Scaling analysis with the first law of scaled turbulence

Scaling has recently undergone a paradigm shift with the arrival of the finite-similitude scaling theory that is applicable to all physics and in principle to all quantitative disciplines. The theory introduces an infinite number of new similitude rules with the simplest rule (the zeroth-order rule) capturing all that is achievable with dimensional analysis. It provides the rules for experimental design, but also establishes a framework for scaling analysis involving additional transport equations presented on a projected space termed the scaling space . Similitude rules of different orders apply to all physical quantities in the scaling space and all physics can be defined there using traditional mathematical representations. It is of interest therefore, and the focus of this paper, to perform scaling analysis on turbulence due to its criticality in many fields of physics and engineering. Turbulence, in principle, can to a large extent be captured on solving the Navier–Stokes equations, which can be shown to satisfy exactly the first-order finite similitude rule . Unresolved turbulence, identified statistically as the unresolved component of a fluid-velocity field, exhibits the astonishing scaling property of adhering to the zeroth-order finite similitude rule . This condition is defined here to be the first law of scaled turbulence as it provides the means to assess all modelling approaches to turbulence. It is shown in the work, through analysis and case studies, that none of the traditional approaches (e.g., LES, RANS) under the action of a gravitational field, possess this critical property. Turbulence eddy viscosity, for example, returned from RANS k−ϵ is first order, but the correct behaviour (as confirmed by the first law) is reciprocal first order. The work presented here shows how scaling analysis provides a new powerful theoretical tool for the assessment of all turbulence models.