be discretized using a finite difference sceme), v and ψ are assumed te be zero on Γ E,u and Γ E,ϑ, respectively, and on Γ E,B it olds tat n C = 0 and s = 0. Let us write f, g ω = ω fg, were f and g are two generic functions defined on a region ω suc tat te integral of teir product is well defined. No subscript will be used wen ω = Ω. Wen f, g L 2 (Ω), we will write f, g = (f, g). Te norm in L 2 (Ω) will be denoted by f = (f, f) /2. Tese symbols will be used for scalars, vectors or second order tensors. Once equations ()-(5) are multiplied by te corresponding test functions, integrated over Ω and second order terms integrated by parts, te resulting variational form of te problem tat we consider is ( t u, v) + A uu (u, u, v) + A ub (B, B, v) + A uϑ (ϑ, v) b u (p, v) = L u (v), (2) b u (q, u) = 0, (3) ( t B, C) + A Bu (u, B, C) + A BB (B, C) + b B (r, C) = L B (C), (4) b B (s, B) = L B2 (s), (5) ( t ϑ, ψ) + A ϑu, (u, ϑ, ψ) + A ϑu,2 (u, u, ψ) + A ϑb (B, B, ψ) + A ϑϑ (ϑ, ψ) = L ϑ (ψ), (6) wic must old for all test functions v, q, C, s and ψ in te functional spaces indicated next. 5

6 Te different multilinear forms appearing in (2)-(6) are given by A uu (u, u 2, v) = v, u u 2 + ν( v, u 2 ), A ub (B, B 2, v) = µ m ρ v, ( B ) B 2, A uϑ (ϑ, v) = β(v, g ϑ), A Bu (u, B, C) = C, (u B), A BB (B, C) = C, B, µ m σ A ϑu, (u, ϑ, ψ) = ψ, u ϑ, A ϑu,2 (u, u 2, ψ) = 2µ f ρc p ψ, S u : S u 2, A ϑb (B, B 2, ψ) = ρc p µ 2 mσ ψ, ( B ) ( B 2 ), A ϑϑ (ϑ, ψ) = k t ρc p ( ψ, ϑ), b u (q, v) = (q, v), ρ b B (s, C) = ( s, C), L u (v) = v, f f + v, t ΓN,u, L B (C) = (C, f m ) + v, J Γ N,B, L B2 (s) = s, B Γ N,B, L ϑ (ψ) = ψ, Q + ψ, q ΓN,T. If we consider te functional spaces V u = {v H (Ω) d v = 0 on Γ E,u }, V p = {q L 2 (Ω) q = 0 if Γ N,u = }, Ω V B = {C H(curl, Ω) n C = 0 on Γ E,B }, V r = {s H (Ω) s = 0 on Γ E,B }, V ϑ = {ψ H (Ω) ψ = 0 on Γ E,ϑ }, all te multilinear forms introduced, except A ϑu,2 (u, u 2, ψ) and A ϑb (B, B 2, ψ), are well defined and continuous for u, u 2, u L 2 (0, T ; V u ), v V u, p D (0, T ; V p ), q V p, B, B 2, B L 2 (0, T ; V B ), C V B, r D (0, T ; V r ), s V r, ϑ L 2 (0, T ; V ϑ ), ψ V ϑ. In tese expressions, L 2 (0, T ; X) denotes te set of mappings defined on Ω (0, T ) suc tat teir X-spatial norm is an L 2 (0, T ) function. Similarly, D (0, T ; X) denotes te set of mappings for wic teir X-spatial norm is a distribution in time. 6

7 If A ϑu,2 (u, u 2, ψ) and A ϑb (B, B 2, ψ) need to be taken into account, one sould require S v W,4 (Ω) d d in te definition of V u and C L 4 (Ω) d in te definition of V B. It is also assumed tat f f V u, f m V B and Q V ϑ (a.e. in time), were X denotes te topological dual of a function space X. To close tis section, let us note tat (2)-(6) can be written as a single variational equation of te form were M( t U, V ) + A(U, V ) = L(V ), (7) U = [u, p, B, r, ϑ] t, V = [v, q, C, s, ψ] t, A(U, V ) := A uu (u, u, v) + A ub (B, B, v) + A uϑ (ϑ, v) b u (p, v) + b u (q, u) + α B [A Bu (u, B, C) + A BB (B, C) + b B (r, C) b B (s, B)] + α ϑ [A ϑu, (u, ϑ, ψ) + A ϑu,2 (u, u, ψ) + A ϑb (B, B, ψ) + A ϑϑ (ϑ, ψ)], L(V ) := L u (v) + α B [L B (C) + L B2 (s)] + α ϑ L ϑ (ψ), M(U, V ) := (u, v) + α B B, C + α ϑ ϑ, ψ. Te scaling coefficients α B and α ϑ need to be introduced to make te semilinear form A(U, V ) and te linear form L(V ) dimensionally consistent. 3 Time integration and linearization 3. Time discretization Consider a nonlinear system of ordinary differential equations of te form ẋ = F (x, t) were x : (0, T ) R n, n, is a vector function. Let us define, for θ [0, ], x n+θ := θx n+ + ( θ)x n, t n+θ := θt n+ + ( θ)t n, F n (x, t) := F (x n, t n ), were t n := nδt, x n is an approximation to x(t n ) and δt is te time step size of a uniform partition of [0, T ]. Tere are two families of one-step time integration metods, namely, te generalized trapezoidal rule and te generalized mid-point rule. Tey read: δt (xn+ x n ) = θf n+ (x, t) + ( θ)f n (x, t), δt (xn+ x n ) = F (x n+θ, t n+θ ). For linear autonomous problems, tat is to say, wen F is linear and time independent, tese two metods coincide. Te generalized mid-point rule is simpler to implement numerically. Its application to te variational problem given by (7) is given by M(δ t U n, V ) + A(U n+θ, V ) = L(V ), (8) were δ t U n = δt (U n+ U n ) = (θδt) (U n+θ U n ). Tis is te time discretization of te problem tat we will consider, altoug oter time integration scemes could be also applied to obtain te final fully discrete and linearized problem. 7

