Scale analysis of miscible density-driven convection in porous media

Abstract

AbstractScale analysis of unstable density-driven miscible convection in porous media is performed. The main conclusions for instabilities in the developed (long time scales) regime are that (i) large-scale structures are responsible for the bulk of the production of concentration variance, (ii) variance dissipation is dominated by the small (diffusive) scales and that (iii) both the production and dissipation rates are independent of the Rayleigh number. These findings provide a strong basis for a new modelling approach, namely, large-mode simulation (LMS), for which closure is achieved by replacing the actual diffusivity with an effective one. For validation, LMS results for vertical flow in a homogeneous rectangular domain are compared with direct numerical simulations (DNS). Some of the analysis is based on the derivation and closure of the concentration mean and variance equations, whereby averaging over the ensemble of all possible initial perturbations is considered. While self-similar solutions are obtained for vertical, statistically one-dimensional fingering, triple correlation of concentration and scalar dissipation rate (rate at which the concentration variance decays due to diffusion) have to be modelled in the general case. For this purpose, an ensemble-averaged Darcy modelling (EADM) approach is proposed.