Convection in a coupled fluid-porous media system

TL;DR

This paper analyzes thermal convection in a coupled fluid-porous system, deriving stability thresholds and transition criteria using linear and nonlinear analysis, and compares different interface conditions through numerics and asymptotics.

Contribution

It introduces a dynamic pressure term in the interface condition, enabling new nonlinear stability results and compares stability thresholds across interface variants.

Findings

Nonlinear stability thresholds closely match linear ones in certain regimes.

The dynamic pressure term allows for energy bounds on nonlinear advection.

The heuristic theory predicts the transition between convection regimes effectively.

Abstract

We perform linear and nonlinear stability analysis for thermal convection in a fluid overlying a saturated porous medium. We use a coupled system, with the Navier-Stokes equations and Darcy's equation governing the free-flow and the porous regions respectively. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid-medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. Within certain regimes, the nonlinear stability thresholds agree closely with the linear ones, and we quantify the differences that exist. We then compare stability thresholds produced by several common variants of the tangential interface conditions, using both numerics and asymptotics in the small Darcy number limit. Finally, we investigate the transition between full convection and fluid-dominated convection using both numerics and a heuristic theory. This heuristic theory is based on comparing the ratio of the Rayleigh number in each domain to its corresponding critical value, and it is shown to agree reasonably well with the numerics regarding how the transition depends on the depth ratio, the Darcy number, and the thermal-diffusivity ratio.