8 3.2 Linealization and block-iterative coupling In tis section we present a linearization for te time discrete problem (8). If fact, we are also interested in uncoupling te calculation of te temperature from te rest of unknowns Linearization of te stationary MHD problem Te first issue we consider is te linearization of te nonlinear terms in (8). To tis end, it is enoug to consider te termally uncoupled and stationary problem, wose differential form is given by (u )u ν u + ρ p µ m ρ ( B) B = f f u = 0, µ m σ ( B) (u B) + r = f m, B = 0. If we introduce now te vector of unknowns U = [u, p, B, r] t and te corresponding test functions V = [v, q, C, s] t, te variational problem can be written as now wit A(U, V ) = L(V ), A(U, V ) = A uu (u, u, v) + A ub (B, B, v) b u (p, v) + b u (q, u) + α B [A Bu (u, B, C) + A BB (B, C) + b B (r, C) b B (s, B)]. Taking te scaling coefficient as α B = /(µ m ρ) we ave A(U, V ) = v, u u + ν( v, u) µ m ρ v, ( B) B ρ (p, v) + (q, u) ρ + [ C, (u B) + ] ( C, B) + ( r, C) ( s, B). (9) µ m ρ µ m σ Te simplest way to linearize te problem is by a fixed point treatment of te quadratic terms. Let us assume we ave an estimate for te velocity and te magnetic field at iteration k, u k and B k, respectively, and we ave to compute tese fields at iteration k +. If e i (k) = k or e i (k) = k + and e i (k) = 2k + e i(k), te approximation of A(U, V ) at iteration k + using te fixed point metod may be written as ( v, u k+) A k+ (U, V ) = v, (u e (k) )u e (k) + ν v, ( B e2(k) ) B e (k) 2 µ m ρ ρ + [ C, (u e3(k) B e 3 (k) ) µ m ρ ( ) ( + r k+, C s, B k+)]. p k+, v + q, u k+ ρ + ( C, B k+) µ m σ 8

9 In order to ave a stable problem at eac iteration, we sould guarantee tat A k+ (U k+, U k+ ) 0, wic leads to conditions u k+ (u e (k) )u e (k) 0, Ω Ω [ ] u k+ ( B e2(k) ) B e 2 (k) B k+ (u e3(k) B e 3 (k) ) 0. Wen u k = 0, B k = 0, tese conditions old only if e (k) = k (as it is well known), e 2 (k) = k + and e 3 (k) = k +. Terefore, calling a u k, u u k+, b B k, B B k+, te only fixed point linearization of te problem tat is stable is (a )u ν u + ρ p µ m ρ ( B) b = f f, u = 0, µ m σ ( B) (u b) + r = f m, B = 0. Tis is te problem for wic te stabilized finite element metod will be presented in Section 4. Note tat te problem needs to be solved in a coupled way, witout te possibility to segregate te fluid mecanics problem from te magnetic one. Te same would appen if instead of a simple fixed-point linearization sceme, te classical Newton-Rapson linearization is used Full coupling and block-iterative coupling Let us consider again te stationary problem, but now accounting also for te termal coupling. Once we ave determined ow to deal wit te nonlinearity arising because of te velocity-magnetic field coupling, te termal coupling is easy to treat, since te temperature term in te momentum equation is linear. We may consider eiter a full coupling or a block iterative coupling. Bot options can be written in a single format as follows. If e(k) = k or e(k) = k + and e (k) = 2k + e(k), we introduce te fully linearized and coupled problem were A uu (u k, u k+, v) + A ub (B k+, B k, v) + A uϑ (ϑ e(k), v) b u (p k+, v) = L u (v), (20) b u (q, u k+ ) = 0, A Bu (u k+, B k, C) + A BB (B k+, C) + b B (r k+, C) = L B (C), b B (s, B k+ ) = L B2 (s), A ϑu, (u e (k), ϑ k+, ψ) + A ϑϑ (ϑ k+, ψ) = L k ϑ (ψ), (2) L k ϑ (ψ) = L ϑ(ψ) A ϑu,2 (u e (k), u e (k), ψ) A ϑb (B e (k), B e (k), ψ). (22) It is clear tat wen e(k) = k + (and tus e (k) = k), te problem needs to be solved for u k+, p n+, B k+, s k+ and ϑ k+ in a coupled way. Te production of eat given by A ϑu,2 (u e (k), u e (k), ψ) and A ϑb (B e (k), B e (k), ψ) needs to be evaluated at te previous iteration (unless a Newton-Rapson-type strategy is used). On te oter and, wen e(k) = k te problem can be solved first for u k+, p n+, B k+ and r k+. Once tese variables are computed, temperature may be updated by solving (2). In 9

