Mathematical Characterization of Bénard-Type Geothermal Scenarios Using Discriminated Non-dimensionalization of the Governing Equations

Abstract

AbstractThe classical non-dimensionalization process of governing equations is a low-cost method commonly applied for a first approximation to the dimensionless numbers that determine the solution patterns in many problems; however, this procedure fails in complex problems, where it is not even possible to define reference quantities – because they are not established in the statement of the problem – to make the dependent or independent variables dimensionless. The application of discrimination corrects this obstacle and allows suitable dimensionless groups to be defined. These, in turn, have two interesting properties: (i) they are of order of magnitude unity, and (ii) they have a clear meaning in terms of balance of physical quantities that counteract each other in a domain or sub-domain of the problem. In this paper, discriminated non-dimensionalization is applied to geothermal scenarios of Bénard-type convective flow, large horizontal boundary sides under a temperature gradient in porous media, to determine the dimensionless groups that control the steady temperature and stream patterns and, from these, the order of magnitude of the main unknowns of the problem. The results were checked numerically for many cases.