Fluid flow - Dimensional analysis. Similarity

The fundamental equations of fluid mechanics and thermal engineering are often difficult to solve. Analytical solutions are rare, and numerical solutions can be cumbersome and time-consuming. In such cases, we can resort to experimental studies, either full-scale or using scale models. We can also replace the resolution of basic equations, which provides local information, with more global models of the problem. These models are based on semi-empirical correlations deduced from experiments carried out under specific conditions, the results of which must be extrapolable to other, similar conditions.

Whether experimental studies are carried out to gain knowledge of a particular situation, or to establish correlations of more general validity, the number of experiments to be carried out must always be kept to a minimum. To achieve this, it's important to know which parameters characterize the phenomenon under study, and how they come into play. The experimenter is aided in this process by dimensional analysis, which makes it easier to formulate semi-empirical relationships that can be used to model the phenomenon under study.

When experimentation on a model is required, mainly because the geometric size of the real problem is not acceptable at laboratory level, the experimenter must respect certain operating conditions linking the study on the model and its transposition to the prototype. These conditions are imposed by the theory of similarity. More generally, these conditions are necessary when we want to apply to a problem the solution obtained for another problem deemed similar. In order to maintain its generality, this solution will always be given in the form of one or more dimensionless equations, in which particular parameters - commonly referred to as dimensionless numbers - appear.

The aim of this article is, on the one hand, to present the process of dimensionally shaping the correlations linking a phenomenon to the parameters that control it, and on the other hand, to explain the conditions necessary for transposing the results of the experimentally studied case (the model) to the practical problem to be solved (the prototype).

For notations and symbols, see end of article.