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Numerical analysis of a porous natural convection system with vorticity and viscous dissipation

Russel Demos, Rashmi Dubey, Ricardo Ruiz-Baier, Segundo Villa-Fuentes

TL;DR

This work addresses numerical analysis for a coupled Brinkman–Darcy flow in porous media with vorticity and viscous dissipation under non-isothermal conditions. It develops a vorticity–velocity–pressure–temperature formulation and a pointwise divergence-free mixed finite element method within a Banach-space framework, decoupling the Brinkman and thermal subproblems via a fixed-point approach. The authors establish well-posedness for both continuous and discrete problems, derive Céa-type quasi-optimal error estimates, and demonstrate optimal convergence rates through 2D and 3D numerical experiments with mixed boundary conditions. The study provides a rigorous foundation for stable, accurate simulations of natural convection in highly permeable porous media and suggests directions for extending to time-dependent problems and energy-conserving formulations.

Abstract

In this paper we propose and analyse a new formulation and pointwise divergence-free mixed finite element methods for the numerical approximation of Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a transport equation for thermal energy with viscous dissipative effect and mixed Navier-type boundary conditions. The solvability analysis of the continuous and discrete problems is significantly more involved than usual as it hinges on Banach spaces needed to properly control the advective and dissipative terms in the non-isothermal energy balance equation. We proceed by decoupling the set of equations and use the Banach fixed-point theorem in combination with the abstract theory for perturbed saddle-point problems. Some of the necessary estimates are straightforward modifications of well-known results, while other technical tools require a more elaborated analysis. The velocity is approximated by Raviart--Thomas elements, the vorticity uses Nédélec spaces of the first kind, the pressure is approximated by piecewise polynomials, and the temperature by continuous and piecewise polynomials of one degree higher than pressure. Special care is needed to establish discrete inf-sup conditions since the curl of the discrete vorticity is not necessarily contained in the discrete velocity space, therefore suggesting to use two different Raviart--Thomas interpolants. A discrete fixed-point argument is used to show well-posedness of the Galerkin scheme. Error estimates in appropriate norms are derived, and a few representative numerical examples in 2D and 3D and with mixed boundary conditions are provided.

Numerical analysis of a porous natural convection system with vorticity and viscous dissipation

TL;DR

This work addresses numerical analysis for a coupled Brinkman–Darcy flow in porous media with vorticity and viscous dissipation under non-isothermal conditions. It develops a vorticity–velocity–pressure–temperature formulation and a pointwise divergence-free mixed finite element method within a Banach-space framework, decoupling the Brinkman and thermal subproblems via a fixed-point approach. The authors establish well-posedness for both continuous and discrete problems, derive Céa-type quasi-optimal error estimates, and demonstrate optimal convergence rates through 2D and 3D numerical experiments with mixed boundary conditions. The study provides a rigorous foundation for stable, accurate simulations of natural convection in highly permeable porous media and suggests directions for extending to time-dependent problems and energy-conserving formulations.

Abstract

In this paper we propose and analyse a new formulation and pointwise divergence-free mixed finite element methods for the numerical approximation of Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a transport equation for thermal energy with viscous dissipative effect and mixed Navier-type boundary conditions. The solvability analysis of the continuous and discrete problems is significantly more involved than usual as it hinges on Banach spaces needed to properly control the advective and dissipative terms in the non-isothermal energy balance equation. We proceed by decoupling the set of equations and use the Banach fixed-point theorem in combination with the abstract theory for perturbed saddle-point problems. Some of the necessary estimates are straightforward modifications of well-known results, while other technical tools require a more elaborated analysis. The velocity is approximated by Raviart--Thomas elements, the vorticity uses Nédélec spaces of the first kind, the pressure is approximated by piecewise polynomials, and the temperature by continuous and piecewise polynomials of one degree higher than pressure. Special care is needed to establish discrete inf-sup conditions since the curl of the discrete vorticity is not necessarily contained in the discrete velocity space, therefore suggesting to use two different Raviart--Thomas interpolants. A discrete fixed-point argument is used to show well-posedness of the Galerkin scheme. Error estimates in appropriate norms are derived, and a few representative numerical examples in 2D and 3D and with mixed boundary conditions are provided.
Paper Structure (28 sections, 16 theorems, 152 equations, 2 figures, 2 tables)

This paper contains 28 sections, 16 theorems, 152 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Scope.
  3. Plan of the paper.
  4. Preliminaries and recurrent notation.
  5. Model problem and its weak formulation
  6. Natural convection with viscous dissipation
  7. Weak formulation
  8. Unique solvability analysis of the continuous formulation
  9. Preliminaries
  10. Properties of bilinear and trilinear forms
  11. Solvability of the decoupled Darcy--Brinkman equations
  12. Solvability of the decoupled thermal energy equations
  13. A fixed-point approach
  14. Galerkin scheme and well-posedness of the discrete problem
  15. Preliminaries
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let $t, \, t' \in (1, \infty)$ be such that $\frac{1}{t} + \frac{1}{t'} = 1$. Then, for each $\boldsymbol{z}\in \mathbf{L}^t(\Omega)$ there hold and therefore $\mathcal{J}_t : \mathbf{L}^t(\Omega) \to \mathbf{L}^{t'}(\Omega)$ and $\mathcal{J}_{t'} : \mathbf{L}^{t'}(\Omega) \to \mathbf{L}^t(\Omega)$ are bijective and inverse to each other.

Figures (2)

  • Figure 6.1: Accuracy tests in 2D and 3D. Approximate solutions (vorticity distribution and streamlines, velocity magnitude and streamlines, pressure profile, and temperature field) computed with the lowest-order methods.
  • Figure 6.2: Example 2. Snapshots at $t=0.2$ (left) and $t=1$ (right) for the the rescaled vorticity (top), velocity magnitude and line integral convolution (second row), pressure profile (third row), and temperature distribution (bottom) for the channel flow past five cylinders with $\mu=10^{-4}$. Here we have used the second-order scheme with $k=1$.

Theorems & Definitions (19)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.1
  • Lemma 3.4
  • Theorem 3.1
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 9 more