Reduced modeling of porous media convection in a minimal flow unit at large Rayleigh number

TL;DR

This paper develops two reduced modeling strategies for high-Rayleigh-number porous media convection, leveraging flow decomposition into boundary layer and interior regions, significantly improving computational efficiency while maintaining key flow features.

Contribution

It introduces novel reduced models that exploit flow structure at large Rayleigh numbers, extending previous eigenbasis approaches to more complex convection scenarios.

Findings

Negligible impact of interior high-wavenumber modes on flow structure

Hybrid model achieves over tenfold computational speedup

Small-scale boundary layer features are accurately captured

Abstract

Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale cellular motions generally are localized within the thermal and vorticity boundary layers near the upper and lower walls, while in the interior, the flow exhibits persistent large-scale structures and only a few low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first strategy, the domain is decomposed into two near-wall regions and one interior region. Our results confirm that suppressing the interior high-wavenumber modes has negligible impact on the essential structural features and transport properties of the flow. In the second strategy, a hybrid reduced model is constructed by using Galerkin projection onto a fully \emph{a priori} eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini \emph{et al.} (\emph{Physica~D}, vol. 240, 2011, pp. 241--248) to large $Ra$. The results indicate that the near-wall upper-bound eigenmodes can economically represent the small-scale rolls within the exquisitely-thin thermal boundary layers. Relative to DNS, the hybrid algorithm enables over an order-of-magnitude increase in computational efficiency with only a modest loss of accuracy.