10 tis case, it is possible to use te variables u k+ and B k+ just computed, tus leading to a Gauss- Seidel-type coupling. Of course, a Jacobi coupling, in wic L k ϑ in (22) is evaluated wit uk and B k, would also be possible. However, te computational effort is te same, and convergence is known to be faster for Gauss-Seidel-type scemes (see [4] for furter discussion). 3.3 Time discrete and linearized sceme Te next step is to consider te time discrete problem using te mid-point rule togeter wit te linearization sceme described previously. Tis leads to te following problem: For n = 0,, 2,..., given u n, p n, B n, r n and ϑ n, find u n+, p n+, B n+, r n+ and ϑ n+ as te converged solutions of te following iterative algoritm: (δ t u n,k+, v) + A uu (u n+θ,k, u n+θ,k+, v) + A ub (B n+θ,k+, B n+θ,k, v) + A uϑ (ϑ n+θ,e(k), v) b u (p n+,k+, v) = L n+θ u (v), (23) b u (q, u n+,k+ ) = 0, (24) (δ t B n,k+, C) + A Bu (u n+θ,k+, B n+θ,k, C) + A BB (B n+θ,k+, C) + b B (r n+,k+, C) = L n+θ B (C), (25) b B (s, B n+,k+ ) = L n+θ B2 (s), (26) (δ t ϑ n,k+, ψ) + A ϑu, (u n+θ,e (k), ϑ n+θ,k+, ψ) + A ϑϑ (ϑ n+θ,k+, ψ) = L n+θ,k ϑ (ψ), (27) wit te obvious definition L n+θ,k ϑ (ψ) = L n+θ ϑ (ψ) A ϑu,2 (u n+θ,e (k), u n+θ,e (k), ψ) A ϑb (B n+θ,e (k), B n+θ,e (k), ψ). In all tese expressions, te first superscript denotes te time step level and te second te iteration counter. For implementation purposes, it is very convenient to write te problem to be solved as a timediscrete system of linear convection-diffusion-reaction equations. Let us consider te case e(k) = k + in (20), te case e(k) = k being similar, and let us call a u n+θ,k, u u n+θ,k+, b B n+θ,k, B B n+θ,k+, ϑ ϑ n+θ,k+ and δ t f n = (θδt) (f f n ) for any of te unknowns f. Te differential equations associated to (23)-(27) are were δ t u + (a )u ν u + ρ p µ m ρ ( B) b + gβϑ = f f, u = 0, δ t B + µ m σ ( B) (u b) + r = f m, Q tot = Q B = 0, δ t ϑ + a ϑ k t ρc p ϑ = Q tot, ρc p µ 2 mσ b 2 2µ f ρc p S a 2. Te problem considered can be written as te vector differential equation Mδ t U + L(U) = F in Ω, (28) 0

11 were M = diag(i, 0, α B I, 0, α ϑ ), I being te d d identity, δ t U = (δt) (U U n ), wit U n known, F = [f f, 0, α B f m, 0, α ϑ Q tot ] t is a known vector of n unk = 2d + 3 components and te (scaled) operator L is given by Tis is an operator of te form a u ν u + ρ p + µ mρb ( B) + gβϑ u ] L(U) = α B [ (u b) + µ ( B) + r mσ. (29) α B B ] α ϑ [a ϑ kt ρc p ϑ L(U) := A i i U i (K ij j U), were A i and K ij are n unk n unk matrices (i, j =,..., d) wose identification is obvious and will be omitted. Here and in wat follows, repeated indexes imply summation up to te number of spatial dimensions and i is te partial derivative wit respect to te Cartesian coordinate x i. Let matrices A i be split as A i = A c i + A f i, were A c i is te part of te convection matrices wic is not integrated by parts and A f i te part tat is integrated by parts. In our case, matrices A f i come from te first order derivatives of te ydrodynamic pressure p. It would be also possible to integrate by parts te first order derivatives corresponding to te terms u u and (u B). Te weak form of te problem supplied wit te appropriate omogeneous boundary conditions can be written again as te time discrete and linearized counterpart of (7): were M(δ t U, V ) + A lin (U, V ) = L(V ), A lin (U, V ) := (V, A c i i U) ((A f i) t i V, U) + ( i V, K ij j U), and L(V ) := (V, F ) plus boundary contributions tat depend on te problem. For te particular case of MHD, taking α B = /(µ m ρ): A lin (U, V ) = v, a u + ν( v, u) µ m ρ v, ( B) b ρ (p, v) + (q, u) ρ + [ C, (u b) + ] ( C, B) + r, C s, B. (30) µ m ρ µ m σ From now on, we will redefine p p/ρ, tat is to say, we will work wit te kinematic pressure. 4 Stabilized formulation for te stationary, linearized and termally uncoupled problem 4. Stability of te Galerkin approximation Consider te linearized uncoupled stationary problem. Its variational form is: Find U W suc tat A lin (U, V ) = L(V ) V W, (3)

12 were W = V u V p V B V r. If we assume tat a = 0, b = 0, A lin satisfies te stability estimate A lin (U, U) = ν u 2 + µ m ρµ m σ B 2. (32) Tis stability estimate and te inf-sup conditions between V u and V p and between V B and V r, given by inf sup q V p\{0} v V u\{0} inf sup s V r\{0} C V B \{0} (q, v) q v β f > 0, (33) ( s, C) s (l C + C ) β m > 0, (34) were β f and β m are constants, are enoug to guarantee tat te linearized problem is well posed. Te parameter l as dimensions of lengt and as been introduced wit te only purpose of aving a correct scaling for te norm in V B H(curl, Ω). Terefore, for eac iteration k, given u k and B k tere is a unique U k+ = (u k+, p k+, B k+, r k+ ), solution of te linearized problem (3). It can be sown tat, under te same condition for wic te nonlinear problem as a unique solution (see [28]), te sequence {U k } k 0 converges (strongly) to te (unique) solution of te nonlinear problem. Te proof of tis result is tecnical, but follows te same strategy as for te stationary Navier- Stokes equations witout magnetic coupling. Remark 2 Since te exact solution of te problem is r = 0, one could replace (4) by l 2 σµ m r + B = 0, were l is again a lengt scale. If tis is done, te term l 2 σ ρ r 2 sould be added to te rigt-and-side of (32), so tat additional stability is obtained on te magnetic pseudo-pressure. We will not use tis possibility ere, altoug it may be needed if a metod able to converge to singular solutions is required (see Remark 3). 4.2 Te subgrid scale framework for a general CDR system of equations Te basic idea of te stabilization metod proposed ere is based on te subgrid scale concept introduced in [8] and tat can be also found in different contexts (not only numerical). Wat follows is a summary of te approac described in [6]. Te starting idea is to split te continuous space as W = W W, were W is te finite element space (and terefore finite dimensional) in wic te approximate solution will be sougt. We call W te space of subscales or subgrid scales. It is readily cecked tat te continuous problem can be written as te system of equations: A lin (U, V ) + A lin (Ũ, V ) = L(V ) V W, (35) A lin (U, Ṽ ) + Alin (Ũ, Ṽ ) = L(Ṽ ) Ṽ W, (36) were U = U + Ũ and U W, Ũ W. It is useful for te following to introduce te notation n el, :=, Ω e,, :=, Ω e, e= were n el is te number of elements of te finite element partition used to build W and Ω e denotes te domain of element number e. n el e= 2

