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

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 a priori eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini et al. (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.

Geometrical dependence of optimal bounds in Taylor–Couette flow

“…A remarkable property of this algorithm is that it eliminates the requirement of numerical continuation (Plasting & Kerswell 2003). As the two-step algorithm can be implemented at any value of the flow parameter, this flexibility has led to wider usage in several other studies of the background method to obtain the optimal bound numerically (Wen et al 2015;Wen & Chini 2018;Ding & Marensi 2019;Lee, Wen & Doering 2019;Souza, Tobasco & Doering 2020). The first step of the algorithm uses a pseudo-time stepping scheme in which the Euler-Lagrange equations (A28a-d) are converted into a time-dependent system of partial differential equations (PDEs) as follows:…”

A short review on porous media convection

“…They find a lower prefactor c for the unit optimal scaling exponent in the F -Ra relationship. Moreover, these upper-bound variational analysis also yields a set of a priori eigenfunctions which can be used to construct low-/reduced-order models for porous media convection [50][51][52].…”

Marginally stable thermal convection in porous media

“…In previous upper bound analysis [33,[49][50][51][52][54][55][56], the optimal background profile τ (z) is determined by minimizing…”