Stochastic-Thermodynamic Modeling of Developed Structured Turbulence

Abstract

The context is devoted to the development of a phenomenological model for the developed turbulence in a compressible homogeneous medium by taking into account nonlinear cooperative processes. The representation of a turbulized fluid motion as a thermodynamic system consisting of two continua, the subsystem of averaged motion and the subsystem of turbulent chaos, which, in turn, is considered as a conglomerate of vortex structures of various spatiotemporal scales, serves as the primary concept. We develop the ideas of a stationary-nonequilibrium state of the dissipatively active subsystem of turbulent chaos emerging due to the influx of negentropy from the external medium (the subsystem of averaged motion) and the appearance of relatively stable coherent vortex structures in the system when varying the flow control parameters. This allows some of the turbulent field rearrangement processes to be considered as self-organization processes in an open system. The methods of the stochastic theory of irreversible processes and extended irreversible thermodynamics are used to derive the defining relations for the turbulent fluxes and forces that close the system of averaged hydrodynamic equations and describe the transport and self-organization processes in the stationary-nonequilibrium case with completeness sufficient for practice.

The original approach to the stochastic-thermohydrodynamic modeling of the subsystem of turbulent chaos we developed is based on the introduction of a set of random variables into the model—fluctuating internal coordinates (like the turbulent energy dissipation rates, the intrinsic vorticities of the field of velocity fluctuations referring to mesoscale vortex structures, etc.) characterizing the structure and temporal evolution of the fluctuating field of hydrodynamic flow parameters. This makes it possible to model the Richardson–Kolmogorov cascade process and to derive the kinetic Fokker–Planck–Kolmogorov (FPK) equations designed to describe the evolution of the probability density function for small-scale turbulence characteristics by thermodynamic methods. In particular, these equations serve as a basis in analyzing the Markovian diffusion processes of the transition from one stationary-nonequilibrium state to another in the space of internal coordinates through a successive loss of stability (and increase in supercriticality) by the subsystem of turbulent chaos far from complete chaos of thermodynamic equilibrium. Such transitions can be described as nonequilibrium “second-order phase transitions” in a vortex continuum, causing the internal coordinates at bifurcation points to change abruptly.

An alternative approach to investigating the mechanisms of such a transition that is based on stochastic Langevin equations closely related to the derived kinetic FPK equations is also considered here. We analyze a cardinal problem of the approach being developed—the possibility of the existence of asymptotically stable stationary-nonequilibrium states in the subsystem of turbulent chaos. We propose a nonequilibrium thermodynamic potential for the stochastic internal coordinates of turbulent chaos that generalizes the well-known Boltzmann–Planck relation for equilibrium states to stationary-nonequilibrium states of the ensemble representing chaos and show that this potential is the Lyapunov function for stationary-nonequilibrium states of the ensemble corresponding to the subsystem of turbulent chaos.

The last section of this chapter is devoted to the thermodynamic derivation of generalized fractional FPK equations describing the evolution of the internal coordinates of the subsystem of turbulent chaos based on fractional dynamics. The introduction of fractional time derivatives into the kinetic FPK equation allows the effects of intermittency in time with which the presence of turbulent bursts against the background of less intense low-frequency background turbulence oscillations are associated to be taken into account in the context of unified mathematical formalism.