13 Integrating by parts all te terms in A lin (Ũ, V ) in (35) and te left-and-side terms of (36) witin eac element domain, we get A lin (U, V ) + Ũ, ni (K ij j V (A f i) t V ) Ũ, + L (V ) = L(V ), (37) Ṽ, ni [K ij j (U + Ũ) Af i(u + Ṽ Ũ)] +, L( Ũ) Ṽ =, F L(U ), (38) were n i is now te i-t component of te exterior normal to Ω e and L is te adjoint operator of L wit omogeneous Diriclet conditions, given by Equation (38) is equivalent to: L (U) := i (A t iu) i (K t ij j U). L(Ũ) = F L(U ) + V,ort in Ω e, (39) Ũ = Ũ ske on Ω e, (40) were V,ort is obtained from te condition tat Ũ must belong to W (and not to te wole space W ) and Ũ ske is a function defined on te element boundaries and suc tat q n := n i [K ij j (U + Ũ) Af i(u + Ũ)] is continuous across interelement boundaries, and terefore te first term in te left-and-side of (38) vanises. Different subgrid scale (SGS) stabilization metods can be devised depending on te way problem (39)-(40) is approximated. Te purpose of tis paper is not to propose a new metodology, but rater to see ow to apply a well establised formulation to te incompressible MHD problem. Tis well known metod can be obtained by approximating te subscales by te algebraic expression Ũ τ P [F L(U )], (4) were τ is a n unk n unk matrix of stabilization parameters, te expression of wic is discussed in te following subsection, and P is te projection onto te space of subscales. Tere are two main possibilities for coosing P and, terefore, for determining te space of subscales. Te most common one is to take P = I, te identity, wen applied to te finite element residual appearing in te rigtand-side of (4). Anoter possibility, wic is actually te one tat we favor, is to take P as te L 2 -projection onto te space ortogonal to te finite element space [7, 8]. For te sake of conciseness, we will restrict ourselves to te first option ( P = I), altoug te second one as clear advantages, particularly in time dependent problems []. Te design of te stabilization parameters, wic is our main concern, is te same using bot approaces. To close te approximation, we neglect te interelement boundary terms in (37), wic can be understood as taking Ũ ske = 0 on te interelement boundaries. Te final problem is: find U W suc tat A lin (U, V ) + Ũ, L (V ) = L(V ) V W, wic, upon substitution of te subscales by (4) wit P = I, yields te following discrete problem: find U W suc tat A lin stab (U, V ) = L stab (V ) V W, (42) were te bilinear form A lin stab and te linear form L stab are now given by A lin stab (U, V ) = A lin (U, V ) L (V ), τ L(U ), (43) L stab (V ) = L(V ) L (V ), τ F. (44) 3

14 4.3 Stabilized finite element approximation for te linearized MHD problem In tis subsection we present a stabilized finite element metod to approximate problem (3). First of all, we recast it as a system of linear convection-diffusion equations. It is in tis general setting tat te finite element approximation will be described. Stabilization for tis problem as several goals. Te first is to avoid te need to satisfy te discrete counterpart of te inf-sup conditions (33) and (34), wic would lead to different interpolation for te variables of te problem. To fix ideas, we will assume equal and continuous interpolation for all te unknowns. Te second goal is to obtain error estimates valid in te limit ν 0 and σµ m, tat is, for convection-dominated flows (bot in te Navier-Stokes equation and te equation for te magnetic field). Finally, te tird objective is to account properly for te coupling of te ydrodynamic and te magnetic problems (case ρµ m 0). Tat tese goals are all satisfied will be seen in te error estimate to be presented. Up to now we ave described te algebraic version of te SGS stabilization in a general setting. Te objective now is to apply it to te MHD problem we are considering. In particular, te adjoint operator of te linearized uncoupled MHD problem L (V ), taking once again α B = /(µ m ρ), is now given by a v ν v q ρµ m b ( C) L v (V ) = ρµ m (v b) + ρµ mσµ m ( C) ρµ m s. (45) ρµ m C To define te metod for te particular MHD problem, an expression for te matrix of stabilization parameters τ needs to be proposed. To our knowledge, tere is no general way to define it for systems of equations [6]. It must be designed for eac particular problem taking into account its stability deficiencies. In te case we are considering, we will see in te following subsection tat stability can be improved maintaining optimal accuracy by taking a simple diagonal expression for τ, wit one scalar component for eac of te equations. In te 3D case, we take τ = diag(τ, τ, τ, τ 2, τ 3, τ 3, τ 3, τ 4 ). (46) Using tis expression and (45), it follows tat te stabilized bilinear form tat we ave to consider in problem (42) is A lin stab (U, V ) = A lin (U, V ) L (V ), τ L(U ) = a u, v + ν( u, v ) (p, v ) + (q, u ) + B, (v b) C, (u b) ρµ m ρµ m + ( B, C ) + ( r, C ) ( s, B ) ρµ m σµ m ρµ m ρµ m + X u (v, q, C ) + ν v, τ (X u (u, p, B ) ν u ) + v, τ 2 ( u ) + X B (s, v ) ρµ m σµ m ( C ), τ 3 (X B (r, u ) + ( B )) ρµ m σµ m + ρ 2 µ 2 C, τ 4 ( B ), (47) m 4

