Linear and Nonlinear Convection in Porous Media between Coaxial Cylinders

In this thesis we develop a mathematical model for describing three-dimensional natural convection in porous media filling a vertical annular cylinder. We apply a linear stability analysis to determine the onset of convection and the preferred convective mode when the annular cylinder is subject to two different types of boundary conditions: heat insulated sidewalls and heat conducting sidewalls. The case of an annular cylinder with insulated sidewalls has been investigated earlier, but our results reveal more details than previously found. We also investigate the case of the radius of the inner cylinder approaching zero and the results are compared with previous work for non-annular cylinders. Using pseudospectral methods we have built a high-order numerical simulator to uncover the nonlinear regime of the convection cells. We study onset and geometry of convection modes, and look at the stability of the modes with respect to different types of perturbations. Also, we examine how variations in the Rayleigh number affects the convection modes and their stability regimes. We uncover an increased complexity regarding which modes that are obtained and we are able to identify stable secondary and mixed modes. We find the different convective modes to have overlapping stability regions depending on the Rayleigh number. The motivation for studying natural convection in porous media is related to geothermal energy extraction and we attempt to determine the effect of convection cells in a geothermal heat reservoir. However, limitations in the simulator do not allow us to make any conclusions on this matter.Master i Anvendt og beregningsorientert matematikkMAMN-MABMAB39

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen-Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen-Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term which makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model

Solution approaches for evaporation-driven density instabilities in a slab of saturated porous media

This work considers the gravitational instability of a saline boundary layer formed by an evaporation-induced flow through a fully-saturated porous slab. Evaporation of saline waters can eventually result in the formation of salt lakes as salt accumulates. As natural convection can impede the accumulation of salt, establishing a relation between its occurrence and the value of physical parameters such as evaporation rate, height of the slab or porosity is crucial. One step towards determining when gravitational instabilities can arise is to compute the ground-state salinity, that evolves due to the uniform upwards flow caused by evaporation. The resulting salt concentration profile exhibits a sharply increasing salt concentration near the surface, which can lead to a gravitationally unstable setting. In this work, this ground state is analytically derived within the framework of Sturm-Liouville theory. Then, the method of linear stability in conjunction with the quasi-steady state approach is employed to investigate the occurrence of instabilities. These instabilities can develop and grow over time depending on the Rayleigh number and the dimensionless height of the porous medium. To calculate the critical Rayleigh number, which can determine the stability of a particular system, the eigenvalues of the linear perturbation equations have to be computed. Here, a novel fundamental matrix method is proposed to solve this eigenvalue problem and shown to coincide with an established Chebyshev-Galerkin method in their shared range of applicability. Finally, a 2-dimensional direct numerical simulation of the full equation system via the finite volume method is employed to validate the time of onset of convective instabilities predicted by the linear theory. Moreover, the fully nonlinear convection patterns are analyzed

Linear and Nonlinear Convection in Porous Media between Coaxial Cylinders

In this thesis we develop a mathematical model for describing three-dimensional natural convection in porous media filling a vertical annular cylinder. We apply a linear stability analysis to determine the onset of convection and the preferred convective mode when the annular cylinder is subject to two different types of boundary conditions: heat insulated sidewalls and heat conducting sidewalls. The case of an annular cylinder with insulated sidewalls has been investigated earlier, but our results reveal more details than previously found. We also investigate the case of the radius of the inner cylinder approaching zero and the results are compared with previous work for non-annular cylinders. Using pseudospectral methods we have built a high-order numerical simulator to uncover the nonlinear regime of the convection cells. We study onset and geometry of convection modes, and look at the stability of the modes with respect to different types of perturbations. Also, we examine how variations in the Rayleigh number affects the convection modes and their stability regimes. We uncover an increased complexity regarding which modes that are obtained and we are able to identify stable secondary and mixed modes. We find the different convective modes to have overlapping stability regions depending on the Rayleigh number. The motivation for studying natural convection in porous media is related to geothermal energy extraction and we attempt to determine the effect of convection cells in a geothermal heat reservoir. However, limitations in the simulator do not allow us to make any conclusions on this matter

A numerical scheme for two-scale phase-field models in porous media

[EN] We consider the flow of two immiscible fluid phases in a porous medium. At the scale of pores, the two fluid phases are separated by interfaces that are transported by the flow. Furthermore, the surface tension at such interfaces depends on the concentration of a surfactant dissolved in one of the fluids. Here we discuss a two-scale model for two-phase porous-media flow, in which concentration-dependent surface tension effects are incorporated. The model is obtained by employing formal homogenization methods and relies on the phase-field approach, in which thin, diffuse interface regions approximate the interfaces. We propose a two-scale numerical scheme and present numerical results revealing the influence of various quantities on the averaged behaviour of the system.This research is supported by the Research Foundation-Flanders (FWO) through the Odysseus programme (Project G0G1316N) and by the German Research Foundation (DFG) through the SFB 1313, Project Number 327154368.Bastidas, M.; Sharmin, S.; Bringedal, C.; Pop, S. (2022). A numerical scheme for two-scale phase-field models in porous media. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 364-373. https://doi.org/10.4995/YIC2021.2021.12571OCS36437

Adaptive and flexible macro-micro coupling software

Many multiscale simulation problems require a many-to-one coupling between different scales. For such coupled problems, researchers oftentimes focus on the coupling methodology, but largely ignore software engineering and high-performance computing aspects. This can lead to inefficient use of hardware resources, on the one hand, but also inefficient use of human resources as solutions to typical technical coupling problems are constantly reinvented. This work proposes a flexible and application-agnostic software framework to couple independent simulation codes in a many-to-one fashion. To this end, we introduce a prototype of a new lightweight software component called Micro Manager, which allows us to reuse the coupling library preCICE for two-scale coupled problems. We demonstrate the applicability of the framework by a two-scale coupled heat conduction problem

Linear and nonlinear convection in porous media between coaxial cylinders

We uncover novel features of three-dimensional natural convection in porous media by investigating convection in an annular porous cavity contained between two vertical coaxial cylinders. The investigations are made using a linear stability analysis, together with high-order numerical simulations using pseudospectral methods to model the nonlinear regime. The onset of convection cells and their preferred planform are studied, and the stability of the modes with respect to different types of perturbation is investigated. We also examine how variations in the Rayleigh number affect the convection modes and their stability regimes. Compared with previously published data, we show how the problem inherits an increased complexity regarding which modes will be obtained. Some stable secondary modes or mixed modes have been identified and some overlapping stability regions for different convective modes are determined