Stabilized finite element for fluid flow

Let us define the function spaces Sv and Sp as follows:

Sv ^def= {v|v ∈ (L₂(Ω))^nsd, divv ∈ L2(Ω), v = 0 on Γ} (3.14)

The classical Galerkin formulation for the flow problem, given by Eqs. (3.1) and (3.2) in Table 3.1, can be stated as follows: Find V ^def= {v, p} ∈ Sv × Sp such that

Here, ⊗ denotes a tensor product. Terms involving gradient of density, ∇ρ, arise due to varying densities in the mushy zone. In introducing the FEM used here, let us first define a modified pressure space S_p^′ as follows:

S_p^{′ def}= {p|p ∈ H¹(Ω), Z

Ω

qdΩ = 0} (3.18)

The stabilized weak form proposed here is the following: Find V = {v, p} ∈ Sv ×Sp^′

such that for all W = {w, q} ∈ Sv × Sp^′ the following holds:

Bstab(W , V ) = Lstab(W ) (3.19)

where:

where v∗ denotes the velocity obtained from the previous iteration. The stabilizing terms in Eqs. (3.20) and (3.21) can be derived by various techniques including a least squares procedure. In addition to those discussed in [85], extra stabilizing terms arise due to change in density in the mushy zone.

Let us now consider a given finite element partition ¯Ω = Snel

e=1Ω¯^e. To avoid high sensitivity of the fluid flow simulator on the variation of ǫ or f , we neglect terms that involve gradients of these fields in the implementation of Eq. (3.19). This leads to simplification of the final weak form of equations and helps in improving the convergence rate. We next define, for the given finite element partition, spaces Sv and S^h p^h′ as follows:

Sv^{h def}= {v^h|v^h ∈ S_v, v^h ∈ (C^o( ¯Ω))^nsd, v^h|_Ω^e ∈ (P (Ω^e))^nsd, e = 1, 2, . . . , nel} (3.24)

S_p^{h′ def}= {p^h|p^h ∈ S_p, p^h ∈ C^o( ¯Ω), p^h|_Ω^e ∈ P (Ω^e), e = 1, 2, . . . , nel} (3.25)

The final FEM is then posed as follows: Find V^h = {v^h, p^h} ∈ Sv × S^h p^h′ such

The stabilizing contributions from the advective, Darcy, viscous and pressure terms are expressed as:

(D) η^h = τ₄^e∇q^h (pressure term)

The form of the stabilizing term δ^h corresponds to the classical SUPG stabilizer along with an additional term arising due to varying density in the mushy zone. The form of η^h corresponds to the classical PSPG stabilizer. ζ^h arises due to varying density in the mushy zone. The stabilizing term, γ^h, which we call DSPG, comes from the Darcy term and it is an important stabilizing term for the generalized Navier-Stokes/Darcy equations. The stabilizing parameters for the advective and pressure terms are selected as follows:

τ₁^e = τ₃^e= min x∈Ω^e



τSU P G, K(ǫ)ρl

ǫµ



for the convective and viscous term, (3.31)

τ₄^e = min x^∈Ωe



τP SP G, K(ǫ)ρl

ǫµ



for the pressure term (3.32) where τ_{SU P G}, τ_{P SP G} are defined as

τSU P G = f h

2kv^hkz(Rev) (3.33)

τP SP G = f h^#

2kV^hkz(ReV ) (3.34)

Here, Rev and ReV are element Reynolds numbers based on the local velocity v^h and a global scaling velocity V^h as shown in Eqs. (2.73) and (2.74), respectively, of Chapter 2. The expressions for element lengths h and h^# along with those of z(Re) are given in detail in references [69, 70, 71, 85] and are not repeated here.

