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Stability analysis for a saline boundary layer formed by uniform upflow using finite elements

Authors
Publisher
Technische Universiteit Eindhoven
Publication Date
Disciplines
  • Mathematics

Abstract

Main.dvi Stability Analysis for a Saline Boundary Layer Formed by Uniform Up ow Using Finite Elements G.J.M. Pieters 2000 ABSTRACT Up owing salty groundwater, evaporating completely at the ground surface, leads to the buildup of a saline boundary layer, usually with solid salt on the surface. The di�usion layer below the surface, if stable, may grow to a �nite thickness at equilibrium between the up owing salt and downward di�usion. Because of the di�erent densities of the uids (low fresh water density in the deeper underground and high salt water density at the boundary layer), gravitation plays a crucial role in the process and therefore it cannot be disregarded. For the instable case, at a critical Rayleigh number, perturbation of the system results in fresh-salt �ngering of the boundary layer. The process described above is mathematically modelled using a scaled, coupled and nonlinear set of partial di�erential equations for the incompressible uid including a convection-di�usion equation and Darcy's law. In this report we �rst give an overview of semi-analytical methods to analyse the stability of the boundary layer. Subsequently we focus on Finite Element (FEM) solutions of the governing equations to validate numerically the semi-analytical stability bounds. Furthermore, we analyse stability with respect to several kinds of perturbations. Contents Preface iii 1 Introduction 1 2 Equations for salt transport 3 2.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The Oberbeck{Boussinesq approximation . . . . . . . . . . . . . . . . . . . . 6 2.4 The pressure formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 The stream function formulation . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Stability criteria for the di�usion layer: a summary of semi-analytical meth- ods 11 3.

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