Closed System of Hydrodynamic Equations to Describe Turbulent Motions of Multicomponent Media

Abstract

Here a closed system of averaged hydrodynamic equations for a turbulized multicomponent chemically active gas mixture designed to describe a wide class of turbulent motions and physical–chemical processes in natural media is derived. The physical meaning of the individual terms in these equations, including the energy transition rates between various energy balance components are analyzed. Authors systematically uses the weighted-mean Favre averaging, which allows the form and analysis of the averaged equations of motion for chemically active gases with variable thermophysical properties to be simplified considerably, along with the traditional probability-theoretic averaging of fluctuating thermohydrodynamic parameters. Assessing the status of the first-order closure problem on the whole, it should be recognized that no general phenomenological theory of turbulent heat conduction and turbulent diffusion for multicomponent reacting mixtures has existed until now. Therefore, in this chapter we consider a thermodynamic approach to the closure of the averaged hydrodynamic equations for a mixture at the level of first-order turbulence models based on the methods of extended irreversible thermodynamics. Special attention is paid to the derivation of closing gradient relations for the Reynolds turbulent stress tensor and the turbulent heat and diffusion fluxes in a multicomponent mixture by thermodynamic methods. The Onsager formalism allows the most general structure of such relations, including those in the form of generalized Stefan–Maxwell relations for multicomponent turbulent diffusion, to be obtained in this case as well. At the closure level under consideration, these relations describe most comprehensively the turbulent heat and mass transport in a multicomponent medium. Both classical models dating back to Prandtl, Taylor, and Karman and more recent second-order closure models based, in particular, on the differential balance equations for the turbulent energy and integral turbulence scale are used to determine the turbulent exchange coefficients. For the convenience of the reader, all calculations are performed comprehensively and can be traced in all details.