A comparison of the modern Lie scaling method to classical scaling techniques

Abstract

Abstract. In the past two decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics including but not limited to groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non dimensionalization, and the Buckingham Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is done with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Numerical validation of novel scaling laws for air entrainment in water

“…Consequently, the Lie group transformations can be used to find the scaling laws of the variables that can guarantee selfsimilarity of a phenomenon. The advantage of this approach is that it gives a complete picture of the requirements that must be satisfied for self-similarity, in contrast with a classical dimensional analysis, based on Buckingham π theorem, which is only applied to the dynamics in the interior of the domain and not to the boundary conditions [26,27].…”