Convection in a Coupled Free Flow-Porous Media System

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.

Stokes–Darcy system, small-Darcy-number behaviour and related interfacial conditions

“…A formal asymptotic expansion for the related Navier–Stokes–Darcy–Boussinesq system was carried out by one of the authors with collaborators albeit using a different non-dimensionalization methodology in the paper by McCurdy et al. (2019). See also the papers of Chen & Chen (1988) and Straughan (2001).…”

“…The mathematical proof of the validity of the approximations will be furnished in Appendix A. A formal asymptotic expansion for the related Navier–Stokes–Darcy–Boussinesq system was carried out by one of the authors with collaborators using a different non-dimensionalization methodology (McCurdy, Moore & Wang 2019).…”

Thermal convection in a linearly viscous fluid overlying a bidisperse porous medium

“…Carr and Straughan [8], Chang et al [11], Hill and Straughan [26,28], Chang et al [12], Samanta [45], Yin et al [60], Tsiberkin [56]. Particular analyses involving the nonlinear theory and bifurcations are given by Han et al [23], McCurdy et al [35], Lyu and Wang [34], and Hill and Straughan [27]. In addition, mathematical analysis of the structural stability of the two layer system has been thoroughly investigated, see e.g.…”

Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics

“…There are a wide range of complex geophysical problems for which the extraction of statistical features is a driving interest. Examples abound from climate science [40,23,39], atmospheric science [32,33,48,8,41], morphology formation [20,9,36,49,13], and thermal convection studies [17,29,25]. The rigorous statistical analysis presented in this paper may ultimately prove valuable for applications such as these.…”