15 were we ave introduced te abbreviations X u (v, q, C ) := a v + q + ρµ m b ( C ), X B (s, v ) := ρµ m (v b) + ρµ m s. Finally, te rigt-and-side of te stabilized problem (44) is now given by L stab (V ) = L(V ) L (V ), τ F = f f, v + ρµ m (f m, C) + X u (v, q, C ) + ν v, τ f f + X B (s, v ) ( C ), τ 3 f ρµ m σµ m m. (48) Te definition of te stabilized finite element metod is now complete up to te expression of te stabilization parameters. Te expressions we propose are te following: a α := c + c ν 2 2, β := c b 3 ρµ m, γ := c 4 ρµ m σµ m 2 ( τ = α + ) 2 β, τ 2 = c 5 (49) αγ τ ( τ 3 = γ + ) β, τ 4 = c 6 ρ 2 µ 2 2 αγ m (50) τ 3 It is understood tat tese expressions are evaluated element by element. Here, a is te maximum norm of te velocity field a computed in te element under consideration. Likewise, b denotes te maximum norm of b in tis element, and te element diameter. Te constants c i, i =, 2, 3, 4, 5, 6, are independent of te pysical parameters of te problem and of te mes discretization. In te numerical calculations we take tem as c = 2, c 2 = 4, c 3 =, c 4 = 4, c 5 =, c 6 =. In te following subsection we justify in detail tis coice from te numerical analysis of te problem. We will proceed in a constructive manner, posing conditions on te stabilization parameters obtained from te requirement tat te metod is stable (coercive) and optimally accurate. Remark 3 At tis point it is important to remark a property of te stabilized formulation we propose concerning convergence to non-regular solutions. In te following analysis we consider te simplest situation, trying to avoid tecnicalities and focusing our attention on te convenience of te formulation designed. We will prove stability of te metod and convergence to smoot solutions, assuming all te regularity we need. However, it is known tat te problem may exibit solutions for te magnetic field wic are strictly in H 0 (curl, Ω), not even in (H (Ω)) d. Tis appens wen Ω as re-entrant corners [2]. Metods based on penalization, wic introduce some sort of control on te divergence of te magnetic field, fail to converge to tese non-regular solutions. Te remedy is eiter to weigt te terms tat give tis divergence control, so as to skip it near singularities [3, 6], or to use mixed interpolations (including te magnetic pseudo-pressure). Te former option does not seem easily applicable to general tree-dimensional problems, wereas te latter requires te discrete version of te inf-sup condition (34). It is not easy to satisfy it (see [28, 7] for possible ways to comply wit it). In particular, we are not aware of continuous interpolations satisfying it. Tus, stabilized metods, as te one proposed ere, seem to be te only alternative to use te simple continuous Lagrangian approximation 5

16 of te magnetic field. However, in te formulation we propose te last term in (47) gives also control on te divergence of B. In order to converge to non-regular solutions, expressions of τ 4 smaller tan te one given by (50) migt be required. 4.4 Numerical analysis and design of te stabilization parameters In tis subsection we proceed to analyze te formulation introduced above and, in particular, to justify te coice (49)-(50). For te sake of simplicity in te notation, we will assume tat a and b are constant. Likewise, we will assume tat te finite element meses are quasi-uniform. In tis case, in (49)- (50) can be taken te same for all te elements (te maximum element diameter), and terefore τ i, i =, 2, 3, 4, are also constant. Moreover, for quasi-uniform meses te following inverse estimates old: v C inv v, v C inv v, (5) for any function v in te finite element space and for a certain constant C inv. Te stability and convergence analysis will be made using te following mes-dependent norm: U 2 := ν u 2 + ρµ m σµ m B 2 + τ a u + p + ρµ m b ( B ) 2 + τ 2 u 2 + τ 3 ρ 2 µ 2 (u b) + r 2 + τ 4 m ρ 2 µ 2 B 2 m ν u 2 + B 2 ρµ m σµ m + τ X u (u, p, B ) 2 + τ 2 u 2 + τ 3 X B (r, u ) 2 + τ 4 ρ 2 µ 2 B 2. (52) m In all wat follows, C will denote a positive constant, not necessarily te same at different appearances Coercivity Let us start by proving stability in te form of coercivity of te bilinear form (47). It is immediately cecked tat A lin stab (U, U ) = A lin (U, U ) L (U ), τ L(U ) = ν u 2 + B 2 ρµ m σµ m + τ X u (u, p, B ) 2 τ ν 2 u 2 + τ 2 u 2 + τ 3 X B (r, u ) 2 τ 3 ρ 2 µ 2 mσ 2 µ 2 B 2 + τ 4 m ρ 2 µ 2 B 2. m Using te second inverse estimate in (5), it is clear tat a sufficient condition for A lin is tat ν τ ν 2 C2 inv 2 αν τ ( α) ν Cinv 2 τ 3 ρµ m σµ m ρ 2 µ 2 mσ 2 µ 2 m 2 2 C 2 inv stab to be coercive, (53) 2 α τ 3 ( α)ρµ m σµ m, (54) ρµ m σµ m C 2 inv 6

