Frontogenesis in turbulent flow through porous media using non-linear Forchheimer equation

Abstract

We present here both one- and two-dimensional models for turbulent flow through heterogeneous unbounded fluid saturated porous media using non-linear Forchheimer extended Darcy (DF) equation in the presence of gravitational field. The fluid is initially at rest and sets in motion due to a uniform horizontal density gradient. It is shown that a purely horizontal motion develops satisfying non-linear DF equation. Analytical solutions of this non-linear Initial Value Problem are obtained and limiting solutions valid for the Darcy regime in the case of laminar flow are derived. A measure of the stability of the flow is discussed briefly using Richardson number. The comparison between the nature of the solutions satisfying the non-linear and linear initial value problems are made. We found that even in the case of turbulent flow the vertical density gradient varies continuously both with space z and time t but the horizontal density gradient remains unchanged. The existence and uniqueness theorem of the Initial Value Problem is proved. The stability of these solutions are discussed and it is shown that the solutions are qualitatively and quantitatively different for z < 1 4 and z > 1 4 in the upper and lower half of the region. In particular, we have shown that the solution which is stable for infinitesimal perturbations is also stable for arbitrary perturbations both in time and space. In the case of two-dimensional motion, a piecewise initial density gradient with continuous distribution of density, stream function formulation is used and the solutions are obtained using time-series analysis. In this case solution shows crowding of the density profiles in the lower-half of the channel reflecting an increase in density gradient and incipient of frontogenesis there, because of the increase in circulation of the flow due to piecewise initial density gradient.