Transient growth in linearly stable gravity-driven flow in porous media

Abstract

In this paper we study gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at the horizontal surface of a porous medium. Van Duijn et al.,[33], derived stability bounds by means of linear stability analysis and an (improved) energy method. These bounds do not coincide, i.e. there exists a subcritical region or stability gap in the system parameter space which is due to the asymmetry of the linear part of the perturbation equations. We show that the linear operator can be symmetrized by means of a similarity transformation. For system parameter values in the stability gap, we show that there exist optimal initial perturbations for which the linearly stable system exhibits transient growth. We show that transient growth is norm dependent by considering weighted norms, which are induced by a one-parameter family of similarity transformations.

Bénard instabilities in a binary-liquid layer evaporating into an inert gas

“…0, the eigenvalue problem presented above is solved using a Tau-Chebyshev method, which is rather classical and thus not recalled here (see for instance [25,26]). Before proceeding to the results, let us stress that in the classical studies of the Rayleigh-Marangoni problems, the control parameter is just the imposed temperature gradient.…”

“…The symbol refers to the ratio of solvent and solute properties, the subscript of indicating the property in question. In (25) is the so lvent to solute molecular mass ratio, while is the same for the air and the solute, where the subscript "a" refers to the air. Note also the notation / which will be used later on.…”

Instability of the salinity profile during the evaporation of saline groundwater

“…In particular, we find an overall minimum for the Rayleigh number corresponding to marginal stability, R s ≈ 9.71 when ρ = 1, so regardless of the value ofρ we expect that infinitesimal perturbations will decay if R s 9.71; conversely we expect instability if R s 14.35, regardless of the value ofρ. (It is, however, possible that some perturbations may grow transiently even below the linear stability boundary, as found for the classical problem by Pieters & van Duijn 2006, and also that subcritical nonlinear instabilities may occur.) An analysis of the steady, horizontally uniform solutions to the model for fresh water ( §2) reveals two main regimes of behaviour.…”

“…The assumption of constant ν a will break down in circumstances where the evaporative flux is high and a humid boundary layer forms above the soil surface, as can frequently be seen, for example, immediately after heavy rain on a hot day; we will comment below on the regimes in which this may occur. For simplicity we follow other studies (Gowing et al 2006;Pieters & van Duijn 2006) by considering the soil to be isothermal, with the same constant system temperature T 0 as the ground surface. This assumption requires some discussion.…”

“…This builds on a substantial body of research on salt transport in groundwater, though few previous studies have investigated travelling evaporation fronts. In particular, Wooding (1960), Wooding, Tyler & White (1997), van Duijn et al (2002) and Pieters & van Duijn (2006) have used Darcian models to investigate instabilities driven by conditions fixed at the soil surface; Yakirevich, Berliner & Sorek (1997) have modelled evaporation from the soil surface and the vertical profiles of soil saturation and ion content beneath it; Gowing, Konukcu & Rose (2006) have used a quasi-steady model based on Richards' equation to predict vertical profiles including a descending evaporation front, and tested it against laboratory experiments. Most relevantly to the current study, Tsypkin & Brevdo (1999) and Tsypkin (2003b) (see also Tsypkin 2003a) have developed one-dimensional models of Darcian flow with salt and heat transport and a moving evaporation front, but the stability of their solutions has not previously been considered.…”