17 wit 0 < α <. Conditions (53) and (54) yield A lin stab (U, U ) C U 2, (55) for a constant C independent of te discretization and of te pysical parameters (it depends only on te constants of te stabilization parameters) Optimal accuracy We ave obtained conditions (53) and (54) on te stabilization parameters by requiring stability. Te rest of conditions will be obtained by imposing tat te stabilized metod proposed is optimally accurate, wic will lead to optimal convergence. For a function v, let π (v) be its optimal finite element approximation. We assume tat te following interpolation estimates old: v π (v) H i (Ω) ε i (v) := C k+ i v H k+ (Ω), i = 0,, (56) were v H q (Ω) is te H q (Ω)-norm of v, tat is, te sum of te L 2 (Ω)-norm of te derivatives of v up to degree q (and tus te H 0 (Ω)-norm coincides wit te L 2 (Ω)-norm), v H q (Ω) te corresponding semi-norm, and k is te degree of te finite element approximation. We will prove in te following tat te error function of te formulation is E() := τ /2 ε 0 (u) + τ /2 2 ε 0 (p) + τ /2 3 ε 0 (B) + τ /2 4 ε 0 (r). (57) Te conditions on te stabilization parameters we will obtain will in fact sow tat tis is indeed te error function and tat tis error function is optimal. Let U be te solution of te continuous problem and π (U) its optimal finite element approximation. Te accuracy estimate tat will be needed to prove convergence later on is A lin stab (U π (U), V ) CE() V, (58) for any finite element function V. Let us prove tis by sowing tat bot te Galerkin and te stabilization terms in A lin stab satisfy estimate (58) for sufficiently smoot solutions of te continuous problem. Starting wit te Galerkin contribution, integrating some terms by parts we ave tat A lin (U π (U), V ) = (u π (u), a v ) + ν( (u π (u)), v ) (p π (p), v ) ( q, u π (u)) (u π (u), b C ) + (B π (B), (v b)) ρµ m ρµ m + ( C, (B π (B))) (r π (r), C ) ρµ m σµ m ρµ m ( s, B π (B)) ρµ ( m C ε 0 (u)τ /2 τ /2 X u (v, q, C ) + ν /2 ε (u)ν /2 v + ε 0 (p)τ /2 2 τ /2 2 v + ε 0 (B)τ /2 3 τ /2 3 X B (s, v ) + + ε 0 (r)τ /2 4 τ /2 4 (ρµ m σµ m ) /2 ε (B) (ρµ m σµ m ) /2 C ) C. (59) ρµ m 7

18 Conditions (53) and (54) and te expression of te interpolation errors imply ν /2 ε (u) Cε 0 (u)τ /2, (ρµ m σµ m ) /2 ε (B) Cε 0 (B)τ /2 3, and terefore from (59) it follows tat te Galerkin contribution A lin (U π (U), V ) to A lin stab (U π (U), V ) can be bounded as indicated in (58). It remains to prove tat also te stabilization terms can be bounded in tis way: L (V ), τ L(U π (U)) = X u (u π (u), p π (p), B π (B)) ν (u π (u)), τ (X u (v, q, C ) + ν v ) + (u π (u)), τ 2 v + X B (r π (r), u π (u)) + (B π (B)), ρµ m σµ m τ 3 (X B (s, v ) C ) ρµ m σµ m + ρ 2 µ 2 (B π (B)), τ 4 C m ( ) C τ /2 X u (u π (u), p π (p), B π (B)) + τ /2 ν (u π (u)) ( ) V + τ /2 ν v + Cτ /2 2 ε (u) V ( + C τ /2 3 X B (r π (r), u π (u)) + τ /2 ) 3 (B π (B)) ρµ m σµ m ( V + τ /2 ) 3 C ρµ m σµ m + Cτ /2 4 ε (B) V. (60) ρµ m Using once again conditions (53) and (54) and te inverse estimates (5) we ave tat τ /2 ν v Cτ /2 ν /2 C inv ν/2 v C V, τ /2 3 C Cτ /2 C inv 3 ρµ m σµ m (ρµ m σµ m ) /2 (ρµ m σµ m ) /2 C C V. So far, we ave not posed any additional conditions on te stabilization parameters oter tan (53) and (54), found from te requirement of coercivity. Te rest of conditions will come from te requirement of optimal accuracy. 8

19 From (60) we ave tat L (V ), τ L(U π (U)) ( C V τ /2 a ε 0(u) + ν 2 ε 0(u) + ε 0(p) + ) b ρµ m ε 0(B) + C V τ /2 2 ε 0(u) ( + C V τ /2 b 3 ρµ m ε 0(u) + σµ m 2 ε 0(B) + ) ε 0(r) + C V τ /2 4 ρµ m ε 0(B) [ ] ( C V τ /2 a + ν ) 2 + τ /2 3 b ρµ m + τ /2 2 ε 0 (u) [ ] + C V τ /2 ε 0 (p) [ ] τ /2 b + C V ρµ m + τ /2 3 ρµ m σµ m 2 + τ /2 4 ε 0 (B) ρµ m + C V [ τ /2 3 ρµ m ] ε 0 (r). From te definitions (49)-(50) of te stabilization parameters it follows tat tese terms can be bounded also as indicated in (58). Remark 4 Te last step provides in fact te crucial design condition for te stabilization parameters. Expressions (49)-(50) follow by imposing tat ( τ /2 a + ν ) 2 + τ /2 3 b ρµ m + τ /2 2 τ /2 τ /2 b ρµ m + τ /2 3 ρµ m σµ m τ /2, τ /2 2, 2 + τ /2 4 ρµ m τ /2 3, τ /2 3 ρµ m τ /2 4, were stands for equality up to constants. Te solution of tese set of conditions is precisely (49)-(50) Convergence As a consequence of te properties of stability and accuracy in te sense of (58), it is trivial to sow tat te metod is optimally convergent. From te ortogonality property A lin stab (U U, V ) = 0 for any finite element function V (a direct consequence of te consistency of te metod), we ave tat C π (U) U 2 A lin stab (π (U) U, π (U) U ) A lin stab (π (U) U, π (U) U ) CE() π (U) U, 9

