On the Nondimensionalization Process in Complex Problems: Application to Natural Convection in Anisotropic Porous Media

I Alhama,F Alhama,M Cánovas

doi:10.1155/2014/796781

Abstract

The nondimensionalization of the equations governing a given problem is a direct, relatively easy, and low-cost way to provide interesting information, the dimensionless groups that rule the problem and define its final patterns of solution. In complex problems, however, this technique frequently does not provide the precise and complete set of monomials we are looking for. The use of discrimination in the process of nondimensionalization, the detailed application of which is explained in this paper, always leads to a minimum set of parameters, which, separately, determine the solution of the problems. In addition, the physical meaning and order of magnitude of these discriminated monomials are also provided by the discrimination. The technique is applied to the coupled problem of natural convection between horizontal plates heated from below, containing an anisotropic porous medium.

Highlights

The search for the minimum group of parameters that rule the solution to a given problem, especially if it is complex and can only be solved numerically, is perhaps the main goal of experimental, numerical, and theoretical research and great effort has been put into the same
A deep understanding of the theory involved in the physical process is required for correct formulation of the complete and unique list of variables; the researcher frequently does not include in this list hidden quantities that are not mentioned in the statement of the problem but which, take part in the balances or conservation laws; this occurs, for example, in 2D scenarios in which one of the lengths extends beyond the limits of where the phenomenon is mainly located
The concept of discrimination as an extension of dimensional analysis is introduced by Madrid and Alhama [10] in the field of convective heat transfer and used by these and other authors for the derivation of the correct dimensionless groups [11,12,13]

Summary

Introduction

The search for the minimum group of parameters that rule the solution to a given problem, especially if it is complex and can only be solved numerically, is perhaps the main goal of experimental, numerical, and theoretical research and great effort has been put into the same. The use of discrimination clearly improves the solution in fundamental aspects: the number of resulting groups is less, their order of magnitude is unity, and (as a consequence) they can be immediately interpreted physically as balances of quantities in the domains where they are applied. Mathematically orientated from the point of view of equations management but without losing the physical understanding of the problem, discriminated nondimensionalization is applied to a coupled, nonlineal, very complex problem involving many physical parameters, in order to demonstrate the formal rules of application, and some complementary and interesting aspects derived from this technique, such as the potential existence of hidden quantities necessary for the definition of dimensionless dependent or independent variables. By making use of the physical meaning of the resulting groups, the order of magnitude of the principal unknown of the problem can be obtained

Governing Equations and Boundary Conditions

Discriminated Dimensionless Form

Resulting Dimensionless Groups and Solutions

Validation

21 Group 9

Conclusions