Computational Techniques for Simulating Natural Convection in Three-Dimensional Enclosures with Tetrahedral Finite Elements

TL;DR

This work develops a tetrahedral finite element framework for three-dimensional natural convection in enclosures, leveraging unstructured meshes and a Marchuk-Yanenko-type operator splitting to decouple pressure, transport, and diffusion. Advection is handled with a wave-equation approach, while backward-Euler discretizations address pressure and diffusion sub-problems; a velocity-pressure macro-element on a finer velocity/temperature mesh ensures stability under the inf-sup condition. The spatial discretization uses piecewise-linear $\mathcal{P}_1$ elements, with a novel macro-element construction that subdivides each pressure tetrahedron into eight velocity tetrahedra, enabling efficient solution on unstructured grids. Numerical experiments for air in a cube with $Pr=0.71$ and $Ra$ up to $10^6$ yield accurate Nusselt-number predictions and reveal key three-dimensional effects and symmetry consistent with prior studies.

Abstract

This article discusses computational techniques for simulating natural convection in three-dimensional domains using finite element methods with tetrahedral elements. These techniques form a new numerical procedure for this kind of problems. In this procedure, the treatment of advection by a wave equation approach is extended to three-dimensional unstructured meshes with tetrahedra. Numerical results of natural convection of an incompressible Newtonian fluid in a cubical enclosure at Rayleigh numbers in the range $10^3$ to $10 ^6$ are obtained and they are in good agreement with those in literature obtained by other methods.

Figures (7)

• Figure 1: Geometry and coordinate system for cubical enclosures, ( $0 \leq x \leq L_x \ ; \  0 \leq y \leq L_y \ ; \  0 \leq z \leq L_z$).
• Figure 2: A typical brick-like macro.
• Figure 3: A systematic division of a brick-like macro into six tetrahedra.
• Figure 4: The six generic tetrahedral elements.
• Figure 5: Typical pressure macro-elements with $8$ sub-tetrahedral.