20 from were π (U) U CE(). Now te triangle inequality implies U U U π U + π (U) U U π U + CE(). A trivial ceck using te expression of te norm given by (52), te interpolation estimates (56) and te stabilization parameters (49)-(50) sows tat U π U CE(), from were U U CE(). (6) Te fact tat tis error estimate is exactly te same as te estimate for te interpolation error U π U CE() justifies wy it as to be considered optimal. Moreover, a simple inspection of wat appens in te limit of dominant second order terms sows tat in tis case te error estimate reduces to te estimate tat could be found using te Galerkin metod using finite element spaces satisfying te discrete form of (33)-(34), but now, owever, using equal interpolation for all te variables. Likewise, in te limit ν 0 and σµ m, te error estimate (6) does not blow up and te result can also be considered optimal. 5 Final numerical sceme Te final numerical sceme tat we propose is obtained by applying te finite element stabilization tecnique described in Section 4 to te time discrete and linearized problem (23)-(27). Te space discretization of tese equations, adding stabilization terms as tose tat appear in (47) and (48) for te stationary termally uncoupled problem, will lead to te following algoritm: For n = 0,, 2,..., given u n, p n, B n, r n and ϑ n, find u n+, p n+, B n+, r n+ and ϑ n+, as te converged solutions of te 20

21 following iterative algoritm: (δ t u n,k+, v ) + A uu (u n+θ,k, u n+θ,k+, v ) + A ub (B n+θ,k+, B n+θ,k, v ) + A uϑ (ϑ n+θ,e(k), v ) b u (p n+,k+, v ) + u n+θ,k v + ν v, τ n+θ,k R n+θ,k+,u + v, τ n+θ,k 2 R n+θ,k+,p + (v B n+θ,k ), τ n+θ,k 3 R n+θ,k+,b = Ln+θ u (v ), (62) b u (q, u n+,k+ ) + q, τ n+θ,k R n+θ,k+,u = 0, (63) (δ t B n,k+, C ) + A Bu (u n+θ,k+ + b B (r n+,k+, C ) + µ m ρ ( C ) B n+θ,k + µ m σ ( C ), τ n+θ,k 3 R n+θ,k+,b + C, τ n+θ,k 4 R n+θ,k+,r, B n+θ,k, C ) + A BB (B n+θ,k+, C ), τ n+θ,k R n+θ,k+,u = Ln+θ B (C ), (64) b B (s, B n+,k+ ) + s, τ n+θ,k 3 R n+θ,k+,b = Ln+θ B2 (s ), (65) (δ t ϑ n,k+, ψ ) + A ϑu, (u n+θ,e (k), ϑ n+θ,k+, ψ ) + A ϑϑ (ϑ n+θ,k+, ψ ) + u n+θ,k ψ + k t ψ, τ n+θ,k 5 R n+θ,k+ ρc,ϑ = L n+θ,k T (ψ ), (66) p were we ave introduced te residuals R,u := δ t u + a u ν u + p µ m ρ ( B ) b + gβϑ f f, R,p := u, R,B := δ t B + µ m σ ( B ) (u b) + r f m, R,r := B, R,ϑ := δ t ϑ + a ϑ k t ρc p ϑ Q tot. Te superscript n + θ, k + in (62)-(66) denotes tat tese residuals are evaluated wit u, p, B, r and ϑ at tis time step and iteration counter, wereas now a u n+θ,k, b B n+θ,k. Te stabilization parameters τ i, i =, 2, 3, 4, are given in (49)-(50) (and computed witin eac element), wereas ( ) a τ 5 = c + c k t 2 ρc p 2. Note tat te termal coupling effect as been neglected in te design of te stabilization terms. Tis approximation is justified in [0]. 2

22 Figure : Flow over a circular cylinder. Domain and finite element mes. 6 Numerical examples In tis section we present tree numerical example designed to sow tat te formulation presented in tis paper satisfies te main requirements posed for its design, namely, it yields smoot solutions for te termally coupled MHD problem in a wide rank of te pysical parameters. To sow tis in te simplest possible setting, tree numerical examples will be discussed. For a convergence test demonstrating optimal accuracy order and for results obtained in te classical Hartmann s problem, see [9]. As a general comment from te pysical point of view, let us mention tat it is well known tat te presence of magnetic fields in flows were oscillations can occur precludes teir onset. Te examples to be presented sow clearly tis beavior. 6. Flow over a circular cylinder Tis numerical simulation was taken from []. Te problem consists in te flow of a conducting fluid around a 2D circular cylinder wile a magnetic field is imposed. From te pysical point of view, te main objective in tis numerical simulation is to observe te vanising of te vortexes sed by te cylinder. Our interest is to sow tat te stabilized formulation we propose yields te correct time response, wit te correct coupling between te ydrodynamic and te magnetic problem. Te domain and te finite element mes used to discretize it are sown in Fig.. Tis mes is made of 4000 linear triangular elements and 200 nodes. Te boundary conditions for tis simulation consist of an imposed constant velocity at te inlet (left boundary), zero normal velocity on te upper and lower boundaries of te domain, zero velocity on te cylinder and free velocity at te outlet. Te boundary conditions for te magnetic field consist of an imposed normal component on te upper and lower boundaries of te domain wit zero current (condition (9)), an imposed tangential component at te inlet and te outlet (condition (8)), and a tangent component also on te cylinder. Numerical simulations ave been carried out for Re = 00 and te following Hartmann numbers: Ha = 0.0,.0, 2.5, and 0.0. Tese Hartmann numbers are obtained increasing te imposed magnetic field. Te magnetic Reynolds Number is Re m = 0 5 and te coupling parameter is taken as S = Ha 2 /Re m Re. Te intended effect of te increment in te Hartmann number is to suppress te vortex sedding beind te cylinder. Tis effect can be observed in Fig. 2, were te contours of te norm of te velocity field at a certain time step are sown for Ha = 0.0, Ha =.0, Ha = 2.5 and Ha = 0. In tis last case, not only te sedding as disappeared, but also te vortices ave been drastically reduced. 22

