A thermodynamical theory of turbulence. I. Basic developments

Abstract

This paper is concerned with the construction of a thermodynamical theory for turbulence based on a continuum model consistent with a wide range of experimental results and observations. A complete theory with appropriate constitutive equations is developed for viscous turbulent flow but the special case of (rate-independent) inviscid turbulent flow is also discussed. The theoretical results obtained readily account for such mechanical aspects of turbulent flow as anisotropy, as well as the energetic effects of turbulent fluctuations, in addition to the more standard thermomechanical effects. More specifically, three different scales of motion and modelling, namely molecular, microscopic and macroscopic, are considered in the construction of the basic theory. Whereas the ordinary thermal effects (such as temperature) on the macroscopic scale represent the manifestation of vibratory motions at the molecular level, similar variables are used to represent the energetic turbulent effects on the macroscopic level that arise from turbulent fluctuations at the microscopic level. The various ingredients of the thermodynamical aspects (both due to thermal and turbulent effects) of the continuum model are incorporated into the theory by means of a recent procedure to thermodynamics by Green & Naghdi ( Proc. R. Soc. Lond. A 357, 253 (1977)). The mechanical aspects of the model for a turbulent fluid requires admission of additional balance laws for eddy concentration and for a kinematical variable which represents the effect of alignment of these eddies (at the microscopic level) along a particular direction on the macroscopic scale, in accordance with observations by Townsend ( The structure of turbulent shear flow , Cambridge University Press (1976)) and others.