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A Finite Element Model for Hydro-thermal Convective Flow in a Porous Medium: Effects of Hydraulic Resistivity and Thermal Diffusivity

S. M. Mallikarjunaiah, Dambaru Bhatta

TL;DR

This work develops a stable, decoupled finite element method for hydro-thermal Darcy-Bénard convection in a porous medium with vertical variations in hydraulic resistivity $\chi(z)$ and thermal diffusivity $\zeta(z)$. A continuous Galerkin discretization using Taylor-Hood elements is paired with a forward-Euler temperature time stepping to achieve an unconditionally stable fully discrete scheme. Validation with a manufactured solution shows optimal convergence; the study analyzes how the Rayleigh number and parameter variations affect heat transfer, quantified by the local and average Nusselt numbers $Nu$, finding that $Nu$ increases with $Ra$ and that the configuration with linear $\chi$ and quadratic $\zeta$ yields the highest heat-transfer rates. The method demonstrates robust performance and potential for extension to 3D and adaptive mesh refinement, making it applicable to engineering problems involving porous-media convection and heat exchange.

Abstract

In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy's law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element discretization for the hydro-thermal convective flow model. The well-posedness of the proposed discrete Galerkin finite element formulation is guaranteed due to the decoupling property and the linearity of the numerical method. Computational experiments confirm the optimal convergence rates for a manufactured solution. Several numerical results are obtained for the variations of the hydraulic resistivity and thermal diffusivity. In the present study, the bottom wall is maintained at a constant higher hot temperature while side vertical walls are thermally insulated and the top wall is maintained at a constant cold temperature. Heat transfer rates at the heated bottom wall are presented in terms of local Nusselt number. A linear variation in hydraulic resistivity and a quadratic variation in thermal diffusivity show an increase in the heat transfer rate.

A Finite Element Model for Hydro-thermal Convective Flow in a Porous Medium: Effects of Hydraulic Resistivity and Thermal Diffusivity

TL;DR

This work develops a stable, decoupled finite element method for hydro-thermal Darcy-Bénard convection in a porous medium with vertical variations in hydraulic resistivity and thermal diffusivity . A continuous Galerkin discretization using Taylor-Hood elements is paired with a forward-Euler temperature time stepping to achieve an unconditionally stable fully discrete scheme. Validation with a manufactured solution shows optimal convergence; the study analyzes how the Rayleigh number and parameter variations affect heat transfer, quantified by the local and average Nusselt numbers , finding that increases with and that the configuration with linear and quadratic yields the highest heat-transfer rates. The method demonstrates robust performance and potential for extension to 3D and adaptive mesh refinement, making it applicable to engineering problems involving porous-media convection and heat exchange.

Abstract

In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy's law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element discretization for the hydro-thermal convective flow model. The well-posedness of the proposed discrete Galerkin finite element formulation is guaranteed due to the decoupling property and the linearity of the numerical method. Computational experiments confirm the optimal convergence rates for a manufactured solution. Several numerical results are obtained for the variations of the hydraulic resistivity and thermal diffusivity. In the present study, the bottom wall is maintained at a constant higher hot temperature while side vertical walls are thermally insulated and the top wall is maintained at a constant cold temperature. Heat transfer rates at the heated bottom wall are presented in terms of local Nusselt number. A linear variation in hydraulic resistivity and a quadratic variation in thermal diffusivity show an increase in the heat transfer rate.
Paper Structure (20 sections, 1 theorem, 38 equations, 14 figures, 11 tables)

This paper contains 20 sections, 1 theorem, 38 equations, 14 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Numerical Method
  4. Time Discretization
  5. Weak Formulation
  6. Fully Discrete Scheme
  7. Solution Strategy for the Linear and Decoupled Equations
  8. Solver for velocity-pressure equations
  9. Linear Solver for Temperature Equation
  10. Numerical Results and Discussion
  11. Convergence study
  12. Comparison with previous studies
  13. Effect of mesh density
  14. Unsteady thermal convection
  15. Case-I: Linear variations in both $\chi(z)$ and $\zeta(z)$ and the effect of $Ra$
  16. ...and 5 more sections

Key Result

Theorem 3.1

For any $\delta t >0$, the solutions to the above fully discrete finite element algorithm are unconditionally stable.

Figures (14)

  • Figure 1: Comparison of the finite element solution and analytical solution for \ref{['sm']}.
  • Figure 2: Solution variables for the DBC problem with heating from left-wall and cooling from right-wall.
  • Figure 3: Solution variables for the DBC problem with heating from left-wall and cooling from right-wall.
  • Figure 4: Results for Case-I
  • Figure 5: Local Nusselt number for different values of Rayleigh number.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Theorem 3.1
  • Remark 3