23 Figure 2: Flow over a circular cylinder. Velocity contours. From te top to te bottom: Ha = 0.0,.0, 2.5 and 0. 23

24 Figure 3: Flow over a circular cylinder. Time evolution of te drag coefficient for Ha = 0.0 (left) and Ha =.0 (rigt) Te suppression of te vortex sedding can also be observed from Fig. 3, were te time evolution of te drag coefficient is plotted for Ha = 0.0 and Ha =.0. Te spontaneous sedding occurs approximately at te same time, but te amplitude is clearly iger for Ha = 0.0 tan for Ha =.0. At Ha = 2.5 it can be seen tat te oscillations in te drag coefficient ave vanised (not sown). 6.2 Clogging in continuous casting of steel Te main objective of tis numerical simulation is to observe te beavior of te flow in a continuous casting nozzle wile a magnetic field is applied. In operations of continuous casting a very serious problem is te clogging of te nozzles. Tis is particularly problematic wen low carbon steels are casted because some deoxidation products (e.g. alumina) get attaced to te walls of te nozzle, forming buildups. Tese buildups can eventually prevent te flow of steel troug te nozzle. Tis can lead to a decrease in te quality of te steel or even to stop te continuous casting operation and diminis te productivity (see [27]). Te origin of te buildups in te nozzle is associated to te appearance of a recirculation zone at te entry of te nozzle. Tis recirculation zone is originated by a deattacment of te flow. Altoug te nozzle can be designed to prevent recirculation, even a small misalignment can originate a deattacment. In order to prevent te recirculation of te flow, te use of a magnetic field as been proposed (see [20, 2]). Te magnetic field used to suppress te recirculation is produced by a coil oriented coaxially wit te flow. Te general effect of te magnetic field is to produce a radial force over te fluid and terefore it tends to attac to te walls of te nozzle. In order to perform te numerical simulation of te process, a 2D model corresponding to a nozzle section as been constructed. Te computational domain considered, as well as te magnetic field prescribed, are sown in Fig. 4 (left). Zero velocity as been prescribed on te walls of te domain. A mes of 8282 linear triangular elements wit 9335 nodes as been used. Te following Hartmann numbers ave been considered: Ha =.0, 0.0, 50.0 and Te Reynolds number tat as been taken is Re and te magnetic Reynolds Re m = Te nature of tis example is purely qualitative because tere is no numerical bencmark to compare wit. Te dimensions and general setting of tis example ave been taken from [2], were te approac to tackle tis problem is purely analytical. Due to te dynamical nature of tis example te Hartmann number used in order to get a uniform velocity field for te fluid is really ig. As it can be seen from 24

25 Figure 4: Clogging in continuous casting. Computational domain (left) and finite element mes (rigt) Figure 5: Clogging in continuous casting. Velocity vectors for different Hartmann numbers. From te left to te rigt and from te top to te bottom: Ha =, Ha = 0, Ha = 50 and Ha =

26 Figure 6: Crystal growt example. Computational domain (top), cross section (bottom left) and boundary conditions (bottom rigt) Fig. 5, te use of magnetic fields in te nozzle precludes te onset of te recirculation zone. Te velocity of te fluid tends to get uniform and terefore te occurrence of buildups is avoided. 6.3 Crystal growt Te main objective of tis numerical simulation is to observe te beavior of te molten silicon inside a crucible, in te so called Czocralski process. Te numerical simulation to be presented is based on te one proposed by Bückle and Scäfer [3]. Te numerical modeling of tis crystal growt process is quite complex because it involves a eat transfer problem togeter wit te MHD problem. In te Czocralski process, te eat convection gives rise to fluid motion wic can be armful for te crystalline structure of te silicon. Basically, convection movements can introduce structural defects in te crystal. By applying an intense magnetic field, tese convection movements are damped inside te crucible and te defects are diminised or completely eliminated. Cylindrical coordinates were used, assuming te solution to be axisymmetric, altoug wit a non-zero azimutal velocity. Tis, togeter wit te axisymmetry of te domain, allowed us to simulate only alf of a cross section. A simplified geometry for tis problem is presented in Fig. 6. Te general setting for tis numerical bencmark is also depicted in Fig. 6. As it can be seen tere, te problem consists in a vertical cylindrical crucible filled wit molted silicon up to a eigt H, wic is rotating wit angular velocity Ω C. Te coaxial crystal on te top of te crucible is also rotating, but in te sense opposite to te crucible and wit angular velocity Ω X. It is assumed tat te crystal and te crucible are isotermal, wit temperatures T X and T C, respectively. Boundary conditions for tis numerical simulation are also indicated in Fig. 6. Te boundary conditions for te fluid velocity consist of non-slip conditions on te crucible walls and on te crystal, and 26

27 Figure 7: Crystal growt example. Velocity vectors. Left: case A, rigt: case C. From te top to te bottom: Ha = 0, Ha = 5 and Ha = 0. 27

28 Figure 8: Crystal growt example. Temperature contours for case C. Top left: Ha = 0, top rigt: Ha = 5, bottom: Ha = 0 00 Steady state evolution Figure 9: Crystal growt example. Convergence towards te steady state (case D2, Ha = 0), measured as te norm of δ t U in time normalized by δ t U in te first time step and in percentage. 28