When the permeability is anisotropic, like in Eqs. (3.8) and (3.9), τ₁^e, τ₃^e and τ₄^e are defined for each direction (x, y and z) according to the component of K(ǫ) in that direction. The presence of stabilizing terms arising due to changes in density and the generalized expression taking into account isotropic or anisotropic permeability is the main difference between the current approach and the one previously described

in Chapter 2 and used by Zabaras and Samanta in [85]. The current numerical scheme is suitable for a wide class of solidification problems with strong convection.

For the Darcy term, we select τ₂^e such that the stabilization takes the form γ^h = −(1 − ǫ)w^h, i.e.

τ₂^e= (1 − ǫ)K(ǫ)ρ_l

ǫµ (3.35)

The stabilization term γ^h induced by the Darcy flow varies linearly with ǫ transi-tioning from γ^h = 0 for a pure fluid phase (ǫ = 1) to γ^h = −w^h towards a pure solid phase (ǫ → 0) similar to that described in Chapter 2.

After applying the stabilized finite element method to discretize governing equa-tions for fluid flow, we obtain the following system of equaequa-tions.

[M + Mδ+ Mγ+ Mζ]{ ˙v} + [N(v) + Nδ(v) + Nγ(v) + Nζ(v)]{v}

+ [K + Kδ+ Kγ+ Kζ]{v} + [D + Dδ+ Dγ+ Dζ]{v}

− [G + Gδ+ Gγ+ Gζ]{p} = {F(T, Cl) + Fδ(T, Cl) + Fγ(T, Cl) + Fζ(T, Cl)}, (3.36) Mη{ ˙v} + (Me + N_η(v) + Kη+ Dη) {v} + Gη{p} = {F_η(T, Cl) + Fη1}, (3.37) The matrices M, N(v), K, D and G are derived from the time-dependent, advective, viscous, Darcy and pressure terms, respectively. All these matrices incorporate effects of varying density in the mushy zone. Fη1 in Eq. (3.37) represents the forcing quantity due to rate of change of density (ρⁿ− ρⁿ⁻¹)/∆t. Subscripts δ, η, γ and ζ identify the SUPG, PSPG, DSPG and viscous stabilizing terms, respectively.

We denote the SUPG contribution as P_α^e, the PSPG contribution in the i^th (i = 1, ..., nsd) direction as E_αi^e , DSPG contribution as D_α^e, where α denotes an elemental node (α = 1, ..., n^e_nodes) in the e^th element. They are defined as follows:

P_α^e = 1

fτ₁^ev^h· ∇N_α^e (3.38)

E_αi^e = τ₄^eN_α,i^e (3.39)

D_α^e = −(1 − ǫ)N_α^e (3.40)

Z_α^e denotes the contribution from viscous terms. After simplifying and neglecting some terms involving second derivatives of shape functions, it takes the following form:

Z_α^e = τ₁^e2µ ρρl

∇ρ · ∇N_α^e (3.41)

Matrices and vectors in Eqs. (3.36) and (3.37) are described in [90]. The time integration of Eqs. (3.36) and (3.37) is performed using a backward Euler scheme with ˙v expressed as

˙v = vⁿ− vⁿ⁻¹

∆t (3.42)

The solution procedure, which is similar to that described in [85], is not repeated here.

Calculation of shrinkage: When densities of the solid and liquid phases are dif-ferent, decrease in volume or shrinkage occurs due to progressive phase change. To feed the resulting volume contraction, a riser is provided near the insulated end.

Our approach here is different from that of [44], where an inlet velocity boundary condition was provided on a portion of the top surface. Feeding velocity in [44] was calculated from the volume required to compensate shrinkage. In our work, liquid from the riser feeds the shrinkage and this leads to an effective reduction in the domain size. The change in volume at each time step is calculated according to the following relation, previously described in [19] :

∆v = Z

Ω

ρs− ρl

ρ_l

∂ǫ

∂tdΩ (3.43)

To account for this change, the top surface representing the riser is moved downwards like a rigid lid whose velocity is determined from ∆v. This approach neglects any

changes in curvature that may occur at the top surface due to surface tension effects.

3.3 Finite Element Scheme for energy and species