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

Abstract

Abstract. In the past 2 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 overland flow, 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 Vaschy-Buckingham-theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made 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.

A Review of Similitude Methods for Structural Engineering

“…In fact, both terms are used interchangeably in the literature although with a slight difference: similarity is closer to the usage in fields of mathematics (for example, self-similarity solutions). An example of such an application is reported by Polsinelli and Levent Kavvas [45] in which Lie scaling methodology is introduced. Such a method performs symmetry analysis of the governing differential equations basing on Lie groups, special structures leading to invariant transformations.…”

Invariance Properties of the Entropy Production, and the Entropic Pairing of Inertial Frames of Reference by Shear-Flow Systems

“…(VI) Lie invariance: The invariance properties or symmetries associated with infinitesimal Lie transformations of a differential or integral equation constitutes a large topic, e.g., [25,26,[55][56][57][58]. For the present study, we restrict the discussion to the oneparameter Lie group of point scaling transformations, e.g., [25][26][27][28][29][30]. For the local entropy production (6), this is defined by the 13-parameter map:…”

“…Transformation gives a rescaled form of (6) rather than a nondimensional equation, while the auxiliary relations (23) provide the relations between conversion factors for the dependent and independent variables. This interpretation is useful, representing a rescaling between a model and a prototype to maintain similarity, e.g., [27][28][29], or rescaling by a change of units. However, since this interpretation can be handled by the more general apparatus of dimensionless invariance (I), it need not be considered further.…”

“…It is readily shown that the major differential and integral conservation equations of fluid mechanics, including of fluid mass, momentum and energy-as well as subsidiary equations such as the Navier-Stokes equations-remain unchanged under Galilean transformation [5], and are, therefore, Galilean invariant. They also exhibit other important invariance properties, including invariance to certain dimensionless transformations [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], invariance to fixed displacements in the time or space coordinates [5,20], invariance to fixed reflections or rotations of the coordinate system [5,20] and, more generally, invariance to the one-parameter Lie group of point transformations, e.g., [25][26][27][28][29][30]. Further invariance properties are also satisfied by some conservation laws: for example, the continuity equation is invariant to reversal of the time coordinate, whereas the momentum and energy equations (including the Navier-Stokes equations) are not.…”

Bi̇r Boyutlu Taşinim Süreçleri̇nde Ölçekleme Anali̇zi̇ Ve Kendi̇ne Benzeşi̇m

“…The traditional approaches lack the process based formal self-similarity analysis. Readers can refer to Polsinelli and Kavvas (2016) for comparison of the modern Lie scaling method with classical scaling techniques. Recently, one-parameter Lie group of point scaling transformations were applied to investigate scale invariance and self-similarity of various hydrologic processes (Haltas and Kavvas 2011), one-dimensional open channel flow process (Ercan et.…”