Stability and finite element approximation of phase change models for natural convection in porous media

Woodfield, James; Alvarez, Mario; Gómez-Vargas, Bryan; Ruiz–Baier, Ricardo

doi:10.1016/j.cam.2019.04.003

Cited by 23 publications

(13 citation statements)

References 40 publications

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“…This is naturally achieved in finite-element methods with dynamical mesh adaptivity (Danaila et al, 2014), while in finite-volume methods with fixed grids, the time step has to be adapted to the space resolution (Ma and Zhang, 2006). Versions of the variable viscosity approach were theoretically studied by Aldbaissy et al (2018); Woodfield et al (2019). A new formulation of the phase-change problem using as variables the pseudo-stress, strain rate and velocity for the Navier-Stokes-Brinkman equations was recently suggested by Alvarez et al (2019).…”

Section: Introductionmentioning

confidence: 99%

“…Versions of the variable viscosity approach were theoretically studied by Aldbaissy et al (2018); Woodfield et al (2019). A new formulation of the phase-change problem using as variables the pseudo-stress, strain rate and velocity for the Navier-Stokes-Brinkman equations was recently suggested by Alvarez et al (2019).…”

Section: Introductionmentioning

confidence: 99%

“…Temperature is discretized using P 2 or P 1 finite elements. The choice of using Taylor-Hood finite elements for the fluid flow and P 2 for the temperature was also made in Woodfield et al (2019); Belhamadia et al (2019). It offers second order accuracy and (inf-sup) stability of the space discretization of the fluid flow.…”

Section: Introductionmentioning

confidence: 99%

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A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Rakotondrandisa

Sadaka

Danaila

2020

Computer Physics Communications

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We present and distribute a new numerical system using classical finite elements with mesh adaptivity for computing two-dimensional liquid-solid phase-change systems involving natural convection. The programs are written as a toolbox for FreeFem++ (www3.freefem.org), a free finite-element software available for all existing operating systems. The code implements a single domain approach. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. This model describes naturally the evolution of the liquid flow which is dominated by convection effects. To make it valid also in the solid phase, a Carman-Kozeny-type penalty term is added to the momentum equations. The penalty term brings progressively (through an artificial mushy region) the velocity to zero into the solid. The energy equation is also modified to be valid in both phases using an enthalpy (temperature-transform) model introducing a regularized latent-heat term. Model equations are discretized using Galerkin triangular finite elements. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretizations are possible. The coupled system of equations is integrated in time using a second-order Gear scheme. Non-linearities are treated implicitly and the resulting discrete equations are solved using a Newton algorithm. An efficient mesh adaptivity algorithm using metrics control is used to adapt the mesh every time step. This allows us to accurately capture multiple solid-liquid interfaces present in the domain, the boundary-layer structure at the walls and the unsteady convection cells in the liquid. We present several validations of the toolbox, by simulating benchmark cases of increasing difficulty: natural convection of air, natural convection of water, melting of a phase-change material, a melting-solidification cycle, and, finally, a water freezing case. Other similar cases could be easily simulated with this toolbox, since the code structure is extremely versatile and the syntax very close to the mathematical formulation of the model.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Rakotondrandisa

Sadaka

Danaila

2020

Computer Physics Communications

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“…Plus, a number of applications including Stokes flow and coupled thermal or thermo-haline effects with Brinkman flows (see e.g. [16,19,22] and [17,18,25], respectively) depend strongly on marked spatial distributions of viscosity.…”

Section: Introductionmentioning

confidence: 99%

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

Anaya

Gómez-Vargas

Mora

et al. 2019

Comptes Rendus. Mathématique

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In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity and pressure with non-constant viscosity. The analysis is performed by the classical Babuška-Brezzi theory, and we state that any inf-sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates which are further confirmed through computational examples.

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“…Here we advocate to the study of well-posedness and stability of a weak formulation (as well as the construction of mixed finite element schemes) for the Boussinesq equations with thermally-dependent viscosity. In this regard, we remark that not many finite element methods that provide analysis for the case of non-constant viscosity are available in the literature (we may basically refer to [2,3,5,34,36,39,[41][42][43], and some of the references therein). For instance, the authors in [36] (see [35] for the continuous analysis) propose an optimally convergent, primal finite element method for the steady-state problem applied to the flow in channels.…”

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confidence: 99%

A new mixed finite element method for the <i>n</i>-dimensional Boussinesq problem with temperature-dependent viscosity

Almonacid¹,

Gatica²,

Oyarzúa³

et al. 2020

Networks &Amp; Heterogeneous Media

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In this paper we propose a new mixed-primal formulation for heatdriven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixedpoint theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the

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Stability and finite element approximation of phase change models for natural convection in porous media

Cited by 23 publications

References 40 publications

A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

A new mixed finite element method for the <i>n</i>-dimensional Boussinesq problem with temperature-dependent viscosity

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